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arXiv:1511.05253 · New Journal of Physics

Bipartite Bell inequalities with three ternary-outcome measurements—from theory to experiments

Sacha Schwarz, Bänz Bessire, André Stefanov, Yeong-Cherng Liang

Published Feb 26, 2016• 15 citations
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Abstract

We explore quantum nonlocality in one of the simplest bipartite scenarios. Several new facet-defining Bell inequalities for the scenario are obtained with their quantum violations analyzed in details. Surprisingly, all these inequalities involving only genuine ternary-outcome measurements can be violated maximally by some two-qubit entangled states, such as the maximally entangled two-qubit state. This gives further evidence that in analyzing the quantum violation of Bell inequalities, or in the application of the latter to device-independent quantum information processing tasks, the commonly held wisdom of equating the local Hilbert space dimension of the optimal state with the number of measurement outcomes is not necessarily justifiable. In addition, when restricted to the minimal qubit subspace, it can be shown that one of these Bell inequalities requires non-projective measurements to attain maximal quantum violation, thereby giving the first example of a facet-defining Bell inequality where a genuine positive-operator-valued measure is relevant. We experimentally demonstrate the quantum violation of this and two other Bell inequalities for this scenario using energy–time entangled photon pairs. Using the obtained measurement statistics, we demonstrate how characterization of the underlying resource in the spirit of device-independence, but supplemented with auxiliary assumptions, can be achieved. In particular, we discuss how one may get around the fact that, due to finite-size effects, raw measurement statistics typically violate the non-signaling condition.

DOI
10.1088/1367-2630/18/3/035001

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We explore quantum nonlocality in one of the simplest bipartite scenarios. Several new facet-defining Bell inequalities for the scenario are obtained with their quantum violations analyzed in details. Surprisingly, all these inequalities involving only genuine ternary-outcome measurements can be violated maximally by some two-qubit entangled states, such as the maximally entangled two-qubit state. This gives further evidence that in analyzing the quantum violation of Bell inequalities, or in the application of the latter to device-independent quantum information processing tasks, the commonly held wisdom of equating the local Hilbert space dimension of the optimal state with the number of measurement outcomes is not necessarily justifiable. In addition, when restricted to the minimal qubit subspace, it can be shown that one of these Bell inequalities requires non-projective measurements to attain maximal quantum violation, thereby giving the first example of a facet-defining Bell inequality where a genuine positive-operator-valued measure is relevant. We experimentally demonstrate the quantum violation of this and two other Bell inequalities for this scenario using energy–time entangled photon pairs. Using the obtained measurement statistics, we demonstrate how characterization of the underlying resource in the spirit of device-independence, but supplemented with auxiliary assumptions, can be achieved. In particular, we discuss how one may get around the fact that, due to finite-size effects, raw measurement statistics typically violate the non-signaling condition.
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.
Bell inequalities are an important tool in device-independent quantum information processing because their violation can serve as a certificate of relevant quantum properties. Probably the best known example of a Bell inequality is due to Clauser, Horne, Shimony and Holt (CHSH), which is defined in the simplest scenario involving two dichotomic measurements and whose all key properties are well understood. There have been many attempts to generalise the CHSH Bell inequality to higher-dimensional quantum systems, however, for most of them the maximal quantum violation---the key quantity for most device-independent applications---remains unknown. On the other hand, the constructions for which the maximal quantum violation can be computed, do not preserve the natural property of the CHSH inequality, namely, that the maximal quantum violation is achieved by the maximally entangled state and measurements corresponding to mutually unbiased bases. In this work we propose a novel family of Bell inequalities which exhibit precisely these properties, and whose maximal quantum violation can be computed analytically. In the simplest scenario it recovers the CHSH Bell inequality. These inequalities involve d measurements settings, each having d outcomes for an arbitrary prime number d ≥ 3 . We then show that in the three-outcome case our Bell inequality can be used to self-test the maximally entangled state of two-qutrits and three mutually unbiased bases at each site. Yet, we demonstrate that in the case of more outcomes, their maximal violation does not allow for self-testing in the standard sense, which motivates the definition of a new weak form of self-testing. The ability to certify high-dimensional MUBs makes these inequalities attractive from the device-independent cryptography point of view.
Facet inequalities play an important role in detecting the nonlocality of a quantum state. The number of such inequalities depends on the Bell test scenario. With the increase in the number of parties, measurement outcomes, or/and the number of measurement settings, there are more nontrivial facet inequalities. For several Bell scenarios, by involving two dichotomic measurement settings for two parties and one dichotomic measurement by other parties, we show that the local polytope has only one nontrivial facet. For three parties, we have three variants of this inequality, depending upon which party is doing one dichotomic measurement. This measurement scenario for a multipartite state may be considered as the minimal scenario involving multipartite correlations that can detect nonlocality. We show that this inequality is violated by all generalized GHZ states. Being the only facet Bell inequality, this inequality is also violated by any entangled three-qubit pure state. We also show that for noisy [Formula: see text] states, our inequality is more effective than the well-known Mermin inequality.
We have determined numerically the maximum quantum violation of over 100 tight bipartite Bell inequalities with two-outcome measurements by each party on systems of up to four-dimensional Hilbert spaces. We have found several cases, including the ones where each party has only four measurement choices, where two-dimensional systems, i.e., qubits, are not sufficient to achieve maximum violation. In a significant proportion of those cases when qubits are sufficient, one or both parties have to make trivial, degenerate ``measurements'' in order to achieve maximum violation. The quantum state corresponding to the maximum violation in most cases is not the maximally entangled one. We also obtain the result that bipartite quantum correlations can always be reproduced by measurements and states which require only real numbers if there is no restriction on the size of the local Hilbert spaces. Therefore in order to achieve maximum quantum violation on any bipartite Bell inequality (with any number of settings and outcomes), there is no need to consider complex Hilbert spaces.
A bipartite Bell inequality is derived which is maximally violated on the two-qubit state space if measurements describable by positive operator valued measure (POVM) elements are allowed, rather than restricting the possible measurements to projective ones. In particular, the presented Bell inequality requires POVMs in order to be maximally violated by a maximally entangled two-qubit state. This answers a question raised by N. Gisin [in Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle: Essays in Honour of Abner Shimony, edited by W. C. Myrvold and J. Christian (Springer, The Netherlands, 2009), pp. 125--138].
While quantum networks have been extensively studied as a natural extension of the standard Bell scenario with a richer correlation structure, general constructions of nonlinear Bell inequalities with self-testing properties are still largely lacking. In this work, we present a general method for constructing such inequalities in the simplest network scenario, in which two independent sources distribute bipartite quantum states to three spatially separated observers. These inequalities allow for arbitrary numbers of binary measurements and are maximally violated by maximally entangled states of the corresponding local dimensions together with sets of pairwise anticommuting Clifford observables. Importantly, their maximal quantum values can be determined analytically, which makes them particularly promising for device-independent applications. In particular, we prove that these Bell inequalities can be used to device-independently certify the underlying quantum network, including both the quantum states produced by the sources and the observables measured by all parties. To the best of our knowledge, this is the first self-testing result for quantum networks that relies solely on the maximal violation of a nonlocality witness.