Abstract
The study of generic properties of quantum states has led to an abundance of insightful results. A meaningful set of states that can be efficiently prepared in experiments are ground states of gapped local Hamiltonians, which are well approximated by matrix product states. In this work, we introduce a picture of generic states within the trivial phase of matter with respect to their non-equilibrium and entropic properties: We do so by rigorously exploring non-translation-invariant matrix product states drawn from a local i.i.d. Haar-measure. We arrive at these results by exploiting techniques for computing moments of random unitary matrices and by exploiting a mapping to partition functions of classical statistical models, a method that has lead to valuable insights on local random quantum circuits. Specifically, we prove that such disordered random matrix product states equilibrate exponentially well with overwhelming probability under the time evolution of Hamiltonians featuring a non-degenerate spectrum. Moreover, we prove two results about the entanglement Renyi entropy: The entropy with respect to sufficiently disconnected subsystems is generically extensive in the system-size, and for small connected systems the entropy is almost maximal for sufficiently large bond dimensions.
Highlights
The application of random matrix theory to the study of interacting quantum many-body systems has proven to be a fruitful endeavor, in various readings
A meaningful set of states that can be efficiently prepared in experiments are ground states of gapped local Hamiltonians, which are well approximated by matrix product states
We arrive at these results by exploiting techniques for computing moments of random unitary matrices and by exploiting a mapping to partition functions of classical statistical models, a method that has lead to valuable insights on local random quantum circuits
Summary
We start by stating the underlying model of random quantum states used throughout this work. Consider an arbitrary MPS vector [32,33] with periodic boundary conditions and n constituents of local dimension d. Such a state vector can be written as. We can construct MPSs with periodic boundary conditions analogously This motivates a very natural probability measure. We denote the resulting measure as μd,n,D Note that this definition can be regarded as drawing the A(i) tensor cores uniformly from the Stiefel manifold of isometries. This probability measure makes a lot of sense: it is a distribution over random disordered, nontranslationinvariant quantum states.
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