Multipartite quantum states over time from two fundamental assumptions

The theory of quantum states over time extends the density operator formalism into the temporal domain, providing a unified of treatment of timelike and spacelike separated systems in quantum theory. Although recent results have characterized quantum states over time involving two timelike separated systems, it remains unclear how to consistently extend the notion of quantum states over time to multipartite temporal scenarios, such as those considered in studies of Leggett-Garg inequalities. In this Letter, we show that two simple assumptions uniquely single out the Markovian multipartite extension of bipartite quantum states over time, namely, linearity in the initial state and a quantum analog of conditionability for multipartite probability distributions. As a direct consequence of our result, we establish a canonical correspondence between multipartite QSOTs and Kirkwood-Dirac type quasiprobability distributions, which we show opens up the possibility of experimentally verifying the temporal correlations encoded in QSOTs via the recent experimental technique of simulating quasiprobability known as quantum snapshotting.

💡 Research Summary

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The paper addresses a fundamental gap in the theory of quantum states over time (QSOT): while bipartite QSOTs have been well studied, a consistent and physically motivated extension to multipartite temporal scenarios has remained elusive. The authors propose that only two elementary assumptions are needed to uniquely determine such an extension.

1. State‑linearity – the QSOT must be convex‑linear in the initial density operator. In other words, if the preparation of the initial state is uncertain (a statistical mixture), the resulting QSOT must be the same mixture of the QSOTs generated from each component. This reflects the convex structure of quantum state space and the linear response of dynamical quantities (work, heat, etc.) to the initial state.

2. Conditionability – analogous to the factorisation of a classical joint probability distribution into an initial marginal and a conditional distribution, the QSOT must admit a decomposition that isolates the contribution of the initial state. Formally, for any n‑chain of channels ((\mathcal{E}_1,\dots,\mathcal{E}n)) there must exist a linear map (\Theta\rho) such that
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