Unitary over bipartite states that can turn a non-product state into a product state

Question

Consider a bipartite quantum state $\rho_{AB}$ over a product of finite-dimensional Hilbert spaces $\mathcal{H}_A \otimes \mathcal{H}_B$. Does there exists a unitary $U$ over $\mathcal{H}_A \otimes \mathcal{H}_B$ such that:

• For any bipartite $\rho_{AB}$, the reduced density matrix over $A$ is preserved $\mathrm{tr}_B{[U(\rho_{AB})U^*]} = \rho_A$.
• For some a priori given $\sigma_{AB}$, the output is a product state: $U(\sigma_{AB})U^* = \sigma_A \otimes \omega_B$ (where $\omega_B$ can be anything).

I'm also interested in the case where such properties might only hold up to some arbitrary $\epsilon >0$ accuracy.

Answer 1

The first condition is satisfied for example by unitaries of the form $U = e^{i\theta}I_A \otimes U_B$ where $I_A$ is identity on subsystem $A$, $U_B$ is any unitary on subsystem $B$ and the phase factor $e^{i\theta}$ is irrelevant.

---

Let us consider the second condition. It turns out that the condition cannot be guaranteed for all states $\sigma_{AB}$. More precisely, there are states $\sigma_{AB}$ such that for every unitary $U$ and every state $\omega_B$ of subsystem $B$ we have $U\sigma_{AB}U^* \ne \sigma_A \otimes \omega_B$. This is a consequence of two facts: that unitary transformations preserve eigenvalues of density matrices and that spectra (sets of eigenvalues) of generic density matrices cannot be reproduced by spectra of product states.

More formally, we can state the first fact by saying that for every unitary $U$, $\lambda$ is an eigenvalue of $\rho$ if and only if it is an eigenvalue of $U\rho U^*$.

In order to show the second fact, first note that the eigenvalues of an $n \times n$ density matrix lie in an $(n-1)$-simplex. Let $n_A = \dim \mathcal{H}$ and $n_B = \dim \mathcal{H}_B$. If $\lambda^A_i$ denotes the eigenvalues of $\sigma_A$ and $\lambda^B_j$ denotes the eigenvalues of $\omega_B$ then the eigenvalues of $\sigma_A \otimes \omega_B$ are the products $\lambda^{AB}_{ij} = \lambda^A_i \lambda^B_j$. Thus, the eigenvalues of $\sigma_A \otimes \omega_B$ lie in the Cartesian product of two simplices that can be described using $(n_A - 1) (n_B - 1)$ real parameters. On the other hand, the eigenvalues of an arbitrary joint density matrix on systems $A$ and $B$ lie in an $(n_An_B - 1)$-simplex. Thus, by a simple parameter counting argument we see that the set of spectra of product states is a measure zero subset of the spectra of arbitrary states.

For a concrete example, suppose that $A$ and $B$ are qubits and that $\sigma_{AB}$ has eigenvalues $0, \frac{1}{4}, \frac{1}{4}, \frac{1}{2}$. Note that there do not exist two sets of numbers $\{\lambda^A_1, \lambda^A_2\}$ and $\{\lambda^B_1, \lambda^B_2\}$ such that

$$ \{\lambda^A_1\lambda^B_1, \lambda^A_1\lambda^B_2, \lambda^A_2\lambda^B_1, \lambda^A_2\lambda^B_2\} = \{0, \frac{1}{4}, \frac{1}{4}, \frac{1}{2}\}. $$

Consequently, there is no unitary $U$ such that $U\sigma_{AB}U^* = \sigma_A \otimes \omega_B$.