Abstract
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.
Highlights
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 evolved according to the prescribed dynamics
Quantum states which violate the Leggett–Garg inequality necessarily exhibit the causal correlations we identify, and recent experimental demonstrations of violations of such inequalities may constitute a limited observation of such causal correlations[14,15,16,17,18]
We note that the general definition of the pseudo-density matrix in terms of expectation values of measurements, given above, does not assume a particular model of underlying dynamics, and a PDM can be reconstructed from experimental results or produced from predictions of other theories without implicitly assuming non-relativistic quantum mechanics
Summary
We note that the general definition of the pseudo-density matrix in terms of expectation values of measurements, given above, does not assume a particular model of underlying dynamics, and a PDM can be reconstructed from experimental results or produced from predictions of other theories without implicitly assuming non-relativistic quantum mechanics. As there is no notion of separate systems inherent in the definition of R, it allows us to describe correlations between measurement events which are not spacelike separated, for example encapsulating the possibility of multiple measurements made at different points in time on a single system This is a generalization of the notion of a quantum state extended across spacetime, rather than the usual restriction to some fixed time present in non-relativistic quantum mechanics.
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