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Sufficiency of Rényi divergences

Niklas Galke, Lauritz van Luijk, Henrik Wilming

TL;DR

The paper investigates when two sets of states (classical or quantum) are interconvertible by channels, linking this question to Rényi divergences through data-processing inequalities. It proves that classical Rényi divergences are sufficient to decide interconvertibility in the classical setting, using a Koashi–Imoto-type normal form and a Laplace-transform framework, while showing that known quantum Rényi divergences are not sufficient due to anti-unitary invariance. The authors propose a conjecture that the minimal quantum Rényi divergence might become sufficient when interconversion is allowed via positive maps, with affirmative results in special cases and a broader discussion of time-reversal detection as a fundamental constraint. They also develop a complete monotonicity approach, deriving an infinite family of RD inequalities and providing numerical evidence supporting the conjecture, which could imply deep links between state convertibility, recovery maps, and symmetry under time-reversal in quantum thermodynamics.

Abstract

A set of classical or quantum states is equivalent to another one if there exists a pair of classical or quantum channels mapping either set to the other one. For dichotomies (pairs of states), this is closely connected to (classical or quantum) Rényi divergences (RD) and the data-processing inequality: If a RD remains unchanged when a channel is applied to the dichotomy, then there is a recovery channel mapping the image back to the initial dichotomy. Here, we prove for classical dichotomies that equality of the RDs alone is already sufficient for the existence of a channel in any of the two directions and discuss some applications. In the quantum case, all families of quantum RDs are seen to be insufficient because they cannot detect anti-unitary transformations. Thus, including anti-unitaries, we pose the problem of finding a sufficient family. It is shown that the Petz and maximal quantum RD are still insufficient in this more general sense and we provide evidence for sufficiency of the minimal quantum RD. As a side result of our techniques, we obtain an infinite list of inequalities fulfilled by the classical, the Petz quantum, and the maximal quantum RDs. These inequalities are not true for the minimal quantum RDs. Our results further imply that any sufficient set of conditions for state transitions in the resource theory of athermality must be able to detect time-reversal.

Sufficiency of Rényi divergences

TL;DR

The paper investigates when two sets of states (classical or quantum) are interconvertible by channels, linking this question to Rényi divergences through data-processing inequalities. It proves that classical Rényi divergences are sufficient to decide interconvertibility in the classical setting, using a Koashi–Imoto-type normal form and a Laplace-transform framework, while showing that known quantum Rényi divergences are not sufficient due to anti-unitary invariance. The authors propose a conjecture that the minimal quantum Rényi divergence might become sufficient when interconversion is allowed via positive maps, with affirmative results in special cases and a broader discussion of time-reversal detection as a fundamental constraint. They also develop a complete monotonicity approach, deriving an infinite family of RD inequalities and providing numerical evidence supporting the conjecture, which could imply deep links between state convertibility, recovery maps, and symmetry under time-reversal in quantum thermodynamics.

Abstract

A set of classical or quantum states is equivalent to another one if there exists a pair of classical or quantum channels mapping either set to the other one. For dichotomies (pairs of states), this is closely connected to (classical or quantum) Rényi divergences (RD) and the data-processing inequality: If a RD remains unchanged when a channel is applied to the dichotomy, then there is a recovery channel mapping the image back to the initial dichotomy. Here, we prove for classical dichotomies that equality of the RDs alone is already sufficient for the existence of a channel in any of the two directions and discuss some applications. In the quantum case, all families of quantum RDs are seen to be insufficient because they cannot detect anti-unitary transformations. Thus, including anti-unitaries, we pose the problem of finding a sufficient family. It is shown that the Petz and maximal quantum RD are still insufficient in this more general sense and we provide evidence for sufficiency of the minimal quantum RD. As a side result of our techniques, we obtain an infinite list of inequalities fulfilled by the classical, the Petz quantum, and the maximal quantum RDs. These inequalities are not true for the minimal quantum RDs. Our results further imply that any sufficient set of conditions for state transitions in the resource theory of athermality must be able to detect time-reversal.
Paper Structure (17 sections, 29 theorems, 125 equations, 2 figures, 1 table)

This paper contains 17 sections, 29 theorems, 125 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Structure of the paper:
  3. Structure of interconvertible states
  4. Dichotomies: Sufficient families of Rényi divergences
  5. The classical case
  6. Quantum case: Insufficiency of known quantum Rényi divergences
  7. Sufficiency and positive maps: A conjecture
  8. Complete monotonicity
  9. Interconvertibility from Rényi Divergences without absolute continuity
  10. Convertibility and interconvertibility for probability vectors
  11. Minimal form.
  12. Interconvertibility.
  13. Lorenz curves.
  14. Convertibility from Rényi divergences for $\alpha>0$ alone.
  15. Proof of Theorem \ref{['thm:interconvertibility']} for infinite-dimensional systems
  16. ...and 2 more sections

Key Result

Theorem 1

Let $(X_i,\mu_i)$, $i=1,2$, be measure spaces and $p_i\ll q_i$ probability measures in $L^1(X_i,\mu_i)$. If there is an open interval $(a,b)\subset\mathbb R$, $a<b$, such that there are stochastic maps between $L^1(X_1,\mu_1)$ and $L^1(X_2,\mu_2)$ interconverting the dichotomies $(p_1,q_1)$ and $(p_2,q_2)$.

Figures (2)

  • Figure 1: The derivatives $(-1)^k \partial_\alpha^k\,g(\alpha|\rho_s,\sigma)$ for $\mathbb D_\alpha = D^{\min}_\alpha$ in the case $\rho_s = \mathrm{e}^{s \sigma_x}/2 \cosh(s),\sigma = \mathrm{e}^{\sigma_z}/\cosh(1)$ and for various values of $s$ (see Eq. \ref{['eq:g']} for the definition of $g(\alpha|\rho,\sigma)$). If $g(\alpha|\rho_s,\sigma)$ was completely monotone, all curves would have to be monotonically decreasing and convex. Note that $[\rho_s,\sigma]=0$ if and only if $s=0$. For $s=0$ the function is completely monotone. As $s$ increases, the commutator-norm $\left\| [\rho_s,\sigma] \right\|$ increases monotonically while lower and lower derivatives signal that the function is not completely monotone. Already for $s=0.3$ one can see that the second derivative is not completely monotone.
  • Figure 2: Koashi-Imoto minimal form from the point of view of Lorenz curves: The blue curve on the left corresponds to $L_{(\mathbf p,\mathbf q)}$. Removing all points from the Lorenz curve that lie in the interior of straight line segments (the blue points) yields the Lorenz curve $L_{(\tilde{\mathbf p},\tilde{\mathbf q})}$ on the right. This corresponds to the map $T$. Obviously, no further "anchoring points" can be removed from the curve without altering it.

Theorems & Definitions (55)

  • Theorem 1: Sufficiency of Rényi divergences
  • Theorem 2
  • Corollary 3
  • Proposition 1
  • Conjecture 4
  • Corollary 5
  • Definition 2
  • Theorem 3: Koashi-Imoto
  • Theorem 4
  • Example 5
  • ...and 45 more