Computable Rényi mutual information: Area laws and correlations

Abstract

The mutual information is a measure of classical and quantum correlations of great interest in quantum information. It is also relevant in quantum many-body physics, by virtue of satisfying an area law for thermal states and bounding all correlation functions. However, calculating it exactly or approximately is often challenging in practice. Here, we consider alternative definitions based on Rényi divergences. Their main advantage over their von Neumann counterpart is that they can be expressed as a variational problem whose cost function can be efficiently evaluated for families of states like matrix product operators while preserving all desirable properties of a measure of correlations. In particular, we show that they obey a thermal area law in great generality, and that they upper bound all correlation functions. We also investigate their behavior on certain tensor network states and on classical thermal distributions.

arXiv comment15 pages, 2 figures, comments welcome

References in corpus (58)

• The density-matrix renormalization group in the age of matrix product states
• The density-matrix renormalization group
• Entanglement Entropy and Quantum Field Theory
• Efficient classical simulation of slightly entangled quantum computations
• Entanglement entropy and conformal field theory
• An Area Law for One Dimensional Quantum Systems
• Matrix product states represent ground states faithfully
• On quantum Renyi entropies: a new generalization and some properties
• Area laws in quantum systems: mutual information and correlations
• On the quantum, classical and total amount of correlations in a quantum state
• Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy
• Min- and Max- Relative Entropies and a New Entanglement Monotone
• Floquet-Magnus Theory and Generic Transient Dynamics in Periodically Driven Many-Body Quantum Systems
• Measuring Renyi Entanglement Entropy with Quantum Monte Carlo
• Exponentially slow heating in periodically driven many-body systems
• Matrix product operator representations
• Quantum Information Processing with Finite Resources -- Mathematical Foundations
• Solving Gapped Hamiltonians Locally
• The power of quantum systems on a line
• Quantum hypothesis testing and the operational interpretation of the quantum Renyi relative entropies
• The Quantum Reverse Shannon Theorem based on One-Shot Information Theory
• Approximating Gibbs states of local Hamiltonians efficiently with PEPS
• alpha-z-relative Renyi entropies
• Quantum Monte Carlo calculation of entanglement Renyi entropies for generic quantum systems
• Infinite matrix product states, Conformal Field Theory and the Haldane-Shastry model
• Quantum Entanglement of Interacting Fermions in Monte Carlo
• Mutual information after a local quench in conformal field theory
• Correlation Detection and an Operational Interpretation of the Renyi Mutual Information
• Relative Entropies in Conformal Field Theory
• Euclidean operator growth and quantum chaos
• Strong converse exponent for classical-quantum channel coding
• Purifications of multipartite states: limitations and constructive methods
• Quantum message compression with applications
• Matrix product operators and states: NP-hardness and undecidability
• On Pinsker's Type Inequalities and Csiszar's f-divergences. Part I: Second and Fourth-Order Inequalities
• On Variational Expressions for Quantum Relative Entropies
• Principles of Quantum Communication Theory: A Modern Approach
• Finite Temperature Critical Behavior of Mutual Information
• Shannon and Rényi mutual information in quantum critical spin chains
• Quantum Corrections to Holographic Mutual Information
• Improved thermal area law and quasi-linear time algorithm for quantum Gibbs states
• Connecting global and local energy distributions in quantum spin models on a lattice
• Subsystem Trace Distance in Quantum Field Theory
• Large deviations in quantum lattice systems: one-phase region
• Universal behavior of the Shannon and Rényi mutual information of quantum critical chains
• Fundamental limitations in the purifications of tensor networks
• Dynamics of quantum information in many-body localized systems
• Efficient description of many-body systems with Matrix Product Density Operators
• Strong converse theorems using Rényi entropies
• Temperature driven quenches in the Ising model: appearance of negative Rényi mutual information
• Defining quantum divergences via convex optimization
• Mutual information area laws for thermal free fermions
• Coding theorems for compound problems via quantum Rényi divergences
• Strong converse exponents for the feedback-assisted classical capacity of entanglement-breaking channels
• Rényi Mutual Information in Holographic Warped CFTs
• Dynamics and statistics in the operator algebra of quantum mechanics
• Locality estimes for complex time evolution in 1D
• Performance of the rigorous renormalization group for first order phase transitions and topological phases

Cited by in corpus (16)

• Quantum many-body systems in thermal equilibrium
• Exponential decay of mutual information for Gibbs states of local Hamiltonians
• Rényi Mutual Information in Quantum Field Theory
• Continuity of quantum entropic quantities via almost convexity
• Quantifying information scrambling via Classical Shadow Tomography on Programmable Quantum Simulators
• Typical Correlation Length of Sequentially Generated Tensor Network States
• Statistical Mechanics of Monitored Dissipative Random Circuits
• General Continuity Bounds for Quantum Relative Entropies
• Rényi mutual information in quantum field theory, tensor networks, and gravity
• Quench dynamics of Rényi negativities and the quasiparticle picture
• Locality, Correlations, Information, and non-Hermitian Quantum Systems
• Thermal Area Law for Lattice Bosons
• Extensive Long-Range Entanglement at Finite Temperatures from a Nonequilibrium Bias
• Exact quench dynamics of symmetry resolved entanglement in a free fermion chain
• Conditional Independence of 1D Gibbs States with Applications to Efficient Learning
• Geometric Rényi mutual information induced by localized particle excitations in quantum field theory