Using spatiotemporal Born rule for testing macroscopic realism: some applications to the pseudo-density matrices and nonclassical temporal correlations

1. Problem Statement

The paper addresses the challenge of certifying nonclassical temporal correlations and testing Macroscopic Realism (MR) within the framework of spatiotemporal quantum states. While spatial quantum correlations (entanglement, Bell nonlocality) are well-understood via density matrices, temporal correlations are complicated by the disturbance caused by sequential measurements (signaling in time).

Key issues include:

• Distinction between MR and Temporal Bell Nonlocality: It is often unclear how violations of Leggett-Garg Inequalities (LGIs) or Temporal Bell Inequalities relate to the fundamental violation of MR. LGIs are necessary but not sufficient conditions for MR violation.
• Lack of a Unified Framework: There is a need for a formalism that treats space and time symmetrically to define temporal entanglement and nonclassicality without relying solely on indirect witnesses.
• Measurement Dependence: Most tests for MR depend heavily on the specific choice of sequential measurements, making it difficult to define a measurement-independent notion of classicality.

2. Methodology

The authors utilize the Pseudo-Density Matrix (PDM) formalism, which extends the concept of density matrices to multiple time steps, treating time steps analogously to spatial subsystems.

• Spatiotemporal Born Rule: The authors employ a generalized Born rule applied to PDMs. For two time steps, this generates the Margenau-Hill (MH) quasiprobability distribution.
• Decomposition of Distributions: They decompose the MH quasiprobability ($Q$) into two parts:
  1. $P$: The standard sequential measurement probability given by the Lüders-von Neumann projection postulate.
  2. $D$: A disturbance term representing the deviation caused by the measurement back-action on the system.
• No-Signaling in Time (NSIT): The paper establishes that the violation of NSIT (the condition that a measurement at time $t_i$ does not affect the statistics at $t_j$) is equivalent to the non-vanishing of the disturbance term $D$.
• Temporal Entanglement Definition: A new definition of temporal entanglement is proposed based on the separability of the PDM. A PDM is "time-separable" if it can be written as a convex sum of product states across time steps (allowing for negative coefficients in the sum, unlike spatial separability).
• Negativity as a Witness: The authors use the negativity of the PDM (defined as $f(R) = |R| - 1$, where $|R| = \sqrt{R^\dagger R}$) as a quantifier for nonclassicality.

3. Key Contributions

A. Equivalence between Disturbance, NSIT, and MR

The central theoretical result is the proof that for a given set of sequential projective measurements:
$$D = 0 \iff \text{NSIT is satisfied} \iff \text{Macroscopic Realism (MR) is satisfied}$$

• Two-Time Steps: The disturbance term $D_{01}$ vanishes if and only if the NSIT condition holds. Since NSIT (combined with the Arrow of Time) is necessary and sufficient for MR, the vanishing of $D$ certifies MR.
• Three-Time Steps: The authors generalize the spatiotemporal Born rule to three time steps. They show that the total disturbance term $D_{012}$ is the sum of four sub-terms. The vanishing of the total disturbance is equivalent to the simultaneous satisfaction of all NSIT conditions for the three-time scenario.

B. Definition and Hierarchy of Temporal Entanglement

• Definition: A spatiotemporal state is temporally entangled if its PDM cannot be decomposed into a time-separable form (Eq. 9).
• Theorem: If a PDM has negativity ($f(R) > 0$), it implies the presence of temporal entanglement.
• Hierarchy: The paper establishes a strict hierarchy for two-time steps:
  1. PDM Negativity $\implies$ Temporal Entanglement.
  2. Temporal Entanglement is necessary (but not sufficient) for the violation of Temporal Bell Inequalities (CHSH).
  3. PDM Negativity is necessary (but not sufficient) for the violation of MR.
  • Crucial Insight: There exist states with temporal entanglement (negative PDM) that satisfy MR (no disturbance) but violate Temporal Bell inequalities. Conversely, there are cases where MR is violated, but Temporal Bell inequalities are not.

C. Measurement-Independent Macroscopic Realism

The authors introduce the concept of measurement-independent MR. If the PDM is positive semi-definite ($f(R) = 0$), then MR is satisfied for any choice of sequential projective measurements. This provides a robust, state-based certification of classicality that does not depend on the experimentalist's choice of observables.

4. Key Results and Examples

• Maximally Mixed State with Trivial Evolution:

  • A qubit with initial state $\rho = I/2$ and trivial evolution ($E=I$) yields a PDM with negative eigenvalues (temporal entanglement).
  • Result: This state maximally violates the Temporal CHSH inequality but satisfies MR (NSIT holds, $D=0$) for all measurements.
  • Implication: Violation of Temporal Bell inequalities does not imply violation of MR. The two concepts probe different facets of nonclassicality.

• Depolarizing Channel:

  • The authors analyze a qubit evolving under a depolarizing channel. They identify a parameter regime where the PDM is negative (temporal entanglement exists), but the Temporal CHSH inequality is not violated, yet MR is still satisfied.
  • This demonstrates that temporal entanglement is a broader concept than temporal Bell nonlocality.

• Thermodynamic Witness:

  • The paper connects PDM negativity to imaginary heat capacity in non-equilibrium thermodynamics. A non-zero imaginary part of the heat capacity is shown to be equivalent to PDM negativity, providing an experimentally accessible thermodynamic witness for temporal entanglement.

• Three-Time Steps vs. LGIs:

  • Using the NSIT violation quantifier ($N_{012}$), the authors show that the region where MR is violated is larger than the region where standard Leggett-Garg Inequalities (LGIs) are violated. This confirms that LGIs are insufficient to fully capture MR violations, whereas the spatiotemporal Born rule approach (via NSIT) is necessary and sufficient.

5. Significance

• Operational Certification of MR: The paper provides a necessary and sufficient condition for testing Macroscopic Realism using the spatiotemporal Born rule, surpassing the limitations of LGIs.
• Clarification of Temporal Nonclassicality: It rigorously disentangles the concepts of Temporal Entanglement, PDM Negativity, Temporal Bell Nonlocality, and Macroscopic Realism, showing they form a strict hierarchy rather than being equivalent.
• Symmetry between Space and Time: By defining temporal entanglement via PDM structure (analogous to spatial entanglement via density matrix separability), the work strengthens the conceptual bridge between spatial and temporal quantum correlations.
• Practical Applications: The connection to thermodynamic quantities (heat capacity) and quantum clocks suggests practical applications in quantum metrology and the study of the quantum-to-classical transition in open systems.

In summary, the paper establishes that PDM negativity is a fundamental signature of temporal entanglement, which is a prerequisite for nonclassical temporal correlations, but distinct from the violation of MR. The disturbance term in the spatiotemporal Born rule serves as the precise operational tool to certify whether a system adheres to Macroscopic Realism.