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arXiv:2406.16202 · Physical Review A

Refined Tsirelson bounds on multipartite Bell inequalities

Rain Lenny, Dana Ben Porath, Eliahu Cohen

Published Sep 23, 2024• 2 citations
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Abstract

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.

DOI
10.1103/physreva.110.032435

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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.
Bell inequalities are mathematical constructs that demarcate the boundary between quantum and classical physics.A new class of multiplicative Bell inequalities originating from a volume maximization game (based on products of correlators within bipartite systems) has been recently proposed.For these new Bell parameters, it is relatively easy to find the classical and quantum, i.e., Tsirelson, limits.Here, we experimentally test the Tsirelson bounds of these inequalities using polarization-entangled photons for a different number of measurements (n), each party can perform.For n = 2, 3, 4, we report the experimental violation of local hidden variable theories.In addition, we experimentally compare the results with the parameters obtained from a fully deterministic strategy, and observe the conjectured nature of the ratio.Finally, utilizing the principle of "relativistic independence" encapsulating the locality of uncertainty relations, we theoretically derive and experimentally test new and richer quantum bounds for both the multiplicative and the additive Bell parameters for n = 2. Our findings strengthen the deep correspondence between local and nonlocal correlations, and pave the way for both theoretical (e.g., better understanding of nonlocal correlations) and practical (e.g., Bell tests and quantum technologies with inefficient detectors) applications.
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.
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 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. Quantum theory has been shown to be compatible with processes violating a definite causal order, but no detailed analysis of the maximum allowed degree of violation has been carried out so far. Here, the authors fill this gap by providing the analogue of Tsirelson’s bound for the violation of causal inequalities.
Two overlapping bipartite binary Bell inequalities cannot be simultaneously violated as this would contradict the usual no-signaling principle. This property is known as monogamy of Bell inequality violations and generally Bell monogamy relations refer to trade-offs between simultaneous violations of multiple inequalities. It turns out that multipartite Bell inequalities admit weaker forms of monogamies that allow for violations of a few inequalities at once. Here we systematically study monogamy relations between correlation Bell inequalities both within quantum theory and under the sole assumption of no signaling. We first investigate the trade-offs in Bell violations arising from the uncertainty relation for complementary binary observables, and exhibit several network configurations in which a tight trade-off arises in this fashion. We then derive a tight trade-off relation which cannot be obtained from the uncertainty relation showing that it does not capture monogamy entirely. The results are extended to Bell inequalities involving different numbers of parties and find applications in device-independent secret sharing and device-independent randomness extraction. Although two multipartite Bell inequalities may be violated simultaneously, we show that genuine multiparty nonlocality, as evidenced by a generalized Svetlichny inequality, does exhibit monogamy property. Finally, using the relations derived we reveal the existence of flat regions in the set of quantum correlations.
The correlations in quantum networks have attracted strong interest with new types of violations of the locality. The standard Bell inequalities cannot characterize the multipartite correlations that are generated by multiple sources. The main problem is that no computationally efficient method is available for constructing useful Bell inequalities for general quantum networks. In this work, we show a significant improvement by presenting new, explicit Bell-type inequalities for general networks including cyclic networks. These nonlinear inequalities are related to the matching problem of an equivalent unweighted bipartite graph that allows constructing a polynomial-time algorithm. For the quantum resources consisting of bipartite entangled pure states and generalized Greenberger-Horne-Zeilinger (GHZ) states, we prove the generic nonmultilocality of quantum networks with multiple independent observers using new Bell inequalities. The violations are maximal with respect to the presented Tsirelson's bound for Einstein-Podolsky-Rosen states and GHZ states. Moreover, these violations hold for Werner states or some general noisy states. Our results suggest that the presented Bell inequalities can be used to characterize experimental quantum networks.