⚡ Pluto.jl ⚡

Problem 2. Consider the density matrix

$ρ = \frac{1}{2} (\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix})$

and another matrix $A$ which is $2 \times 2$ real valued and symmetric. Assuming that the $tr (ρ A) = - 1$ and $tr (ρ A^{2}) = 1$ . Recosntruct the matrix from this information.

16.1 μs

Solution 2.

16.8 μs

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using SymPy

350 μs

Setting up some usefull sympy functions.

10.4 μs

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begin
    ⋅(X,Y) = sympy.MatMul(X,Y,evaluate=true)
    𝚝𝚛(X) = sympy.trace(X)
end;

64.5 μs

Define the the matrices $ρ$ , $A$ , and $A^{2}$ , recall $A$ is real and symmetric.

20.6 μs

x
 
begin
    ρ = 1/2 * sympy.Matrix([ 1 -1;
                            -1  1]);
    A₁₁,A₁₂,A₂₂ = sympy.symbols("A₁₁,A₁₂,A₂₂",real=true)
    A₂₁ = A₁₂
    A = sympy.Matrix([A₁₁ A₁₂;
                      A₂₁ A₂₂])
    A² = A⋅A
end;

202 ms

The matrix $ρ A$ then looks like:

14.9 μs

$[\begin{matrix} 0.5 A ₁ ₁ - 0.5 A ₁ ₂ & 0.5 A ₁ ₂ - 0.5 A ₂ ₂ \\ - 0.5 A ₁ ₁ + 0.5 A ₁ ₂ & - 0.5 A ₁ ₂ + 0.5 A ₂ ₂ \end{matrix}]$

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ρ⋅A

3.4 ms

Now defining expressions corresponding to the conditions $tr (ρ A) = - 1$ and $tr (ρ A^{2}) = 1$ .

17.7 μs

expr1

$0.5 A ₁ ₁ - 1.0 A ₁ ₂ + 0.5 A ₂ ₂ + 1$

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expr1 = 𝚝𝚛(ρ⋅A) + 1

6.4 ms

expr2

$0.5 A ₁ ₁^{2} - 1.0 A ₁ ₁ A ₁ ₂ + 1.0 A ₁ ₂^{2} - 1.0 A ₁ ₂ A ₂ ₂ + 0.5 A ₂ ₂^{2} - 1$

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expr2 =𝚝𝚛(ρ⋅A²) - 1

20.8 ms

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solution = solve([expr1,expr2],A,dict=true); 

139 ms

The solutions for the matrix $A$ is then:

$A = [\begin{matrix} A ₂ ₂ & A ₂ ₂ + 1.0 \\ A ₂ ₂ + 1.0 & A ₂ ₂ \end{matrix}]$

where $A_{22}$ is some real valued constant.

6.5 ms

Run this command to plug solution into original symbolic variable.

A.subs(solution...)

8.9 μs