Matrix Product Representation of Locality Preserving Unitaries

Abstract

The matrix product representation provides a useful formalism to study not only entangled states, but also entangled operators in one dimension. In this paper, we focus on unitary transformations and show that matrix product operators that are unitary provides a necessary and sufficient representation of 1D unitaries that preserve locality. That is, we show that matrix product operators that are unitary are guaranteed to preserve locality by mapping local operators to local operators while at the same time all locality preserving unitaries can be represented in a matrix product way. Moreover, we show that the matrix product representation gives a straight-forward way to extract the GNVW index defined in Ref.\cite{Gross2012} for classifying 1D locality preserving unitaries. The key to our discussion is a set of `fixed point' conditions which characterize the form of the matrix product unitary operators after blocking sites. Finally, we show that if the unitary condition is relaxed and only required for certain system sizes, the matrix product operator formalism allows more possibilities than locality preserving unitaries. In particular, we give an example of a simple matrix product operator which is unitary only for odd system sizes, does not preserve locality and carries a `fractional' index as compared to their locality preserving counterparts.

arXiv comment14 pages

References in corpus (22)

• Efficient classical simulation of slightly entangled quantum computations
• Entanglement spectrum of a topological phase in one dimension
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Symmetry protection of topological order in one-dimensional quantum spin systems
• Classifying quantum phases using Matrix Product States and PEPS
• 2D symmetry protected topological orders and their protected gapless edge excitations
• Matrix product operator representations
• Classification of Interacting Topological Floquet Phases in One Dimension
• Classification of topological phases in periodically driven interacting systems
• Phase Structure of 1d Interacting Floquet Systems I: Abelian SPTs
• 1D Many-body localized Floquet systems II: Symmetry-Broken phases
• Anyons and matrix product operator algebras
• Index theory of one dimensional quantum walks and cellular automata
• Matrix Product Unitaries: Structure, Symmetries, and Topological Invariants
• A Matrix-Product-Operator Approach to the Nonequilibrium Steady State of Driven-Dissipative Quantum Arrays
• Abelian Floquet symmetry-protected topological phases in one dimension
• Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods
• Out of equilibrium dynamics with Matrix Product States
• Matrix product operators for symmetry-protected topological phases: Gauging and edge theories
• Twisted Injectivity in PEPS and the Classification of Quantum Phases
• Matrix Product State and mean field solutions for one-dimensional systems can be found efficiently

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