One-way deficit and quantum phase transitions in $XX$ Model

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by  Yao-Kun Wang

One-way deficit and quantum phase transitions in XX Model
Yao-Kun Wang1,2,∗
1College of Mathematics, Tonghua Normal University, Tonghua, Jilin 134001, P. R. China
2Institute of Physics, Chinese Academy of Sciences, Beijing 100190, P. R. China
Quantumcorrelationsincludingentanglementandquantumdiscordhasdrawnmuchattentionin
characterizing quantum phase transitions. Quantum deficit originates in questions regarding work
extraction from quantum systems coupled to a heat bath [Phys. Rev. Lett. 89, 180402 (2002)].
It links quantum thermodynamics with quantum correlations and provides a new standpoint for
understandingquantumnon-locality. Inthispaper,weevaluatetheone-waydeficitoftwoadjacent
spins in the bulk for the XX model. In the thermodynamic limit, the XX model undergoes a first
order transition from fully polarized to a critical phase with quasi-long-range order with decrease
6
of quantum parameter. We find that the one-way deficit becomes nonzero after the critical point.
1
0 Therefore, theone-way deficit characterizes thequantumphase transition in theXX model.
2
PACSnumbers:
n
a
J I. INTRODUCTION the quantum phase transition point in the XX model.

2
1
The recent development in quantum information the-
h] otrrayns[1it]iohnass[p2r].oviEdsepdecmiaulclyh, iunssiingghtthinetoquqaunatnutmumcoprrhealsae- II. ONE-WAY DEF0I<CIλT<I1N XX MODEL FOR
p
tionstoresearchquantumphasetransitionshasattracted
t- much attention and has been successful in characteriz- One-waydeficitbyvonNeumannmeasurementonone
n ing a large number of critical phenomena of great inter- side is given by [21]
a
u est. In this family, entanglement was the first and most
q outstanding member to detect variousofquantumphase ∆→(ρab)= minS( ΠkρabΠk)−S(ρab). (1)
[ transitions, see [3–7]. Quantum discord is a measure of {Πk} Xk

the difference between the mutualinformation andmax-
1 We investigate the Spin-1 XX model with the Hamil-
v imum classicalmutual information, is also used to study 2
tonian as
0 quantum phase transitions [8, 9]. Another indication of
5 quantumness thatis alsofound its applications forprob- N N
1
7 ing quantum phases and quantum phase transition [10– H =− σxσx +σyσy −λ σz, (2)
02 13]. xx 2Xi=1(cid:2) i i+1 i i+1(cid:3) Xi=1 i
Besides entanglement and quantum discord, quantum
.
1 deficit[14–16] originates on asking how to use nonlocal wheretheinteractionconstantistakenastheenergyunit
0 operation to extract work from a correlated system cou- and λ represents the strength of the external magnetic
6 pled to a heat bath [14]. Oppenheim et al. defined the field and σx,σy,σz are the usual Pauli matrices. Its
1
work deficit [14] is a measure of the difference between quantum phase transition is like that in one phase the
:
v the information of the whole system and the localizable system is gapped while in the whole region of the other
Xi information[17, 18]. Recently, by means of relative en- phase the system is critical. The open boundary condi-
tropy, Streltsov et al. [19, 20] give the definition of the tions are assumed, i.e. N +1≡0. The phase diagramis
r
a one-way information deficit which is also called one-way symmetricinrespectofλ[22,23],thereforeweonlycon-
deficit, which uncovers an important role of quantum sider the case of positive λ. For λ > 1 the ground state
deficitasaresourceforthedistributionofentanglement. is polarized. At λ=1 the systemundergoesa first order
In this paper, we will endeavor to calculate the one- quantum transition. In the region λ < 1 the system is
way deficit of two adjacent spins in the bulk of the XX critical. This model canbe solvedanalytically[23] using
model. In the thermodynamic limit, it undergoes a first thefollowingJordan-WignerandFouriertransformations
order transition from fully polarized to a critical phase
N l−1
with quasi-long-range order with decrease of quantum 2 πkl
d = sin σz σ−, (3)
parameter λ. We find that the one-way deficit becomes k rN +1 N +1 m l
Xl=1 (cid:0) (cid:1)mY=1
nonzero after the critical point. Therefore, the one-way
deficit characterizesthe quantumphase transitionin the whichturntheHamiltonianintothediagonalizedformin
XX model. That is, we can employ the deficit to detect terms of fermion operator as H = N Λ d†d +Nλ
xx k=1 k k k
with Λ =2 cos πk −λ . P
k N+1
In the phahse w(cid:0)here 0(cid:1)≤λi<1, the ground state corre-
∗Electronicaddress: [email protected] sponds to that having fermions occupied at negative Λk
2
only. The reduced density matrix for two spins at l and Let
l+1 has been obtained in [23] as
ρ = a |↑↑ih↑↑|+a |↓↓ih↓↓|+b |↑↓ih↑↓|
l,l+1 + − +
+b |↓↑ih↓↑|+e(|↑↓ih↓↑|+|↓↑ih↑↓|) (4)
−
with
1
a± = 4[1±hσlzi±hσlz+1i+hσlzσlz+1i], c=c1 = c2 =hσlxσlx+1i,
b = 1[1±hσzi∓hσz i−hσzσz i], c3 = hσlzσlz+1i,
± 4 l l+1 l l+1 r = s=hσzi. (7)
l
1
e= hσxσx i. (5)
2 l l+1
Inthethermodynamiclimititcanbeshownthatforbulk
spins we have[24]
hσzσz i = 1− 2arccos(λ) 2− 4 (1−λ2),
l l+1 π π2
(cid:0) (cid:1)
2
hσxσx i = − sin(arccos(λ)),
l l+1 π
2arccos(λ)
hσzi = hσz i=1− . (6) The density matrix in Eq. (4) is rewritten as
l l+1 π
1+2r+c 0 0 0
3
1 0 1−c 2c 0
ρ =  3 
l,l+1 4 0 2c 1−c3 0
 0 0 0 1−2r+c 
 3 
1+r+s+c 0 0 c −c
3 1 2
1 0 1+r−s−c c +c 0
=  3 1 2  . (8)
4 0 c1+c2 1−r+s−c3 0
 c −c 0 0 1−r−s+c 
 1 2 3 
The state in Eq. (8) has the following form Next, we evaluate the one-way deficit of the X states
in Eq. (8). Let {Π = |kihk|,k = 0,1} be the local
3 k
ρab = 1(I ⊗I+rσ ⊗I+I ⊗sσ + c σ ⊗σ ), (9) measurement for the party b along the computational
3 3 i i i
4 base |ki; then any von Neumann measurement for the
Xi=1
party b can be written as
where σ ,σ ,σ is Pauli matrix.
1 2 3
The eigenvalues of the X states in Eq. (8) are given {B =VΠ V† :k =0,1} (11)
k k
by
for some unitary V ∈U(2). For any unitary V,
1
u = (1−c ±2|c|),
± 3
4
t+y i y +y i
1 V =tI +i~y·~σ = 3 2 1 . (12)
v± = (1+c3±2|r|). (cid:18)−y2+y1i t−y3i (cid:19)
4
The entropy is given by with t∈R, ~y =(y ,y ,y )∈R3, and
1 2 3
1
S(ρ) = 2− [(1−c +2|c|)log(1−c +2|c|) t2+y2+y2+y2 =1, (13)
4 3 3 1 2 3
+(1−c3−2|c|)log(1−c3−2|c|) after the measurement Bk, the state ρab will be changed
+(1+c3+2|r|)log(1+c3+2|r|) into the ensemble {ρk,pk} with
+(1+c3−2|r|)log(1+c3−2|r|). 1
ρ := (I ⊗B )ρ(I ⊗B ), p =tr(I ⊗B )ρ(I ⊗B )(.14)
(10) k p k k k k k
k
3
To evaluate ρ and p , we write By using z2+z2+z2 = 1, the eigenvalues of the states
k k 1 2 3
in the equation (17) are
p ρ = (I ⊗B )ρ(I ⊗B )
k k k k
1 1
= 4(I⊗V)(I ⊗Πk)[I +rσ3⊗I +sI⊗V†σ3V† ω1,2 = 4(cid:18)1−rz3±qr2−2rc3z3+c2+(c23−c2)z32(cid:19),
3
+ c σ ⊗(V†σ V)](I ⊗Π )(I ⊗V†).
j j j k 1
Xj=1 ω3,4 = 4(cid:18)1+rz3±qr2+2rc3z3+c2+(c23−c2)z32(cid:19).
By the relations [25]
The entropy of Π ρabΠ is S( Π ρabΠ ) =
k k k k
V†σ1V = (t2+y12−y22−y32)σ1+2(ty3+y1y2)σ2 Pk Pk
4
+2(−ty2+y1y3)σ3, − ωilogωi. When λ is fixed, r,c,c3 is constant. It
V†σ2V = 2(−ty3+y1y2)σ1+(t2+y22−y12−y32)σ2 coniP=v1erts the problem about minS( Π ρabΠ ) to the
k k
+2(ty1+y2y3)σ3, {Πk} Pk
V†σ V = 2(ty +y y )σ +2(−ty +y y )σ problem about the function of one variable z3 for mini-
3 2 1 3 1 1 2 3 2
mum. That is
+(t2+y2−y2−y2)σ ,
3 1 2 3
minS( Π ρabΠ )=minS( Π ρabΠ ). (18)
k k k k
and {Πk} Xk {z3} Xk
Π σ Π =Π ,Π σ Π =−Π ,Π σ Π =0,
0 3 0 0 1 3 1 1 j k j By Eqs. (1), (10), (18), the one-way deficit of the X
(15) states in Eq. (8) is given by
for j =0,1,k=1,2, we obtain ∆→(ρab)
1 = minS( Π ρabΠ )−S(ρab)
p ρ = [I+rz I+cz σ +cz σ +(r+c z )σ ] k k
0 0 4 3 1 1 2 2 3 3 3 {Πk} Xk
⊗(VΠ V†), 1
0 = min{− [ 1−rz + r2−2rc z +c2+(c2−c2)z2
1 z3 4 (cid:18) 3 q 3 3 3 3(cid:19)
p ρ = [I−rz I−cz σ −cz σ +(r−c z )σ ]
1 1 3 1 1 2 2 3 3 3
4
·log 1−rz + r2−2rc z +c2+(c2−c2)z2
⊗(VΠ1V†), (cid:18) 3 q 3 3 3 3(cid:19)
where + 1−rz − r2−2rc z +c2+(c2−c2)z2
(cid:18) 3 q 3 3 3 3(cid:19)
z = 2(−ty +y y ),
1 2 1 3
z2 = 2(ty1+y2y3), ·log(cid:18)1−rz3−qr2−2rc3z3+c2+(c23−c2)z32(cid:19)
z = t2+y2−y2−y2.
3 3 1 2
+ 1+rz + r2+2rc z +c2+(c2−c2)z2
Then, we will evaluate the eigenvalues of ΠkρabΠk by (cid:18) 3 q 3 3 3 3(cid:19)
Pk
·log 1+rz + r2+2rc z +c2+(c2−c2)z2
ΠkρabΠk =p0ρ0+p1ρ1, (16) (cid:18) 3 q 3 3 3 3(cid:19)
Xk
+ 1+rz − r2+2rc z +c2+(c2−c2)z2
and (cid:18) 3 q 3 3 3 3(cid:19)
p ρ +p ρ ·log 1+rz − r2+2rc z +c2+(c2−c2)z2 ]}
0 0 1 1 (cid:18) 3 q 3 3 3 3(cid:19)
1
= (I +rσ3)⊗I 1
4 + [(1−c +2|c|)log(1−c +2|c|)
3 3
4
1
+4(rz3I +cz1σ1+cz2σ2+c3z3σ3)⊗Vσ3V†. +(1−c3−2|c|)log(1−c3−2|c|)
+(1+c +2|r|)log(1+c +2|r|)
The eigenvalues of p ρ + p ρ are the same with the 3 3
0 0 1 1
eigenvalues of the states (I ⊗V†)(p ρ +p ρ )(I ⊗V), +(1+c3−2|r|)log(1+c3−2|r|)].
0 0 1 1
and (19)
(I⊗V†)(p0ρ0+p1ρ1)(I ⊗V) Note that ∆→(ρab) is an even function for the variable
1 z3, so we can focus on z3 ∈[0,1] instead of [−1,1].
= (I +rσ )⊗I
4 3 For example, we set λ = 0.6, then r = 0.409666,c =
1 −0.509296,c3 = −0.0915564, and obtain that the value
+ (rz3I +cz1σ1+cz2σ2+c3z3σ3)⊗σ3. (17) of the one-way deficit is 0.418314.
4
4
III. ONE-WAY DEFICIT IN XX MODEL FOR InFig. 1,wedrawnthecurveofone-waydeficitoftwo
λ>1 adjacent spins in the bulk of XX model in λ∈[0,1.5].
We find that the one-way deficit is nonzero in the do-
When λ > 1, Λ is negative for all k. Therefore the main λ ∈ [0,1) and then becomes zero when λ ≥ 1.
k
ground state has N fermions, which is translated as all As the XX model undergoes a first order transition at
spins up in spin language. It is straightforward to see the critical point λ = 1 from fully polarized to a criti-
that in this case the one-way deficit of the ground state cal phase with quasi-long-range order, we conclude that
or that of any part of its reduced density operator is one-way deficit can be used to detect quantum phase of
always zero. the XX model, and moreover, may reveal the insight of
In fact, for λ>1 the ground state is polarized, whose phase transition by quantum correlations.
reduced density matrix for two spins at l and l+1 has
turned into
0.5
ρ =|↑↑ih↑↑| (20)
l,l+1
0.4
The density matrix in Eq. (20) is rewritten as
t0.3
ci
ρl,l+1 =|0ih0|⊗|0ih0| (21) fi
e
D0.2
Next, we evaluate the one-way deficit of the states
in Eq. (21). Let {Π = |kihk|,k = 0,1} be the local
k 0.1
measurement for the party b along the computational
base |ki. By (11), (12), (14), after the measurement B ,
k
0
the state in Eq. (21) will be changed into the ensemble 0 0.5 1 1.5
{ρ¯ ,p¯ }. After some algebraic calculus, we obtain λ
k k
p¯ ρ¯ =(t2+y2)|0ih0|⊗V|0ih0|V†,
0 0 3 FIG. 1: (Color online) One-way deficit of two adjacent spins
p¯1ρ¯1 =(y12+y22)|0ih0|⊗V|1ih1|V† inthebulkfortheXX modelinthethermodynamiclimitas
a function of thequantum parameter λ.
Then, we will evaluate the eigenvalues of Π ρ Π
k l,l+1 k
Pk
by
Π ρ Π =p¯ ρ¯ +p¯ ρ¯ , (22) IV. CONCLUSION
k l,l+1 k 0 0 1 1
Xk
Wehavegivenamethodtoevaluatetheone-waydeficit
and
two adjacent spins in the bulk for the XX model in
t2+y2 0 the thermodynamic limit. We have drawn the curve of
p¯ ρ¯ +p¯ ρ¯ =|0ih0|⊗V 3 V†. (23)
0 0 1 1 (cid:18) 0 y12+y22 (cid:19) one-way deficit of the XX model. We find that we can
use one-way deficit to detect quantum phase of the XX
The eigenvalues of p¯ ρ¯ +p¯ ρ¯ are t2+y2,y2+y2,0,0. model. We find that the one-way deficit becomes zero
0 0 1 1 3 1 2
The entropy of Π ρ Π is when λ ≥ 1. Therefore, the one-way deficit can charac-
k l,l+1 k
Pk terizes the quantum phase transition in the XX model.
This may shed lights on the study of properties of quan-
S( Πkρl,l+1Πk) tum correlations in different quantum phases.
Xk
= −(t2+y2)log(t2+y2)
3 3
−(y2+y2)log(y2+y2). (24) Acknowledgments
1 2 1 2
By Eq. (13), we know that S( Πkρl,l+1Πk) is binary We are thankful to Yu-Ran Zhang and Jin-Ju Chen
Pk for fruitful discussions. This work was supported by the
entropy function. When t = 0,y = 0,y2 +y2 = 1 or
3 1 2 ScienceandTechnologyResearchPlanProjectoftheDe-
y =0,y =0,t2+y2 =1,thefunctionS( Π ρ Π )
1 2 3 k l,l+1 k partment of Education of Jilin Province in the Twelfth
Pk
reaches the minimum 0. By the entropy of ρ in Eq. Five-Year Plan, the National Natural Science Founda-
l,l+1
(21) being zero and Eq. (1), the one-way deficit of the tion of China under grant Nos. 11175248, 11275131,
states in Eq. (21) is 0. 11305105.
5
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