We prove the sharp uniform continuity bound for quantum conditional entropy. If two bipartite states are at trace distance at most $δ$ and $d=\dim A$ , the optimal dimension-only modulus of continuity is $h_2(δ)+δ\log(d^2-1)$ up to $δ=1-d^{-2}$ and $2\log d$ thereafter, where $h_2$ denotes the binary entropy. When $\dim B\ge d$ , this bound is tight for every $δ\in[0,1]$ . 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.We prove the sharp uniform continuity bound for quantum conditional entropy. If two bipartite states are at trace distance at most $δ$ and $d=\dim A$, the optimal dimension-only modulus of continuity is $h_2(δ)+δ\log(d^2-1)$ up to $δ=1-d^{-2}$ and $2\log d$ thereafter, where $h_2$ denotes the binary entropy. When $\dim B\ge d$, this bound is tight for every $δ\in[0,1]$. 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.

quant-pharXiv:2607.24687Jul 28, 20260 citations

Optimal uniform continuity bound for conditional entropy of classical–quantum states

Mark M. Wilde

arXiv:1909.01755Quantum Information ProcessingJan 2, 202015 citations

Sharp continuity bounds for entropy and conditional entropy

ZhiHua Chen, Zhihao Ma, Ismail Nikoufar +1 more

arXiv:1701.02398Science China Physics Mechanics and AstronomyNov 24, 201619 citations

A Tight Uniform Continuity Bound for the Arimoto-Rényi Conditional Entropy and its Extension to Classical-Quantum States

Michael G. Jabbour, Nilanjana Datta

We prove a tight uniform continuity bound for Arimoto’s version of the conditional$\alpha $-Rényi entropy for the range$ \alpha \in [0, 1 $). This definition of the conditional$ \alpha $-Rényi entropy is the most natural one among the multiple forms which exist in the literature, since it satisfies two desirable properties of a conditional entropy, namely, the fact that conditioning reduces entropy, and that the associated reduction in uncertainty cannot exceed the information gained by conditioning. Furthermore, it has found interesting applications in various information theoretic tasks such as guessing with side information and sequential decoding. This conditional entropy reduces to the conditional Shannon entropy in the limit$ \alpha \to 1 $, and this in turn allows us to recover the recently obtained tight uniform continuity bound for the latter from our result. Finally, we apply our result to obtain a tight uniform continuity bound for the conditional$ \alpha $-Rényi entropy of a classical-quantum state, for$ \alpha $in the same range as above. This again yields the corresponding known bound for the conditional entropy of the state in the limit$ \alpha \to 1$.We prove a tight uniform continuity bound for Arimoto’s version of the conditional$\alpha $-Rényi entropy for the range$\alpha \in [0, 1$). This definition of the conditional$\alpha $-Rényi entropy is the most natural one among the multiple forms which exist in the literature, since it satisfies two desirable properties of a conditional entropy, namely, the fact that conditioning reduces entropy, and that the associated reduction in uncertainty cannot exceed the information gained by conditioning. Furthermore, it has found interesting applications in various information theoretic tasks such as guessing with side information and sequential decoding. This conditional entropy reduces to the conditional Shannon entropy in the limit$\alpha \to 1$, and this in turn allows us to recover the recently obtained tight uniform continuity bound for the latter from our result. Finally, we apply our result to obtain a tight uniform continuity bound for the conditional$\alpha $-Rényi entropy of a classical-quantum state, for$\alpha $in the same range as above. This again yields the corresponding known bound for the conditional entropy of the state in the limit$\alpha \to 1$.

arXiv:2007.05049IEEE Transactions on Information TheoryJan 13, 202211 citations

Purity based continuity bounds for quantum information measures

Komal Kumar, Nirman Ganguly

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.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.

arXiv:2306.16631Jun 29, 20230 citations

Quantum entropies, relative entropies, and related preserver problems

Dániel Virosztek

Jan 1, 20164 citations

We prove the sharp uniform continuity bound for quantum conditional entropy. If two bipartite states are at trace distance at most

δ

and

d=\dim A

, the optimal dimension-only modulus of continuity is

h_2(δ)+δ\log(d^2-1)

up to

δ=1-d^{-2}

and

2\log d

thereafter, where

h_2

denotes the binary entropy. When

\dim B\ge d

, this bound is tight for every

δ\in[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.

We prove a tight uniform continuity bound for Arimoto’s version of the conditional$\alpha

-Rényi entropy for the range

\alpha \in [0, 1

). This definition of the conditional

\alpha

-Rényi entropy is the most natural one among the multiple forms which exist in the literature, since it satisfies two desirable properties of a conditional entropy, namely, the fact that conditioning reduces entropy, and that the associated reduction in uncertainty cannot exceed the information gained by conditioning. Furthermore, it has found interesting applications in various information theoretic tasks such as guessing with side information and sequential decoding. This conditional entropy reduces to the conditional Shannon entropy in the limit

\alpha \to 1

, and this in turn allows us to recover the recently obtained tight uniform continuity bound for the latter from our result. Finally, we apply our result to obtain a tight uniform continuity bound for the conditional

\alpha

-Rényi entropy of a classical-quantum state, for

\alpha

in the same range as above. This again yields the corresponding known bound for the conditional entropy of the state in the limit

\alpha \to 1$.