Characterizing the transformation and classification of multipartite entangled states is a basic problem in quantum information. We study the problem under the two most common environments, local operations and classical communications (LOCC), stochastic LOCC and two more general environments, multicopy LOCC (MCLOCC), and multicopy SLOCC (MCSLOCC). We show that two transformable multipartite states under LOCC or SLOCC are also transformable under MCLOCC and MCSLOCC. What is more, these two environments are equivalent in the sense that two transformable states under MCLOCC are also transformable under MCSLOCC, and vice versa. Based on these environments we classify the multipartite pure states into a few inequivalent sets and orbits, between which we build the partial order to decide their transformation. In particular, we investigate the structure of SLOCC-equivalent states in terms of tensor rank, which is known as the generalized Schmidt rank. Given the tensor rank, we show that Greenberger-Horne-Zeilinger states can be used to generate all states with a smaller or equivalent tensor rank under SLOCC, and all reduced separable states with a cardinality smaller than or equivalent to the tensor rank under LOCC. Using these concepts, we extended the concept of the ``maximally entangled state'' in the multipartite system.
A basic question regarding quantum entangled states is whether one can be probabilistically converted to another through local operations and classical communication exclusively. While the answer for bipartite systems is known, we show that for tripartite systems, this question encodes some of the most challenging open problems in mathematics and computer science. In particular, we show that there is no easy general criterion to determine the feasibility, and in fact, the problem is NP hard. In addition, we find obtaining the most efficient algorithm for matrix multiplication to be precisely equivalent to determining the maximum rate to convert the Greenberger-Horne-Zeilinger state to a triangular distribution of three EPR states. Our results are based on connections between multipartite entanglement and tensor rank (also called Schmidt rank), a key concept in algebraic complexity theory.
Hoping to simplify the classification of pure entangled states of multi $(m)\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ quantum systems, we study exactly and asymptotically (in
n) reversible transformations among
nth tensor powers of such states (i.e.,
n copies of the state shared among the same
m parties) under local quantum operations and classical communication (LOCC). For exact transformations, we show that two states whose marginal one-party entropies agree are either locally unitarily equivalent or else LOCC incomparable. In particular we show that two tripartite Greenberger-Horne-Zeilinger states are LOCC incomparable to three bipartite Einstein-Podolsky-Rosen (EPR) states symmetrically shared among the three parties. Asymptotic transformations yield a simpler classification than exact transformations; for example, they allow all pure bipartite states to be characterized by a single parameter---their partial entropy---which may be interpreted as the number of EPR pairs asymptotically interconvertible to the state in question by LOCC transformations. We show that $m\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ pure states having an $m\ensuremath{-}\mathrm{w}\mathrm{a}\mathrm{y}$ Schmidt decomposition are similarly parametrizable, with the partial entropy across any nontrivial partition representing the number of standard quantum superposition or ``cat'' states $|{0}^{\ensuremath{\bigotimes}m}〉+|{1}^{\ensuremath{\bigotimes}m}〉$ asymptotically interconvertible to the state in question. For general $m\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ states, partial entropies across different partitions need not be equal, and since partial entropies are conserved by asymptotically reversible LOCC operations, a multicomponent entanglement measure is needed, with each scalar component representing a different kind of entanglement, not asymptotically interconvertible to the other kinds. In particular we show that the
m=4 cat state is not isentropic to, and therefore not asymptotically interconvertible to, any combination of bipartite and tripartite states shared among the four parties. Thus, although the
m=4 cat state can be prepared from bipartite EPR states, the preparation process is necessarily irreversible, and remains so even asymptotically. For each number of parties
m we define a minimal reversible entanglement generating set (MREGS) as a set of states of minimal cardinality sufficient to generate all $m\ensuremath{-}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}$ pure states by asymptotically reversible LOCC transformations. Partial entropy arguments provide lower bounds on the size of the MREGS, but for
m>2 we know no upper bounds. We briefly consider several generalizations of LOCC transformations, including transformations with some probability of failure, transformations with the catalytic assistance of states other than the states we are trying to transform, and asymptotic LOCC transformations supplemented by a negligible
[o(n)] amount of quantum communication.
Understanding multipartite entanglement is a key goal in quantum information. Entanglement in pure states can be characterized by considering transformations under local operations assisted by classical communication (LOCC). However, it has been shown that, for $n\ensuremath{\ge}5$ parties, multipartite pure states are generically isolated, i.e., they can be neither reached nor transformed under LOCC. Nonetheless, in any real laboratory, one never deterministically transforms a pure initial state exactly to a pure target state. Instead, one transforms a mixed state near the initial state to an ensemble that is on average close to the target state. This motivates studying approximate LOCC transformations. After reviewing in detail the known results in the bipartite case, we present the gaps that remain open in the multipartite case. While the analysis of the multipartite setting is much more technically involved due to the existence of different stochastic LOCC (SLOCC) classes, certain features simplify in the approximate setting. In particular, we show that it is sufficient to consider pure initial states, that it is sufficient to consider LOCC protocols with finitely many rounds of communication, and that approximate transformations can be approximated by ensemble transformations within an SLOCC class. Then we formally define a hierarchy of different forms of approximate transformations that are relevant from a physical point of view. Whereas this hierarchy collapses in the bipartite case, we show that this is not the case for the multipartite setting, which is fundamentally richer. To wit, we show that optimal multipartite approximate transformations are not generally deterministic, that ensemble transformations within an SLOCC class can achieve a higher fidelity than deterministic transformations within an SLOCC class, and that there are approximate transformations with no deterministic transformations nearby.
Abstract Classification of multipartite pure states via the local unitary (LU) equivalence or the stochastic local operations and classical communication (SLOCC) equivalence is a fundamental task in investigating multipartite entanglement. It is shown in the seminal paper (2000 Phys . Rev . A 62 062314) that pure states of three qubits can be partitioned into six SLOCC classes. We explore here a necessary and sufficient condition for LU equivalence of pure states for any SLOCC class via Acín et al ’s Schmidt coefficients (ASC) of three qubits in (2000 Phys. Rev. Lett. 85 1560). We show that the LU equivalent states in W SLOCC class have the same ASC irrespective of phases while there are generally two different ASCs for the LU equivalent states in GHZ SLOCC class (irrespective of phases for some subclass). Consequently, we obtain a complete LU classification of three qubit pure states in the sense of ASC. Moreover, we discuss the choice among the different ASCs and then propose an approach to find a unique ASC for any given three qubit pure state. At last we analyse the entanglement/separability of the marginal states of the two inequivalent genuine entangled SLOCC classes. We find that any bipartite marginal states of the W SLOCC class are entangled, and that of the GHZ SLOCC class has eight different cases which are depend on the ASCs.
Multipartite pure states are equivalent under stochastic local operations and classical communication (SLOCC) whenever they can be mapped into one another by invertible local operations. It is shown that this is equivalent to the interconvertibility through finite sequences of elementary local operations. A multipartite version of the Gauss-Jordan elimination strategy is then obtained, enabling the analytical SLOCC classification of previously unknown examples. It is argued that the problem of SLOCC classification is equivalent to the problem of classifying the multipartite fully reduced forms that the coefficient matrix of a state can assume after being subjected to this Gauss-Jordan elimination procedure. The method is applied to examples of so-called locally maximally entangleable and hypergraph states, showing different SLOCC equivalences. Moreover, possible physical implications for states with a Dicke-like structure are sketched.
We introduce a notion of the entanglement transformation rate to characterize the asymptotic comparability of two multipartite pure entangled states under stochastic local operations and classical communication (SLOCC). For two well known SLOCC inequivalent three-qubit states |GHZ⟩=(1/2)(|000⟩+|111⟩) and |W⟩=(1/3)(|100⟩+|010⟩+|001⟩), we show that the entanglement transformation rate from |GHZ⟩ to |W⟩ is exactly 1. That means that we can obtain one copy of the W state from one copy of the Greenberg-Horne-Zeilinger (GHZ) state by SLOCC, asymptotically. We then apply similar techniques to obtain a lower bound on the entanglement transformation rates from an N-partite GHZ state to a class of Dicke states, and prove the tightness of this bound for some special cases which naturally generalize the |W⟩ state. A new lower bound on the tensor rank of the matrix permanent is also obtained by evaluating the tensor rank of Dicke states.