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arXiv:2610.25299

Fully quantum fidelity-based entropic continuity bounds

Matthew Simon Tan, Ernest Y. -Z. Tan

quant-phSubmitted Oct 18, 2026• Announced Oct 20, 2026• 0 citations
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Abstract

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.

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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.

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Continuity inequalities for sandwiched Rényi and Tsallis conditional entropies with application to the channel entropy continuity

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For the sandwiched Rényi entropy the conditional entropy can be defined two ways: $\widetilde{H}^\downarrow_α(A|B)_ρ, \widetilde{H}^\uparrow_α(A|B)_ρ$. In the limiting case, $α=1$, both definitions consolidate into conditional entropy $H(A|B)=S(AB)-S(B)$. The continuity inequality for conditional entropy $H(A|B)$, 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 $\widetilde{H}^\uparrow_α$ 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 $\widetilde{H}^\downarrow_α(A|B)_ρ, \widetilde{T}^\downarrow_α(A|B)_ρ$ 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.

math-phCross-listedarXiv:2512.01745•Journal of Mathematical Physics•Dec 2, 2025•0 citations

Sharp Continuity of Petz and Sandwiched Rényi Conditional Entropies

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We determine the sharp modulus of continuity, in trace distance, of the optimized Petz and sandwiched Rényi conditional entropies for every order $α\in[\frac12,1)$. If two bipartite states are within trace distance $δ$, then both conditional entropies differ by at most $\frac{1}{1-α} \log[(1-\varepsilon)^α +(D-1)^{1-α}\varepsilon^α]$, where $\varepsilon := \min\{δ,1-1/D\}$ and $D$ is the effective dimension, given by the dimension of the first subsystem times the largest possible Schmidt rank. For every distance constraint $δ\in[0,1]$, the bound is attained by an isotropic pair with a maximally entangled anchor. Taking $α\uparrow1$ recovers the recent sharp continuity bound of quantum conditional entropy by Berta et al. [arXiv:2607.24687]. The proof linearizes the relevant concave Rényi functional at a comparison point dictated by the isotropic equality family. Schmidt-rank domination extends the equality geometry to an arbitrary anchor state, after which trace-distance duality and a noncommutative calibration estimate control the perturbation and anchor term without weakening the sharp constant. The latter estimate requires matrix analysis and is assisted by ChatGPT 5.6 Sol.

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We prove a variety of improved uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality. We then employ this classical result to derive a tight energy-constrained continuity bound for the von Neumann entropy. To deal with more general entropies in infinite dimensions,e.g.$\alpha $-Rényi and$\alpha $-Tsallis entropies, we develop a novel approximation scheme based on operator Hölder continuity estimates. Finally, we settle an open problem raised by Shirokov regarding the characterisation of states with finite entropy.

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Abstract In this work, we prove uniform continuity bounds for entropic quantities related to the sandwiched Rényi divergences such as the sandwiched Rényi conditional entropy. We follow three different approaches: The first one is the “almost additive approach”, which exploits the sub-/superadditivity and joint concavity/convexity of the exponential of the divergence. In our second approach, termed the “operator space approach”, we express the entropic measures as norms and utilize their properties for establishing the bounds. These norms draw inspiration from interpolation space norms. We not only demonstrate the norm properties solely relying on matrix analysis tools but also extend their applicability to a context that holds relevance in resource theories. By this, we extend the strategies of Marwah and Dupuis as well as Beigi and Goodarzi employed in the sandwiched Rényi conditional entropy context. Finally, we merge the approaches into a mixed approach that has some advantageous properties and then discuss in which regimes each bound performs best. Our results improve over the previous best continuity bounds or sometimes even give the first continuity bounds available. In a separate contribution, we use the ALAFF method, developed in a previous article by some of the authors, to study the stability of approximate quantum Markov chains.

arXiv:2308.12425•Annales Henri Poincaré•Dec 26, 2024•3 citations

Tightening Continuity Bounds for Entropies and Bounds on Quantum Capacities

Michael G. Jabbour, Nilanjana Datta

Uniform continuity bounds on entropies are generally expressed in terms of a single distance measure between probability distributions or quantum states, typically, the total variation- or trace distance. However, if an additional distance measure is known, the continuity bounds can be significantly strengthened. Here, we prove a tight uniform continuity bound for the Shannon entropy in terms of both the local- and total variation distances, sharpening an inequality in (Sason, 2013). We also obtain a uniform continuity bound for the von Neumann entropy in terms of both the operator norm- and trace distances. We then apply our results to compute upper bounds on channel capacities. We first refine the concept of approximate degradable channels by introducing$(\varepsilon ,\nu)$–degradable channels. These are$\varepsilon $–close in diamond norm and$\nu $–close in completely bounded spectral norm to their complementary channel when composed with a degrading channel. This leads to improved upper bounds to the quantum- and private classical capacities of such channels. Moreover, these bounds can be further improved by considering certain unstabilized versions of the above norms. We show that upper bounds on the latter can be efficiently expressed as semidefinite programs. As an application, we obtain a new upper bound on the quantum capacity of the qubit depolarizing channel.

arXiv:2310.17329•IEEE Journal on Selected Areas in Information Theory•Jan 1, 2024•3 citations
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.
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 determine the sharp modulus of continuity, in trace distance, of the optimized Petz and sandwiched Rényi conditional entropies for every order α∈[12,1)α\in[\frac12,1)α∈[21​,1). If two bipartite states are within trace distance δδδ, then both conditional entropies differ by at most 11−αlog⁡[(1−ε)α+(D−1)1−αεα]\frac{1}{1-α} \log[(1-\varepsilon)^α +(D-1)^{1-α}\varepsilon^α]1−α1​log[(1−ε)α+(D−1)1−αεα], where ε:=min⁡{δ,1−1/D}\varepsilon := \min\{δ,1-1/D\}ε:=min{δ,1−1/D} and DDD is the effective dimension, given by the dimension of the first subsystem times the largest possible Schmidt rank. For every distance constraint δ∈[0,1]δ\in[0,1]δ∈[0,1], the bound is attained by an isotropic pair with a maximally entangled anchor. Taking α↑1α\uparrow1α↑1 recovers the recent sharp continuity bound of quantum conditional entropy by Berta et al. [arXiv:2607.24687]. The proof linearizes the relevant concave Rényi functional at a comparison point dictated by the isotropic equality family. Schmidt-rank domination extends the equality geometry to an arbitrary anchor state, after which trace-distance duality and a noncommutative calibration estimate control the perturbation and anchor term without weakening the sharp constant. The latter estimate requires matrix analysis and is assisted by ChatGPT 5.6 Sol.
We prove a variety of improved uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality. We then employ this classical result to derive a tight energy-constrained continuity bound for the von Neumann entropy. To deal with more general entropies in infinite dimensions,e.g.$\alpha −Reˊnyiand-Rényi and−Reˊnyiand\alpha $-Tsallis entropies, we develop a novel approximation scheme based on operator Hölder continuity estimates. Finally, we settle an open problem raised by Shirokov regarding the characterisation of states with finite entropy.
Abstract In this work, we prove uniform continuity bounds for entropic quantities related to the sandwiched Rényi divergences such as the sandwiched Rényi conditional entropy. We follow three different approaches: The first one is the “almost additive approach”, which exploits the sub-/superadditivity and joint concavity/convexity of the exponential of the divergence. In our second approach, termed the “operator space approach”, we express the entropic measures as norms and utilize their properties for establishing the bounds. These norms draw inspiration from interpolation space norms. We not only demonstrate the norm properties solely relying on matrix analysis tools but also extend their applicability to a context that holds relevance in resource theories. By this, we extend the strategies of Marwah and Dupuis as well as Beigi and Goodarzi employed in the sandwiched Rényi conditional entropy context. Finally, we merge the approaches into a mixed approach that has some advantageous properties and then discuss in which regimes each bound performs best. Our results improve over the previous best continuity bounds or sometimes even give the first continuity bounds available. In a separate contribution, we use the ALAFF method, developed in a previous article by some of the authors, to study the stability of approximate quantum Markov chains.
Uniform continuity bounds on entropies are generally expressed in terms of a single distance measure between probability distributions or quantum states, typically, the total variation- or trace distance. However, if an additional distance measure is known, the continuity bounds can be significantly strengthened. Here, we prove a tight uniform continuity bound for the Shannon entropy in terms of both the local- and total variation distances, sharpening an inequality in (Sason, 2013). We also obtain a uniform continuity bound for the von Neumann entropy in terms of both the operator norm- and trace distances. We then apply our results to compute upper bounds on channel capacities. We first refine the concept of approximate degradable channels by introducing(ε,ν)(\varepsilon ,\nu)(ε,ν)–degradable channels. These are$\varepsilon –closeindiamondnormand–close in diamond norm and–closeindiamondnormand\nu $–close in completely bounded spectral norm to their complementary channel when composed with a degrading channel. This leads to improved upper bounds to the quantum- and private classical capacities of such channels. Moreover, these bounds can be further improved by considering certain unstabilized versions of the above norms. We show that upper bounds on the latter can be efficiently expressed as semidefinite programs. As an application, we obtain a new upper bound on the quantum capacity of the qubit depolarizing channel.