Fidelity-Based Smooth Min-Relative Entropy: Properties and Applications

TL;DR

The paper provides a comprehensive study of the fidelity-based smooth min-relative entropy, establishing its data-processing property and connecting it to smooth min-relative entropy and smooth sandwiched Renyi relative entropies. It derives second-order asymptotics for the smooth F-min-relative entropy and related divergences, revealing that the leading term is the quantum relative entropy and the second-order term involves its variance, thereby unifying several smooth divergences in the i.i.d. setting. Leveraging these results, the authors obtain one-shot and second-order bounds for randomness distillation in general resource theories, including explicit LOCC and 1W-LOCC scenarios and a gamma-based upper bound, with precise results for certain cq states. They also provide practical SDP and bilinear-program formulations to compute the smooth F-min-relative entropy and related quantities, enabling numerical exploration and tighter bounds. The work broadens the toolbox for quantum resource transformations and offers concrete pathways toward understanding the limits of distillation tasks in mixed-state target scenarios.

Abstract

The fidelity-based smooth min-relative entropy is a distinguishability measure that has appeared in a variety of contexts in prior work on quantum information, including resource theories like thermodynamics and coherence. Here we provide a comprehensive study of this quantity. First we prove that it satisfies several basic properties, including the data-processing inequality. We also establish connections between the fidelity-based smooth min-relative entropy and other widely used information-theoretic quantities, including smooth min-relative entropy and smooth sandwiched Rényi relative entropy, of which the sandwiched Rényi relative entropy and smooth max-relative entropy are special cases. After that, we use these connections to establish the second-order asymptotics of the fidelity-based smooth min-relative entropy and all smooth sandwiched Rényi relative entropies, finding that the first-order term is the quantum relative entropy and the second-order term involves the quantum relative entropy variance. Utilizing the properties derived, we also show how the fidelity-based smooth min-relative entropy provides one-shot bounds for operational tasks in general resource theories in which the target state is mixed, with a particular example being randomness distillation. The above observations then lead to second-order expansions of the upper bounds on distillable randomness, as well as the precise second-order asymptotics of the distillable randomness of particular classical-quantum states. Finally, we establish semi-definite programs for smooth max-relative entropy and smooth conditional min-entropy, as well as a bilinear program for the fidelity-based smooth min-relative entropy, which we subsequently use to explore the tightness of a bound relating the last to the first.

Figures (5)

• Figure 1: One way local operations and classical communication protocol from $A$ to $B$. First, the channel $\mathcal{E}_{A \to ML}$ with classical outputs $L$ and $M$ is applied by Alice. Then, system $L$ is communicated to Bob via a noiseless classical channel. Bob applies the decoding channel $\mathcal{D}_{LB \to M'}$ to get the classical output $M'$. At the end of the protocol, the output state shared by Alice and Bob is $\omega_{MM'}$, and it should be close to a maximally classically correlated state. In the above figure, classical systems are denoted by double lines.
• Figure 2: General local operations and classical communication protocol from $A$ to $B$: First the channel $\mathcal{E}^{(1)}_{A \to A_1L_1}$ with classical output $L_1$ is applied by Alice. Then, system $L_1$ is communicated to Bob via a noiseless classical channel. Next Bob applies the channel $\mathcal{D}^{(2)}_{L_1 B\to L_2 B_2}$, and the classical output $L_2$ is communicated to Alice. This procedure is continued for $k-1$ rounds. During the final round Alice performs $\mathcal{E}^{(k)}_{A_{k-2} L_{k-1}\to M L_k}$ where both the output systems are classical and communicates $L_k$ to Bob. Bob completes the protocol by applying the channel $\mathcal{D}^{(k+1)}_{L_k B_{k-1} \to M'}$. At the end of the protocol, the output state shared by Alice and Bob is $\omega_{MM'}$, and it should be close to a maximally classically correlated state.
• Figure 3: For fixed $\varepsilon=0.0001$, the plots depicts upper and lower bounds on the randomness distillation rate of an isotropic state $\rho_{AB}= (1-p)\Phi^d_{AB} +p \frac{I_{AB}}{d^2}$ with $p=0.3$ and $d=2$. The horizontal lines depict asymptotic values of the upper and lower bounds, while the curves depict the lower bound from \ref{['prop:second-order-expansion-CR-some-cq-lower']} and the upper bound from \ref{['thm:Upper-Bound-Second-Order-general']}.
• Figure 4: This plot shows the tightness of the lower bound on $D^\varepsilon_{\min,F}(\rho \Vert \sigma)$ obtained from running the seesaw algorithm for 10 iterations for each $\varepsilon$, where $\rho,\sigma$ are random quantum states in a Hilbert space of dimension two. By changing $\delta$, the upper bound $D_{\max}^{1-\varepsilon-\delta}(\rho\Vert\sigma)+ \log_{2} \!\left( \frac{1}{1-f(\varepsilon,\delta)}\right)$ is also shown.
• Figure 5: This plot shows the tightness of the lower bound on $D^\varepsilon_{\min,F}(\rho \Vert \sigma)$ obtained from running the seesaw algorithm for ten iterations for each $\varepsilon$, where $\rho,\sigma$ are random quantum states in a Hilbert space of dimension four. By changing $\delta$, the upper bound $D_{\max}^{1-\varepsilon-\delta}(\rho\Vert\sigma)+ \log_{2} \!\left( \frac{1}{1-f(\varepsilon,\delta)}\right)$ is also shown.

Theorems & Definitions (43)

• Definition 1: Fidelity-based smooth min-relative entropy
• Remark 1: Inequality constraint in the definition of smooth $F$-min-relative entropy
• Lemma 1: Unitary Invariance
• Theorem 1
• Remark 2: On the choice of subnormalized states
• Lemma 2
• Corollary 1
• Theorem 2
• Definition 2
• Remark 3: Inequality constraint in smooth sandwiched Rényi relative entropy definition