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arXiv:2106.09359 · Annalen der Physik

The Best Approximation of an Objective State With a Given Set of Quantum States

Li‐qiang Zhang, Nan‐nan Zhou, Chang‐shui Yu

Published Dec 30, 2021• 2 citations
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Abstract

Abstract Approximating a quantum state by the convex mixing of some given states has strong experimental significance and provides alternative understandings of quantum resource theory. It is essentially a complex optimal problem which, up to now, has only partially solved for qubit states. Here, the most general case is focused on that the approximation of a d‐dimensional objective quantum state by the given state set consisting of any number of (mixed‐) states. The problem is thoroughly solved with a closed solution of the minimal distance in the sense of l2 norm between the objective state and the set. In particular, the minimal number of states in the given set is presented to achieve the optimal distance. The validity of this closed solution is further verified numerically by several randomly generated quantum states.

DOI
10.1002/andp.202100407

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Optimal convex approximation of quantum channels based on ααα-affinity
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We consider the problem of optimally approximating an unavailable quantum state $\ensuremath{\rho}$ by the convex mixing of states drawn from a set of available states ${{\ensuremath{\nu}}_{i}}$. The problem is recast to look for the least distinguishable state from $\ensuremath{\rho}$ among the convex set ${\ensuremath{\sum}}_{i}{p}_{i}{\ensuremath{\nu}}_{i}$, and the corresponding optimal weights pi{{p}_{i}}pi​ provide the optimal convex mixing. We present the complete solution for the optimal convex approximation of a qubit mixed state when the set of available states comprises the three bases of the Pauli matrices.