The Stabilizer Dimension of n-Qubit Symmetric States
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DOI: 10.1007/s10773-013-1847-1
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- Li, B., Li, J. & Wang, Z. Int J Theor Phys (2014) 53: 612. doi:10.1007/s10773-013-1847-1
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Abstract
We consider local unitary transformations acting on a multiparty symmetric pure state and determine the stabilizer dimension of any pure symmetric state.
Keywords
Pure symmetric state Stabilizer dimension Local unitary transformation1 Introduction
Entanglement is considered a key resource in quantum information. It is a fundamental theoretical problem to classify and quantify entanglement in multiparty quantum systems. Despite its importance for the field of quantum information theory, the properties of entangled states are not fully explored yet [1]. One of the first and most natural proposals for classifying entanglement is to consider two states to have the same entanglement type if they can be transformed to each other by a local unitary transformation [2, 3]. Mathematically, the local unitary Lie group acts on the manifold of quantum states, partitioning it into orbits, each orbit represents a type of entanglement. Generally speaking, the number of orbits is infinite, so we consider the dimension of the orbits.
The study of stabilizer dimension was initiated in Ref. [4], much progress has been made in understanding orbits and orbit dimension these years. Sudbery [5] gave six polynomial invariants for 3-qubit pure states; Makhlin [6] gave 18 polynomial invariants that separate orbits for 2-qubit mixed states; recently, Lyons and Walck changed the problem of orbits into a study the stabilizer dimension of pure multipartite state [7, 8, 9, 10, 11]. In [7, 8], the authors obtain minimum orbit dimension and its classification for n-qubit pure states. In [9], they find the maximum stabilizer dimension of nonproduct pure state is n−1, and obtain that for n≥3,n≠4 only generalized n-qubit GHZ states and their local unitary equivalents have maximum stabilizer dimension [10].
While it is an elementary computation to find the stabilizer dimension for a particular given state, it is very difficult to determine stabilizer dimension for general classes of states. Very recently, Zhang et al. in [12] gave a satisfactory answer for graph states. In this paper, we will determine the stabilizer dimension of general n-qubit symmetric pure states.
2 Symmetric States and Lie Algebra Action
3 Stabilizer Dimension of Dicke States
Lemma 1
For any symmetric Dicke state\(|\psi_{n}^{(m)}\rangle \), X∈su(2)^{n}, \(\theta\in\mathbb {R}\), if (4) is satisfied, thenu_{1}=u_{2}=⋯=u_{n}=0.
Proof
Theorem 1
The stabilizer dimension of\(|\psi_{n}^{(0)}\rangle \), \(|\psi_{n}^{(n)}\rangle \)and its LU equivalents is 1; the stabilizer dimension of\(|\psi_{n}^{(m)}\rangle \)and its LU equivalents isn, where 1≤m≤n−1.
Proof
4 Stabilizer Dimension of Pure Symmetric States
We first determine the stabilizer dimension of a class of particular states.
Proposition 1
Let\(|\varPhi_{0}\rangle =\sum_{0 \leq m\leq \frac{n}{2}}(-1)^{m}e^{-im\delta}|\psi_{n}^{(2m)}\rangle \), \(|\varPhi_{1}\rangle =\sum_{0 \leq m\leq \frac{n-1}{2}}(-1)^{m}e^{-im\delta}| \psi_{n}^{(2m+1)}\rangle \)be unnormalized symmetric states, whereδ∈[0,2π] is a parameter, then the stabilizer dimension of |Φ_{0}〉, |Φ_{1}〉 and their LU equivalents aren−1.
Proof
It is interesting for us to note that the result of Ref. [10], we find that |Φ_{0}〉 and |Φ_{1}〉 (possibly after local unitary transformation) has the maximum stabilizer dimension n−1, therefore, |Φ_{0}〉, |Φ_{1}〉 must LU equivalent to the generalized n-qubit GHZ state. In fact, |Φ_{0}〉 and |Φ_{1}〉 have stabilizers conjugate to the stabilizer of the GHZ state. □
Lemma 2
For any pure symmetric state |ψ〉, X∈su(2)^{n}, \(\theta\in\mathbb{R}\), if |ψ〉 satisfies the following conditions: (i) X⋅|ψ〉=iθ|ψ〉, (ii) |ψ〉 is not local unitary equivalent to |Φ_{0}〉, |Φ_{1}〉, thenu_{1}=u_{2}=⋯=u_{n}=0.
Proof
We separate the argument into four cases.
Theorem 2
For anyn-qubit pure symmetric state\(|\psi \rangle =\sum_{m=0}^{n}d_{m}|\psi_{n}^{(m)}\rangle \), the stabilizer dimension of |ψ〉 can only be four cases, i.e., 0, 1, n−1 andn.
Proof
(1) If |ψ〉 is LU equivalent to \(|\psi_{n}^{(0)}\rangle \) or \(|\psi_{n}^{(n)}\rangle \), by Theorem 1, then the stabilizer dimension is n.
(2) If |ψ〉 is LU equivalent to \(|\psi_{n}^{(m)}\rangle \) where 1≤m≤n−1, by Theorem 1 the stabilizer dimension is 1.
(3) If |ψ〉 is LU equivalent to the generalized n-qubit Greenberger-Horne-Zeilinger states, there are only two equations which have the form in (28), and it is easy to know that the dimension is n−1. From Proposition 1 we see that the stabilizer dimension of |Φ_{0}〉, |Φ_{1}〉 are also n−1.
Four kinds of possible stabilizer dimension for n-qubit symmetric states
Case | Stabilizer dimension |
---|---|
The LU equivalents of \(|\psi_{n}^{(0)}\rangle \) or \(|\psi_{n}^{(n)}\rangle \) | n |
The LU equivalents of \(|\psi_{n}^{(m)}\rangle \), where 1≤m≤n−1 | 1 |
The LU equivalents of the generalized GHZ state | n−1 |
Other | 0 |
5 Conclusions
The entanglement classification and some other relevant properties for symmetric states have been studied by many authors [13, 16, 17, 18]. In this paper, we analyzed the stabilizer dimension of n-qubit pure symmetric states. We show the stabilizer dimension for any pure symmetric state can only be four different cases. We further provide the states (under local unitary transformation) which have the corresponding dimension. We hope our findings will shed new light on the characterization of entanglement of multiparty pure symmetric state.
Acknowledgements
The authors thank Professor Shaoming Fei for his advice. Bo Li is supported by Natural Science Foundation of China (Grants No. 11305105), the Natural Science Foundation of Jiangxi Province (Grants No. 20132BAB212010). Jiao-jiao Li is supported by Youth Foundation of Henan Normal University (12QK02) and Zhixi Wang is supported by KZ201210028032.