Local hidden variable values without optimization procedures

Dardo Goyeneche1, Wojciech Bruzda2, Ondřej Turek3, Daniel Alsina4, and Karol Życzkowski2,5

1Departamento de Física, Facultad de Ciencias Básicas, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile
2Institute of Theoretical Physics, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland
3Department of Mathematics, Faculty of Science, University of Ostrava, 701 03 Ostrava, Czech Republic
4School of Electronic and Electrical Engineering, University of Leeds, Leeds LS2 9JT, UK
5Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland

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Abstract

The problem of computing the local hidden variable (LHV) value of a Bell inequality plays a central role in the study of quantum nonlocality. In particular, this problem is the first step towards characterizing the LHV polytope of a given scenario. In this work, we establish a relation between the LHV value of bipartite Bell inequalities and the mathematical notion of excess of a matrix. Inspired by the well developed theory of excess, we derive several results that directly impact the field of quantum nonlocality. We show infinite families of bipartite Bell inequalities for which the LHV value can be computed exactly, without needing to solve any optimization problem, for any number of measurement settings. We also find tight Bell inequalities for a large number of measurement settings.

Quantum non-locality is a fundamental property of nature, inequivalent to quantum entanglement. These two non-classical resources have several practical applications. For instance, quantum non-locality is useful to generate genuine random numbers and to define protocols that can be certified without relying on the measurement devices, so-called device independent quantum protocols. Bell inequalities, associated with a given multipartite physical system and a measurement scheme, provide rigorous criteria allowing us to detect quantum non-locality. Any such inequality can be characterized by two positive real numbers $C$ and $Q$, defined as the maximal value that the analyzed function can take when local strategies are implemented in the classical and quantum setups, respectively. The number $C$ is called the local hidden variable (LHV) value, as it corresponds to local theories. By construction one has $C \le Q$. In several cases $C$ is strictly smaller than $Q$, and confirming this in an experiment provides an evidence that the quantum theory is essentially non-local. The task of finding the LHV value $C$ is hard in general, as it requires a large computational power. In this work, we shed some light on this issue by establishing a one-to-one correspondence between the LHV value and the notion of $\textit{maximal excess}$ of a matrix — a well-developed concept in mathematics, extensively studied for the last five decades. As a consequence, we determine the LHV value for infinitely many Bell inequalities, for any number of measurement settings per party. We also generalize the mathematical notion of maximal excess and extend the validity of some properties of the excess of Hadamard matrices to a larger class of complex matrices of any order.

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[1] Ravishankar Ramanathan, "Violation of all two-party facet Bell inequalities by almost-quantum correlations", Physical Review Research 3 3, 033100 (2021).

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