⚡ Pluto.jl ⚡

Problem 1. Let $| 0 ⟩$ and $| 1 ⟩$ be the standard basis in $C^{2}$ , consider the Bell state corresponding to two qubits:

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

If we now define two projection operators/matrices:

${\hat{P}}_{1} = (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}) {\hat{P}}_{2} = (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}) .$

Calculate the measurement probability of qubit 1 corresponding to ${\hat{P}}_{1}$ . Furthermore show what the post-measurment state is and the probability for qubit 2 corresponding to ${\hat{P}}_{1}$ in the post-measured state.

11.8 μs

Solution.

12.9 μs

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begin
    using SymPy
end

12.6 s

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begin
    halfsqrt = 1/sympy.sqrt(2)
    ⋅ = *
    ⊗(X,Y) = kron(X,Y)
end;

169 μs

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begin
    b₀ = [1;
          0]
    b₁ = [0;
          1]
    ψ = halfsqrt ⋅ (b₀ ⊗ b₀ + b₁ ⊗ b₁)
    ψ′= ψ.adjoint()
end;

1.1 ms

$| ψ ⟩ = [\begin{matrix} \frac{\sqrt{2}}{2} \\ 0 \\ 0 \\ \frac{\sqrt{2}}{2} \end{matrix}]$

1.8 ms

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begin
    P̂₁ = sympy.Matrix([ 1 0;
                        0 0])
    P̂₂ = sympy.Matrix([ 0 0;
                        0 1])
    I₂ = sympy.eye(2,2)
end;

25.3 ms

Notice that ${\hat{P}}_{1} \cdot {\hat{P}}_{2}$ is:

7.8 μs

$[\begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array}]$

557 μs

The measurment probability of qubit 1 having an outcome corresponding to ${\hat{P}}_{1}$ is given as:

10.6 μs

𝓞

$[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

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𝓞 = P̂₁ ⊗ I₂

255 μs

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𝓞′ = 𝓞.adjoint();

1.4 ms

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p₁ = ψ′ ⋅ ( 𝓞′ ⋅ 𝓞 ) ⋅ ψ;

3.1 ms

which is $p_{1} (0) = 1 / 2$ and noting that:

9.1 μs

${({\hat{P}}_{1} \otimes I_{2})}^{2} = {\hat{P}}_{1} \otimes I_{2}$

8.6 μs

The resulting state from the projective measurement is then:

11.1 μs

$[\begin{array}{r} 1 \\ 0 \\ 0 \\ 0 \end{array}]$

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ϕ = 1/√p₁[1] ⋅ 𝓞 ⋅ ψ |> sympy.Matrix

1.9 ms

This resulting state is no longer entangled (i.e. it collapsed in the parlance of measurment interpretation). The measurement probability on the projective measurment state for the outcome of qubit 2 corresponding to the operaor ${\hat{P}}_{1}$ is such:

8.0 μs

x
 
begin
    ϕ′ = ϕ.adjoint()
    p₂ = ϕ′ ⋅ ( 𝓞′ ⋅ 𝓞 ) ⋅ ϕ
end;

3.6 ms

$p_{2} (0) = 1$

9.2 μs