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

Naim Elias Comar, Lucas C. Céleri, Mia Stamatova, Vlatko Vedral, Aditya Varna Iyer, Rafael Chaves

Abstract

We show that, given an evolving quantum system and the quasiprobability distribution generated by the spatiotemporal generalization of the Born rule in pseudo density-matrices (PDMs), this distribution deviates from the sequential measurements probability distribution, given by the Lüders von-Neumann distribution, if and only if the non-signaling in time (NSIT) is violated; equivalently, if and only if the macroscopic realism (MR) is violated. Furthermore, we propose a definition of temporal entanglement according to the structure of the PDMs that is analogous to the definition of spatial entanglement in density matrices, showing that temporal entanglement is necessary for the violation of temporal Bell inequalities and the violation of MR. We employ our results to study the relationship between the negativity of the PDM, temporal entanglement, violation of temporal Bell inequalities, and MR.

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

Abstract

We show that, given an evolving quantum system and the quasiprobability distribution generated by the spatiotemporal generalization of the Born rule in pseudo density-matrices (PDMs), this distribution deviates from the sequential measurements probability distribution, given by the Lüders von-Neumann distribution, if and only if the non-signaling in time (NSIT) is violated; equivalently, if and only if the macroscopic realism (MR) is violated. Furthermore, we propose a definition of temporal entanglement according to the structure of the PDMs that is analogous to the definition of spatial entanglement in density matrices, showing that temporal entanglement is necessary for the violation of temporal Bell inequalities and the violation of MR. We employ our results to study the relationship between the negativity of the PDM, temporal entanglement, violation of temporal Bell inequalities, and MR.
Paper Structure (23 sections, 12 theorems, 83 equations, 5 figures)

This paper contains 23 sections, 12 theorems, 83 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. The pseudo-density matrix (PDM)
  4. Temporal entanglement
  5. Macroscopic realism, Leggett-Garg inequalities, and no-signaling in time
  6. Temporal Bell nolocality
  7. Margenau-Hill quasiprobability distribution generated by the PDM
  8. Main results
  9. Two-time steps
  10. Three-time steps
  11. Different types of temporal non-classicality
  12. Discussion for two-time steps
  13. Discussion for three-time steps
  14. Discussion and Conclusions
  15. The Choi-Jamiolkovski matrix for a CPTP map $\mathcal{E}$
  16. ...and 8 more sections

Key Result

Theorem 1

where the subindex $012$ explicitly refers to the three-time steps scenario, $(t_0,t_1,t_2)$. Note that it is also valid for the two time steps scenario $(t_0, t_1)$.

Figures (5)

  • Figure 1: Two-time steps sequential measurements representing the Lüdders von-Neumann projection postulate (Eq. \ref{['Ludders_vN']}).
  • Figure 2: Three-time steps sequential measurements representing the Lüdders von-Neumann projection postulate (Eq. \ref{['Ludders_nV_3points']}).
  • Figure 3: Relation between different notions of temporal correlations for two-time steps.
  • Figure 4: LGI correlations term $\mathcal{K}_3$, NSIT violation quantifier $\mathcal{N}_{012},$ LGI bound ($\mathcal{K}_3=1$, with the orange shaded area representing the region of LGI violation), and the NSIT condition ($\mathcal{N}_{012 }=0$), in function of $\Omega t$.
  • Figure 5: $\mathcal{B}_{\text{max}}$, temporal CHSH bound (the gray shaded area represents the region were temporal CHSH is violated), negativity of the PDM $R_{01}$, and the absolute violation of NSIT $\mathcal{N}_{01}$, for the case of initial pure state $\ket{0} \bra{0}$ and depolarizing channel, with measurements chosen with angles $\theta_1 = \pi/2$ and $\theta_2 = \pi$.

Theorems & Definitions (21)

  • Theorem 1: Adapted from Ref. Clemente_2015
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Lemma 2
  • Lemma 3
  • ...and 11 more