add partial_transpose

TODO

@@ -1,4 +1,4 @@
-MA - Partial transpose
+MA - Partial transpose (LP)
 MA - Permute systems
 MA - DPS hierarchy
 MA - See-saw

src/partial_tra.jl

@@ -99,3 +99,67 @@ Takes the partial trace of matrix `X` with subsystem dimensions `dims` over the
 """
 partial_trace(X::AbstractMatrix, remove::Integer, dims::Vector{<:Integer}) = partial_trace(X, [remove], dims)
 export partial_trace
+
+"""
+    partial_trasnpose(X::AbstractMatrix, transp::Vector, dims::Vector)
+
+Takes the partial transpose of matrix `X` with subsystem dimensions `dims` on the subsystems in `transp`.
+"""
+function partial_transpose(X::AbstractMatrix{T}, transp::Vector{<:Integer}, dims::Vector{<:Integer}) where {T}
+    isempty(transp) && return X
+    length(transp) == length(dims) && return LA.transpose(X)
+
+    keep = Vector{eltype(transp)}(undef, length(dims) - length(transp))  # Systems kept 
+    counter = 0
+    for i = 1:length(dims)
+        if !(i in transp)
+            counter += 1
+            keep[counter] = i
+        end
+    end
+
+    dY = prod(dims)                             # Dimension of the final output Y    
+
+    Y = Matrix{T}(undef, (dY, dY))              # Final output Y
+
+    tXi = Vector{Int64}(undef, length(dims))    # Tensor indexing of X for row 
+    tXj = Vector{Int64}(undef, length(dims))    # Tensor indexing of X for column
+
+    tYi = Vector{Int64}(undef, length(dims))    # Tensor indexing of Y for row 
+    tYj = Vector{Int64}(undef, length(dims))    # Tensor indexing of Y for column
+
+    @views tXikeep = tXi[keep]
+    @views tXitransp = tXi[transp]
+    @views tXjkeep = tXj[keep]
+    @views tXjtransp = tXj[transp]
+
+    @views tYikeep = tYi[keep]
+    @views tYitransp = tYi[transp]
+    @views tYjkeep = tYj[keep]
+    @views tYjtransp = tYj[transp]
+
+    for i in 1:dY
+        _tidx!(tXi, i, dims)
+        for j in 1:i
+            _tidx!(tXj, j, dims)
+
+            tYikeep .= tXikeep
+            tYitransp .= tXjtransp
+
+            tYjkeep .= tXjkeep
+            tYjtransp .= tXitransp
+
+            Yi, Yj = _idx(tYi, dims), _idx(tYj, dims)
+            Y[Yi, Yj] = X[i, j]
+            Y[Yj, Yi] = X[j, i]
+        end
+    end
+    return Y
+end
+"""
+    partial_trasnpose(X::AbstractMatrix, transp::Vector, dims::Vector)
+
+Takes the partial transpose of matrix `X` with subsystem dimensions `dims` on the subsystem `transp`.
+"""
+partial_transpose(X::AbstractMatrix, transp::Integer, dims::Vector{<:Integer}) = partial_transpose(X, [transp], dims)
+export partial_transpose

test/partial_tra.jl

@@ -24,4 +24,32 @@
         end
     end
 end
+
+@testset "Partial Transpose" begin
+    d1, d2, d3 = 2, 3, 4
+    for R in [Float64, Double64, Float128, BigFloat]
+        for T in [R, Complex{R}]
+            a = randn(T, d1, d1)
+            b = randn(T, d2, d2)
+            c = randn(T, d3, d3)
+            ab = kron(a, b)
+            ac = kron(a, c)
+            bc = kron(b, c)
+            abc = kron(ab, c)
+            @test partial_transpose(ab, [1, 2], [d1, d2]) ≈ transpose(ab)
+            @test partial_transpose(ab, 2, [d1, d2]) ≈ kron(a, transpose(b))
+            @test partial_transpose(ab, 1, [d1, d2]) ≈ kron(transpose(a), b)
+            @test partial_transpose(ab, Int64[], [d1, d2]) ≈ ab
+            @test partial_transpose(abc, [1, 2, 3], [d1, d2, d3]) ≈ transpose(abc)
+            @test partial_transpose(abc, [2, 3], [d1, d2, d3]) ≈ kron(a, transpose(b), transpose(c))
+            @test partial_transpose(abc, [1, 3], [d1, d2, d3]) ≈ kron(transpose(a), b, transpose(c))
+            @test partial_transpose(abc, [1, 2], [d1, d2, d3]) ≈ kron(transpose(a), transpose(b), c)
+            @test partial_transpose(abc, 3, [d1, d2, d3]) ≈ kron(ab, transpose(c))
+            @test partial_transpose(abc, 2, [d1, d2, d3]) ≈ kron(a, transpose(b), c)
+            @test partial_transpose(abc, 1, [d1, d2, d3]) ≈ kron(transpose(a), bc)
+            @test partial_transpose(abc, Int64[], [d1, d2, d3]) ≈ abc
+        end
+    end
+end
+
 #TODO add test with JuMP variables