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Entanglement Reference

These APIs support Schmidt decompositions and entropy calculations for vectors and MPS objects.

EDKit.ent_S — Function

ent_S(v::AbstractVector, Aind::AbstractVector{<:Integer}, b::AbstractBasis; α::Real=1, cutoff::Real=1e-20)

Compute the bipartite entanglement entropy of a state represented in basis b.

The singular values returned by ent_spec are squared into Schmidt probabilities before being passed to entropy.

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ent_S(v, Aind, L; α=1, cutoff=1e-20)

Convenience overload that infers a TensorBasis from the vector length and system size L.

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ent_S(ψ::MPS, b::Integer)

Return entanglement entropy between site b and b+1.

Arguments:

  • ψ: MPS
  • b: Link index

Returns:

  • The entropy associated with the cut at bond b.
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ent_specs(ψ::MPS)

Return entanglement entropies along the full chain.

Arguments:

  • ψ: MPS

Returns:

  • S: Vector of entropy
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EDKit.ent_spec — Function

ent_spec(v::AbstractVector, Aind::AbstractVector{<:Integer}, b::AbstractBasis)

Compute the Schmidt singular values of a state across a bipartition.

Returns:

  • The singular values of the Schmidt matrix produced by schmidt.
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EDKit.entropy — Function

entropy(s::AbstractVector{<:Real}; α::Real=1, cutoff::Real=1e-20)

Compute the entropy of Schmidt values.

Inputs:

  • s : Schmidt values.
  • α : Renyi index.
  • cutoff: Cutoff of the Schmidt values.

Outputs:

  • S: Entanglement entropy.
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entropy(dm::DensityMatrix; α=1, cutoff=1e-20)

Compute the entropy of a density matrix from its eigenvalues.

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EDKit.schmidt — Function
schmidt(v, Ainds, b::AbelianBasis; B1=nothing, B2=nothing)

Construct the Schmidt matrix of a state represented in an AbelianBasis.

Unlike onsite bases, each basis coefficient must be expanded over the entire Abelian orbit with the correct symmetry phase before it contributes to the bipartite decomposition.

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schmidt(v::AbstractVector, Ainds::AbstractVector{<:Integer}, b::AbstractOnsiteBasis)

Schmidt decomposition of state v, with respect to given lattice bipartition.

Inputs:

  • v : State represented by a (abstract) vector.
  • Ainds: List of indices in subsystem A, the remaining indices are regarded as subsystem B.
  • b : Basis.

Outputs:

  • S: Matrix S in the decomposition: |v⟩ = Sᵢⱼ |Aᵢ⟩|Bⱼ⟩.
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Schmidt decomposition specialized to TranslationalBasis.

Each reduced-basis coefficient is expanded across the full translation orbit with the appropriate momentum phase before contributing to the bipartite matrix.

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Schmidt decomposition specialized to FlipBasis.

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Schmidt decomposition specialized to ParityBasis.

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Schmidt decomposition specialized to ParityFlipBasis.

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