The Bell-type (spatial), Kochen-Specker (contextuality) or Leggett-Garg (temporal) inequalities are based on classically plausible but otherwise quite distinct assumptions. For any of these inequalities, satisfaction is equivalent to a joint probability distribution for all observables in the experiment. This implies a joint distribution for all pairs of observables, and is indifferent to whether or not they commute in the theory. This indifference underpins a unification of the above inequalities into a general framework of correlation inequalities. When the physical scenario is such that the correlated pairs are all compatible, the resulting correlation is nonsignaling, which may be local or multi-particle, corresponding to contextuality or Bell-type inequalities. If the pairs are incompatible, the resulting correlation corresponds to Leggett-Garg (LG) inequalities. That quantum mechanics (QM) violates all these inequalities suggests a close connection between the local, spatial and temporal properties of the theory. As a concrete manifestation of the unification, we extend the method due to Roy and Singh ( J. Phys. A , 11 (1978) L167) to derive and study a new class of hybrid spatio-temporal inequalities, where the correlated pairs in the experiment are both compatible or incompatible. The implications for cryptography and monogamy inequalities of the unification are briefly touched upon.
The Bell-type (spatial), Kochen-Specker (contextuality) or Leggett-Garg (temporal) inequalities are based on classically plausible but otherwise quite distinct assumptions. For any of these inequalities, satisfaction is equivalent to a joint probability distribution for all observables in the experiment. This implies a joint distribution for all pairs of observables, and is indifferent to whether or not they commute in the theory. This indifference underpins a unification of the above inequalities into a general framework of correlation inequalities. When the physical scenario is such that the correlated pairs are all compatible, the resulting correlation is nonsignaling, which may be local or multi-particle, corresponding to contextuality or Bell-type inequalities. If the pairs are incompatible, the resulting correlation corresponds to Leggett-Garg (LG) inequalities. That quantum mechanics (QM) violates all these inequalities suggests a close connection between the local, spatial and temporal properties of the theory. As a concrete manifestation of the unification, we extend the method due to Roy and Singh ( J. Phys. A , 11 (1978) L167) to derive and study a new class of hybrid spatio-temporal inequalities, where the correlated pairs in the experiment are both compatible or incompatible. The implications for cryptography and monogamy inequalities of the unification are briefly touched upon.
We highlight the existence of a joint probability distribution as the common underpinning assumption behind Bell-type, contextuality, and Leggett-Garg-type tests. We then present a procedure to translate contextual scenarios into temporal Leggett-Garg-type and spatial Bell-type ones. To demonstrate the generality of this approach we construct a family of spatial Bell-type inequalities. We show that in the Leggett-Garg scenario a necessary condition for contextuality in time is given by a violation of consistency conditions in the consistent histories approach to quantum mechanics.
Introduction Our aim in this chapter is to extend the Bell inequality discussion in Chapter 17 to its temporal analogue, known as the Leggett–Garg (LG) inequality . Bell inequalities involve spatial nonlocality, that is, signal observations distributed over space. It was shown by Leggett and Garg that analogous inequalities involving temporal nonlocality can be formulated (Leggett and Garg, 1985). We shall discuss one of these, known as the LG inequality . We saw in Chapter 17 that Bell inequalities are based on certain classical mechanics (CM) assumptions about the nature of reality. Likewise, the LG inequality is based on two CM principles that are not incorporated into quantum mechanics (QM). Definition 19.1 The principle of macrorealism asserts that if a macroscopic (that is, large-scale) system under observation (SUO) can be observed to be in one of two or more macroscopically distinct states, then it will always be in one or another of those states at any given time, not in a quantum superposition of those states, even when it is not being observed. This principle is a foundational principle in CM but is incompatible with QM in at least two ways. First, it is vacuous, as it asserts the truth of a proposition in the absence of empirical validation: how can we define the concept of always , without introducing counterfactuality? Second, it is violated in quantum theory, for instance, in path integral calculations. This principle in embedded in the nexus of issues explored in this book. The quantized detector network (QDN) version of it takes the form “If any SUO can be observed in any number of possible states, then it will be observed in one of them in any run.” This is a near tautology. The classical version says the same thing but makes an additional assertion about something going on in the absence of observation, which is the vacuous element that QM cannot accept. Definition 19.2 The principle of noninvasive measurability asserts that the actual state an SUO is in can always be determined cost-free, that is, without having any effect on that state or on its subsequent dynamical evolution. This principle is subtle. We experience its apparent validity all the time in our ordinary lives: we look at objects and they appear not to change by those acts of observation.
Abstract The relations between Bell nonlocality and Kochen–Specker contextuality have been subject of research from many different perspectives in the last decades. Recently, some interesting results on these relations have been explored in the so-called generalized Bell scenarios, that is, scenarios where Bell spatial separation (or agency independence) coexist with (at least one of the) parties’ ability to perform compatible measurements at each round of the experiment. When this party has an n-cycle compatiblity setup, it was first claimed that Bell nonlocality could not be concomitantly observed with contextuality at this party’s local experiment. However, by a more natural reading of the definition of locality, it turns out that both Bell nonlocality and local contextuality can, in fact, be jointly present. In spite of it, in this work we prove that in the simplest of those scenarios there cannot be arbitrary amounts of both of these two resources together. That is, in these cases we show that the violation of any Bell inequality limits the possible violations of any local noncontextuality inequality. We also explore this trade-off relation using quantifiers of nonlocality and contextuality, discussing how such a relation can be understood in terms of a ‘global’ notion of contextuality, and we study possible extensions of this result to other scenarios.
Contextuality means non-existence of a joint distribution for random variables recorded under mutually incompatible conditions, subject to certain constraints imposed on how the identity of these variables may change across these conditions. In simple quantum systems contextuality is indicated by violations of Bell-type or Leggett-Garg-type inequalities. These inequalities, however, are predicated on the assumption of no-signaling, defined as invariance of the distributions of measurement results with respect to other (e.g., earlier in time) measurements' settings. Signaling makes the inequalities inapplicable: a non-signaling system with any degree of contextuality, however high, loses any relation to this concept as soon as it exhibits any degree of signaling, however small. This is unsatisfactory. We describe a principled way of defining and measuring contextuality in arbitrary systems with random outputs, whether signaling is absent or present.
Nonlocality and contextuality are at the root of conceptual puzzles in quantum mechanics, and they are key resources for quantum advantage in information-processing tasks.Bell nonlocality is best understood as the incompatibility between quantum correlations and the classical theory of causality, applied to relativistic causal structure.Contextuality, on the other hand, is on a more controversial foundation.In this work, I provide a common conceptual ground between nonlocality and contextuality as violations of classical causality.First, I show that Bell inequalities can be derived solely from the assumptions of no signaling and no fine-tuning of the causal model.This removes two extra assumptions from a recent result from Wood and Spekkens and, remarkably, does not require any assumption related to independence of measurement settings-unlike all other derivations of Bell inequalities.I then introduce a formalism to represent contextuality scenarios within causal models and show that all classical causal models for violations of a Kochen-Specker inequality require fine-tuning.Thus, the quantum violation of classical causality goes beyond the case of spacelike-separated systems and already manifests in scenarios involving single systems.
Leggett-Garg (LG) inequalities test whether correlations measured at different times admit a positive joint probability distribution over histories. A violation requires negative values in any joint quasiprobability distribution reproducing those correlations, but such distributions can have negative values even without a violation. Spacetime density kernels extend the density-matrix description to multiple times and are used to study timelike entanglement. We establish a universal bound that relates LG violations to their negative eigenvalue weight. For LG tests involving any number of measurements, we determine the least compatible kernel negativity through an exact relation valid for any system dimension, initial state, dynamics, or measurement rule. This minimum is nonzero if and only if at least one LG inequality is violated. Three-time Lüders LG tests certify at most one eighth of negativity, whereas measurements resolving individual spin levels can approach one half. Including additional incompatible qubit observables gives a further exact minimum that can exceed the LG-only ceiling without increasing the LG violation.