Abstract
We explore quantum nonlocality in one of the simplest bipartite scenarios. Several new facet-defining Bell inequalities for the {[3 3 3] [3 3 3]} 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.
Highlights
In our article [1], we claimed to have given the first example of a facet-defining Bell inequality where a genuine positive-operator-valued (POVM) measure is relevant
Using the obtained measurement statistics, we demonstrate how characterization of the underlying resource in the spirit of deviceindependence, but supplemented with auxiliary assumptions, can be achieved
In this work, starting from the Bell inequality I3+ presented in Ref. [69] and analyzing its quantum violation for a family of two-qutrit states, we obtain several novel facet-defining Bell inequalities for the Bell scenario {[3 3 3] [3 3 3]}. All these newly obtained Bell inequalities that are not reducible to simpler Bell scenarios can already be violated maximally using entangled two-qubit states, including a Bell state. Some of these Bell inequalities that are reducible to Bell inequalities involving fewer number of outcomes require entangled two-qutrit states to achieve maximal quantum violation
Summary
In the classic paper where Schrodinger [1] introduced the term quantum entanglement, he remarked that this is not one but rather the characteristic trait of quantum mechanics that forces its entire departure from a classical line of thought. The paradigm of device-independent quantum information [3, 12] — where the analysis of quantum information is based solely on the observed correlations — has been applied in the context of randomness ex- For all these tasks, an imperative step is to certify that the observed correlation is not Bell-local — a task that is often achieved through the violation of Bell inequalities [2]. We present in this work several novel facet-defining Bell inequalities for this scenario Some of these newly obtained Bell inequalities — despite being ternary-outcome and irreducible to one having fewer measurement outcomes — can already be violated maximally via local measurements on entangled two-qubit states. [33, 39, 41], showing that in determining the quantum state that maximally violates a given Bell inequality, optimal choice of the local Hilbert space dimension is not necessarily correlated with the (maximal) number of measurement outcomes involved. Technical details related to numerical optimizations and certain results obtained thereof are relegated to the appendices
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