LOCC

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. 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. 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 r ≥ 1 {displaystyle rgeq 1} one can define L O C C r {displaystyle LOCC_{r}} , the set of LOCC operations that can be achieved with r {displaystyle r} rounds of classical communication. The set becomes strictly larger whenever r {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. A one-round LOCC L O C C 1 {displaystyle LOCC_{1}} is a quantum instrument { E x } {displaystyle left{{mathcal {E}}_{x} ight}} , for which the trace-non-increasing completely positive maps (CPMs) E x {displaystyle {mathcal {E}}_{x}} are local for all measurement results x {displaystyle x} , i.e., E x = ⨂ j ( E x j ) {displaystyle {mathcal {E}}_{x}=igotimes _{j}({cal {E}}_{x}^{j})} and there is one site j = K {displaystyle j=K} such that only at K {displaystyle K} the map E x K {displaystyle {mathcal {E}}_{x}^{K}} E x = ⨂ j ≠ K ( T j x ) ⊗ E K {displaystyle {mathcal {E}}_{x}=igotimes _{j ot =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 K {displaystyle K} applying the (local) instrument { E x K } {displaystyle left{{mathcal {E}}_{x}^{K} ight}} and communicating the classical result x {displaystyle x} to all other parties, which then each perform (conditioned on x {displaystyle x} ) trace-preserving (deterministic) local quantum operations T x j {displaystyle {cal {T}}_{x}^{j}} . Then L O C C r {displaystyle LOCC_{r}} are defined recursively as those operations that can be realized by following up a operation L O C C r − 1 {displaystyle LOCC_{r-1}} with a L O C C 1 {displaystyle 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 allow also 'coarse-graining', i.e., discarding some of the classical information encoded in the measurement results (of all rounds). The union of all L O C C r {displaystyle LOCC_{r}} operations is denoted by L O C C N {displaystyle LOCC_{mathbb {N} }} and contains instruments that can be approximated better and better with more LOCC rounds. It topological closure L O C C N ¯ {displaystyle {overline {LOCC_{mathbb {N} }}}} contains all such operations. It can be shown that all these sets are different: The set of all LOCC operations is contained in the set S E P {displaystyle SEP} of all separable operations. S E P {displaystyle SEP} contains all operations that can be written using Kraus operators that have all product form, i.e., with ∑ l K 1 l ⊗ K 2 l ⋯ ⊗ K N ( K 1 l ⊗ K 2 l ⋯ ⊗ K N ) † = 1 {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 S E P {displaystyle SEP} are LOCC, i.e., there are examples that cannot be implemented locally even with infinite rounds of communication.