The stabilizer for n-qubit symmetric states

The stabilizer group for an n-qubit state |ϕ〉 is the set of all invertible local operators (ILO) g=g_1⊗ g_2⊗...⊗ g_n, g_i∈GL(2,C) such that |ϕ〉=g|ϕ〉. Recently, G. Gour et al. GKW presented that almost all n-qubit state |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 the trivial stabilizer group. At last, 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 GKW partly.

Authors

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I Introduction

Quantum entanglement HHH is a valuable resource for a variety of tasks that cannot be finished by classical resource. Among the most popular tasks are quantum teleportation BB and quantum superdense coding BS . 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 grows, 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), separable operations (SEP).
Two n-partite states and are SLOCC equivalent WGJ if and only if there exists n invertible local operations (ILO) such that

 |ϕ⟩=A1⊗A2⊗⋯⊗An|ψ⟩. (1)

Then the classification for multi-qubit pure states under SLOCC attracts much attention MBDM ; CC ; L ; ZZH ; LL ; GW . However, there are uncountable number of SLOCC inequivalent classes in -qubit systems when

so it is a formidable task to classify multipartite states under SLOCC.

SEP is simple to describe mathematically and contains LOCC strictly, as there exists pure state transformations belonging to SEP, but cannot be achieved by LOCC. The authors in GRW presented that the existence of transformations under separable operations between two pure states depends largely on the stabilizer of the state. Recently, Gour showed that almost all of the stabilizer group for 5 or more qubits pure states contains only the identity GKW . And the authors in SW generalized this result to n-qudit systems when
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 TSPM ; PSM ; PR ; PJM ; BBM . The authors in PSM proved if and are -qubit symmetric pure states, and there exists n invertible operations such that

then there exists an invertible matrix A such that

 |ψ⟩=A⊗n|ϕ⟩. (2)

Moreover, P. Migdal PJM generalized the results from qubit systems to qudit systems.
In the article, we consider the problem on the stabilizer groups for -qubit symmetric states. This article is organized as follows. In section II, we present preliminary knowledge on -qubit symmetric pure states. In section III, we present our main results. First, we present the stabilizer group for an -qubit symmetric state is nontrivial when , then we present a class of -qubit symmetric states whose stabilizer group is trivial when , at last, when the pure state owns a nontrivial stabilizer group, when there exists only one case when owns a nontrivial stabilizer group, when the stabilizer group of almost all -qubit symmetric pure state is trivial. In section IV, we will end with a summary.

Ii Preliminary Knowledge

In this section, we will first recall the definition of symmetric states, and then we present the Majorana representation for an -qubit symmetric pure state.
A pure state can be represented by a point on a Bloch sphere geometrically as here two parameters

We call an -partite pure state symmetric state if it is invariant under permuting the particles. That is, for any permutation operator Generally, there are two main characterizations for an -qubit symmetric pure state , Majorana representation M and Dicke representation . The Majorana representation for an n-qubit symmetric pure state is that there exists single particles such that

 |ψ⟩=eiθ√K∑σ∈perm|ϕσ(1)⟩|ϕσ(2)⟩⋯|ϕσ(n)⟩, (3)

where the sum runs over all distinct permutations , is a normalization prefactor and the are single qubit states we would denote as below. An n-qubit symmetric pure state can also be characterized as the sum of the Dicke states , that is,

 |ψ⟩=∑xk|D(n,k)⟩. (4)

Here the Dicke states are defined as

 |D(n,k)⟩=1√Ckn∑|0⋅⋅⋅0n−k1⋅⋅⋅1k⟩, (5)

where the sum runs over all the permutations of the qubits. Up to a global phase factor, the parameters in a pure state are the roots of the polynomial here denotes the binomial coefficient of and
Next we introduce an isometric linear map

 M:L(H1,H2)⟶H1⊗H2 (6) |b⟩⟨a|⟶|b⟩|a⟩ (7)

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 -qubit symmetric state. Now we introduce some properties of this map:
Assume we have that

 A⊗BM(X)=M(AXBT). (8)

Assume then

 M(|ϕ⟩⟨ψ|)=|ϕ⟩|¯ψ⟩. (9)

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

 f(z)=az+bcz+d, (10)

with From the above equality, we see that when this function when this function The transformation owns the following properties:
transformation map circles to circles.
transformation are conformal.
If two points are symmetric with respect to a circle, then their images under a transformation are symmetric with respect to the image circle. This is called the ”Symmetry Principle.”
With the exception of the identity mapping, a transformation has at most two fixed points.
There exists a unique transformation sending any three points to any other three points.
The unique transformation sending three points to any other three points is given by

 (M(z)−q′)(r′−s′)(M(z)−s′)(r′−q′)=(z−q)(r−s)(z−s)(r−q).

The transformation forms a group, transformation is isotropic to the projective linear group
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, it preserves the angels at where curves meet. By transforming the majorana points of a pure state to an extended complex plane, we may get the following proposition MW .

Lemma 1.

Assume are two pure symmetric states, if and are SLOCC-equivalent iff there exists a transformation between their Majorana points.

At last, we recall two parameters defined in TSPM , diversity degree and degeneracy configuration of an -qubit symmetric pure state. Both two parameters can be used to identify the SLOCC entanglement classes of all -qubit symmetric pure state. Assume is an -qubit symmetric pure state, up to a global phase factor, two states and are identical if and only if and we define their number the degeneracy number. Then we define the degeneracy configuration of a symmetric state as the list of its degeneracy numbers ordered in decreasing order. We denote the number of the elements in the set as the diversity degree of the symmetric state, it stands for the number of distinct in the Eq. For example, a -qubit GHZ state we have is roots of the degeneracy number of is 3, the degeneracy configuration of is

Iii Main Results

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

Theorem 1.

Assume that is a -qubit symmetric pure state, then the stabilizer group for the state is nontrivial.

Proof.

Assume and and then we have according to the equality we have that

 AX=X(BT)−1. (11)

here we assume that and then we can obtain the following equation set:

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩c0(c−a)=c1(b−g)c0f−c2b=c1(a−h)c0c−c2f=c1(d−e)c1(f−c)=c2(d−h) (12)

As there are four equations and eight variables, then we know that the rank of the solution vectors is more than 1, that is, the stabilizer group for the state

contains more than the identity. ∎

Now we present a lemma to show an -qubit symmetric pure state owns a nontrivial stabilizer group when .

Lemma 2.

Assume and are symmetric pure states, and is SLOCC equivalent to if there exists a nontrivial ILO such that then the stabilizer group for is nontrivial.

Proof.

As and are symmetric states, then there exists an ILO such that that is, the stabilizer group for is nontrivial. ∎

In TSPM , the authors presents a three-qubit symmetric pure state is SLOCC equivalent to or As we know, when we choose From Lemma 2, we see that the stabilizer group for all three-qubit pure symmetric states is nontrivial. And 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 for the state we have for the state we choose an ILO for the state we can choose an ILO and for the last element in the set S, we can also choose an ILO then due to the lemma 2, we have the stabilizer group for all four-qubit symmetric pure state is nontrivial. Note that for four-qubit pure states, the authors in BBM also proposed the similar results.
Here we denote

 G≡SL(2,C)⊗SL(2,C)⊗⋯⊗SL(2,C) K≡SU(2)⊗SU(2)⊗⋯⊗SU(2) ~G≡GL(2,C)⊗GL(2,C)⊗⋯⊗GL(2,C) ~K≡U(2)⊗U(2)⊗⋯⊗U(2) (13)

Next we will use the method proposed in GKW to give a class of symmetric states with its stabilizer group containing only the identity. First we introduce the definition of -invariant polynomials. A polynomial is -invariant if here we denote and . is a -invariant polynomial with degree 2, which is defined as here is the Pauli operator with its matrix representation . Due to the property of we have that when

is odd,

Another polynomial, is a polynomial with degree , it is defined as here we assume that Below we denote the stabilizer group for a pure state as

Example 1.

Assume is an -partite symmetric pure state , here we denote that is the greatest common divisor of and Then

Proof.

First we denote as and let then

 g1⊗h|ψ⟩ =|0⟩h[a1(xk|D(n−1,k)⟩+xl|D(n−1,l)⟩)+b1(xk|D(n−1,k−1)⟩+xl|D(n−1,l−1)⟩ +xn|D(n−1,n−1)⟩)]+|1⟩h[c1(xk|D(n−1,k)⟩+xl|D(n−1,l)⟩) +d1(xk|D(n−1,k−1)⟩+xl|D(n−1,l−1)⟩+xn|D(n−1,n−1)⟩)] =|0⟩(xk|D(n−1,k)⟩+xl|D(n−1,l)⟩) +|1⟩(xk|D(n−1,k−1)⟩+xl|D(n−1,l−1)⟩+xn|D(n−1,n−1)⟩) (14)

Apply to the left hand side (LHS) and the right hand side (RHS) of the above equality, and we denote that then

 h[a1|ζ1⟩+b1|ζ2⟩% ]=xk|D(n−1,k)⟩+xl|D(n−1,l)⟩. (15)

Assume is odd, then by using to the equality we have As is a symmetric state, we may assume due to and through simple computation, we have that Applying to the equality we have as we have that is
When is even, we use to the equality , we have From the same method as when is odd, we have that

Then we present a class of symmetric critical states with . First we present the definition of critical states and a meaningful characterization of critical states. The set of critical states is defined as:

 Crit(Hn)={ψ|⟨ψ|X|ψ⟩=0,X∈Lie(G)}. (16)

Here is the Lie algebra of 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.

Lemma 3.

(The Kempf-Ness theorem)
1. if and only if for all denotes the Euclidean norm of
2. If then with equality if and only if Moreover, if g is positive semidefinite then the equality condition holds if and only if
3. If then is closed in if and only if

Due to this lemma, Gour and Wallach in GNW showed that is critical if and only if all the local density matrices of are proportional to the identity. And by the lemma in GKW and the theorem 2.12 in W , for a class of pure states if we could show the set contains only the identity, then

Example 2.

Assume or is an -qubit symmetric critical pure state with then

Proof.

First we prove When assume satisfies then

 (U⊗U)ρ1,2(U⊗U)+=ρ1,2, (17)

the equality can be changed as

 (U⊗U)[(n−2)|a1|2n|00⟩⟨00|+|a1|2n(|01⟩+|10⟩)(⟨01|+⟨10|)+|an|2|11⟩⟨11|](U⊗U)+ = |a1|2(n−2)n|00⟩⟨00|+|a1|2n(|01⟩+|10⟩)(⟨01|+⟨10|)+|an|2|11⟩⟨11| (18)

As the formula on the RHS of the equality above can be seen as a Schmidt decomposition, and cannot increase the Schmidt rank, then as is a unitary operator, then At last, due to then we have The other case is similar. ∎

Here we present another proof on is defined in the equality of the article GKW . Note that the examples above tells us 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 SW .
At last, I would like to apply transformation to show when the diversity number of an -qubit symmetric pure state is 5 or 6, the stabilizer group of is trivial, when under a conjecture we make, the stabilizer group of is trivial.
Assume a pure symmetric state can be represented in terms of Majorana representation:

 |ψ⟩=eiα√K∑σ∈perm|ϕσ(1)⟩|ϕσ(2)⟩⋯%$|ϕσ(n)⟩$, (19)

where the sum takes over all the permutations and K is the normalization for the state Due to the main results proposed by Mathonet PSM , we see that if then there exists an ILO such that if we could prove then the stabilizer group of is trivial. From the Eq.

 g⊗n|ψ⟩= |ψ⟩, (20) ∑σ∈perm⨂ig|ϕσ(i)⟩= ∑σ∈perm⨂i|ϕσ(i)⟩. (21)

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 such that

 g|ϕi⟩=λi|ϕσ(i)⟩ (22)

with That is,

Theorem 2.

Assume is an -qubit symmetric pure state, then the stabilizer group of is nontrivial if and only if there exists nontrivial transformation that permutes the Majorana roots of

Assume the diversity number of a symmetric state is , the divergence configuration for the state is with and Then from the Lemma and the property of transformation, we have:

Corollary 1.

Assume is an -qubit symmetric state, its degeneracy configuration is if there exists and such that these three values are unequal to the residual elements in the set, then the stabilizer group for the state contains only the identity.

At last, we talk about the stabilizer group for an -qubit symmetric pure state with the increase of the diversity number of when the stabilizer group for is nontrivial, when under a conjecture we make, the stabilizer group for is trivial.
m=1 (separable states): When a symmetric state is separable, then it can be represented as

 |ψ⟩=|ϵϵ⋯ϵ⟩, (23)

it is easy to see when an ILO satisfies that is an eignvector of is the stabilizer for the state
In this case, the state can be represented as

 |ψ⟩=eiθ√K∑|ϵ1⋯ϵ1k1ϵ2⋯ϵ2n−k1⟩, (24)

where the sum takes over all the permutation of and in this case, when first we denote a local operator this means if then can be written as And when the ILO can also be written as .
When due to Lemma 3, by searching a nontrivial Mbius transformation between the Majorana points of the pure state we can see whether a pure state owns a nontrivial stabilizer group.
When assume the degeneracy configuration of a pure state is due to the property (5), we see that except when , owns a nontrivial stabilizer group.
Here we assume that each is not as if there exists such that we can always make be not by Mbius transformation. When assume the degeneracy configuration of the pure state is when all the four number are different from each other, the stabilizer group for is trivial. First we show when the stabilizer group is nontrivial, there exists a nontrivial Mbius transformation that can permute Let from the property (6) of the Mbius transformation, we see exists.
Here we note that when is a four qubit symmetric pure state, always owns a nontrivial stabilizer group. As the diversity configuration can only be or from the above analysis, we see that the stabilizer group is nontrivial.
Next we consider a general case. Assume the stabilizer group of the state is nontrivial, then we have a nontrivial Mbius transformation which permutes the majorana points, that is, we could assume , if here we denote that is the greatest common divisor of and , then we could have , that is, is trivial, when , , so it is invalid. However, we cannot prove the case when is invalid. When , then we have

 (z1−z2)(z3−z4)(z1−z4)(z3−z2)=(z2−z1)(z4−z5)(z2−z5)(z4−z1),,

that is,

 (z3−z4)(z2−z5)=(z3−z2)(z4−z5), (z3−z5)(z2−z4)=0.

This case is invalid.
Assume is an -qubit symmetric pure state with its degeneracy configuration , and we divide the degeneracy configuration into parts according to whether are equal. Assume is a set with its elements , here , is the number of each part. Due to the above analysis, only when the transformation is represented as