Reference

The structure of a finite dimensional C*-algebra.

• For example, [1 2; 3 4] corresponds to S₀ = M₁⊗I₂ ⊕ M₃⊗I₄.
• For an n-dimensional non-commutative graph use [1 n] for S₀ = Iₙ.
• For an n-vertex classical graph use ones(Integer, n, 2) for S₀ = diagonals.

S0Graph(sig::AlgebraShape, S::Subspace{ComplexF64, 2})

S0Graph(g::AbstractGraph)

Represents an S₀-graph as defined in arxiv:1002.2514.

• n::Integer

  Dimension of Hilbert space A such that S ⊆ L(A)

• sig::Array{var"#s12",2} where var"#s12"<:Integer

  Structure of C*-algebra S₀

• S::Subspaces.Subspace{Complex{Float64},2}

  Subspace that defines the graph

• S0::Subspaces.Subspace{Complex{Float64},2}

  C*-algebra S₀

• S1::Subspaces.Subspace{Complex{Float64},2}

  Commutant of C*-algebra S₀

• D::Array{Float64,2}

  Block scaling array D from definition 23 of arxiv:2101.00162

complement(g::S0Graph) -> S0Graph

Returns the complement graph perp(S) + S₀.

create_S0_S1(sig::AlgebraShape) -> Tuple{Subspace, Subspace}

Create a C*-algebra and its commutant with the given structure.

dsw(g::S0Graph, w::AbstractArray{var"#s13",2} where var"#s13"<:Number; use_diag_optimization, eps, verbose) -> NamedTuple{(:λ, :x, :Z),_A} where _A<:Tuple

Compute weighted θ using theorem 14 of arxiv:2101.00162.

Returns optimal λ, x, and Z values in a named tuple. If use_diag_optimization=true (the default) then x ⪰ w and x is in the commutant of S₀. By theorem 29 of arxiv:2101.00162, θ(g, w) = θ(g, x).

dsw_schur(g::S0Graph) -> NamedTuple{(:λ, :w, :Z),Tuple{Convex.AbstractExpr,Convex.AbstractExpr,Convex.AbstractExpr}}

Schur complement form of weighted θ from theorem 14 of arxiv:2101.00162.

Returns λ, w, and Z variables (for Convex.jl) in a named tuple.

See also: dsw_schur2.

dsw_schur2(g::S0Graph) -> NamedTuple{(:λ, :w, :Z),Tuple{Convex.AbstractExpr,Convex.AbstractExpr,Convex.AbstractExpr}}

Schur complement form of weighted θ from theorem 14 of arxiv:2101.00162, optimized for the case S₀ ≠ ℂI, at the cost of w being constrained to S₁ (the commutant of S₀).

Returns λ, w, and Z variables (for Convex.jl) in a named tuple.

See also: dsw_schur2.

dsw_via_complement(g::S0Graph, w::AbstractArray{var"#s17",2} where var"#s17"<:Number; use_diag_optimization, eps, verbose) -> NamedTuple{(:λ, :x, :y, :Z),_A} where _A<:Tuple

Compute weighted θ via the complement graph, using theorem 29 of arxiv:2101.00162.

θ(S, w) = max{ tr(w x) : x ⪰ 0, y = Ψ(x), θ(Sᶜ, y) ≤ 1 }

Returns optimal λ, x, y, and Z in a named tuple.

If w is in the commutant of S₀ then the weights w and y saturate the inequality in theorem 32 of arxiv:2101.00162.

empty_S0Graph(sig::AlgebraShape) -> S0Graph

Creates an empty S₀-graph (i.e. S=S₀) with S₀ having the given structure.

forget_S0(g::S0Graph) -> S0Graph

Returns an S₀-graph with S₀=ℂI.

random_S0Graph(sig::AlgebraShape) -> S0Graph

Creates a random S₀-graph with S₀ having the given structure.

Returns a random density operator in S₀.

Returns a random unitary in S₀.

Returns a random density operator in the commutant of S₀.

Returns a random unitary in the commutant of S₀.

vertex_graph(g::S0Graph) -> S0Graph

Returns the S₀-graph with S=S₀.

Ψ(g::S0Graph, w::Union{Convex.Variable, AbstractArray{var"#s13",2} where var"#s13"<:Number}) -> Any

Block scaling superoperator Ψ from definition 23 of arxiv:2101.00162