This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (January 2019) (Learn how and when to remove this message)

LOCC, or local operations and classical communication, is a method in quantum information theory where a local (product) operation is performed on part of the system, and where the result of that operation is "communicated" classically to another part where usually another local operation is performed conditioned on the information received.

Mathematical properties

The formal definition of the set of LOCC operations is complicated due to the fact that later local operations depend in general on all the previous classical communication and due to the unbounded number of communication rounds. For any finite number ${\displaystyle r\geq 1}$ one can define ${\displaystyle \operatorname {LOCC} _{r}}$, the set of LOCC operations that can be achieved with ${\displaystyle r}$ rounds of classical communication. The set becomes strictly larger whenever ${\displaystyle r}$ is increased and care has to be taken to define the limit of infinitely many rounds. In particular, the set LOCC is not topologically closed, that is there are quantum operations that can be approximated arbitrarily closely by LOCC but that are not themselves LOCC.^[1]

A one-round LOCC ${\displaystyle \operatorname {LOCC} _{1}}$ is a quantum instrument ${\displaystyle \left\{{\mathcal {E}}_{x}\right\}}$, for which the trace-non-increasing completely positive maps (CPMs) ${\displaystyle {\mathcal {E}}_{x}}$ are local for all measurement results ${\displaystyle x}$, i.e., ${\displaystyle {\mathcal {E}}_{x}=\bigotimes _{j}({\cal {E}}_{x}^{j})}$ and there is one site ${\displaystyle j=K}$ such that only at ${\displaystyle K}$ the map ${\displaystyle {\mathcal {E}}_{x}^{K}}$ ${\displaystyle {\mathcal {E}}_{x}=\bigotimes _{j\not =K}({\cal {T_{j}^{x}}})\otimes {\cal {E}}_{K}}$ is not trace-preserving. This means that the instrument can be realized by the party at site ${\displaystyle K}$ applying the (local) instrument ${\displaystyle \left\{{\mathcal {E}}_{x}^{K}\right\}}$ and communicating the classical result ${\displaystyle x}$ to all other parties, which then each perform (conditioned on ${\displaystyle x}$) trace-preserving (deterministic) local quantum operations ${\displaystyle {\cal {T}}_{x}^{j}}$.

Then ${\displaystyle \operatorname {LOCC} _{r}}$ are defined recursively as those operations that can be realized by following up an operation ${\displaystyle \operatorname {LOCC} _{r-1}}$ with a ${\displaystyle \operatorname {LOCC} _{1}}$-operation. Here it is allowed that the party which performs the follow-up operations depends on the result of the previous rounds. Moreover, we also allow "coarse-graining", i.e., discarding some of the classical information encoded in the measurement results (of all rounds).

The union of all ${\displaystyle \operatorname {LOCC} _{r}}$ operations is denoted by ${\displaystyle \operatorname {LOCC} _{\mathbb {N} }}$ and contains instruments that can be approximated better and better with more LOCC rounds. Its topological closure ${\displaystyle {\overline {\operatorname {LOCC} }}_{\mathbb {N} }}$ contains all such operations.

It can be shown that all these sets are different:^[1]

${\displaystyle \operatorname {LOCC} _{r}\subset \operatorname {LOCC} _{r+1}\subset \operatorname {LOCC} _{\mathbb {N} }\subset {\overline {\operatorname {LOCC} }}_{\mathbb {N} }}$

The set of all LOCC operations is contained in the set ${\displaystyle \operatorname {SEP} }$ of all separable operations. ${\displaystyle \operatorname {SEP} }$ contains all operations that can be written using Kraus operators that have all product form, i.e.,

${\displaystyle {\cal {E}}(\rho )=\sum _{l}K_{1}^{l}\otimes K_{2}^{l}\dots \otimes K_{N}\rho (K_{1}^{l}\otimes K_{2}^{l}\dots \otimes K_{N})^{\dagger },}$

with ${\displaystyle \sum _{l}K_{1}^{l}\otimes K_{2}^{l}\dots \otimes K_{N}(K_{1}^{l}\otimes K_{2}^{l}\dots \otimes K_{N})^{\dagger }=1}$. Not all operations in ${\displaystyle \operatorname {SEP} }$ are LOCC,

${\displaystyle {\overline {\operatorname {LOCC} }}_{\mathbb {N} }\subset \operatorname {SEP} ,}$

i.e., there are examples that cannot be implemented locally even with infinite rounds of communication.^[1]

LOCC are the "free operations" in the resource theories of entanglement: Entanglement cannot be produced from separable states with LOCC and if the local parties in addition to being able to perform all LOCC operations are also furnished with some entangled states, they can realize more operations than with LOCC alone.

Examples

LOCC operations are useful for state preparation, state discrimination, and entanglement transformations.

State preparation

Alice and Bob are given a quantum system in the product state ${\displaystyle |00\rangle =|0\rangle _{A}\otimes |0\rangle _{B}}$. Their task is to produce the separable state ${\displaystyle \rho ={\frac {1}{2}}|00\rangle \langle 00|+{\frac {1}{2}}|11\rangle \langle 11|}$. With local operations alone this cannot be achieved, since they cannot produce the (classical) correlations present in ${\displaystyle \rho }$. However, with LOCC (with one round of communication) ${\displaystyle \rho }$ can be prepared: Alice throws an unbiased coin (that shows heads or tails each with 50% probability) and flips her qubit (to ${\displaystyle }$Zdroj:https://en.wikipedia.org?pojem=LOCC
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.