Abstract
The standard formalism of quantum theory treats space and time in fundamentally different ways. In particular, a composite system at a given time is represented by a joint state, but the formalism does not prescribe a joint state for a composite of systems at different times. If there were a way of defining such a joint state, this would potentially permit a more even-handed treatment of space and time, and would strengthen the existing analogy between quantum states and classical probability distributions. Under the assumption that the joint state over time is an operator on the tensor product of single-time Hilbert spaces, we analyse various proposals for such a joint state, including one due to Leifer and Spekkens, one due to Fitzsimons, Jones and Vedral, and another based on discrete Wigner functions. Finding various problems with each, we identify five criteria for a quantum joint state over time to satisfy if it is to play a role similar to the standard joint state for a composite system: that it is a Hermitian operator on the tensor product of the single-time Hilbert spaces; that it represents probabilistic mixing appropriately; that it has the appropriate classical limit; that it has the appropriate single-time marginals; that composing over multiple time steps is associative. We show that no construction satisfies all these requirements. If Hermiticity is dropped, then there is an essentially unique construction that satisfies the remaining four criteria.
Highlights
Quantum theory, as usually formalized, contains a fundamental asymmetry between space and time
If a composite system consists of several components existing at a given time, the formalism specifies that the joint state is given by a density matrix acting on the tensor product of the Hilbert spaces associated with the components. (This is the case regardless of whether the components are spatially separated systems or different degrees of freedom of a single system.) But in principle, there is another sort of composite system besides these, namely one where the components are time-like separated
Given that a joint probability distribution is the appropriate way to describe an agent’s incomplete information of a composite system, regardless of the spatio-temporal relations that hold among its components, it is natural to ask whether there is a notion of a joint quantum state that is applicable to an arbitrary composite and which coincides with the standard notion when the composite consists of a set of systems considered at a given time
Summary
As usually formalized, contains a fundamental asymmetry between space and time. Part of the reason for this is that each proposal allows for the possibility that an agent situated at that region of space–time can intervene upon the system, with the intervention corresponding to a quantum instrument: the quantum instrument is a set of trace non-increasing completely positive maps, one for each classical outcome of the intervention, that mediate the incoming and outgoing Hilbert spaces In such approaches, the main criterion of success is whether the formalism can be used to compute the joint probability distribution over the outcome variables for a given set of interventions. We show that if Hermiticity is retained, but an assumption of associativity is dropped, the FJV construction satisfies the remaining criteria
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