# Operational measures

There are link building service entanglement measures which are defined by certain task which should be achieved optimally by means of local operations and classical communication. They are therefore called operational measures. The most common loan modification operational measures are Entanglement of distillation, Entanglement cost.

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Typically and operational measure involves:

- input state

- class of allowed operations by means of which input state should be transformed which is the class of local operations and classical communication.

- output state

The typical **task** is to obtain the greatest amount of output states given certain number of input states. Then the operational measure equals the **optimal rate** of number of output states that can be obtained from input states via LOCC, per number of input states (in asymptotic limit).

Formally one defines operational measure as follows: Let *ρ* and *σ* be the input and output state. Consider a protocol i.e. sequence of LOCC operations *P* = {*P**n*} such, that *P**n*(*ρ* ⊗ *n*) = *τ**n* for each n. If lim*n* → ∞∣∣*τ**n* − *σ* ⊗ *m*∣∣ = 0, we say that the protocol P achieves rate given by

$R_P(\rho \rightarrow \sigma):=\limsup_{n,m \rightarrow \infty} {m\over n}$.

Then the operational measure *E**o**p* is defined as *E**o**p*(*ρ*) = sup*P**R**P*(*ρ* → *σ*)

In place of the input state and output state in the above definition one can consider the set of input state and the set of output states respectively. In such case the supremum in definition of *E**o**p* is taken also over input and output states.

Moreover the task may by modified, so that the otput state maximises certain function (see Distillable key).

Category: Handbook of Quantum Information Category: Entanglement