Continuity of quantum entropic quantities via almost convexity

TL;DR

This work introduces the almost locally affine (ALAFF) method, which allows us to prove a great variety of continuity bounds for the derived entropic quantities and proves almost concavity for the Umegaki relative entropy and the BS-entropy.

Abstract

Based on the proofs of the continuity of the conditional entropy by Alicki, Fannes, and Winter, we introduce in this work the almost locally affine (ALAFF) method. This method allows us to prove a great variety of continuity bounds for the derived entropic quantities. First, we apply the ALAFF method to the Umegaki relative entropy. This way, we recover known almost tight bounds, but also some new continuity bounds for the relative entropy. Subsequently, we apply our method to the Belavkin-Staszewski relative entropy (BS-entropy). This yields novel explicit bounds in particular for the BS-conditional entropy, the BS-mutual and BS-conditional mutual information. On the way, we prove almost concavity for the Umegaki relative entropy and the BS-entropy, which might be of independent interest. We conclude by showing some applications of these continuity bounds in various contexts within quantum information theory.

Figures (6)

• Figure 1: A flow chart demonstrating how convexity and almost concavity of a divergence can be used to obtain uniform continuity and explicit continuity bounds on entropic quantities derived from that divergence. The subscripts of the functions $f_{D, 1/2}$ and $f_{DB}$ stand for divergence first, second argument and divergence bound respectively.
• Figure 2: In this flow chart we collect the main results from this chapter, starting with the almost concavity of the relative entropy, which together with the ALAFF method outputs a collection of continuity bounds for related entropic quantities. For the convexity and almost concavity, we are setting $\rho= p \rho_1 + (1-p) \rho_2$ and $\sigma= p \sigma_1 + (1-p) \sigma_2$, with $p \in [0,1]$. We denote by $\widetilde{m}_\sigma$ the minimal non-zero eigenvalue of $\sigma$. The specific bounds obtained for the relative entropy fixing the first argument and in the general case (modifying both arguments) are omitted due to their technicality.
• Figure 3: Two plots comparing the divergence bounds from \ref{['tab:divergence_bounds_comparison']}.
• Figure 4: In this flow chart we collect the main results from this section, starting with the almost concavity for the BS-entropy, which together with the ALAFF method outputs a plethora of continuity bounds for related entropic quantities. For the convexity and almost concavity of the BS-entropy we are setting $\rho = p \rho_1 + (1-p) \rho_2$ and $\sigma = p \sigma_1 + (1-p) \sigma_2$, with $p \in [0,1]$. We denote by $m_\sigma$ the minimal eigenvalue of $\sigma$. In the almost concavity bound, $\hat{c}_0$ is the maximum of $\left\Vert\sigma_1^{-1}\right\Vert_\infty$ and $\left\Vert\sigma_2^{-1}\right\Vert_\infty$. Additionally, we assume in all the continuity bounds that $m \leq \left\Vert\eta^{-1}\right\Vert_\infty$, for $\eta= \sigma, \rho$.
• Figure 5: We investigate the dependence of the almost convex remainder term of the BS-conditional entropy on the minimal eigenvalue of the involved states. For the minimal eigenvalues $10^{-4}, 10^{-8}, 10^{-16}, 10^{-32}$ we sampled five hundred pairs of qubits $(\rho, \sigma)$ both of them with controlled eigenvalues. We then sampled for every state pair ten values of $p$, the convex interpolation parameter, and plotted the remainder. As can be seen from the plot, the remainder appears to be independent of the minimal eigenvalue and the shape suggests a binary entropy or Gini impurity. The result shows a similar pattern if the dimension is increased.

Theorems & Definitions (90)

• Definition 4.1: Almost (joint) concavity of a divergence
• Remark 4.2
• Definition 4.3: Perturbed $\Delta$-invariant subset
• Remark 4.4
• Definition 4.5: Almost locally affine (ALAFF) function
• Theorem 4.6: Almost locally affine (ALAFF) method
• proof : Proof
• Remark 4.7
• Theorem 5.1: Almost concavity of the relative entropy
• proof : Proof.