Counterfactual definiteness (CFD) means that if some property is measured in some context, then the outcome of the measurement would have been the same had this property been measured in a different context. A context includes all other measurements made together with the one in question, and the spatiotemporal relations among them. The proviso for CFD is non-disturbance: any physical influence of the contexts on the property being measured is excluded by the laws of nature, so that no one measuring this property has a way of ascertaining its context. It is usually claimed that in quantum mechanics CFD does not hold, because if one assigns the same value to a property in all contexts it is measured in, one runs into a logical contradiction, or at least contravenes quantum theory and experimental evidence. We show that this claim is not substantiated if one takes into account that only one of the possible contexts can be a factual context, all other contexts being counterfactual. With this in mind, any system of random variables can be viewed as satisfying CFD. The concept of CFD is closely related to but distinct from that of noncontextuality, and it is the latter property that may or may not hold for a system, in particular being contravened by some quantum systems.
Counterfactual definiteness (CFD) means that if some property is measured in some context, then the outcome of the measurement would have been the same had this property been measured in a different context. A context includes all other measurements made together with the one in question, and the spatiotemporal relations among them. The proviso for CFD is non-disturbance: any physical influence of the contexts on the property being measured is excluded by the laws of nature, so that no one measuring this property has a way of ascertaining its context. It is usually claimed that in quantum mechanics CFD does not hold, because if one assigns the same value to a property in all contexts it is measured in, one runs into a logical contradiction, or at least contravenes quantum theory and experimental evidence. We show that this claim is not substantiated if one takes into account that only one of the possible contexts can be a factual context, all other contexts being counterfactual. With this in mind, any system of random variables can be viewed as satisfying CFD. The concept of CFD is closely related to but distinct from that of noncontextuality, and it is the latter property that may or may not hold for a system, in particular being contravened by some quantum systems.
Contextuality is usually defined as absence of a joint distribution for a set of measurements (random variables) with known joint distributions of some of its subsets. However, if these subsets of measurements are not disjoint, contextuality is mathematically impossible even if one generally allows (as one must) for random variables not to be jointly distributed. To avoid contradictions one has to adopt the Contextuality‐by‐Default approach: measurements made in different contexts are always distinct and stochastically unrelated to each other. Contextuality is reformulated then in terms of the (im)possibility of imposing on all the measurements in a system a joint distribution of a particular kind: such that any measurements of one and the same property made in different contexts satisfy a specified property, . In the traditional analysis of contextuality means “are equal to each other with probability 1”. However, if the system of measurements violates the “no‐disturbance principle”, due to signaling or experimental biases, then the meaning of has to be generalized, and the proposed generalization is “are equal to each other with maximal possible probability” (applied to any set of measurements of one and the same property). This approach is illustrated on arbitrary systems of binary measurements, including most of quantum systems of traditional interest in contextuality studies (irrespective of whether the “no‐disturbance” principle holds in them).
This paper deals with three traditional ways of defining contextuality: (C1) in terms of (non)existence of certain joint distributions involving measurements made in several mutually exclusive contexts; (C2) in terms of relationship between factual measurements in a given context and counterfactual measurements that could be made if one used other contexts; and (C3) in terms of (non)existence of 'hidden variables' that determine the outcomes of all factually performed measurements. It is generally believed that the three meanings are equivalent, but the issues involved are not entirely transparent. Thus, arguments have been offered that C2 may have nothing to do with C1, and the traditional formulation of C1 itself encounters difficulties when measurement outcomes in a contextual system are treated as random variables. I show that if C1 is formulated within the framework of the Contextuality-by-Default (CbD) theory, the notion of a probabilistic coupling, the core mathematical tool of CbD, subsumes both counterfactual values and 'hidden variables'. In the latter case, a coupling itself can be viewed as a maximally parsimonious choice of a hidden variable. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.
In quantum physics, the term ‘contextual’ can be used in more than one way. One usage, here called ‘Bell contextual’ since the idea goes back to Bell, is that if A , B and C are three quantum observables, with A compatible (i.e. commuting) with B and also with C , whereas B and C are incompatible, a measurement of A might yield a different result (indicating that quantum mechanics is contextual) depending upon whether A is measured along with B (the { A , B } context) or with C (the { A , C } context). An analysis of what projective quantum measurements measure shows that quantum theory is Bell non-contextual: the outcome of a particular A measurement when A is measured along with B would have been exactly the same if A had, instead, been measured along with C . A different definition, here called ‘globally (non)contextual’ refers to whether or not there is (non-contextual) or is not (contextual) a single joint probability distribution that simultaneously assigns probabilities in a consistent manner to the outcomes of measurements of a certain collection of observables, not all of which are compatible. A simple example shows that such a joint probability distribution can exist even in a situation where the measurement probabilities cannot refer to properties of a quantum system, and hence lack physical significance, even though mathematically well defined. It is noted that the quantum sample space, a projective decomposition of the identity, required for interpreting measurements of incompatible properties in different runs of an experiment using different types of apparatus, has a tensor product structure, a fact sometimes overlooked. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.
The circumstances of measurement have more direct significance in quantum than in classical physics, where they can be neglected for well-performed measurements. In quantum mechanics, the dispositions of the measuring apparatus-plus-environment of the system measured for a property are a non-trivial part of its formalization as the quantum observable. A straightforward formalization of context, via equivalence classes of measurements corresponding to sets of sharp target observables, was recently given for sharp quantum observables. Here, we show that quantum contextuality, the dependence of measurement outcomes on circumstances external to the measured quantum system, can be manifested not only as the strict exclusivity of different measurements of sharp observables or valuations but via quantitative differences in the property statistics across simultaneous measurements of generalized quantum observables, by formalizing quantum context via coexistent generalized observables rather than only its subset of compatible sharp observables. Here, the question of whether such quantum contextuality follows from basic quantum principles is then addressed, and it is shown that the Principle of Indeterminacy is sufficient for at least one form of non-trivial contextuality. Contextuality is thus seen to be a natural feature of quantum mechanics rather than something arising only from the consideration of impossible measurements, abstract philosophical issues, hidden-variables theories, or other alternative, classical models of quantum behavior.
An analysis using classical stochastic processes is used to construct a consistent system of quantum counterfactual reasoning. When applied to a counterfactual version of Hardy's paradox, it shows that the probabilistic character of quantum reasoning together with the ``one framework'' rule prevents a logical contradiction, and there is no evidence within this approach for any nonlocal influences. Counterfactual reasoning can support a realistic interpretation of standard quantum theory (measurements reveal what is actually there) under appropriate circumstances.
Bell-type inequalities are proven using oversimplified probabilistic models and/or counterfactual definiteness (CFD). If setting-dependent variables describing measuring instruments are correctly introduced, none of these inequalities may be proven. In spite of this, a belief in a mysterious quantum nonlocality is not fading. Computer simulations of Bell tests allow people to study the different ways in which the experimental data might have been created. They also allow for the generation of various counterfactual experiments' outcomes, such as repeated or simultaneous measurements performed in different settings on the same "photon-pair", and so forth. They allow for the reinforcing or relaxing of CFD compliance and/or for studying the impact of various "photon identification procedures", mimicking those used in real experiments. Data samples consistent with quantum predictions may be generated by using a specific setting-dependent identification procedure. It reflects the active role of instruments during the measurement process. Each of the setting-dependent data samples are consistent with specific setting-dependent probabilistic models which may not be deduced using non-contextual local realistic or stochastic hidden variables. In this paper, we will be discussing the results of these simulations. Since the data samples are generated in a locally causal way, these simulations provide additional strong arguments for closing the door on quantum nonlocality.