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Multipartite quantum states over time from two fundamental assumptions

Seok Hyung Lie, James Fullwood

Abstract

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.

Multipartite quantum states over time from two fundamental assumptions

Abstract

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.

Paper Structure

This paper contains 7 sections, 7 theorems, 51 equations, 1 figure.

Table of Contents

  1. Holistic spatiotemporal products
  2. Equivalence between QSOT and quasiprobability distributions
  3. Non-Markovianity of quasiprobability distribution
  4. Derivation of Eq. \ref{['eqn:snapshot']}
  5. Some preliminary results
  6. Broadcasting, conditionability and decomposability
  7. Proof of Main Theorem

Key Result

Theorem 1

A $\star$-product which is state-linear and conditionable is uniquely determined by its restriction to 1-chains. In particular, for every $n$-chain $\bm{\mathcal{E}}=(\mathcal{E}_1,\ldots,\mathcal{E}_n)$ with initial state $\rho$,

Figures (1)

  • Figure 1: (a) The Markovian extension of a QSOT uniquely characterized in Theorem \ref{['thm:iterfrcond']} allows for sampling observables $O_i$ at time $t_i$ through temporally localized interventions, i.e., quantum snapshotting. (b) Contrarily, non-Markovian QSOTs do not have such a simple decomposition and sampling an observable at each time requires global access to multiple time-steps.

Theorems & Definitions (21)

  • Definition 1: Quantum state over time
  • Definition 2: Conditionability
  • Theorem 1: Unique multi-partite extension of QSOTs
  • Definition 3
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 4
  • ...and 11 more