# Quantum Bell inequalities from Information Causality – tight for Macroscopic Locality

Mariami Gachechiladze1, Bartłomiej Bąk2, Marcin Pawłowski3, and Nikolai Miklin3

1Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany
2Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland
3International Centre for Theory of Quantum Technologies (ICTQT), University of Gdansk, 80-308 Gdańsk, Poland

### Abstract

In a Bell test, the set of observed probability distributions complying with the principle of local realism is fully characterized by Bell inequalities. Quantum theory allows for a violation of these inequalities, which is famously regarded as Bell nonlocality. However, finding the maximal degree of this violation is, in general, an undecidable problem. Consequently, no algorithm can be used to derive quantum analogs of Bell inequalities, which would characterize the set of probability distributions allowed by quantum theory. Here we present a family of inequalities, which approximate the set of quantum correlations in Bell scenarios where the number of settings or outcomes can be arbitrary. We derive these inequalities from the principle of Information Causality, and thus, we do not assume the formalism of quantum mechanics. Moreover, we identify a subspace in the correlation space for which the derived inequalities give the necessary and sufficient conditions for the principle of Macroscopic Locality. As a result, we show that in this subspace, the principle of Information Causality is strictly stronger than the principle of Macroscopic Locality.

Quantum technologies have advanced rapidly in recent decades, both in academia and industry. The capabilities of quantum theory to boost computational speed, communication privacy, and measurement precision are dazzling and may appear unbounded. However, quantum theory has limitations, and understanding them is critical from a technological and fundamental standpoint. Constraints that do not allow for the infinite power of quantum correlations stem from the mathematical formalism of quantum mechanics, making them difficult to derive and give a physical interpretation. Another approach to the same problem is to constrain the set of quantum correlations by physical principles such as the ”no faster-than-light communication” principle. In this paper, we look at two seemingly unrelated physical principles and show that they lead to the same set of constraints, which we refer to as ”quantum Bell inequalities.” Rather than being coincidental, our constraints result from systematic methods that we develop for both principles, allowing us to derive them for correlations of any size.
The two principles that we consider are information causality and macroscopic locality. The former is a generalization of the ”no faster-than-light communication” principle and bounds the strength of correlations that two distant parties can establish if some amount of communication is allowed. The latter postulates that correlations obtained from measuring many microscopic systems at once must obey the laws of classical mechanics. We first derive an infinite family of quantum Bell inequalities from the information causality and subsequently show that, surprisingly, these inequalities constitute the necessary and sufficient conditions for the macroscopic locality.
Our inequalities are expected to be useful for security proofs in quantum cryptography. Furthermore, the techniques developed can help derive other inequalities from the considered principles.

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### Cited by

[1] Wojciech Bruzda, "Structured Unitary Matrices and Quantum Entanglement", arXiv:2204.12470.

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