W-States have achieved the status of the standard fully symmetric entangled states, for many entanglement application purposes. Z-States are a generalization of W-States that display an elegant algebra, enabling short paths to desired results. This paper describes Z-States algebra starting from neat definitions and laying down explicitly some fundamental theorems on composition and distillation, needed for applications. These theorems are synthesized into a generic tunable Entanglement-Distillation Protocol. Applications are readily developed based upon the tunable protocol. A few examples are provided to illustrate the approach generality. A concomitant graphical representation allows fast comprehension of the protocol inputs, operations and outcomes.
Whenever one needs to illustrate standard multi-qubit entangled states, it has been customary to have recourse to W-States or to GHZ-States (GHZ standing for Greenberger, Horne and Zeilinger [12]).
W-States have N terms with N entangled qubits, where in each term there is exactly one qubit with a value |1> and all other qubits with value |0>. Thus:
W-States have been shown to maximize the bipartite Entanglement Formation measure [17]. This work deals with Z-States, a generalization of W-States with more degrees of freedom, viz. they have k qubits with value |1> in each term, instead of exactly one.
We develop in this paper some fundamental results of the Z-States algebra. These refer, first of all, to the ability to compose larger Z-States from smaller ones. As a consequence, we gain the capability to distill composed Z-States involving only a determined number of local operations.
The fundamental theorems are then synthesized into a generic tunable Entanglement-Distillation Protocol. The generality of the protocol is illustrated by examples showing how one readily distillates desirable Z-States. These are further depicted by a convenient graphical representation, allowing fast comprehension of the inputs, operations and outcomes.
As a matter of terminology, we shall use state and vector interchangeably (quantum states are of course rays in the Hilbert space but we shall be careful not to be misguided by terminology freedom). Moreover, all the paper deals with real valued vectors and thus we will use real valued vector spaces 1 .
In the introduction we refer to related work on W-States, entanglement distillation from various points of view and previous applications of Z-States.
The prominence of GHZ-States [12] and W-States as multi-qubit entangled states has been noted by Dur et al. [9]. In particular, they observe that W-States with N=3 retain maximally bipartite entanglement when any one of the three qubits is traced out. This property is suitably generalized to any value of N. Yan et al. [22] deal with maximally entangled states of N qubits, an interesting property of N-GHZ states. They prove that two local observables are sufficient to characterize such states. Fortescue and Lo [10] define random distillation of multiparty entangled states as conversion of such states into entangled states shared between fewer parties, where those parties are not predetermined. They discuss distillation protocols for W, GHZ and particular cases of Z-States2 . Cui et al. [5] describe ways of converting an N-qubit W-State into maximum entanglement shared between two random parties.
Experimental entanglement distillation has been achieved by several techniques. Dong et al. [8] describe a mesoscopic distillation of deterministically prepared entangled light pulses that have undergone non-Gaussian noise. It employs linear optical components and global classical communication. Miyake and Briegel [18] propose a multipartite distillation scheme by complementary stabilizer measurements; they construct a recurrence protocol for the 3-qubit W-state (see also Cao and Yang [4]). Fujii et al. [11] devised a scalable experimental scheme to generate N-qubit W-States by using separated cavity-QED systems and linear optics, with post-selection. Initially they generate the four-qubit W state |W4>. Then they symmetrically breed couples of |W N > states into |W (2N-2) >, a particular symmetric case of our Entanglement-Distillation protocol. Z-States have been observed experimentally in the particular case of four entangled photons with two excitations (cf. Kiesel et al. [16]). Toth [20] provides conditions to detect Z-States with N qubits, in the vicinity of N/2 qubits with value |1>.
Entanglement characterization has been done by Stockton et al. [19] for the entanglement of symmetric states, such as GHZ and Z-States by calculations of several entanglement measures. Among the latter they obtain values for reduced entropy and entanglement of formation. Hein et al. [13] characterize the entanglement of graph states. A graph state is a special pure multi-party quantum state of a distributed quantum system. These states are mathematical graphs where the vertices of the graph are quantum spin systems and edges represent Ising interactions. Graph states are a possible generalization of the standard multi-party entangled states, say GHZ states. A more complex example is the graph associated with the quantum Fourier transform.
Distillation protocols can be designed by different approaches. Hostens et al. [15] show the equivalence of two such approaches, one based upon local operations yielding permutations of tensor products of Bell states (see also Dehaene et al. [6]) and another using stabilizer codes.
Various properties and applications of distillation are considered in the literature. Horodecki and Horodecki [14] analyse distillation of mixed states in higher dimensional systems. They provide a separability criterion for the total d
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