Extended necessary condition for local operations and classical communication: Tight bound for all measurements

We give a necessary condition that a separable measurement can be implemented by local quantum operations and classical communication (LOCC) in any finite number of rounds of communication, generalizing and strengthening a result obtained previously. That earlier result involved a bound that is tight when the number of measurement operators defining the measurement is relatively small. The present results generalize that bound to one that is tight for any finite number of measurement operators, and we also provide an extension which holds when that number is infinite. We apply these results to the famous example on a $3times3$ system known as domino states, which were the first demonstration of nonlocality without entanglement. Our new necessary condition provides an additional way of showing that these states cannot be perfectly distinguished by (finite-round) LOCC. It directly shows that this conclusion also holds for their cousins, the rotated domino states. This illustrates the usefulness of the present results, since our earlier necessary condition, which these results generalize, is not strong enough to reach a conclusion about the domino states.

On the structure of LOCC: finite vs. infinite rounds

Every measurement that can be implemented by local quantum operations and classical communication (LOCC) using an infinite number of rounds is the limit of a sequence of measurements each of which requires only a finite number of rounds. This rather obvious and well-known fact is nonetheless of interest as it shows that these infinite-round measurements can be approximated arbitrarily closely simply by using more and more rounds of communication. Here we demonstrate the perhaps less obvious result that (at least) for bipartite systems, the reverse relationship also holds. Specifically, we show that every finite-round bipartite LOCC measurement is the limit of a continuous sequence of LOCC measurements, where each measurement in that sequence can be implemented by LOCC, but only with the use of an infinite number of rounds. Thus, the set of LOCC measurements that require an infinite number of rounds is dense in the entirety of LOCC, as is the set of finite-round LOCC measurements. This means there exist measurements that can only be implemented by LOCC by using an infinite number of rounds, but can nonetheless be approximated closely by using one round of communication, and actually in some cases, no communication is needed at all. These results follow from a new necessary condition for finite-round LOCC, which is extremely simple to check, is very easy to prove, and which can be violated by utilizing an infinite number of rounds.

A class of unambiguous state discrimination problems achievable by separable measurements but impossible by local operations and classical communication

We consider an infinite class of unambiguous quantum state discrimination problems on multipartite systems, described by Hilbert space $cal{H}$, of any number of parties. Restricting consideration to measurements that act only on $cal{H}$, we find the optimal global measurement for each element of this class, achieving the maximum possible success probability of $1/2$ in all cases. This measurement turns out to be both separable and unique, and by our recently discovered necessary condition for local quantum operations and classical communication (LOCC), it is easily shown to be impossible by any finite-round LOCC protocol. We also show that, quite generally, if the input state is restricted to lie in $cal{H}$, then any LOCC measurement on an enlarged Hilbert space is effectively identical to an LOCC measurement on $cal{H}$. Therefore, our necessary condition for LOCC demonstrates directly that a higher success probability is attainable for each of these problems using general separable measurements as compared to that which is possible with any finite-round LOCC protocol.

Necessary condition for local quantum operations and classical communication with extensive violation by separable operations

We give a conceptually simple necessary condition such that a separable quantum operation can be implemented by local operations on subsystems and classical communication between parties (LOCC), a condition which follows from a novel approach to understanding LOCC. This necessary condition holds for any number of parties and any finite number of rounds of communication and as such, also provides a completely general sufficient condition that a given separable operation cannot be exactly implemented by LOCC. Furthermore, it demonstrates an extremely strong difference between separable operations and LOCC, in that there exist examples of the former for which the condition is extensively violated. More precisely, the violation by separable operations of our necessary condition for LOCC grows without limit as the number of parties increases.

Local quantum protocols for separable measurements with many parties

In a recent paper cite{mySEPvsLOCC}, we showed how to construct a quantum protocol for implementing a bipartite, separable quantum measurement using only local operations on subsystems and classical communication between parties (LOCC) within any fixed number of rounds of communication, whenever such a protocol exists. Here, we generalize that construction to one that applies for any number of parties. One important observation is that the construction automatically determines the ordering of the parties measurements, overcoming a significant apparent difficulty in designing protocols for more than two parties. We also present various other results about LOCC, including showing that if, in any given measurement operator of the separable measurement under consideration, the local parts for two different parties are rank-1 operators that are not repeated in any other measurement operator of the measurement, then this separable measurement cannot be exactly implemented by LOCC in any finite number of rounds.

All unitaries having operator Schmidt rank 2 are controlled unitaries

We prove that every unitary acting on any multipartite system and having operator Schmidt rank equal to 2 can be diagonalized by local unitaries. This then implies that every such multipartite unitary is locally equivalent to a controlled unitary with every party but one controlling a set of unitaries on the last party. We also prove that any bipartite unitary of Schmidt rank 2 is locally equivalent to a controlled unitary where either party can be chosen as the control, and at least one party can control with two terms, which implies that each such unitary can be implemented using local operations and classical communication (LOCC) and a maximally entangled state on two qubits. These results hold regardless of the dimensions of the systems on which the unitary acts.

Brauer algebras of type B

For each n>0, we define an algebra having many properties that one might expect to hold for a Brauer algebra of type Bn. It is defined by means of a presentation by generators and relations. We show that this algebra is a subalgebra of the Brauer algebra of type Dn+1 and point out a cellular structure in it. This work is a natural sequel to the introduction of Brauer algebras of type Cn, which are subalgebras of classical Brauer algebras of type A2n-1 and differ from the current ones for n>2.

Fast protocols for local implementation of bipartite nonlocal unitaries

In certain cases the communication time required to deterministically implement a nonlocal bipartite unitary using prior entanglement and LOCC (local operations and classical communication) can be reduced by a factor of two. We introduce two such fast protocols and illustrate them with various examples. For some simple unitaries, the entanglement resource is used quite efficiently. The problem of exactly which unitaries can be implemented by these two protocols remains unsolved, though there is some evidence that the set of implementable unitaries may expand at the cost of using more entanglement.

The Birman-Murakami-Algebras Algebras of Type Dn

The Birman-Murakami-Wenzl algebra (BMW algebra) of type Dn is shown to be semisimple and free of rank (2^n+1)n!!-(2^(n-1)+1)n! over a specified commutative ring R, where n!! is the product of the first n odd integers. We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type Dn is the image af an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the polynomial ring Z with delta and its inverse adjoined. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley-Lieb algebra of type Dn is a subalgebra of the BMW algebra of the same type.

Brauer algebras of type C

For each natural number n greater than 1, we define an algebra satisfying many properties that one might expect to hold for a Brauer algebra of type Cn. The monomials of this algebra correspond to scalar multiples of symmetric Brauer diagrams on 2n strands. The algebra is shown to be free of rank the number of such diagrams and cellular, in the sense of Graham and Lehrer.