Sandwiched (quantum) α-Rényi divergence has been recently defined in the independent works of Wilde et al. [“Strong converse for the classical capacity of entanglement-breaking channels,” preprint arXiv:1306.1586 (2013)] and Müller-Lennert et al. [“On quantum Rényi entropies: a new definition, some properties and several conjectures,” preprint arXiv:1306.3142v1 (2013)]. This new quantum divergence has already found applications in quantum information theory. Here we further investigate properties of this new quantum divergence. In particular, we show that sandwiched α-Rényi divergence satisfies the data processing inequality for all values of α > 1. Moreover we prove that α-Holevo information, a variant of Holevo information defined in terms of sandwiched α-Rényi divergence, is super-additive. Our results are based on Hölder's inequality, the Riesz-Thorin theorem and ideas from the theory of complex interpolation. We also employ Sion's minimax theorem.
Sandwiched (quantum) α-Rényi divergence has been recently defined in the independent works of Wilde et al. [“Strong converse for the classical capacity of entanglement-breaking channels,” preprint arXiv:1306.1586 (2013)] and Müller-Lennert et al. [“On quantum Rényi entropies: a new definition, some properties and several conjectures,” preprint arXiv:1306.3142v1 (2013)]. This new quantum divergence has already found applications in quantum information theory. Here we further investigate properties of this new quantum divergence. In particular, we show that sandwiched α-Rényi divergence satisfies the data processing inequality for all values of α > 1. Moreover we prove that α-Holevo information, a variant of Holevo information defined in terms of sandwiched α-Rényi divergence, is super-additive. Our results are based on Hölder's inequality, the Riesz-Thorin theorem and ideas from the theory of complex interpolation. We also employ Sion's minimax theorem.
The doubly minimized Petz Rényi mutual information of order α is defined as the minimization of the Petz divergence of order α of a fixed bipartite quantum state relative to any product state. The doubly minimized sandwiched Rényi mutual information is defined analogously using the sandwiched divergence in place of the Petz divergence. In this work, we establish several properties of these two types of Rényi mutual information. In particular, for the Petz case, we prove additivity for α ∈ [1/2, 2]. For the sandwiched case, we establish a novel duality relation for α ∈ [2/3,∞] via Sion’s minimax theorem, and we subsequently use this duality relation to prove additivity for the same range of α. Previously, additivity for the sandwiched case was known only for α ∈ [1,∞], but it had been conjectured to hold for α ∈ [1/2,∞].
The doubly minimized Petz Rényi mutual information of order α is defined as the minimization of the Petz divergence of order α of a fixed bipartite quantum state relative to any product state. The doubly minimized sandwiched Rényi mutual information is defined analogously using the sandwiched divergence in place of the Petz divergence. In this work, we establish several properties of these two types of Rényi mutual information. In particular, for the Petz case, we prove additivity for α ∈ [1/2, 2]. For the sandwiched case, we establish a novel duality relation for α ∈ [2/3,∞] via Sion’s minimax theorem, and we subsequently use this duality relation to prove additivity for the same range of α. Previously, additivity for the sandwiched case was known only for α ∈ [1,∞], but it had been conjectured to hold for α ∈ [1/2,∞].
We introduce a new quantum Rényi divergence Dα# for α∈(1,∞) defined in terms of a convex optimization program. This divergence has several desirable computational and operational properties such as an efficient semidefinite programming representation for states and channels, and a chain rule property. An important property of this new divergence is that its regularization is equal to the sandwiched (also known as the minimal) quantum Rényi divergence. This allows us to prove several results. First, we use it to get a converging hierarchy of upper bounds on the regularized sandwiched α -Rényi divergence between quantum channels for α>1 . Second it allows us to prove a chain rule property for the sandwiched α -Rényi divergence for α>1 which we use to characterize the strong converse exponent for channel discrimination. Finally it allows us to get improved bounds on quantum channel capacities.
We provide a transparent, simple, and unified treatment of recent results on the equality conditions for the data-processing inequality of the sandwiched quantum Rényi divergence, including the statement that the equality in the data-processing implies recoverability via the Petz recovery map for the full range of the Rényi parameter α recently proven by Jenčová [J. Phys. A: Math. Theor. 50, 085303 (2017)]. We also obtain a new set of equality conditions, generalizing a previous result by Leditzky et al. [Lett. Math. Phys. 107, 61 (2017)].
The α -sandwiched Rényi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for α≥1/2 . In this article, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality for the α -sandwiched Rényi divergence for all α≥1/2 . For the range α∈[1/2,1) , our result provides the only condition for equality obtained thus far. To prove our result, we first consider the special case of partial trace and derive a condition for equality based on the original proof of the data processing inequality by Frank and Lieb (J Math Phys 54(12):122201, 2013) using a strict convexity/concavity argument. We then generalize to arbitrary quantum operations via the Stinespring Representation Theorem. As applications of our condition for equality in the data processing inequality, we deduce conditions for equality in various entropic inequalities. We formulate a Rényi version of the Araki–Lieb inequality and analyze the case of equality, generalizing a result by Carlen and Lieb (Lett Math Phys 101(1):1–11, 2012) about equality in the original Araki–Lieb inequality. Furthermore, we prove a general lower bound on a Rényi version of the entanglement of formation and observe that it is attained by states saturating the Rényi version of the Araki–Lieb inequality. Finally, we prove that the known upper bound on the entanglement fidelity in terms of the usual fidelity is saturated only by pure states.
Abstract The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f -divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz–Rényi relative entropies. In this paper, I introduce the optimized quantum f -divergence as a related generalization of quantum relative entropy. I prove that it satisfies the data processing inequality, and the method of proof relies upon the operator Jensen inequality, similar to Petz’s original approach. Interestingly, the sandwiched Rényi relative entropies are particular examples of the optimized f -divergence. Thus, one benefit of this paper is that there is now a single, unified approach for establishing the data processing inequality for both the Petz–Rényi and sandwiched Rényi relative entropies, for the full range of parameters for which it is known to hold. This paper discusses other aspects of the optimized f -divergence, such as the classical case, the classical-quantum case, and how to construct optimized f -information measures.