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Local Invariance of Divergence-based Quantum Information Measures

Christopher Popp, Tobias C. Sutter, Beatrix C. Hiesmayr

TL;DR

This work develops a unified approach to quantum information measures built from generalized divergences, applicable to both asymptotic and one-shot settings, and proves their invariance under local transformations. By introducing a reversal channel for local isometries and leveraging the data-processing inequality, the authors establish precise invariance results for several divergence-based quantities, distinguishing which types remain invariant under all local isometries and which require a unitary on one subsystem. The framework defines multiple information quantity types using a smoothing environment $B^{\varepsilon}(\rho)$ and the sine-distance based smoothing $B_P^{\varepsilon}(\rho)$, and shows that these invariances hold without committing to a particular divergence form. These invariances can simplify computation and optimization of quantum information protocols, enabling state reductions and protocol design that preserve key invariant measures.

Abstract

Quantum information quantities, such as mutual information and entropies, are essential for characterizing quantum systems and protocols in quantum information science. In this contribution, we identify types of information measures based on generalized divergences and prove their invariance under local isometric or unitary transformations. Leveraging the reversal channel for local isometries together with the data processing inequality, we establish invariance for information quantities used in both asymptotic and one-shot regimes without relying on the specific functional form of the underlying divergence. These invariances can be applied to improve the computation of such information quantities or optimize protocols and their output states whose performance is determined by some invariant measure. Our results improve the capability to characterize and compute many operationally relevant information measures with application across the field of quantum information processing.

Local Invariance of Divergence-based Quantum Information Measures

TL;DR

This work develops a unified approach to quantum information measures built from generalized divergences, applicable to both asymptotic and one-shot settings, and proves their invariance under local transformations. By introducing a reversal channel for local isometries and leveraging the data-processing inequality, the authors establish precise invariance results for several divergence-based quantities, distinguishing which types remain invariant under all local isometries and which require a unitary on one subsystem. The framework defines multiple information quantity types using a smoothing environment and the sine-distance based smoothing , and shows that these invariances hold without committing to a particular divergence form. These invariances can simplify computation and optimization of quantum information protocols, enabling state reductions and protocol design that preserve key invariant measures.

Abstract

Quantum information quantities, such as mutual information and entropies, are essential for characterizing quantum systems and protocols in quantum information science. In this contribution, we identify types of information measures based on generalized divergences and prove their invariance under local isometric or unitary transformations. Leveraging the reversal channel for local isometries together with the data processing inequality, we establish invariance for information quantities used in both asymptotic and one-shot regimes without relying on the specific functional form of the underlying divergence. These invariances can be applied to improve the computation of such information quantities or optimize protocols and their output states whose performance is determined by some invariant measure. Our results improve the capability to characterize and compute many operationally relevant information measures with application across the field of quantum information processing.

Paper Structure

This paper contains 5 sections, 6 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Local invariance of information quantities
  3. Notation and setting
  4. Generalized divergence-based types of information quantities
  5. Conclusion

Key Result

Proposition 1

Any information quantity of type $\mathbf{I}_1, \mathbf{I}_2$ and $\mathbf{I}^{\varepsilon}_3$ as in Definition def:typesMutInf is invariant under local isometric transformations of the form $\mathcal{V}_{AB}(\cdot) = V_A \otimes V_B ~(\cdot)~V_A^\dagger \otimes V_B^\dagger$.

Figures (1)

  • Figure 1: Schematic visualization of the smoothing environment $B^\varepsilon$ and its image $B_{\mathcal{V}}^\varepsilon$ and pre-image $B_{\mathcal{R_V}}^\varepsilon$ under the isometric channel $\mathcal{V}$ and its reversal channel $\mathcal{R_V}$.

Theorems & Definitions (19)

  • Definition 1: Generalized divergence, data-processing inequality
  • Definition 2: Fidelity, sine distance
  • Definition 3: Smoothing environment
  • Definition 4: Types of generalized mutual information
  • Definition 5: Types of generalized conditional entropies
  • Proposition 1
  • Proposition 2
  • Definition 6: Reversal channel
  • Definition 7
  • Lemma 1
  • ...and 9 more