⚡ Pluto.jl ⚡

Problem 1. Consider the density matrix:

$ρ = (\begin{matrix} 5 / 12 & 1 / 6 & 1 / 6 \\ 1 / 6 & 1 / 6 & 1 / 6 \\ 1 / 6 & 1 / 6 & 5 / 12 \end{matrix})$

show that we have a mixed state. Find the von Neumann entropy.

15.0 μs

Solution.

14.1 μs

x
 
begin 
    using LinearAlgebra
end

46.6 ms

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begin
    ⋅ = *
    𝚝𝚛 = tr
    entropy(X::Array{T,1}) where T<: Number = begin
        χ = 0.00e0
        for i=1:length(X)
            χ -= X[i]*log(X[i])
        end
        return χ
    end
            
    entropy(X::Array{T,2}) where T<: Number = -1⋅𝚝𝚛(X⋅log(X))
end;

62.0 μs

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begin
    ρ = [5//12 1//6 1//6;
         1//6  1//6 1//6;
         1//6  1//6 5//12]
end;

4.1 μs

First we check if the density matrix is idempotent, i.e. $ρ^{2} = ρ$ , if true the matrix is pure else its mixed.

8.2 μs

false

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ρ⋅ρ == ρ

7.2 μs

So the matrix is not a pure state and indeed a mixed.

The von Neumann entropy can be calculated in two equivalent ways:

$S (ρ) = - tr (ρ \log (ρ))$

$S (ρ) = \sum_{i}^{N} λ_{i} \log (λ_{i})$

where $λ_{i}$ are the eigenvalues. both calculations are given below.

11.2 μs

0.7724450450599503

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entropy(ρ)

70.5 μs

x
 
𝐞ᵢ = eigen(ρ).values;

36.9 μs

0.7724450450599518

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entropy(𝐞ᵢ)

431 ns