[Paper Review] Monotonic multi-state quantum $f$-divergences

Keiichiro Furuya, Nima Lashkari+1 more|arXiv (Cornell University)|Mar 17, 2021

Quantum Information and Cryptography54 references4 citations

TL;DR

This paper introduces a broad class of multi-state quantum f-divergences using Tomita-Takesaki modular theory and Kubo-Ando operator means, proving they satisfy the data processing inequality across general von Neumann algebras. The construction generalizes known Rényi divergences and f-divergences, including Petz’s and Matsumoto’s measures, and conjectures operational interpretations in asymmetric quantum state discrimination.

ABSTRACT

We use the Tomita-Takesaki modular theory and the Kubo-Ando operator mean to write down a large class of multi-state quantum $f$-divergences and prove that they satisfy the data processing inequality. For two states, this class includes the $(α,z)$-Rényi divergences, the $f$-divergences of Petz, and the measures in \cite{matsumoto2015new} as special cases. The method used is the interpolation theory of non-commutative $L^p_ω$ spaces and the result applies to general von Neumann algebras including the local algebra of quantum field theory. We conjecture that these multi-state Rényi divergences have operational interpretations in terms of the optimal error probabilities in asymmetric multi-state quantum state discrimination.

Motivation & Objective

• To develop a general framework for multi-state quantum f-divergences that extend classical and quantum distinguishability measures.
• To prove that these divergences satisfy the data processing inequality, a fundamental requirement for physically meaningful distinguishability measures.
• To unify and generalize existing quantum Rényi divergences, including Petz’s and Matsumoto’s f-divergences, within a single operator-theoretic framework.
• To extend the theory to general von Neumann algebras, including local algebras in quantum field theory.
• To conjecture that these divergences have operational significance in asymmetric multi-state quantum state discrimination.

Proposed method

• Utilizes Tomita-Takesaki modular theory to define non-commutative Lp spaces and modular operators for states on von Neumann algebras.
• Applies Kubo-Ando operator means to construct a family of multi-state f-divergences parameterized by operator convex functions and weights.
• Employs interpolation theory of non-commutative Lp spaces to derive the divergences and prove monotonicity under quantum channels.
• Constructs a one-parameter family of divergences via weighted geometric means of modular operators, generalizing Rényi-type divergences.
• Uses the limit of these divergences as the parameter ε→0 to recover the quantum relative entropy as a special case.
• Applies the Riesz-Thorin interpolation theorem to prove boundedness of channel maps on non-commutative Lp spaces, ensuring data processing monotonicity.

Experimental results

Research questions

• RQ1 Can a unified framework be constructed for multi-state quantum f-divergences that generalizes known Rényi and f-divergences?
• RQ2 Do these generalized divergences satisfy the data processing inequality under arbitrary quantum channels?
• RQ3 What is the operational meaning of these multi-state divergences in quantum information tasks like state discrimination?
• RQ4 How do these divergences reduce to known quantities like relative entropy or log-fidelity in limiting cases?
• RQ5 Can the construction be extended to general von Neumann algebras, including those in algebraic quantum field theory?

Key findings

• The proposed multi-state f-divergences satisfy the data processing inequality for all quantum channels, ensuring their physical consistency.
• The class includes the $(\alpha,z)$-Rényi divergences, Petz’s f-divergences, and Matsumoto’s measures as special cases.
• In the limit $\epsilon \to 0$, the divergences converge to a convex combination of quantum relative entropies, $\sum_i \beta_i S(\psi_i \| \omega)$.
• The construction is valid for general von Neumann algebras, including the local algebras of quantum field theories.
• The method relies on interpolation theory of non-commutative $L^p_\omega$ spaces and modular theory, ensuring mathematical rigor.
• The paper proves that the $(p\to q)$-norm of a channel on non-commutative $L^p$ spaces is bounded by 1, which underlies the data processing result.