II. INDEX p AND VCM 2 For example, the N -qubit GHZ state In this section, we briefly review the method of de- tecting superposition of macroscopically distinct states in quantum many-body states [5, 14–16]. Let us consider an N -site lattice (1 ≪ N < ∞ ) where the dimension of the Hilbert space on each site is an N -independent constant, such as a chain of N spin-1/2 particles. Throughout this paper, f ( N ) = O ( N $^{k}$) means For a given pure state | ψ 〉 , the index p (1 ≤ p ≤ 2) is defined by where the maximum is taken over all Hermitian additive operators ˆ A . Here, an additive operator is a sum of local operators { ˆ a ( l ) } N l $_{=1}$, where ˆ a ( l ) is a local operator acting on site l . For example, if the system is a chain of N spin-1 / 2 particles, ˆ a ( l ) is a linear combina- tion of three Pauli operators, ˆ σ$_{x}$ ( l ) , ˆ σ$_{y}$ ( l ) , ˆ σ$_{z}$ ( l ), and the identity operator ˆ 1( l ) acting on site l . In this case, the x -component of the total magnetization which obviously contains superposition of macroscopi- cally distinct states, has p = 2, since It was shown in Ref. [5] that a state having p = 2 is unstable against a local noise from the environment and a local measurement, whereas a state having p = 1 is stable. There is an efficient method of calculating index p [14, 15]. For simplicity, we assume that the Hilbert space on each site is two-dimensional one. Generalizations to higher dimensional cases are immediate. For a given pure state | ψ 〉 , let us define the 3 N × 3 N Hermitian matrix called the variance-covariance matrix (VCM) by where α, β = x, y, z ; l, l ′ = 1 , 2 , · · · , N ; ˆ σ$_{x}$ ( l ), ˆ σ$_{y}$ ( l ), and ˆ σ$_{z}$ ( l ) are Pauli operators on site l . Since the VCM is Hermitian, all eigenvalues are real. Let e$_{1}$ be the largest eigenvalue of the VCM. Then is satisfied [14, 15], which means that we have only to calculate e$_{1}$ to obtain the value of index p . Since a ma- trix of a polynomial size can be diagonalized within a polynomial steps, e$_{1}$ is obtained efficiently by numerical calculations. and the z -component of the total staggard magnetization are, for example, additive operators. The index p takes the minimum value 1 for any product state where | φ$_{l}$ 〉 is a state of site l (this means that p > 1 is an entanglement witness for pure states). If p takes the maximum value 2, the state contains superposition of macroscopically distinct states because in this case the relative fluctuation of an additive operator does not van- ish in the thermodynamic limit: Let us first study the zero-temperature case. We con- sider the one-dimensional periodic chain of N qubits. The code space of the Z$_{2}$ quantum memory is stabilized by the “bond operators” ( l = 1 , 2 , ..., N ) [8] which act on the “virtual qubits” (or “dual qubits”) em- bedded on bonds, where ˆ σ$_{z}$ ( N + 1) = ˆ σ$_{z}$ (1). Since stabilizers of the Z$_{2}$ quantum memory are generated by N − 1 bond operators { ˆ B$_{1}$, ˆ B$_{2}$, ..., ˆ B$_{N}$$_{−}$$_{1}$ } , which means that the code space is 2 N − ( N − 1) = 2 1 dimensional sub- space. The centralizers of these stabilizers are generated by and because the fluctuation of an observable in a pure state means the existence of a superposition of eigenvec- tors of that observable corresponding to different eigen- values. III. ZERO TEMPERATURE | GHZ 〉 ≡ 1 √$_{2}$ ( | 0 ⊗ $^{N}$〉 + | 1 ⊗ $^{N}$〉 ) , 〈 GHZ | ˆ M 2 z | GHZ 〉 − 〈 GHZ | ˆ M$_{z}$ | GHZ 〉 2 = O ( N $^{2}$) . lim N →∞ f ( N ) N k = const. = 0 . max ˆ A [ 〈 ψ | ˆ A $^{2}$| ψ 〉 − 〈 ψ | ˆ A | ψ 〉 $^{2}$] = O ( N $^{p}$) , ˆ A = N ∑ l =1 ˆ a ( l ) V$_{αl,βl}$ ′ ≡ 〈 ψ | ˆ σ$_{α}$ ( l )ˆ σ$_{β}$ ( l $^{′}$) | ψ 〉 − 〈 ψ | ˆ σ$_{α}$ ( l ) | ψ 〉〈 ψ | ˆ σ$_{β}$ ( l $^{′}$) | ψ 〉 , e$_{1}$ = O ( N p − $^{1}$) ˆ M$_{x}$ ≡ N ∑ l =1 ˆ σ$_{x}$ ( l ) ˆ M st z ≡ N ∑ l =1 ( − 1) $^{l}$ˆ σ$_{z}$ ( l ) N ⊗ l =1 | φ$_{l}$ 〉 , ˆ B$_{l}$ ≡ ˆ σ$_{z}$ ( l )ˆ σ$_{z}$ ( l + 1) , ˆ B$_{N}$ = N − 1 ∏ l =1 ˆ B$_{l}$, lim N →∞ √ 〈 ψ | ˆ A $^{2}$| ψ 〉 − 〈 ψ | ˆ A | ψ 〉 2 N = 0 , ˆ X ≡ N ∏ l =1 ˆ σ$_{x}$ ( l ) , (1)