⚡ Pluto.jl ⚡

Problem 2. Consider the standard basis $| 0 ⟩$ , $| 1 ⟩$ , and the state:

$| ψ ⟩ = \frac{1}{\sqrt{2}} (| 0 ⟩ \otimes | 0 ⟩ + | 1 ⟩ \otimes | 1 ⟩)$

Given the projection matrices:

$P_{1} = \frac{1}{2} (I_{2} - σ_{x}) P_{2} = \frac{1}{2} (I_{2} + σ_{z})$

Find the expectation value for the projection matrices on the state vector in the $C^{2}$ Hilbert space

14.3 μs

Solution.

12.7 μs

xxxxxxxxxx
 
begin
    using SymPy
end

201 μs

xxxxxxxxxx
 
begin
    𝑖 = sympy.I
    sqrthalf = 1/sympy.sqrt(2)
    ⋅ = *
    ⊗(X,Y) = kron(X,Y)
end;

157 μs

xxxxxxxxxx
 
begin
    b₀ = [1;
          0]
    b₁ = [0;
          1]
    ψ = sqrthalf ⋅ (b₀ ⊗ b₀ + b₁ ⊗ b₁)
    ψᵃ = ψ.adjoint()
end;

244 ms

xxxxxxxxxx
 
begin
    I₂ = sympy.eye(2,2)
    σx = sympy.Matrix([0 1;
                       1 0])
    σz = sympy.Matrix([1 0;
                       0 -1])
end;

837 μs

xxxxxxxxxx
 
begin
    P₁ = 1/2 ⋅ (I₂ - σx)
    P₂ = 1/2 ⋅ (I₂ + σz)
    P₁₂ = P₁ ⊗ P₂
end;

1.1 ms

The projection matrix over the $C^{2}$ space is then:

$P_{12} = P_{1} \otimes P_{2} =$

$[\begin{matrix} 0.5 & 0 & - 0.5 & 0 \\ 0 & 0 & 0 & 0 \\ - 0.5 & 0 & 0.5 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]$

1.9 ms

xxxxxxxxxx
 
expectation = ψᵃ⋅P₁₂⋅ψ;

1.0 ms

The expectation, $⟨ ψ | P_{1} \otimes P_{2} | ψ ⟩$ , is:

$0.250000000000000$

10.2 μs