Quantum paradoxes show that the outcomes of different quantum measurements cannot be described by a single measurement-independent reality. Any theoretical description of a quantum measurement implies the selection of a specific measurement context. Here, we investigate generalised quantum measurements, in order to identify the mechanism by which this specific context is selected. We show that external quantum fluctuations, represented by the initial state of the measurement apparatus, play an essential role in the selection of the context. This has the non-trivial consequence that, when considering measurements other than just idealised projection-valued measures, different outcomes of a single measurement setup can represent different measurement contexts. We further show this result underpins recent claims that contextuality can occur in scenarios without measurement incompatibility.
Quantum paradoxes show that the outcomes of different quantum measurements cannot be described by a single measurement-independent reality. Any theoretical description of a quantum measurement implies the selection of a specific measurement context. Here, we investigate generalised quantum measurements, in order to identify the mechanism by which this specific context is selected. We show that external quantum fluctuations, represented by the initial state of the measurement apparatus, play an essential role in the selection of the context. This has the non-trivial consequence that, when considering measurements other than just idealised projection-valued measures, different outcomes of a single measurement setup can represent different measurement contexts. We further show this result underpins recent claims that contextuality can occur in scenarios without measurement incompatibility.
The quantum fluctuations of a physical property can be observed in the measurement statistics of any measurement that is at least partially sensitive to that physical property. Quantum theory indicates that the effective distribution of values taken by the physical property depends on the specific measurement context based on which these values are determined and weak values have been identified as the contextual values describing this dependence of quantum fluctuations on the measurement context. Here, the relation between classical statistics and quantum contextuality is explored by considering systems entangled with a quantum reference. The quantum fluctuations of the system can then be steered by precise projective measurements of the reference, resulting in different contextual values of the quantum fluctuations depending on the effective state preparation context determined by the measurement of the reference. The results show that mixed-state statistics are consistent with a wide range of potential contexts, indicating that the precise definition of a context requires maximal quantum coherence in both state preparation and measurement.
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.
In quantum theory, a measurement context is defined by an orthogonal basis in a Hilbert space, where each basis vector represents a specific measurement outcome. The precise quantitative relation between two different measurement contexts can thus be characterized by the inner products of nonorthogonal states in that Hilbert space. Here, we use measurement outcomes that are shared by different contexts to derive specific quantitative relations between the inner products of the Hilbert space vectors that represent the different contexts. It is shown that the probabilities that describe the paradoxes of quantum contextuality can be derived from a very small number of inner products, revealing details of the fundamental relations between measurement contexts that go beyond a basic violation of noncontextual limits. The application of our analysis to a product space of two systems reveals that the nonlocality of quantum entanglement can be traced back to a local inner product representing the relation between measurement contexts in only one system. Our results thus indicate that the essential nonclassical features of quantum mechanics can be traced back to the fundamental difference between quantum superpositions and classical alternatives.
The origin and basis of the notion of quantum contextuality is identified in the Copenhagen approach to quantum mechanics, where context is automatically invoked by its requirement that the experimental arrangement involved in any measurements or set of measurements be taken into account while, in general, the outcome of a measurement may depend on other measurements immediately preceding or jointly performed on the same system. For Bohr, the specification of the experimental situation of any measurement is essential to its significance in light of complementarity and the omnipresence of the quantum of action in physics; for Heisenberg, the incompatibility of pairs of sharp measurements belonging to different situations coheres with both the completeness of the quantum state as an objective physical description and the principle of indeterminacy. Here, context in the Copenhagen approach is taken to be the equivalence class of experimental arrangements corresponding to a set of compatible measurements of quantum observables in standard quantum mechanics; the associated form of contextuality in quantum mechanics arises via the non-commutativity in general of sharp observables, proven by von Neumann, that can appear, providing different contexts. This notion is related to theoretical situations explored later by Bell, by Kochen and Specker, and by others in relation to the classification of hidden-variables theories and elsewhere in physics. 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’.
Quantum mechanics describes seemingly paradoxical relations between the outcomes of measurements that cannot be performed jointly. In Hilbert space, the outcomes of such incompatible measurements are represented by nonorthogonal states. In this paper we investigate how the relation between outcomes represented by nonorthogonal quantum states differs from the relations suggested by a joint assignment of measurement outcomes that do not depend on the actual measurement context. The analysis is based on a well-known scenario where three statements about the impossibilities of certain outcomes would seem to make a specific fourth outcome impossible as well, yet quantum theory allows the observation of that outcome with a nonvanishing probability. We show that the Hilbert space formalism modifies the relation between the four measurement outcomes by defining a lower bound of the fourth probability that increases as the total probability of the first three outcomes drops to zero. Quantum theory thus not only makes the violation of noncontextual consistency between the measurement outcomes possible, but actually requires it as a necessary consequence of the Hilbert space inner products that describe the contextual relation between the outcomes of different measurements.