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Conditional mutual information: A generalization of causal inference in quantum systems

Anupam Ghosh

TL;DR

This work extends causal inference into the quantum domain by defining an asymmetric quantum conditional mutual information (QCMI) as a directional causal index that incorporates quantum interventions. It shows that, unlike the symmetric quantum mutual information, the asymmetric QCMI can reveal causal directionality from an intervention site to a distant target conditioned on intermediate spins, and it recovers the classical CCMI in the appropriate diagonal limit. The authors apply this framework to spin-chain dynamics, revealing finite-speed propagation of causal influence and establishing an operational arrival time and an effective propagation velocity that respect the Lieb–Robinson bound. They illustrate the approach with GHZ-like states and explicit Ising and XX-chain models, connecting the measured propagation speeds to both group velocities and LR bounds. Overall, the paper provides a rigorous foundation for quantum causality and demonstrates its applicability to quantum many-body systems, with potential implications for quantum simulation and communication.

Abstract

The concept of causality is fundamental to numerous scientific explanations; however, its extension to the quantum regime has yet to be rigorously explored. This letter introduces the development of a quantum causal index, a novel extension of the classical causal inference framework, tailored to learn the causal relationships inherent in quantum systems. Our study focuses on the asymmetric quantum conditional mutual information (QCMI), incorporating the von Neumann entropy, as a directional metric of causal influence in quantum many-body systems. We analyze spin chains using the QCMI, implementing a projective measurement on one site as the intervention and monitoring its effect on a distant site conditioned on intermediate spins. Additionally, we study the effective causal propagation velocity, which is the speed at which QCMI becomes significant at distant sites. These findings indicate the presence of finite-speed propagation of causal influence, along with the emergence of coherent oscillations.

Conditional mutual information: A generalization of causal inference in quantum systems

TL;DR

This work extends causal inference into the quantum domain by defining an asymmetric quantum conditional mutual information (QCMI) as a directional causal index that incorporates quantum interventions. It shows that, unlike the symmetric quantum mutual information, the asymmetric QCMI can reveal causal directionality from an intervention site to a distant target conditioned on intermediate spins, and it recovers the classical CCMI in the appropriate diagonal limit. The authors apply this framework to spin-chain dynamics, revealing finite-speed propagation of causal influence and establishing an operational arrival time and an effective propagation velocity that respect the Lieb–Robinson bound. They illustrate the approach with GHZ-like states and explicit Ising and XX-chain models, connecting the measured propagation speeds to both group velocities and LR bounds. Overall, the paper provides a rigorous foundation for quantum causality and demonstrates its applicability to quantum many-body systems, with potential implications for quantum simulation and communication.

Abstract

The concept of causality is fundamental to numerous scientific explanations; however, its extension to the quantum regime has yet to be rigorously explored. This letter introduces the development of a quantum causal index, a novel extension of the classical causal inference framework, tailored to learn the causal relationships inherent in quantum systems. Our study focuses on the asymmetric quantum conditional mutual information (QCMI), incorporating the von Neumann entropy, as a directional metric of causal influence in quantum many-body systems. We analyze spin chains using the QCMI, implementing a projective measurement on one site as the intervention and monitoring its effect on a distant site conditioned on intermediate spins. Additionally, we study the effective causal propagation velocity, which is the speed at which QCMI becomes significant at distant sites. These findings indicate the presence of finite-speed propagation of causal influence, along with the emergence of coherent oscillations.

Paper Structure

This paper contains 8 sections, 52 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Quantum conditional mutual information (QCMI)
  3. Advantage of using asymmetric QCMI for Causal Analysis
  4. Application of Asymmetric QCMI for Causal Analysis in Spin Chains
  5. Conclusion
  6. Classical Conditional Mutual Information (CCMI)
  7. Classical reduction of the QCMI
  8. Lieb--Robinson velocity for the XX chain

Figures (3)

  • Figure 1: The asymmetric QCMI $I(A;B|C)$ has been calculated for the $3$-qubit spin chain, and it is plotted as a function of time ($t$). The corresponding parameters are: $N=3$ and $J = h = 1.0$ in Eq. \ref{['eq:ham_three']}.
  • Figure 2: The asymmetric QCMI $I(A;B|C)$ has been plotted as a function of time ($t$) for the four-site XX spin chain. The corresponding parameters are: $N=4$, $J = 1.0$, and $h = 0.5$ in Eq. \ref{['eq:ham_xx']}.
  • Figure 3: The arrival time $t_\mathrm{arr}(d)$ has been plotted as a function of distance $d$. The effective causal propagation velocity $v_{\mathrm{eff}}$ can be calculated from the slope of this plot using Eq. \ref{['eq:slope']}. The associated parameters in Eq. \ref{['eq:ham_xx']} are: $N=8$, $J = 1.0$, and $h = 0.3$.