The determinisitc entanglement concentration (DEC) protocol under local operations and classical communication (LOCC) for three-copy partially entangled states in bipartite systems is presented. The explicit elements of the operators used are calculated due to the general construction of the generalized measurement operator and the corresponding permutation matrices. The measurement is implemented based on the direct sum extension under the motivation to extend the initial state space with minimum number of ancillary dimensions. Morever, the concentration protocol is generalized to the DEC for n-copy multi-partite GHZ-class states. It is also pointed out that the formation of the measurement operator and the algorithms used to calculate its elements can be utilized in more general cases achieving the feasible entanglement transformation that is clarified by Nielsen's theorem, such as the transformation between arbitrary bipartite states in multicopy and high-dimensional cases and so does multi-partite GHZ-class states.
The determinisitc entanglement concentration (DEC) protocol under local operations and classical communication (LOCC) for three-copy partially entangled states in bipartite systems is presented. The explicit elements of the operators used are calculated due to the general construction of the generalized measurement operator and the corresponding permutation matrices. The measurement is implemented based on the direct sum extension under the motivation to extend the initial state space with minimum number of ancillary dimensions. Morever, the concentration protocol is generalized to the DEC for n-copy multi-partite GHZ-class states. It is also pointed out that the formation of the measurement operator and the algorithms used to calculate its elements can be utilized in more general cases achieving the feasible entanglement transformation that is clarified by Nielsen's theorem, such as the transformation between arbitrary bipartite states in multicopy and high-dimensional cases and so does multi-partite GHZ-class states.
The schematic optical realization of deterministic multicopy (three-copy) entanglement concentration under local operations and classical communication (LOCC) is presented, which can resort to the linear optical elements, controlled-NOT gate that uses weak cross-Kerr nonlinearity effect and entangled qudits with different degrees of freedom. This concentration is based on Nielsen's theorem, which in principle quantifies the unit success probability. In addition, the required positive operator-valued measurement (POVM) is implemented through direct sum extension, because the least number of ancillary dimensions is demanded. The construction of POVM is systematically formulated, which results in the feasibility to generate the protocol to n-copy (n > 3) cases.
Entanglement concentration can be achieved in a deterministic fashion [Morikoshi, Phys. Rev. Lett. 84, 3189 (2000); Phys. Rev. A 64, 022316 (2001)]. We present a protocol for deterministic extraction of an Einstein-Podolsky-Rosen pair from two partially entangled pairs via local operations and classical communication. This protocol consists of a generalized measurement described by a positive operator-valued measure (POVM), one-way classical communication, and a corresponding unitary operation or a choice between the two pairs. Based on the theory of majorization, we explicitly construct the required POVM, which can be realized by performing a unitary operation in the extended space and a conventional Von Neuman orthogonal measurement. By decomposing the evolution process from the initial state to the final state, we construct the quantum circuits for realizing the unitary operation with quantum Toffoli gates, and thus provide a physical means to realize the deterministic entanglement concentration. Our method for constructing quantum circuits differs from the usual method based on decomposition of unitary matrices, and is convenient for a large class of quantum processes involving generalized measurements.
A necessary and sufficient condition of the possibility of a deterministic local operations and classical communication (LOCC) transformation of three-qubit pure states is given. The condition shows that the three-qubit pure states are a partially ordered set parametrized by five well-known entanglement parameters and a novel parameter; the five are the concurrences CAB, CAC, CBC, the tangle τABC and the fifth parameter J5 of Acín et al. (2000) Ref. [19], while the other new one is the entanglement charge Qe. The order of the partially ordered set is defined by the possibility of a deterministic LOCC transformation from a state to another state. In this sense, the present condition is an extension of Nielsen’s work (Nielsen (1999) [14]) to three-qubit pure states. We also clarify the rules of transfer and dissipation of the entanglement which is caused by deterministic LOCC transformations. Moreover, the minimum number of times of measurements to reproduce an arbitrary deterministic LOCC transformation between three-qubit pure states is given.
Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically motivated extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC transformations acting on a finite number of entangled pure states. We study both multipartite and bipartite multi-state transformations. In the multipartite case, we demonstrate that one can change the stochastic LOCC (SLOCC) class of the individual initial states by only applying Local Unitaries (LUs). We show that, by transferring entanglement from one state to the other, one can perform state conversions not possible in the single copy case; provide examples of multipartite entanglement catalysis; and demonstrate improved probabilistic protocols. In the bipartite case, we identify numerous non-trivial LU transformations and show that the source entanglement is not additive. These results demonstrate that multi-state LOCC has a much richer landscape than single-state LOCC.
Entanglement is known to be a promising resource which enables us to execute various quantum tasks. Its quantification has been successful for bipartite pure states. However, the quantification of entanglement for multipartite states has been in a difficult situation. The difficulty is due to the difference of structures between bipartite and multipartite states. In the present thesis, we introduce a new approach for this problem. We give a necessary and sufficient condition of the possibility of a deterministic LOCC transformation of three-qubit pure states. This condition is expressed as a transformation law of six entanglement parameters jAB, jAC , jBC , jABC , J5 and Qe, where jAB, jAC and jBC are bipartite entanglements, jABC is a tripartite entanglement, J5 is a tripartite parameter which means a kind of phase, and Qe is a new tripartite parameter which means a kind of charge. This fact shows that three-qubit pure states are a partially ordered set parametrized by the six entanglement parameters. The order of the partially ordered set is defined by the possibility of a deterministic LOCC transformation from a state to another state. In this sense, the present condition is an extension of Nielsen’s work [14] to three-qubit pure states. We also clarify the rules of transfer and dissipation of entanglement. These rules guarantee that the tripartite entanglement can be transformed into bipartite entanglements, but that the bipartite entanglements cannot be transformed into the tripartite entanglement. This implies that the tripartite entanglement is a higher entity than the bipartite entanglements. With a new combination of the six entanglement parameters, the present condition can be simplified enough to determine easily whether a deterministic LOCC transformation from an arbitrary state |ψ〉 to another arbitrary state |ψ′〉 is possible or not. Moreover, the minimum number of times of measurements to reproduce an arbitrary deterministic LOCC transformation is given. This is an extension of Horodecki et al.’s work [24] to three-qubit pure states.
The states of three-qubit systems split into two inequivalent types of genuine tripartite entanglement, namely, the Greenberger-Horne-Zeilinger (GHZ) type and the W 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 W states, under the local measurement operations carried out by a single party and p (p=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.