Necessary condition for local quantum operations and classical communication with extensive violation by separable operations

We give a conceptually simple necessary condition such that a separable quantum operation can be implemented by local operations on subsystems and classical communication between parties (LOCC), a condition which follows from a novel approach to understanding LOCC. This necessary condition applies to all LOCC protocols involving any number of parties and any finite number of rounds of communication. Furthermore, it demonstrates an extremely strong difference between separable operations and LOCC, in that there exist examples of the former for which the condition is extensively violated. More precisely, the violation by separable operations of our necessary condition for LOCC grows without limit as the number of parties increases.

Figure 1

Illustration of the two types of removals used in pruning a canonical LOCC tree. (a) A type-1 removal, where the parent $n_p$ is removed along with the maximal keeperless subtree $T$. This type of removal is used when there is at least one ${\widehat{\mathcal{K}}}_j$ leaf in $T$ whose corresponding keeper is not in $T$'s sibling subtree, $T^{\prime }$. The root of $T^{\prime }$ is $n_c$, which may be the only node in $T^{\prime }$, in which case it turns out that $n_c$ is itself a keeper leaf. (b) A type-2 removal where $n_c$, which is the root of the sibling subtree of $T$, is removed with $T$. This type of removal is used when every leaf in $T$ has its corresponding keeper leaf in either $T_1$ or $T_2$. Under these circumstances, $n_c$ cannot be a leaf.

Figure 2

A type-1 removal creates a new sibling pair, ${\tilde{n}}_c^{\prime }$ and ${\tilde{n}}_p^{\prime }$, but this pair will never be involved in a subsequent type-2 removal, showing that type-2 removals always involve siblings that were siblings in the original tree.