Can a quantum state over time resemble a quantum state at a single time?

TL;DR

This paper investigates whether a joint quantum state over multiple times can be consistently defined, revealing that no such state can satisfy all desired properties, thus requiring some axioms to be relaxed.

Contribution

It demonstrates the impossibility of constructing a joint quantum state over time that meets all five proposed conditions, highlighting the need to relax certain axioms.

Findings

No construction satisfies all five conditions simultaneously.

Dropping Hermiticity leads to a unique construction up to ordering.

The results challenge the notion of a symmetric joint state over multiple times.

Abstract

Standard quantum theory represents a composite system at a given time by a joint state, but it does not prescribe a joint state for a composite of systems at different times. If a more even-handed treatment of space and time is possible, then such a joint state should be definable, and one might expect it to satisfy the following five conditions: 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 an even-handed treatment of space and time is possible, therefore, one or more axioms must be dropped. In particular, if Hermiticity is dropped, then we show that the construction is fixed uniquely up to an ordering convention.