Necessary and sufficient conditions for local manipulation of multipartite pure quantum states

Suppose several parties jointly possess a pure multipartite state, |\psi>. Using local operations on their respective systems and classical communication (i.e. LOCC) it may be possible for the parties to transform deterministically |\psi> into another joint state |\phi>. In the bipartite case, Nielsen majorization theorem gives the necessary and sufficient conditions for this process of entanglement transformation to be possible. In the multipartite case, such a deterministic local transformation is possible only if both the states in the same stochastic LOCC (SLOCC) class. Here we generalize Nielsen majorization theorem to the multipartite case, and find necessary and sufficient conditions for the existence of a local separable transformation between two multipartite states in the same SLOCC class. When such a deterministic conversion is not possible, we find an expression for the maximum probability to convert one state to another by local separable operations. In addition, we find necessary and sufficient conditions for the existence of a separable transformation that converts a multipartite pure state into one of a set of possible final states all in the same SLOCC class. Our results are expressed in terms of (1) the stabilizer group of the state representing the SLOCC orbit, and (2) the associate density matrices (ADMs) of the two multipartite states. The ADMs play a similar role to that of the reduced density matrices, when considering local transformations that involves pure bipartite states. We show in particular that the requirement that one ADM majorize another is a necessary condition but in general far from being also sufficient as it happens in the bipartite case.Comment: Published version. Abstract and introduction revised significantl

Classification of multipartite entanglement of all finite dimensionality.

We provide a systematic classification of multiparticle entanglement in terms of equivalence classes of states under stochastic local operations and classical communication (SLOCC). We show that such a SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under local action of the special linear group. We then construct the complete set of all such SL-invariant polynomials (SLIPs). Our construction is based on Schur-Weyl duality and applies to any number of qudits in all (finite) dimensions. In addition, we provide an elegant formula for the dimension of the homogenous SLIPs space of a fixed degree as a function of the number of qudits. The expressions for the SLIPs involve in general many terms, but for the case of qubits we also provide much simpler expressions

Transformations among Pure Multipartite Entangled States via Local Operations Are Almost Never Possible

Local operations assisted by classical communication (LOCC) constitute the free operations in entanglement theory. Hence, the determination of LOCC transformations is crucial for the understanding of entanglement. We characterize here almost all LOCC transformations among pure multipartite multilevel states. Combined with the analogous results for qubit states shown by Gour \emph{et al.} [J. Math. Phys. 58, 092204 (2017)], this gives a characterization of almost all local transformations among multipartite pure states. We show that nontrivial LOCC transformations among generic, fully entangled, pure states are almost never possible. Thus, almost all multipartite states are isolated. They can neither be deterministically obtained from local-unitary-inequivalent (LU-inequivalent) states via local operations, nor can they be deterministically transformed to pure, fully entangled LU-inequivalent states. In order to derive this result, we prove a more general statement, namely, that, generically, a state possesses no nontrivial local symmetry. We discuss further consequences of this result for the characterization of optimal, probabilistic single copy and probabilistic multi-copy LOCC transformations and the characterization of LU-equivalence classes of multipartite pure states.Comment: 13 pages main text + 10 pages appendix, 1 figure; close to published versio

All Maximally Entangled Four Qubits States

We find an operational interpretation for the 4-tangle as a type of residual entanglement, somewhat similar to the interpretation of the 3-tangle. Using this remarkable interpretation, we are able to find the class of maximally entangled four-qubits states which is characterized by four real parameters. The states in the class are maximally entangled in the sense that their average bipartite entanglement with respect to all possible bi-partite cuts is maximal. We show that while all the states in the class maximize the average tangle, there are only few states in the class that maximize the average Tsillas or Renyi $\alpha$-entropy of entanglement. Quite remarkably, we find that up to local unitaries, there exists two unique states, one maximizing the average $\alpha$-Tsallis entropy of entanglement for all $\alpha\geq 2$, while the other maximizing it for all $0<\alpha\leq 2$ (including the von-Neumann case of $\alpha=1$). Furthermore, among the maximally entangled four qubits states, there are only 3 maximally entangled states that have the property that for 2, out of the 3 bipartite cuts consisting of 2-qubits verses 2-qubits, the entanglement is 2 ebits and for the remaining bipartite cut the entanglement between the two groups of two qubits is 1ebit. The unique 3 maximally entangled states are the 3 cluster states that are related by a swap operator. We also show that the cluster states are the only states (up to local unitaries) that maximize the average $\alpha$-Renyi entropy of entanglement for all $\alpha\geq 2$.Comment: 15 pages, 2 figures, Revised Version: many references added, an appendix added with a statement of the Kempf-Ness theore

Closed formula for the relative entropy of entanglement in all dimensions

The relative entropy of entanglement is defined in terms of the relative entropy between an entangled state and its closest separable state (CSS). Given a multipartite-state on the boundary of the set of separable states, we find a closed formula for all the entangled state for which this state is a CSS. Quite amazing, our formula holds for multipartite states in all dimensions. In addition we show that if an entangled state is full rank, then its CSS is unique. For the bipartite case of two qubits our formula reduce to the one given in Phys. Rev. A 78, 032310 (2008).Comment: 8 pages, 1 figure, significantly revised; theorem 1 is now providing necessary and sufficient conditions to determine if a state is CS

Extensive Entropy Bounds

It is shown that, for systems in which the entropy is an extensive function of the energy and volume, the Bekenstein and the holographic entropy bounds predict new results. More explicitly, the Bekenstein entropy bound leads to the entropy of thermal radiation (the Unruh-Wald bound) and the spherical entropy bound implies the "causal entropy bound". Surprisingly, the first bound shows a close relationship between black hole physics and the Stephan-Boltzmann law (for the energy and entropy flux densities of the radiation emitted by a hot blackbody). Furthermore, we find that the number of different species of massless fields is bounded by $\sim 10^{4}$.Comment: 8 pages, revtex, To appear in Phys. Rev.

Entanglement of subspaces in terms of entanglement of superpositions

We investigate upper and lower bounds on the entropy of entanglement of a superposition of bipartite states as a function of the individual states in the superposition. In particular, we extend the results in [G. Gour, arxiv.org:0704.1521 (2007)] to superpositions of several states rather than just two. We then investigate the entanglement in a subspace as a function of its basis states: we find upper bounds for the largest entanglement in a subspace and demonstrate that no such lower bound for the smallest entanglement exists. Finally, we consider entanglement of superpositions using measures of entanglement other than the entropy of entanglement.Comment: 7 pages, no figure