add schmidt decomposition

docs/src/api.md

@@ -26,6 +26,12 @@ binary_relative_entropy
 conditional_entropy
 ```
 
+## Entanglement
+
+```@docs
+schmidt_decomposition
+```
+
 ## Measurements
 
 ```@docs

src/Ket.jl

@@ -14,6 +14,7 @@ include("measurements.jl")
 include("entropy.jl")
 include("norms.jl")
 include("partial_tra.jl")
+include("entanglement.jl")
 
 import Requires
 function __init__()

src/entanglement.jl

@@ -0,0 +1,25 @@
+"""
+    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.
+"""
+function schmidt_decomposition(ψ::AbstractVector, dims::AbstractVector{<:Integer})
+    length(dims) != 2 && throw(ArgumentError("Two subsystem sizes must be specified."))
+    m = transpose(reshape(ψ, dims[2], dims[1])) #necessary because the natural reshaping would be row-major, but Julia does it col-major
+    U, λ, V = LA.svd(m)
+    return λ, U, conj(V)
+end
+"""
+    schmidt_decomposition(ψ::AbstractVector, dims::AbstractVector{<:Integer})
+    
+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.
+"""
+function schmidt_decomposition(ψ::AbstractVector)
+    n = length(ψ)
+    d = isqrt(n)
+    d^2 != n && throw(ArgumentError("Subsystems are not equally-sized, please specify sizes."))
+    return schmidt_decomposition(ψ, [d, d])
+end
+export schmidt_decomposition

test/basic.jl

@@ -51,7 +51,7 @@
             @test gell_mann(Complex{R}, 3, 1) == [0 0 -im; 0 0 0; im 0 0]
             @test gell_mann(Complex{R}, 3, 2) == [0 0 0; 0 0 -im; 0 im 0]
         end
-        @test gell_mann(3, 3) == Diagonal([1/sqrt(3), 1/sqrt(3), -2/sqrt(3)])
+        @test gell_mann(3, 3) == Diagonal([1, 1, -2] / sqrt(3))
         @test gell_mann(1, 1, 4) == Matrix{Float64}(I, 4, 4)
     end
     @testset "Cleanup" begin

test/entanglement.jl

@@ -0,0 +1,11 @@
+@testset "Entanglement       " begin
+    for R in [Float64, Double64, Float128, BigFloat]
+        T = Complex{R}
+        ψ = random_state_vector(T, 6)
+        λ, U, V = schmidt_decomposition(ψ, [2, 3])
+        @test vec(Diagonal(λ)) ≈ kron(U', V') * ψ
+        ψ = random_state_vector(T, 4)
+        λ, U, V = schmidt_decomposition(ψ)
+        @test vec(Diagonal(λ)) ≈ kron(U', V') * ψ
+    end
+end

test/runtests.jl

@@ -7,6 +7,7 @@ using Test
 
 include("basic.jl")
 include("entropy.jl")
+include("entanglement.jl")
 include("measurements.jl")
 include("nonlocal.jl")
 include("norms.jl")