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arXiv:1005.4656 · Physical Review A

Generalized X states of N qubits and their symmetries

Sai Vinjanampathy, A. R. P. Rau

Published Sep 29, 2010• 30 citations
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Abstract

Several families of states such as Werner states, Bell-diagonal states, and Dicke states are useful in understanding multipartite entanglement. Here we present a $({2}^{N+1}\ensuremath{-}1)$-parameter family of $N$-qubit ``$X$ states'' that embraces all those families, generalizing previously defined states for two qubits. We also present the algebra of the operators that characterize the states and an iterative construction for this algebra, a subalgebra of su$({2}^{N})$. We show how a variety of entanglement witnesses can detect entanglement in such states. Connections are also made to structures in projective geometry.

DOI
10.1103/physreva.82.032336

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Several families of states such as Werner states, Bell-diagonal states, and Dicke states are useful in understanding multipartite entanglement. Here we present a (2N+1\ensuremath−1)({2}^{N+1}\ensuremath{-}1)(2N+1\ensuremath−1)-parameter family of NNN-qubit ``XXX states'' that embraces all those families, generalizing previously defined states for two qubits. We also present the algebra of the operators that characterize the states and an iterative construction for this algebra, a subalgebra of su(2N)({2}^{N})(2N). We show how a variety of entanglement witnesses can detect entanglement in such states. Connections are also made to structures in projective geometry.
A class of two-qubit states called X-states are increasingly being used to discuss entanglement and other quantum correlations in the field of quantum information. Maximally entangled Bell states and "Werner" states are subsets of them. Apart from being so named because their density matrix looks like the letter X, there is not as yet any characterization of them. The su(2) X su(2) X u(1) subalgebra of the full su(4) algebra of two qubits is pointed out as the underlying invariance of this class of states. X-states are a seven-parameter family associated with this subalgebra of seven operators. This recognition provides a route to preparing such states and also a convenient algebraic procedure for analytically calculating their properties. At the same time, it points to other groups of seven-parameter states that, while not at first sight appearing similar, are also invariant under the same subalgebra. And it opens the way to analyzing invariant states of other subalgebras in bipartite systems.
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.
Multipartite entanglement is a key concept in quantum mechanics for which, despite the experimental progress in entangling three or more quantum devices, there is still no general quantitative theory that exists. In order to characterize the robustness of multipartite entanglement, one often employs generalized Werner states, that is, mixtures of a Greenberger-Horne-Zeilinger (GHZ) state and the completely unpolarized state. While two-qubit Werner states have been instrumental for various important advancements in quantum information, as of now there is no quantitative account for such states of more than two qubits. By using the GHZ symmetry introduced recently, we find exact results for tripartite entanglement in three-qubit generalized Werner states and, moreover, the entire family of full-rank mixed states that share the same symmetries.
We introduce Transition states (T states), denoted by ∣Tkn⟩\ket{T_k^n}∣Tkn​⟩, as a class of multipartite entangled states characterized by a fixed number of state transitions between adjacent qubits. These states form equal-amplitude superpositions over all states with a specified transition count. Unlike Bell states based on two-qubit correlations, GHZ states characterized by global correlations among all qubits, and W and Dicke states based on fixed numbers of qubit excitations, T states are defined by transition counts along an ordered sequence of qubits. We prove that T states are unitarily equivalent to Dicke states through a chain of CX (controlled-X) operations, thereby establishing a direct correspondence between transition-based and excitation-based representations of multipartite entanglement.
We present a method to detect entanglement partitions of multipartite quantum systems, by exploiting their inherent symmetries. Structures like genuinely multipartite entanglement, mmm-separability and entanglement depth are detected as very special cases. This formulation enables us to characterize all the entanglement partitions of all three- and four- partite states and witnesses with unitary and permutation symmetry. In particular, we find and parametrize a complete set of bound entangled states therein. For larger systems, we provide a large family of analytical witnesses detecting many-body states of arbitrary size where none of the parties is separable from the rest. This method relies on weak Schur sampling with projective measurements, and thus can be implemented in a quantum computer. Beyond physics, our results extend to the mathematical literature: we establish new inequalities between matrix immanants, and characterize the set of such inequalities for matrices of size three and four.
Abstract We present two results on multiqubit Werner states, defined to be those states that are invariant under the collective action of any given single-qubit unitary that acts simultaneously on all the qubits. Motivated by the desire to characterize entanglement properties of Werner states, we construct a basis for the real linear vector space of Werner invariant Hermitian operators on the Hilbert space of pure states; it follows that any mixed Werner state can be written as a mixture of these basis operators with unique coefficients. Continuing a study of ‘polygon diagram’ Werner states constructed in earlier work, with a goal to connect diagrams to entanglement properties, we consider a family of multiqubit states that generalize the singlet, and show that their 2-qubit reduced density matrices are separable.