Operational measures

There are 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 operational measures are Entanglement of distillation, Entanglement cost.

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 = {Pn} such, that P_n(\rho^{\otimes n}) = \tau_n for each n. If \lim_{n\rightarrow \infty} ||\tau_n - \sigma^{\otimes 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 Eop is defined as E_{op}(\rho)=\sup_P R_P(\rho\rightarrow \sigma)

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 Eop 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).

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