diff --git a/src/basic.jl b/src/basic.jl index d2c40c3..fbee23f 100644 --- a/src/basic.jl +++ b/src/basic.jl @@ -198,7 +198,7 @@ export pauli gellmann([T=ComplexF64,], d::Integer = 3) Constructs the set `G` of generalized Gell-Mann matrices in dimension `d` such that -`G[1] = I` and `Tr(G[i]*G[j]) = 2 δ_ij`. +`G₁ = I` and `Tr(GᵢGⱼ) = 2 δᵢⱼ`. Reference: [Generalizations of Pauli matrices](https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices) """ diff --git a/src/entanglement.jl b/src/entanglement.jl index 9888d76..58aaaac 100644 --- a/src/entanglement.jl +++ b/src/entanglement.jl @@ -139,14 +139,14 @@ end Upper bound on the random robustness of `ρ` such that it has a Schmidt number `s`. If a state ``ρ`` with local dimensions ``d_A`` and ``d_B`` has Schmidt number ``s``, then there is -a PSD matrix ``ω`` in the extended space ``AA′B′B``, where ``A′`` and ``B^′`` have dimension ``s``, -such that ``ω / s`` is separable against ``AA′|B′B`` and ``Π† ω Π = ρ``, where ``Π = 1_A ⊗ s ψ^+ ⊗ 1_B``, +a PSD matrix ``ω`` in the extended space ``AA'B'B``, where ``A'`` and ``B'`` have dimension ``s``, +such that ``ω / s`` is separable against ``AA'|B'B`` and ``Π^† ω Π = ρ``, where ``Π = 1_A ⊗ s ψ^+ ⊗ 1_B``, and ``ψ^+`` is a non-normalized maximally entangled state. Separabiity is tested with the DPS hierarchy, -with `n` controlling the how many copies of the ``B′B`` subsystem are used. +with `n` controlling the how many copies of the ``B'B`` subsystem are used. References: - Hulpke, Bruss, Lewenstein, Sanpera [arXiv:quant-ph/0401118](https://arxiv.org/abs/quant-ph/0401118)\ -Weilenmann, Dive, Trillo, Aguilar, Navascués [arXiv:1912.10056](https://arxiv.org/abs/1912.10056) +- Hulpke, Bruss, Lewenstein, Sanpera [arXiv:quant-ph/0401118](https://arxiv.org/abs/quant-ph/0401118) +- Weilenmann, Dive, Trillo, Aguilar, Navascués [arXiv:1912.10056](https://arxiv.org/abs/1912.10056) """ function schmidt_number( ρ::AbstractMatrix{T}, @@ -240,8 +240,7 @@ export random_robustness Constrains state `ρ` of dimensions `dims` in JuMP model `model` to respect the DPS constraints of level `n`. -References: - Doherty, Parrilo, Spedalieri [arXiv:quant-ph/0308032](https://arxiv.org/abs/quant-ph/0308032) +Reference: Doherty, Parrilo, Spedalieri, [arXiv:quant-ph/0308032](https://arxiv.org/abs/quant-ph/0308032) """ function _dps_constraints!( model::JuMP.GenericModel{T}, diff --git a/src/multilinear.jl b/src/multilinear.jl index 5691b79..b5f81b1 100644 --- a/src/multilinear.jl +++ b/src/multilinear.jl @@ -276,7 +276,7 @@ end 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. +`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. """ function permute_systems(X::AbstractMatrix, perm::AbstractVector{<:Integer}, dims::Matrix{<:Integer}) perm == 1:length(perm) && return X diff --git a/src/nonlocal.jl b/src/nonlocal.jl index 63e9856..88bcfa9 100644 --- a/src/nonlocal.jl +++ b/src/nonlocal.jl @@ -407,9 +407,9 @@ export tensor_probability """ tensor_correlation(p::AbstractArray{T, N2}, behaviour::Bool = false; marg::Bool = true) -Converts a 2x...x2xmx...xm probability array into -- a mx...xm correlation array (no marginals) -- a (m+1)x...x(m+1) correlation array (marginals). +Converts a 2 x ... x 2 x m x ... x m probability array into +- an m x ... x m correlation array (no marginals) +- an (m+1) x ... x (m+1) correlation array (marginals). If `behaviour` is `true` do the transformation for behaviours. Doesn't assume normalization. Also accepts the arguments of `tensor_probability` (state and measurements) for convenience. diff --git a/src/states.jl b/src/states.jl index 69c45f9..3e1046b 100644 --- a/src/states.jl +++ b/src/states.jl @@ -71,7 +71,7 @@ export state_bell """ state_phiplus_ket([T=ComplexF64,] d::Integer = 2) -Produces the ket of the maximally entangled state Φ⁺ of local dimension `d`. +Produces the ket of the maximally entangled state ϕ⁺ of local dimension `d`. """ function state_phiplus_ket(::Type{T}, d::Integer = 2; kwargs...) where {T<:Number} return state_ghz_ket(T, d, 2; kwargs...) @@ -82,7 +82,7 @@ export state_phiplus_ket """ state_phiplus([T=ComplexF64,] d::Integer = 2; v::Real = 1) -Produces the maximally entangled state Φ⁺ of local dimension `d` with visibility `v`. +Produces the maximally entangled state ϕ⁺ of local dimension `d` with visibility `v`. """ function state_phiplus(::Type{T}, d::Integer = 2; v::Real = 1) where {T<:Number} rho = ketbra(state_phiplus_ket(T, d; coeff = one(T)))