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arXiv:2401.16769 · New Journal of Physics

Tracing quantum correlations back to collective interferences

Ming Ji, Jonte R. Hance, Holger Friedrich Hofmann

Published Jun 1, 2024• 3 citations
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Abstract

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.

DOI
10.1088/1367-2630/ad5619

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•Jun 19, 2024•0 citations
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.
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.
Abstract Basic quantum effects are often illustrated using single particle interferences in two-path interferometers. A wider range of non-classical phenomena can be illustrated using three-path interferometers, but the increased complexity of quantum statistics in a three-dimensional Hilbert space makes it difficult to identify a representative set of observable properties that could be used to characterize specific phenomena. Here, I propose a characterization of pure states based on a five-stage interferometer recently introduced to demonstrate the relation between different measurement contexts (Optica Quantum 1, 63 (2023)). It is shown that the orthogonality relations between the states representing the different measurement contexts can be used to classify pure states within the three-dimensional Hilbert space according to the non-classical correlations between different contexts expressed by negative Kirkwood–Dirac distributions.
Quantum contextuality describes scenarios in which it is impossible to explain the experimental evidence in terms of a measurement-independent reality. Here a three-path interferometer is introduced in which all five contexts needed for a demonstration of contextuality are realized in sequence. It is then possible to observe a paradoxical situation where the paths connecting input ports to their corresponding output ports appear to be blocked by destructive interference. It is shown that the conditional currents observed in weak measurements provide a consistent explanation of the paradox, indicating that weak values might help to bridge the gap between wave-like propagation effects and local particle detection.
We analyze nonclassical correlations between outcomes of measurements conducted on two spatial radiation modes. These correlations cannot be simulated with statistical mixtures of coherent states or, more generally, with non-negative phase-space functions of quantum states and measurements. We argue that nonclassical correlations are naturally related to Bell nonlocality, the former being a more general class of quantum correlations. Indeed, it is known that local realistic as well as noncontextual models correspond to non-negative solutions to a system of linear equations for the joint probability distributions of all observables. We demonstrate that nonclassical correlations correspond to a particular solution to this system, which may have negative values even if local realism is not violated. A modification of Bell inequalities enables us to test such correlations. At the same time, our approach leads to a formulation of Bell inequalities applicable also to continuous variables. The results are illustrated with two-mode squeezed vacuum states and with hybrid entangled states (Schr\"odinger-Cat states), one mode being analyzed by balanced and the other one by unbalanced homodyne detection.
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.
The presence of an absorber in one of the paths of an interferometer changes the output statistics of that interferometer in a fundamental manner. Since the individual quantum particles detected at any of the outputs of the interferometer have not been absorbed, any non-trivial effect of the absorber on the distribution of these particles over these paths is a counterfactual effect. Here, we quantify counterfactual effects by evaluating the information about the presence or absence of the absorber obtained from the output statistics, distinguishing between classical and quantum counterfactual effects. We identify the counterfactual gain which quantifies the advantage of quantum counterfactual protocols over classical counterfactual protocols, and show that this counterfactual gain can be separated into two terms: a semi-classical term related to the amplitude blocked by the absorber, and a Kirkwood-Dirac quasiprobability assigning a joint probability to the blocked path and the output port. A negative Kirkwood-Dirac term between a path and an output port indicates that inserting the absorber into that path will have a focussing effect, increasing the probability of particles arriving at that output port, resulting in a significant enhancement of the counterfactual gain. We show that the magnitude of quantum counterfactual effects cannot be explained by a simple removal of the absorbed particles, but originates instead from a well-defined back-action effect caused by the presence of the absorber in one path, on particles in other paths.