⚡ Pluto.jl ⚡

Problem 1. Consider the Hadamard basis in $C^{2}$ :

$| ψ_{+} ⟩ = \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ 1 \end{matrix}) | ψ_{-} ⟩ = \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ - 1 \end{matrix})$

Apply the Kronecker (tensor) product to find a basis in a new $C^{4}$ .

12.6 μs

Solution 1

6.4 μs

This is a simple permutation of tensor products for the basis states

4.5 μs

xxxxxxxxxx
 
using SymPy

10.9 s

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

92.5 ms

xxxxxxxxxx
 
begin
    ψₚ = sqrthalf ⋅[1;
                    1]
    ψₘ = sqrthalf ⋅[1;
                   -1]
end;

85.3 ms

There are a total of $2^{\otimes N}$ basis states, $N = 2$ in this problem.

18.7 μs

xxxxxxxxxx
 
begin
    ψ₁ = ψₚ ⊗ ψₚ
    ψ₂ = ψₚ ⊗ ψₘ
    ψ₃ = ψₘ ⊗ ψₚ
    ψ₄ = ψₘ ⊗ ψₘ
end;

1.0 s

xxxxxxxxxx
 
Ψ = [ψ₁ ψ₂ ψ₃ ψ₄];

3.5 μs

The new basis set is:

9.5 μs

31.3 ms

$| ψ_{3} ⟩ = [\begin{matrix} \frac{1}{2} \\ \frac{1}{2} \\ - \frac{1}{2} \\ - \frac{1}{2} \end{matrix}]$

530 ms

And we can check they are orthonormal via inner product:

10.7 μs

75.1 ms

$⟨ ψ_{1} | ψ_{2} ⟩ = 0$

1.0 ms

Main.workspace33.factor

99.4 ms