We investigate quantum correlations in time in different approaches. We assume that temporal correlations should be treated in an even-handed manner with spatial correlations. We compare the pseudo-density matrix formalism with several other approaches: indefinite causal structures, consistent histories, generalised quantum games, out-of-time-order correlations(OTOCs), and path integrals. We establish close relationships among these space-time approaches in non-relativistic quantum theory, resulting in a unified picture. With the exception of amplitude-weighted correlations in the path integral formalism, in a given experiment, temporal correlations in the different approaches are the same or operationally equivalent.
We investigate quantum correlations in time in different approaches. We assume that temporal correlations should be treated in an even-handed manner with spatial correlations. We compare the pseudo-density matrix formalism with several other approaches: indefinite causal structures, consistent histories, generalised quantum games, out-of-time-order correlations(OTOCs), and path integrals. We establish close relationships among these space-time approaches in non-relativistic quantum theory, resulting in a unified picture. With the exception of amplitude-weighted correlations in the path integral formalism, in a given experiment, temporal correlations in the different approaches are the same or operationally equivalent.
In ordinary, non-relativistic, quantum physics, time enters only as a parameter and not as an observable: a state of a physical system is specified at a given time and then evolved according to the prescribed dynamics. While the state can, and usually does, extend across all space, it is only defined at one instant of time. Here we ask what would happen if we defined the notion of the quantum density matrix for multiple spatial and temporal measurements. We introduce the concept of a pseudo-density matrix (PDM) which treats space and time indiscriminately. This matrix in general fails to be positive for measurement events which do not occur simultaneously, motivating us to define a measure of causality that discriminates between spatial and temporal correlations. Important properties of this measure, such as monotonicity under local operations, are proved. Two qubit NMR experiments are presented that illustrate how a temporal pseudo-density matrix approaches a genuinely allowed density matrix as the amount of decoherence is increased between two consecutive measurements.
Space-time is one of the most essential, yet most mysterious concepts in physics. In quantum mechanics it is common to understand time as a marker of instances of evolution and define states around all the space but at one time; while in general relativity space-time is taken as a combinator, curved around mass. Here we present a unified approach on both space and time in quantum theory, and build quantum states across spacetime instead of only on spatial slices. We no longer distinguish measurements on the same system at different times with measurements on different systems at one time and construct spacetime states upon these measurement statistics. As a first step towards non-relativistic quantum field theory, we consider how to approach this in the continuous-variable multi-mode regime. We propose six possible definitions for spacetime states in continuous variables, based on four different measurement processes: quadratures, displaced parity operators, position measurements and weak measurements. They are motivated by the pseudo-density matrix formulation among indefinite causal structures and the path integral formalism. We show that these definitions lead to desirable properties, and raise the differences and similarities between spatial and temporal correlations. An experimental proposal for tomography is presented, construing the operational meaning of the spacetime states.
Recent efforts to formulate quantum mechanics in a way that treats space and time on a more equal footing have led to a large variety of spacetime-oriented approaches. In this work we present a detailed study of spacetime states, the objects that play the role of quantum states in the recently introduced framework of spacetime quantum mechanics, and show that the main proposals in the literature are different manifestations of the same underlying object. Path integrals, quantum states over time, pseudo-density matrices, the Page and Wootters mechanism, superdensity operators, and timelike-entanglement proposals all arise from spacetime states through particular evaluations, reduced information, linear maps, or quantum channels. This unification provides explicit mathematical representations of these formalisms, reveals relations among them, and clarifies the spacetime information each one captures. We also study the broader relevance of the spacetime-state point of view for Leggett-Garg inequalities, OTOCs, temporal tensor networks, fermionic systems, relativistic QFTs, quantum reference frames, and classical physics, together with additional insights and perspectives revealed by the common unifying framework.
Space-time is one of the most essential, yet most mysterious concepts in physics. In quantum mechanics it is common to understand time as a marker of instances of evolution and define states around all the space but at one time; while in general relativity space-time is taken as a combinator, curved around mass. Here we present a unified approach on both space and time in quantum theory, and build quantum states across spacetime instead of only on spatial slices. We no longer distinguish measurements on the same system at different times with measurements on different systems at one time and construct spacetime states upon these measurement statistics. As a first step towards non-relativistic quantum field theory, we consider how to approach this in the continuous-variable multi-mode regime. We propose six possible definitions for spacetime states in continuous variables, based on four different measurement processes: quadratures, displaced parity operators, position measurements and weak measurements. The basic idea is to treat different instances of time as different quantum modes. They are motivated by the pseudo-density matrix formulation among indefinite causal structures and the path integral formalism. We show that these definitions lead to desirable properties, and raise the differences and similarities between spatial and temporal correlations. An experimental proposal for tomography is presented, construing the operational meaning of the spacetime states.
While spatial quantum correlations have been studied in great detail, much less is known about the genuine quantum correlations that can be exhibited by temporal processes. Employing the quantum comb formalism, processes in time can be mapped onto quantum states, with the crucial difference that temporal correlations have to satisfy causal ordering, while their spatial counterpart is not constrained in the same way. Here, we exploit this equivalence and use the tools of multipartite entanglement theory to provide a comprehensive picture of the structure of correlations that (causally ordered) temporal quantum processes can display. First, focusing on the case of a process that is probed at two points in time -- which can equivalently be described by a tripartite quantum state -- we provide necessary as well as sufficient conditions for the presence of bipartite entanglement in different splittings. Next, we connect these scenarios to the previously studied concepts of quantum memory, entanglement breaking superchannels, and quantum steering, thus providing both a physical interpretation for entanglement in temporal quantum processes, and a determination of the resources required for its creation. Additionally, we construct explicit examples of W-type and GHZ-type genuinely multipartite entangled two-time processes and prove that genuine multipartite entanglement in temporal processes can be an emergent phenomenon. Finally, we show that genuinely entangled processes across multiple times exist for any number of probing times.
Quantum correlations have a very fundamental significance in the field of quantum
computation and quantum information theory. Though it is very well defined in the
study of spatially separated systems, quantum mechanics allows for the full range of
correlation between temporally separated systems too. In this project, our aim is to
study it's notion in the time domain. Recently Joseph F. Fitzsimon et al. has given a
new approach to the study of quantum correlation in the time domain by introducing
the concept of Pseudo-Density Matrix (PDM) whereas, a very fundamental development
on the temporal correlation has been done by Legget- Garg in 1985( which can't be
accounted by any macro-realist theory). Here, we set out to understand the PDM
notion for different cases and the significance of it to hold the correlations in time
domain using a quantifying tool. This study is extended to find out the correspondences
or incompatibility of it with Legget-Garg inequality and Concurrence - a measure of
quantum entanglement.