LOCC
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.
is increased and care has to be taken to define the limit of infinitely many rounds.
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).
are LOCC, i.e., there are examples that cannot be implemented locally even with infinite rounds of communication.
Alice and Bob are given a quantum system in the product state
With local operations alone this cannot be achieved, since they cannot produce the (classical) correlations present in
can be prepared: Alice throws an unbiased coin (that shows heads or tails each with 50% probability) and flips her qubit (to
She then sends the result of the coin-flip (classical information) to Bob who also flips his qubit if he receives the message "tails".
As a simple example, consider the two Bell states Let's say the two-qubit system is separated, where the first qubit is given to Alice and the second is given to Bob.
Without communication, Alice and Bob cannot distinguish the two states, since for all local measurements all measurement statistics are exactly the same (both states have the same reduced density matrix).
E.g., assume that Alice measures the first qubit, and obtains the result 0.
Since this result is equally likely to occur (with probability 50%) in each of the two cases, she does not gain any information on which Bell pair she was given and the same holds for Bob if he performs any measurement.
But now let Alice send her result to Bob over a classical channel.
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.
Nielsen [3] 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.
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 the other.
the probability of any arbitrary state being convertible into another via LOCC becomes negligible.
[5] 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).
When the table presents both red and green color, the states are not convertible.
If correlations between the system and the catalyst are allowed, catalytic transformations between bipartite pure states are characterized via the entanglement entropy.
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.
[8][9] 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.
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.
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.
