⚡ Pluto.jl ⚡

Problem 1. GIven the standard basis $e_{1}, e_{2}, e_{3}, e_{4}$ in $C^{4}$ and the untary matrices:

$U_{C N O T} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix}), V_{H} = H \otimes I_{2}$

where $H$ is the Hadamard matrix $\frac{1}{\sqrt{2}} (\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix})$ . Show that the unitary matrices acting on the standard basis generates Bell states. That is:

$U_{C N O T} V_{H} | e_{i} ⟩$

9.9 μs

Solution. This is just a simple demonstration of unitary operators acting on a basis set, therefore we just go through the matrix algebra as shown below.

9.2 μs

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using SymPy

8.4 s

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begin
    sqrthalf = 1/sympy.sqrt(2)
    ⋅(x,y) = x*y
    ⊗(x,y) = kron(x,y)
end;

75.6 ms

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begin
    𝐞₁ = [1;
          0;
          0;
          0]
    𝐞₂ = [0;
          1;
          0;
          0]
    𝐞₃ = [0;
          0;
          1;
          0]
    𝐞₄ = [0;
          0;
          0;
          1]
    𝐞 = [𝐞₁ 𝐞₂ 𝐞₃ 𝐞₄];
end;
          

3.0 μs

x
 
begin
    U = [1 0 0 0;
         0 1 0 0;
         0 0 0 1;
         0 0 1 0];
    H = sqrthalf ⋅ [ 1  1;
                     1 -1];
    I₂ = sympy.eye(2,2) #could also use sympy.Identity(2)
    V = H ⊗ I₂
end;

512 ms

13.3 ms

$U \cdot H \otimes I | e_{4} ⟩ = [\begin{matrix} 0 \\ \frac{\sqrt{2}}{2} \\ - \frac{\sqrt{2}}{2} \\ 0 \end{matrix}]$

6.4 ms