Can every bipartite state be written as $\rho_{AB} = \sum_{ij} c_{ij}\sigma_A^i\otimes \omega_B^j$?

Question

Can every bipartite quantum state (including entangled ones) be written in the following way

$$\rho_{AB} = \sum_{ij} c_{ij}\sigma_A^i\otimes \omega_B^j$$

where $\sigma_A^i$ and $\omega_B^j$ are density matrices? And if $\rho_{AB}$ is a quantum state, must it also be the case that $c_{ij}$ are nonnegative? If the answers are yes, how can one show this?

I ask this because of a claim that the set of product states $\rho_A\otimes\rho_B$ span the vector space of all operators acting on $H_A\otimes H_B$ in an answer here - https://quantumcomputing.stackexchange.com/a/5066/4831.

Answer 1

If all $c_{ij}$ are non-negative, this is the definition of a separable state. But you can argue this by giving a protocol that lets you construct, using LOCC, the state $\rho_{AB}$.

You probably could use this description to cover entangled states if you let the $c_{ij}$ be negative. Here's a simple argument for bipartite states of qubits: the Pauli matrices form a basis, meaning that $$ \rho_{AB}=\sum q_{ij}\sigma_i\otimes\sigma_j $$ (I'm using $\sigma$ here for Pauli matrices, NOT density matrices). Since $\rho_{AB}$ is Hermitian, the $q_{ij}$ are real. Now, I can always rewrite this as $$ \rho_{AB}=\left(q_{00}-\sum_{(i,j)\neq (0,0)}|q_{ij}|\right)I+ \sum_{(i,j)\neq (0,0)} |q_{ij}|(I+\text{sgn}(q_{ij})\sigma_i\otimes\sigma_j). $$ Now note that every term $I\pm\sigma_i\otimes\sigma_j$ is non-negative, with a separable basis, i.e. is a separable state. Indeed, the coefficient in front of every term is positive except, perhaps, that of the identity term.

(I should perhaps clarify that the term in front of the identity being positive is sufficient for the state to be separable, but is not necessary. In some cases there are better separable decompositions than the one I've given.)

There's a similar basis of Hermitian matrices for any Hilbert space dimension, so the result holds just as well for any bipartite state, not just qubits.

Answer 2

I know the question is already answered, but there was some question on my comment and I wanted to elaborate on that.

First, let us consider one system only. The $\mathbb{R}$-span of all states $\rho$ is the space of Hermitian operators. Indeed, by the spectral decomposition, already the set of pure states is enough. This also implies that the $\mathbb{C}$-span contains all linear operators, since we can decompose any operator into a Hermitian and anti-Hermitian part and expand those in states.

For two (or more) systems, the dimension of $L(H_A\otimes H_B)=L(H_A)\otimes L(H_B)$ is the product of the individual dimensions, $\dim L(H_A) \times \dim L(H_B)$. This implies that taking tensor products of generating sets of $L(H_A)$ and $L(H_B)$ has to yield a generating set, too. Just reduce the generating set to a basis of $L(H_A)$ and $L(H_B)$ and you'll get a product basis which lies in the product of the generating sets.

Note that this also shows that this is not as special property of states. Any generating set will do.

What is special is that states are Hermitian and trace-one. Thus, they all lie in the according affine space in the real vector space of Hermitian operators. Hence, any other Hermitian trace-one operator is not any arbitrary (real) linear combination of states, but an affine one (take the trace on both sides) $$ A = \sum_{i,j} c_{i,j} \rho_i \otimes \rho_{j} \quad \Rightarrow \quad \sum_{ij} c_{ij} = 1. $$ As stated before, if we restrict to coefficients to be non-negative, we get the convex hull of product states, which is the set of separable states.

BTW, this is the geometric background for the robustness of entanglement which is an entanglement monotone defined as the minimum negativity in an affine decomposition of a state $\rho$ into product states.