Bipartite quantum measurements with optimal single-sided distinguishability

TL;DR

This paper investigates optimal bipartite quantum bases with maximal single-sided distinguishability, providing explicit constructions for dimensions where SIC measurements exist, and demonstrates their application in local quantum state tomography.

Contribution

It introduces a general construction of optimal bases in bipartite systems for all dimensions with known SIC measurements, extending previous work beyond two-qubit systems.

Findings

Explicit construction of optimal bases for N=3 and higher dimensions.

Demonstration of local quantum state tomography using the optimal basis.

Validation of the tomography scheme on IBM quantum computers.

Abstract

We analyse orthogonal bases in a composite $N\times N$ Hilbert space describing a bipartite quantum system and look for a basis with optimal single-sided mutual state distinguishability. This condition implies that in each subsystem the $N^2$ reduced states form a regular simplex of a maximal edge length, defined with respect to the trace distance. In the case $N=2$ of a two-qubit system our solution coincides with the elegant joint measurement introduced by Gisin. We derive explicit expressions of an analogous constellation for $N=3$ and provide a general construction of $N^2$ states forming such an optimal basis in ${\cal H}_N \otimes {\cal H}_N$. Our construction is valid for all dimensions for which a symmetric informationally complete (SIC) generalized measurement is known. Furthermore, we show that the one-party measurement that distinguishes the states of an optimal basis of the composite system leads to a local quantum state tomography with a linear reconstruction formula. Finally, we test the introduced tomographical scheme on a complete set of three mutually unbiased bases for a single qubit using two different IBM machines.