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Space-time generalization of mutual information

Paolo Glorioso, Xiao-Liang Qi, Zhenbin Yang

TL;DR

The paper introduces the space-time mutual information (STMI), a quantum-information–theoretic generalization of mutual information that applies to space-time regions A and B separated in time. Defined via a quantum-hypothesis-testing task and ancilla-assisted couplings within a QUALM framework, STMI provides an operational upper bound on all dynamical correlations between A and B and reduces to ordinary MI for spacelike separation. It establishes a suite of properties, including a decomposition into a W-connection term and a B-occupation term, monotonicity under local channels, and bounds on two-point functions; it also connects STMI to quantum channel discrimination and explores additivity in the pure-state initial case. The authors analyze STMI in simple and many-body contexts—single-qubit channels (depolarizing and dephasing), many-body localization versus thermalization—and provide a classical analogue, highlighting STMI as a versatile diagnostic for temporal correlations and information propagation in quantum dynamics. They also propose an ansatz to simplify optimization for factorized initial states and discuss open questions, such as additivity in general settings and the role of entanglement in the initial state.

Abstract

The mutual information characterizes correlations between spatially separated regions of a system. Yet, in experiments we often measure dynamical correlations, which involve probing operators that are also separated in time. Here, we introduce a space-time generalization of mutual information which, by construction, satisfies several natural properties of the mutual information and at the same time characterizes correlations across subsystems that are separated in time. In particular, this quantity, that we call the \emph{space-time mutual information}, bounds all dynamical correlations. We construct this quantity based on the idea of the quantum hypothesis testing. As a by-product, our definition provides a transparent interpretation in terms of an experimentally accessible setup. We draw connections with other notions in quantum information theory, such as quantum channel discrimination. Finally, we study the behavior of the space-time mutual information in several settings and contrast its long-time behavior in many-body localizing and thermalizing systems.

Space-time generalization of mutual information

TL;DR

The paper introduces the space-time mutual information (STMI), a quantum-information–theoretic generalization of mutual information that applies to space-time regions A and B separated in time. Defined via a quantum-hypothesis-testing task and ancilla-assisted couplings within a QUALM framework, STMI provides an operational upper bound on all dynamical correlations between A and B and reduces to ordinary MI for spacelike separation. It establishes a suite of properties, including a decomposition into a W-connection term and a B-occupation term, monotonicity under local channels, and bounds on two-point functions; it also connects STMI to quantum channel discrimination and explores additivity in the pure-state initial case. The authors analyze STMI in simple and many-body contexts—single-qubit channels (depolarizing and dephasing), many-body localization versus thermalization—and provide a classical analogue, highlighting STMI as a versatile diagnostic for temporal correlations and information propagation in quantum dynamics. They also propose an ansatz to simplify optimization for factorized initial states and discuss open questions, such as additivity in general settings and the role of entanglement in the initial state.

Abstract

The mutual information characterizes correlations between spatially separated regions of a system. Yet, in experiments we often measure dynamical correlations, which involve probing operators that are also separated in time. Here, we introduce a space-time generalization of mutual information which, by construction, satisfies several natural properties of the mutual information and at the same time characterizes correlations across subsystems that are separated in time. In particular, this quantity, that we call the \emph{space-time mutual information}, bounds all dynamical correlations. We construct this quantity based on the idea of the quantum hypothesis testing. As a by-product, our definition provides a transparent interpretation in terms of an experimentally accessible setup. We draw connections with other notions in quantum information theory, such as quantum channel discrimination. Finally, we study the behavior of the space-time mutual information in several settings and contrast its long-time behavior in many-body localizing and thermalizing systems.
Paper Structure (19 sections, 1 theorem, 98 equations, 12 figures)

This paper contains 19 sections, 1 theorem, 98 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Definition of space-time mutual information
  3. General intuition
  4. Definition
  5. Properties of Space-time Mutual Information
  6. Bound on space-time correlation functions
  7. Markov property
  8. Quantum channel discrimination and additivity
  9. Additivity of the STMI for initial pure states
  10. STMI as a constrained quantum channel discrimination
  11. An ansatz for factorized initial states
  12. Examples
  13. Single-qubit system
  14. MBL and Thermalization
  15. Classical space-time mutual information
  16. ...and 4 more sections

Key Result

Theorem 1

The STMI bounds all two-point correlation functions between subsystems $A$ and $B$. Explicitly, for any $N\geq 1$, we have the following bounds on symmetric and retarded correlation functions: where $\mathcal{O}_{B}(t)$ is a (Heisenberg) operator supported in subregion ${B}$, and similarly for $\mathcal{O}_{A}$, $||\cdot||_{\infty}$ denotes the operator norm, $(\cdots)_c$ the connected component

Figures (12)

  • Figure 1: (a) System with initial state $\rho_{\text{in}}$ undergoing evolution with unitary $U$. The upper cap stands for tracing over the corresponding region. (b) Coupling between subsystem $A$ and ancilla $W$ giving rise to the connected state $\rho_{BW}$. (c) Disconnected state $\rho_{B,0}\otimes \rho_{W}$, where $\rho_{B,0}$ is the unperturbed evolved state reduced to subsystem $B$.
  • Figure 2: Definition of $J_{N}(A:B)$ in Eq. (\ref{['eq:def1']}) for the uncorrelated and correlated case.
  • Figure 3: Mutual information of the Choi state corresponding to the unitary $U$.
  • Figure 4: Illustration of the two situations involving three regions $ABC$, for the discussion of Markovian condition in Sec. \ref{['sec:conditional STMI']}.
  • Figure 5: (a) Definition of $\mathcal{N}_N(\rho_{A W})$. (b) By specifying the $\mathcal{A}^{(i)}$ to suitable swaps, $\mathcal{N}_N(\rho_{A W})$ reduces to our (\ref{['eq:def2']}).
  • ...and 7 more figures

Theorems & Definitions (1)

  • Theorem 1