General Local Operations Research Articles

Complete Set of Operational Measures for the Characterization of Three-Qubit Entanglement

We characterize the entanglement contained in a pure three-qubit state via operational entanglement measures. To this end, we derive a new decomposition for arbitrary three-qubit states which is characterized by five parameters (up to local unitary operations). We show that these parameters are uniquely determined by bipartite entanglement measures. These quantities measure the entanglement required to generate the state following a particular preparation procedure and have a clear physical meaning. Moreover, we show that the classification of states obtained in this way is strongly related to the one obtained when considering general local operations and classical communication.

Rescaling multipartite entanglement measures for mixed states

A relevant problem regarding entanglement measures is the following: Given an arbitrary mixed state, how does a measure for multipartite entanglement change if general local operations are applied to the state? This question is nontrivial as the normalization of the states has to be taken into account. Here we answer it for pure-state entanglement measures which are invariant under determinant 1 local operations and homogeneous in the state coefficients, and their convex-roof extension which quantifies mixed-state entanglement. Our analysis allows to enlarge the set of mixed states for which these important measures can be calculated exactly. In particular, our results hint at a distinguished role of entanglement measures which have homogeneous degree 2 in the state coefficients.

Entanglement of Three-Qubit Greenberger-Horne-Zeilinger–Symmetric States

The first characterization of mixed-state entanglement was achieved for two-qubit states in Werner's seminal work [Phys. Rev. A 40, 4277 (1989)]. A physically important extension concerns mixtures of a pure entangled state [such as the Greenberger-Horne-Zeilinger (GHZ) state] and the unpolarized state. These mixed states serve as benchmark for the robustness of multipartite entanglement. They share the symmetries of the GHZ state. We call such states GHZ symmetric. Here we give a complete description of the entanglement in the family of three-qubit GHZ-symmetric states and, in particular, of the three-qubit generalized Werner states. Our method relies on the appropriate parametrization of the states and on the invariance of entanglement properties under general local operations. An application is the definition of a symmetrization witness for the entanglement class of arbitrary three-qubit states.

Local unitary equivalence and entanglement of multipartite pure states

The necessary and sufficient conditions for the equivalence of arbitrary n-qubit pure quantum states under Local Unitary (LU) operations derived in [B. Kraus Phys. Rev. Lett. 104, 020504 (2010)] are used to determine the different LU-equivalence classes of up to five-qubit states. Due to this classification new parameters characterizing multipartite entanglement are found and their physical interpretation is given. Moreover, the method is used to derive examples of two n-qubit states (with n>2 arbitrary) which have the properties that all the entropies of any subsystem coincide, however, the states are neither LU-equivalent nor can be mapped into each other by general local operations and classical communication.

Classification of qubit entanglement: SL(2,ℂ) versus SU(2) invariance

The role of SU(2) invariants for the classification of multiparty entanglement is discussed and exemplified for the Kempe invariant I 5 of pure three-qubit states. It is found being an independent invariant only in presence of both W-type entanglement and three-tangle. In this case, constant I 5 allows for a wide range of both three-tangle and concurrences. This means that I 5 provides no information on the entanglement in the system in addition to that contained already in the tangles (concurrences and three-tangle) themselves. Furthermore, norm-preserving SL ⊗3 orbits of states with equal tangles but continuously varying I 5 are shown to exist. As a consequence, I 5 can be increased (and decreased) by general local operations. Nevertheless, numerical analysis of random SLOCC’s has not shown any violation of the monotone property of I 5. In case I 5 finally turned out to being an entanglement monotone, this would imply that both SU(2) invariance and the stronger monotone property are too weak requirements for the characterization and quantification of entanglement for systems of three qubits, and that SL(2,ℂ) invariance is required. This conclusion can be extended to general multipartite systems (including higher local dimension) because the entanglement classes of three-qubit systems appear as subclasses.

All Bipartite Entangled States Display Some Hidden Nonlocality

One of the most significant and well-known properties of entangled states is that they may lead to violations of Bell inequalities and are thus inconsistent with any local-realistic theory. However, there are entangled states that cannot violate any Bell inequality, and in general the precise relationship between entanglement and observable nonlocality is not well understood. We demonstrate that a violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality can be demonstrated in a certain kind of Bell experiment for all entangled states. Our proof of the result consists of two main steps. We first provide a simple characterization of the set of states that do not violate the CHSH inequality even after general local operations and classical communication. Second, we prove that for each entangled state sigma, there exists another state rho not violating the CHSH inequality, such that rhomultiply sign in circlesigma violates the CHSH inequality.

Analytically solvable two-qubit entanglement monotone

A so-called quasientanglement measure was obtained [J. Zhang et al., Phys. Rev. A 73, 022319 (2006)] by extending Jaeger's Minkowskian norm entanglement measure to general multipartite mixed states. The obtained quasientanglement measure has good properties, but it is still left open whether it is nonincreasing under general local operations and classical communications (LOCC). We show that the obtained quasientanglement measure is indeed nonincreasing on average under LOCC at least for two-qubit states. Thus, it is a good candidate for the two-qubit entanglement monotone. Further, it is shown that this entanglement measure is less than the concurrence for a large class of mixed entangled states.

Asymptotic Violation of Bell Inequalities and Distillability

A multipartite quantum state violates a Bell inequality asymptotically if, after jointly processing by general local operations an arbitrarily large number of copies of it, the result violates the inequality. In the bipartite case we show that asymptotic violation of the Clauser-Horne-Shimony-Holt inequality is equivalent to distillability. Hence, bound entangled states do not violate it. In the multipartite case we consider the complete set of full-correlation Bell inequalities with two dichotomic observables per site. We show that asymptotic violation of any of these inequalities by a multipartite state implies that pure-state entanglement can be distilled from it, although the corresponding distillation protocol may require that some of the parties join into several groups. We also obtain the extreme points of the set of distributions generated by measuring N quantum systems with two dichotomic observables per site.

Entanglement transformations of pure Gaussian states

We present a theory of entanglement transformations of Gaussian pure states with local Gaussian operations and classical communication. This is the experimentally accessible set of operations that can be realized with optical elements such as beam splitters, phase shifts and squeezers, together with homodyne measurements. We provide a simple necessary and sufficient condition for the possibility to transform a pure bipartite Gaussian state into another one. We contrast our criterion with what is possible if general local operations are available.

Optimal simulation of two-qubit Hamiltonians using general local operations

We consider the simulation of the dynamics of one nonlocal Hamiltonian by another, allowing arbitrary local resources but no entanglement or classical communication. We characterize notions of simulation, and proceed to focus on deterministic simulation involving one copy of the system. More specifically, two otherwise isolated systems A and B interact by a nonlocal Hamiltonian $H\ensuremath{\ne}{H}_{A}{+H}_{B}.$ We consider the achievable space of Hamiltonians ${H}^{\ensuremath{'}}$ such that the evolution ${e}^{\ensuremath{-}{\mathrm{iH}}^{\ensuremath{'}}t}$ can be simulated by the interaction H interspersed with local operations. For any dimensions of A and B, and any nonlocal Hamiltonians H and ${H}^{\ensuremath{'}},$ there exists a scale factor s such that for all times t the evolution ${e}^{\ensuremath{-}{\mathrm{iH}}^{\ensuremath{'}}\mathrm{st}}$ can be simulated by H acting for time t interspersed with local operations. For two-qubit Hamiltonians H and ${H}^{\ensuremath{'}},$ we calculate the optimal s and give protocols achieving it. The optimal protocols do not require local ancillas, and can be understood geometrically in terms of a polyhedron defined by a partial order on the set of two-qubit Hamiltonians.

Approximate quantum cloning and the impossibility of superluminal information transfer

We show that nonlocality of quantum mechanics cannot lead to superluminal transmission of information, even if most general local operations are allowed, as long as they are linear and trace preserving. In particular, any quantum mechanical approximate cloning transformation does not allow signalling. On the other hand, the no-signalling constraint on its own is not sufficient to prevent a transformation from surpassing the known cloning bounds. We illustrate these concepts on the basis of some examples.