⚡ Pluto.jl ⚡

Problem 1. Given the Pauli matrices

$σ_{x} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), σ_{y} = (\begin{matrix} 0 & i \\ - i & 0 \end{matrix}), σ_{z} = (\begin{matrix} 0 \\ 0 & - 1 \end{matrix})$

and a hermitian matrix that has dimensions 8x8 and written as:

$H = (σ_{x} \otimes σ_{x} + σ_{y} \otimes σ_{y} + σ_{z} \otimes σ_{z}) \otimes σ_{x} .$

Find the eigenvalues and and eigenvectors.

26.5 μs

Solution 1. The interesting thing to note here is that the hermitian matrix can be written as the direct sum:

$H = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) \oplus (\begin{matrix} 0 & - 1 & 0 & 2 \\ - 1 & 0 & 2 & 0 \\ 0 & 2 & 0 & - 1 \\ 2 & 0 & - 1 & 0 \end{matrix}) \oplus (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$

This means that the eigenspectrum can be calculated from the $2 \times 2$ and $4 \times 4$ matrices.

19.3 μs

xxxxxxxxxx
 
begin
    using SymPy
end

11.7 s

x
 
begin
    𝑖 = sympy.I
    ⊗(x,y)=kron(x,y)
    ⊗(x::Array{Sym,2},y::Array{Sym,2}) = kron(x,y)
    𝐓(x::Array{Sym,2}) = sympy.Matrix(sympy.transpose(x))
    
    """Direct sum of vector space.
    """ 
    ⊕(x,y) = begin
        xdim = size(x);
        ydim = size(y);
        M = [           x           zeros(Int,xdim[1],ydim[2]);
              zeros(Int,ydim[1],xdim[2])          y           ];
        return M
    end
end;

658 μs

xxxxxxxxxx
 
begin
    σ₁ = [0 1;
          1 0]
    σ₂ = [0 -𝑖;
          𝑖 0]
    σ₃ = [1 0;
          0 -1]   
end;

168 μs

The hermitian matrix given by the expression above is then:

14.5 μs

$[\begin{array}{rrrrrrrr} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 & 2 & 0 & 0 \\ 0 & 0 & - 1 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 & 0 & - 1 & 0 & 0 \\ 0 & 0 & 2 & 0 & - 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{array}]$

x
 
H = (σ₁ ⊗ σ₁ + σ₂ ⊗ σ₂ + σ₃ ⊗ σ₃) ⊗ σ₁

12.7 ms

Now just checking the direct sum comparison.

15.1 μs

x
 
begin
    D = Sym[ 0 -1  0  2;
            -1  0  2  0;
             0  2  0 -1;
             2  0 -1  0];
    H′ = σ₁ ⊕ D ⊕ σ₁
end

65.8 ms

x
 
@assert H == H′ "Direct sum doesn't equal hermitian matrix"

341 μs

Now finding the eigenvalues and eigenvectors

14.6 μs

x
 
begin
    ℍ = H.eigenvects(chop=true)
    #unpack :e - eigenvalue, :m - multiplicty (degeneracy)
    eₕ = [Dict(:e=>e[1],:m=>e[2]) for e in ℍ]
    vₕ = [v[end][1:end] for v in ℍ]
end;

252 ms

99.4 μs

$H : [λ_{1} = - 3, λ_{2} = - 1, λ_{3} = 1, λ_{4} = 3]$

102 ms

Let see how these eigenvalues compare to the eigenvalues of the matrix $D$ .

15.8 μs

xxxxxxxxxx
 
begin
    𝔻 = D.eigenvects(chop=true)
    eᵨ = [Dict(:e=>e[1],:m=>e[2]) for e in 𝔻]
    vᵨ = [v[end][1:end] for v in 𝔻]
end;

246 ms

$D : [λ_{1} = - 3, λ_{2} = - 1, λ_{3} = 1, λ_{4} = 3]$

we find that the eigenvalues are the same but the degeneracy is now removed. We can look at the eigenvectors the the $D$ matrix:

75.1 ms

83.1 μs

eigenvector index 1

14.3 μs

$[\begin{matrix} - 1 \\ - 1 \\ 1 \\ 1 \end{matrix}]$

2.3 ms