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Wikipedia / 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.

Mathematical properties

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 is a quantum instrument, for which the trace-non-increasing completely positive maps.

-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).

contains all such operations.

It can be shown that all these sets are different:

of all separable operations contains all operations that can be written using Kraus operators that have all product form, i.e.,

are LOCC,

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

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

. In general, all separable states (and only these) can be prepared from a product states with LOCC operations alone.

State discrimination

on a bi- or multipartite Hilbert space it is. As a simple example, consider the two Bell states

Let's say the two-qubit. Thus with LOCC and two measurements these two states can be distinguished perfectly. Note that with global (nonlocal or entangled) measurements, a single measurement (on the joint Hilbert space) is sufficient to distinguish these two (mutually orthogonal) states.

There are quantum states that cannot be distinguished with LOCC operations.

Entanglement transformations

While LOCC cannot generate entangled states out of product states, they can be used to transform entangled states into other entangled states. The restriction to LOCC severely limits which transformations are possible.

Entanglement conversion

Nielsen has derived a general condition to determine whether one pure state of a bipartite quantum system may be transformed into another using only LOCC. Full details may be found in the paper referenced earlier, the results are sketched out here.

Consider two particles in a Hilbert space with Schmidt decompositions

's are known as Schmidt coefficients

In more concise notation:

This is a more restrictive condition than that local operations cannot increase entanglement measures have the same amount of entanglement but converting one into the other is not possible and even that conversion in either direction is impossible because neither set of Schmidt coefficients majorises if all Schmidt coefficients are non-zero then the probability of one set of coefficients majorising the probability of any arbitrary state being convertible into another via LOCC becomes negligible.

The operations described so far are deterministic, i.e., they succeed with probability 100%. If one is satisfied by probabilistic transformations, many more transformations are possible using LOCC. These operations are called stochastic LOCC (SLOCC). In particular for multi-partite states the convertibility under SLOCC is studied to gain a qualitative insight into the entanglement properties of the involved states.

Going beyond LOCC: Catalytic conversion

If entangled states are available as a resource, these together with LOCC allow a much larger class of transformations. This is the case even if these resource states are not consumed in the process (as they are, for example, in quantum teleportation). Such transformations are called entanglement catalysis. and requiring that this state is still available at the end of the conversion process. I.e., the catalyst state is left unchanged by the conversion and can then be removed, leaving only the desired final state. Consider the states,

{| class="wikitable"

by LOCC if the color are all red or white. When the table presents both red and green color, the states are not convertible.

：

Similarly, we make up the table:

{| class="wikitable"

by the LOCC. The catalyst by the LOCC.

If correlations between the system and the catalyst are allowed, catalytic transformations between bipartite pure states are characterized via the entanglement entropy. via catalytic LOCC if and only if

is the von Neumann entropy are the reduced states, respectively. In general, the conversion is not exact, but can be performed with an arbitrary accuracy. The amount of correlations between the system and the catalyst can also be made arbitrarily small.

A broader class of transformations can be achieved through the use of quantum batteries. These are ancillary systems introduced during the transformation process, akin to quantum catalysts, with the condition that they retain their entanglement upon completion of the procedure. The incorporation of quantum batteries significantly extends the scope of LOCC, enhancing the ability to manipulate quantum states. In the asymptotic regime, where a large number of identical copies of the initial state are available, this approach enables reversible interconversion between any two entangled states. This leads to the formulation of a second law of entanglement manipulation, analogous to the second law of thermodynamics. Previously, the existence of a framework leading to the second law of entanglement manipulation has been identified as a major open problem in quantum information science.