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arXiv:1203.5547 · Journal of Mathematical Physics

Physical transformations between quantum states

Zejun Huang, Chi-Kwong Li, Edward Poon, Nung-Sing Sze

Published Oct 1, 2012• 20 citations
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Abstract

Given two sets of quantum states {A1, …, Ak} and {B1, …, Bk}, represented as sets as density matrices, necessary and sufficient conditions are obtained for the existence of a physical transformation T, represented as a trace-preserving completely positive map, such that T(Ai) = Bi for i = 1, …, k. General completely positive maps without the trace-preserving requirement, and unital completely positive maps transforming the states are also considered.

DOI
10.1063/1.4755846

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Given two sets of quantum states {A1, …, Ak} and {B1, …, Bk}, represented as sets as density matrices, necessary and sufficient conditions are obtained for the existence of a physical transformation T, represented as a trace-preserving completely positive map, such that T(Ai) = Bi for i = 1, …, k. General completely positive maps without the trace-preserving requirement, and unital completely positive maps transforming the states are also considered.
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.
Comparison of statistical models (experiments) is an important branch of mathematical statistics, which gives deep insights in many aspects of foundation of statistics. So far, there are two quantum versions of the concept: Comparison with respect to classical tasks and full quantum tasks. In the latter version, a quantum statistical model is more informative than another if and only if a trace preserving positive map sends the former to the latter. On the other hand, in the former version, the existence of a trace preserving positive map sending the former to the latter is a useful sufficient condition. A natural question is whether this is necessary or not. We resolve this question negatively by giving a counter example.
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.
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.