Bipartite quantum measurements with optimal single-sided distinguishability

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 ${\mathcal{H}}_N\otimes {\mathcal{H}}_N$. 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.

arXiv comment14 + 9 pages, 9 figures

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• Tristochastic operations and convolutions of quantum states
• Non-locality of conjugation symmetry: characterization and examples in quantum network sensing
• Symmetric Localizable Multipartite Quantum Measurements from Pauli Orbits
• The Multiqubit Elegant Joint Measurement
• Experimental genuine quantum nonlocality in the triangle network
• Localization of joint quantum measurements on ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$ by entangled resources with Schmidt number at most $d$