Matrix product states and the decay of quantum conditional mutual information

Abstract

A uniform matrix product state defined on a tripartite system of spins, denoted by $ABC,$ is shown to be an approximate quantum Markov chain when the size of subsystem $B,$ denoted $|B|,$ is large enough. The quantum conditional mutual information (QCMI) is investigated and proved to be bounded by a function proportional to $\exp (-q(|B|-K)+2K\ln |B|)$, with $q$ and $K$ computable constants. The properties of the bounding function are derived by a new approach, with a corresponding improved value given for its asymptotic decay rate $q$. We show the improved value of $q$ to be optimal. Numerical investigations of the decay of QCMI are reported for a collection of matrix product states generated by selecting the defining isometry with respect to Haar measure.

16+10 pages, 10 figures

References in corpus (13)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Area laws in quantum systems: mutual information and correlations
• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Sequential Generation of Matrix-Product States in Cavity QED
• Random Unitaries Give Quantum Expanders
• Generating random quantum channels
• Mixed s-sourcery: Building many-body states using bubbles of Nothing
• Universality of spectra for interacting quantum chaotic systems
• On orthogonal bases in the Hilbert-Schmidt space of matrices
• Decay of quantum conditional mutual information for purely generated finitely correlated states
• Matrix Product Density Operators: when do they have a local parent Hamiltonian?
• Markovian Matrix Product Density Operators : Efficient computation of global entropy

Cited by in corpus (5)

• Reviving the Lieb-Schultz-Mattis Theorem in Open Quantum Systems
• Detecting Quantum Anomalies in Open Systems
• Many-body entropies and entanglement from polynomially-many local measurements
• Universal Characterization of Quantum Many-Body States through Local Information
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures