Quantum Computing Asked on July 27, 2020

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

$$sum_k (A_kotimes B_k)rho_{AB}(A_k^daggerotimes B_k^dagger)tag A$$

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 $Phi_{mathrm{LGM}}(rho)=sum_{ij}(A_iotimes B_j)rho(A_i^dagger otimes B_j^dagger)$, as any such $Phi_{mathrm{LGM}}$ should be by definition writable as composition/tensor product of two local operations: $Phi_{mathrm{LGM}}=Phi_AotimesPhi_B$, 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 $Phi$.

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 $N_i = A_iotimes B_i$.

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 $Iotimes A^{i_1}_{i_2}$, 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 $N_i=A_iotimes B_i$ with double index $iequiv{i_1,i_2}$.

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 $N_i=otimes B_iotimes cdots$.

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).

Correct answer by Norbert Schuch on July 27, 2020

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