Layer Cake Representations for Quantum Divergences

TL;DR

The paper introduces a layer cake representation to define quantum analogues of classical divergences, providing a unified framework that recovers known integral representations and yields new forms.It proves key equivalences to recent integral-based definitions and derives diverse representations, including trace, Riemann–Stieltjes integral, and variational dual forms, across Rényi and f-divergences.The results enable new error-exponent bounds for quantum hypothesis testing and deepen understanding of noncommutative likelihood ratios, with α→1 limits connecting to the Umegaki relative entropy and Frenkel's integral formula.Overall, the layer cake approach offers versatile, technically rich tools for analyzing quantum distinguishability and information-theoretic quantities.

Abstract

Defining suitable quantum extensions of classical divergences often poses a challenge due to the non-commutative nature of quantum information. In this work, we propose a new approach via what we call the layer cake representation. The resulting quantum Rényi and $f$-divergences are then proven to be equivalent to those recently defined via integral representations. Nevertheless, the approach can provide several insights. We give an alternative proof of the integral representation of the relative entropy by Frenkel and prove a conjecture regarding a trace expression for the Rényi divergence. Additionally, we give applications to error exponents in hypothesis testing, a new Riemann-Stieltjes type integral representation and a variational representation.

Figures (6)

• Figure 1: Schematic relations between the known formulas of the quantum relative entropy $D(\rho\Vert\sigma)$
• Figure 2: Schematic relations between the known formulas of the quantum $f$-divergence $D_{f}(\rho\Vert\sigma)$
• Figure 3: A numerical demonstration of the piecewise analytic map $\gamma \mapsto \mathop{\mathrm{Tr}}\nolimits\left[ \sigma \{ \rho > \gamma \sigma\} \right]$ in blue color. Each dashed vertical line in red corresponds to a jump at some $\gamma$ such that $\rho - \gamma \sigma$ is singular. Let $\{(\lambda_i(\gamma), |v_i(\gamma)\rangle)\}_i$ be analytic eigen-pairs for $\rho-\gamma \sigma$; see \ref{['eq:analytic-eigen']}. Each dotted line in orange color is an analytic map $\gamma \mapsto \mathop{\mathrm{Tr}}\nolimits\left[ \sigma \sum_{i\in\mathcal{I}} |v_i(\gamma)\rangle \langle v_i(\gamma) | \right]$ for some $\gamma$-independent index set $\mathcal{I}$ (including the empty set), where the caldinality $|\mathcal{I}|$ plays the role of the rank of the projection. Each dotted orange line covers one portion of the blue piecewise curve $\mathop{\mathrm{Tr}}\nolimits[\sigma \left\{\rho>\gamma \sigma\right\}] = \mathop{\mathrm{Tr}}\nolimits[ \sigma \sum_i \mathbf{1}_{\{ \lambda_i(\gamma) > 0 \}} |v_i(\gamma)\rangle \langle v_i(\gamma)| ]$ for a certain range of $\gamma$.
• Figure 4: A numerical demonstration of the piecewise analytic map $\gamma \mapsto \mathop{\mathrm{Tr}}\nolimits\left[ \sigma \{ \rho \leq \gamma \sigma\} \right]$ in blue color. The interpretation of the plot is similar to that of Figure \ref{['figure:RS-integral_quantum']}. The map $\gamma \mapsto \mathop{\mathrm{Tr}}\nolimits\left[ \sigma \{ \rho \leq \gamma \sigma\} \right]$ is a right-continuous monotone increasing map from $0$ to $1$.
• Figure 5: An illustration of the Riemann--Stieltjes integral representation for $D(\rho\Vert\sigma)= -\int_0^\infty \log \gamma \,\mathrm{d} \text{Tr}\left[\rho\{ \rho > \gamma \sigma \}\right]$ (Proposition \ref{['prop:relative_entropy_formula2']}) for the commuting case: $\rho = \sum_i \lambda_i |i\rangle \langle i|$ and $\sigma = \sum_i \mu_i |i\rangle \langle i|$. The plotted function in blue is $\gamma\mapsto \mathop{\mathrm{Tr}}\nolimits[\rho\{\rho>\gamma \sigma\}]$. The indices $i_1, i_2,\ldots$ are chosen such that ${\lambda_{i_1}}/{\mu_{i_1}} \leq {\lambda_{i_2}}/{\mu_{i_2}} \leq \cdots$. The equation above each strip (partition) is $- \log \gamma_{i_k} \left( \text{Tr}\left[\rho\{ \rho > \gamma_{i_k} \sigma \}\right] - \text{Tr}\left[\rho\{ \rho > \gamma_{i_{k-1}} \sigma \}\right] \right) = \lambda_{i_k} \log \gamma_{i_k}$, where $\gamma_{i_k} = \lambda_{i_k}/\mu_{i_k}$ for some $k$. The overall sum over each partition is $\sum_k \lambda_{i_k} \log ({\lambda_{i_k}}/{\mu_{i_k}}) = D(\rho\Vert \sigma)$.

Theorems & Definitions (28)

• proof
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• Definition 1: Quantum $f$-divergence
• Definition 2: Quantum Rényi divergence
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• proof : Proof of Lemma \ref{['lemm:alpha-representation']}
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