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arXiv:1103.2557

Structural unification of space and time correlations in quantum theory

S. Marcovitch, Benni Reznik

Published Mar 13, 2011• 4 citations
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Abstract

We suggest a natural mapping between bipartite states and quantum evolutions of local states, which is a Jamiolkowski map. It is shown that spatial correlations of weak measurements in bipartite systems precisely coincide with temporal correlations of local systems. This mapping has several practical and conceptual implications on the correspondence between Bell and Leggett-Garg inequalities, the statistical properties of evolutions in large systems, temporal decoherence and computational gain, in evaluation of spatial correlations of large systems.

DOI
10.48550/arxiv.1103.2557

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We suggest a natural mapping between bipartite states and quantum evolutions of local states, which is a Jamiolkowski map. It is shown that spatial correlations of weak measurements in bipartite systems precisely coincide with temporal correlations of local systems. This mapping has several practical and conceptual implications on the correspondence between Bell and Leggett-Garg inequalities, the statistical properties of evolutions in large systems, temporal decoherence and computational gain, in evaluation of spatial correlations of large systems.
The Hilbert space formalism of quantum theory manifests a map between bipartite states and time evolutions, known as Jamiolkowski isomorphism. We extend this map in a physical setting to prove the equality of spatial correlations in bipartite systems and temporal correlations in local systems. We show that these correlations can be observed using weak measurements. This result has several practical and conceptual implications such as manifestation of state independent decoherence, the correspondence between Bell and Leggett-Garg inequalities, multipartite systems, the statistical properties of evolutions in large systems, and computational gain, in evaluation of spatial correlations in large systems.
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Abstract The formalism of generalized quantum histories allows a symmetrical treatment of space and time correlations, by taking different traces of the same history density matrix. In this framework, the characterization of spatial and temporal entanglement is revisited. An operative protocol is presented, to map a history state into the ket of a static composite system. It is demonstrated, by examples, how the Leggett–Garg and the temporal Clauser‐Horne‐Shimony‐Holt (CHSH) inequalities can be violated in this approach.
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.
The traditional formalism of nonrelativistic quantum theory allows the state of a quantum system to extend across space, but only restricts it to a single instant in time, leading to distinction between theoretical treatments of spatial and temporal quantum correlations. Here we unify the geometrical description of two-point quantum correlations in space-time. Our study presents the geometry of correlations between two sequential Pauli measurements on a single qubit undergoing an arbitrary quantum channel evolution together with two-qubit spatial correlations under a common framework. We establish a symmetric structure between quantum correlations in space and time. This symmetry is broken in the presence of nonunital channels, which further reveals a set of temporal correlations that are indistinguishable from correlations found in bipartite entangled states.