A Matrix Convexity Approach to Some Celebrated Quantum Inequalities

Abstract

Some of the important inequalities associated with quantum entropy are immediate algebraic consequences of the Hansen-Pedersen-Jensen inequality. A general argument is given using matrix perspectives of operator convex functions. A matrix analogue of Maréchal's extended perspectives provides additional inequalities, including a $p+q\le 1$ result of Lieb.

8 pages

References in corpus (4)

• Jensen's Operator Inequality
• Multiplicativity of completely bounded p-norms implies a new additivity result
• Another Short and Elementary Proof of Strong Subadditivity of Quantum Entropy
• Extensions of Lieb's concavity theorem

Cited by in corpus (49)

• User-friendly tail bounds for sums of random matrices
• Quantum f-divergences and error correction
• Different quantum f-divergences and the reversibility of quantum operations
• Semidefinite approximations of the matrix logarithm
• Renyi generalizations of the conditional quantum mutual information
• A Unified Treatment of Convexity of Relative Entropy and Related Trace Functions, with Conditions for Equality
• Optimal Quantum Thermometry with Coarse-grained Measurements
• Device-independent lower bounds on the conditional von Neumann entropy
• Metric adjusted skew information: Convexity and restricted forms of superadditivity
• Rényi generalizations of quantum information measures
• Semi-definite programming and quantum information
• Approximate reversal of quantum Gaussian dynamics
• Operator extension of strong subadditivity of entropy
• Free Convex Algebraic Geometry
• Remarks on on Kim's Strong Subadditivity Matrix Inequality: Extensions and Equality Conditions
• Recoupling coefficients and quantum entropies
• Modulus of convexity for operator convex functions
• An inequality for expectation of means of positive random variables
• Geometric relative entropies and barycentric Rényi divergences
• Classical and nonclassical randomness in quantum measurements
• A new perspective on low-rank optimization
• Convexity of quantum ${\chi }^2$-divergence
• Perspectives and completely positive maps
• Convexity of quasi-entropy type functions: Lieb's and Ando's convexity theorems revisited
• Poincaré type inequalities for group measure spaces and related transportation cost inequalities
• Circular Free Spectrahedra
• Operator means in JB-algebras
• The Ramsey property for Operator spaces and noncommutative Choquet simplices
• T product Tensors Part I: Inequalities
• Pusz--Woronowicz's functional calculus revisited
• Tomita-Takesaki Modular Theory vs. Quantum Information Theory
• Yet Another Proof of the Joint Convexity of Relative Entropy
• Ruminations on Matrix Convexity and the Strong Subadditivity of Quantum Entropy
• Parameterized coherence measure
• Quantum Fokker-Planck Dynamics
• Noncommutative Perspectives of Operator Monotone Functions in Hilbert Spaces
• On trace-convex noncommutative polynomials
• Noncommutative partially convex rational functions
• Choi-Davis-Jensen's inequality without convexity
• Jointly convex mappings related to the Lieb's functional and Minkowski type operator inequalities
• Some operator convex functions of several variables
• Regular operator mappings and multivariate geometric means
• Operator log-convex functions and f-divergence functional
• Non-commutative f-divergence functional
• Operator Entropy Inequalities
• Hanner's Inequality For Positive Semidefinite Matrices
• Log-majorizations between quasi-geometric type means for matrices
• Ando-Hiai type inequalities for operator means and operator perspectives
• Variational Representations related to Quantum Rényi Relative Entropies