using a unitary matrix to transpose

Question

A unitary matrix U is a matrix such that the conjugate transpose of U, when multiplied on the right with U, yields identity. My question is, is it possible to obtain the transpose of any density matrix using some unitary operation? (I read somewhere that the transpose operation is an 'anti-unitary' operation, but I don't think that means that a unitary operation can NOT simulate a transpose operation).

Answer 1

The transpose $\rho^T$ of a density matrix $\rho$ is again a density matrix, furthermore it has the same eigenvalues as $\rho$ (including multiplicities). Therefore $\rho^T$ is unitarily equivalent to $\rho$, i.e. there exists a unitary matrix $U_\rho$ such that $\rho^T= U^*_\rho \rho U_\rho$.

But this matrix depends on $\rho$. If there existed a unitary matrix $U$ such that $\rho^T=U^*\rho U$ for all density matrices, then this would imply that the map $\rho\mapsto\rho^T$ is completely positive, which it is not (as Juan already pointed out.

Note also that taking the transpose depends on the choice of a basis!

Answer 2

The transpose is not a unitary map. One simple way to understand this is by recalling that the transpose is not a completely positive map. More precisely, let $T$ be the transpose map. Then the state $\rho_1=(\hat{I}\otimes T)\cdot\rho$ in general will not be a valid density natrix. However, for maps of the form $\hat{I}\otimes U$ for some unitary $U$, the state $\rho_2=(\hat{I}\otimes U)\cdot\rho$ will always be a valid density matrix. Therefore, the transposition map is not unitary.

In some cases, the transpose map might be equivalent to some unitary map for a particular state, but this does not mean that the map itself can be represented as a unitary.