⚡ Pluto.jl ⚡

Problem 2.

$K_{1} = (\begin{matrix} 0 & 0 & - i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{matrix}) K_{2} = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & - 1 \end{matrix})$

and the normalized state in $C^{3}$ ,

$| ψ ⟩ = \frac{1}{\sqrt{3}} (\begin{matrix} 1 \\ 1 \\ 1 \end{matrix})$

Find the left and right-hand side of the entropi inequality.

12.9 μs

Solution.

The entropic inequality is given by:

$S (ρ) \leq \log (n)$

where $n$ is the dimensions of $ρ$ .

In this case we are interested in the Shannon information entropy associated with the hermitian matrices. This requires determining the spectral decomposition, i.e., eigenvectors. Moreover we actuall what to calculate the entropic uncertainty relation for the two hermitian matrices acting on this Hilbert space.

13.0 μs

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begin
    using SymPy
end

8.9 s

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begin
    𝑖 = sympy.I
    ⋅(x,y) = x*y
    ⋅(x::Array{Sym,2},y) = x.dot(y)
    norm(x,y) = begin
        braket = x.adjoint()⋅x
        sympy.sqrt(braket[1])
    end
end;

69.4 μs

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begin
    K₁ = [0 0 -𝑖;
          0 0  0;
          𝑖 0  0];
    K₂ = sympy.Matrix([1 0  0;
                       0 0  0;
                       0 0 -1]);
    ψ = 1/sympy.sqrt(3) ⋅ [1;
                           1;
                           1];
end;

1.1 ms

The Shannon information entropy for the hermitian matrix, $O$ , is given by:

$S_{| ψ ⟩} (O) = - \sum_{i = 1}^{n} p_{i} (O, | ψ ⟩) \cdot \ln (p_{i} (O, | ψ ⟩))$

where the probability $p_{i}$ is obtained from:

$p_{i} (O, | ψ ⟩) = | ⟨ ψ | α_{i} ⟩ |^{2}$

with $| α_{i} ⟩$ being the $i^{th}$ eigenvector.

Therefore we first need to find the eigenspectrum for the matrices $K_{1}$ and $K_{2}$ "

25.9 μs

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begin
    𝐞₁ = K₁.eigenvects()
    𝐞₂ = K₂.eigenvects()
end;

7.7 ms

Now normalize the eigenvectors since we want the probabilities: $p (K, | ψ ⟩) = | ⟨ ψ | α ⟩ |^{2}$

9.8 μs

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begin
    # There are 3 eigenvectors
    𝐯₁ = Array{Array{Sym}}(undef,3)
    𝐯₂ = Array{Array{Sym}}(undef,3)
    for i=1:3 
        ν₁,ν₂ = 𝐞₁[i][3][1],𝐞₂[i][3][1]
        n₁,n₂ = norm(ν₁,ν₁),norm(ν₂,ν₂)
        𝐯₁[i] = ν₁/n₁
        𝐯₂[i] = ν₂/n₂
    end
end

8.6 ms

Calculating the Shannon entropy for each hermitian matrice:

7.9 μs

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begin
    S₁ = 0.00
    S₂ = 0.00
    for i=1:3
        p¹ᵢ = sympy.Abs(ψ.adjoint()⋅𝐯₁[i])^2 
        p²ᵢ = sympy.Abs(ψ.adjoint()⋅𝐯₁[i])^2
        S₁ -= p¹ᵢ ⋅ sympy.log(p¹ᵢ)
        S₂ -= p²ᵢ ⋅ sympy.log(p²ᵢ)
    end
end

8.2 ms

Now we want to calculate the terms on the right and left sides of the following inequality:

$S_{| ψ ⟩} (K_{1}) + S_{| ψ ⟩} (K_{2}) \geq - 2 \ln (max_{1 \leq i, j \leq n} | ⟨ α_{i} | β_{j} ⟩ |)$

The left hand side is then just:

$2 \log (3)$ .

Calculating the right hand side:

359 μs

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rhs = -2⋅sympy.log(maximum([sympy.Abs(𝐯₁[i].adjoint()⋅𝐯₂[j]) for i=1:3,j=1:3]));

49.4 ms

$2 \log (3) \geq 0$

The right hand side of the inequality makes sense since both $K_{1}$ and $K_{2}$ share a common eigenvector:

$(\begin{matrix} 0 \\ 1 \\ 0 \end{matrix})$

462 μs