Observable and computable entanglement in time

TL;DR

This work introduces a spacetime density matrix $T_{AB}$ to formalize entanglement in time for timelike-separated subsystems, generalizing standard entanglement measures to dynamical settings. It defines two timelike information measures, $\mathrm{Tr}\,T_{AB}^n$ and $||T_{AB}-T_{AB}^\dagger||_p$, derives bounds on time-ordered correlators, and establishes a Lieb-Robinson-type bound for the imaginary part of $T_{AB}$ that signals causal influence. The authors validate the framework through analytic results in relativistic QFT and 1+1D CFT, holography, and numerical studies in Ising chains and free fermions, complemented by experimental measurement protocols implemented on IBM quantum hardware. The work also connects timelike entanglement to timelike pseudoentropy in certain regimes and opens pathways for probing time correlations in quantum many-body dynamics, with potential implications for quantum information flow and holographic duality.

Abstract

We propose a novel family of entanglement measures for time-separated subsystems. Our definitions are applicable to any quantum system, continuous or discrete. To illustrate their utility, we derive upper and lower bounds on time-separated correlation functions, akin to the bound on spatially separated correlators in terms of the mutual information. In certain cases our bounds are tight. For relativistic quantum field theories our definition agrees with the analytic continuation from spacelike to timelike separated regions. We provide relevant measurement protocols and execute them on the IBM quantum device ibm_sherbrooke for a simple qubit system. Also we perform explicit computations for an Ising spin chain, free fermions, (1+1)-dimensional conformal field theories and holographic theories. Finally we explain how the proposed entanglement in time provides a microscopic definition for the recently introduced timelike pseudoentropy.

Figures (13)

• Figure 1: (a) Here $A,B$ are spacelike separated. There are many existing tools in RQFT to compute $\mathop{\mathrm{Tr}}\limits \rho_{AB}^n$. (b) Timelike separated $A,B$. In RQFT this configuration can be continuously deformed to (a).
• Figure 2: Numerical demonstration of inequalities (\ref{['eq:th1']}), (\ref{['lg7']}), (\ref{['b1']}) for the Hamiltonian eq. \ref{['eq:ising-ham']} and Pauli Y operators acting on the first and sixth qubits at different times. The plot was obtained for a quantum Ising model with 11 spins and parameters $J = 1, h = -1.05, B_z = 0.5$, and a thermal initial state $\rho_0$ with the temperature $100 J$.
• Figure 3: Real part of the mutual information $I_{vN} = S_{vN}(AB) - S_{vN}(A) - S_{vN}(B)$ for free fermions. Here $S_{vN}(A), S_{vN}(B)$ are the conventional von Neumann entropies for intervals $A,B$: $(c/3) \log L/\epsilon$. Dots are the numerical lattice result obtained from taking the principal branch of $\log T$, and the solid line is the CFT prediction $(1/3) \operatorname{Re} \log(1-L^2/t^2)$ obtained from eq.(\ref{['eq:two_int_renyi']}). We used two intervals $[0,L]$ of length $L=40$ but separated in time by $t$.
• Figure 4: The real and imaginary parts of $\mathop{\mathrm{Tr}}\limits T^2$ for free fermions. Dots are the numerical lattice result, solid lines are CFT predictions; see eq.(\ref{['eq:cft_tt']}). Subsystems $A$ and $B$ both have length $L=50$, but $A$ is the interval $[0,50]$ and subsystem $B$ is the interval $[70,120]$ which is moved forward in time by $t$.
• Figure 5: Two timelike separated twist operators.

Theorems & Definitions (9)

• Theorem 1
• Conjecture 1
• Theorem 2
• Conjecture 2
• Theorem 3
• Theorem 4
• Corollary 4.1
• Corollary 4.2
• Theorem 5