Low-temperature coherence properties of Z_2 quantum memory

0.2 MB·English

by  Tomoyuki Morimae

Low-temperature coherence properties of Z quantum memory
2
Tomoyuki Morimae∗
Department of Basic Science, University of Tokyo, 3-8-1 Komaba, Tokyo 153-8902, Japan and
Laboratoire Paul Painlev´e, Universit´e Lille 1, 59655 Villeneuve d’Ascq C´edex, France
(Dated: January 8, 2010)
Weinvestigatelow-temperaturecoherencepropertiesoftheZ2 quantummemorywhichiscapable
of storing the information of a single logical qubit. We show that the memory has superposition
of macroscopically distinct states for some values of a control parameter and at sufficiently low
temperature, and that the code states of this memory have no instability except for the inevitable
one. However, we also see that the coherence power of this memory is limited by space and time.
WealsobrieflydiscusstheRVBmemory,whichisanimprovementoftheZ2 quantummemory,and
therelations of our results to the obscured symmetry breaking in statistical physics.
0
1 PACSnumbers: 03.65.Aa,03.67.-a,03.65.Ta,05.30.-d
0
2
n I. INTRODUCTION means that logical operations are non-local.
a Although Kitaev’s toric code is highly sophisticated
J Quantum memory [1, 2], which stores the coherentin- and indeed has inspired plenty of successive studies [2,

8 formation of logical qubits, is an essential ingredient of 6–13], it is not easy to implement a scalable toric code
quantuminformationprocessings,andplayscrucialroles in a laboratory since non-local operations are required.
]
h in almost all fields of quantum information science [3]. A complementary approach is therefore also important
p Basically,aquantummemoryconsistsofmacroscopically from the practical point of view.
- many physical qubits which encode the state of few log- Inthispaper,westudythelesselaboratebutmorefea-
t
n ical qubits. Each logical basis should be encoded on siblequantummemory,namelytheZ2quantummemory.
a macroscopically distinct physical states since otherwise Although the Z2 quantum memory is too primitive to
u theindistinguishabilityoflogicalbasesiseasilydestroyed be a complete and universal quantum memory, it is still
q
by local errors. On the other hand, in order to store valuable to study it, since the simple structure of this
[
quantum coherent superposition of these logical bases, memory means feasibility in a laboratory and the pos-

1 thememorymustbeabletohavesuperpositionofmacro- sibility of capturing the essence of theoretical aspects of
v scopically distinct states of physicalqubits. To maintain quantum memory. In general, a quantum memory must
5
such superposition is very difficult. First, if the size of have a large coherence for some values of a control pa-
9
2 thememoryisinfinite,suchmacroscopicsuperpositionis rameterinordertostoreacoherentinformationoflogical
1 impossible since superposition of different phases is just qubits. Therefore, we analyze the coherent properties of
. a classical mixture of them in an infinite system [4–6]. the Z2 quantum memory at low temperature, by using
1
Second, even in finite systems, such macroscopic super- themethodofdetectingsuperpositionofmacroscopically
0
0 position is usually very unstable [5]. Therefore, how to distinct states developed in Refs. [5, 14–16]. We show
1 balance those two contradictory demands (i.e., classical that the Z2 quantum memory can have superposition of
: informationmustbeencodedonmacroscopicallydistinct macroscopically distinct states for some values of a con-
v
i states but we also need superposition of them) is one of trol parameter and at sufficiently low temperature, and
X the most challenging problem in the implementation of that the code states have no instability except for the
r quantum memory. inevitable one. However, we also see that the power of
a Thecodespaceofaquantummemoryisoftenrealized superposition of macroscopically distinct states in this
as the lowest-energy eigenspace of a many-body Hamil- memory is limited by space and time. These results sug-
tonian. One of the most beautiful examples is Kitaev’s gest that the Z2 quantum memory is of limited use as a
toric code [1]. Kitaev introduced a four-body Hamilto- prototype of a small quantum memory.
nian which exhibits a topological phase transition. The This paper is organized as follows. In the next sec-
ground states are degenerated and energetically isolated tion, we briefly review the method of detecting super-
fromthe excitedstates. Eachofthese groundstates cor- position of macroscopically distinct states. In Sec. III,
responds to different topological phases in the thermo- we study the zero-temperature case. We next study the
dynamic limit, and therefore logical bases encoded on finite-temperature case in Sec. IV. Finally, in Sec. V, we
these degenerate ground states are immune to local er- briefly discuss the RVB quantum memory, which is an
rors. Logicalqubitoperationsareperformedbyapplying improvement of the Z2 quantum memory, and the rela-
long strings of Pauli operators on physical qubits, which tions of our results to the concept of symmetry breaking
in statistical physics in Sec. VI.
∗[email protected]
2
II. INDEX p AND VCM For example, the N-qubit GHZ state
1
In this section, we briefly review the method of de- GHZ (0⊗N + 1⊗N ),
| i≡ √2 | i | i
tecting superposition of macroscopically distinct states
in quantum many-body states [5, 14–16]. which obviously contains superposition of macroscopi-
Let us consider an N-site lattice (1 N < ) where cally distinct states, has p=2, since
≪ ∞
the dimension of the Hilbert space on each site is an
N-independent constant, such as a chain of N spin-1/2 hGHZ|Mˆz2|GHZi−hGHZ|Mˆz|GHZi2 =O(N2).
particles. Throughoutthis paper, f(N)=O(Nk) means
It was shown in Ref. [5] that a state having p = 2 is
f(N) unstable against a local noise from the environment and
lim =const.=0. a local measurement, whereas a state having p = 1 is
N→∞ Nk 6
stable.
For a given pure state ψ , the index p (1 p 2) is Thereis anefficientmethod ofcalculatingindex p [14,
| i ≤ ≤
defined by 15]. For simplicity, we assume that the Hilbert space
on each site is two-dimensional one. Generalizations to
max[ ψ Aˆ2 ψ ψ Aˆψ 2]=O(Np),
Aˆ h | | i−h | | i higher dimensional cases are immediate.
For a given pure state ψ , let us define the 3N 3N
where the maximum is takenoverallHermitian additive Hermitian matrix called t|hei variance-covariance m×atrix
operators Aˆ. Here, an additive operator (VCM) by
Aˆ= N aˆ(l) Vαl,βl′ ≡hψ|σˆα(l)σˆβ(l′)|ψi−hψ|σˆα(l)|ψihψ|σˆβ(l′)|ψi,
X
l=1 where α,β = x,y,z; l,l′ = 1,2, ,N; σˆx(l), σˆy(l), and
···
σˆ (l) are Pauli operators on site l. Since the VCM is
isasumoflocaloperators aˆ(l) N ,whereaˆ(l)isalocal z
operator acting on site l. {For e}xl=am1 ple, if the system is Hermitian, all eigenvalues are real. Let e1 be the largest
eigenvalue of the VCM. Then
a chain of N spin-1/2 particles, aˆ(l) is a linear combina-
tion of three Pauli operators, σˆx(l),σˆy(l),σˆz(l), and the e1 =O(Np−1)
identity operator ˆ1(l) acting on site l. In this case, the
is satisfied [14, 15], which means that we have only to
x-component of the total magnetization
calculate e to obtain the value of index p. Since a ma-
1
N trix of a polynomial size can be diagonalized within a
Mˆx ≡Xσˆx(l) polynomial steps, e1 is obtained efficiently by numerical
l=1 calculations.
andthez-componentofthetotalstaggardmagnetization
N III. ZERO TEMPERATURE
Mˆst ( 1)lσˆ (l)
z ≡X − z
l=1 Let us first study the zero-temperature case. We con-
sider the one-dimensional periodic chain of N qubits.
are, for example, additive operators. The index p takes
The code space of the Z quantum memory is stabilized
the minimum value 1 for any product state 2
by the “bond operators” (l=1,2,...,N) [8]
N
φ , Bˆ σˆ (l)σˆ (l+1),
O| li l ≡ z z
l=1
which acton the “virtualqubits” (or “dual qubits”)em-
where |φli is a state of site l (this means that p > 1 bedded on bonds, where σˆz(N +1)=σˆz(1). Since
is an entanglement witness for pure states). If p takes
the maximum value 2, the state contains superposition N−1
Bˆ = Bˆ ,
ofmacroscopicallydistinctstatesbecauseinthiscasethe N Y l
relativefluctuationofanadditive operatordoesnotvan- l=1
ish in the thermodynamic limit: stabilizers of the Z quantum memory are generated by
2
N 1 bond operators Bˆ ,Bˆ ,...,Bˆ , which means
1 2 N−1
q ψ Aˆ2 ψ ψ Aˆψ 2 tha−t the code space is 2{N−(N−1) = 21 di}mensional sub-
lim h | | i−h | | i =0,
N→∞ N 6 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- N
tors of that observable corresponding to different eigen- Xˆ ≡Yσˆx(l), (1)
values. l=1
3
and
11
Zˆ σˆz(1). 10
≡
They work as the logical bit flip and logical phase, re- 9
spectively. The codespaceisalsospecifiedasthe lowest- 8
energy eigenspace of the two-body Hamiltonian
7
1
N e
6
Hˆ = Bˆ .
0 −X l 5
l=1
4
Itisobviousthattwodegenerateseparablegroundstates
of this Hamiltonian are macroscopically distinct with 3
eachother, and therefore the indistinguishability of logi-
2
cal bases, ˜0 and ˜1 , is not destroyed by local errors. 3 4 5 6 7 8 9 10 11
| i | i
Ideally, the logical Hadamard operation N
1
˜0 ˜0 + ˜1
| i→ √2(cid:16)| i | i(cid:17) FIG. 1. (Color online) e1 versus N for |E0(λ)i with various
1 λ. From the bottom, λ =1.5, 1.4, 1.3, 1.2, 1.1, 1.0, 0.9, 0.8,
˜1 ˜0 ˜1 ,
| i→ √2(cid:16)| i−| i(cid:17) 0.7, 0.6, and 0.5, respectively. Lines are guides to theeye.
which is the essential ingredient of various quantum in-
formation processings, is realized by using the logical Xˆ 0.45
operation, Eq. (1). However, such non-local operation is 0.4
not easy to experimentally implement. Therefore, it is
0.35
reasonableto try to manipulate the quantummemoryin
0.3
the local way:
)z 0.25
N M
Hˆ =Hˆ +λ σˆ (l), (2) P( 0.2
0 X x
l=1 0.15
where λ is an external control parameter. 0.1
From the Perron-Frobenius theorem [17], the ground 0.05
state of this Hamiltonian is non-degenerate if λ = 0. It
6 0
is also known that this Hamiltonian exhibits the quan- -15 -10 -5 0 5 10 15
tum phase transition at λ = 1 [18]. Let us denote the M
z
exact ground state of Hˆ corresponding to the external
parameter λ by E (λ) , and evaluate index p of E (λ)
0 0
| i | i
for various λ. In Fig. 1, the largest eigenvalue e of the
VCM versus N is plotted by changing the valu1e of λ. FIG. 2. The probability distribution P(Mz) of Mˆz in
This figure shows that |E0(0.5)i with N = 13. Similar structures are obtained for
other valuesof N.
E (λ 1) has p<2
0
• | ≥ i
E (λ<1) has p=2, By seeing the eigenvector of the VCM corresponding
0
• | i to the largest eigenvalue e , we can also know that the
1
whichmeansthatthe Z2 quantummemoryhassuperpo- additive operator which gives the maximum fluctuation
sition of macroscopically distinct states for values λ< 1 in E (λ<1) is
0
| i
of the control parameter λ.
In order to see the structure of the superposition, the N
probabilitydistributionP(Mz)ofMˆz in E0(0.5) isplot- Mˆz ≡Xσˆz(l).
| i
ted in Fig. 2 for N = 13. This figure suggests that the l=1
groundstateis,inaroughpicture,asuperpositionoftwo
According to Ref. [5], this means that E (λ<1) is un-
0
macroscopically distinct states: | i
stableagainstthelocalnoisedescribedbytheinteraction
Hamiltonian
E (0.5) φ + φ , (3)
0 + −
| i≃| i | i
N
wrehspereect|φiv±eliya.resomestatessatisfyinghφ±|Mˆz|φ±i≃±N, Hˆint ≡Xf(l)σˆz(l), (4)
l=1
4
where f(l) is a noise parameter of a long wavelength.
-2
Although this noise is inevitable since we need the su-
perposition of macroscopically distinct logical bases, we -3
can still show that E0(λ < 1) has no other instability -4
| i
than this inevitable one. In Fig. 3, we plot the second
-5
largesteigenvalue e of the VCM versus N for E (0.5) .
From this figure we2obtain | 0 i E)0 -6
-1
e2 ≤O(N0), ln(E -7
-8
which means that E (0.5) is stable againstallnoises of
0
| i -9
the type
-10
N
Hˆ f(l)aˆ(l) -11
int ≡X 2 4 6 8 10 12 14
l=1
N
except for the case of Eq. (4).
FIG. 4. ln∆E versusN for λ=0.5.
1.35
Indeed, we plot e versus N for E′ in Fig. 5. This
1.3 figure shows that e1 = O(N0), wh|ic0himeans that the
1
superposition of macroscopically distinct states in quan-
1.25 tummemorydisappearsforsufficientlylargesystemsize.
e2 Therefore, the Z2 quantum memory is of limited use as
a small quantum memory.
1.2
2.003
1.15
2.0025
1.1
2 4 6 8 10 12 14
2.002
N
1 2.0015
e
FIG. 3. e2 versusN for |E0(0.5)i.
2.001
In short, we have seen that the Z quantum memory
2
canhavesuperpositionofmacroscopicallydistinct states 2.0005
and it is stable against any local noise except for the
inevitable one. 2
3 4 5 6 7 8 9 10 11 12
As is seeninFig.4, the energygap∆E E E be-
1 0
≡ − N
tween the exact ground state and the first excited state
for λ=0.5 decays exponentially fast as N . There-
→∞
fore,it is physicallyallowedto takea linear combination ′
FIG. 5. e1 versus N for |E0i
of the exact ground state and the first excited state for
sufficiently large N.
It is worth mentioning that such a fast decay of the
From Eq. (3) and the analogy with the physics of the
energygapisanaturalconsequenceofthe localityofthe
single two-level atom system, the rough picture of the
Hamiltonian. According to Eqs. (3) and (5), the differ-
first excited state E (0.5) is expected to be
1
| i encebetweentheexactgroundstateandthefirstexcited
E (0.5) φ φ , (5) stateisthe relativephaseof φ . Iftheenergygapdoes
1 + − ±
| i≃| i−| i | i
not go to 0 for large N, we can distinguish the exact
and therefore the superposition
groundstateandthefirstexcitedstatebymeasuringthe
1 energy. If the Hamiltonian is a local one, i.e., it does
|E0′i≡ √2(cid:16)|E0(0.5)i+|E1(0.5)i(cid:17) not contain many-point correlation, this means that we
canknowthe relativephasebymeasuringonlyfew-point
of the exact ground state and the first excited state is correlations,which obviouslycontradicts to the common
expectedtohavenosuperpositionofmacroscopicallydis- sense of decoherence [19].
tinct states.
5
In addition to the limit of the space, the exponential Indeed, let us note that
decayofthe energygapalsogivesthe limitto the opera-
tiontime. AssumethatweperformthelogicalHadamard Aˆ,ρˆ 2 =Tr Aˆ,ρˆ † Aˆ,ρˆ
gatebyadiabaticallyincreasingthe controlparameterλ. (cid:13)(cid:13)(cid:2) (cid:3)(cid:13)(cid:13)2 (cid:16)(cid:2) (cid:3) (cid:2) (cid:3)(cid:17)
(cid:13) (cid:13)
Then,accordingtothetheoryofadiabaticquantumcom- =Tr ρˆ Aˆ, Aˆ,ρˆ
(cid:16) (cid:2) (cid:2) (cid:3)(cid:3)(cid:17)
putation[20,21],theexponentialdecayoftheenergygap
means the exponentially long operation time: Tr(ρˆ[Aˆ,[Aˆ,ρˆ]])
≤ ρˆ
1 k k∞
T ≃ (∆E)2, (cid:13) Aˆ, Aˆ,ρˆ (cid:13) ,
≤(cid:13)(cid:2) (cid:2) (cid:3)(cid:3)(cid:13)1
(cid:13) (cid:13)
where ∆E is the minimum energy gap. Therefore, the
large Z2 quantum memory also has the limit of the op- where kXˆk2 ≡ qTr(Xˆ†Xˆ) is the 2-norm, kXˆk∞ is the
eration time. operator norm. If we define the 3N 3N Hermitian
×
matrix W by
IV. FINITE TEMPERATURE Wα,l,β,l′ ≡Tr(cid:16)(cid:2)ρˆ,σˆα(l)(cid:3)(cid:2)σˆβ(l′),ρˆ(cid:3)(cid:17), (6)
Next let us study the Z quantum memory at finite where α,β =x,y,z andl,l′ =1,2,...,N,it is easyto see
2
temperature. that the order of
At finite temperature T, a system is generally in the 2
max [Aˆ,ρˆ]
equilibrium state: (cid:13) (cid:13)
Aˆ (cid:13) (cid:13)2
(cid:13) (cid:13)
e−Hˆ/kT with respect to N is equal to that of e N, where e is
ρˆ= . 1 1
Tr(e−Hˆ/kT) the largest eigenvalue of W.
In Fig. 6, we plot the largest eigenvalue e of the ma-
1
Ifthe stateisnotnecessarilypure,index pcannotdetect trix W, Eq. (6), versus kT for the equilibrium state of
superposition of macroscopically distinct states since a the Hamiltonian Eq. (2), for λ = 0.5 and N = 8. This
fluctuation is not necessarilyequivalent to the coherence figure shows that superposition of macroscopically dis-
in mixed states. tinct states at zero temperature persists at sufficiently
In order to detect superposition of macroscopically low temperature.
distinct states in mixed states, index q was proposed
in Ref. [22]. For a given many-body state ρˆ, index q
8
(1 q 2) is defined by
≤ ≤
7
max N,max [Aˆ,[Aˆ,ρˆ]] =O(Nq),
(cid:16) (cid:13) (cid:13) (cid:17)
Aˆ (cid:13) (cid:13)1 6
(cid:13) (cid:13)
where Xˆ Tr Xˆ†Xˆ isthe1-norm,andmax means 5
k k1 ≡ p Aˆ
the maximum over all Hermitian additive operators Aˆ. 1 4
e
As detailedin Ref. [22], q takesthe minimum value 1 for
3
any separable state,
2
N
XλiO|φ(li)ihφ(li)|, 1
i l=1
0
where φ(i) is a state of site l. On the other hand, if 0 0.0005 0.001 0.0015 0.002 0.0025 0.003
q takes|thleimaximum value 2, the state contains super- kT
positionofmacroscopicallydistinctstates. Inparticular,
for pure states, p=2 q =2.
⇐⇒ FIG. 6. e1 versuskT/J for N =8 (J =1).
Unlike the case of index p, there is no method of effi-
ciently calculating index q at the time of writing. How-
ever, we can calculate a lower bound of the value of q.
V. RVB MEMORY
Inthispaper,wehaveseenthattheZ quantummem-
2
ory can have superposition of macroscopically distinct
states for values λ < 1 of the control parameter λ, and
6
this superposition of macroscopically distinct states is where
stable against any local noise except for the inevitable
tˆ l,l+1 l,l+1
one. Let us briefly discuss a possibility of avoiding this l,l+1
≡| ih |
inevitable instability without introducing too much so-
is the projection operator on the singlet state of sites l
phisticated methods.
and l+1. Then, we can show that the state Ψ has su-
One of the most simple ways would be to use RVB | i
perpositionofmacroscopicallydistinctstatesinthesense
(Resonating Valence Bond) states:
of
1 N/2 N/2 ΨTˆ2 Ψ ΨTˆ Ψ 2 =O(N2). (8)
Ψ 2l 1,2l + 2l,2l+1 h | | i−h | | i
| i≡ q2+4(−21)N/2hOl=1| − i Ol=1| ii A proof is given in Appendix.
In summary, we have seen that the stability of the Z
N/2 N/2 2
1 1 quantum memory can be improved by introducing local
2l 1,2l + 2l,2l+1
≃ √2O| − i √2O| i entanglement between nearest neighbor sites. The RVB
l=1 l=1
state thus created still has superposition of macroscopi-
1 1
VB + VB , cally distinct states.
1 2
≡ √2| i √2| i
where i,j represents the singlet pair between sites i
| i VI. CONCLUSION AND DISCUSSION
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- In this paper, we have investigated the low-
Ghosh model [23]. temperature coherence properties of the Z2 quantum
memory. We have shown that (i) the memory can have
superposition of macroscopically distinct states for the
valuesλ<1ofthecontrolparameterλandatsufficiently
N 1 2 N 1 2 lowtemperature,(ii)thecodestatesofthismemoryhave
3 3 no instability except for the inevitable one, and (iii) the
power of superpositionof macroscopicallydistinct states
4 4
in this memory is limited by space and time. We have
alsobrieflydiscussedhowtheRVBmemoryimprovesthe
Z quantum memory.
2
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-
FIG. 7. (Color online) The nearest-neighbor RVB state on tal concepts in modern physics [24–26]. According to
a one-dimensional periodic lattice. Red circles represent sin- Landau-Ginzburg theory [27], the state of the phase is a
gletpairs. Twomacroscopically distinctVBstatesaresuper- local minimum of the effective potential as a function of
posed.
the order parameter: if the temperature is high, the po-
tential has the unique minimum at the origin and there-
Ψ is a superposition of two macroscopically distinct forethesystemisinthesymmetricphase,whereasatsuf-
| i
symmetry-broken states, VB1 and VB2 . The advan- ficientlylowtemperature,theeffectivepotentialbecomes
| i | i
tage of this state is that, as is shown in Appendix, there the double-well type whose local minima correspond to
is no long-range two-point correlationin this state: the symmetry-brokenphases.
Although such intuitive picture of the symmetry-
Ψaˆ(l)aˆ(l′)Ψ Ψaˆ(l)Ψ Ψaˆ(l′)Ψ =0, (7)
h | | i−h | | ih | | i breaking has contributed to the progress of physics for
foranypairoflocaloperatorsaˆ(l)andaˆ(l′)with l l′ long time, it has been often pointed out by many re-
2. Therefore, Ψ hasp=1andthismeansthatt|he−st|at≥e searchers that such picture is not always correct if the
is stable again|stiany local noise of the type system is of a finite volume [28–33]. For example, if the
many-body Hamiltonian Hˆ and the orderoperator Oˆ do
N not commute with each other [Hˆ,Oˆ] = 0 (typical exam-
Hˆ = f(l)aˆ(l). 6
int X plesincondensedmatterphysicsarethetransverseIsing
l=1 model,theHeisenbergantiferromagnet,andtheHubbard
Inordertodetectsuperpositionofmacroscopicallydis- model), the exact ground state of a finite volume is of-
tinct states in Ψ , we must consider the sum of bilocal ten non-degenerate and therefore symmetric. This sym-
| i
operators. Indeed, let us consider the operator metric exact ground state is completely different to the
symmetry-broken “mean-field ground states”, which are
N inherently separable since the mean-field approximation
Tˆ ( 1)ltˆ ,
≡X − l,l+1 neglects the correlations among sites [34].
l=1
7
Such symmetric exact ground state E often has the as potassium dihydrogen phosphate (KH PO ) crys-
0 2 4
| i
peculiarpropertythattherelativefluctuationofamacro- tals [36].)
scopic observable Aˆ, which is mostly the order operator First, by numerical calculations, we have explicitly
Oˆ, does not vanish even in the thermodynamic limit: shown for the first time how the macroscopic coherence
properties of the exact ground state changes when the
q E0 Aˆ2 E0 E0 AˆE0 2 transverse magnetic field is changed (Fig. 1). We have
lim h | | i−h | | i =0, alsovisualizedthe structureofthemacroscopicsuperpo-
N→∞ N 6
sition in the exact ground state (Fig. 2), and seen that
where N is the number oftotalsites (aliasthe volumeof the exact ground state is approximately an equal weight
thesystem). Sincethismeansthatamacroscopicobserv- superposition of two symmetry-broken phases.
able does not have a definite value even in the thermo- Second,wehaveshownthatonlyMˆ fluctuatesmacro-
z
dynamic limit, E is an anomalous state [5] from the scopically in the λ< 1 phase (Fig. 3). Since the ground
0
| i
view point of thermodynamics where any macroscopic state is symmetric, this means that the second moment
observable is supposed to have definite value [5, 27]. In of Mˆ is of O(N2) in that phase. In terms of statistical
z
terms of index p, |E0i has p = 2 and therefore contains physics,thismeansthatMˆz isthe uniqueorderoperator
superpositionofmacroscopicallydistinctstates. Inother in this phase, whichis, to the author’sknowledge,a new
words, the symmetry of the ground state is “obscured” result.
by such large quantum fluctuation. This effect is often Third,wehaveshownthattheequal-weightsuperposi-
called “obscured symmetry breaking” [28]. tion of the exact ground state and the first excited state
However,whentheexactgroundstatehassuchanoma- has p=1 (Fig. 5). This means that the superposition of
lous property, it is often the case that the energy gap the exact ground state and the first excited state is er-
between the exact ground state and the low-lying eigen- godic. Although similar results have been obtained, the
states decays very fast as N [28, 29]. Then, it is advantagesof our result are (i) instead of the trialstate,
→ ∞
physicallyallowedtotakealinearcombinationoftheex- we have directly used the first excited state to show the
act ground state and some of the low-lying eigenstates ergodicity, (ii) our result that the fluctuation is of O(N)
tsotrufocrtmeda|En0′aipaprerobxeimlieavteedgtrooubnredaksttahtees|yEm0′im.etTryhuasndcobne- iesrgsotdroicnigtyerfotrhaannyEaqd.di(t9iv),eaonpdera(itioi)r,wwehehraevaes osnhloywtnratnhse-
“ergodic” in the sense that lationally invariant additive operators are considered in
the previous studies. (A disadvantage of our result is
lim qhE0′|Aˆ2|E0′i−hE0′|Aˆ|E0′i2 =0 tlahtaiotnits.i)sIlensssugmenmeraarly,sitnocecownesihdaevreinudseexdnpufmoretrhicealgcroaulcnud-
N→∞ N
states of many-body Hamiltonians in condensed matter
for any macroscopic observable Aˆ [28, 29]. In terms of physics is very useful for the study of the foundation of
index p, this means that E′ has no superposition of statistical physics.
| 0i
macroscopically distinct states since p<2.
Indeed, Horsch and Linden [29] introduced the trial
state Oˆ E which approximates the first excited state, ACKNOWLEDGMENTS
0
| i
and showed that the energy gap between the exact
groundstateandthetrialstatedecaysasfastasorfaster The author thanks A. Shimizu and Y. Matsuzaki for
than 1/N. They also showed that a linear combination usefuldiscussions. This workwaspartiallysupportedby
of the exactgroundstate and the trial state exhibits the Japan Society for the Promotion of Science.
desired Z symmetry breaking.
2
KomaandTasaki[28]rigorouslyshowedthatthelinear
combination ψ of the exact ground state and the trial Appendix
| i
state is also the ergodic state in the sense that
1. Proof of Eq. (7)
1
ψ Aˆ2 ψ ψ Aˆψ 2 =0 (9)
N2hh | | i−h | | i i
Notethat Ψ isasimultaneouseigenvectorofMˆ ,Mˆ ,
x y
for any translationally invariant Aˆ. andMˆ corre|spiondingtotheeigenvalue0,sinceasinglet
z
As is pointed out in Ref. [10], the low-temperature isasimultaneouseigenvectorofx,y,andz componentof
coherence properties of quantum memory is closely re- the total magnetization corresponding to the eigenvalue
lated to the obscured symmetry breaking in statistical
physics. Indeed, our results in this paper are considered
as improvement of the previous results, since Hamilto-
nian Eq. (2) is equivalent to that of the transverse Ising
model [35] in condensed matter physics. (For exam-
ple, this model was used to describe the order-disorder
transitioninsomedouble-wellferroelectricsystems,such
8
0. Since each of the states and
1
σˆx(l)|Ψi, h◦◦••|tˆ2,3|◦◦••i=−2h◦◦••|◦••◦i
σˆ (l)Ψ ,
y | i 1
σˆ (l)σˆ (l′)Ψ , = .
x x | i 4
σˆ (l)σˆ (l′)Ψ ,
x y | i Third, by iterating the swap process,
σˆ (l)σˆ (l′)Ψ ,
x z
| i
σˆ (l)σˆ (l′)Ψ , N/2−1
y y | i VB =( 2)N/2−1 tˆ VB ,
σˆ (l)σˆ (l′)Ψ , | 2i − Y 2l,2l+1| 1i
y z
| i l=1
for l l′ 2 has no component in the eigenspace of and therefore
| − | ≥
Mˆ corresponding to the eigenvalue M = 0, they are
z z
N/2−1
orthogonalto Ψ. In the same way,
h | VB VB =( 2)N/2−1 VB tˆ VB ,
h 2| 2i − h 2| Y 2l,2l+1| 1i
σˆz(l)Ψ , l=1
| i
σˆz(l)σˆz(l′)Ψ , which gives
| i
for l l′ 2areorthogonalto Ψ sinceeachofthemhas 1 N/2−1
no c|o−mp|o≥nent in the eigenspache o|f Mˆx corresponding to hVB2|VB1i=(cid:16)− 2(cid:17) ≃0.
the eigenvalue M =0. Hence we have shown Eq. (7).
x Let us define
N
φ Tˆ VB + VB = tˆ VB
2. Proof of Eq. (8) | 1i≡ | 1i 2 | 1i X l,l+1| 1i
l=even
N
Before showing the equation, let us note some useful φ Tˆ VB VB = tˆ VB .
relations. First, the projectionoperatortˆ2,3 “swaps”the | 2i≡ | 2i− 2 | 2i −l=Xodd l,l+1| 2i
entanglement
Then,
1
tˆ = , N
2,3|◦◦••i −2|◦••◦i VBi φj δi,j( 1)i+1 ,
h | i≃ − 8
where singlet pairs are schematically represented: sites
which gives
representedbythecircleofthesamecolormakeasinglet
pair (see Fig. 8). 3N
VB Tˆ VB δ ( 1)i .
i j i,j
h | | i≃ − 8
Therefore,
t
2,3
ΨTˆ Ψ 0.
h | | i≃
Also, we can show
N2 3N
1 2 3 4 φ φ δ + ,
h i| ji≃ i,j(cid:16)64 32 (cid:17)
which gives
9N2 3N
VB Tˆ2 VB δ + .
h i| | ji≃ i,j(cid:16) 64 32 (cid:17)
Therefore,
1 2 3 4
ΨTˆ2 Ψ =O(N2).
h | | i
FIG.8. (Coloronline)Theprojectionoperatortˆ2,3 swapsthe Hence we have shown Eq. (8).
entanglement. Redlines represent singlet pairs.
Second, by using this equation, we obtain
= tˆ
2,3
h◦••◦|◦◦••i h◦••◦| |◦◦••i
1
=
−2h◦••◦|◦••◦i
1
=
−2
9
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