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

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 $$\Phi$$.

• I don't have a rigorous answer or references, so I'm just going to comment. My guess is that the quantum state cannot be changed using classical communication, the communication can be sent by Alice and received by Bob, who will perform some local operation on his system based on Alice's message. In this sense the local operation applied by Bob will be some probabilistic mixture of operations, with the distribution given by the possible messages of Alice. But I'm interested too in a formal definition of CC as an operation on a bipartite system. – user2723984 Aug 12 at 16:35
• @user2723984 I was thinking along the same line, but I see two major issues in this line of reasoning: 1) does this actually encompass all possible LOCC scenarios? I can imagine a more complex scenario with multiple rounds of CC, like Alice does a partial measurement, tells the results to Bob which operates accordingly on his system, and then communicates the results to Alice which then performs another local measurement/operation on her side conditionally to Bob's results. Are all scenarios like this captured in the simple "A talks to B" scenario? – glS Aug 12 at 17:32
• 2) it seems to me that this reasoning relies upon the assumption that I can interpret $\sum_k A_k \rho A_k^\dagger$ as "the state $\rho$ is acted upon by one of the operators $A_k$ (up to some constant), I just don't know which one". But is this way of understand the Kraus decomposition of a channel correct? That's not obvious to me given that the $A_k$ are in general not unitary. – glS Aug 12 at 17:35
• on your second point, not all Kraus operators can be interpreted like this (I have posted a question on that point) but a probabilistic mixture of unitaries ia a valid Kraus operators set, moreover the ones that can't be expressed like this might simply be local non unitary transformations that Bob does to his system – user2723984 Aug 12 at 17:54

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 $$N_i = A_i\otimes B_i$$.
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 $$N_i=\otimes B_i\otimes \cdots$$.
• right, this makes a lot of sense. To translate into the formalism of maps, I guess the idea is that given a POVM $\sum_a F_a=I$ you find operators $A_a$ st $F_a=A_a^\dagger A_a$, and then the map corresponding to the POVM is $\Phi(X)=\sum_a A_a X A_a^\dagger$. The reasoning we are doing here with POVM would then seamlessly translate into the Kraus repr of the map. This suggests that one can understand the Kraus representation of a map as describing the mixture of post-measurement states with their probabilities. Is this interpretation always sensible that you know of? – glS Aug 14 at 9:26