The stabilizer for..-qubit symmetric states

The stabilizer group for an n-qubit state ｜Φ〉 is the set of all invertible local operators（ILO） g = g1g2…gn,gi 2 GL（2,C） such that ｜Φ〉= g｜Φ〉. Recently, Gour et al. [Gour G, Kraus B and Wallach N R 2017 J. Math. Phys. 58092204] presented that almost all n-qubit states jyi own a trivial stabilizer group when n≥5. In this article, we consider the case when the stabilizer group of an n-qubit symmetric pure state jyi is trivial. First we show that the stabilizer group for an n-qubit symmetric pure state ｜Φ〉 is nontrivial when n≤4. Then we present a class of n-qubit symmetric states ｜Ψ〉 with a trivial stabilizer group when n≥5. Finally, we propose a conjecture and prove that an n-qubit symmetric pure state owns a trivial stabilizer group when its diversity number is bigger than 5 under the conjecture we make, which confirms the main result of Gour et al. partly.

Entanglement and Volume Monogamy Features of Permutation Symmetric N-Qubit Pure States with N-Distinct Spinors: GHZ and States

We explore the entanglement features of pure symmetric N-qubit states characterized by N-distinct spinors with a particular focus on the Greenberger-Horne-Zeilinger (GHZ) states and , an equal superposition of W and obverse W states. Along with a comparison of pairwise entanglement and monogamy properties, we explore the geometric information contained in them by constructing their canonical steering ellipsoids. We obtain the volume monogamy relations satisfied by states as a function of number of qubits and compare with the maximal monogamy property of GHZ states.

Generation and classification of robust remote symmetric Dicke states

In this paper, we present an approach to generating arbitrary symmetric Dicke states with distant trapped ions and linear optics. Distant trapped ions can be prepared in the symmetric Dicke states by using two photon-number-resolving detectors and a polarization beam splitter. The atomic symmetric Dicke states are robust against decoherence, for atoms are in a metastable level. We discuss the experimental feasibility of our scheme with current technology. Finally, we discuss the classification of arbitrary n-qubit symmetric Dicke states under statistical local operation and classical communication and prove the existence of [n/2] inequivalent classes of genuine entanglement of n-qubit symmetric Dicke states.

Monogamous nature of symmetric N-qubit states of the W class: Concurrence and negativity tangle

Using Majorana representation of symmetric N-qubit pure states, we have examined the monogamous nature of the family of states with two-distinct spinors, the W class of states. We have evaluated the N-concurrence tangle and showed that all the states in this family have vanishing concurrence tangle. The negativity tangle for the W class of states is shown to be non-zero, illustrating the fact that the concurrence tangle is always lesser than or equal to the negativity tangle in a pure N-qubit state.

Mixedness of the N-qubit states with exchange symmetry

The mixedness of the N-qubit quantum states with exchange symmetry has been studied, and the results show that the linear entropy of the single qubit reduced density matrix （RDM）, which can describe the mixedness, is completely determined by the expectation values 〈Sz〉 and 〈S±〉 for both the pure and the mixed states. The mixedness of the pure states can be used to describe the bipartite entanglement, as an example we have calculated the mixedness of the Dicke state and the spin squeezed Kitagawa-Ueda state. For the mixed states, we determine the mixedness properties of both the ground states and the thermal states in mean-field clusters of spin-1/2 particles interacting via the anisotropy Heisenberg XXZ interaction, and found for the ferromagnetic case （J 〈 0）, the mixedness will approximate to the pairwise entanglement when the anisotropic parameter △ 〉 △c.

Revised Geometric Measure of Entanglement for Multipartite and Continuous Variable Systems

Based on the revised geometric measure of entanglement （RGME） proposed by us [J. Phys. A： Math. Theor. 40 （2007） 3507], we obtain the RGME of multipartite state including three-qubit GHZ state, W state, and the generalized Smolin state （GSS） in the presence of noise and the two-mode squeezed thermal state. Moreover, we compare their RGME with geometric measure of entanglement （GME） and relative entropy of entanglement （RE）. The results indicate RGME is an appropriate measure of entanglement. Finally, we define the Gaussian GME which is an entangled monotone.

Cubature Kalman filters: Derivation and extension

This paper focuses on the cubature Kalman filters （CKFs） for the nonlinear dynamic systems with additive process and measurement noise. As is well known, the heart of the CKF is the third-degree spherical–radial cubature rule which makes it possible to compute the integrals encountered in nonlinear filtering problems. However, the rule not only requires computing the integration over an n-dimensional spherical region, but also combines the spherical cubature rule with the radial rule, thereby making it difficult to construct higher-degree CKFs. Moreover, the cubature formula used to construct the CKF has some drawbacks in computation. To address these issues, we present a more general class of the CKFs, which completely abandons the spherical–radial cubature rule. It can be shown that the conventional CKF is a special case of the proposed algorithm. The paper also includes a fifth-degree extension of the CKF. Two target tracking problems are used to verify the proposed algorithm. The results of both experiments demonstrate that the higher-degree CKF outperforms the conventional nonlinear filters in terms of accuracy.

Deterministic quantum communication using the symmetric W state

This study proposes a new coding function for the symmetric W state. Based on the new coding function, a theoretical protocol of deterministic quanama communication （DQC） is proposed. The sender can use the proposed coding function to encode his/her message, and the receiver can perform the imperfect Bell measurement to obtain the sender＇s message. In comparison to the existing DQC protocols that also use the W class state, the proposed protocol is more efficient and also more practical within today＇s technology. Moreover, the security of this protocol is analyzed to show that any eavesdropper will be detected with a very high probability under both the ideal and the noisy quantum channel.

Single-Qubit Operation Sharing with Bell and W Product States

Two tripartite schemes are put forward with shared entanglements and Local Operation and Classical Communication (LOCC) for sharing an operation on a remote target sate.The first scheme uses a Bell and a symmetric W states as quantum channels,while the second replaces the symmetric W state by an asymmetric one.Both schemes are treated and compared from the aspects of quantum resource consumption,operation complexity,classical resource consumption,success probability and efficiency.It is found that the latter scheme is better than the former one.Particularly,the sharing can be achieved only probabilistically with the first scheme deterministically with the second one.