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The spatiotemporal Born rule is quasiprobabilistic

James Fullwood, Zhihao Ma, Zhen Wu

TL;DR

The paper develops a spatiotemporal Born rule that unifies spatial and temporal quantum correlations within a quasiprobabilistic framework. It introduces a unique operator $\varrho_{AB}$ encoding the TMH quasiprobabilities for sequential measurements, with $Q(i,j)=\operatorname{Tr}[\varrho_{AB}(P_i\otimes Q_j)]$ and a concrete construction via $\varrho_{AB}=\tfrac{1}{2}\{\rho\otimes \mathds{1}, \mathscr{J}[\mathcal{E}]\}$. For qubits, temporal correlations from LvN and TMH converge to $\operatorname{Tr}[\varrho_{AB}(\sigma_A\otimes\sigma_B)]$, linking to the pseudo-density matrix formalism and enabling direct experimental access. The work also develops a quantum Bayes’ rule that yields an operational time-reversal symmetry for open systems and provides necessary and sufficient conditions for when a LvN-based spatiotemporal Born rule can exist, tying the existence to the equality $\boldsymbol{P}(i,j)=\boldsymbol{Q}(i,j)$. Overall, the results offer a covariant, bi-additive description of space-time quantum correlations, with implications for quantum information, thermodynamics, and foundational questions about temporal causality.

Abstract

Contrary to general relativity, quantum theory treats space and time in fundamentally different ways. In particular, while joint probabilities associated with spacelike separated measurements are defined in terms of the Born rule, joint probabilities associated with measurements performed in sequence are defined in terms of the state-update rule. In this work, we show that one obtains a more unified perspective of space and time in quantum theory by embracing a quasiprobabilistic description of sequential measurements. More precisely, we show that there exists a unique \emph{pseudo}-density operator encoding canonical quasiprobabilities associated with sequential measurements in precisely the same manner that a density operator encodes joint probabilities associated with spacelike separated measurements, thus providing a natural extension of the Born rule into the temporal domain. As an application, we show how such a spatiotemporal Born rule combined in conjunction with a quantum Bayes' rule yields an operational notion of time-reversal symmetry for sequential measurements on an \emph{open} quantum system.

The spatiotemporal Born rule is quasiprobabilistic

TL;DR

The paper develops a spatiotemporal Born rule that unifies spatial and temporal quantum correlations within a quasiprobabilistic framework. It introduces a unique operator encoding the TMH quasiprobabilities for sequential measurements, with and a concrete construction via . For qubits, temporal correlations from LvN and TMH converge to , linking to the pseudo-density matrix formalism and enabling direct experimental access. The work also develops a quantum Bayes’ rule that yields an operational time-reversal symmetry for open systems and provides necessary and sufficient conditions for when a LvN-based spatiotemporal Born rule can exist, tying the existence to the equality . Overall, the results offer a covariant, bi-additive description of space-time quantum correlations, with implications for quantum information, thermodynamics, and foundational questions about temporal causality.

Abstract

Contrary to general relativity, quantum theory treats space and time in fundamentally different ways. In particular, while joint probabilities associated with spacelike separated measurements are defined in terms of the Born rule, joint probabilities associated with measurements performed in sequence are defined in terms of the state-update rule. In this work, we show that one obtains a more unified perspective of space and time in quantum theory by embracing a quasiprobabilistic description of sequential measurements. More precisely, we show that there exists a unique \emph{pseudo}-density operator encoding canonical quasiprobabilities associated with sequential measurements in precisely the same manner that a density operator encodes joint probabilities associated with spacelike separated measurements, thus providing a natural extension of the Born rule into the temporal domain. As an application, we show how such a spatiotemporal Born rule combined in conjunction with a quantum Bayes' rule yields an operational notion of time-reversal symmetry for sequential measurements on an \emph{open} quantum system.

Paper Structure

This paper contains 7 sections, 5 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Two-point sequential measurement scenarios
  3. The spatiotemporal Born rule
  4. 2-time correlators for systems of qubits
  5. Time-reversal symmetry and a spatiotemporal Bayes' rule
  6. Existence of a spatiotemporal Born rule for the Lüders-von Neumann distribution
  7. Discussion

Key Result

Proposition 1

Let $(\rho,\{P_i\},\mathcal{E},\{Q_j\})$ be a TPSM scenario, and let $\bold{D}$ be the function given by where $\rho_i=P_i\rho P_i+(\mathds{1}-P_i)\rho(\mathds{1}-P_i)$. Then the following statements hold.

Figures (1)

  • Figure 1: Schematic representation of the Born rules for the distributions $\bold{P}_{\text{space}}$ and $\bold{Q}$. (a) For $\bold{P}_{\text{space}}$, spacelike separated systems $A$ and $B$ which are in a joint spatial state $\rho_{AB}$ are measured in parallel. (b) For $\bold{Q}$, timelike separated systems $A$ and $B$ which are in a joint spatiotemporal state $\varrho_{AB}$ are measured in sequence.

Theorems & Definitions (14)

  • Definition 1
  • Proposition 1
  • proof
  • Example 1
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • ...and 4 more