Why can any LOCC operation be written as $\sum_k (A_k\otimes B_k)\rho(A_k^\dagger\otimes B_k^\dagger)$?

Question

This statement can be found in Vedral et al. 1997, eq. (1).

More precisely, given a bipartite state $\rho_{AB}$, they claim that any operation on $\rho_{AB}$ involving local operations plus classical communication can be written as $$\sum_k (A_k\otimes B_k)\rho_{AB}(A_k^\dagger\otimes B_k^\dagger)\tag A$$ for some operators $A_k, B_k$. This is a seminal result, used for example to prove the existence of bound entangled states.

I don't have any problem with the previous statement they make about general local operations being writable as $\Phi_{\mathrm{LGM}}(\rho)=\sum_{ij}(A_i\otimes B_j)\rho(A_i^\dagger \otimes B_j^\dagger)$, as any such $\Phi_{\mathrm{LGM}}$ should be by definition writable as composition/tensor product of two local operations: $\Phi_{\mathrm{LGM}}=\Phi_A\otimes\Phi_B$, and then if $A_i$ and $B_j$ are the Kraus operators of $\Phi_A$ and $\Phi_B$ we get the result.

However, when we allow classical communication it seems less obvious what a generic operation should look like. The Kraus decomposition of such a map $\Phi$ would a priori be written $\Phi(\rho_{AB})=\sum_k A_k \rho_{AB} A_k^\dagger$ where $A_k$ acts nonlocally on $AB$, but then I'm not sure how to translate the LOCC condition into a constraint for $\Phi$.

Answer 1

The form (A) above is known as a separable superoperator. The effect of any LOCC protocol can be described as a separable superoperator, or as a separable POVM with POVM elements $N_i = A_i\otimes B_i$.

This can be seen as follows (adapted from this answer - I will focus on the POVM case, the superoperator is obtained by ignoring the final outcome of the POVM):

Denote the parties by Alice and Bob. Without loss of generality, we can start with an action of Alice. Alice's first measurement has POVM elements $A_{i_1}\otimes I$. Alice then communicates her outcome $i_1$ to Bob. Bob's subsequent measurement has elements $I\otimes A^{i_1}_{i_2}$, where $i_2$ enumerates Bob's outcomes, and $A^{i_1}$ indicates that Bob's POVM can depend on Alice's outcome. The total POVM of both has then elements $$ N_{i_1,i_2}=A_{i_1}\otimes B^{i_1}_{i_2}\ , $$ which is a separable POVM $N_i=A_i\otimes B_i$ with double index $i\equiv{i_1,i_2}$.

Clearly, this can be iterated to an arbitrary number of rounds, and generalized to an arbitrary number of parties, and will always have POVM elements of the form $N_i=\otimes B_i\otimes \cdots$.

Conversely, not every separable POVM can be written as a LOCC POVM. A counterexample is given in Bennett et al., Quantum Nonlocality without Entanglement, Phys. Rev. A. 59, 1070 (1999).