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I am trying to understand (Local operations and Classical communication) LOCC operations, but there is something I do not know.

My first question is, what is the difference between a LOCC operation and a LOCC protocol?

Second, suppose a state $\rho$, if a LOCC operation applied to it you get an ensemble {$p_i$,$\sigma_i$}, $\rho \overset{\text{LOCC}}{\rightarrow} \{p_i,\sigma_i\}$ ($p_i$ is the probability to obtain $\sigma_i$). How can I describe this $\sigma_i$ in a formula? Is this dependent on the LOCC protocol which was used? Is it correct when I say that the index $i$ is generally an index family, because it depends on the number of communication rounds?

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I guess the terminology is not completely standardised, but I see it in the following way:

An LOCC operation is an element of the class LOCC, which contains all local quantum operations and classical communication. In other words, an LOCC operation would be doing a local quantum operation or doing some classical communication.

An LOCC protocol is just a sequence of such operations. Usually, a protocol implements a task, such as distillation of entanglement or the like.

How to describe the set of LOCC operations and/or protocols? The set of quantum operations, for instance, is the set of completely positive trace preserving maps, which has a nice description in terms of Kraus operators for instance. The set LOCC in contrast has no such easy mathematical description. Therefore, specifying $\rho \stackrel{\mathrm{LOCC}}{\rightarrow} \{p_i,\sigma_i\}$ is harder and usually, people consider special protocols and/or special subsets of operations and only then try to write down a general form of such operations.

I would recommend having a look at the following paper, which explains a lot about the set of LOCC operations: https://arxiv.org/abs/1210.4583

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