Stabilizer formalism

When using the stabilizer formalism to specify a quantum state |φi, the state is described by a set of operators that have the eigenstate |φi with the eigenvalue 1. This set of operators is called the stabilizer which stabilize |φi. Then |φi is called the stabilizer state.

For example, a N-qubit cluster state |LNi [31] is stabilized by the group of stabilizer given by

GLN = hS^1,LN, S2,LN, ..., SN,LNi , (3.1)

where

S1,LN = X1Z2, SN,LN = Z_{N −1}XN, Sk,LN = Z_k−1XkZk+1 (3.2)

for k = 2, 3, ..., N − 1, are the generators of the group.

Theorem 3. For some stabilizer state |φi, every operator g^φ ∈ G^φ with the general form, gφ= ±Nm

k=1Vbk, where bVk ∈ {Z, X, Y }, can be specified by the correlator operators:

gφ= ˆC0φ+ ˆC1φ, (3.3)

with DCb0φ

E

> 0 and D Cb1φ

E

> 0, (3.4)

for |φi, which implies that some subsystem of n qubits for n < m is dependent on the one composed of (m − n) qubits [121].

Proof. The Pauli operator bV_k can be expressed explicitly by bV_k = P1

v_k=0(−1)^vkvˆ_kk^′ for k^′ ∈ {x, y, z} which denote the type of Pauli operator where ˆv^kk′ = |v^kik^′k^′hv^k| and k is used to number the qubits. Then, we have Nn

k=n^′+1Vbk = ˆ0_(n−n^′₎ − ˆ1(n−n^′), where

ˆ0_(n−n^′₎ (ˆ1_(n−n^′₎) is the sum of all (n − n^′)-qubit operators, ⊗ⁿk=n^′+1vˆkk^′, with even (odd) and deduce that the correlators for the operator

Cˆ0φ = (ˆ0_n− ˆ1ⁿ)ˆ0_(m−n), ˆC1,φ = (ˆ1_n− ˆ0ⁿ)ˆ1_(m−n). (3.7)

are all positive for |φi. For g^φ = −Nm

k=1Vk, by the same approach proposed above, we have positive values of correlators corresponding to the operators

Cˆ0φ = (ˆ0_n− ˆ1ⁿ)ˆ1_(m−n), ˆC1,φ = (ˆ1_n− ˆ0ⁿ)ˆ0_(m−n). (3.8)

Hence, we know that the subsystem of n qubits is dependent on the one composed of (m − n) qubits by the first theorem given in Sec. 2.2.

Given a m-qubit operator belonging to some stabilizer with the form like the one in

the third theorem, it specifies the dependence characters between any two subsystems with k qubits and (m − k) ones. Each operator g^φ gives nφ sets of correlators where

nφ= to the sets of correlator operators for specifying the correlation of dependence between any two subsystems with k qubits and (m − k) ones respectively.

Through the third theorem, we could view the group of stabilizer as the set which con-tains all correlators for specifying the dependence between qubits of the N-qubit system under different measurement directions. In what follows, we will discuss the correlators derived from the stabilizer under a given measurement setting.

(a) Cluster state. First, lest us consider a concrete example involved the following generators of the six-qubit cluster state: S1,L6 = X1Z2 , S3,L6 = Z2X3Z4, and S5,L6 = Z4X5Z6. S1,L6 shows that the first qubit is dependent of the second one. S3,L6 identifies the correlations: [2|3, 4], [3|2, 4], and [4|2, 3] , and S^5,L6 identifies the ones: [4|5, 6], [5|4, 6], and [6|4, 5] . We know that the six-qubit system possesses the multipartite correlation by these information featured in correlators. However, when measuring qubits under the setting where the odd (even)-number qubits are measured along x (z) direction, the correlation conditions given by the generators S1,L6, S3,L6, and S5,L6 are incomplete. From an observation, we know that the measured directions involved in the products of the generators are the same as the ones of the generators, and we could acquire more criteria of correlations from these products of generators. For instance, S1,L6S3,L6 = X1X3Z4 let

us know the condition for {[1, 3, 4]}, S^1,L6S5,L6 = X1Z2Z4X5Z6 for {[1, 2, 4, 5, 6]}, and S3,L6S5,L6 = Z2X3X5Z6 for {[2, 3, 5, 6]}. Hence a complete condition of correlations can be provided from the subgroup of the stabilizer generated by S1,L6, S3,L6 , and S5,L6: GL6,1 = hS^1,L6, S3,L6, S5,L6i, if disregarding the identify operator, i.e.,

G˜L6,1 = {X¹X3Z4, X1Z2Z4X5Z6, Z2X3X5Z6, X1X3X5Z6, X1Z2, Z2X3Z4, Z4X5Z6}. (3.11)

Similarly, under another measurement setting where the odd (even)-number qubits are measured along z (x) direction, the set of operators

G˜L6,2 = {Z¹X2Z3Z5X6, Z3X4X6, Z1X2X4X6, Z1X2X4Z5, Z5X6, Z1X2Z3, Z3X4Z5}, (3.12)

generated by S2,L6 = Z1X2Z3, S4,L6 = Z3X4Z5, S6,L6 = Z5X6, also gives an identification of multipartite correlation of |L⁶i. Furthermore, for the N -qubit cluster state, both the subgroups G_LN,1= hSk,LN : for all odd ki and GLN,2 = hSk,LN : for all even ki give com-plete descriptions of N-qubit correlation of |φdi by ˜G_LN,1 and ˜G_LN,2, under two different measurement settings.

(b) Greenberger-Horne-Zeilinger (GHZ) state [113]. An N-qubit GHZ state is specified by the stabilizer

G_GHZN = hS1,GHZN, S_k,GHZN : for k = 2, ..., Ni , (3.13)

where

S1,GHZN = ON

k=1

Xk, Sk,GHZN = Z_k−1Zk, (3.14)

for k = 2, ..., N. Let us discuss S1,GHZ_N first. By Theorem 3 and the related discussions, we know that there are nGHZN sets of correlators to describe the correlations between subsystems with {[1, 2, ..., N]}, where n^GHZN is defined by Eq. (3.10). The feature of truly

multipartite correlation is shown via these correlators under the x-direction measurements.

On the other hand, the operators gGHZN produced by generators Sk,GHZN for k ≥ 2 specify two kinds of criterion of correlation including the dependence between each qubit, [i, j], and the correlations between subsystems with {[1, 2, ..., m]} where m is even. To investigate the correlation between the k1th and k2th qubits for k2 > k1, we can utilize the product of the generators, Qk2−1

k1+1Sk,GHZN, to have the operator Zk1Zk2 and know that these qubits are dependent. For the second type of criterion, it is given byN

k∈T^eZ_k where Tedenotes the set which contains even number of qubits. For example, we have the same correlators as S1,GHZN by QN/2

k=1S2k,GHZN =NN

k=1Zk where N is even. Thus, under two local measurement settings, a complete knowledge of correlation between qubits is included in ˜GGHZN,1 = {S^1,GHZN} and the set of operators ˜GGHZN,2that is derived from the subgroup of stabilizer, GGHZN,2 = hS^k,GHZN : for k = 2, ..., Ni, and in which the identity operator is disregarded.

(c) Graph state. A N-qubit graph state [122], |RNi, is specified by a graph described in terms of N vertices and some edges connecting them and is defined explicitly by the stabilizing operators:

Sk,RN|R^Ni = |R^Ni , (3.15)

where

Sk,RN = Xk

O

i∈Nk

Zi (3.16)

and Nk denotes the set of vertices i for which vertices k and i are adjacent. Through Theorem 3, we realize that the vertex k is dependent on the ones in the neighborhood Nk. Furthermore, we can identify the correlation between two vertices that are not adjacent via the correlators. For instance, a four-qubit box-cluster state is specified by the following

stabilizing operators:

S1,R4 = X1Z2Z4, S2,R4 = X2Z1Z3, S3,R4 = X3Z2Z4, S4,R4 = X4Z1Z3. (3.17)

Although the first qubit and the third one are not adjacent, we can identify that they are dependent via S1,R4S3,R4 = X1X3. Similarly, the second qubit and the fourth one are correlated by S2,R4S4,R4 = X2X4. Therefore, the sets

G˜R4,1 = {X¹Z2Z4, X3Z2Z4, X1X3}, ˜GR4,2= {X²Z1Z3, X4Z1Z3, X2X4}, (3.18)

can describe the correlation inherent in the state |R^Ni under two different measurement settings.