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Quantum relative entropy for unravelings of master equations

Marcos Ruibal Ortigueira, Robert de Keijzer, Luke Visser, Oliver Tse, Servaas Kokkelmans

TL;DR

This paper defines a quantum relative entropy for unravelings of open quantum dynamics by minimizing the KL divergence over ensembles of pure states that realize two faithful states. It shows that the minimum is achieved by ensembles supported on a common (potentially non-orthogonal) basis and that this unraveling-relative entropy coincides with the Belavkin-Staszewski entropy, providing a physical interpretation of BS as the classical KL of pure-state ensembles. The authors derive a constructive common-basis algorithm, prove equivalence results, and establish a contraction principle for the Lindblad flow together with a large-deviations perspective for empirical pure-state measures. Together, these results deepen the connection between classical and quantum divergences in unravelings and suggest practical implications for quantum dynamics and hypothesis testing. The framework also offers avenues for generalization to broader $f$-divergences and non-faithful states, with potential links to quantum discrimination tasks.

Abstract

This work explores connections between the quantum relative entropy of two faithful states $ρ,σ$ (i.e. full-rank density matrices) and the Kullback-Leibler divergences of classical measures $μ,ν$. Here, $μ$ and $ν$ are measures on the space of pure states, realizing $ρ$ and $σ$ respectively. The motivation for this result is to establish a notion of quantum relative entropy in the space of pure state distributions, which are the resulting objects of unravelings of the Lindblad equation, such as the stochastic Schrödinger equation. Our results show that the measures that achieve the minimal KL divergence are those supported on a (possibly non-orthogonal) common basis between $ρ$ and $σ$. Using the classical and quantum data-processing inequalities, our notion of quantum relative entropy is shown to be equivalent to the Belavkin-Staszewski entropy on states, revealing new insights on this quantity. Furthermore, the common basis is used to provide a novel proof of contraction of the relative entropy under Lindblad flow and offers insights into results from large deviation theory.

Quantum relative entropy for unravelings of master equations

TL;DR

This paper defines a quantum relative entropy for unravelings of open quantum dynamics by minimizing the KL divergence over ensembles of pure states that realize two faithful states. It shows that the minimum is achieved by ensembles supported on a common (potentially non-orthogonal) basis and that this unraveling-relative entropy coincides with the Belavkin-Staszewski entropy, providing a physical interpretation of BS as the classical KL of pure-state ensembles. The authors derive a constructive common-basis algorithm, prove equivalence results, and establish a contraction principle for the Lindblad flow together with a large-deviations perspective for empirical pure-state measures. Together, these results deepen the connection between classical and quantum divergences in unravelings and suggest practical implications for quantum dynamics and hypothesis testing. The framework also offers avenues for generalization to broader -divergences and non-faithful states, with potential links to quantum discrimination tasks.

Abstract

This work explores connections between the quantum relative entropy of two faithful states (i.e. full-rank density matrices) and the Kullback-Leibler divergences of classical measures . Here, and are measures on the space of pure states, realizing and respectively. The motivation for this result is to establish a notion of quantum relative entropy in the space of pure state distributions, which are the resulting objects of unravelings of the Lindblad equation, such as the stochastic Schrödinger equation. Our results show that the measures that achieve the minimal KL divergence are those supported on a (possibly non-orthogonal) common basis between and . Using the classical and quantum data-processing inequalities, our notion of quantum relative entropy is shown to be equivalent to the Belavkin-Staszewski entropy on states, revealing new insights on this quantity. Furthermore, the common basis is used to provide a novel proof of contraction of the relative entropy under Lindblad flow and offers insights into results from large deviation theory.

Paper Structure

This paper contains 15 sections, 19 theorems, 93 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Classical probability measures
  4. Quantum relative entropies
  5. Unraveling Relative Entropy
  6. Common basis construction
  7. Equivalence of $D_\mathtt{BS}$ and $D_\mathtt{UNR}$
  8. Minimization of the KL divergence
  9. Consequences of Theorem \ref{['thm:maintheorem']}
  10. Contraction along the Lindblad flow
  11. Large deviations perspective
  12. Conclusion
  13. Proofs of Lemmas supporting Theorem \ref{['thm:maintheorem']}
  14. $f$-divergences
  15. Calculation KL divergence derivative under SSE evolution

Key Result

Proposition 1

Let $\mathfrak{h}$ be finite-dimensional Hilbert space with inner product $\braket{\cdot|\cdot}$. Then the set of pure states $\mathscr{P}(\mathfrak{h})$ is compact under the Fubini-Study metric

Figures (3)

  • Figure 1: Classical to quantum relative entropies. The unraveling entropy (UNR) proposed in this work is equivalent to the BS entropy, which is the limit of the geometric Rényi entropy.
  • Figure 2: Approximate probability density functions of $D_\mathtt{BS}(\rho\|\sigma), D_\mathtt{KL}(\mu_{\mathtt{CB}}\| \nu_{\mathtt{CB}})$, and $D_\mathtt{UNR}(\rho\|\sigma)$ constructed from $10^5$ random pair of states $(\rho,\sigma)$ sampled according to the Haar measure.
  • Figure 3: Common basis construction on 1-qubit state. Two ensembles of pure states $\{\psi_\omega\}_\omega$ (circles) encode for states $\rho$ (stars). The unique common basis $\{\psi_{\mathtt{CB},i}\}_{i=1}^n$ of pure states (red crosses) is constructed by finding the intersection of the Bloch sphere and the line connecting both $\rho$'s (red dashed). Note that the states $\{\psi_{\mathtt{CB},i}\}_{i=1}^n$ are not orthogonal.

Theorems & Definitions (42)

  • Proposition 1: Compactness of $\mathscr{P}(\mathfrak{h})$
  • Definition 2: KL divergence
  • Definition 3: Markov kernel
  • Definition 4: Composition measure
  • Proposition 5: DPI for KL divergence
  • Proposition 6: Sanov's theorem
  • Proposition 7: Contraction Principle
  • Definition 8: Realization of states
  • Definition 9: Common basis
  • Definition 10: Quantum relative entropy
  • ...and 32 more