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arXiv:2606.21626

Systematic derivation of Tsirelson bounds in arbitrary dimensions

Lorenzo Coccia, Matteo Padovan, Giuseppe Vallone

quant-phSubmitted Jun 19, 2026• Announced Jun 22, 2026• 0 citations
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Abstract

The study of Bell nonlocality and the bounds of quantum correlations, the so-called Tsirelson bounds, is fundamental to quantum information science and the exploration of the limits of quantum theory. While quantum bounds for qubit systems have been extensively characterized, determining tight quantum bounds for correlations attainable with high-dimensional quantum states and measurements remains a significant challenge. In this work, we propose a systematic derivation of bipartite Tsirelson and local bounds written in terms of sum-of-squares decompositions. Using this method, we discover novel bounds and recover established results for maximally entangled states of qubits and qu$d$its.

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48 pages, 2 figures

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The study of Bell nonlocality and the bounds of quantum correlations, the so-called Tsirelson bounds, is fundamental to quantum information science and the exploration of the limits of quantum theory. While quantum bounds for qubit systems have been extensively characterized, determining tight quantum bounds for correlations attainable with high-dimensional quantum states and measurements remains a significant challenge. In this work, we propose a systematic derivation of bipartite Tsirelson and local bounds written in terms of sum-of-squares decompositions. Using this method, we discover novel bounds and recover established results for maximally entangled states of qubits and qudddits.
Quantum mechanics is nonlocal, meaning it cannot be described by any classical local hidden variable model. In this thesis we study two aspects of quantum nonlocality. Part I addresses the question of what classical resources are required to simulate nonlocal quantum correlations. We start by constructing new local models for noisy entangled quantum states. These constructions exploit the connection between nonlocality and Grothendieck's inequality, first noticed by Tsirelson. Next, we consider local models augmented by a limited amount of classical communication. After generalizing Bell inequalities to this setting, we show that (i) one bit of communication is sufficient to simulate the correlations of projective measurements on a maximally entangled state of two qubits, and (ii) five bits of communication are sufficient to simulate the joint correlation of two-outcome measurements on any bipartite quantum state. The latter result can be interpreted as a stronger (constrained) version of Grothendieck's inequality. In part II, we investigate the monogamy of nonlocal correlations. In a setting where three parties, A, B, and C, share an entangled quantum state of arbitrary dimension, we: (i) bound the trade-off between AB's and AC's violation of the CHSH inequality, obtaining an intriguing generalization of Tsirelson's bound, and (ii) demonstrate that forcing B and C to be classically correlated prevents A and B from violating certain Bell inequalities. We study not only correlations that arise within quantum theory, but also arbitrary correlations that do not allow signaling between separate groups of parties. These results are based on new techniques for obtaining Tsirelson bounds, or bounds on the quantum value of a Bell inequality, and have applications to interactive proof systems and cryptography.
Despite their importance, there is an ongoing challenge characterizing multipartite quantum correlations. The Seevinck-Svetlichny (SS) and Mermin-Klyshko (MK) inequalities present constraints on correlations in multipartite systems, a violation of which allows to classify the correlations by using the nonseparability property. In this work we present refined Tsirelson (quantum) bounds on these inequalities, derived from inequalities stemming from a fundamental constraint, tightly akin to quantum uncertainty. Unlike the original, known inequalities, our bounds do not consist of a single constant point but rather depend on correlations in specific subsystems (being local correlations for our bounds on the SS operators and bipartite correlations for our bounds on the MK operators). We analyze concrete examples in which our bounds are strictly tighter than the known bounds, i.e., lie beneath the previously found constants, thus better characterizing the set of allowed quantum correlations. We interpret the results as complementarity relations between multipartite and local correlations, as well as multipartite and bipartite correlations.
Quantum theory is in principle compatible with processes that violate causal inequalities, an analogue of Bell inequalities that constrain the correlations observed by sets of parties operating in a definite causal order. Since the introduction of causal inequalities, determining their maximum quantum violation, analogue to Tsirelson's bound for Bell inequalities, has remained an open problem. Here we provide a general method for bounding the violation of arbitrary causal inequalities, establishing limits to the correlations achievable by arbitrary local experiments and by arbitrary quantum processes with indefinite causal order. We prove that the maximum violation is generally smaller than the algebraic maximum of the corresponding correlation, and determine Tsirelson-like bounds for a class of causal inequalities including some of the most paradigmatic examples. Our results motivate a search for physical principles characterizing the boundary of the set of quantum correlations with indefinite causal order.
Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest (2,2,2)(2,2,2)(2,2,2) scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the (2,2,3)(2,2,3)(2,2,3) scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.
The future progress of semi-device independent quantum information science depends crucially on our ability to bound the strength of the nonlocal correlations achievable with finite dimensional quantum resources. In this work, we characterize quantum nonlocality under local dimension constraints via a complete hierarchy of semidefinite programming relaxations. In the bipartite case, we find that the first level of the hierarchy returns non-trivial bounds in all cases considered, allowing to study nonlocality scenarios with four measurement settings on one side and twelve (12) on the other in a normal desktop. In the tripartite case, we apply the hierarchy to derive a Bell-type inequality that can only be violated when each of the three parties has local dimension greater than two, hence certifying three-dimensional tripartite entanglement in a device independent way. Finally, we show how the new method can be trivially modified to detect non-separable measurements in two-qubit scenarios.