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.Comment: Version 2 has new title along with an added discussion about using an enlarged Hilbert space and why this is not helpfu

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 $3\times3$ 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.Comment: 6 pages, no figures, comments welcome. Version 2 fixes some issues with the case of an infinite number of measurement operators. Version 3 has a minor change to the title and an added footnote about the fact that using an enlarged Hilbert space is not helpfu

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.Comment: Comments welcom

When a quantum measurement can be implemented locally ... and when it cannot

Local operations on subsystems and classical communication between parties (LOCC) constitute the most general protocols available on spatially separated quantum systems. Every LOCC protocol implements a separable generalized measurement -- a complete measurement for which every outcome corresponds to a tensor product of operators on individual subsystems -- but it is known that there exist separable measurements that cannot be implemented by LOCC. A longstanding problem in quantum information theory is to understand the difference between LOCC and the full set of separable measurements. In this paper, we show how to construct an LOCC protocol to implement an arbitrary separable measurement, except that with those measurements for which no LOCC protocol exists, the method shows explicitly that this is the case.Comment: 21 pages, 7 figures. Extensively revised to include details of all arguments, explicitly proving all results in full rigor. Version 3 has sections reordered and other restructuring, but otherwise contains the same discussion as version

Computing in unipotent and reductive algebraic groups

The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on the maximal unipotent subgroup of a split reductive group and show how this improves computation in the reductive group itself.Comment: 22 page