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.
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.
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.
Quantum contextuality describes situations where the statistics observed in different measurement contexts cannot be explained by a measurement of the independent reality of the system. The most simple case is observed in a three-dimensional Hilbert space, with five different measurement contexts related to each other by shared measurement outcomes. The quantum formalism defines the relations between these contexts in terms of well-defined relations between operators, and these relations can be used to reconstruct an unknown quantum state from a finite set of measurement results. Here, I introduce a reconstruction method based on the relations between the five measurement contexts that can violate the bounds of non-contextual statistics. A complete description of an arbitrary quantum state requires only five of the eight elements of a Kirkwood-Dirac quasiprobability, but only an overcomplete set of eleven elements provides an unbiased description of all five contexts. A set of five fundamental relations between the eleven elements reveals a deterministic structure that links the five contexts. As illustrated by a number of examples, these relations provide a consistent description of contextual realities for the measurement outcomes of all five contexts.
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’.
We introduce a contextual quantum system comprising mutually complementary observables organized into two or more collections of pseudocontexts with the same probability sums of outcomes. These pseudocontexts constitute nonorthogonal bases within the Hilbert space, featuring a state-independent sum of probabilities. In other words, regardless of the initial-state preparation, the total probability remains constant but may be distinct from unity. The measurement contextuality in this setup arises from the quantum realizations of the hypergraph, which adhere to a specific bound on the linear combination of probabilities. In contrast, classical realizations can surpass this bound. The violation of quantum bounds stems from the inability of classical ontological models, specifically the set-theoretic representation of the hypergraph corresponding to the quantum observables' collections, to adhere to and explain the observed statistics.
Quantum mechanics describes the relation between different measurement contexts in terms of superpositions of the potential measurement outcomes. This relation between measurement contexts makes it impossible to determine context independent realities. Here, I illustrate the problem using three path interferences that implement the three box paradox and show that the representation of the final measurement outcome as a superposition of intermediate paths describes well-defined coherences between seemingly empty paths.
Abstract In this paper, we investigate the possibility of explaining nonclassical correlations between two quantum systems in terms of quantum interferences between collective states of the two systems. We achieve this by mapping the relations between different measurement contexts in the product Hilbert space of a pair of two-level systems onto an analogous sequence of interferences between paths in a single-particle interferometer. The relations between different measurement outcomes are then traced to the distribution of probability currents in the interferometer, where paradoxical relations between the outcomes are identified with currents connecting two states that are orthogonal and should therefore exclude each other. We show that the relation between probability currents and correlations can be represented by continuous conditional (quasi)probability currents through the interferometer, given by weak values; the violation of the noncontextual assumption is expressed by negative conditional currents in some of the paths. Since negative conditional currents correspond to the assignment of negative conditional probabilities to measurements results in different measurement contexts, the necessity of such negative probability currents represents a failure of noncontextual local realism. Our results help to explain the meaning of nonlocal correlations in quantum mechanics, and support Feynman’s claim that interference is the origin of all quantum phenomena.