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Bell violations from arbitrary joint measurability structures

Shiv Akshar Yadavalli, Nikola Andrejić, Ravi Kunjwal

Published Aug 23, 2020• 1 citation
OpenAlex

Abstract

Measurement incompatibility is necessary but not sufficient for violation of a Bell inequality. The structure of (in)compatibility relations among a set of measurements can be represented by a joint measurability structure, i.e., a hypergraph with its vertices representing measurements and its hyperedges representing all (and only) compatible sets of measurements. We show that for any joint measurability structure that admits at least a pair of incompatible vertices (i.e., a non-trivial joint measurability structure), there exists a quantum realization with a set of measurements that enables a Bell violation. In other words, we identify, for any joint measurability structure, an incompatible set of measurements that is sufficient for a Bell violation.

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Measurement incompatibility is necessary but not sufficient for violation of a Bell inequality. The structure of (in)compatibility relations among a set of measurements can be represented by a joint measurability structure, i.e., a hypergraph with its vertices representing measurements and its hyperedges representing all (and only) compatible sets of measurements. We show that for any joint measurability structure that admits at least a pair of incompatible vertices (i.e., a non-trivial joint measurability structure), there exists a quantum realization with a set of measurements that enables a Bell violation. In other words, we identify, for any joint measurability structure, an incompatible set of measurements that is sufficient for a Bell violation.
Measurements in quantum theory can fail to be jointly measurable. Like entanglement, this incompatibility of measurements is necessary but not sufficient for violating Bell inequalities. The (in)compatibility relations among a set of measurements can be represented by a joint measurability structure, i.e., a hypergraph whose vertices denote measurements and hyperedges denote all and only compatible sets of measurements. Since incompatibility is necessary for a Bell violation, the joint measurability structure on each wing of a Bell experiment must necessarily be non-trivial, i.e., it must admit a subset of incompatible vertices. Here we show that for any non-trivial joint measurability structure with a finite set of vertices, there exists a quantum realization with a set of measurements that enables a Bell violation, i.e., given that Alice has access to this incompatible set of measurements, there exists a set of measurements for Bob and an entangled state shared between them such that they can jointly violate a Bell inequality. Hence, a non-trivial joint measurability structure is not only necessary for a Bell violation, but also sufficient.
We discuss the connection between the incompatibility of quantum measurements, as captured by the notion of joint measurability, and the violation of Bell inequalities. Specifically, we explicitly present a given set of non-jointly-measurable positive-operator-value measures (POVMs) MA{\mathcal{M}}_{A}MA​ with the following property. Considering a bipartite Bell test where Alice uses MA{\mathcal{M}}_{A}MA​, then for any possible shared entangled state $\ensuremath{\rho}$ and any set of (possibly infinitely many) POVMs NB{\mathcal{N}}_{B}NB​ performed by Bob, the resulting statistics admits a local model and can thus never violate any Bell inequality. This shows that quantum measurement incompatibility does not imply Bell nonlocality in general.
Measurements in quantum theory exhibit incompatibility, i.e., they can fail to be jointly measurable.An intuitive way to represent the (in)compatibility relations among a set of measurements is via a hypergraph representing their joint measurability structure: its vertices represent measurements and its hyperedges represent (all and only) subsets of compatible measurements.Projective measurements in quantum theory realize (all and only) joint measurability structures that are graphs.On the other hand, general measurements represented by positive operator-valued measures (POVMs) can realize arbitrary joint measurability structures.Here we explore the scope of joint measurability structures realizable with qubit POVMs.We develop a technique that we term marginal surgery to obtain nontrivial joint measurability structures starting from a set of compatible measurements.We show explicit examples of marginal surgery on a special set of qubit POVMs to construct joint measurability structures such as N-cycle and N-Specker scenarios for any integer N 3.We also show the realizability of various joint measurability structures with N ∈ {4, 5, 6} vertices.In particular, we show that all possible joint measurability structures with N = 4 vertices are realizable.We conjecture that all joint measurability structures are realizable with qubit POVMs.This contrasts with the unbounded dimension required in R. Kunjwal et al. [Phys. Rev. A 89, 052126 (2014)].Our results also render this previous construction maximally efficient in terms of the required Hilbert space dimension.We also obtain a sufficient condition for the joint measurability of any set of binary qubit POVMs which powers many of our results and should be of independent interest.
We study the compatibility (or joint measurability) of quantum observables in a setting where the experimenter has access to multiple copies of a given quantum system, rather than performing the experiments on each individual copy separately. We introduce the index of incompatibility as a quantifier of incompatibility in this multi-copy setting, as well as the notion of the compatibility stack representing various compatibility relations present in a given set of observables. We then prove a general structure theorem for multi-copy joint observables and use it to prove that all abstract compatibility stacks with three vertices have realizations in terms of quantum observables.
Incompatibility of observables, or measurements, is one of the key features of quantum mechanics, related, among other concepts, to Heisenberg's uncertainty relations and Bell nonlocality. In this manuscript we show, however, that even though incompatible measurements are necessary for the violation of any Bell inequality, some relevant Bell-like inequalities may be obtained if compatibility relations are assumed between the local measurements of one (or more) of the parties. Hence, compatibility of measurements is not necessarily a drawback and may, however, be useful for the detection of Bell nonlocality and device-independent certification of entanglement.