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arXiv:1505.07387

Coherence convertibility for mixed states

Xiaofei Qi, Zhaofang Bai, Shuanping Du

Published May 27, 2015• 7 citations
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Abstract

In this paper, by providing a class of coherence measures in finite dimensional systems, a sufficient and necessary condition for the existence of coherence transformations that convert one probability distribution of any pure states into another one is obtained.
DOI
10.48550/arxiv.1505.07387

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Coherence manipulation is fundamental in the study of the resource theory of quantum coherence. It is aimed at identifying when it is possible to convert a given coherent state to another using only incoherent operations. In this paper, it is shown that a finite number of measure conditions are not sufficient to characterize coherence manipulation on general mixed states. Further, a finite number of measure conditions to classify coherence manipulation on subspace-independent states are found. For the input pure state, we also furnish a structural characterization of coherence manipulation in terms of a finite number of measure conditions.
We derive a necessary condition for the existence of a completely-positive, linear, trace-preserving map which deterministically transforms one finite set of pure quantum states into another. This condition is also sufficient for linearly-independent initial states. We also examine the issue of quantum coherence, that is, when such operations maintain the purity of superpositions. If, in any deterministic transformation from one linearly-independent set to another, even a single, complete superposition of the initial states maintains its purity, the initial and final states are related by a unitary transformation.
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It is shown that a finite number of conditions are not sufficient to determine the locality of transformations between two probability distributions of pure states as well as the locality of transformations between two $d\ifmmode\times\else\texttimes\fi{}d$ mixed states with $d\ensuremath{\geqslant}4$. As an example, an infinite, but minimal, set of necessary and sufficient conditions for the existence of a local procedure that converts one probability distribution of two pure pair of qubits into another one is found.
Let A = {rho_1,...,rho_n} be a given set of quantum states. We consider the problem of finding necessary and sufficient conditions on another set B = {sigma_1,...,sigma_n} that guarantee the existence of a physical transformation taking rho_i to sigma_i for all i. Uhlmann has given an elegant such condition when both sets comprise pure states. We give a simple proof of this condition and develop some consequences. Then we consider multi-probabilistic transformations between sets of pure states which leads to conditions for the problem of transformability between A and B when one set is pure and the other is arbitrary.