Observable and computable entanglement in time

This paper introduces a novel family of entanglement measures specifically designed for subsystems separated in time. These measures are applicable to both continuous and discrete quantum systems.

Significance:

The primary significance lies in extending the concept of entanglement beyond spatial separation to encompass temporal separation. This opens a new avenue for investigating the dynamics of quantum systems, enabling the probing of past events and their influence on the present state.

Key Innovations:

1. Timelike Density Matrix: The central innovation is a generalization of the density matrix, termed the “timelike density matrix” (TAB), tailored to characterize timelike separated subsystems. The authors prove properties analogous to those of conventional density matrices.
2. Entanglement Imagitivity: The paper introduces a new measure called “entanglement imagitivity,” denoted as ||TAB - TAB||p, which is non-zero if and only if one subsystem can influence the other at a later time.
3. Bounds on Correlation Functions: Using the timelike density matrix, the paper derives upper and lower bounds on dynamical (Wightman) correlation functions, analogous to bounds on spatially separated correlators in terms of mutual information.
4. Experimental Protocols: The authors provide experimentally feasible protocols for measuring entanglement in time, including a demonstration on the IBM quantum device ibm_sherbrooke for a simple qubit system.
5. Connection to RQFT: The framework is motivated by, and consistent with, relativistic quantum field theory (RQFT), where the timelike entanglement measures can be obtained via analytic continuation from spacelike separated regions. The paper relates this to the recently introduced timelike pseudoentropy.

Prior Ingredients Needed:

1. Entanglement Theory: An understanding of entanglement as a property of quantum states, with applications in quantum information theory and condensed matter physics.
2. Density Matrices: Familiarity with the concept of density matrices and their role in describing mixed quantum states.
3. Quantum Field Theory (QFT): Exposure to the basics of QFT, particularly the concepts of correlation functions, operator ordering, and the relation between spacelike and timelike separated regions.
4. Von Neumann and Rényi Entropies: Knowledge of how to calculate entanglement entropies like the von Neumann and Rényi entropies
5. Conformal Field Theory (CFT): Familiarity with conformal field theory in 1+1 dimensions, particularly the use of twist operators to calculate entanglement entropies.
6. Holography/AdS/CFT: Familiarity with AdS/CFT correspondence and how the entanglement entropy can be calculated in terms of extremal surfaces in dual gravitational theories.
7. Tensor Networks: Understanding of matrix product states and operator for describing low-lying quantum states of gapped local Hamiltonians.