LOCC

Definition

Let $H_A$ and $H_B$ be finite-dimensional complex Hilbert spaces, and write $B(H)$ for the operators on a space $H$. A local instrument of party $A$ is a finite family $(\mathcal{E}_j)_j$ of completely positive maps on $B(H_A)$ whose sum $\sum_j \mathcal{E}_j$ preserves the trace. Local instruments of party $B$ are defined in the same way on $B(H_B)$. The set of LOCC channels on $B(H_A \otimes H_B)$ is the smallest set of maps with these properties:

1. For all quantum channels $\mathcal{E}$ on $B(H_A)$ and $\mathcal{F}$ on $B(H_B)$, the map $\mathcal{E} \otimes \mathcal{F}$ is an LOCC channel.
2. If $(\mathcal{E}_j)_j$ is a local instrument of $A$ and each $\Lambda_j$ is an LOCC channel, then $\sum_j \Lambda_j \circ (\mathcal{E}_j \otimes \id)$ is an LOCC channel.
3. If $(\mathcal{F}_j)_j$ is a local instrument of $B$ and each $\Lambda_j$ is an LOCC channel, then $\sum_j \Lambda_j \circ (\id \otimes \mathcal{F}_j)$ is an LOCC channel.

In clauses 2 and 3, one party measures, announces the outcome $j$, and the protocol continues with $\Lambda_j$.