On Dirac-type correlations
James Fullwood, Boyu Yang
TL;DR
This work introduces Dirac measures on the space of separable projectors and shows they correspond to local-density operators that encode correlations across possibly non-spacelike separated quantum systems. It proves a Gleason-like theorem establishing a one-to-one map between Dirac measures and local-density operators, enabling a bilinear correlation functional that generalizes Born-rule representations to spacetime correlations. By surveying canonical examples (Kirkwood-Dirac, Leifer-Spekkens, Margenau-Hill) and deriving a quantum Bayes' rule, the paper unifies temporal and spatial quantum correlations within a single operational framework. The results provide a foundation for background-independent quantum theory and broaden the interpretation of quantum states over time as fundamental objects describing spacetime correlations.
Abstract
Quantum correlations often defy an explanation in terms of fundamental notions of classical physics, such as causality, locality, and realism. While the mathematical theory underpinning quantum correlations between spacelike separated systems has been well-established since the 1930s, the mathematical theory for correlations between non-spacelike separated systems is much less developed. In this work, we develop the theory of what we refer to as "local-density operators", which we view as joint states for possibly non-spacelike separated quantum systems. Local-density operators are unit trace operators whose marginals are genuine density operators, which we show not only subsumes the notion of density operator, but also several extensions of the notion of density operator into the spatiotemporal domain, such as pseudo-density operators and quantum states over time. More importantly, we prove a result which establishes a one-to-one correspondence between local-density operators and what we refer to as "Dirac measures", which are complex-valued measures on the space of separable projectors associated with two quantum systems. In the case that one of the systems is the trivial quantum system with a one-dimensional Hilbert space, our result recovers the fundamental result known as Gleason's Theorem, which implies that the Born rule from quantum theory is the only way in which one may assign probabilities to the outcomes of measurements performed on quantum systems in a non-contextual manner. As such, our results establish a direct generalization of Gleason's Theorem to measurements performed on possibly non-spacelike separated systems, thus extending the mathematical theory of quantum correlations across space to quantum correlations across space and time.