add tests for entanglement entropy

Project.toml

@@ -27,14 +27,16 @@ JuMP = "1.22"
 LinearAlgebra = "1"
 Nemo = "0.39 - 0.45"
 Quadmath = "0.5.10"
+Random = "1"
 Requires = "1"
 Test = "1"
 julia = "1.9"
 
 [extras]
 DoubleFloats = "497a8b3b-efae-58df-a0af-a86822472b78"
 Quadmath = "be4d8f0f-7fa4-5f49-b795-2f01399ab2dd"
+Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [targets]
-test = ["CyclotomicNumbers", "DoubleFloats", "Quadmath", "Test"]
+test = ["CyclotomicNumbers", "DoubleFloats", "Quadmath", "Random", "Test"]

src/entanglement.jl

@@ -86,7 +86,7 @@ function entanglement_entropy(ρ::AbstractMatrix{T}, dims::AbstractVector) where
     JuMP.set_optimizer(model, Hypatia.Optimizer{Rs})
     JuMP.set_attribute(model, "verbose", false)
     JuMP.optimize!(model)
-    return JuMP.objective_value(model), JuMP.value.(σ)
+    return JuMP.objective_value(model), LA.Hermitian(JuMP.value.(σ))
 end
 
 """
@@ -113,3 +113,38 @@ function _svec(M::AbstractMatrix, ::Type{R}) where {R} #the weird stuff here is
     end
     return v
 end
+
+"""
+    _test_entanglement_entropy_qubit(h::Real, ρ::AbstractMatrix, σ::AbstractMatrix)
+
+Tests whether `ρ` is indeed a entangled state whose closest separable state is `σ`.
+
+Reference: Miranowicz and Ishizaka, [arXiv:0805.3134](https://arxiv.org/abs/0805.3134)
+"""
+function _test_entanglement_entropy_qubit(h, ρ, σ)
+    R = typeof(h)
+    λ, U = LA.eigen(σ)
+    g = zeros(R, 4, 4)
+    for j = 1:4
+        for i = 1:j-1
+            g[i, j] = (λ[i] - λ[j]) / log(λ[i] / λ[j])
+        end
+        g[j, j] = λ[j]
+    end
+    g = LA.Hermitian(g)
+    σT = partial_transpose(σ, 2, [2, 2])
+    λ2, U2 = LA.eigen(σT)
+    phi = partial_transpose(ketbra(U2[:, 1]), 2, [2, 2])
+    G = zero(U)
+    for i = 1:4
+        for j = 1:4
+            G += g[i, j] * ketbra(U[:, i]) * phi * ketbra(U[:, j])
+        end
+    end
+    G = LA.Hermitian(G)
+    x = real(LA.pinv(vec(G)) * vec(σ - ρ))
+    ρ2 = σ - x * G
+    ρ_matches = isapprox(ρ2, ρ; rtol = sqrt(Base.rtoldefault(R)))
+    h_matches = isapprox(h, relative_entropy(ρ2, σ); rtol = sqrt(Base.rtoldefault(R)))
+    return ρ_matches && h_matches
+end

test/entanglement.jl

@@ -1,11 +1,29 @@
 @testset "Entanglement       " begin
-    for R in [Float64, Double64, Float128, BigFloat]
-        T = Complex{R}
-        ψ = random_state_ket(T, 6)
-        λ, U, V = schmidt_decomposition(ψ, [2, 3])
-        @test vec(Diagonal(λ)) ≈ kron(U', V') * ψ
-        ψ = random_state_ket(T, 4)
-        λ, U, V = schmidt_decomposition(ψ)
-        @test vec(Diagonal(λ)) ≈ kron(U', V') * ψ
+    @testset "Schmidt decomposition" begin
+        for R in [Float64, Double64, Float128, BigFloat]
+            T = Complex{R}
+            ψ = random_state_ket(T, 6)
+            λ, U, V = schmidt_decomposition(ψ, [2, 3])
+            @test vec(Diagonal(λ)) ≈ kron(U', V') * ψ
+            ψ = random_state_ket(T, 4)
+            λ, U, V = schmidt_decomposition(ψ)
+            @test vec(Diagonal(λ)) ≈ kron(U', V') * ψ
+        end
+    end
+    @testset "Entanglement entropy" begin
+        for R in [Float64, Double64] #Float128 and BigFloat take too long
+            Random.seed!(8) #makes all states entangled
+            ψ = random_state_ket(R, 6)
+            @test entanglement_entropy(ψ, [2, 3]) ≈ entanglement_entropy(ketbra(ψ), [2, 3])[1] atol = 1e-3 rtol = 1e-3
+            ρ = random_state(R, 4)
+            h, σ = entanglement_entropy(ρ)
+            @test Ket._test_entanglement_entropy_qubit(h, ρ, σ)
+            T = Complex{R}
+            ψ = random_state_ket(T, 6)
+            @test entanglement_entropy(ψ, [2, 3]) ≈ entanglement_entropy(ketbra(ψ), [2, 3])[1] atol = 1e-3 rtol = 1e-3
+            ρ = random_state(T, 4)
+            h, σ = entanglement_entropy(ρ)
+            @test Ket._test_entanglement_entropy_qubit(h, ρ, σ)
+        end
     end
 end