Min- and Max-Relative Entropies and a New Entanglement Monotone

By N. Datta. IEEE Transactions on Information Theory, 2008

481 citations recorded by the source corpus.

Two new relative entropy quantities, called the min- and max-relative entropies, are introduced and their properties are investigated. The well-known min- and max-entropies, introduced by Renner, are obtained from these. We define a new entanglement monotone, which we refer to as the max-relative entropy of entanglement, and which is an upper bound to the relative entropy of entanglement. We also generalize the min- and max-relative entropies to obtain smooth min-and max-relative entropies. These act as parent quantities for the smooth Renyi entropies (ETH Zurich, Ph.D. dissertation, 2005), and allow us to define the analogues of the mutual information, in the smooth Renyi entropy framework. Further, the spectral divergence rates of the information spectrum approach are shown to be obtained from the smooth min- and max-relative entropies in the asymptotic limit.

Subfields: Artificial Intelligence, Electrical and Electronic Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7c1df2356b339e6cf2500076f50f0426f679ed8a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Omar Fawzi
2. Renato Renner
3. Gilad Gour
4. Marco Tomamichel
5. Mario Berta
6. Matthias Christandl
7. Fernando G. S. L. Brandão
8. Frédéric Dupuis

In Electrical and Electronic Engineering

1. Renato Renner
2. Omar Fawzi
3. Marco Tomamichel
4. Mario Berta
5. Gilad Gour
6. Matthias Christandl

In Statistical and Nonlinear Physics

1. Omar Fawzi
2. Renato Renner
3. Marco Tomamichel
4. Gilad Gour
5. Heng Fan
6. Mario Berta
7. Matthias Christandl
8. Robert W. Spekkens

Citations in the corpus

• Quantum coherence and geometric quantum discord (2017, background): “It is also an important concept in quantum information science (Brandão and Datta, 2011; Buscemi and Datta, 2010; Datta, 2009b,c).”
• The Quantum Reverse Shannon Theorem Based on One-Shot Information Theory (2009, background): “For a more detailed discussion of the smooth entropy formalism we refer to [12, 17, 18, 38, 39].”
• On quantum Rényi entropies: A new generalization and some properties (2013, background): “It is given as e Moreover, the specialization α = is related to the fidelity, where The expression lim α → 0 D α ( ρ σ ) = − log Tr(Π ρ σ ), where Π ρ denotes the projector to the support of ρ is introduced by Datta [9, Def.”
• Exponential Decay of Correlations Implies Area Law (2012, unknown): “[63] we define the max-relative entropy as Smax(ρ||σ) := min{λ : ρ ≤ 2σ} (B3) and the ε-smooth relative entropy as S max(ρ||σ) := min ρ̃∈Bε(ρ) Smax(ρ̃||σ).”
• Quantum Conditional Mutual Information and Approximate Markov Chains (2014, methodology): “he marginal on C. Appendices A One-shot relative entropies In this appendix, we briefly review the generalised relative entropy introduced in [54, 22] and the smooth max-relative entropy introduced in [20] (we will use a slightly modified variant defined in [51, 50]). Definition A.1. For any two non-neg”
• The resource theory of informational nonequilibrium in thermodynamics (2013, background): “Finally, in various places (in particular in our discussion of measures of nonuniformity) we make use of work concerning the zoo of generalized entropies (Datta, 2009; van Erven and Harremoës, 2010; Gorban et al., 2010; Renner, 2008; Renner and Wolf, 2004).”
• Leftover Hashing Against Quantum Side Information (2010, unknown): “We follow [31] and first introduce the max relative entropy.”