Man-Duen Choi

Bio: Man-Duen Choi is an academic researcher from University of Toronto. The author has contributed to research in topics: Quantum error correction & Matrix (mathematics). The author has an hindex of 27, co-authored 70 publications receiving 5932 citations. Previous affiliations of Man-Duen Choi include University of California, Berkeley & Indian Statistical Institute.

The completely positive lifting problem for C*-algebras

Abstract: (1.1)~~~~~~~~~~~~~~~~~~~1 where A, B are C*-algebras (resp., unital C*-algebras), J is a closed twosided ideal in B, 7 is the quotient map, and A, * are contractive (resp., unital) completely positive maps. The lifting problem for q is to determine whether or not one can find * so that the diagram commutes. In this paper, we will show that this is the case if A is separable, and either A, B/J, or B

Quantum Computation and Quantum Information

Abstract: Part I. Fundamental Concepts: 1. Introduction and overview 2. Introduction to quantum mechanics 3. Introduction to computer science Part II. Quantum Computation: 4. Quantum circuits 5. The quantum Fourier transform and its application 6. Quantum search algorithms 7. Quantum computers: physical realization Part III. Quantum Information: 8. Quantum noise and quantum operations 9. Distance measures for quantum information 10. Quantum error-correction 11. Entropy and information 12. Quantum information theory Appendices References Index.

Quantum entanglement

Abstract: All our former experience with application of quantum theory seems to say: {\it what is predicted by quantum formalism must occur in laboratory} But the essence of quantum formalism - entanglement, recognized by Einstein, Podolsky, Rosen and Schr\"odinger - waited over 70 years to enter to laboratories as a new resource as real as energy This holistic property of compound quantum systems, which involves nonclassical correlations between subsystems, is a potential for many quantum processes, including ``canonical'' ones: quantum cryptography, quantum teleportation and dense coding However, it appeared that this new resource is very complex and difficult to detect Being usually fragile to environment, it is robust against conceptual and mathematical tools, the task of which is to decipher its rich structure This article reviews basic aspects of entanglement including its characterization, detection, distillation and quantifying In particular, the authors discuss various manifestations of entanglement via Bell inequalities, entropic inequalities, entanglement witnesses, quantum cryptography and point out some interrelations They also discuss a basic role of entanglement in quantum communication within distant labs paradigm and stress some peculiarities such as irreversibility of entanglement manipulations including its extremal form - bound entanglement phenomenon A basic role of entanglement witnesses in detection of entanglement is emphasized

On the Generators of Quantum Dynamical Semigroups

Abstract: The notion of a quantum dynamical semigroup is defined using the concept of a completely positive map. An explicit form of a bounded generator of such a semigroup onB(ℋ) is derived. This is a quantum analogue of the Levy-Khinchin formula. As a result the general form of a large class of Markovian quantum-mechanical master equations is obtained.

Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization

Abstract: In the first part of this thesis, we introduce a specific class of Linear Matrix Inequalities (LMI) whose optimal solution can be characterized exactly. This family corresponds to the case where the associated linear operator maps the cone of positive semidefinite matrices onto itself. In this case, the optimal value equals the spectral radius of the operator. It is shown that some rank minimization problems, as well as generalizations of the structured singular value ($mu$) LMIs, have exactly this property. In the same spirit of exploiting structure to achieve computational efficiency, an algorithm for the numerical solution of a special class of frequency-dependent LMIs is presented. These optimization problems arise from robustness analysis questions, via the Kalman-Yakubovich-Popov lemma. The procedure is an outer approximation method based on the algorithms used in the computation of hinf norms for linear, time invariant systems. The result is especially useful for systems with large state dimension. The other main contribution in this thesis is the formulation of a convex optimization framework for semialgebraic problems, i.e., those that can be expressed by polynomial equalities and inequalities. The key element is the interaction of concepts in real algebraic geometry (Positivstellensatz) and semidefinite programming. To this end, an LMI formulation for the sums of squares decomposition for multivariable polynomials is presented. Based on this, it is shown how to construct sufficient Positivstellensatz-based convex tests to prove that certain sets are empty. Among other applications, this leads to a nonlinear extension of many LMI based results in uncertain linear system analysis. Within the same framework, we develop stronger criteria for matrix copositivity, and generalizations of the well-known standard semidefinite relaxations for quadratic programming. Some applications to new and previously studied problems are presented. A few examples are Lyapunov function computation, robust bifurcation analysis, structured singular values, etc. It is shown that the proposed methods allow for improved solutions for very diverse questions in continuous and combinatorial optimization.