Quantum Doeblin coefficients: A simple upper bound on contraction coefficients

TL;DR

This work introduces the quantum Doeblin coefficient $\alpha(\mathcal N)=\sup\{\varepsilon: \mathcal E_{\varepsilon} \succeq_{\operatorname{deg}} \mathcal N\}$ as an efficiently computable upper bound on contraction coefficients for quantum channels, via SDP. It proves general bounds $\eta_f(\mathcal N) \le 1-\alpha(\mathcal N)$ (and $\eta_{\mathrm{Tr}}(\mathcal N) \le 1-\alpha(\mathcal N)$) and develops several enhancements: a transpose-degradability variant $\alpha^T(\mathcal N)$, a Hermitian-relaxation variant $\alpha^H(\mathcal N)$, and reverse Doeblin coefficients $\widehat\alpha(\mathcal N)$ to bound expansion, all with SDP formulations. The paper also defines reverse transpose versions, generalized depolarizing channels for tightened bounds, and analyzes their impact on trace-distance contraction and expansion across several channels (notably depolarizing and generalized amplitude-damping channels). Applications to resource theories and information-theoretic bounds show the practical utility of these coefficients for assessing data-processing ranges and channel noise. Overall, the work provides a versatile, computable toolkit for bounding contraction and expansion in quantum information processing, with several directions for tightening and broader applicability.

Abstract

Contraction coefficients give a quantitative strengthening of the data processing inequality. As such, they have many natural applications whenever closer analysis of information processing is required. However, it is often challenging to calculate these coefficients. As a remedy we discuss a quantum generalization of Doeblin coefficients. These give an efficiently computable upper bound on many contraction coefficients. We prove several properties and discuss generalizations and applications. In particular, we give additional stronger bounds. One especially for PPT channels and one for general channels based on a constraint relaxation. Additionally, we introduce reverse Doeblin coefficients that bound certain expansion coefficients.

Figures (8)

• Figure 1: Plot of the Doeblin coefficient $\alpha({\cal A}_{p,\eta})$ for $p\in[0,1]$ and $\eta\in[0,1]$.
• Figure 2: Plot of the Doeblin coefficients $\alpha({\cal D}_p)$ (blue) and $\alpha^T({\cal D}_p)$ (red) for $p\in[0,\frac{4}{3}]$. Recall that the qubit depolarizing channel is PPT for $p\geq\frac{2}{3}$.
• Figure 3: Plot of $1-\alpha({\cal A}_{p,\eta})$ (dashed) and $1-\alpha^H({\cal A}_{p,\eta})$ (solid) over $p$ for $\eta=0.5$ (black), $0.6$ (red), $0.7$ (blue) and $0.8$ (green).
• Figure 4: Plot of the reverse Doeblin coefficients $\widecheck\alpha({\cal D}_p)$ (blue) and $\widecheck\alpha^T({\cal D}_p)$ (red) for $p\in[0,\frac{4}{3}]$.
• Figure 5: Plot of the reverse Doeblin coefficient $\widecheck\alpha({\cal A}_{p,\eta})$ for $p\in[0,1]$ and $\eta\in[0,1]$.

Theorems & Definitions (34)

• Proposition 2.1: Proposition II.5, hirche2022contraction
• Corollary 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Corollary 3.4
• proof
• Lemma 3.5