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arXiv:0907.3960 · Physical Review A

Deterministic transformations of multipartite entangled states with tensor rank 2

S. Turgut, Y. Gül, Namik Kemal Pak

Published Jan 21, 2010• 35 citations
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Abstract

Transformations involving only local operations assisted with classical communication are investigated for multipartite entangled pure states having tensor rank 2. All necessary and sufficient conditions for the possibility of deterministically converting truly multipartite, rank-2 states into each other are given. Furthermore, a chain of local operations that successfully achieves the transformation has been identified for all allowed transformations. The identified chains have two nice features: (1) each party needs to carry out at most one local operation and (2) all of these local operations are also deterministic transformations by themselves. Finally, it is found that there are disjoint classes of states, all of which can be identified by a single real parameter, which remain invariant under deterministic transformations.

DOI
10.1103/physreva.81.012317

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Transformations involving only local operations assisted with classical communication are investigated for multipartite entangled pure states having tensor rank 2. All necessary and sufficient conditions for the possibility of deterministically converting truly multipartite, rank-2 states into each other are given. Furthermore, a chain of local operations that successfully achieves the transformation has been identified for all allowed transformations. The identified chains have two nice features: (1) each party needs to carry out at most one local operation and (2) all of these local operations are also deterministic transformations by themselves. Finally, it is found that there are disjoint classes of states, all of which can be identified by a single real parameter, which remain invariant under deterministic transformations.
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.
The transformations of W-type entangled states by using local operations assisted with classical communication are investigated. For this purpose, a parametrization of the W-type states which remains invariant under local unitary transformations is proposed and a complete characterization of the local operations carried out by a single party is given. These are used for deriving the necessary and sufficient conditions for deterministic transformations. A convenient upper bound for the maximum probability of distillation of arbitrary target states is also found.
We study conditions for the deterministic transformation $\ensuremath{\mid}\ensuremath{\psi}⟩\ensuremath{\rightarrow}\ensuremath{\mid}\ensuremath{\phi}⟩$ of a bipartite entangled state by a separable operation. If the separable operation is a local operation with classical communication (LOCC), Nielsen's majorization theorem provides necessary and sufficient conditions. For the general case, we derive a necessary condition in terms of products of Schmidt coefficients, which is equivalent to the Nielsen condition when either of the two factor spaces is of dimension 2, but is otherwise weaker. One implication is that no separable operation can reverse a deterministic map produced by another separable operation, if one excludes the case where the Schmidt coefficients of $\ensuremath{\mid}\ensuremath{\psi}⟩$ are the same as those of $\ensuremath{\mid}\ensuremath{\phi}⟩$. The question of sufficient conditions in the general separable case remains open. When the Schmidt coefficients of $\ensuremath{\mid}\ensuremath{\psi}⟩$ are the same as those of $\ensuremath{\mid}\ensuremath{\phi}⟩$, we show that the Kraus operators of the separable transformation restricted to the supports of $\ensuremath{\mid}\ensuremath{\psi}⟩$ on the factor spaces are proportional to unitaries. When that proportionality holds and the factor spaces have equal dimension, we find conditions for the deterministic transformation of a collection of several full Schmidt rank pure states $\ensuremath{\mid}{\ensuremath{\psi}}_{j}⟩$ to pure states $\ensuremath{\mid}{\ensuremath{\phi}}_{j}⟩$.
The main subject of this thesis is the investigation of the transformations of pure multipartite entangled states having Schmidt rank 2 by using only local operations assisted with classical communications (LOCC). A new parameterization is used for describing the entangled state of p particles distributed to p distant, spatially separated persons. Product, bipartite and truly multipartite states are identified in this new parametrization. Moreover, alternative parameterizations of local operations carried out by each party are provided. For the case of a deterministic transformation to a truly multipartite final state, one can find an analytic expression that determines whether such a transformation is possible. In this case, a chain of measurements by each party for carrying out the transformation is found. It can also be seen that, under deterministic LOCC transformations, there are some quantities that remain invariant. For the purpose of applying the results of this thesis in the context of the quantum information and computation, brief reviews of the entanglement purification, measurement based quantum computation and quantum codes are given.
The states of three-qubit systems split into two inequivalent types of genuine tripartite entanglement, namely, the Greenberger-Horne-Zeilinger (GHZ) type and the WWW type. A state belonging to one of these classes can be stochastically transformed only into a state within the same class by local operations and classical communications. We provide local quantum operations, consisting of the most general two-outcome measurement operators, for the deterministic transformations of three-qubit pure states in which the initial and the target states are in the same class. We explore these transformations, originally having standard GHZ and standard WWW states, under the local measurement operations carried out by a single party and ppp (p=2,3p=2,3p=2,3) parties (successively). We find a notable result that the standard GHZ state cannot be deterministically transformed to a GHZ-type state in which all its bipartite entanglements are nonzero, i.e., a transformation can be achieved with unit probability when the target state has at least one vanishing bipartite concurrence.
We consider generic pure n-qubit states and a general class of pure states of arbitrary dimensions and arbitrarily many subsystems. We characterize those states which can be reached from some other state via local operations assisted by finitely many rounds of classical communication (LOCC_{N}). For n qubits with n>3, we show that this set of states is of measure zero, which implies that the maximally entangled set is generically of full measure if restricted to the practical scenario of LOCC_{N}. Moreover, we identify a class of states for which any LOCC_{N} protocol can be realized via a concatenation of deterministic steps. We show, however, that in general there exist state transformations which require a probabilistic step within the protocol, which highlights the difference between bipartite and multipartite LOCC.