QuIGS Seminar Talk

Normal compressions and higher rank numerical ranges

Abstract

The classical numerical range has been studied extensively by mathematicians interested in the areas of functional analysis and matrix analysis since the 1930's. The higher rank numerical range generalizes the classical numerical range of an operator and is a much newer discovery which was first introduced by Choi, Kribs and Zyczkowski in 2006. Research into higher rank numerical ranges has originally been motivated by problems in quantum information theory, in particular the Knill-Laflamme representation of quantum data error correcting code. In this talk, we will discuss some of the basic structure and theory of higher rank numerical ranges and discuss their geometric representation in the complex plane. We will introduce a general program to describe normal compressions of normal (diagonal) operators and discuss an interesting pair matching problem that arises from 2 by 2 compressions. Finally we discuss several continuity results attached to this pair matching problem.

This work was done in collaboration of Dr. J. Holbrook and Dr. R. Pereira of the University of Guelph.

QuIGS Seminar Talk

Dilation theory in finite dimensions

Abstract

The celebrated Sz.-Nagy Dilation Theorem states that every contraction on a Hilbert space can be realized as "a part of" a unitary operator acting on an infinite dimensional Hilbert space. This result is both useful and inspiring, and it has led to a whole theory known today as dilation theory. In this talk I will discuss some attempts to develop a version of dilation theory that does not involve infinite dimensional spaces. This version - which to my knowledge is still rather undeveloped - provides results which are perhaps more effective, but on the other hand its development presents some incredible difficulties.

QuIGS Seminar Talk

Quantum gate fidelity in terms of choi matrices

Abstract

We provide new results for computing and comparing the quantum gate fidelity of quantum channels via their Choi matrices. We extend recent work that showed there exist non-dual pairs of quantum channels with equal gate fidelity by providing an explicit characterization of all such channels. We use our characterization to show that when the dimension is 2 (or 3, under slightly stronger hypotheses), the gate fidelity of two channels is equal if and only if their difference equals the difference of some unital map and its dual � a fact that has been shown to be false when the dimension is 4 or larger. We also present a formula for the minimum gate fidelity of a channel in terms of a well-studied norm on a compression of its Choi matrix. As a consequence, several new ways of bounding and approximating the minimum gate fidelity follow, including a simple semidefinite program to compute it for qubit channels.

QuIGS Seminar Talk

An introduction to noncommutative maxima and the distinguishability of quantum states

Abstract

The field of Quantum Information Theory naturally exposes many interesting structures in operator theory and in operator algebras. One such structure is a matrix-valued generalization of the notion of the maximum l=max(r_1,...,r_m) of several real numbers r_k. In particular, given a finite collection of self-adjoint matrices (or trace-class operators) rho_k, there is a unique matrix L=(rho_1,...,rho_m) of minimal trace for which L>=rho_k for all k in the natural partial order on self-adjoint operators (A>=B iff A-B is positive semidefinite.)

The purpose of this talk is to give a brief introduction to non-commutative maxima and their connection to the closely-related Minimum-error Quantum Detection Problem: Suppose that an integer k from the set {1,...,m} is encoded or stored in the state of a physical system. With what probability P may the value of k be successfully recovered by an observer limited only by the laws of quantum mechanics? The necessary concepts from Quantum theory will be briefly explained.

If time permits I will introduce estimates of quantum state distinguishability, or equivalently estimates of the trace Tr (max (rho_1,...\rho_m) where the rho_k are density operators representing quantum states. (These estimates may be found at http://arxiv.org/abs/0907.3386)

QuIGS Seminar Talk

Minimax estimation for mixtures of wishart distributions

Abstract

The space of positive definite symmetric matrices has been studied extensively as a means of understanding dependence in multivariate data along with the accompanying problems in statistical inference. Many books and papers have been written on this subject, and more recently there has been considerable interest in high- dimensional random matrices with particular emphasis on the distribution of certain eigenvalues. Our present paper is motivated by modern data acquisition technology, particularly, by the availability of diusion tensor-magnetic resonance data. With the availability of such data acquisition capabilities, smoothing or nonparametric techniques are required that go beyond those applicable only to data arising in Euclidean spaces. Accordingly, we present a Fourier method of minimax Wishart mixture density estimation on the space of positive definite symmetric matrices.

QuIGS Seminar Talk

Negative curvature, the cartan centroid, and the law of large numbers

Abstract

Some years ago (with Rajendra Bhatia) we studied the Cartan centroid interpreted as the geometric mean of several positive definite matrices, identifying the question of monotonicy for this notion of geometric mean. In a striking development, it turns out that monotonicity follows from an elegant probabilistic argument (Lawson-Lim, Sturm, Bhatia-Karandikhar). We'll describe this development, giving the necessary background. As time permits, we'll mention some applications and survey related binary procedures that converge to various matrix geometric means (Ando-Li-Mathias, Bini et al., Palfia)

QuIGS Seminar Talk

Does gm play dice with the universe?

Abstract

Although this talk is a continuation of last week's seminar, I'll do my best to make it stand alone. I'll explain some aspects of doing statistics in a space of non-positive curvature (defined by a semi-parallelogram law), leading up to a version of the law of large numbers. This is just what we need to establish that the "Cartan centroid" definition of geometric mean for matrices A_k, ie

GM(A_1,�,A_m),

is monotone. Then I'll discuss the question of whether this stochastic approach is really necessary (does GM play dice?), or whether some deterministic procedure converges to GM.

QuIGS Seminar Talk

Careful quantum state estimation

Abstract

Quantum state estimation is of central importance to quantum information processing. The knowledge that a state of interest is of the intended form at each stage of a quantum information processing task is required when the requisite tools (quantum computer, quantum cryptographic device, etc.) are being developed and tested.

The current de facto standard for performing quantum state estimation is the use of maximum-likelihood estimation (MLE) with the Cholesky decomposition used as the parametrization of the density matrix being fit. This method suffers from technical problems when the true density matrix is less than full rank. We present an alternative parametrization and discuss options for its implementation.

QuIGS Seminar Talk

Introduction to coherently controlled quantum channels

Abstract

In this talk, we discuss the notion of Hamiltonian controllability of qubit quantum channels. By allowing for such controls, we explicitly determine the tangent cone structure of the subsemigroups of well known unital time dependent quantum channels. These tangent cones contain all possible directions a controlled quantum channel can be steered to and hence give valuable insight to the reachable set of density matrices. Finally, we demonstrate the scalability of this approach and discuss its generalizations to multi-qubit systems. (ArXiV:1103.2703)

QuIGS Seminar Talk

Quantum channels arising from abstract harmonic analysis

Abstract

The recent representation theory surrounding locally compact groups has initiated several new connections between harmonic analysis and quantum information. In this talk, we will demonstrate how one can use this theory to generate two intriguing classes of quantum channels for every locally compact group. Focusing mainly on finite groups, we will explore several concrete examples and discuss their applications to quantum error correction and to the convex structure of bistochastic channels. This is joint work with Matthias Neufang.

QuIGS Seminar Talk

Complete unitary-similarity invariants for complex unicellular matrices

Abstract

An old and generally non-tractable problem in algebra concerns the determination of when two complex matrices are unitarily similar. That is, what are necessary and sufficient conditions on complex matrices A and B so that B has the form B=U*AU for some unitary matrix U? The best known solutions to the problem are due to Specht (1940s) and Littlewood (1950s), but these solutions are not so useful from the point of view of analysis. In this lecture, I will give a brief overview of the unitary-similarity problem from the analytic point of view and describe recent joint work with Tanya Gerasimova and Nadya Shvai that gives some new, complete invariants for matrices with only finitely many invariant subspaces.

QuIGS Seminar Talk

Network based clustering and field data: building decision support tools for detection of subclinical disease in cattle

Abstract

Livestock represent an important source of wealth and livelihood for much of the world. Poor access to trained veterinary professionals and easy and inexpensive access to mobile phone technology suggests the development of accurate decision support tools for early diagnosis of disease in livestock could have far reaching impacts. We look at clustering methods, focusing on network based clustering, for the exploration and profiling of subclinical diseased states in livestock using field collected observations from small holder held cattle in western Kenya.

QuIGS Seminar Talk

Local transformations of quantum states requiring infinite rounds of classical communication

Abstract

The class of Local Operations with Classical Communication (LOCC) is a fundamental object in quantum communication and entanglement theory. However, despite its importance, LOCC still lacks a clear understanding from both a physics and math perspective. For instance, it is unknown the extent to which more rounds of measurement and communication can enhance the ability to perform certain tasks.

In this talk I will introduce the class of LOCC and describe the subtleties in formulating its precise mathematical characterization. We will then investigate the question of LOCC transformations between certain multipartite states and find that, indeed, the class of LOCC becomes strictly more powerful as more rounds of classical communication are permitted. Specially, for every n, there always exists an n-round protocol that is impossible to implement in n - 2 rounds. Furthermore, I will describe certain entanglement transformations that are possible if and only if the protocol uses an infinite (unbounded) number of rounds. Interestingly, the number of rounds required to distill bipartite entanglement from particular multipartite states depends discontinuously on the amount of entanglement distilled.

QuIGS Seminar Talk

Non-distillable entanglement guarantees distillable entanglement

Abstract

The monogamy of entanglement is one of the basic quantum mechanical features, which says that when two partners Alice and Bob are more entangled then either of them has to be less entangled with the third party. Here we qualitatively present the converse monogamy of entanglement: given a tripartite pure system and when Alice and Bob are entangled and non-distillable, then either of them is distillable with the third party. Our result leads to the classification of tripartite pure states based on bipartite reduced density operators, which is a novel and effective way to this long-standing problem compared to the means by stochastic local operations and classical communications. Furthermore we systematically indicate the structure of the classified states and generate them. We also extend our results to multipartite states. This is joint work with Masahito Hayashi

QuIGS Seminar Talk

Private quantum channels I

QuIGS Seminar Talk

Private quantum channels II

QuIGS Seminar Talk

Quantifying entanglement

QuIGS Seminar Talk

Quantum data analysis I

Abstract

The emergence of quantum information processing is gaining tremendous momentum, particularly so because of the very real sense of the creation of a quantum computer and currently available applications of quantum cryptography. Consequently development of quantum data analysis is quickly becoming ever important, where state estimation, or, quantum tomography has been an area of active research in quantum information processing. The idea is that probabilities being one dimensional scalars, are determined by several quantum parameters. Often, quantum experiments are performed under the hypothesis that the prepared state is pure, although no statistical verification that the actual data comes from a pure state is statistically tested. In this afternoon of talks we develop statistical tests for purity of a state and apply it to some recently performed quantum experiments.

QuIGS Seminar Talk

Quantum data analysis II

Abstract

The emergence of quantum information processing is gaining tremendous momentum, particularly so because of the very real sense of the creation of a quantum computer and currently available applications of quantum cryptography. Consequently development of quantum data analysis is quickly becoming ever important, where state estimation, or, quantum tomography has been an area of active research in quantum information processing. The idea is that probabilities being one dimensional scalars, are determined by several quantum parameters. Often, quantum experiments are performed under the hypothesis that the prepared state is pure, although no statistical verification that the actual data comes from a pure state is statistically tested. In this afternoon of talks we develop statistical tests for purity of a state and apply it to some recently performed quantum experiments.

QuIGS Seminar Talk

Quantum data analysis III

Abstract

The emergence of quantum information processing is gaining tremendous momentum, particularly so because of the very real sense of the creation of a quantum computer and currently available applications of quantum cryptography. Consequently development of quantum data analysis is quickly becoming ever important, where state estimation, or, quantum tomography has been an area of active research in quantum information processing. The idea is that probabilities being one dimensional scalars, are determined by several quantum parameters. Often, quantum experiments are performed under the hypothesis that the prepared state is pure, although no statistical verification that the actual data comes from a pure state is statistically tested. In this afternoon of talks we develop statistical tests for purity of a state and apply it to some recently performed quantum experiments.

QuIGS Seminar Talk

Quantum data analysis IV

Abstract

The emergence of quantum information processing is gaining tremendous momentum, particularly so because of the very real sense of the creation of a quantum computer and currently available applications of quantum cryptography. Consequently development of quantum data analysis is quickly becoming ever important, where state estimation, or, quantum tomography has been an area of active research in quantum information processing. The idea is that probabilities being one dimensional scalars, are determined by several quantum parameters. Often, quantum experiments are performed under the hypothesis that the prepared state is pure, although no statistical verification that the actual data comes from a pure state is statistically tested. In this afternoon of talks we develop statistical tests for purity of a state and apply it to some recently performed quantum experiments.

QuIGS Seminar Talk

Special classes of completely positive linear maps

Abstract

Completely positive linear mappings between C*-algebras were originally developed in the 1950's as a special case of positive linear operators between matrix algebras. Within the last two decades, mathematical physicists have determined that completely positive maps play a crucial role in quantum information theory as structures which model information transfer between quantum systems. In this thesis we analyze two main classes of completely positive linear maps: the Schur maps which arise from the Schur matrix product and maps which are equal to their adjoint. In the analysis of Schur maps, we prove that many of the geometric properties of the convex set of correlation matrices may be derived from analysis of the set of Schur maps. We then give several necessary and sufficient conditions on characterizing self-dual completely positive linear maps. In doing so, we completely characterize the extreme points of the convex set of 2 X 2 unital self-dual completely positive trace-preserving linear maps. Finally, we use the concept of a conditional expectation to provide a general framework for some of the special classes of completely positive linear maps.

Advisory Committee:
R. Pereira (advisor)
D. Kribs

Examining Committee:
M. Cojocaru (Chair)
R. Pereira
D. Kribs
P. Fischer

QuIGS Seminar Talk

Bounded convex monoids

Abstract

We call a bounded convex subset of a Banach algebra which is closed under multiplication and contains a multiplicative identity a bounded convex monoid. Stochastic and doubly stochastic matrices, the unit ball of a Banach algebra, correlation matrices under the Schur product, and completely positive linear maps under composition are all examples of bounded convex monoids. These diverse examples, from many different domains, provide motivation for studying the theory of bounded and compact convex monoids. We present several results on the extreme points of certain classes of bounded convex monoids, and we give sufficient conditions for a compact convex monoid to contain an absorbing element. Results are presented on the convex sets held invariant under the action of certain convex monoids.

Advisory Committee:
R. Pereira (advisor)
D. Kribs (co-advisor)

Examining Committee:
M. Cojocaru (Chair)
R. Pereira
D. Kribs
D. Ashlock

QuIGS Seminar Talk

Higher rank numerical ranges of normal operators

Abstract

Research in higher rank numerical ranges has originally been motivated by problems in quantum information theory. The higher rank numerical range generalizes the classical numerical range of an operator. In this thesis, we present the basic structure and theory of higher-rank numerical ranges for unitary and normal operators and discuss some of their interesting geometric properties. We also introduce a general program to describe normal compressions of a normal (diagonal) operator and discuss an interesting pair matching problem related to 2 x 2 compressions.

Advisory Committee
J. Holbrook (advisor)
R. Pereira (co-advisor)
D. Kribs
P. Fischer

Examining Committee:
A. Lawniczak (Chair)
J. Holbrook
R. Pereira
M. Cojocaru

QuIGS Seminar Talk

Spectral theory of beurling-fourier algebras on compact groups

Abstract

For a compact group G, I will define the Beurling-Fourier algebras A?(G) on G, for weights ? on G. The classical Fourier algebra of G corresponds to the case where ? is the constant weight 1. When G is abelian, A?(G)=l1(G,?), the classical Beurling algebra on G. To describe the spectrum of A?(G), we require an abstract Lie theory which is built from Krein-Tannaka duality, and was formalized separately by McKennon, and Cartwright and McMullen, in the `70s. This Lie theory allows us to develop the complexification GC, even for non-Lie G. The Gelfand spectrum G? can always be realized as a subset of GC.

We consider the following questions. When, for a symmetric weight ?, is A?(G) symmetric? When is A?(G) regular on G? Can we gain information on sets of spectral synthesis in G??

This is part of joint work, with Jean Ludwig and Lyudmila Turowska. These algebras have also been developed, along different lines, by H.H.Lee and E. Samei.

This colloquium is jointly organized with the Department of Mathematics & Statistics colloquim series.

QuIGS Seminar Talk

What's wrong with qubits?

Abstract

We will talk about the structure of the ground space of two-body frustration-free Hamiltonians of qubit systems. It will be shown that there is always a product of single- or two-qubit states in the ground space. Furthermore, we give a characterization of the whole ground space in terms of span of product states. These results establish a no-go theorem for measurement based quantum computing with natural qubit state and a proof that the counting version of Quantum 2-SAT problem is #P-Complete.

QuIGS Seminar Talk

Linear preserver problems in quantum information theory

Abstract

The theory of linear preserver problems on complex matrices is introduced. We will investigate how three basic types of linear preserver problems are used in quantum information theory -- the problems of characterizing linear maps that (a) preserve certain matrix norms, (b) preserve the set of positive semidefinite operators, and (c) preserve the rank of matrices on which they act. We will use classical results on those linear preserver problems to prove that any operator that preserves the set of separable pure states is necessarily either a local unitary or the composition of a local unitary with the swap operator. We also extend this result to the natural Schmidt rank k generalization as well as the multipartite setting. Our results are used to characterize the isometries of a family of norms that have recently been investigated in quantum information theory, as well as the operators that preserve the geometric measure of entanglement.

QuIGS Seminar Talk

Quantum information: an algebraic geometric approach (part 1)

Abstract

We explore quantum entanglement, quantum channel capacity and quantum gate discrimination from an algebraic geometric point of view. Many intrinsic concepts in quantum information theory have their natural counterpoints in algebraic geometry. For example, the set of entangled states is the complement of Segre variety, the set of pure states with bounded Schmidt rank is isomorphic to some determintant variety, etc. From this point of view, a series of suprising results are derived.

QuIGS Seminar Talk

Quantum codes give counterexamples to the unique pre-image conjecture of the n-representability problem

Abstract

It is well known that the ground state energy of many-particle Hamiltonians involving only 2-body interactions can be obtained using constrained optimizations over density matrices which arise from reducing an N-body state. While determining which 2-body density matrices are "N-representable" is a computationally hard problem, all known extreme N-representable 2-body reduced density matrices arise from a unique N-body pre-image, satisfying a conjecture established in 1972. We present explicit counterexamples to this conjecture through giving Hamiltonians with 2-body interactions which have degenerate ground states that cannot be distinguished by any 2-body operator. We relate the existence of such counterexamples to quantum error correction codes and topologically ordered spin systems.

QuIGS Seminar Talk

Minimal and maximal operator spaces and operator systems in entanglement theory

Abstract

We examine k-minimal and k-maximal operator spaces and operator systems, and investigate their relationships with the separability problem in quantum information theory. We show that the matrix norms that define the k-minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k-positive linear maps and bound entanglement. Similarly, we investigate the k-super minimal and k-super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k-block positive operators and (unnormalized) states with Schmidt number no greater than k, respectively.

QuIGS Seminar Talk

Quantum information: an algebraic geometric approach (part 2)

Abstract

We explore quantum entanglement, quantum channel capacity and quantum gate discrimination from an algebraic geometric point of view. Many intrinsic concepts in quantum information theory have their natural counterpoints in algebraic geometry. For example, the set of entangled states is the complement of Segre variety, the set of pure states with bounded Schmidt rank is isomorphic to some determinant variety, etc. From this point of view, a series of surprising results are derived.

QuIGS Seminar Talk

Recent results and open problems in the analytic theory of polynomials

Abstract

Polynomials are omnipresent in mathematics. The analytic theory of polynomials is the study of polynomials using techniques in analysis. Despite being over 200 years old, it is still an active area of research. We will look at some of the recent progress in this area over the past decade. We will survey some of the work which connects the majorization order to polynomial inequalities and Mahler's measure. Sendov's conjecture and some of its generalizations will also be explored. Some ideas and problems in mathematical physics involving the relationship between polynomials and their zero sets on the Riemann sphere such as the Majorana representation and Atiyah's conjecture will be discussed. The relationship between these new results and classical concepts such as apolarity will be explored.

QuIGS Seminar Talk

On MDS error-correcting codes

Abstract

It will be shown that there is a close connection between the existence of higher dimensional stochastic (0,1) matrices and the existence of the corresponding MDS codes. Further results about the existence of MDS codes will also be presented. It is a joint work with J. P. R. Christensen.

QuIGS Seminar Talk

Bootstrapping persistence diagrams

Abstract

Multivariate topological data analysis has been gaining popularity as a way to find local clusters in high dimensional data. In this talk we will discuss ideas for incorporating bootstrap methods for obtaining confidence regions in persistent diagrams.

QuIGS Seminar Talk

Efficiency for van trees inequality

Abstract

After recalling some basic facts on Fisher information, the density information and Cramer- Rao inequality, we concentrate on an elegant generalization of the latter, apparently more familiar to engineers than to statisticians, which is due to van Trees (1968). We emphasize the multivariate case. The estimators for which equality is reached either for Cramer Rao or for van Trees inequalities are quite interesting to study. The ubiquity of the concepts of exponential family and Gaussian distribution appears there once again.

QuIGS Seminar Talk

Quantum Communications

Abstract

Shannon's information theory, which mathematically formalizes the problem of communication in the presence of noise, underlies the reliability of all modern communication technologies. Despite its unqualified engineering successes, information theory as originally formulated has a glaring foundational problem: while the theory is entirely classical, the underlying physics of communication devices is ultimately quantum mechanical. Far from a purely theoretical issue, the quantum nature of communications devices has practical consequences. For example, current quantum cryptography technologies enable private communication with a level of security that is classically impossible. I will give an overview of the recent rapid progress in developing a fully quantum theory of information and communications. Questions I will consider include: How much noise can we tolerate when trying to coherently transmit a quantum state? What is the relationship between entanglement and privacy? What are the fundamental resources in the quantum theory of information? And finally, can two useless resources be useful together? DR. GRAEME SMITH IS A CANDIDATE FOR THE QUANTUM INFORMATION POSITION Coffee and Donuts will be served at 2:30 p.m. in SCIE 1511. For further information, please contact: Prof. Brian Allen (519) 824-4120, ext. 56556, email: ballen@uoguelph.ca Prof. David Kribs (519) 824-4120, ext. 53038, email: dkribs@uoguelph.ca

QuIGS Seminar Talk

Quantum Error Correction via Codes Over GF(3)

Abstract

``Quantum Error Correction via Codes Over GF(4)" is one of the seminal papers in quantum coding theory. This 1996 work transforms the problem of finding binary quantum-error-correcting codes into the problem of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Ever since then, these codes, called additive codes or stabilizer codes, have dominated the research on quantum coding theory. There is now a rich theory of stabilizer codes, and a thorough understanding of their properties. Nevertheless, there are a few known examples of nonadditive codes which outperform any possible stabilizer code. In previous work, my colleagues and I introduced the codeword stabilized quantum codes framework for understanding additive and nonadditive codes. This has allowed us to find, using exhaustive or random search, good new nonadditive codes. However, these new codes have no obvious structure to generalize to other cases. A systematical understanding of constructing nonadditive quantum codes is still lacking. In this talk, we introduce the idea of constructing binary quantum- error-correcting codes via classical codes over the field GF(3). Quantum codes directly constructed this way are nonadditive codes adapted to the amplitude damping channel. These codes have higher performance than all the previous codes for the amplitude damping channel. We then further generalize this GF(3) idea to the case of constructing 'usual' binary quantum codes for the depolarizing channel, which leads to a systematical way of constructing good nonadditive codes that outperform best additive codes. Using this method, many good new codes are found. Particularly, we construct families of good nonadditive codes which not only ourperform any additive codes, but also asymptotically achieve the quantum Hamming bound. Generalization to nonbinary case is straightforward and families of good nonadditive nonbinary quantum codes are found. What is more, our new method can also be used to construct additive codes, and additive codes with better parameters than previous known are found. Based on joint work with Markus Grassl, Peter Shor, Graeme Smith, and John Smolin. DR. BEI ZENG IS A CANDIDATE FOR THE QUANTUM INFORMATION POSITION Coffee and Donuts will be served at 2:30 p.m. in SCIE 1511. For further information, please contact: Prof. Brian Allen (519) 824-4120, ext. 56556, email: ballen@uoguelph.ca Prof. David Kribs (519) 824-4120, ext. 53038, email: dkribs@uoguelph.ca

QuIGS Seminar Talk

Harnessing Decoherence: Simulation, Computation, and Transport in Open Quantum Systems

Abstract

It is a generally accepted fact that the laws of quantum physics could provide radically novel and more powerful methods of information processing with respect to any conceivable classically operating device. During the last two decades, Universal Quantum Computation has emerged as a standard paradigm to process information at quantum scales via constructing non-trivial unitary transformations. However, our ability to perform arbitrary quantum computation on a quantum system is restricted to its naturally available interactions. The other major problem is the extremely demanding physical requirements for performing fault-tolerant quantum error-correction to fight with decoherence. Here, I illustrate an alternative approach for utilizing "dynamical decoherence" in order to enhance our ability to measure, manipulate, and transport information within open quantum systems. This can be achieved by identifying and quantifying the limitations and computational capabilities of open quantum dynamics to generate a desired observable-induced target process. In particular, I will address the following questions: What are the efficient ways for characterizing the relevant parameters of an unknown quantum Hamiltonian system via minimal quantum measurement and classical data processing resources? How can we harness decoherence to optimally guide our open system to a desired target operation/state? What are the fundamental computational/simulation power of realistic open quantum systems? What are the natural/engineering limits for enhancement of quantum transport (e.g., energy transfer) in biological and nano-scale systems by exploiting an effective interplay of free Hamiltonian and the environmental effects? DR. MASOUD MOHSENI IS A CANDIDATE FOR THE QUANTUM INFORMATION POSITION Coffee and Donuts will be served at 2:30 p.m. in MACN 201. For further information, please contact: Prof. Brian Allen (519) 824-4120, ext. 56556, email: ballen@uoguelph.ca Prof. David Kribs (519) 824-4120, ext. 53038, email: dkribs@uoguelph.ca

QuIGS Seminar Talk

Universal Blind Quantum Computation

Abstract

When the technology to build quantum computers becomes available, it is likely that it will only be accessible to a handful of centres around the world. Much like today's rental system of supercomputers, users will probably be granted access to the computers in a limited way. This scenario prompts two important questions: "How can a user of a quantum computer be convinced that the output is correct?" and "Can this be done while keeping the user's input private?". We answer both questions by giving a protocol that allows a client Alice (who does not have any quantum computational resources or quantum memory) to interact with a server Bob (who has a quantum computer) in order for Alice to obtain the outcome of her target computation such that she is convinced that the output is correct and her privacy is preserved. This means that Bob learns nothing about Alice's inputs, outputs, or desired computation. The privacy is perfect, does not rely on any computational assumptions, and holds no matter what actions a cheating Bob undertakes. Alice only needs to be able to prepare single qubits randomly chosen from a finite set and send them to the server, who has the balance of the required quantum computational resources. After this initial preparation, Alice and Bob use two-way classical communication which enables Alice to drive the computation by giving single-qubit measurement instructions to Bob, depending on previous measurement outcomes. Ideally, we would want Alice's resources to be completely classical; we give a two-server version of our protocol which achieves exactly this.

DR. BROADBENT IS A CANDIDATE FOR THE QUANTUM INFORMATION POSITION

Coffee and Donuts will be served at 2:30 p.m. in SCIE 1511.

For further information, please contact:
Prof. Brian Allen (519) 824-4120, ext. 56556, email: ballen@uoguelph.ca
Prof. David Kribs (519) 824-4120, ext. 53038, email: dkribs@uoguelph.ca

QuIGS Seminar Talk

Quantum tomography with applications

QuIGS Seminar Talk

Schmidt norms for quantum states, Pt. 1

Abstract

In this talk we consider a family of vector and operator norms which we refer to as Schmidt norms. We build up their basic mathematical properties and use them to investigate several connections within quantum information theory. In particular, we will see that they can be used to help classify k-positive linear maps (and hence entanglement witnesses), and they are useful for approaching the problem of finding non-positive partial transpose bound entangled states. We show that the vector norms can be explicitly calculated, and we derive several inequalities in order to bound the operator norms and compute them in special cases. We show that one particular entangled Werner state is bound entangled if and only if a certain norm inequality holds on a given family of projections, and we use our inequalities to study that family of projections. We also develop a family of semidefinite programs that can be used to further bound the operator norms.

QuIGS Seminar Talk

Schmidt norms for quantum states, Pt. 2

Abstract

In this talk we consider a family of vector and operator norms which we refer to as Schmidt norms. We build up their basic mathematical properties and use them to investigate several connections within quantum information theory. In particular, we will see that they can be used to help classify k-positive linear maps (and hence entanglement witnesses), and they are useful for approaching the problem of finding non-positive partial transpose bound entangled states. We show that the vector norms can be explicitly calculated, and we derive several inequalities in order to bound the operator norms and compute them in special cases. We show that one particular entangled Werner state is bound entangled if and only if a certain norm inequality holds on a given family of projections, and we use our inequalities to study that family of projections. We also develop a family of semidefinite programs that can be used to further bound the operator norms.

QuIGS Seminar Talk

Schmidt norms for quantum states, Pt. 3

Abstract

In this talk we consider a family of vector and operator norms which we refer to as Schmidt norms. We build up their basic mathematical properties and use them to investigate several connections within quantum information theory. In particular, we will see that they can be used to help classify k-positive linear maps (and hence entanglement witnesses), and they are useful for approaching the problem of finding non-positive partial transpose bound entangled states. We show that the vector norms can be explicitly calculated, and we derive several inequalities in order to bound the operator norms and compute them in special cases. We show that one particular entangled Werner state is bound entangled if and only if a certain norm inequality holds on a given family of projections, and we use our inequalities to study that family of projections. We also develop a family of semidefinite programs that can be used to further bound the operator norms.

QuIGS Seminar Talk

Graph concatenation for quantum codes

Abstract

Graphs are closely related to quantum error-correcting codes: every stabilizer code is locally equivalent to a graph code, and every codeword stabilized code can be described by a graph and a classical code. For the construction of good quantum codes of relatively large block length, concatenated quantum codes and their generalizations play an important role. We develop a systematic method for constructing concatenated quantum codes based on "graph concatenation", where graphs representing the inner and outer codes are concatenated via a simple graph operation called "generalized local complementation." Our method applies to both binary and non-binary concatenated quantum codes as well as their generalizations. Salman Beigi, Isaac Chuang, Markus Grassl, Peter Shor, Bei Zeng

http://arxiv.org/abs/0910.4129

QuIGS Seminar Talk

Sampling theory, curved manifolds and function spaces with a finite spatial density of degrees of freedom

Abstract

Sampling theory can be described as the the study of the special reconstruction and interpolation properties of certain function spaces. For example, a reproducing kernel Hilbert space R is said to have a finite spatial density of degrees of freedom, or the 'sampling property', if there exists a countable set  := {xn}n2Z of spatial points with finite 'density' (and no finite accumulation point) such that any function f in R is uniquely determined and perfectly reconstructible from the values {f(xn)}. The classic example of such a function space is the Paley-Wiener space PW of -bandlimited functions. In information theory, sampling theory was first applied by Shannon to provide a crucial connection between discrete and continuous representations of information. In this talk I will discuss various generalizations of the classical sampling theory for Paley-Wiener spaces and their applications to quantum information theory and physics on curved manifolds. In particular, the problem of determining necessary and sufficient conditions for a subspace of L2(M), M a smooth manifold, to be a reproducing kernel Hilbert space with the sampling property will be examined.

QuIGS Seminar Talk

Some applications of results on the diagonals of matrices and matrix algebras

Abstract

We will look at the structure of unital self-adjoint matrix algebras (von Neumann algebras over finite dimensional Hilbert spaces) consisting entirely of matrices with uniform main diagonal entries. We will also examine some classical necessary and sufficient conditions that characterize the main diagonal entries of certain special classes of matrices (Hermitian or normal matrices with fixed spectrum, orthogonal matrices etc.) Applications of these results to quantum entanglement and completely positive linear maps will be discussed.

QuIGS Seminar Talk

Quantum walks on the hypercube

Abstract

The mixing time of a discrete-time quantum walk on the hypercube is considered. The mean probability distribution of a Markov chain on a hypercube is known to mix to a uniform distribution in time O(n log n). We show that the mean probability distribution of a discrete-time quantum walk on a hypercube mixes to a (generally non-uniform) distribution pi(x) in time O(n) and the stationary distribution is determined by the initial state of the walk. An explicit expression for pi(x) is derived for the particular case of a symmetric walk. These results are consistent with those obtained previously for a continuous-time quantum walk. The effect of decoherence due to randomly breaking links between connected sites in the hypercube is also considered.

QuIGS Seminar Talk

Topological Graphs and their C*-algebras

Abstract

The talk will be a survey of the speaker's work on topological graphs and their C*-algebras with emphasis on the relations with dynamical systems, Markov shifts, Cuntz-Krieger algebras and groupoids.

QuIGS Seminar Talk

Sparse Inertially Arbitrary Sign Patterns

Abstract

The inertia of a real matrix $A$ is the ordered triple $i(A)=(n_+,n_-,n_0)$ where $n_+$ is the number of eigenvalues of $A$ with postive real part, $n_-$ is the number of eigenvalues of $A$ with negative real part, and $n_0$ is the number with zero real part. A sign pattern $S$ is a matrix of order $n$ whose entries whose entries are in $\{+,-,0\}$. A real matrix $A$ has sign pattern $S$ is $sgn(A)=S$. A sign pattern $S$ is \emph{inertially arbitrary} if for every ordered triple $(n_1,n_2,n_3)$, with $n_1+n_2+n_3=n$, there is some real matrix $A$ with $i(A)=(n_1,n_2,n_3)$ having sign pattern $S$. We will describe some known inertially arbitrary patterns. We will describe some techniques used to determine that a pattern is inertially arbitrary and introduce some sparse inertially arbitrary sign patterns. We note some patterns which are inertially but not spectrally arbitrary.

QuIGS Seminar Talk

Algebraic Geometry, Convex Polytopes and Entanglement

Abstract

This talk will explore connections between convex polytopes with integral vertices and the geometry of varieties in C^n. No specialized background knowledge will be assumed. A connection with entanglement in quantum mechanics will be discussed.

QuIGS Seminar Talk

Diffusions on Euclidean motion groups (I)

Abstract

In this talk we review the representation theory and Fourier analysis on the Euclidean motion groups, give a convergent rate result about the nonparametric estimate of the density functions on the Euclidean motion groups.

QuIGS Seminar Talk

Diffusions on Euclidean motion groups (II)

Abstract

The connections between convolutions and diffusions on the Euclidean motion groups will be explored. Some numerical and stochastic simulation about the solution of the diffusions are also given.

QuIGS Seminar Talk

Coverings of skew-products and crossed products by coactions

QuIGS Seminar Talk

The multiplicative domain in quantum error correction

QuIGS Seminar Talk

Gaussian White Noise on Manifolds, Statistical Morse Theory and Topological Data Analysis

Abstract

We examine the estimation of a signal embedded in white noise on a compact manifold. A sharp asymptotic minimax bound is determined under the sup-norm risk over Holder classes of functions and generalizes similar results available for spheres in various dimensions. The estimation allows for the development of a statistical Morse theory using the level sets of the estimated function and together with the sup-norm bound allows the bounding of the Hausdorff distance in a persistence diagram in computational algebraic topology.

QuIGS Seminar Talk

Converses to Cauchy's interlacing theorem with applications to QIT

Abstract

Cauchy's interlacing theorem along with its converse (see for example Fan and Pall 1957) says that for strictly increasing real sequences a_1,a_2,�,a_n and b_1,b_2,�,b_m we have a_k less than b_k less than a_{k+(n-m)} (k=1,2,�,m) IFF the a_k's are eigenvalues of a Hermitian nxn matrix A and the b_k's are eigenvalues of an mxm principal submatrix of A. In this talk we'll consider versions of this theorem for normal matrices, applications to higher-rank numerical ranges, and consequences for QIT.

QuIGS Seminar Talk

Introductory Quantum Mechanics with a View to Quantum Computing Applications

Abstract

An introduction to quantum mechanics will be presented, to provide a foundation for quantum computing material that will be presented in future seminars.

QuIGS Seminar Talk

Identification of correctable codes in NMR quantum information processing

Abstract

Finding correctable encodings that protect against a physical quantum process is in general hard. Two main obstacles are that an exponential number of experiments are needed to gain complete information about the quantum process, and known algorithmic methods for finding correctable encodings involve operations on exponentially large matrices. This talk will introduce the concept of twirling a quantum channel over a set of unitary operations and in particular the case of twirling over the set of Pauli operators and qubit permutations. The resulting quantum channel will be shown to be characterized by a polynomial number of parameters that can be efficiently found experimentally. It will be shown that in some cases correctable codes for the original channel can be found using this limited information via a simple sorting algorithm.

QuIGS Seminar Talk

Concept Quantum Computers

Abstract

This is the second talk in a series of three. This talk will cover the various experimental approaches that physicists are taking with regard to constructing a working quantum computing device. Nuclear magnetic resonance based quantum computing will be covered in detail, from an experimental view point.

QuIGS Seminar Talk

Heat kernel smoothing and its application to brain imaging

Abstract

We present a novel data filtering/smoothing technique for manifold data. Gaussian kernel smoothing has been widely used in 3D brain images to smooth out noise. The Gaussian kernel weights an observation according to its Euclidean distance. When the observation lies on a highly convoluted brain surface, it is more natural to assign the weights based on the geodesic distance along the surface. On a curved manifold, the straight line between two points may not be the shortest distance so one may incorrectly assign less weight to closer observations. Therefore, smoothing data residing on manifolds requires constructing a convolution kernel that is isotropic along the geodesic curves. With this motivation in mind, we construct a heat kernel and develop heat kernel smoothing. We have validated the proposed method on a unit sphere, where the analytic form of heat kernel is explicitly given in terms of spherical harmonics. As an illustration, we present a case study of detecting abnormal structural anomaly in autism. The methodology is currently applied in smoothing noisy functional data for constructing a persistency diagram.

QuIGS Seminar Talk

NMR Techniques for Quantum Control and Computation

Abstract

This is the third talk in a series of three. This talk will cover the mathematics and physics behind nuclear magnetic resonance based quantum computing. Control, pulses techniques and experimental limitations will also be discussed.

QuIGS Seminar Talk

Deconvolution on the Euclidean motion group

Abstract

In this talk we review Fourier analysis on the real line R and compact manifolds such as the circle, sphere and special orthogonal group SO(3). Some results about the optimal rates of convergence for deconvolution density estimation in those cases will be discussed. We extend those results and obtain sharp minimax results on the non-commutative and non-compact Lie group SE(3), the 3-dimensional Euclidean motion group.

QuIGS Seminar Talk

Quantum entanglement and its properties

Abstract

Entanglement is one of the most mysterious features of quantum theory. Many years ago it was called by Schroedinger "the characteristic trait of quantum mechanics", since it does not have a direct classical analogue. Quantum entanglement is a key resource in the theory of quantum information, including quantum cryptography and quantum computation. Thus in recent years it became a subject of an active research. In this talk recent developments in the investigation of quantum entanglement will be reviewed. In particular we will analyze geometry of the set of entangled states for bipartite quantum systems and present some open mathematical problems on foliation of complex projective spaces and characterization of the set of positive maps closely related to quantum entanglement.

QuIGS Seminar Talk

Completely positive maps and passive quantum error correction

Abstract

MSc Defence. We begin with an examination of the assumptions made in the theory of passive quantum error correction. Physical motivation is discussed for the use of completely positive trace-preserving maps as a description of the evolution of quantum systems. We introduce the notions of noiseless subsystems and decoherence-free subspaces. Building on a generalization of the noiseless subsystem approach, we discuss recent results that provide a robust procedure for finding noiseless subsystems given a particular noise model. Unitarily correctable subsystems are introduced, which form a type of active quantum error correction that does not require a measurement. The key result for these systems is that we can apply our procedure for finding noiseless subsystems to the problem of finding unitarily correctable subsystems, in the case of a unital noise model.

QuIGS Seminar Talk

Adaptive estimation for inverse problems with noisy operators

Abstract

We consider an inverse problem with random noise where we want to estimate a function f. Moreover, we suppose that the operator A that we need to invert is not completely known. We observe its singular values but with some noise. To construct our estimator, we minimize a modification of an unbiased risk estimator. We obtain some non asymptotic exact oracle inequality. Then, we obtain asymptotic minimax results on some standard classes of coefficients.

QuIGS Seminar Talk

Ill-posed Inverse Problems on the Euclidean Motion Group

Abstract

Ill-posed deconvolution problems are defined on the Euclidean motion group (SE(2)), which is the semi-direct product of R^2 and SO(2). Rate calculations for the mean integrated square risk (MISE) for a number of models are presented. Complications of working on a non-compact, non-commutative group are discussed.

QuIGS Seminar Talk

A mathematical model for parallel computing

Abstract

As practical constraints have for now halted the exponential increase in processor speed, improvements in computational hardware are instead coming from an increase in parallelism. Parallel computing presents important practical and theoretical challenges, such as detecting deadlocks and determining the essential schedules. I will survey a mathematical model for concurrent parallel computing and some ideas for solving these problems.

QuIGS Seminar Talk

Geodesic Principal Component Analysis

Abstract

A general framework for principal component analysis (PCA) on quotient spaces that result from an isometric Lie group action on a complete Riemannian manifold is laid out. A forestry application to the study of shapes of tree stems is demonstrated.

QuIGS Seminar Talk

Ill-posed Inverse Problems on the Euclidean Motion Group - Part II

Abstract

Ill-posed inverse problems are defined on the Euclidean motion group (SE(2)), which is the semi-direct product of R^2 and SO(2). Rate calculations for the mean integrated square risk (MISE) for a number of models are presented. Complications of working on a non-compact, non-commutative group are discussed. This is a continuation of the initial talk given on Aug 15.

QuIGS Seminar Talk

Human Vibration Data and the Euclidean Motion Group

Abstract

Professor Dickey will present the Euclidean motion group data gathered in his lab. Time permitting, we will go to his lab and see a physical demonstration. Subject volunteers will be sought.

QuIGS Seminar Talk

Geometric Interpretation of Quantum Operations on the Bloch Sphere

Abstract

The Bloch sphere picture for representing single-qubit quantum states is reviewed. An algorithm for decomposing quantum operations (completely positive trace-preserving maps) into a set of four geometric transformations on the Bloch sphere is presented. Several examples are provided.

QuIGS Seminar Talk

Completely Bounded Norms in Quantum Information

Abstract

Completely bounded norms provide a natural measure for completely bounded and completely positive maps, and play an increasingly important role in quantum information science. We give a brief introduction to the mathematical theory underlying completely bounded maps and norms. We also present an algorithm for computing and estimating CB-norms, and describe an implementation of the algorithm on Maple. This talk is based on joint work with Vern Paulsen and David Kribs.

QuIGS Seminar Talk

Local and Global Mixture Geometry

Abstract

This talk looks at some geometric ideas which have found a rich application in statistical inference. In general the inference problem with mixture models can be very challenging and very different from the gold-standard of inference in exponential families. However a simplifying modelling assumption, which often holds in practical situations, exploits the underlying geometry of the mixture model and ensures that the resulting inference problem is tractable. This simplification gives rise to a class of models which I have called local mixtures. The talk describes these local mixture models and then investigates if the idea of local mixtures can be extended to much more general families.

QuIGS Seminar Talk

Faces of amenability

Abstract

I will discuss relationships between various aspects of amenability, from groups, to Banach algebras to operator spaces and maybe quantum groups. The talk will be themed around the classical theorem of B. Johnson: $G$ is an amenable group if, and only if, the group algebra $L^1(G)$ is an amenable Banach algebra.

QuIGS Seminar Talk

Quantum Reference Frames and the Classification of Rotationally-Invariant Maps

Abstract

We analyze, in general, the evolution of a quantum directional reference frame when it is used as a resource for performing quantum operations. We introduce the "moments" of a quantum reference frame, which serve as a complete description of its properties as a frame. We give a convenient representation for any map which is covariant with respect to an irreducible representation of SU(2). We use these results to investigate how many times a quantum directional reference frame represented by a spin-j system can be used to perform a certain quantum operation with a given probability of success. From our results follows a classification of the dynamics of spin-j system under the repeated action of any covariant map with respect to SU(2). Joint work with Lana Sheridan, Martin Laforest and Stephen Bartlett.

QuIGS Seminar Talk

Recasting the Elliott conjecture

Abstract

A C*-algebra is a norm-closed *-subalgebra of the bounded linear operators on a Hilbert space. In the mid 1970s, Elliott discovered that a large class of these algebras were determined up to isomorphism by their K-theory. He later initiated a program to investigate this phenomenon in depth, and conjectured that simple separable amenable C*-algebras would be classified up to *-isomorphism by their K-theory and traces. In this talk we will trace the history of Elliott's program, and discuss some recent developments which have dramatically changed our view of the structure theory of separable amenable C*-algebras. The talk will be accessible to a general mathematical audience.

QuIGS Seminar Talk

Codeword Stabilized Quantum Codes

Abstract

I will describe an approach to quantum error correcting code design that encompasses additive (stabilizer) codes, as well as all known examples of nonadditive codes with good parameters. Our quantum codes are described by a single graph state, together with a classical binary code. The graph state may be thought of as mapping quantum errors to an exotic classical error model, which we must then arrange to correct with our classical code. Stabilizer codes arise naturally as those quantum codes with a linear classical code. I will show how to use this framework to generate new codes with superior parameters to any previously known, as well construct encoding circuits for all codes within this family. Based on arXiv: 0708.1021, which is joint work with Andrew Cross, John Smolin, and Bei Zeng.

QuIGS Seminar Talk

Shrinkage Estimation with Monotone Incomplete Multivariate Normal Data

Abstract

This talk concerns shrinkage estimators for the mean vector of a normal distribution with monotone incomplete data. The Stein phenomenon was revolutionary at the time that it first appeared: As measured by squared-error loss, one could improve on the natural estimator. Today, the phenomenon has been widely studied, enormous numbers of papers have been written, along with many books and monographs. The maximum likelihood estimator (MLE) for the mean vector with monotone incomplete multivariate normal data has a very simple form and has many nice properties, but its distribution is very complicated. We introduce its distribution which is obtained by Chang and Richards and show that the MLE is inadmissible by considering a dominating shrinkage estimator.

QuIGS Seminar Talk

On the quantum computation of partition functions for classical statistical physics

Abstract

I will discuss some recent results on the quantum computation of the Potts model partition function. I will outline the scheme and point out how we used connections between classical coding theory, classical statistical physics and Gauss sums. I will also give a peek at a new result about a certain class of quantum circuits that can be simulated classically. This result relies on a recently discovered connection between instances of the Ising model and quantum circuits.

QuIGS Seminar Talk

Complementarity in Quantum Cryptography and Quantum Error Correction

Abstract

In this talk, I'll introduce work which shows that basic notions in quantum cryptography and quantum error correction are complementary to each other. Error-correcting codes for quantum channels are the key vehicles used to avoid noise such as decoherence induced by physical attempts to build quantum computers. Private codes for quantum channels play a central role in the development of private quantum communication networks designed to prevent adversarial attacks by eavesdroppers. It turns out that a code is private for a channel precisely when it is correctable for a complementary channel, and there is a straightforward algebraic recipe that allows to move between the two perspectives. Moreover, an approximate version of the relationship can be quantified in terms of diamond (or completely bounded) norms for channels. I'll begin with an introductory look at the two notions, then formulate the main result and discuss potential cross-fertilization between the disciplines motivated by the work. This talk is based on joint work with Dennis Kretschmann (TU Braunschweig) and Robert Spekkens (Cambridge).

QuIGS Seminar Talk

Statistical Estimation on Groups of Rigid Motions

Abstract

There are a number of applications where the data obtained is best described as contained in a (low dimensional) group of rigid motions. Analysis of this type of data has often ignored this fact and failed to take advantage of the properties inherent in the motion groups. Through some examples, I justify this perspective and provide some results that suggest this is a more statistically sound method of analysis for this type of data. Density estimation techniques, as well as error rate estimates will be discussed briefly.

QuIGS Seminar Talk

Higher-rank Numerical Ranges, A Survey

Abstract

The higher-rank numerical ranges, created in connection with quantum error correction, have taken on a life of their own. There is now a substantial body of results, including a striking generalization of the classical Toeplitz-Hausdorff theorem about the ordinary numerical range: all the higher-rank numerical ranges are convex. We shall see how this generalization is proved and give a survey of other recent results.

QuIGS Seminar Talk

C*-algebras, elementary operators and completely positive linear maps

Abstract

We will give an introduction to C*-algebras and will discuss an interesting bijection between the set of completely positive elementary operators on a prime C*-algebra and the set of Hilbert subspaces of the C*-algebra. This bijection is used to classify all the elementary operators which are also projections onto a C*-subalgebra. We will also discuss some possible relationships of relevance in quantum information between the cone of positive linear maps on a C*-algebra, the cone of completely positive linear maps on the C*-algebra and the cone generated by the unitary similarities on the C*-algebra.

QuIGS Seminar Talk

Determinantal Representations of Polynomials

Abstract

A determinantal representation of a homogeneous polynomial p(x_0,...,x_n) (or a hypersurface it defines) is given by matrices A_i such that Det(sum A_i x_i) = p. We will discuss their existence and parametrization. Special emphasis will be on determinantal representations of cubic curves and cubic surfaces.

QuIGS Seminar Talk

Concentric polygons in the plane, rank 2 numerical ranges, and quantum error correction

Abstract

A key result in quantum error correction characterizes those subspaces C yielding correctable codes (for a quantum channel E) as C such that for each pair a,b there exists a complex ?_ab with P_c E_a* E_b P_c = ?_ab P_c where the E_a are the error operators defining E. This motivates the study of the �rank k numerical ranges� of a given N � N matrix T: ?_k(T) = {? ? C : PTP = ?P where P is a projection of rank k}. Choi, Kribs, and Zyczkowski initiated work on these higher�rank numerical ranges, and they have made valuable advances in the theory and applications of such ranges. Note that ?_1(T) is the familiar numerical range W(T) = {(Tu,u) : ||u|| = 1}. We clarify the structure of the sets ?_k(T), with a focus on the case where k = 2 and T is unitary. This allows the analysis of bi�unitary channels on a state space of arbitrary dimension N. In the course of this investigation we encounter some curious properties of convex polygons in the plane.

QuIGS Seminar Talk

Universal quantum computation on graph states

Abstract

Quantum computation is commonly phrased as (1) prepare "zero" states, (2) apply an appropriate unitary transformation (as a quantum circuit) and (3) apply single-qubit quantum measurements. More recently, alternative prescription was proposed by Raussendorf and Briegel as (1') prepare certain initial "graph-state," and (2') apply adaptive single-qubit quantum measurements. In this seminar, a simple and systematic derivation of the above "graph-state model of quantum computation" will be given. Questions on universal classes of random graph states will be discussed. Joint work with Panos Aliferis, Andrew Childs, and Michael Nielsen.

QuIGS Seminar Talk

Statistical Inverse Problems on the Euclidean Motion Group

Abstract

The estimation of an indirect signal embedded in white noise on the Euclidean Motion Group, SE(2), is investigated. The linear minimax risk is calculated and related to the overall minimax risk.

QuIGS Seminar Talk

Additivity and p-norms in quantum information

Abstract

A few years ago, it was realized that several long-standing conjectures about entanglement of formation and the additivity under tensor products of one type of capacity, or the minimal entropy, of a quantum channel are equivalent. Moreover, they would follow from a related conjecture about the multiplicativity under tensor products of norms which measure the optimal output purity of a channel. The term channel refers to a completely positive, trace-preserving map acting on algebra of observables or, equivalently for finite dimensional systems, the set of all states; this map models the noise acting on a quantum system. The conjectures about p-norm multiplicativity have also proved challenging, but led to several new developments. These include applications of operator spaces to quantum systems, and the notion of a conjugate channel. This talk will try to explain these concepts and, give a feeling for the flavor of the subject, and a brief summary of what is known about these conjectures.

QuIGS Seminar Talk

Random quantum states and transformations

Abstract

In this talk I will give an overview of how randomly drawn quantum states and quantum operations (from the appropriate "uniform" measures) are useful in various quantum information protocols. I will then discuss how a generalization of "spherical t-designs" to the quantum setting provides a useful classification scheme for the degree of randomization provided by different subsets of states or transformations. Time permitting I will describe some explicit constructions.

QuIGS Seminar Talk

Quantum Causal Histories

Abstract

The goal of this talk is to give an introduction to the "Quantum Causal History" formalism from quantum cosmology and gravity.

QuIGS Seminar Talk

Fourier analysis on vector bundles

QuIGS Seminar Talk

Sharp adaptation for statistical inverse problems on vector bundles over a manifold

QuIGS Seminar Talk

The equivalence of measurement-based quantum computation and fault-tolerance via teleportation

Abstract

It has been known since 1993 that, by sharing a maximally entangled quantum state, two parties can teleport an unkown quantum state perfectly after exchanging only two bits of classical information. Since then, variants of this protocol have been used to demonstrate how to: (1) construct fault-tolerant quantum circuits; (2) perform computation using only graph state preparation and single qubit measurements; (3) extract error syndrome information from encoded quantum states. I will review these results, and demonstrate how they can be assembled together to give us a natural view of teleportation-based fault-tolerant computation as computation using modified graph states and single qubit measurements. Joint work with V. Danos, E. Kashefi and H. Ollivier.

QuIGS Seminar Talk

Stabilizer formalism for quantum error correction I

QuIGS Seminar Talk

Stabilizer formalism for quantum error correction II

QuIGS Seminar Talk

On higher rank numerical ranges and quantum error correction

QuIGS Seminar Talk

Discussion of higher rank numerical ranges and quantum error correction

QuIGS Seminar Talk

I. The Toeplitz-Hausdorff Theorem II. Quantum error correcting subsystems

QuIGS Seminar Talk

On noiseless subsystems

QuIGS Seminar Talk

Stabilizer formalism for operator quantum error correction

QuIGS Seminar Talk

Operator Algebra Quantum Error Correction

Abstract

Operator algebras are used in physics to represent quantum observables. We characterize the algebras of observables conserved by a quantum channel and show that they define all the noiseless subsystems. The resulting theory of error correction contains all previously known error correction schemes, with bonus.

QuIGS Seminar Talk

Quantum Dynamics as Generalized Conditional Probabilities

Abstract

In this talk I will describe an extension of the Choi-Jamiolkowski isomorphism between bipartite quantum states and completely positive maps that is closely analogous to the relationship between joint probabilities, conditional probabilities and stochastic dynamics in classical probability theory. This can be used to prove direct correspondences between the various types of no-cloning/no-broadcasting theorem and statements about the monogamy of entanglement, both of which are thought to be fundamental to our understanding of how quantum information differs from its classical counterpart. If time permits, I will discuss a further application to the question of when the dynamics of a system initially correlated with its environment can be described by a Completely Positive map. This talk is based on quant-ph/0606022.

QuIGS Seminar Talk

From Passive to Active Quantum Error Correction

Abstract

Recent advances in quantum computing include the development of a structure theory for passive quantum error correction. The theory has led to a computational algorithm that can be applied to arbitrary error models. Passive error correction may be regarded as a special case of active error correction. Many of the fundamental problems in active error correction are simply unsolvable in complete generality. Thus, the possibility of extending the passive theory to special cases of the active theory is of immediate interest, for both theoretical and experimental reasons. I will discuss such an extension discovered recently in joint work with Robert Spekkens (U. of Cambridge).

QuIGS Seminar Talk

Deconvolution density estimation on SE(2)

Abstract

This talk will discuss deconvolution density estimation on the Euclidean motion group SE(2).

QuIGS Seminar Talk

Arbitrage and Geometry

Abstract

Arbitrage is a fundamental notion in mathematical finance, and making the "no free lunch" assumption that arbitrage opportunities in the marketplace are unavailable has played a fundamental role in financial economics. The Arbitrage Theorem, an important example of a theorem of the alternative, will be explained for the case of an m x n payoff matrix corresponding to m scenarios and n investments. For an appropriate definition of a random payoff matrix, we use geometric reasoning to show what the probability of an arbitrage opportunity is. As a corollary, we conclude that if the number of scenarios m is even, and the number of available investments is m/2 then the probability of an arbitrage opportunity is 1/2. (This is joint work with Edward R. Scheinerman)

QuIGS Seminar Talk

Convexity of higher-rank numerical ranges

Abstract

The rank-k numerical range of an NxN matrix T is the set of complex z such that PTP=zP for some rank-k orthogonal projection P. Convexity results for these numerical ranges extend the Toeplitz-Hausdorff Theorem for the classical numerical range W(T), which corresponds to k=1. We discuss a variety of such results. The notion of higher-rank numerical range was suggested by problems of error correction in quantum computing, and our results have applications to that field.

QuIGS Seminar Talk

Exact inference for multivariate normal populations with monotone incomplete data

Abstract

This talk is based on joint research with Wan-Ying Chang, Washington Department of Fish and Wildlife, Olympia, WA. It is well known that incomplete data in multivariate data complicates the ensuing likelihood inference. The resulting formulas, confidence levels, and distribution theory are more complicated, and unbiasedness properties of hypothesis tests on mean vectors and covariance matrices generally are believed to be beyond reach. We shall show that many of the familiar formulas which comprise classical multivariate analysis for normal data can be generalized to the setting of monotone incomplete data. In that setting, we shall deduce generalizations of the usual normality of the sample mean vector and the Wishart distribution; derive elliptical confidence regions for the population mean vector, and bounds bounds on the corresponding confidence level; obtain simultaneous confidence intervals for the components of the population mean vector; determine the unbiasedness of some likelihood ratio tests on the population mean vector and covariance matrix; and explain how a host of other results may be obtained.

QuIGS Seminar Talk

Graphical Markov Models: An Introduction

Abstract

Graphical Markov models represent a set on distributions compatible to a graph comprised of vertices (variables) and edges. In short, these graphs encode the (conditional) independencies that hold among a set of variables. They were first introduced as path diagrams in the social sciences, and as structural equation models in econometrics. Today, we will take a look at how conditional independence relations are encoded by such graphs, and how they can be used to understand the effect of interventions on a system.

QuIGS Seminar Talk

Computational and Convexity Problems for Higher-Rank Numerical Ranges with Applications to Quantum Error Correction

Abstract

MSc Project Seminar

QuIGS Seminar Talk

How big are 2x2 matrices?

Abstract

The smallest matrices that are worthy of the name are 2x2, but these nevertheless present several surprisingly difficult problems. We focus here on three such problems, associated with three well-known and natural ways of measuring the size of a matrix: the operator norm, the numerical radius, and the Schur(multiplier) norm.

QuIGS Seminar Talk

A unified and generalized approach to quantum error correction

Abstract

The possibility of protecting quantum information against undesireable noise has been a major breakthrough for the field of quantum computing, and has opened the door for potential practical applications. I shall discuss recent work with Raymond Laflamme, David Poulin and Maia Lesosky that introduces a new paradigm for quantum error correction -- called "operator quantum error correction". This approach brings the known techniques -- i.e. the standard model, the method of decoherence-free subspaces, and the noiseless subsystem method -- under a single umbrella. It also opens up new possibilities for the error correction of quantum operations, most importantly it includes a generalized notion of noiseless subsystems.

QuIGS Seminar Talk

The Universe as a Quantum Computer

Abstract

I will give an introduction to quantum gravity and current, network-based models of Planck scale spacetime. The big open problem is to relate these background-independent models to the familiar spacetime and known physics. I then show how these models can be mapped to quantum superpositions of a quantum computer. This suggests that quantum information theory is ideally suited to address questions in a background-independent situation. I conclude by exploring some such possibilities.

QuIGS Seminar Talk

Wave propagation in curved spacetime

Abstract

I will first review the theory of Green's functions for the wave equation in four-dimensional curved spacetime, which partially goes back to Hadamard. I will then present some applications, including the power-law decay of fields in asymptotically-flat spacetimes, the hereditary effect in radiation theory, and the quasi-normal oscillations of black holes.

QuIGS Seminar Talk

Shannon sampling: introduction and applications from information technology to cosmology

Abstract

Shannon sampling theory shows that, in certain circumstances, functions defined (for example) on IR are fully determined by their values at discrete sets of points with finite spacing. Shannon's basic sampling theorem is in fact one of the most frequently applied theorems: it is what allows for example CD players to reconstruct the continuous music signal from its discrete amplitude samples that were recorded on a CD. I will review the basic concepts and theorems of sampling theory along with some traditional applications in information technology. I will also address recent applications in mathematical physics that range from quantum information theory to cosmology.

QuIGS Seminar Talk

An Introduction to Morse Theory

Abstract

Morse theory can be considered a global Calculus of Variations. The main Morse Lemma provides a concrete local expression for multivariate real valued functions at nondegenerate critical points. Although the proof is relatively straightforward, the interpretation provides insights into global topological properties based entirely on local features of critical points and has been used to classify compact manifolds. This will be an introductory talk and will be aimed toward a general mathematical audience.

QuIGS Seminar Talk

Some reflexive Lie semigroup operator algebras

Abstract

The Beurling-Lax theorem describes the lattice of invariant subspaces of the operator algebra $H^\infty$ in terms of inner functions. Sarason proved in 1966 that this operator algebra is reflexive, meaning that its invariant subspace lattice is sufficiently rich that we can recover the algebra from it. More recently, Katavolos and Power coined the term "Lie semigroup operator algebra" and studied two examples related to $H^\infty$ which also have interesting invariant subspace lattices. We will introduce these operator algebras and subspace lattices and outline a proof that they too are reflexive.

QuIGS Seminar Talk

The Stinespring dilation theorem and von Neumann's inequality

Abstract

I will begin this talk with a direct proof of von Neumann's classical inequality for polynomials and contraction operators on Hilbert space. Next, I will cover the Stinespring dilation theorem which states that every completely positive map can be viewed as a compression of a *-homomorphism. This theorem is the mathematical basis for the description of evolution of open quantum dynamical systems. As a corollary to this theorem, an additional derivation of the von Neumann inequality will be given.

QuIGS Seminar Talk

Geometry Statistics and Weyl I

Abstract

This will be the first of four talks where I will bring together my research which interlinks geometry with statistics. The upcoming talk, will be an introduction to manifolds, spectral geometry, Weyl's asymptotic eigenvalue calculations and some new results on asymptotic sums of powers of eigenvalues and sums of powers of exponentials of eigenvalues.

QuIGS Seminar Talk

Geometry Statistics and Weyl II

Abstract

This will be the second of four talks where I will bring together my research which interlinks geometry with statistics. The upcoming talk, will continue with Weyl's asymptotic eigenvalue calculations and asymptotic sums of powers of eigenvalues and sums of powers of exponentials of eigenvalues. We will then introduce the statistical problem and minimax risk.

QuIGS Seminar Talk

Bioinformatics, Microarrays, Multiclustering

Abstract

This talk will introduce the problem of microarray data analysis and present a novel clustering algorithm used in such analysis. A micro-array is any of a number of technological platforms capable of reading the activity level of thousands of genes at the same time. This technology produces data in quantities that challenge standard statistical and exploratory methods. Clustering of the data is a standard technique used to reduce the effective size of data. This talk will explain some popular clustering methods and then outline a novel method called multiclustering.

QuIGS Seminar Talk

Geometry Statistics and Weyl III

Abstract

This will be the third of four talks where I will bring together my research which interlinks geometry with statistics. The upcoming talk, will introduce the statistical problem and begin the calculation of the minimax risk.

QuIGS Seminar Talk

Passive Quantum Error Correction

Abstract

In this talk, I will discuss some of my recent work on error correction problems in quantum computing. I'll focus on the fundamental passive technique, which involves "decoherence-free subspaces" and "noiseless subsystems". In this method, symmetries within the noise of quantum channels are used to encode initial states in sectors of the system Hilbert space that are immune to the errors of the channel. The underlying mathematics involves completely positive maps and operator algebras.

QuIGS Seminar Talk

25 Years After

Abstract

The Weyl conjecture (1911) about the second term in the asymptotics of eigenvalue distribution function for Laplace operator in a domain with the boundary was proven by the speaker in 1979. In this talk I will consider the history of this topic - until modern development.

QuIGS Seminar Talk

Two-Uniform Frames, Graphs and Deletion Errors

Abstract

The central topic of this talk is the use of two-uniform (equiangular) frames for the expansion of vectors in a finite-dimensional Hilbert space. Holmes and Paulsen showed that, when they exist, these frames are optimal for minimizing the norm of the reconstruction error when up to two of the frame coefficients are set to zero (deleted). Here, we discuss a graph-theoretic haracterization of equivalence classes of two-uniform frames and use it to derive error bounds when more than two deletions occur. In certain cases, we can identify equivalence classes of optimal two-uniform frames for higher numbers of deletions.

QuIGS Seminar Talk

Normal Dilations

Abstract

How could complicated phenomena often become tractable events in a larger environmenent with better views? Along these lines, we see that normal matrices may have all sorts of non-normal submatrices. But, for better purposes, we always want to dilate non-normal matrices to appropriate normal matrices (of larger sizes) with elegant features.

QuIGS Seminar Talk

Sharp Adaptation for Statistical Inverse Problems on Manifolds with Application to Medical Imaging

Abstract

This talk will examine the estimation of an indirect signal embedded in white noise over a compact manifold. It will be shown that the sharp minimax bound is determined by the degree to which the indirect signal is embedded in the linear operator. Thus when the linear operator has polynomial decay, recovery of the signal is polynomial, whereas if the linear operator has exponential decay, recovery of the signal is logarithmic. The constants are determined for both of these classes and adaptive sharp estimation is also carried out. In the polynomial case a blockwise shrinkage estimator is needed while in the exponential case, a straight projection estimator will suffice. Some of the results depend on aspects of spectral geometry and in particular, the asymptotic eigenvalue calculations associated with H. Weyl. The framework of this analysis include applications to medical imaging where the manifold is taken to be the two-dimensional sphere. The first application, cone-beam image reconstruction, is associated with single photon emission computed tomography (SPECT). The statistical formulation of the problem involves Compton scattering through the Klein-Nishina distribution. A second application, and also dealing with medical imaging, is that associated with diffusion magnetic resonance imaging (MRI). The data comes as three-dimensional euclidean Fourier transforms, however, of particular interest is the angular (directional) portion since medically, this can clinically reveal certain anomalies in the brain arising from trauma for example. This angular composition can also be structured as a statistical inverse problem on the sphere.

QuIGS Seminar Talk

On the classification of C*-algebras of type AF, AI and AT

Abstract

I will give an overview of some of the work that has been done on the classification of C*-algebras along the lines of Elliott's classification program. I will start with Elliott's result from 1976 on the classification of approximately finite (AF) C*-algebras and then move on to more general classes of C*-algebras. In each case I will sketch the main ideas of the proof.

QuIGS Seminar Talk

On the origin of quantum advantages in information processing

Abstract

In order to understand why there exist information processing tasks for which quantum protocols outperform their classical counterparts, it is useful to have a proper conception of the differences between quantum and classical theories. Although the formalism of quantum theory closely approximates the formalism one uses to describe a classical theory with a constraint on how much information can be acquired by an observer, there are subtle differences between the two. These differences are captured by so-called "no-go theorems" that establish the impossibility of modelling quantum theory by a noncontextual hidden variable theory. I shall discuss recent progress in this area and speculate about its possible significance for information processing.

QuIGS Seminar Talk

Surrounded by operator algebras

Abstract

In this talk we will give a brief introduction to the study of operator algebras. We will outline a number of simple examples from both the finite and infinite dimensional settings and, time permitting, make connections with wavelet analysis, quantum computing and quantum gravity.

QuIGS Seminar Talk

Schur norms and Toeplitz matrices

Abstract

This talk is related to (but not dependent on) my seminar last October: The Schur norm of a matrix. I'll review the definition and basic properties of Schur norms and illustrate their role in perturbation problems and error analysis. Computing the Schur norm of an arbitrary matrix may be a formidable task, but for Toeplitz matrices (matrices with constant diagonals) there are several helpful techniques. We'll treat in particular the Schur norm corresponding to upper-triangular truncation of a matrix.

QuIGS Seminar Talk

Surrounded by operator algebras II

Abstract

This will be a continuation of my introductory talk from two weeks ago. I will present more examples of operator algebras, this time focusing on the infinite dimensional case. Time permitting, connections will be made with function theory, quantum gravity, wavelet analysis and dynamical systems.

QuIGS Seminar Talk

Nonsmooth Analysis of Singular Values

Abstract

The singular values of a rectangular matrix are nonsmooth functions of its entries. In this work we study the nonsmooth analysis of absolutely symmetric functions of singular values. (Absolutely symmetric functions are invariant under permutations and sign changes of its arguments.) In particular we give simple formulae for the regular, the limiting, the horizon, the proximal and the Clarke ubdifferentials of such functions, where in the last case we consider both cases when the function is either locally Lipschitz or just lower semicontinuous. We illustrate the results by calculating the various subdifferentials of individual singular values. In additional applications we derive von Neumann's trace inequality for singular values and give a nonsmooth proof of Lidskii's theorem for weak majorization. (Based on joint work with Adrian Lewis, SFU.)

QuIGS Seminar Talk

Commutators and projections in C*-algebras

Abstract

Let A be a norm closed selfadjoint algebra of continuous linear operators on a Hilbert space H. A commutator in A is an element of the form [x,y] := x y - y x. A projection is a selfadjoint idempotent. There are many instances when it is easy to show that the linear span of the projections in A forms a dense subset of A. It can be much more difficult to determine whether or not the linear span is the whole algebra. A second question one might ask is: suppose A admits a trace functional. Is every element with trace zero a commutator, as is the case for the algebra of n x n matrices? In this talk we shall discuss these problems, and show how a positive answer to the second can be useful in answering the first.

QuIGS Seminar Talk

Quantum error detection and correction

Abstract

In this talk I will give an introduction to the basic methods of error detection and error correction in quantum computing. A 'quantum code' is a subspace of Hilbert space, or equivalently, a projection operator on the space. The errors, or noise, associated with a quantum transmission are given by operators on the space. Error detection and correction for a fixed code may be presented in terms of simple relations satisfied by the code projection and the noise operators for a transmission. I will outline two of the main theorems in this setting and discuss some examples.

QuIGS Seminar Talk

Decoherence-full subsystems and the cryptographic power of private shared reference frames

Abstract

In this talk, I will discuss ways in which parties exchanging quantum systems can protect information from an eavesdropper by making use of secret correlations that the eavesdropper does not share. Such correlations can come in the form of a classical secret key or quantum entanglement. They may also come in the form of a private shared reference frame, which is the case that I will consider in detail. An example of a shared reference frame is a pair of synchronized clocks, or a pair of gyroscopes that are aligned. I will show how the eavesdropper's lack of information about the private reference frame can be modelled by a superoperator that has a very simple form. For a given such superoperator, quantum information theorists have found it useful to introduce the concept of a decoherence-free subystem. This is a factor space of a subspace for which every state is invariant under the action of the superoperator. I shall introduce a dual concept, that of a *decoherence-full* subsystem, which is a factor space of a subspace for which every state is mapped to the completely mixed state under the action of the superoperator. I will show how identifying these subsystems helps to solve the problem of optimizing the amount of information that can be hidden from the eavesdropper.

QuIGS Seminar Talk

Operator algebras generated by isometries

Abstract

I will give an overview of the study of special operator algebras generated by isometries, from work of Beurling in the 1950s to some of my own work on free semigroup algebras over the past decade.

QuIGS Seminar Talk

Statistics and Quantum Computing

Abstract

There has recently been interest among statisticians with respect to the statistics of quantum computing. The idea is that one can present the density matrix as a parameterized statistical model and thereby relate it to classical statistical methods which includes such things as exponential models and various notions of information, among others. This talk will give a brief survey of this approach as well as a possible data analytic way of statistically analyzing quantum encryption.

QuIGS Seminar Talk

Finite dimensional C*-algebras and quantum computing

QuIGS Seminar Talk

Introduction to directed graph operator algebras

Abstract

Every directed graph naturally determines a family of operators, a projection for each vertex and a partial isometry for each directed edge, that satisfy relations suggested by the graph. The study of the associated operator algebras, so-called 'directed graph operator algebras', was initiated in the 1980's as an attempt to study 'topological Markovian flows' in dynamical systems. Through the intervening years the study of these algebras has turned into an entire subdiscipline of functional analysis. In this talk I'll give an introduction to directed graph operator algebras. After describing how operators can be naturally associated with graphs, I'll outline a number of examples from the fundamental class of 'non-selfadjoint' graph algebras. Examples from this class include analytic function algebras, matrix function algebras, crossed product algebras, algebras with free behaviour and examples that mix these possibilities. I shall finish by touching on aspects of the theory for these algebras that has been developed to this point.

QuIGS Seminar Talk

Quantum Computers and Decoherence: Exorcising the Demon from the Machine

Abstract

Recently discovered algorithms indicate that quantum computers may one day enable exponentially faster computation than is fundamentally possible using their classical counterparts. The realization of this promise hinges above all on the ability to protect quantum computers against the deleterious effect of the interaction with their environment, leading to decoherence. A decohered quantum computer is equivalent to a badly malfunctioning classical computer. In this talk the what's, why's, how's and problems of quantum computation will first be briefly reviewed. This will be followed by a proposal for a solution of the decoherence problem, with applications to atomic and solid-state quantum computing. Two key ingredients of the proposed solution are "decoherence-free subspaces" and "dynamical symmetrization". In the decoherence-free subspace approach, a symmetry in the system environment interaction is sought. Provided such a symmetry exists quantum information appears as a conserved quantity, that cannot decohere. Dynamical symmetrization is a method that uses strong and fast pulses (similar to the spin-echo effect) in order to enforce such a symmetry on the system-environment interaction. The combination of decoherence-free subspaces and dynamical symmetrization is a powerful tool for fighting decoherence, and is fully compatible with quantum computation in a wide range of proposed implementations of quantum computers. References: - "Concatenated Dynamical Decoupling", K. Khodjasteh and D.A.L., quant-ph/0408128. - "Creating Decoherence-Free Subspaces with Strong and Fast Pulses, L.-A. Wu and D.A.L., Phys. Rev. Lett. 88, 207902 (2002) - "Reducing Constraints on Quantum Computer Design by Encoded Selective Recoupling", D.A.L. and L.-A. Wu, Phys. Rev. Lett. 88, 017905 (2002). - "Universal Fault-Tolerant Quantum Computation on Decoherence-Free Subspaces", D. Bacon , J. Kempe, D.A.L., and K.B. Whaley, Phys. Rev. Lett. 85, 1758 (2000). - Review paper: "Decoherence-Free Subspaces and Subsystems", D.A.L. and K.B. Whaley, quant-ph/0301032.

QuIGS Seminar Talk

On homomorphisms of Fourier algebras

Abstract

The 'Fourier algebra' of a group, frequently a locally compact topological group, is an object that in some sense gneralises the duality theory which underlies classical Fourier analysis. For example, if $\mathbb{T}$ is the unit circle in the complex plane, then the Fourier algebra on $A(\mathbb{T})$ represents exactly the set of continuous functions on $\mathbb{T}$ whose Fourier coefficients are summable. In another description, $A(\mathbb{T})$ is isomorphic to $\ell^1(\Zee)$, where $\ell^1(\Zee)$ is the space of Laurent series which is an extension of the group algebra on $\Zee$. In 1959, P.J. Cohen published a famous result characterising continuous homomorphisms between group algebras of locally compact abelian groups. With M. Ilie, I have extended this characterisation to Fourier algebras of all amenable locally compact groups. I present some evidence to suggest that our result is the best possible.

QuIGS Seminar Talk

Some remarks about polynomial dynamical systems and their fibre bundle structure.

Abstract

In this talk we overview our very recent developments in determining/writing a fibre bundle structure of the set of quadratic systems with the help of differential 1-forms and pencils and nets of cubics.

QuIGS Seminar Talk

Geometry and means

Abstract

Positive definite matrices play a significant role in various mathematical settings: as covariance matrices in statistics, as transfer matrices in systems engineering, as factors in the polar decomposition of tensors used in continuum physics, and as density matrices in quantum mechanics, for example. It is often necessary to consider the average or "mean" of several of such matrices. In general the matrices do not commute and this makes it difficult to define a good notion of geometric mean (GM). In fact, there are a number of competing definitions for the GM. We'll discuss a natural Riemannian geometry on the space of positive definite matrices and consider variants of the GM that are based on that non-Euclidian geometry.

QuIGS Seminar Talk

A mathematical model for concurrent systems

Abstract

Coordinating the actions of concurrent processes using shared resources is a problem which is of increasing practical importance. Spaces with a partial order model such systems. It would be useful to be able to replace a given model with a smaller representation which nevertheless preserves the relevant computer-scientific information. I will show that the obvious equivalences are too coarse, but that one can define a good notion of equivalence by taking into account the context. I will apply this idea to analyze many simple examples.

QuIGS Seminar Talk

Purity and information loss in open quantum evolutions

Abstract

I will discuss, in a simple mathematical fashion, some aspects of the dynamical evolutions in open quantum systems relevant to the field of Quantum Information Processing. The talk will be an informal black-board presentation mostly aimed to address an audience of non-experts.

QuIGS Seminar Talk

Transients and attractors in epidemics

Abstract

Historical records of childhood disease incidence reveal complex dynamics. For measles, a simple model has indicated that epidemic patterns represent attractors of a nonlinear dynamical system and that transitions between different attractors are driven by slow changes in birth rates and vaccination levels. The same analysis can explain the main features of chickenpox dynamics, but fails for rubella and whooping cough. We show that an additional (perturbative) analysis of the model, together with knowledge of the population size in question, can account for all the observed incidence patterns by predicting how stochastically sustained transient dynamics should be manifested in these systems.

QuIGS Seminar Talk

Noncommutative Orlicz spaces in Quantum Information Geometry

Abstract

Classical information geometry concerns the study of families of probability distributions. Parametric models are viewed as finite dimensional manifolds and probabilistic concepts acquire a geometric meaning, such as the Fisher information metric, which is regarded as a Riemannian metric. Nonparametric models require the extension of the theory to infinite dimensional manifolds, where special Orlicz spaces are used as generalized coordinates. Quantum information geometry is its noncommutative counterpart. Parametric models are replaced by finite dimensional manifolds of density matrices and several well known quantum mechanical quantities, such as the Bogoliubov Kubo-Mori correlations, can be reinterpreted in light of differential geometry. The extension to infinite dimensional manifolds of states on von Neumann algebras is technically challenging in part due to the lack of a well developed theory of Quantum Orlicz spaces. In this talk, I will review some recent results relating to noncommutative Orlicz spaces and present some concrete applications to quantum information manifolds of density operators on Hilbert spaces.

QuIGS Seminar Talk

Two innovations in the Hessenberg QR algorithm

Abstract

The Hessenberg QR algorithm is the standard method for computing the eigenvalues and optionally the Schur decomposition of a non-symmetric matrix. It was introduced in 1961 by Francis. Since then there have been theoretical advances, for example, Batterson proved that there are regions where the algorithm will not converge without exceptional shifts. However, there have been no significant changes in the implemention of the algorithm. I will present two innovations. The first shows how to implement the algorithm in a way where the bulk of the computations are in performing matrix multiplication -- a very efficient operation on high performance computers. The problem of finding such a ``BLAS-3'' algorithm had been open for about 15 years. The second shows how, given a Hessenberg matrix, one can often deflate eigenvalues without doing a QR sweep. Each of these ideas produces a significant improvement in speed. This is joint work with Ralph Byers and Karen Braman (both U. Kansas).

QuIGS Seminar Talk

Quantum Computing

Abstract

Advances in computing are revolutionizing our world. Present day computers advance at a rapid pace toward the barrier defined by the laws of quantum physics. The quantum computation program short-circuits that constraint by exploiting the quantum laws to advantage rather than regarding them as obstacles. Quantum computer accepts any superposition of its inputs as an input, and processes the components simultaneously, performing a sophisticated interference experiment of classical inputs. This �quantum parallelism� allows one to explore exponentially many trial solutions with relatively modes means, and to select the correct one. This has a particularly dramatic effect on factoring of large integers, which is at the core of the present day encryption strategies (public key) used in diplomatic communication, and (increasingly) in business. As demonstrated approximately five years ago, quantum computers could yield the most commonly used encryption protocol obsolete. Since then, it was also realized that quantum computation can lead to breakthroughs elsewhere, including simulations of quantum systems, implementation of novel encryption strategies (quantum cryptography), as well as more mundane applications such as sorting. I will describe recent work done in quantum computation, in particular the discovery and implementation of methods to make quantum information robust against corruption, both in theory and experiments. I will end with speculations about the field.

QuIGS Seminar Talk

Introduction to Quantum Circuits and Gates

Abstract

I shall describe the basic Hilbert space notation and nomenclature used in quantum information theory, and introduce the notions of quantum operations, circuits and gates.

QuIGS Seminar Talk

Shor's Algorithm

Abstract

I will present Shor's factoring algorithm from a phase estimation perspective. After a brief introduction of quantum computation and quantum circuits, I will illustrate the notion of phase kick back. Then, I will generalize this notion to phase estimation which can be applied to the problem of order finding to which factoring can be reduced classically.

QuIGS Seminar Talk

An Introduction to Quantum Error Correction

Abstract

In this tutorial, we will explain the basics of quantum error correction. We will start by considering simple errors and the 9 qubit Shor code. This will naturally lead us to CSS codes and their robustness against general errors.

QuIGS Seminar Talk

Introduction to Quantum Channels

Abstract

The study of quantum channels (or quantum evolutions) is central to quantum information theory. Mathematically, a quantum channel is given by a completely positive trace preserving map which acts on the set of operators on a finite dimensional Hilbert space. I shall discuss how quantum channels arise in quantum computing, outline some of the basic properties of completely positive maps, and give several examples.

QuIGS Seminar Talk

Introduction to Quantum Cryptography

Abstract

I will give an overview of quantum cryptography, with an emphasis on quantum key distribution. I will discuss the BB84 and B92 protocols, and briefly discuss some of the techniques used in security proofs. I will also give an overview of other types of quantum cryptographic protocols and impossibility results, including the impossibility of quantum bit commitment.

QuIGS Seminar Talk

Quantum Channels as Operator Algebras

Abstract

Every quantum channel is determined by a set of noise operators. The 'interaction algebra' associated with a channel is the operator algebra generated by the noise operators and their adjoints, and the set of all operators which commute with this algebra is called the 'noise commutant'. Understanding the structure of these operator algebras is an important problem in quantum error correction, for the 'noiseless subsystem method' in particular. I will discuss how this structure can be identified with operator algebra techniques and show how the noise commutant can be used to produce codes which are immune to the noise of a channel.

QuIGS Seminar Talk

Making Algorithms Exact Using Quantum Computer Algorithms

Abstract

Quantum algorithms seek to exploit quantum interference to perform computations much faster than any known classical algorithm. The quantum algorithm must carefully choreograph the interference pattern so that the ``good'' or correct outcomes experience constructive interference and the ``bad'' or wrong outcomes experience destructive interference, thereby giving a significant probability amplitude to successful outcomes. It is indeed possible in some cases to engineer the interference so that destructive interference entirely cancels out ALL of the wrong outcomes, leaving us with an algorithm that exactly (or ``deterministically'' or ``with no error'') solves the problem at hand. In other words, there is no randomness or uncertainty in the outcome of the computation. In this talk I will describe the basic techniques behind making algorithms exact using quantum computers, and survey the algorithms to which we can apply these methods. (Much of this work is joint work with Christof Zalka).

QuIGS Seminar Talk

Collective Noise Channels

Abstract

A quantum channel is a completely positive map that serves as a mathematical model for the transmission of quantum information. The associated "noise operators" determine how that information may be distorted during transmission. Operators that commute with the noise operators are transmitted without modification, so that the calculation of the commutant is an important step in developing noise-free coding methods. We will review the basic ideas here and present several approaches to the computation of the commutant for a class of quantum channels known as collective noise channels.