API reference

AlternatingSeparableLMO{T, N, MB <: AbstractMatrix{Complex{T}}} <: SeparableLMO{T, N}

AlternatingSeparableLMO implements compute_extreme_point(lmo, direction) which returns a pure product state used in Frank-Wolfe algorithms. The method used is an alternating algorithm starting from random pure states on each party and alternatively optimizing each reduced state via an eigendecomposition.

Type parameters:

• T: element type of the correlation tensor
• N: number of parties
• MB: type of the matrix basis

Fields:

• dims: dimensions of the reduced state on each party
• matrix_basis: matrix basis of the correlation tensor
• max_iter: maximum number of alternation steps
• threshold: threshold to stop the alternation
• nb: number of random rounds to find the possible global optimal
• workspace: contains fields pre-allocated performance-critical functions
• tmp: temporary vector for fast scalar products of bipartite tensors

CrossPolytopeSubdivision{T} <: ApproximationQuantumStates{T}

Approximates the complex sphere in dimension d via an edgewise subdivision of the cross-polytope on the real sphere in dimension 2d-1.

EnumeratingSeparableLMO{T, N, MB <: AbstractMatrix{Complex{T}}} <: SeparableLMO{T, N}

EnumeratingSeparableLMO implements compute_extreme_point(lmo, direction) which returns a pure product state used in Frank-Wolfe algorithms. The method used is the enumeration of a fine approximation of the sets of states on all parties except the first one, for which an eigendecomposition suffices.

Type parameters:

• T: element type of the correlation tensor
• N: number of parties
• MB: type of the matrix basis

Fields:

• dims: dimensions of the reduced state on each party
• matrix_basis: matrix basis of the correlation tensor
• workspace: contains fields pre-allocated performance-critical functions
• tmp: temporary vector for fast scalar products of bipartite tensors

KSeparableLMO{T, N, LMO} <: SeparableLMO{T, N}

KSeparableLMO implements compute_extreme_point(lmo::LMO, direction) which returns a pure product state by considering all K-partite partitions by iterating over all LMO{T, K}.

Type parameters:

• T: element type of the correlation tensor
• N: number of parties
• MB: type of the matrix basis
• LMO: type of the LMO for k-partite separability

Fields:

• dims: dimensions of the reduced state on each party
• matrix_basis: matrix basis of the correlation tensor
• lmos: list of LMOs for all K-partitions
• partitions: list of K-partite partitions

PolytopePhasedComplexSphere{T} <: ApproximationQuantumStates{T}

Approximates the complex sphere in dimension d via a polytope on the real sphere in dimension d and d-1 phases.

PolytopeReducedComplexSphere{T} <: ApproximationQuantumStates{T}

Approximates the complex sphere in dimension d via a polytope on the real sphere in dimension 2d-1.

PureState{T, N} <: AbstractArray{T, N}

Represents a pure product state. Each subsystem is a pure state stored as a tensor PureState.tensors[n].

Workspace{T, N}

Structure for initial pre-allocation of performance-critical functions.

_correlation_tensor_ket!(tensor::Vector{T}, φ::Vector{Complex{T}}, matrix_basis)

Convert a ket φ to a correlation tensor Vector{T}, with same subspace dimension d.

_eigmin!(ket::Vector, matrix::Matrix)

Computes the minimal real eigenvalue and updates ket in place The variable matrix of size d × d also gets overwritten. For BLAS-compatible types, uses LAPACK.syev! for d ≤ 5 and LAPACK.syevr! otherwise. For other types, falls back to eigen!.

_reduced_tensor!(tensor::Vector{T}, pure_tensors::NTuple{N, Vector{T}}, dir::Array{T, N}, j::Int, s::Vector{Int} = setdiff(1:N, j)) where {T <: Real, N}

Computes the correlation tensor of the j-th subsystem (the tensor-version of the partial trace). When N=2, j=1, computes ⟨ϕ2|dir|ϕ2⟩ for dir ∈ H₁ ⊗ H₂

correlation_tensor(ρ::Matrix{T}, dims::NTuple{N, Int})

Convert a dimension-(a)symmetry density matrix ρ to a correlation tensor Array{T, N}, with subspace dimensions dims.

function density_matrix(tensor::Vector{T}, dims::NTuple{N,Int}, matrix_basis = _gellmann(T, dims)) where {T <: Real} where {N}

Convert tensors of pure states to density matrix (for eigendecomposition).

entangled_bound_with_white_noise(ρ::AbstractMatrix{CT}, dims::NTuple{N, Int}; atol = 1e-4) where {CT <: Number, N}

Given a quantum state ρ, using PPT criterion calculate the upper bound for the amount of white noise it can tolerate such that ρ = (1-p) * ρ + p * I/d is entangled. Note this is corresponding to the fully separability. Returns the upper bound p.

entanglement_detection(ρ::AbstractMatrix{T}, dims::NTuple{N, Int}; measure, algorithm, kwargs...) where {T <: Number, N}

Given a quantum state ρ with subsystem dimensions as dims, make a judgment about whether it is entangled or separable. If the argument dims is omitted equally-sized subsystems are assumed, which is solving on the symmetry bipartite separable space.

Returns a named tuple (ent, atoms, witness) with:

• ent true/false/nothing, the judgment as entangled/separable/can't tell
• witness the witness for the entangled ρ, nothing for separable ρ
• decompose a tuple with weights and pure separable states, the best separable decomposition for separable ρ or for the closest separable state of entangled ρ

entanglement_robustness(ρ_p::Function, p_list::Array{Vector{T}}, dims::NTuple{N, Int}; monotone, robust_monitor, kwargs...)

Decide the entanglement and separable regions for a family of quantum state ρ_p.

Inputs:

• ρ_p the function for a family of quantum states
• p_list the parameter region for ρ_p(p)
• monotone the monotony of the function if with only one parameter (default: true)
• robust_monitor the monitor for the calculation of the robustness problem

Returns a named tuple (ent_range, nan_range, sep_range) with:

• ent_range the parameter range for p such that ρ_p(p) is entangled.
• nan_range the parameter range for p such that ρ_p(p) can not be decided.
• sep_range the parameter range for p such that ρ_p(p) is separable.

entanglement_witness(ρ::AbstractMatrix, σ::AbstractMatrix; dims, kwargs...)

Given a direction from a state σ in the separable space to a entangled state ρ, find the best witness that detect the entanglement of the state along this direction:

W = (σ-ρ + Tr[σ(ρ-σ)]*I)/||ρ-σ||

which satisfies ∀σ ∈ SEP, Tr(Wσ) ≥ 0, and ∃ρ, Tr(Wρ) < 0.

Returns a named tuple (W, α, ε, σ, ϕ) with:

• W the witness operator
• α = Tr[ϕ(ρ-σ)]/||ρ-σ||, the overlap
• ε the ε-net, normalized by ||ρ-σ||
• β = Tr[σ(ρ-σ)]/||ρ-σ||, the overlap
• σ decide the direction from σ to ρ
• ϕ the closest pure separable state

entanglement_witness(ρ::AbstractMatrix{T}; dims, kwargs...)

Given a quantum state ρ, find the best witness that detect the entanglement of the state, defined by

W = (σ-ρ + Tr[σ(ρ-σ)]*I)/||ρ-σ||

which satisfies ∀σ ∈ SEP, Tr(Wσ) ≥ 0, and ∃ρ, Tr(Wρ) < 0.

Returns a tuple (W, σ, α) with:

• W the witness operator
• σ the closest pure separable state
• α = Tr[σ(ρ-σ)], the overlap

separable_ball_criterion(ρ, σ, dims; kwargs...)

Given a quantum state ρ and a separable state σ, define a quantum state

ρ' = (1+t)/t * ρ - 1/t * σ, s.t. ρ = t/(1-t) * ρ' + 1/(1-t) * σ.

By confirming ρ' inside a separable ball to confirm the separability of ρ.

Returns a named tuple (sep, atoms, witness) with:

• sep true/false, the judgment of separability
• p = t/(1-t), the weight for ρ'

separable_ball_criterion(ρ, dims; kwargs...)

Given a quantum state ρ with subsystem dimensions as dims, define a quantum state

ρ' = (1+t)/t * ρ - 1/t * σ, s.t. ρ = t/(1-t) * ρ' + 1/(1-t) * σ.

where σ is the closest separable pure state. By confirming ρ' inside a separable ball to confirm the separability of ρ.

Returns a named tuple (sep, atoms, witness) with:

• sep true/false, the judgment of separability
• p = t/(1-t), the weight for ρ'

separable_ball_membership(ρ::AbstractMatrix{T}, dims::NTuple{N, Int}; kwargs...)

Judges whether a given quantum state ρ in a separable ball with subsystem dimensions as dims.

separable_ball_radius(::Type{T}, dims::NTuple{N, Int})

Lower bound of the radius of a separable ball, centered around the maximally mixed state.

Reference:

• N-qubits
• N-qudits
• general case

separable_bound_with_white_noise(::Type{T}, dims::NTuple{N, Int}, ent_bound::T, distance::T) where {T <: Real, N}

Consider a quantum state mixed with white noise as ρ(p) = (1-p) * ρ + p * I/d for the separability problem.

Given

• dims for the structure of the separable space
• ent_bound the noise level such that ρ(ent_bound) could be entangled (not necessary)
• distance = min{σ ∈ SEP}||ρ(`entbound`) - σ||, the distance between the state and the separable space

Return the noise level such that ρ(sep_bound) must be separable.

separable_distance(ρ::AbstractMatrix{CT}, dims::NTuple{N, Int}, lmo::LMO=AlternatingSeparableLMO(float(real(CT)), dims); measure, fw_algorithm, kwargs...)
separable_distance(C::Array{T, N}; matrix_basis, measure, fw_algorithm, kwargs...)

Computes the distance between the quantum density matrix ρ and the separable space under a specific measure via a specific fw_algorithm:

f(ρ) = min_{σ ∈ SEP} g(ρ,σ)

For the density matrix ρ, if the argument dims is omitted equally-sized subsystems are assumed, which is solving on the symmetry bipartite separable space. For the correlation tensor C, if the argument matrix_basis is omitted, Gell-Mann matrix is assumed, which is Pauli basis for qubit systems.

The quantum state can also be given by a correlation tensor C corresponding to the experimental data from a set of (over-)completed matrix_basis. If the argument matrix_basis is omitted the generalized Gell-Mann basis are assumed.

The measure g(ρ,σ) can be set as

• "2-norm",
• "relative-entropy",
• squared "Bures metric".

The fw_algorithm can be used as

• FrankWolfe.frank_wolfe
• FrankWolfe.lazified_conditional_gradient
• FrankWolfe.away_frank_wolfe
• FrankWolfe.blended_pairwise_conditional_gradient

Returns a named tuple (σ, v, primal) with:

• σ the closest density matrix in the separable space
• v the closest pure separable state ket on the boundary of the separable space
• primal primal value f(x), the distance to the separable space
• active_set all the pure separable states, which combined to the closest separable state σ
• lmo the structure for related computation

function whitenoiserobustness(ρ::AbstractMatrix{CT}, dims::NTuple{N, Int}, lmo::LMO; entproof, sepsearch, noise_atol, kwargs...) where {CT <: Number, N, LMO <: SeparableLMO}

Given a quantum state ρ, calculate the upper and lower bounds for the amount of white noise it can tolerate to be separable.

• ent_proof: true/false use the enumerate method to guarantee the entanglement
• sep_search: true/false use the linear search method to find a possible better separable bound
• noise_atol: the numerical accuracy for the noise level

Returns a tuple (ent_bound, sep_bound).