⚡ Pluto.jl ⚡

Problem 1. $| ψ ⟩$ is a given state in Hilbert space $C^{n}$ . If $X$ and $Y$ are $n \times n$ hermitian matrices. The correlation for the state $| p s i ⟩$ is given by:

$C_{X Y} (| ψ ⟩) := ⟨ ψ | X Y | ψ ⟩ - ⟨ ψ | X | ψ ⟩ ⟨ ψ | Y | ψ ⟩$

If $n = 4$ and $X$ and $Y$ are:

$X = (\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}) Y = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{matrix})$

and we consider the state:

$| ψ ⟩ = \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ 0 \\ 0 \\ 1 \end{matrix})$

. Evaluate the correlation measure.

20.5 μs

Solution. This is a fairly straightfoward calculation that can be done by hand. The correlation is 0.

8.1 μs

xxxxxxxxxx
 
begin
    using SymPy
end

8.3 s

x
 
begin
    ⋅(X,Y) = X * Y
    ⋅(X::Array{T,1},Y::Array{T,1}) where T<:SymPy.Sym = X[1] ⋅ Y[1]
    halfsqrt = 1/sympy.sqrt(2)
end;

233 μs

x
 
begin
    ψ = halfsqrt ⋅ [1;
                    0;
                    0;
                    1];
    ψ′ = ψ.adjoint()
end;

5.6 ms

xxxxxxxxxx
 
begin
    X = [0 0 0 1;
         0 0 1 0;
         0 1 0 0;
         1 0 0 0]
    Y = [1 0 0 0;
         0 0 0 1;
         0 0 1 0;
         0 1 0 0]
end;

27.6 ms

x
 
cor = (ψ′ ⋅ X ⋅ ψ) ⋅ (ψ′ ⋅ Y ⋅ ψ) .- (ψ′ ⋅ (X ⋅ Y) ⋅ ψ)' ;

427 ms

$C_{X Y} (| ψ ⟩) = 0$

13.9 μs