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arXiv:2210.04874 · Physical Review A

Lipschitz continuity of quantum-classical conditional entropies with respect to angular distance and related properties

Michael Liaofan Liu, Florian Kanitschar, Amir Arqand, Ernest Y.-Z. Tan

Published Feb 21, 2023• 0 citations
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Abstract

We derive a Lipschitz continuity bound for quantum-classical conditional entropies with respect to angular distance, with a Lipschitz constant that is independent of the dimension of the conditioning system. This bound is sharper in some situations than previous continuity bounds, which were either based on trace distance (where Lipschitz continuity is not possible), or based on angular distance but did not include a conditioning system. However, we find that the bound does not directly generalize to fully quantum conditional entropies. To investigate possible counterexamples in that setting, we study the characterization of states which saturate the Fuchs--van de Graaf inequality and thus have angular distance approximately equal to trace distance. We give an exact characterization of such states in the invertible case. For the noninvertible case, we show that the situation appears to be significantly more elaborate, and seems to be strongly connected to the question of characterizing the set of fidelity-preserving measurements.

DOI
10.1103/physreva.107.022426

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We derive a Lipschitz continuity bound for quantum-classical conditional entropies with respect to angular distance, with a Lipschitz constant that is independent of the dimension of the conditioning system. This bound is sharper in some situations than previous continuity bounds, which were either based on trace distance (where Lipschitz continuity is not possible), or based on angular distance but did not include a conditioning system. However, we find that the bound does not directly generalize to fully quantum conditional entropies. To investigate possible counterexamples in that setting, we study the characterization of states which saturate the Fuchs--van de Graaf inequality and thus have angular distance approximately equal to trace distance. We give an exact characterization of such states in the invertible case. For the noninvertible case, we show that the situation appears to be significantly more elaborate, and seems to be strongly connected to the question of characterizing the set of fidelity-preserving measurements.
We present a continuity bound for quantum conditional entropies in terms of fidelity, with no dependence on the dimension of the conditioning system. Our results encompass both the von Neumann entropy and one version of sandwiched Rényi entropy, which is particularly relevant for applications in quantum key distribution (QKD). In such applications, this bound should provide improved results as compared to previous continuity bounds based on trace distance. We also show that the bounds can be sharpened as expected for classical-quantum states, and verify that they hold when the conditioning system has countably infinite dimension. Finally, we present several variants tailored for exponential functions of the Rényi entropies instead, which should be relevant for QKD applications.
In quantum information theory, communication capacities are mostly given in terms of entropic formulas. Continuity of such entropic quantities are significant, as they ensure uniformity of measures against perturbations of quantum states. Traditionally, continuity bounds have been provided in terms of the trace distance, which is a bonafide metric on the set of quantum states. In the present contribution we derive continuity bounds for various information measures based on the difference in purity of the concerned quantum states. In a finite-dimensional system, we establish continuity bounds for von Neumann entropy which depend only on purity distance and dimension of the system. We then obtain uniform continuity bounds for conditional von Neumann entropy in terms of purity distance which is free of the dimension of the conditioning subsystem. Furthermore, we derive the uniform continuity bounds for other entropic quantities like relative entropy distance, quantum mutual information and quantum conditional mutual information. As an application, we investigate the variation in squashed entanglement with respect to purity. We also obtain a bound to the quantum conditional mutual information of a quantum state which is arbitrarily close to a quantum Markov chain.
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.
For the sandwiched Rényi entropy the conditional entropy can be defined two ways: H~α↓(A∣B)ρ,H~α↑(A∣B)ρ\widetilde{H}^\downarrow_α(A|B)_ρ, \widetilde{H}^\uparrow_α(A|B)_ρHα↓​(A∣B)ρ​,Hα↑​(A∣B)ρ​. In the limiting case, α=1α=1α=1, both definitions consolidate into conditional entropy H(A∣B)=S(AB)−S(B)H(A|B)=S(AB)-S(B)H(A∣B)=S(AB)−S(B). The continuity inequality for conditional entropy H(A∣B)H(A|B)H(A∣B), called the Alicki-Fannes-Winter (AWF) inequality, shows that if the states are close in trace-distance, then the conditional entropies are also close. Having the AWF inequality for conditional entropy, we show that the channel entropy defined through the relative entropy is continuous with respect to the diamond-distance between channels. Inspired by this, similar continuity inequalities for the Rényi conditional entropy H~α↑\widetilde{H}^\uparrow_αHα↑​ were obtained in the work [A. Marwah and F. Dupuis, J. Math. Phys. 63, 052201 (2022)]. We provide continuity bounds for the sandwiched Rényi and Tsallis conditional entropies H~α↓(A∣B)ρ,T~α↓(A∣B)ρ\widetilde{H}^\downarrow_α(A|B)_ρ, \widetilde{T}^\downarrow_α(A|B)_ρHα↓​(A∣B)ρ​,Tα↓​(A∣B)ρ​ for states with the same marginal on the conditioning system. Similar to the previous bounds, our bound depends only on the dimension of the conditioning system. We apply this result to prove continuity of the channel entropy for Rényi and Tsallis channel entropies defined through the sandwiched Rényi and Tsallis relative entropies.
We derive a measurement-independent asymptotic continuity bound on the observational entropy for general positive operator valued measures measurements, making essential use of its property of bounded concavity. The same insight is used to obtain continuity bounds for other entropic quantities, including the measured relative entropy distance to a convex set of states under a general set of measurements. As a special case, we define and study conditional observational entropy, which is an observational entropy in one (measured) subsystem conditioned on the quantum state in another (unmeasured) subsystem. We also study continuity of relative entropy with respect to a jointly applied channel, finding that observational entropy is uniformly continuous as a function of the measurement. But we show by means of an example that this continuity under measurements cannot have the form of a concrete asymptotic bound.
We prove the sharp uniform continuity bound for quantum conditional entropy. If two bipartite states are at trace distance at most δδδ and d=dim⁡Ad=\dim Ad=dimA, the optimal dimension-only modulus of continuity is h2(δ)+δlog⁡(d2−1)h_2(δ)+δ\log(d^2-1)h2​(δ)+δlog(d2−1) up to δ=1−d−2δ=1-d^{-2}δ=1−d−2 and 2log⁡d2\log d2logd thereafter, where h2h_2h2​ denotes the binary entropy. When dim⁡B≥d\dim B\ge ddimB≥d, this bound is tight for every δ∈[0,1]δ\in[0,1]δ∈[0,1]. The key proof idea was developed with the assistance of ChatGPT 5.6 Sol, building on and adapting the tight classical proof of Alhejji \& Smith [IEEE ISIT (2020)], which follows a conceptually different approach.