6 this superposition of macroscopically distinct states is stable against any local noise except for the inevitable one. Let us briefly discuss a possibility of avoiding this inevitable instability without introducing too much so- phisticated methods. One of the most simple ways would be to use RVB (Resonating Valence Bond) states: where is the projection operator on the singlet state of sites l and l + 1. Then, we can show that the state | Ψ 〉 has su- perposition of macroscopically distinct states in the sense of A proof is given in Appendix. In summary, we have seen that the stability of the Z$_{2}$ quantum memory can be improved by introducing local entanglement between nearest neighbor sites. The RVB state thus created still has superposition of macroscopi- cally distinct states. where | i, j 〉 represents the singlet pair between sites i and j , and the periodic boundary condition is assumed (Fig. 7). This state is realized as a ground state of the one-dimensional spin ladder model or the Majumdar- Ghosh model [23]. In this paper, we have investigated the low- temperature coherence properties of the Z$_{2}$ quantum memory. We have shown that (i) the memory can have superposition of macroscopically distinct states for the values λ < 1 of the control parameter λ and at sufficiently low temperature, (ii) the code states of this memory have no instability except for the inevitable one, and (iii) the power of superposition of macroscopically distinct states in this memory is limited by space and time. We have also briefly discussed how the RVB memory improves the Z$_{2}$ quantum memory. To conclude this paper, let us briefly discuss the rela- tions of our results to symmetry breaking in statistical physics. Symmetry breaking is one of the most fundamen- tal concepts in modern physics [24–26]. According to Landau-Ginzburg theory [27], the state of the phase is a local minimum of the effective potential as a function of the order parameter: if the temperature is high, the po- tential has the unique minimum at the origin and there- fore the system is in the symmetric phase, whereas at suf- ficiently low temperature, the effective potential becomes the double-well type whose local minima correspond to the symmetry-broken phases. | Ψ 〉 is a superposition of two macroscopically distinct symmetry-broken states, | VB$_{1}$ 〉 and | VB$_{2}$ 〉 . The advan- tage of this state is that, as is shown in Appendix, there is no long-range two-point correlation in this state: for any pair of local operators ˆ a ( l ) and ˆ a ( l $^{′}$) with | l − l $^{′}$|  ≥ 2. Therefore, | Ψ 〉 has p = 1 and this means that the state is stable against any local noise of the type Although such intuitive picture of the symmetry- breaking has contributed to the progress of physics for long time, it has been often pointed out by many re- searchers that such picture is not always correct if the system is of a finite volume [28–33]. For example, if the many-body Hamiltonian ˆ H and the order operator ˆ O do not commute with each other [ ˆ H, ˆ O ] = 0 (typical exam- ples in condensed matter physics are the transverse Ising model, the Heisenberg antiferromagnet, and the Hubbard model), the exact ground state of a finite volume is of- ten non-degenerate and therefore symmetric. This sym- metric exact ground state is completely different to the symmetry-broken “mean-field ground states”, which are inherently separable since the mean-field approximation neglects the correlations among sites [34]. In order to detect superposition of macroscopically dis- tinct states in | Ψ 〉 , we must consider the sum of bilocal operators. Indeed, let us consider the operator ˆ T ≡ N ∑ l =1 ( − 1) $^{l}$ˆ t$_{l,l}$$_{+1}$, ˆ H$_{int}$ = N ∑ l =1 f ( l )ˆ a ( l ) . 〈 Ψ | ˆ a ( l )ˆ a ( l $^{′}$) | Ψ 〉 − 〈 Ψ | ˆ a ( l ) | Ψ 〉〈 Ψ | ˆ a ( l $^{′}$) | Ψ 〉 = 0 , (7) FIG. 7. (Color online) The nearest-neighbor RVB state on a one-dimensional periodic lattice. Red circles represent sin- glet pairs. Two macroscopically distinct VB states are super- posed. | Ψ 〉 ≡ 1 √ 2 + 4( − 1 $_{2}$) N/ $_{2}$[$^{N/}$$^{2}$ ⊗ l =1 | 2 l − 1 , 2 l 〉 + N/ 2 ⊗ l =1 | 2 l, 2 l + 1 〉 ] ≃ 1 √$_{2}$ N/ 2 ⊗ l =1 | 2 l − 1 , 2 l 〉 + 1 √$_{2}$ N/ 2 ⊗ l =1 | 2 l, 2 l + 1 〉 ≡ 1 √$_{2}$ | VB$_{1}$ 〉 + 1 √$_{2}$ | VB$_{2}$ 〉 , ˆ t$_{l,l}$$_{+1}$ ≡ | l, l + 1 〉〈 l, l + 1 | 〈 Ψ | ˆ T $^{2}$| Ψ 〉 − 〈 Ψ | ˆ T | Ψ 〉 2 = O ( N $^{2}$) . (8) VI. CONCLUSION AND DISCUSSION