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arXiv:1407.6604

Projection methods in quantum information science

Yuen-Lam Cheung, Dmitriy Drusvyatskiy, Chi-Kwong Li, Diane Christine Pelejo, Henry Wolkowicz

Published Jul 24, 2014• 0 citations
arXivHTMLPDF

Abstract

We consider the problem of constructing quantum operations or channels, if they exist, that transform a given set of quantum states $\{ρ_1, \dots, ρ_k\}$ to another such set $\{\hatρ_1, \dots, \hatρ_k\}$. In other words, we must find a {\em completely positive linear map}, if it exists, that maps a given set of density matrices to another given set of density matrices. This problem, in turn, is an instance of a positive semi-definite feasibility problem, but with highly structured constraints. The nature of the constraints makes projection based algorithms very appealing when the number of variables is huge and standard interior point-methods for semi-definite programming are not applicable. We provide emperical evidence to this effect. We moreover present heuristics for finding both high rank and low rank solutions. Our experiments are based on the \emph{method of alternating projections} and the \emph{Douglas-Rachford} reflection method.

DOI
10.48550/arxiv.1407.6604

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We consider the problem of constructing quantum operations or channels, if they exist, that transform a given set of quantum states {ρ1,…,ρk}\{ρ_1, \dots, ρ_k\}{ρ1​,…,ρk​} to another such set {ρ^1,…,ρ^k}\{\hatρ_1, \dots, \hatρ_k\}{ρ^​1​,…,ρ^​k​}. In other words, we must find a {\em completely positive linear map}, if it exists, that maps a given set of density matrices to another given set of density matrices. This problem, in turn, is an instance of a positive semi-definite feasibility problem, but with highly structured constraints. The nature of the constraints makes projection based algorithms very appealing when the number of variables is huge and standard interior point-methods for semi-definite programming are not applicable. We provide emperical evidence to this effect. We moreover present heuristics for finding both high rank and low rank solutions. Our experiments are based on the \emph{method of alternating projections} and the \emph{Douglas-Rachford} reflection method.
An ensemble of quantum states can be described by a Hermitian, positive semidefinite and unit trace matrix called density matrix. Thus, the study of methods for optimizing a certain function (energy, entropy) over the set of density matrices has a direct application to important problems in quantum information and computation. We propose a projected gradient method for solving such problems. By exploiting the geometry of the feasible set, which is the intersection of the cone of Hermitian positive semidefinite matrices with the hyperplane defined by the unit trace constraint, we describe an efficient procedure to compute the projection onto this set using the Frobenius norm. Some important applications, such as quantum state tomography, are described and numerical experiments illustrate the effectiveness of the method when compared to previous methods based on fixed-point iterations or semidefinite programming.
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For a given set of input-output pairs of quantum states or observables, we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output. The physical maps on quantum states are trace-preserving completely positive maps, but we also consider variants of these requirements. We generalize the definition of complete positivity to linear maps defined on arbitrary subspaces, then formulate this notion as a semidefinite program, and relate it by duality to approximative extensions of this map. This gives a characterization of the maps which can be approximated arbitrarily well as the restriction of a map that is completely positive on the whole algebra, also yielding the familiar extension theorems on operator spaces. For quantum channel extensions and extensions by probabilistic operations we obtain semidefinite characterizations, and we also elucidate the special case of Abelian inputs or outputs. Finally, revisiting a theorem by Alberti and Uhlmann, we provide simpler and more widely applicable conditions for certain extension problems on qubits, and by using a semidefinite programming formulation we exhibit counterexamples to seemingly reasonable but false generalizations of the Alberti-Uhlmann theorem.
The immense power of computing has been continually growing, yet we still want more speed and computing capacity. If, as Moores Law states, the number of transistors on a microprocessor continues to double every other year, we will find the circuits on a microprocessor measured on an atomic scale in a couple of decades. Then, we would start to consider quantum computers, which will make use of the power of atoms to perform memory and processing tasks. The mathematics behind quantum computers will be the focus of my presentation. The information contained in qubits (quantum bits), which are analogues of classical bits, can be represented in a matrix form, namely a density matrix. It has been found that a density matrix is positive semi-definite and of trace 1. The operators acting on density matrices need to preserve the trace and positivity so that the resulting output after processing the information is still a density matrix. Therefore, these operators can be thought of as a trace-preserving positive map. In this presentation, the definitions of such maps and properties will be introduced along with the theorems that are related to completely positive maps.
This work is concerned with the issue of applications of the semi-definite programming (SDP) in the field of quantum information science. Our results of the analysis of certain quantum information protocols using this optimization technique are presented, and an implementation of a relevant numerical tool is introduced. The key method used is NPA discovered by Navascues et al. [Phys. Rev. Lett. 98, 010401 (2007)]. In chapter 1 a brief overview of mathematical methods used in this work is presented. In chapter 2 an introduction to quantum information science is given. Chapter 3 concerns the device-independent (DI) and semi-device-independent (SDI) approaches in quantum cryptography. In chapter 4 our results regarding quantum information protocols which we developed using SDP are described. The results include a new type of quantum key distribution protocol based on Hardy's paradox; several protocols for quantum random number expansion; a method of formulation of SDP relaxations of SDI protocols; and a DI Santha-Vazirani source of randomness amplification protocol. In chapter 5 an introduction to interior point methods is given. In chapter 6 the implementation of our numerical tool dedicated to problems occurring in chapter 4 are discussed and compared with other solutions.
We study sequences (both cyclic and randomized) of idempotent completely positive trace-preserving quantum maps, and show how they asymptotically converge to the intersection of their fixed point sets via alternating projection methods, highlighting the robustness features of the protocol against randomization. The general results are then specialized to stabilizing entangled states in finite-dimensional multipartite quantum systems subject to locality constraints, a problem of key interest for quantum information applications.