Problems in the implementation of Density matrix.

[alex180500]

In Density.doit() there is a mistake for the _generate_outer_prod() method when returning return Mul(*c_part1)*Mul(*c_part2) * op. The commutative part 2 should be the complex conjugate, so the return should be return Mul(*c_part1)*conjugate(Mul(*c_part2))*op.

Example:

c = sp.Matrix(sp.symbols("c_:2:2", complex=True, real=False)).reshape(2, 2)
state = sum([el * Qubit(f"{bin(count)[2:].zfill(2)}") for count, el in enumerate(c)])

$$c_{00}\ket{00} + c_{01}\ket{01} + c_{10}\ket{10} + c_{11}\ket{11}$$

What should be the density matrix:

rho = state * q.Dagger(state)
q.represent(rho)

$$\left[\begin{matrix}c_{00} \overline{c_{00}} & c_{00} \overline{c_{01}} & c_{00} \overline{c_{10}} & c_{00} \overline{c_{11}}\\c_{01} \overline{c_{00}} & c_{01} \overline{c_{01}} & c_{01} \overline{c_{10}} & c_{01} \overline{c_{11}}\\c_{10} \overline{c_{00}} & c_{10} \overline{c_{01}} & c_{10} \overline{c_{10}} & c_{10} \overline{c_{11}}\\c_{11} \overline{c_{00}} & c_{11} \overline{c_{01}} & c_{11} \overline{c_{10}} & c_{11} \overline{c_{11}}\end{matrix}\right]$$

What the density matrix is using the Density class:

q.represent(Density([state, 1]))

$$\left[\begin{matrix}c_{00}^{2} & c_{00} c_{01} & c_{00} c_{10} & c_{00} c_{11}\\c_{00} c_{01} & c_{01}^{2} & c_{01} c_{10} & c_{01} c_{11}\\c_{00} c_{10} & c_{01} c_{10} & c_{10}^{2} & c_{10} c_{11}\\c_{00} c_{11} & c_{01} c_{11} & c_{10} c_{11} & c_{11}^{2}\end{matrix}\right]$$

[alex180500]

Other than that also evaluating the partial trace over a Density matrix gets a wrong result (even assuming the wrong matrix):
Example:

rho = Density([state, 1])
q.represent(rho)

$$\left[\begin{matrix}c_{00}^{2} & c_{00} c_{01} & c_{00} c_{10} & c_{00} c_{11}\\c_{00} c_{01} & c_{01}^{2} & c_{01} c_{10} & c_{01} c_{11}\\c_{00} c_{10} & c_{01} c_{10} & c_{10}^{2} & c_{10} c_{11}\\c_{00} c_{11} & c_{01} c_{11} & c_{10} c_{11} & c_{11}^{2}\end{matrix}\right]$$

Partial trace calculates:

q.represent(Tr(rho, [0]).doit())

$$\left[\begin{matrix}c_{00}^{2} + c_{01}^{2} & 0\\0 & c_{10}^{2} + c_{11}^{2}\end{matrix}\right]$$

But instead we should get:

$$\left[\begin{matrix}c_{00}^{2} + c_{01}^{2} & c_{00} c_{10} + c_{01} c_{11}\\c_{00} c_{10} + c_{01} c_{11} & c_{10}^{2} + c_{11}^{2}\end{matrix}\right]$$

[alex180500]

Other than that also evaluating the partial trace over a Density matrix gets a wrong result (even assuming the wrong matrix):

q.represent(Tr(rho, [0]).doit())

Thanks a lot @amishhaa for the PR, for this other problem I create another issue?

[amishhaa]

@alex180500 I think it's best to create another issue so it comes to attention faster and can be fixed ASAP.