An ergodic theorem for quantum processes with applications to matrix product states

Abstract

Any discrete quantum process is represented by a sequence of quantum channels. We consider ergodic quantum processes obtained by a map that takes the points along the trajectory of a discrete ergodic dynamical system to the space of quantum channels. Under natural irreducibility conditions, we obtain a theorem showing that the state under such a process converges exponentially fast to an ergodic sequence depending on the process, but independent of the initial state. As an application, we describe the thermodynamic limit of ergodic matrix product states and prove that the 2-point correlations of local observables in such states decay exponentially with their distance in the bulk.

18 pages + 1 figure + references. Lemma 2.1 added, and some text revised. To appear in Commun. Math. Phys

References in corpus (4)

• Double-slit photoelectron interference in strong-field ionization of the neon dimer
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
• Stability of Periodically Driven Topological Phases against Disorder

Cited by in corpus (8)

• Typical Correlation Length of Sequentially Generated Tensor Network States
• Law of large numbers and central limit theorem for ergodic quantum processes
• Classical restrictions of generic matrix product states are quasi-locally Gibbsian
• Quenched large deviations of Birkhoff sums along random quantum measurements
• Asymptotic Purification of Quantum Trajectories under Random Generalized Measurements
• Limit theorems for Quantum Trajectories
• The product structure of MPS-under-permutations
• A Multiplicative Ergodic Theorem for Bistochastic Ergodic Quantum Processes with Applications to Entanglement