A generalization of the injectivity condition for Projected Entangled Pair States

Abstract

We introduce a family of tensor network states that we term semi-injective Projected Entangled-Pair States (PEPS). They extend the class of injective PEPS and include other states, like the ground states of the AKLT and the CZX models in square lattices. We construct parent Hamiltonians for which semi-injective PEPS are unique ground states. We also determine the necessary and sufficient conditions for two tensors to generate the same family of such states in two spatial dimensions. Using this result, we show that the third cohomology labeling of Symmetry Protected Topological phases extends to semi-injective PEPS.

63 pages

References in corpus (42)

• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Persistent entanglement in arrays of interacting particles
• String-net condensation: A physical mechanism for topological phases
• Entanglement spectrum of a topological phase in one dimension
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Thermodynamic limit of the density matrix renormalization for the spin-1 Heisenberg chain
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Classifying quantum phases using Matrix Product States and PEPS
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• DMRG and periodic boundary conditions: a quantum information perspective
• Criticality, the area law, and the computational power of PEPS
• Stripe order in the underdoped region of the two-dimensional Hubbard model
• Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
• Competing states in the t-J model: uniform d-wave state versus stripe state
• Exact and Asymptotic Measures of Multipartite Pure State Entanglement
• 2D symmetry protected topological orders and their protected gapless edge excitations
• Valence Bond Solids for Quantum Computation
• PEPS as ground states: degeneracy and topology
• Tensor-entanglement renormalization group approach to 2D quantum systems
• Solving Gapped Hamiltonians Locally
• Anyons and matrix product operator algebras
• Resonating valence bond states in the PEPS formalism
• Approximating Gibbs states of local Hamiltonians efficiently with PEPS
• Explicit tensor network representation for the ground states of string-net models
• Tensor renormalization of quantum many-body systems using projected entangled simplex states
• Simultaneous dimerization and SU(4) symmetry breaking of 4-color fermions on the square lattice
• A quantum version of Wielandt's inequality
• The spectral gap for some spin chains with discrete symmetry breaking
• Matrix Product States: Symmetries and Two-Body Hamiltonians
• Stochastic Error Cancellation in Analog Quantum Simulation
• Matrix Product Density Operators: Renormalization Fixed Points and Boundary Theories
• A canonical form for Projected Entangled Pair States and applications
• Matrix product operators for symmetry-protected topological phases: Gauging and edge theories
• Rigorous RG algorithms and area laws for low energy eigenstates in 1D
• Twisted Injectivity in PEPS and the Classification of Quantum Phases
• Chiral topological spin liquids with projected entangled pair states
• Chiral projected entangled-pair state with topological order
• Characterizing Topological Order with Matrix Product Operators
• Fundamental limitations in the purifications of tensor networks
• Entanglement, fractional magnetization and long-range interactions
• Tensor-entanglement renormalization group approach to topological phases

Cited by in corpus (27)

• Matrix Product States and Projected Entangled Pair States: Concepts, Symmetries, and Theorems
• Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter
• Fermionic projected entangled-pair states and topological phases
• Classifying phases protected by matrix product operator symmetries using matrix product states
• Normal projected entangled pair states generating the same state
• Universal measurement-based quantum computation in a one-dimensional architecture enabled by dual-unitary circuits
• Mathematical open problems in Projected Entangled Pair States
• An architecture for two-qubit encoding in neutral ytterbium-171 atoms
• Tensor network representations from the geometry of entangled states
• Locally Purified Density Operators for Symmetry-Protected Topological Phases in Mixed States
• The minimal canonical form of a tensor network
• Multi wavefunction overlap and multi entropy for topological ground states in (2+1) dimensions
• Matrix Product Density Operators: when do they have a local parent Hamiltonian?
• Higher Berry Curvature from the Wave function II: Locally Parameterized States Beyond One Dimension
• Charting the space of ground states with tensor networks
• A $H^3(G,\mathbb{T})$-valued index of symmetry protected topological phases with on-site finite group symmetry for two-dimensional quantum spin systems
• An invariant of symmetry protected topological phases with on-site finite group symmetry for two-dimensional Fermion systems
• The resource theory of tensor networks
• Tangent space generators of matrix product states and exact Floquet quantum scars
• Gauging quantum states with non-anomalous matrix product operator symmetries
• A generic quantum Wielandt's inequality
• Fractional domain wall statistics in spin chains with anomalous symmetries
• The Tensor as an Informational Resource
• Non-onsite symmetries and quantum teleportation in split-index matrix product states
• Les Houches Lecture Notes on Tensor Networks
• Tensor network representations of fermionic crystalline topological phases on two-dimensional lattices
• Equivalence of residual entropy of hexagonal and cubic ices from tensor network methods