The stabilizer for n-qubit symmetric states
Shi Xian1, 2, 3, †
Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
University of Chinese Academy of Sciences, Beijing 100049, China
UTS-AMSS Joint Research Laboratory for Quantum Computation and Quantum Information Processing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

 

† Corresponding author. E-mail: shixian01@gmail.com

Project partially supported by the National Key Research and Development Program of China (Grant No. 2016YFB1000902), the National Natural Science Foundation of China (Grant Nos. 61232015 and 61621003), the Knowledge Innovation Program of the Chinese Academy of Sciences (CAS), and Institute of Computing Technology of CAS.

Abstract

The stabilizer group for an n-qubit state |ϕ⟩ is the set of all invertible local operators (ILO) g = g1g2 ⊗ ⋯ ⊗ gn, such that |ϕ⟩ = g|ϕ⟩. Recently, Gour et al. [Gour G, Kraus B and Wallach N R 2017 J. Math. Phys. 58 092204] presented that almost all n-qubit states |ψ⟩ 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 |ψ⟩ 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 |ϕ⟩ 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.

1. Introduction

Quantum entanglement[1] is a valuable resource for a variety of tasks that cannot be finished by classical resource. Among the most popular tasks are quantum teleportation[2] and quantum superdense coding.[3] Due to the importance of quantum entanglement, the classification of quantum entanglement states is a big issue for the quantum information theory.

Entanglement theory is a resource theory with its free transformation is local operations and classical communication (LOCC). As LOCC is hard to deal with mathematically, and with the number of the parties of the quantum systems growing, the classification of all entanglement states under the LOCC restriction becomes very hard. A conventional way is to consider other operations, such as stochastic LOCC (SLOCC), local unitary operations (LU), and separable operations (SEP).

Two n-partite states |ϕ⟩ and |ψ⟩ are SLOCC equivalent[4] if and only if there exists n invertible local operations (ILO) Ai, i = 1, 2,..., n such that

The classification for multi-qubit pure states under SLOCC is attracting a great deal of attention.[510] However, there are an uncountable number of SLOCC inequivalent classes in n-qubit systems when n ≥ 4, so it is a formidable task to classify multipartite states under SLOCC.

SEP is simple to describe mathematically and contains LOCC strictly, as pure state transformations exist belonging to SEP, but they cannot be achieved by LOCC. The authors in Ref. [11] presented that the existence of transformations under separable operations between two pure states depends largely on the stabilizer of the state. Recently, Gour et al. showed that almost all of the stabilizer group for 5 or more qubits pure states contains only the identity.[12] The authors in Ref. [13] generalized this result to n-qudit systems when n > 3, d > 2.

Symmetric states belong to the space that is spanned by the pure states invariant under particle exchange, and there are some results done on the classifications under SLOCC limited to symmetric states.[1418] The authors in Ref. [15] proved if |ψ⟩ and |ϕ⟩ are n-qubit symmetric pure states, and n invertible operations Ai, i = 1, 2,..., n exist such that |ψ = A1A2 ⊗ ⋯ ⊗ An |ϕ⟩, then there exists an invertible matrix A such that

Moreover, Migdal et al.[16] generalized the results from qubit systems to qudit systems.

In this article, we consider the problem on the stabilizer groups for n-qubit symmetric states. This article is organized as follows. In Section 2, we present preliminary knowledge on n-qubit symmetric pure states. In Section 3, we present our main results. First, we present the stabilizer group for an n-qubit symmetric state is nontrivial when n ≤ 4, then we present a class of n-qubit symmetric states whose stabilizer group is trivial n ≥ 5, at last, based on the Majorana representation for symmetric states, we apply the Möbius transformation to present when the diversity number m of a pure symmetric state |ψ⟩ is 5 or 6, it owns a trivial stabilizer group, when m = 2, the pure state |ψ⟩ owns a nontrivial stabilizer group, 5 when m = 4, there exists only one case when |ψ⟩ owns a nontrivial stabilizer group, when m ≥ 7, under a conjecture we make, a symmetric pure state |ψ⟩ owns a trivial stabilizer group. In Section 4, we will end with a summary.

2. Preliminary knowledge

In this section, we will first recall the definition of symmetric states, and then we present the Majorana representation for an n-qubit symmetric pure state.

A pure state can be represented by a point on a Bloch sphere geometrically as |ψ⟩ = cos(θ/2)|0⟩ + eiϕ sin(θ/2)|1⟩, here two parameters θ ∈ [0, π], ϕ ∈ [0, 2π).

Fig. 1. Bloch sphere.

We call an n-partite pure state |ψ⟩ symmetric state if it is invariant under permuting the particles. That is, for any permutation operator Pπ, Pπ |ψ⟩ = |ψ⟩. Generally, there are two main characterizations for an n-qubit symmetric pure state |ψ⟩, Majorana representation[19] and Dicke representation. The Majorana representation for an n-qubit symmetric pure state is that single particles |ϕi⟩, i = 1, 2,..., n, exist such that

where the sum runs over all distinct permutations σ, K is a normalization prefactor, and the |ϕi⟩ are single qubit states |ϕi⟩ = cos(θi/2)|0⟩ + eiϕi sin(θi/2)1⟩, i = 1, 2,... n, we would denote |ϕi⟩ as |ϕi⟩ = ai|0⟩ + bi|1⟩ below. An n-qubit symmetric pure state |ψ⟩ can also be characterized as the sum of the Dicke states |D(n, k)⟩, that is,

Here the Dicke states |D(n, k)⟩ are defined as

where the sum runs over all the permutations of the qubits. Up to a global phase factor, the parameters ai/bi in a pure state |ϕi⟩ = ai|0⟩ + bi|1⟩ are the roots of the polynomial , here denotes the binomial coefficient of n and k.

Next we introduce an isometric linear map

here we denote that is the set of all linear maps from the Hilbert space to the Hilbert space . This map is useful for considering the stabilizer group for a 2-qubit symmetric state. Now we introduce some properties of this map:

(i) Assume , we have that

(ii) Assume , then

Then we recall the definition and some important properties of the Möbius transformation, which is useful for the last part of this article. The Möbius transformation is defined on the extended complex plane onto itself,[20] it can be represented as

with a, b, c, d ∈ ℂ, adbc ≠ 0. From the above equality, we see that when c ≠ 0, this function f : ℂ − {−d/c} → ℂ − {a/c}, f(−d/c) = ∞, f(∞) = a/c, when c = 0, this function f : ℂ → ℂ, f(∞) = ∞. The Möbius transformation owns the following properties:

Möbius transformation maps circles to circles.

Möbius transformations are conformal.

If two points are symmetric with respect to a circle, then their images under a Möbius transformation are symmetric with respect to the image circle. This is called the “Symmetry Principle”.

With the exception of the identity mapping, a Möbius transformation has at most two fixed points.

There exists a unique Möbius transformation sending any three points to any other three points.

The unique Möbius transformation zM(z) sending three points q, r, s to any other three points q′, r′, s′ is given by

The Möbius transformation forms a group, Möbius transformation is isotropic to the projective linear group PSL(2, ℂ) ≅ SL(2, ℂ)/{I, −I}.

As we know, the stereographic projection is a mapping that projects a sphere onto a plane. This projection is defined on the whole sphere except a point, and this map is smooth and bijective. It is conformal, i.e., it preserves the angles at where curves meet. By transforming the majorana points of a pure state |ψ⟩ to an extended complex plane, we may get the following proposition.[21]

At last, we recall two parameters defined in Ref. [14], the diversity degree and degeneracy configuration of an n-qubit symmetric pure state. Both parameters can be used to identify the SLOCC entanglement classes of all n-qubit symmetric pure states. Assume |ψ⟩ is an n-qubit symmetric pure state,

up to a global phase factor, two states |ϕi⟩ and |ϕj⟩ are identical if and only if aibjajbi = 0, and we define their number as the degeneracy number. Then we define the degeneracy configuration {ni} of a symmetric state |ψ⟩ as the list of its degeneracy numbers ni ordered in decreasing order. We denote the number of the elements in the set {ni} as the diversity degree m of the symmetric state, it stands for the number of distinct |ϕi⟩ in Eq. (3). For example, for a 3-qubit GHZ state , we have |ϕ1⟩ = (|0⟩ + w|1⟩)/2, |ϕ2⟩ = (|0⟩ + w2|1⟩)/2, |ϕ3⟩ = (|0⟩ + |1⟩)/2, w is roots of P(z) = 1 − z3, the degeneracy number of |GHZ⟩ is 3, and the degeneracy configuration of |GHZ⟩ is {1, 1, 1}.

3. Main Results

First we present that the stabilizer group for a two-qubit symmetric pure state is nontrivial.

Now we present a lemma to show an n-qubit symmetric pure state owns a nontrivial stabilizer group when n = 3, 4.

In Ref. [14], the authors presented that a three-qubit symmetric pure state is SLOCC equivalent to |W⟩ or |GHZ⟩. As we know, when we choose

g⊗3|W⟩ = |W⟩, . From Lemma 2, we see that the stabilizer group for all three-qubit pure symmetric states is nontrivial. A four-qubit symmetric state is SLOCC equivalent to one of the elements in

then we present a nontrivial stabilizer for the elements in the set S, for the state |D(4, 0)⟩, we have , for the state |D(4, 1)⟩, we choose an ILO

for the state |D(4,2)⟩, we can choose an ILO g = σz, and for the last element in the set S, we can also choose an ILO g = σz, then due to the Lemma 2, we have the stabilizer group for all four-qubit symmetric pure states is nontrivial. Note that for four-qubit pure states, the authors in Ref. [18] also proposed similar results.

Here we denote

Next we will use the method proposed in Ref. [12] to give a class of symmetric states with its stabilizer group containing only the identity. First we introduce the definition of SL-invariant polynomials. A polynomial is SL-invariant if f(g|ψ⟩) = f(|ψ⟩), ∀gSL(2, ℂ) ⊗ SL(2, ℂ) ⋯ ⊗ SL(2, ℂ), here we denote and

f2 is an SL-invariant polynomial with degree 2, which is defined as , here σy is the Pauli operator with its matrix representation

Due to the property of σy, we have that when n is odd, f2(·) = 0. Another polynomial, f4(|ψ〉), is a polynomial with degree 4, it is defined as

here we assume that |ψ⟩ = |0⟩|ψ0⟩ + |1⟩|ψ1⟩. Below we denote the stabilizer group for a pure state |ψ⟩ as

Apply ⟨0| ⊗ I⊗(n−1) to the left-hand side (LHS) and the right-hand side (RHS) of the above equality, and we denote that |ζ1⟩ = xk|D(n − 1, k)⟩ + xl|D(n − 1, l)⟩, |ζ2⟩ = xk|D(n − 1, k − 1)⟩ + xl|D(n − 1, l − 1)⟩ + xn|D(n − 1,n − 1)⟩, then

Assume n is odd, then by using f2(·) to the equality (15), we have , b1 = 0. As |ψ⟩ is a symmetric state, we may assume

due to g1g2 ⊗ ⋯ ⊗ gn|ψ⟩ = |ψ⟩ and through simple computation, we have ci = 0. Applying

to the equality (14), we have di = λai, Πdi = 1, λk = λl = 1, as gcd(k, l) = 1, we have λ = 1, that is .

When n is even, we use f4(·) to the equality (15), we have , b1 = 0. From the same method as the case when n is odd, we have that .

Then we present a class of symmetric critical states |ψ⟩ with Gψ = {I}. First we present the definition of critical states and a meaningful characterization of critical states. The set of critical states is defined as:

Here Lie(G) is the Lie algebra of G. The critical set is valuable, as many important states in quantum information theory, such as the Bell states, GHZ states, cluster states, and graph states, belong to the set of critical states. Then we present a fundamental characterization of critical states.

Here we present another proof on GLn = {I}, |Ln⟩ is defined in the equality (8) of the article.[12] Note that the examples above tell us Gψ = {I}, however, . It seems that this result is simple, however, this method is very useful to present nontrivial examples of states in n-qudit systems with nontrivial stabilizer groups.[13]

Finally, I would like to apply Möbius transformation to show when the diversity number m of an n-qubit symmetric pure state |ψ⟩ is 5 or 6, the stabilizer group of |ψ⟩ is trivial, when m ≥ 7, under a conjecture we make, the stabilizer group of |ψ⟩ is trivial.

Assume a pure symmetric state |ψ⟩ can be represented in terms of Majorana representation:

where the sum takes over all the permutations and K is the normalization for the state |ψ⟩. Due to the main results proposed by Mathonet et al.,[15] we see that if |ψ⟩ = g1g2 ⊗ ⋯ ⊗ gn|ψ⟩, then there exists an ILO g such that |ψ⟩ = gn|ψ⟩, i.e., if we could prove {g|gn|ψ⟩ = |ψ⟩} = {I}, then the stabilizer group of |ψ⟩ is trivial. From Eq. (19),

Due to the uniqueness of the Majorana representation for a symmetric state and according to the equality (21), we see that there is a permutation σ such that

Assume the diversity number of a symmetric state |ψ⟩ is m, the divergence configuration for the state |ψ⟩ is {k1, k2,..., km} with k1k2 · ≥ km and Σiki = n. Then from the Lemma 1 and the property iv) of Möbius transformation, we have:

Here we only consider the first two cases, the other two are similar to the second one. In case I), from property vi) of Möbius transformation f, we have

from the above equalities, we have

then from Eqs. (26) and (27), we have

i.e., z1 = z4, z2 = z3, z4 = z2, or z3 = z1, however, each of these four conditions is invalid, that is, case I) is invalid. For case II), we may also from property vi) of Möbius transformation f have

from the above equalities, we have

this is invalid. Then we see the above four conditions are invalid. When k1 = k2k3k4, similar to the above analyses, the stabilizer group for the pure states is trivial. Then we see that only when k1 = k2, k3 = k4 the stabilizer group is nontrivial.

Here we note that when |ψ⟩ is a four-qubit symmetric pure state, |ψ⟩ always owns a nontrivial stabilizer group. As the diversity configuration can only be {1, 1, 1, 1}, {2, 1, 1}, {3, 1}, or {4}, from the above analyses, we see that the stabilizer group is nontrivial.

When m = 5, assume the degeneracy configuration of the pure state |ψ⟩ is {k1, k2, k3, k4, k5}. First we analyze the case when k1 = k2 = k3 = k4 = k5, if the stabilizer group is nontrivial, then there exists a nontrivial Möbius transformation f that can permute zi, i = 1, 2, 3, 4, 5, and we need to consider the following cases,

Case a) f(z1) = z2, f(z2) = z3, f(z3) = z4, f(z4)=z5, and f(z5)=z1;

Case b) f(z1) = z2, f(z2) = z3, f(z3) = z4, f(z4) = z1, and f(z5) = z5;

Case c) f(z1) = z2, f(z2) = z3, f(z3) = z1, f(z4) = z4, and f(z5) = z5;

Case d) f(z1) = z2, f(z2) = z3, f(z3)=z1, f(z4) = z5, and f(z5) = z4;

Case e) f(z1) = z2, f(z2) = z1, f(z3) = z4, f(z4) = z3, and f(z5) = z5.

Similar to the analysis when m = 4, we have that the above case c) is invalid. For case d), we can consider the Möbius transformation g = ff, as g(z1) = z3, g(z2) = z1, g(z3) = z2, g(z4) = z4, g(z5) = z5, so Case d) is invalid. For Case e), we can get

Then Case e) is invalid. For Case b), similar to the analysis when m = 4, we see this is invalid. For Case a), we have

from the equalities (36) and (37), we have

combining Eqs. (41) and (38), we have

combining Eqs. (42) and (39), we have

similarly, we have

combining Eqs. (43) and (44), we have z1z2 = z2z3, then combining Eq. (40) and the above equality, we have (z2z4)(z3z1) = 0, then we see this is invalid. Due to the above analyses, we see that when m = 5, the pure state |ψ⟩ owns a trivial stabilizer.

When m = 6, assume the degeneracy configuration of |ψ⟩ is {k1, k2,..., k6} with k1 = k2 = ⋯ = k6, here we first present that there cannot exist a Möbius transformation f such that f(z1) = z2, f(z2) = z3,..., f(z5) = z6, f(z6) = z1. Here we also apply the method reductio ad absurdum. If there exists a Möbius transformation f satisfying f(z1) = z2, f(z2) = z3,..., f(z5) = z6, f(z6) = z1, due to the property vi) of Möbius transformation, we have

that is,

Due to Eq. (49) and z4z6, if we could prove (z1z2)(z3z5) + (z1z5)(z3z2) = 0 is invalid, then this case is invalid. Due to the Möbius transformation f, we have

from Eq. (45) and (z1z2)(z3z5) + (z1z5)(z3z2) = 0, we have

combining Eq. (50) and Eq. (52), we have z4z6 = 0, which is invalid.

Combining the results when m = 5, 6, we may conjecture that:

4. Conclusion

In this article, we consider the stabilizer group for a symmetric state |ψ⟩. First we present that the stabilizer group for an n-qubit symmetric state |ψ⟩ contains more than the identity when n = 2, 3, 4; then similar to the method presented in Ref. [12], we give a class of states whose stabilizer group contains only the identity; we also propose a class of states |ψ⟩ with , ; finally, we apply the Möbius transformation to present when the diversity number m of a symmetric pure state |ψ⟩ is 5 and 6, the pure state |ψ⟩ owns a trivial stabilizer group, and when m ≥ 7, it is valid under the conjecture we make. This may confirm the main result in Ref. [12] partly when considering the n-qubit symmetric pure states.

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