Sharp continuity bounds for entropy and conditional entropy

Abstract

Dear Editor,The von Neumann entropy captures many operational quantities in the quantum information theory such as quantum capacity of the communication channel. Von Neumann entropy is continuous and is represented by Fannes inequality, which was originally given in [1]. Quantum correlations such as entanglement and quantum discord , et al., are important resources in quantum information processing. In the last year enormous progress on the generation, concentration, detection and quantification of entanglement has been achieved [2]. Fannes inequality has many applications in the quantum information theory, such as the investigation of continuity of entanglement measures, including entanglement of formation, relative entropy of entanglement, squashed entanglement and conditional entanglement of mutual information, and the continuity of quantum channel capacities [3]. Recently Fannes inequality was improved to get a sharp one and it was also generalized to Tsallis entropy [4,5].However, in non-asymptotic settings, the natural quantities that arise are Rényi entropies [6] and the properties of Rényi entropies were also investigated in many papers, such as [7]. Rényi entropies have many applications, as in the case of one-shot problems, typically arising in cryptographic settings, the min-and max-entropies are widely used [6]. In [8], the authors found that Rényi-2 entropy was a proper measure of information for any multimode Gaussian state, and they defined and analyzed the measures of Gaussian entanglement and quantum correlation by using Rényi-2 entropy, and found its properties such as monogamy. In our work, we study the continuity property of Rényi-α entropy, which includes Rényi-2 entropy as a special case. Our result is also useful in studying the continuity of the entanglement measure of the Gaussian state in quantum harmonic systems.On the other hand, the authors found that Tsalli-2 entropy (i.e., linear entropy) was natural to define the measure of quantum correlation for the the discrete system [9]. They called this measure as linear discord and used conditional linear en-tropy to define the linear discord. They found that the linear discord has deep connection with the original discord defined by von Neumann entropy. Moreover, they gave the analytical formula for arbitrary 2 ⊗ d state of the linear discord. However, a question still remains open: if two states are close, is their linear discord also close to each other? In other words, is the linear discord continuous? For the original discord, the answer is affirmative, see [10]. For the linear discord, there is no answer yet. Hence it is worthwhile to study the continuity of conditional linear entropy.We have two aims in this work: first, we study the continuity estimation of the Rényi entropy and present a tight inequality relating the Rényi entropy difference of two quantum states to their trace norm distance, which includes the sharp Fannes inequality for von Neumann entropy as a special case. Second, we study the continuity of conditional linear en...

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