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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 

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: 

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. 

## II. INDEX p AND VCM 

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For example, the N -qubit GHZ state 

which obviously contains superposition of macroscopi- cally distinct states, has p = 2, since 

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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 

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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. 

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 

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## III. ZERO TEMPERATURE 

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