diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 2f64109..2247256 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-07-24T13:29:09","documenter_version":"1.5.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-07-24T16:47:16","documenter_version":"1.5.0"}} \ No newline at end of file diff --git a/dev/api/index.html b/dev/api/index.html index 25643a2..5626522 100644 --- a/dev/api/index.html +++ b/dev/api/index.html @@ -1,3 +1,3 @@ -List of functions · Ket.jl

List of functions

Basic

Ket.ket — Function
ket([T=ComplexF64,] i::Integer, d::Integer = 2)

Produces a ket of dimension d with nonzero element i.

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Ket.ketbra — Function
ketbra(v::AbstractVector)

Produces a ketbra of vector v.

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Ket.proj — Function
proj([T=ComplexF64,] i::Integer, d::Integer = 2)

Produces a projector onto the basis state i in dimension d.

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Ket.pauli — Function
pauli([T=ComplexF64,], ind::Vector{<:Integer})

Constructs the Pauli matrices: 0 or "I" for the identity, 1 or "X" for the Pauli X operation, 2 or "Y" for the Pauli Y operator, and 3 or "Z" for the Pauli Z operator. Vectors of integers between 0 and 3 or strings of I, X, Y, Z automatically generate Kronecker products of the corresponding operators.

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Ket.gell_mann! — Function
gell_mann!(res::AbstractMatrix{T}, i::Integer, j::Integer, d::Integer = 3)

In-place version of gell_mann.

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Ket.partial_trace — Function
partial_trace(X::AbstractMatrix, remove::AbstractVector, dims::Vector)

Takes the partial trace of matrix X with subsystem dimensions dims over the subsystems in remove.

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partial_trace(X::AbstractMatrix, remove::Integer, dims::Vector)

Takes the partial trace of matrix X with subsystem dimensions dims over the subsystem remove.

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partial_trace(X::AbstractMatrix, remove::Integer)

Takes the partial trace of matrix X over the subsystems remove assuming two equally-sized subsystems.

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Ket.partial_transpose — Function
partial_transpose(X::AbstractMatrix, transp::AbstractVector, dims::Vector)

Takes the partial transpose of matrix X with subsystem dimensions dims on the subsystems in transp.

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partial_transpose(X::AbstractMatrix, transp::Integer, dims::Vector)

Takes the partial transpose of matrix X with subsystem dimensions dims on the subsystem transp.

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partial_transpose(X::AbstractMatrix, transp)

Takes the partial transpose of matrix X on the subsystems in transp assuming two equally-sized subsystems.

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Ket.permute_systems! — Function
permute_systems!(X::AbstractVector, perm::Vector, dims::Vector)

Permutes the order of the subsystems of vector X with subsystem dimensions dims in-place according to the permutation perm.

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permute_systems!(X::AbstractVector, perm::Vector)

Permutes the order of the subsystems of vector X, which is composed by two subsystems of equal dimensions, in-place according to the permutation perm.

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Ket.permute_systems — Function
permute_systems(X::AbstractMatrix, perm::Vector, dims::Vector)

Permutes the order of the subsystems of the square matrix X, which is composed by square subsystems of dimensions dims, according to the permutation perm.

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permute_systems(X::AbstractMatrix, perm::Vector)

Permutes the order of the subsystems of the square matrix X, which is composed by two square subsystems of equal dimensions, according to the permutation perm.

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permute_systems(X::AbstractMatrix, perm::Vector, dims::Matrix)

Permutes the order of the subsystems of the matrix X, which is composed by subsystems of dimensions dims, according to the permutation perm. dims should be a n x 2 matrix where dims[i, 1] is the number of rows of subsystem i, and dims[i,2] is its number of columns.

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Ket.cleanup! — Function
cleanup!(M::AbstractArray{T}; tol = Base.rtoldefault(real(T)))

Zeroes out real or imaginary parts of M that are smaller than tol.

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Ket.symmetric_projection — Function
symmetric_projection(dim::Integer, n::Integer; partial::Bool=true)

Orthogonal projection onto the symmetric subspace of n copies of a dim-dimensional space. By default (partial=true) it returns an isometry (say, V) encoding the symmetric subspace. If partial=false, then it returns the actual projection V * V'.

Reference: Watrous' book, Sec. 7.1.1

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Ket.orthonormal_range — Function
orthonormal_range(A::AbstractMatrix{T}; mode::Integer=nothing, tol::T=nothing, sp::Bool=true) where {T<:Number}

Orthonormal basis for the range of A. When A is sparse (or mode = 0), uses a QR factorization and returns a sparse result, otherwise uses an SVD and returns a dense matrix (mode = 1). Input A will be overwritten during the factorization. Tolerance tol is used to compute the rank and is automatically set if not provided.

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Ket.permutation_matrix — Function
permutation_matrix(dims::Union{Integer,Vector{<:Integer}}, perm::Vector{<:Integer})

Unitary that permutes subsystems of dimension dims according to the permutation perm. If dims is an Integer, assumes there are length(perm) subsystems of equal dimensions dims.

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Entropy

Ket.entropy — Function
entropy([base=2,] ρ::AbstractMatrix)

Computes the von Neumann entropy -tr(ρ log ρ) of a positive semidefinite operator ρ using a base base logarithm.

Reference: von Neumann entropy.

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entropy([base=2,] p::AbstractVector)

Computes the Shannon entropy -Σᵢpᵢlog(pᵢ) of a non-negative vector p using a base base logarithm.

Reference: Entropy (information theory).

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Ket.relative_entropy — Function
relative_entropy([base=2,] ρ::AbstractMatrix, σ::AbstractMatrix)

Computes the (quantum) relative entropy tr(ρ (log ρ - log σ)) between positive semidefinite matrices ρ and σ using a base base logarithm. Note that the support of ρ must be contained in the support of σ but for efficiency this is not checked.

Reference: Quantum relative entropy.

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relative_entropy([base=2,] p::AbstractVector, q::AbstractVector)

Computes the relative entropy D(p||q) = Σᵢpᵢlog(pᵢ/qᵢ) between two non-negative vectors p and q using a base base logarithm. Note that the support of p must be contained in the support of q but for efficiency this is not checked.

Reference: Relative entropy.

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Ket.binary_relative_entropy — Function
binary_relative_entropy([base=2,] p::Real, q::Real)

Computes the binary relative entropy D(p||q) = p log(p/q) + (1-p) log((1-p)/(1-q)) between two probabilities p and q using a base base logarithm.

Reference: Relative entropy.

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Ket.conditional_entropy — Function
conditional_entropy([base=2,] pAB::AbstractMatrix)

Computes the conditional Shannon entropy H(A|B) of the joint probability distribution pAB using a base base logarithm.

Reference: Conditional entropy.

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conditional_entropy([base=2,], rho::AbstractMatrix, csys::AbstractVector, dims::AbstractVector)

Computes the conditional von Neumann entropy of rho with subsystem dimensions dims and conditioning systems csys, using a base base logarithm.

Reference: Conditional quantum entropy.

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Entanglement

Ket.schmidt_decomposition — Function
schmidt_decomposition(ψ::AbstractVector, dims::AbstractVector{<:Integer})

Produces the Schmidt decomposition of ψ with subsystem dimensions dims. Returns the (sorted) Schmidt coefficients λ and isometries U, V such that kron(U', V')*ψ is of Schmidt form.

Reference: Schmidt decomposition.

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schmidt_decomposition(ψ::AbstractVector)

Produces the Schmidt decomposition of ψ assuming equally-sized subsystems. Returns the (sorted) Schmidt coefficients λ and isometries U, V such that kron(U', V')*ψ is of Schmidt form.

Reference: Schmidt decomposition.

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Ket.entanglement_entropy — Function
entanglement_entropy(ψ::AbstractVector, dims::AbstractVector{<:Integer})

Computes the relative entropy of entanglement of a bipartite pure state ψ with subsystem dimensions dims.

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entanglement_entropy(ψ::AbstractVector)

Computes the relative entropy of entanglement of a bipartite pure state ψ assuming equally-sized subsystems.

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entanglement_entropy(ρ::AbstractMatrix, dims::AbstractVector)

Lower bounds the relative entropy of entanglement of a bipartite state ρ with subsystem dimensions dims.

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entanglement_entropy(ρ::AbstractMatrix)

Lower bounds the relative entropy of entanglement of a bipartite state ρ assuming equally-sized subsystems.

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Ket.entanglement_dps — Function
entanglement_dps(ρ::AbstractMatrix{T}, n::Integer = 2, d1::Union{Integer,Nothing} = nothing; sn::Integer = 1, ppt::Bool = true, verbose::Bool = false,
-solver = Hypatia.Optimizer{_solver_type(T)}, witness::Bool = true, noise::Union{AbstractMatrix,Nothing} = nothing) where {T<:Number}

Test whether ρ has a symmetric extension with n copies of the second subsystem. If yes, returns true and the extension found. Otherwise, returns false and a witness W such that tr(W * ρ) = -1 and tr(W * σ) is nonnegative in any symmetrically-extendable σ.

When witness = false, it returns the maximum visibility of ρ such that it has a symmetric extension when mixed with noise (default is white noise). If this visibility is < 1, then the state is entangled, otherwise the test is inconclusive. No witness is returned in this case.

For sn > 1, it tests whether the state has a Schmidt number greater than sn. For example, if sn = 1 (default) and the state does not have an extension, then it has Schmidt number > 1 (is entangled). Likewise, if sn = 2 and it does not have an extension, then ρ has Schidt number at least 3. The witness parameter works as above. No witness is returned in this case.

References: Doherty, Parrilo, Spedalieri arXiv:quant-ph/0308032 Hulpke, Bruss, Lewenstein, Sanpera arXiv:quant-ph/0401118 Weilenmann, Dive, Trillo, Aguilar, Navascués arXiv:1912.10056

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Measurements

Ket.sic_povm — Function
sic_povm([T=ComplexF64,] d::Integer)

Constructs a vector of d² vectors |vᵢ⟩ such that |vᵢ⟩⟨vᵢ| forms a SIC-POVM of dimension d. This construction is based on the Weyl-Heisenberg fiducial.

Reference: Appleby, Yadsan-Appleby, Zauner, arXiv:1209.1813

source
Ket.test_sic — Function
test_sic(vecs)

Tests whether vecs is a vector of d² vectors |vᵢ⟩ such that |vᵢ⟩⟨vᵢ| forms a SIC-POVM of dimension d.

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Ket.test_povm — Function
test_povm(A::Vector{<:AbstractMatrix{T}})

Checks if the measurement defined by A is valid (hermitian, semi-definite positive, and normalized).

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Ket.dilate_povm — Function
dilate_povm(vecs::Vector{Vector{T}})

Does the Naimark dilation of a rank-1 POVM given as a vector of vectors. This is the minimal dilation.

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dilate_povm(E::Vector{<:AbstractMatrix})

Does the Naimark dilation of a POVM given as a vector of matrices. This always works, but is wasteful if the POVM elements are not full rank.

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Ket.povm — Function
povm(B::Vector{<:AbstractMatrix{T}})

Creates a set of (projective) measurements from a set of bases given as unitary matrices.

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povm(A::Array{T, 4}, n::Vector{Int64})

Converts a set of measurements in the common tensor format into a matrix of matrices. The second argument is fixed by the size of A but can also contain custom number of outcomes.

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Ket.mub — Function
mub([T=ComplexF64,] d::Integer)

Construction of the standard complete set of MUBs. The output contains 1+minᵢ pᵢ^rᵢ bases, where d = ∏ᵢ pᵢ^rᵢ.

Reference: Durt, Englert, Bengtsson, Życzkowski, arXiv:1004.3348.

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Nonlocality

Ket.chsh — Function
chsh([T=Float64,] d::Integer = 2)

CHSH-d nonlocal game in full probability notation. If T is an integer type the game is unnormalized.

Reference: Buhrman and Massar, arXiv:quant-ph/0409066.

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Ket.local_bound — Function
local_bound(G::Array{T,4})

Computes the local bound of a bipartite Bell functional G, written in full probability notation as a 4-dimensional array.

Reference: Araújo, Hirsch, and Quintino, arXiv:2005.13418.

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Ket.tsirelson_bound — Function
tsirelson_bound(CG::Matrix, scenario::Vector, level::Integer)

Upper bounds the Tsirelson bound of a bipartite Bell funcional game CG, written in Collins-Gisin notation. scenario is vector detailing the number of inputs and outputs, in the order [oa, ob, ia, ib]. level is an integer determining the level of the NPA hierarchy.

This function requires Moment. It is only available if you first do "import MATLAB" or "using MATLAB".

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Ket.correlation_tensor — Function
correlation_tensor(p::AbstractArray{T, N2}; marg::Bool = true)

Applies N sets of measurements onto a state rho to form a probability array. Convert a 2x...x2xmx...xm probability array into

  • a mx...xm correlation array (no marginals)
  • a (m+1)x...x(m+1) correlation array (marginals).
source
Ket.probability_tensor — Function
probability_tensor(rho::LA.Hermitian, all_Aax::Vector{Measurement}...)

Applies N sets of measurements onto a state rho to form a probability array.

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Ket.fp2cg — Function
fp2cg(V::Array{T,4}) where {T <: Real}

Takes a bipartite Bell functional V in full probability notation and transforms it to Collins-Gisin notation.

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Norms

Ket.kyfan_norm — Function
kyfan_norm(X::AbstractMatrix, k::Integer, p::Real = 2)

Computes Ky-Fan (k,p) norm of matrix X.

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Ket.diamond_norm — Function
diamond_norm(J::AbstractMatrix, dims::AbstractVector)

Computes the diamond norm of the supermap J given in the Choi-Jamiołkowski representation, with subsystem dimensions dims.

Reference: Diamond norm

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diamond_norm(K::Vector{<:AbstractMatrix})

Computes the diamond norm of the CP map given by the Kraus operators K.

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Random

Ket.random_state — Function
random_state([T=ComplexF64,] d::Integer, k::Integer = d)

Produces a uniformly distributed random quantum state in dimension d with rank k.

Reference: Życzkowski and Sommers, arXiv:quant-ph/0012101.

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Ket.random_unitary — Function
random_unitary([T=ComplexF64,] d::Integer)

Produces a Haar-random unitary matrix in dimension d. If T is a real type the output is instead a Haar-random (real) orthogonal matrix.

Reference: Stewart, doi:10.1137/0717034.

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Ket.random_povm — Function
random_povm([T=ComplexF64,] d::Integer, n::Integer, r::Integer)

Produces a random POVM of dimension d with n outcomes and rank min(k, d).

Reference: Heinosaari et al., arXiv:1902.04751.

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States

Ket.state_phiplus_ket — Function
state_phiplus_ket([T=ComplexF64,] d::Integer = 2)

Produces the vector of the maximally entangled state Φ⁺ of local dimension d.

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Ket.state_phiplus — Function
state_phiplus([T=ComplexF64,] d::Integer = 2; v::Real = 1)

Produces the maximally entangled state Φ⁺ of local dimension d with visibility v.

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Ket.isotropic — Function
isotropic(v::Real, d::Integer = 2)

Produces the isotropic state of local dimension d with visibility v.

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Ket.state_psiminus_ket — Function
state_psiminus_ket([T=ComplexF64,] d::Integer = 2)

Produces the vector of the maximally entangled state ψ⁻ of local dimension d.

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Ket.state_psiminus — Function
state_psiminus([T=ComplexF64,] d::Integer = 2; v::Real = 1)

Produces the maximally entangled state ψ⁻ of local dimension d with visibility v.

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Ket.state_ghz_ket — Function
state_ghz_ket([T=ComplexF64,] d::Integer = 2, N::Integer = 3; coeff = 1/√d)

Produces the vector of the GHZ state local dimension d.

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Ket.state_ghz — Function
state_ghz([T=ComplexF64,] d::Integer = 2, N::Integer = 3; v::Real = 1, coeff = 1/√d)

Produces the GHZ state of local dimension d with visibility v.

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Ket.state_w_ket — Function
state_w_ket([T=ComplexF64,] N::Integer = 3; coeff = 1/√d)

Produces the vector of the N-partite W state.

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Ket.state_w — Function
state_w([T=ComplexF64,] N::Integer = 3; v::Real = 1, coeff = 1/√d)

Produces the N-partite W state with visibility v.

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Ket.white_noise — Function
white_noise(rho::AbstractMatrix, v::Real)

Returns v * rho + (1 - v) * id, where id is the maximally mixed state.

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Ket.white_noise! — Function
white_noise!(rho::AbstractMatrix, v::Real)

Modifies rho in place to tranform in into v * rho + (1 - v) * id where id is the maximally mixed state.

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Supermaps

Ket.choi — Function
choi(K::Vector{<:AbstractMatrix})

Constructs the Choi-Jamiołkowski representation of the CP map given by the Kraus operators K. The convention used is that choi(K) = ∑ᵢⱼ |i⟩⟨j|⊗K|i⟩⟨j|K'

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Internal functions

Ket._partition — Function
partition(n::Integer, k::Integer)

If n ≥ k partitions the set 1:n into k parts as equally sized as possible. Otherwise partitions it into n parts of size 1.

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Ket._fiducial_WH — Function
_fiducial_WH([T=ComplexF64,] d::Integer)

Computes the fiducial Weyl-Heisenberg vector of dimension d.

Reference: Appleby, Yadsan-Appleby, Zauner, arXiv:1209.1813 http://www.gerhardzauner.at/sicfiducials.html

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Ket._idx — Function
_idx(tidx::Vector, dims::Vector)

Converts a tensor index tidx = [i₁, i₂, ...] with subsystems dimensions dims to a standard index.

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Ket._tidx — Function
_tidx(idx::Integer, dims::Vector)

Converts a standard index idx to a tensor index [i₁, i₂, ...] with subsystems dimensions dims.

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Ket._idxperm — Function
_idxperm(perm::Vector, dims::Vector)

Computes the index permutation associated with permuting the subsystems of a vector with subsystem dimensions dims according to perm.

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+List of functions · Ket.jl

List of functions

Basic

Ket.ket — Function
ket([T=ComplexF64,] i::Integer, d::Integer = 2)

Produces a ket of dimension d with nonzero element i.

source
Ket.ketbra — Function
ketbra(v::AbstractVector)

Produces a ketbra of vector v.

source
Ket.proj — Function
proj([T=ComplexF64,] i::Integer, d::Integer = 2)

Produces a projector onto the basis state i in dimension d.

source
Ket.pauli — Function
pauli([T=ComplexF64,], ind::Vector{<:Integer})

Constructs the Pauli matrices: 0 or "I" for the identity, 1 or "X" for the Pauli X operation, 2 or "Y" for the Pauli Y operator, and 3 or "Z" for the Pauli Z operator. Vectors of integers between 0 and 3 or strings of I, X, Y, Z automatically generate Kronecker products of the corresponding operators.

source
Ket.gell_mann! — Function
gell_mann!(res::AbstractMatrix{T}, i::Integer, j::Integer, d::Integer = 3)

In-place version of gell_mann.

source
Ket.partial_trace — Function
partial_trace(X::AbstractMatrix, remove::AbstractVector, dims::Vector)

Takes the partial trace of matrix X with subsystem dimensions dims over the subsystems in remove.

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partial_trace(X::AbstractMatrix, remove::Integer, dims::Vector)

Takes the partial trace of matrix X with subsystem dimensions dims over the subsystem remove.

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partial_trace(X::AbstractMatrix, remove::Integer)

Takes the partial trace of matrix X over the subsystems remove assuming two equally-sized subsystems.

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Ket.partial_transpose — Function
partial_transpose(X::AbstractMatrix, transp::AbstractVector, dims::Vector)

Takes the partial transpose of matrix X with subsystem dimensions dims on the subsystems in transp.

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partial_transpose(X::AbstractMatrix, transp::Integer, dims::Vector)

Takes the partial transpose of matrix X with subsystem dimensions dims on the subsystem transp.

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partial_transpose(X::AbstractMatrix, transp)

Takes the partial transpose of matrix X on the subsystems in transp assuming two equally-sized subsystems.

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Ket.permute_systems! — Function
permute_systems!(X::AbstractVector, perm::Vector, dims::Vector)

Permutes the order of the subsystems of vector X with subsystem dimensions dims in-place according to the permutation perm.

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permute_systems!(X::AbstractVector, perm::Vector)

Permutes the order of the subsystems of vector X, which is composed by two subsystems of equal dimensions, in-place according to the permutation perm.

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Ket.permute_systems — Function
permute_systems(X::AbstractMatrix, perm::Vector, dims::Vector)

Permutes the order of the subsystems of the square matrix X, which is composed by square subsystems of dimensions dims, according to the permutation perm.

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permute_systems(X::AbstractMatrix, perm::Vector)

Permutes the order of the subsystems of the square matrix X, which is composed by two square subsystems of equal dimensions, according to the permutation perm.

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permute_systems(X::AbstractMatrix, perm::Vector, dims::Matrix)

Permutes the order of the subsystems of the matrix X, which is composed by subsystems of dimensions dims, according to the permutation perm. dims should be a n x 2 matrix where dims[i, 1] is the number of rows of subsystem i, and dims[i,2] is its number of columns.

source
Ket.cleanup! — Function
cleanup!(M::AbstractArray{T}; tol = Base.rtoldefault(real(T)))

Zeroes out real or imaginary parts of M that are smaller than tol.

source
Ket.symmetric_projection — Function
symmetric_projection(dim::Integer, n::Integer; partial::Bool=true)

Orthogonal projection onto the symmetric subspace of n copies of a dim-dimensional space. By default (partial=true) it returns an isometry (say, V) encoding the symmetric subspace. If partial=false, then it returns the actual projection V * V'.

Reference: Watrous' book, Sec. 7.1.1

source
Ket.orthonormal_range — Function
orthonormal_range(A::AbstractMatrix{T}; mode::Integer=nothing, tol::T=nothing, sp::Bool=true) where {T<:Number}

Orthonormal basis for the range of A. When A is sparse (or mode = 0), uses a QR factorization and returns a sparse result, otherwise uses an SVD and returns a dense matrix (mode = 1). Input A will be overwritten during the factorization. Tolerance tol is used to compute the rank and is automatically set if not provided.

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Ket.permutation_matrix — Function
permutation_matrix(dims::Union{Integer,Vector{<:Integer}}, perm::Vector{<:Integer})

Unitary that permutes subsystems of dimension dims according to the permutation perm. If dims is an Integer, assumes there are length(perm) subsystems of equal dimensions dims.

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Entropy

Ket.entropy — Function
entropy([base=2,] ρ::AbstractMatrix)

Computes the von Neumann entropy -tr(ρ log ρ) of a positive semidefinite operator ρ using a base base logarithm.

Reference: von Neumann entropy.

source
entropy([base=2,] p::AbstractVector)

Computes the Shannon entropy -Σᵢpᵢlog(pᵢ) of a non-negative vector p using a base base logarithm.

Reference: Entropy (information theory).

source
Ket.relative_entropy — Function
relative_entropy([base=2,] ρ::AbstractMatrix, σ::AbstractMatrix)

Computes the (quantum) relative entropy tr(ρ (log ρ - log σ)) between positive semidefinite matrices ρ and σ using a base base logarithm. Note that the support of ρ must be contained in the support of σ but for efficiency this is not checked.

Reference: Quantum relative entropy.

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relative_entropy([base=2,] p::AbstractVector, q::AbstractVector)

Computes the relative entropy D(p||q) = Σᵢpᵢlog(pᵢ/qᵢ) between two non-negative vectors p and q using a base base logarithm. Note that the support of p must be contained in the support of q but for efficiency this is not checked.

Reference: Relative entropy.

source
Ket.binary_relative_entropy — Function
binary_relative_entropy([base=2,] p::Real, q::Real)

Computes the binary relative entropy D(p||q) = p log(p/q) + (1-p) log((1-p)/(1-q)) between two probabilities p and q using a base base logarithm.

Reference: Relative entropy.

source
Ket.conditional_entropy — Function
conditional_entropy([base=2,] pAB::AbstractMatrix)

Computes the conditional Shannon entropy H(A|B) of the joint probability distribution pAB using a base base logarithm.

Reference: Conditional entropy.

source
conditional_entropy([base=2,], rho::AbstractMatrix, csys::AbstractVector, dims::AbstractVector)

Computes the conditional von Neumann entropy of rho with subsystem dimensions dims and conditioning systems csys, using a base base logarithm.

Reference: Conditional quantum entropy.

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Entanglement

Ket.schmidt_decomposition — Function
schmidt_decomposition(ψ::AbstractVector, dims::AbstractVector{<:Integer})

Produces the Schmidt decomposition of ψ with subsystem dimensions dims. Returns the (sorted) Schmidt coefficients λ and isometries U, V such that kron(U', V')*ψ is of Schmidt form.

Reference: Schmidt decomposition.

source
schmidt_decomposition(ψ::AbstractVector)

Produces the Schmidt decomposition of ψ assuming equally-sized subsystems. Returns the (sorted) Schmidt coefficients λ and isometries U, V such that kron(U', V')*ψ is of Schmidt form.

Reference: Schmidt decomposition.

source
Ket.entanglement_entropy — Function
entanglement_entropy(ψ::AbstractVector, dims::AbstractVector{<:Integer})

Computes the relative entropy of entanglement of a bipartite pure state ψ with subsystem dimensions dims.

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entanglement_entropy(ψ::AbstractVector)

Computes the relative entropy of entanglement of a bipartite pure state ψ assuming equally-sized subsystems.

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entanglement_entropy(ρ::AbstractMatrix, dims::AbstractVector)

Lower bounds the relative entropy of entanglement of a bipartite state ρ with subsystem dimensions dims.

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entanglement_entropy(ρ::AbstractMatrix)

Lower bounds the relative entropy of entanglement of a bipartite state ρ assuming equally-sized subsystems.

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Ket.entanglement_dps — Function
entanglement_dps(ρ::AbstractMatrix{T}, n::Integer = 2, d1::Union{Integer,Nothing} = nothing; sn::Integer = 1, ppt::Bool = true, verbose::Bool = false,
+solver = Hypatia.Optimizer{_solver_type(T)}, witness::Bool = true, noise::Union{AbstractMatrix,Nothing} = nothing) where {T<:Number}

Test whether ρ has a symmetric extension with n copies of the second subsystem. If yes, returns true and the extension found. Otherwise, returns false and a witness W such that tr(W * ρ) = -1 and tr(W * σ) is nonnegative in any symmetrically-extendable σ.

When witness = false, it returns the maximum visibility of ρ such that it has a symmetric extension when mixed with noise (default is white noise). If this visibility is < 1, then the state is entangled, otherwise the test is inconclusive. No witness is returned in this case.

For sn > 1, it tests whether the state has a Schmidt number greater than sn. For example, if sn = 1 (default) and the state does not have an extension, then it has Schmidt number > 1 (is entangled). Likewise, if sn = 2 and it does not have an extension, then ρ has Schidt number at least 3. The witness parameter works as above. No witness is returned in this case.

References: Doherty, Parrilo, Spedalieri arXiv:quant-ph/0308032 Hulpke, Bruss, Lewenstein, Sanpera arXiv:quant-ph/0401118 Weilenmann, Dive, Trillo, Aguilar, Navascués arXiv:1912.10056

source

Measurements

Ket.sic_povm — Function
sic_povm([T=ComplexF64,] d::Integer)

Constructs a vector of d² vectors |vᵢ⟩ such that |vᵢ⟩⟨vᵢ| forms a SIC-POVM of dimension d. This construction is based on the Weyl-Heisenberg fiducial.

Reference: Appleby, Yadsan-Appleby, Zauner, arXiv:1209.1813

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Ket.test_sic — Function
test_sic(vecs)

Tests whether vecs is a vector of d² vectors |vᵢ⟩ such that |vᵢ⟩⟨vᵢ| forms a SIC-POVM of dimension d.

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Ket.test_povm — Function
test_povm(A::Vector{<:AbstractMatrix{T}})

Checks if the measurement defined by A is valid (hermitian, semi-definite positive, and normalized).

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Ket.dilate_povm — Function
dilate_povm(vecs::Vector{Vector{T}})

Does the Naimark dilation of a rank-1 POVM given as a vector of vectors. This is the minimal dilation.

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dilate_povm(E::Vector{<:AbstractMatrix})

Does the Naimark dilation of a POVM given as a vector of matrices. This always works, but is wasteful if the POVM elements are not full rank.

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Ket.povm — Function
povm(B::Vector{<:AbstractMatrix{T}})

Creates a set of (projective) measurements from a set of bases given as unitary matrices.

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povm(A::Array{T, 4}, n::Vector{Int64})

Converts a set of measurements in the common tensor format into a matrix of matrices. The second argument is fixed by the size of A but can also contain custom number of outcomes.

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Ket.mub — Function
mub([T=ComplexF64,] d::Integer)

Construction of the standard complete set of MUBs. The output contains 1+minᵢ pᵢ^rᵢ bases, where d = ∏ᵢ pᵢ^rᵢ.

Reference: Durt, Englert, Bengtsson, Życzkowski, arXiv:1004.3348.

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Nonlocality

Ket.chsh — Function
chsh([T=Float64,] d::Integer = 2)

CHSH-d nonlocal game in full probability notation. If T is an integer type the game is unnormalized.

Reference: Buhrman and Massar, arXiv:quant-ph/0409066.

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Ket.local_bound — Function
local_bound(G::Array{T,4})

Computes the local bound of a bipartite Bell functional G, written in full probability notation as a 4-dimensional array.

Reference: Araújo, Hirsch, and Quintino, arXiv:2005.13418.

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Ket.tsirelson_bound — Function
tsirelson_bound(CG::Matrix, scenario::Vector, level::Integer)

Upper bounds the Tsirelson bound of a bipartite Bell funcional game CG, written in Collins-Gisin notation. scenario is vector detailing the number of inputs and outputs, in the order [oa, ob, ia, ib]. level is an integer determining the level of the NPA hierarchy.

This function requires Moment. It is only available if you first do "import MATLAB" or "using MATLAB".

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Ket.correlation_tensor — Function
correlation_tensor(p::AbstractArray{T, N2}; marg::Bool = true)

Applies N sets of measurements onto a state rho to form a probability array. Convert a 2x...x2xmx...xm probability array into

  • a mx...xm correlation array (no marginals)
  • a (m+1)x...x(m+1) correlation array (marginals).
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Ket.probability_tensor — Function
probability_tensor(rho::LA.Hermitian, all_Aax::Vector{Measurement}...)

Applies N sets of measurements onto a state rho to form a probability array.

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Ket.fp2cg — Function
fp2cg(V::Array{T,4}) where {T <: Real}

Takes a bipartite Bell functional V in full probability notation and transforms it to Collins-Gisin notation.

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Norms

Ket.kyfan_norm — Function
kyfan_norm(X::AbstractMatrix, k::Integer, p::Real = 2)

Computes Ky-Fan (k,p) norm of matrix X.

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Ket.diamond_norm — Function
diamond_norm(J::AbstractMatrix, dims::AbstractVector)

Computes the diamond norm of the supermap J given in the Choi-Jamiołkowski representation, with subsystem dimensions dims.

Reference: Diamond norm

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diamond_norm(K::Vector{<:AbstractMatrix})

Computes the diamond norm of the CP map given by the Kraus operators K.

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Random

Ket.random_state — Function
random_state([T=ComplexF64,] d::Integer, k::Integer = d)

Produces a uniformly distributed random quantum state in dimension d with rank k.

Reference: Życzkowski and Sommers, arXiv:quant-ph/0012101.

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Ket.random_unitary — Function
random_unitary([T=ComplexF64,] d::Integer)

Produces a Haar-random unitary matrix in dimension d. If T is a real type the output is instead a Haar-random (real) orthogonal matrix.

Reference: Stewart, doi:10.1137/0717034.

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Ket.random_povm — Function
random_povm([T=ComplexF64,] d::Integer, n::Integer, r::Integer)

Produces a random POVM of dimension d with n outcomes and rank min(k, d).

Reference: Heinosaari et al., arXiv:1902.04751.

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States

Ket.state_phiplus_ket — Function
state_phiplus_ket([T=ComplexF64,] d::Integer = 2)

Produces the vector of the maximally entangled state Φ⁺ of local dimension d.

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Ket.state_phiplus — Function
state_phiplus([T=ComplexF64,] d::Integer = 2; v::Real = 1)

Produces the maximally entangled state Φ⁺ of local dimension d with visibility v.

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Ket.isotropic — Function
isotropic(v::Real, d::Integer = 2)

Produces the isotropic state of local dimension d with visibility v.

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Ket.state_psiminus_ket — Function
state_psiminus_ket([T=ComplexF64,] d::Integer = 2)

Produces the vector of the maximally entangled state ψ⁻ of local dimension d.

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Ket.state_psiminus — Function
state_psiminus([T=ComplexF64,] d::Integer = 2; v::Real = 1)

Produces the maximally entangled state ψ⁻ of local dimension d with visibility v.

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Ket.state_ghz_ket — Function
state_ghz_ket([T=ComplexF64,] d::Integer = 2, N::Integer = 3; coeff = 1/√d)

Produces the vector of the GHZ state local dimension d.

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Ket.state_ghz — Function
state_ghz([T=ComplexF64,] d::Integer = 2, N::Integer = 3; v::Real = 1, coeff = 1/√d)

Produces the GHZ state of local dimension d with visibility v.

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Ket.state_w_ket — Function
state_w_ket([T=ComplexF64,] N::Integer = 3; coeff = 1/√d)

Produces the vector of the N-partite W state.

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Ket.state_w — Function
state_w([T=ComplexF64,] N::Integer = 3; v::Real = 1, coeff = 1/√d)

Produces the N-partite W state with visibility v.

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Ket.white_noise — Function
white_noise(rho::AbstractMatrix, v::Real)

Returns v * rho + (1 - v) * id, where id is the maximally mixed state.

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Ket.white_noise! — Function
white_noise!(rho::AbstractMatrix, v::Real)

Modifies rho in place to tranform in into v * rho + (1 - v) * id where id is the maximally mixed state.

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Supermaps

Ket.choi — Function
choi(K::Vector{<:AbstractMatrix})

Constructs the Choi-Jamiołkowski representation of the CP map given by the Kraus operators K. The convention used is that choi(K) = ∑ᵢⱼ |i⟩⟨j|⊗K|i⟩⟨j|K'

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Internal functions

Ket._partition — Function
partition(n::Integer, k::Integer)

If n ≥ k partitions the set 1:n into k parts as equally sized as possible. Otherwise partitions it into n parts of size 1.

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Ket._fiducial_WH — Function
_fiducial_WH([T=ComplexF64,] d::Integer)

Computes the fiducial Weyl-Heisenberg vector of dimension d.

Reference: Appleby, Yadsan-Appleby, Zauner, arXiv:1209.1813 http://www.gerhardzauner.at/sicfiducials.html

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Ket._idx — Function
_idx(tidx::Vector, dims::Vector)

Converts a tensor index tidx = [i₁, i₂, ...] with subsystems dimensions dims to a standard index.

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Ket._tidx — Function
_tidx(idx::Integer, dims::Vector)

Converts a standard index idx to a tensor index [i₁, i₂, ...] with subsystems dimensions dims.

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Ket._idxperm — Function
_idxperm(perm::Vector, dims::Vector)

Computes the index permutation associated with permuting the subsystems of a vector with subsystem dimensions dims according to perm.

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diff --git a/dev/index.html b/dev/index.html index 90a7c2f..7713450 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,2 +1,2 @@ -Home · Ket.jl

Ket.jl

Dev

Toolbox for quantum information, nonlocality, and entanglement.

Highlights are the functions mub and sic_povm, that produce respectively MUBs and SIC-POVMs with arbitrary precision, local_bound that uses a parallelized algorithm to compute the local bound of a Bell inequality, and partial_trace and partial_transpose, that compute the partial trace and partial transpose in a way that can be used for optimization with JuMP. Also worth mentioning are the functions to produce uniformly-distributed random states, unitaries, and POVMs: random_state, random_unitary, random_povm. And the eponymous ket, of course.

For the full list of functions see the documentation.

+Home · Ket.jl

Ket.jl

Dev

Toolbox for quantum information, nonlocality, and entanglement.

Highlights are the functions mub and sic_povm, that produce respectively MUBs and SIC-POVMs with arbitrary precision, local_bound that uses a parallelized algorithm to compute the local bound of a Bell inequality, and partial_trace and partial_transpose, that compute the partial trace and partial transpose in a way that can be used for optimization with JuMP. Also worth mentioning are the functions to produce uniformly-distributed random states, unitaries, and POVMs: random_state, random_unitary, random_povm. And the eponymous ket, of course.

For the full list of functions see the documentation.