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Quantum Information and Convex Optimization

Michael Reimpell

Published Sep 25, 2007• 6 citations
Publisher

Abstract

This thesis is concerned with convex optimization problems in quantum information theory. It features an iterative algorithm for optimal quantum error correcting codes, a postprocessing method for incomplete tomography data, a method to estimate the amount of entanglement in witness experiments, and it gives necessary and sufficient criteria for the existence of retrodiction strategies for a generalized mean king problem.

DOI
10.24355/dbbs.084-200803130100-3

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This thesis is concerned with convex optimization problems in quantum information theory. It features an iterative algorithm for optimal quantum error correcting codes, a postprocessing method for incomplete tomography data, a method to estimate the amount of entanglement in witness experiments, and it gives necessary and sufficient criteria for the existence of retrodiction strategies for a generalized mean king problem.
We show that the problem of designing a quantum information error correcting procedure can be cast as a bi-convex optimization problem, iterating between encoding and recovery, each being a semidefinite program. For a given encoding operator the problem is convex in the recovery operator. For a given method of recovery, the problem is convex in the encoding scheme. This allows us to derive new codes that are locally optimal. We present examples of such codes that can handle errors which are too strong for codes derived by analogy to classical error correction techniques. 1
Convexity arises naturally in the study of quantum information. As a result, many useful tools from convex analysis can be used to give important results regarding aspects of quantum information. This thesis builds up methods using core concepts from convex analysis, including convex optimization problems, convex roof constructions, and conic programming, to study mathematical problems related to quantum entanglement. In this thesis, I develop a method for solving convex optimization problems that arise in quantum information theory by analyzing the corresponding converse problem. That is, given an element in a convex set, I determine a family of convex functions that are minimized at this point. This method is used find explicit formulas for the relative entropy of entanglement, as well as other important quantities used to quantify entanglement, and allows one to show important relationships between them. I also construct a practical algorithm that can be used to compute these quantities. This thesis also presents a method to compute convex roofs of arbitrary entanglement measures evaluated on highly symmetric bipartite states. I also establish a framework for completely characterizing quantum resource theories that are convex. For resource theories with a simple mathematical structure, this gives rise to a complete set of resource monotones that can be computed in practice using semidefinite programs. This has applications to the study of entanglement transformations.
There exists, in general, a convex set of quantum state estimators that maximize the likelihood for informationally incomplete data. We propose an estimation scheme, catered to measurement data of this kind, to search for the exact maximum-likelihood-maximum-entropy estimator using semidefinite programming and a standard multi-dimensional function optimization routine. This scheme can be used to infer the expectation values of a set of entanglement witnesses that can be used to verify the entanglement of the unknown quantum state for composite systems. Next, we establish an alternative numerical scheme that is more computationally robust for the sole purpose of maximizing the likelihood and entropy.
There exists, in general, a convex set of quantum state estimators that maximize the likelihood for informationally incomplete data. We propose an estimation scheme, catered to measurement data of this kind, to search for the exact maximum-likelihood-maximum-entropy estimator using semidefinite programming and a standard multi-dimensional function optimization routine. This scheme can be used to infer the expectation values of a set of entanglement witnesses that can be used to verify the entanglement of the unknown quantum state for composite systems. Next, we establish an alternative numerical scheme that is more computationally robust for the sole purpose of maximizing the likelihood and entropy.
In this PhD thesis we study the effects of quantum entanglement, one of quantum mechanics most peculiar features, in nonlocal games and communication problems in zero-error information theory. A nonlocal game is a thought experiment in which two cooperating players, who are forbidden to communicate, want to perform a certain task. Zero-error information theory is the mathematical field that studies communication problems where no error is tolerated. The unifying link among the various scenarios we consider is their combinatorial nature and in particular their reformulations as graph theoretical problems, mainly concerning the chromatic and stability numbers and some quantum generalizations thereof. In this thesis we propose a novel approach to the study of these quantum graph parameters using the paradigm of conic optimization. For that, we introduce and study the completely positive semidefinite cone, a new matrix cone consisting of all symmetric matrices that admit a Gram representation by positive semidefinite matrices. Furthermore, we investigate whether entanglement allows for better-than-classical communication schemes in some well-known problems from zero-error information theory. For example we study the channel coding problem, which asks a sender to transmit data reliably to a receiver in the presence of noise, as well as some of its generalizations.
Convex optimization problems arise naturally in quantum information theory, often in terms of minimizing a convex function over a convex subset of the space of hermitian matrices. In most cases, finding exact solutions to these problems is usually impossible. As inspired by earlier investigations into the relative entropy of entanglement (REE) (Miranowicz and Ishizaka 2008 Phys. Rev. A 78 032310 ), we introduce a general method to solve the converse problem rather than find explicit solutions. That is, given a matrix in a convex set, we determine a family of convex functions that are minimized at this point. This method allows us find explicit formulae for the REE and the Rains bound, two well-known upper bounds on the distillable entanglement, and yields interesting information about these quantities, such as the fact that they coincide in the case where at least one subsystem of a multipartite state is a qubit.