⚡ Pluto.jl ⚡

Problem 2. Consider a $6 \times 6$ matrix $A = a_{i j}$ , then calculate the partial trace in the basis:

$(\begin{matrix} 1 \\ 0 \end{matrix}) \otimes I_{3}, (\begin{matrix} 0 \\ 1 \end{matrix}) \otimes I_{3} .$

Also calculate the partial trace with the basis:

$(\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}) \otimes I_{2}, (\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}) \otimes I_{2}, (\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}) \otimes I_{2}$

11.5 μs

Solution. The key thing to recall is ta the partial trace defines an operator of a subspace so:

$A := {tr}_{B} (A \otimes B) \equiv \sum_{i = 1}^{N_{B}} (I_{A} \otimes ⟨ ϕ_{i} |) (A \otimes B) (I_{A} \otimes | ϕ_{i} ⟩)$

where $| ϕ_{i} ⟩$ define an orthonormal basis in $B$ subspace.

11.6 μs

xxxxxxxxxx
 
using SymPy

8.4 s

x
 
begin
    ⊗(x,y)=kron(x,y)
    ⊗(x::Array{Sym,2},y::Array{Sym,2}) = kron(x,y)
    ⊗(x::Sym,y) = sympy.Matrix(kron(x.as_explicit(),y))
    ⊗(x,y::Sym) = sympy.Matrix(kron(x,y.as_explicit()))
    𝐓(x::Array{Sym,2}) = sympy.Matrix(sympy.transpose(x))
    𝚝𝚛(x) = sympy.trace(x)
    ⋅(x,y) = x*y
    ⋅(x::Union{Sym,SymMatrix},y) = sympy.MatMul(x.as_explicit(),y,evaluate=true)
    ⋅(x,y::Union{Sym,SymMatrix}) = sympy.MatMul(x,y.as_explicit(),evaluate=true)
end; 

115 μs

x
 
begin
    A = sympy.MatrixSymbol("A",6,6)
    
    b₁ = [1;
          0]
    
    b₂ = [0;
          1]
    
    I₃ = sympy.Identity(3)
end;

170 μs

Calculate the matrices associated with subspace basis.

8.3 μs

x
 
begin
    n₁,n₁ᵀ = b₁⊗I₃,𝐓(b₁⊗I₃)
    n₂,n₂ᵀ = b₂⊗I₃,𝐓(b₂⊗I₃)
end;

6.0 ms

Now calculate the partial trace in this basis using the equation above

11.1 μs

$[\begin{matrix} A_{0, 0} + A_{3, 3} & A_{0, 1} + A_{3, 4} & A_{0, 2} + A_{3, 5} \\ A_{1, 0} + A_{4, 3} & A_{1, 1} + A_{4, 4} & A_{1, 2} + A_{4, 5} \\ A_{2, 0} + A_{5, 3} & A_{2, 1} + A_{5, 4} & A_{2, 2} + A_{5, 5} \end{matrix}]$

x
 
n₁ᵀ ⋅ A ⋅ n₁ + n₂ᵀ ⋅ A ⋅ n₂

20.1 ms

Now looking at the other basis:

7.8 μs

xxxxxxxxxx
 
begin
    c₁ = [1;
          0;
          0]
    
    c₂ = [0;
          1;
          0]
    
    c₃ = [0;
          0;
          1]
    
    I₂ = sympy.Identity(2)
end;

121 μs

x
 
begin
    m₁,m₁ᵀ = c₁⊗I₂,𝐓(c₁⊗I₂)
    m₂,m₂ᵀ = c₂⊗I₂,𝐓(c₂⊗I₂)
    m₃,m₃ᵀ = c₃⊗I₂,𝐓(c₃⊗I₂)
end;

6.9 ms

$[\begin{matrix} A_{0, 0} + A_{2, 2} + A_{4, 4} & A_{0, 1} + A_{2, 3} + A_{4, 5} \\ A_{1, 0} + A_{3, 2} + A_{5, 4} & A_{1, 1} + A_{3, 3} + A_{5, 5} \end{matrix}]$

x
 
m₁ᵀ ⋅ A ⋅ m₁ + m₂ᵀ ⋅ A ⋅ m₂ + m₃ᵀ ⋅ A ⋅ m₃

13.9 ms