# Bipartite quantum measurements with optimal single-sided distinguishability

Jakub Czartowski1 and Karol Życzkowski1,2,3

1Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland
2Centrum Fizyki Teoretycznej PAN, Al. Lotników 32/46, 02-668 Warszawa, Poland
3National Quantum Information Center (KCIK), University of Gdańsk, Poland

### 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.

Quantum nonlocality has long been among the most unsettling properties of quantum mechanics, as exemplified by the now famous Bell inequality. Recent studies extended these investigations for a network settings giving rise to so-called nonlinear Bell inequalities. Gisin and collaborators have identified a distinguished basis for two-qubit system, called Elegant Joint Measurement (EJM), which yields particularly high breaking of Bell inequality related to the triangular network of three parties with one randomness source for each pair among them.

In our work we extend the prior results by first analysing the properties of EJM in terms of entangling power and typicality of quantum gates, demonstrating its optimality. Furthermore, we introduce a family of similar bases for two systems of arbitrary identical dimension. Their existence depends on another widely studied set of quantum states, so called Symmetric Informationally Complete generalized measurements (SIC). We conjecture that their existence may be interdependent, with one existing only if the other exists. Finally, we demonstrate the usefulness of the newly introduced family of quantum measurements by showing its ability to distinguish between orthogonal bipartite states states basing on the information from one side only. These theoretical results are supplemented by real world implementation on quantum computers provided by IBM to the public domain.

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### Cited by

[1] Armin Tavakoli, Alejandro Pozas-Kerstjens, Ming-Xing Luo, and Marc-Olivier Renou, "Bell nonlocality in networks", Reports on Progress in Physics 85 5, 056001 (2022).

[2] Cen-Xiao Huang, Xiao-Min Hu, Yu Guo, Chao Zhang, Bi-Heng Liu, Yun-Feng Huang, Chuan-Feng Li, Guang-Can Guo, Nicolas Gisin, Cyril Branciard, and Armin Tavakoli, "Entanglement Swapping and Quantum Correlations via Symmetric Joint Measurements", Physical Review Letters 129 3, 030502 (2022).

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