Continuity of entropies via integral representations (2408.15226v2)

Abstract: We show that Frenkel's integral representation of the quantum relative entropy provides a natural framework to derive continuity bounds for quantum information measures. Our main general result is a dimension-independent semi-continuity relation for the quantum relative entropy with respect to the first argument. Using it, we obtain a number of results: (1) a tight continuity relation for the conditional entropy in the case where the two states have equal marginals on the conditioning system, resolving a conjecture by Wilde in this special case; (2) a stronger version of the Fannes-Audenaert inequality on quantum entropy; (3) better estimates on the quantum capacity of approximately degradable channels; (4) an improved continuity relation for the entanglement cost; (5) general upper bounds on asymptotic transformation rates in infinite-dimensional entanglement theory; and (6) a proof of a conjecture due to Christandl, Ferrara, and Lancien on the continuity of 'filtered' relative entropy distances.

• The paper introduces an integral representation framework that sharpens continuity bounds for quantum relative entropy.
• It refines key results such as the Fannes–Audenaert inequality and resolves conjectures on conditional entropy stability.
• The approach enhances analyses of quantum channel capacities and entanglement costs under small perturbations.

Continuity of Entropies via Integral Representations

Continuity bounds for quantum information measures are critical for understanding the stability of information-theoretic tasks under small perturbations. The paper "Continuity of entropies via integral representations" by Berta, Lami, and Tomamichel introduces a general framework to derive such bounds, leveraging Frenkel's integral representation of quantum relative entropy. This approach provides new insights and tighter bounds for various quantum information measures, resolving several open conjectures and enhancing the current state-of-the-art results.

Main Contributions

Integral Representation of Quantum Relative Entropy

The core idea of the paper is the integral representation of the quantum relative entropy due to Frenkel. This formula expresses the quantum relative entropy as: $$D(\rho\|\sigma) = (\log e) \int_1^\infty d\gamma \left(\frac{1}{\gamma} E_\gamma(\rho\|\sigma) + \frac{1}{\gamma^2} E_\gamma(\sigma\|\rho)\right),$$ where $E_\gamma$ are the hockey-stick divergences. This representation is instrumental in deriving dimension-independent semi-continuity bounds, particularly for the quantum relative entropy with respect to its first argument.

Dimension-Independent Semi-Continuity Relation

One of the key results is a dimension-independent semi-continuity bound for the quantum relative entropy: $$D(\rho\|\omega) - D(\sigma\|\omega) \leq \log(M-1) + h_2(),$$ assuming $$\frac12 \|\rho - \sigma\|_1 \leq \leq 1 - \frac{1}{M}$$ and $\rho \leq M \omega$. This bound is tight and provides a robust tool for analyzing the stability of quantum information measures.

Applications and Implications

The paper showcases several implications of this general result across different areas of quantum information theory:

1. Improved Fannes--Audenaert Inequality: The integral representation approach leads to a refined version of the celebrated Fannes--Audenaert inequality, which states that: $$\left| S(\rho) - S(\sigma)\right| \leq \log(d\lambda_{\max}(\sigma) - 1) + h_2(),$$ where $\lambda_{\max}(\sigma)$ is the largest eigenvalue of $\sigma$. This tighter bound is particularly useful for mixed states.
2. Continuity of Conditional Entropy: The paper resolves a conjecture by Wilde in the special case where two states have equal marginals on the conditioning system. For $\rho_{AB} = \sigma_{AB}$ having equal marginals on $B$, it is shown that: $$\left| H(A|B)_\rho - H(A|B)_\sigma \right| \leq \log\left(|A|\min\{|A|,|B|\} - 1\right) + h_2(),$$ given $\frac12 \|\rho_{AB} - \sigma_{AB}\|_1 \leq$.
3. Continuity of Quantum Capacities: Quantum channel capacities, including the quantum capacity, can be analyzed using these continuity bounds. For channels $M_{A\to B}$ and $N_{A\to B}$ with $\frac12\|M - N\|_\diamond \leq$, the quantum capacity obeys: $$|Q(M) - Q(N)| \leq 2\left(\log(|B|^2 - 1) + h_2()\right).$$
4. Continuity of Entanglement Costs: The entanglement cost's continuity is improved as well. For $\frac12 \|\rho_{AB} - \sigma_{AB}\|_1 \leq$, the bound is: $$|E_c(\rho_{AB}) - E_c(\sigma_{AB})| \leq \delta \log(d^2 -1) + h_2(\delta),$$ with $\delta = \sqrt{(2-) }$.
5. Approximate Degradability: For approximately degradable channels, the quantum capacity bound is refined. For an $\epsilon$-degradable channel: $$Q(N) \leq U_\Theta(N) + \epsilon\log(|E|^2 - 1) + h_2(\epsilon).$$
6. Filtered Entropy Distances: The paper resolves a conjecture regarding the continuity of filtered relative entropy distances: $$\Big| D^L(\rho\|F) - D^L(\sigma\|F) \Big| \leq \log D + h_2() + g(),$$ where $g()$ is a function with $\lim_{\to 0} g() = 0$.

Technical Innovations and Future Directions

The integral representation approach provides a method that works directly at the level of quantum entropies, avoiding reductions to the classical case. This fully quantum method enables the derivation of tighter and more general continuity bounds.

Future research could explore further extensions of these bounds, such as the continuity bounds for the squashed entanglement and tight bounds for the conditional entropy for general bipartite states, which remain open problems. The techniques developed can also be applied to other areas in quantum information theory where stability under perturbations is crucial.

Conclusion

This paper significantly advances the understanding of continuity bounds in quantum information theory by introducing a general framework based on integral representations of quantum relative entropy. The results derived have broad implications, improving existing bounds and resolving several conjectures, thereby providing a powerful tool for future research in quantum information stability.