All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed set of operational equivalences (original) (raw)

Characterising and bounding the set of quantum behaviours in contextuality scenarios

Quantum, 2021

The predictions of quantum theory resist generalised noncontextual explanations. In addition to the foundational relevance of this fact, the particular extent to which quantum theory violates noncontextuality limits available quantum advantage in communication and information processing. In the first part of this work, we formally define contextuality scenarios via prepare-and-measure experiments, along with the polytope of general contextual behaviours containing the set of quantum contextual behaviours. This framework allows us to recover several properties of set of quantum behaviours in these scenarios, including contextuality scenarios and associated noncontextuality inequalities that require for their violation the individual quantum preparation and measurement procedures to be mixed states and unsharp measurements. With the framework in place, we formulate novel semidefinite programming relaxations for bounding these sets of quantum contextual behaviours. Most significantly, ...

What is the appropriate notion of noncontextuality for unsharp measurements in quantum theory?

In order to claim that one has experimentally tested whether a noncontextual ontological model could underlie certain measurement statistics in quantum theory, it is necessary to have a notion of noncontextuality that applies to unsharp measurements, i.e., those that can only be represented by positive operator-valued measures rather than projection-valued measures. This is because any realistic measurement necessarily has some nonvanishing amount of noise and therefore never achieves the ideal of sharpness. Assuming a generalized notion of noncontextuality that applies to arbitrary experimental procedures, it is shown that the outcome of a measurement depends deterministically on the ontic state of the system being measured if and only if the measurement is sharp. Hence for every unsharp measurement, its outcome necessarily has an indeterministic dependence on the ontic state. We defend this proposal against alternatives. In particular, we demonstrate why considerations parallel to Fine's theorem do not challenge this conclusion.

The Status of Determinism in Proofs of the Impossibility of a Noncontextual Model of Quantum Theory

Foundations of Physics, 2014

Minimal state-dependent proof of measurement contextuality for a qubit

Physical Review A, 2014

ABSTRACT We show that three unsharp dichotomic qubit measurements are enough to violate a generalized-noncontextual inequality in a state-dependent manner. For the case of trine spin axes we calculate the optimal quantum violation of this inequality. We conjecture this to be the optimal quantum violation obtainable from qubit measurements. Besides, we show that unsharp qubit measurements do not allow a state-independent violation of this inequality. We thus provide a minimal state-dependent proof of measurement contextuality requiring one qubit and three unsharp measurements. This is a novel no-go theorem for generalized-noncontextual models of these measurements.

From statistical proofs of the Kochen-Specker theorem to noise-robust noncontextuality inequalities

Physical Review A

The Kochen-Specker theorem rules out models of quantum theory wherein projective measurements are assigned outcomes deterministically and independently of context. This notion of noncontextuality is not applicable to experimental measurements because these are never free of noise and thus never truly projective. For nonprojective measurements, therefore, one must drop the requirement that an outcome is assigned deterministically in the model and merely require that it is assigned a distribution over outcomes in a manner that is context-independent. By demanding context-independence in the representation of preparations as well, one obtains a generalized principle of noncontextuality that also supports a quantum no-go theorem. Several recent works have shown how to derive inequalities on experimental data which, if violated, demonstrate the impossibility of finding a generalized-noncontextual model of this data. That is, these inequalities do not presume quantum theory and, in particular, they make sense without requiring an operational analogue of the quantum notion of projectiveness. We here describe a technique for deriving such inequalities starting from arbitrary proofs of the Kochen-Specker theorem. It extends significantly previous techniques that worked only for logical proofs, which are based on sets of projective measurements that fail to admit of any deterministic noncontextual assignment, to the case of statistical proofs, which are based on sets of projective measurements that do admit of some deterministic noncontextual assignments, but not enough to explain the quantum statistics.

Robust Self-Testing of Quantum Systems via Noncontextuality Inequalities

Physical Review Letters

Characterising unknown quantum states and measurements is a fundamental problem in quantum information processing. In this Letter, we provide a novel scheme to self-test local quantum systems using non-contextuality inequalities. Our work leverages the graph-theoretic framework for contextuality introduced by Cabello, Severini, and Winter, combined with tools from mathematical optimisation that guarantee the unicity of optimal solutions. As an application, we show that the celebrated Klyachko-Can-Binicioğlu-Shumovsky inequality and its generalisation to contextuality scenarios with odd n-cycle compatibility relations admit robust self-testing.

Sub-poissonian statistics as an experimental test for the contextuality of quantum theory

Physics Letters A, 1984

It is argued that the phenomenon of sub-poissonian statistics can be regarded as experimental evidence for the contextual character of quantum theory. To this end, it is shown that the statistics predicted by non-contextual hidden-variable theories must satisfy certain inequalities which are a kind of local counterparts of the famous Bell inequalities and which exclude sub-poissonian statistics. 1. Introduction. Most recent work on the interpretation of quantum mechanics hinges on the question whether local hidden-variable theories are possible which reproduce the statistical predictions of quantum mechanics. The issue is amenable to experimental investigation since such local hidden variable theories must statisfy inequalities of the Bell-type, whereas it can be shown that the quantum mechanical formalism leads to predictions which, in certain circumstances, violate those inequalities. The experimental results [ 1 ] thus far strongly favour quantum mechanics; i.e. it seems that experiment excludes the validity of any local hidden-variable theory. The common emphasis on the importance of the notion of locality tends to obscure the fact that a more genera ! point is at issue here. In a number of socalled no-hidden-variable-proofs by Gleason [2], Kochen and Specker [3], and others [4], it has been demonstrated that the predictions of quantum mechanics are incompatible with non-contextual hidden variable theories, i.e. theories in which the result of an individual measurement of a physical quantity is independent of which other physical quantities are measured along with it. The proofs by Gleason and by Kochen and Specker show that it is already impossible to fulfil the requirement that the measurement results do not depend on which other physical quantities are also measured if these other quantities correspond to quantum mechanical observables commuting with the

Uniqueness of all fundamental noncontextuality inequalities

Physical Review Research

Contextuality is one way of capturing the nonclassicality of quantum theory. The contextual nature of a theory is often witnessed via the violation of noncontextuality inequalities-certain linear inequalities involving probabilities of measurement events. Using the exclusivity graph approach (one of the two main graph theoretic approaches for studying contextuality), it was shown [Cabello et al. Phys. Rev. A 88, 032104 (2013); Chudnovsky et al. Ann. Math. 164, 51 (2006)] that a necessary and sufficient condition for witnessing contextuality is the presence of an odd number of events (greater than three) which are either cyclically or anticyclically exclusive. Thus, the noncontextuality inequalities the underlying exclusivity structure of which is as stated, either cyclic or anticyclic, are fundamental to quantum theory. We show that there is a unique noncontextuality inequality for each nontrivial cycle and anticycle. In addition to the foundational interest, we expect this to aid the understanding of contextuality as a resource to quantum computing and its applications to local self-testing.

Discrimination of measurement contexts in quantum mechanics

Physics Letters A, 2011

We demonstrate that it is possible to discern the way that has been followed to measure a quantum observable that can be expressed in terms of different products of observables, whereas no such discrimination is possible by assigning predetermined values. Specifically we show how to distinguish different routes (contexts) to measure C = AB = A ′ B ′ , when C, A, B and C, A ′ , B ′ commute with each other, but A and B do not commute with A ′ and B ′ .

All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed set of operational equivalences (original) (raw)

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