One-way deficit and quantum phase transitions detection in XY model

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by  Yao-Kun Wang

One-way deficit and quantum phase transitions detection in XY model
Yao-Kun Wang1 and Yu-Ran Zhang2,∗
1College of Mathematics, Tonghua Normal University, Tonghua, Jilin 134001, P. R. China
2Institute of Physics, Chinese Academy of Sciences, Beijing 100190, P. R. China
Quantumdeficitoriginatesinquestionsregardingworkextractionfromquantumsystemscoupled
to a heat bath [Phys. Rev. Lett. 89, 180402 (2002)]. It is a kind of quantum correlations
besides entanglement and quantum discord, and links quantum thermodynamics with quantum
correlations. Inthispaper,weevaluatetheone-waydeficitoftwoadjacentspinsinthebulkforthe
XY model. Wefindthattheone-waydeficitcharacterizesthequantumphasetransitionintheXX
model and thetransverse field Ising model that theXY model reducesto for specified parameters.
This study may enlighten extensive studies of quantum phase transitions from the perspective
7 of quantum information and quantum correlations, including finite-temperature phase transitions,
1 phasetransitions of otherquantum many-bodymodels or even topological phasetransitions.
0
2
PACSnumbers:

n
a
J I. INTRODUCTION deficit of nearby two spins arrives its maximal value al-
most at the critical point of transversefield Ising model.
6
Quantum deficit is a kind of nonclassical correlation Our results will enlighten extensive studies of the quan-

] besides entanglement and quantum discord. Quantum tum information properties of ground states in different
h
phases of critical systems. On the contrary, it will also
p deficit [1–3]originatesonasking how to use nonlocalop-
benefitmanyapplicationssuchasto detectthe quantum
- erationtoextractworkfromacorrelatedsystemcoupled
t phasetransitionandtoevaluatethecapacityofquantum
n to a heat bath only in the case of pure states [1]. In the
computations in critical systems.
a general case, the advantage is related to more general
u formsofquantumcorrelations. Oppenheimetal. defined
q
theworkdeficit[1]isameasureofthedifferencebetween
[ II. ONE-WAY DEFICIT IN XY MODEL
the information of the whole system and the localizable
1 information[4,5]. Recently,Streltsovetal. [6,7]givethe
v definitionof the one-wayinformationdeficit by meansof One-waydeficitbyvonNeumannmeasurementonone
9 relativeentropy,which is alsocalled one-waydeficit that side is given by [21]
2
uncovers an important role of quantum deficit as a re-
016 souMrcaenyfodretvheelodpismtreinbtustiionnqoufaenntutamngilnefmoremnta.tion process- ∆→(ρab)={mΠikn}S(Xk ΠkρabΠk)−S(ρab), (1)
. ing [8] has provided much insight into quantum phase
1 where S(·) denotes to the von Neumann entropy. As a
transitions[9]. Especially,quantumcorrelationshasbeen
0 kind of quantum correlations besides entanglement and
7 successfulincharacterizingalargenumberofcriticalphe- quantumdiscord,one-waydeficitlinksquantumthermo-
1 nomena of great interest. In particular, entanglement
dynamicswithquantumcorrelationsanddeservefurther
: was the first and most outstanding member to detect a
v investigations in critical systems.
numberofcriticalpoints,see[9–14]. Furthermore,quan-
i Then, we will consider the XY model [22] in the zero-
X tum discord is an outstanding quantum correlation, is
temperature case. The Hamiltonian of our model is as
r also used to study quantum phase transitions [15]. An-
a follows [23]:
other indication of quantumness that is also found its
applications for probing quantum phases and quantum L−1 (1+γ)σiσi+1+(1−γ)σiσi+1
phase transition [16–20]. H =− 1 1 2 2 +hσi
(cid:20) 2 3(cid:21)
In this paper, we analytically calculate the one-way Xi=0
deficit of the thermal ground state of two adjacent spins (2)
in the bulk of the XY model. We find that the one-way with L being the number of spins in the chain, σi the
n
deficitmay characterizethe quantumphasetransitionin ith spin Pauli operator in the direction n = 1,2,3 and
the XX model and the transverse field Ising model that periodic boundary conditions assumed. The XX model
the XY model reduces to for specified parameters. In and transverse field Ising model thus correspond to the
details, we find the phenomenon of sudden death of the specialcasesforthisgeneralclassofmodels. Forthecase
one-way deficit in XX model as the strength of trans- thatγ →0,ourmodelreducestoXX model. Whenγ =
verse field increases to the critical point; the one-way 1, the model reduces to transverse field Ising model [24].
In fact, there exists additional structure of interest in
phase space beyond the breaking of phase flip symmetry
ath=1,whichisthecriticalpointbetweentwoquantum
∗Electronicaddress: [email protected] phases. It is worth noting that there exists a quater of
2
for some unitary V ∈U(2). For any unitary V,
t+y i y +y i
V =tI +i~y·~σ = 3 2 1 . (8)
(cid:18)−y2+y1i t−y3i (cid:19)
with t∈R, ~y =(y ,y ,y )∈R3, and
1 2 3
t2+y2+y2+y2 =1, (9)
1 2 3
after the measurement B , the state ρab will be changed
k
into the ensemble {ρ ,p } with
k k
1
ρ = (I ⊗B )ρ(I ⊗B ), (10)
k k k
FIG. 1: (Color online) One-way deficit of two adjacent spins pk
in thebulkfortheXY modelinthethermodynamiclimit as p = tr(I ⊗B )ρ(I ⊗B ). (11)
k k k
a function of the quantumparameter h,γ.
To evaluate ρ and p , we write
k k
circle, h2 +γ2 = 1, on which the ground state is fully pkρk = (I ⊗Bk)ρ(I ⊗Bk)
separable. 1
= (I ⊗V)(I ⊗Π )[I +rσ ⊗I+sI ⊗V†σ V†
For the thermal ground state of XY model written as 4 k 3 3
Equation (2), the Bloch representation of the reduced 3
density matrix for two nearby spins at positions i and + c σ ⊗(V†σ V)](I ⊗Π )(I ⊗V†). (12)
j j j k
i+1 has been obtained in [25] as Xj=1
3 Using the relations [26]
1
ρab = (I ⊗I+rσ ⊗I+sI ⊗σ + c σ ⊗σ ), (3)
3 3 i i i
4 Xi=1 V†σ1V = (t2+y12−y22−y32)σ1+2(ty3+y1y2)σ2
+2(−ty +y y )σ , (13)
whereI istheidentity, r=s=hσii, c =hσiσi+1i, c = 2 1 3 3
3 1 1 1 2 V†σ V = 2(−ty +y y )σ +(t2+y2−y2−y2)σ
hσiσi+1i,andc =hσiσi+1i.InthethermallimitT →0, 2 3 1 2 1 2 1 3 2
2 2 3 3 3
we have hσii = −1 π dφ(1 + cosφ/h), hσiσi+1i = +2(ty1+y2y3)σ3, (14)
hσii2 − G G3 , hσiσπiR+01iωφ= G and hσiσi+31i3= G V†σ3V = 2(ty2+y1y3)σ1+2(−ty1+y2y3)σ2
wh3ere we h1av−e1 1 1 −1 2 2 1 +(t2+y32−y12−y22)σ3, (15)
1 π dφ and
G ≡ − [cos(φ)(1+cosφ/h)
±1
π Z0 ωφ
Π σ Π =Π ,Π σ Π =−Π ,Π σ Π =0, (16)
0 3 0 0 1 3 1 1 j k j
∓γsin(φ)sin(φ)/h]. (4)
for j =0,1,k=1,2, we obtain
and ω = (γh−1sinφ)2+(1+h−1cosφ)2.
φ
TheeigepnvaluesoftheX statesinEq.(3)aregivenby 1
p ρ = [I +sz I+c z σ +c z σ +(r+c z )σ ]
0 0 3 1 1 1 2 2 2 3 3 3
4
λ1,2 = 1(1−c3±|c1+c2|), ⊗(VΠ0V†), (17)
4
1
1 p ρ = [I −sz I−c z σ −c z σ +(r−c z )σ ]
λ = [1+c ± (2r)2+(c −c )2]. (5) 1 1 4 3 1 1 1 2 2 2 3 3 3
3,4 3 1 2
4
p ⊗(VΠ V†), (18)
1
with which the entropy is given by
where
4
S(ρ)=− λilogλi. (6) z1 =2(−ty2+y1y3), z2 =2(ty1+y2y3),
Xi=1 z3 =t2+y32−y12−y22. (19)
Next, we evaluate the one-way deficit of the X states Then,wewillevaluatetheeigenvaluesof Π ρabΠ by
k k k
in Eq. (3). Let {Πk = |kihk|,k = 0,1} be the local kΠkρabΠk =p0ρ0+p1ρ1, and P
measurement for the party b along the computational
P
base |ki; then any von Neumann measurement for the 1
p ρ +p ρ = (I +rσ )⊗I
party b can be written as 0 0 1 1 4 3
1
{Bk =VΠkV† :k =0,1} (7) +4(sz3I +c1z1σ1+c2z2σ2+c3z3σ3)⊗Vσ3V†. (20)
3
(a) 0.5 (b) 0.5 of XY model as a function of h,γ. When γ is a fixed
γ=0 γ=1
value, we observe that as the transverse field strength h
ficit 0.25 ficit0.25 increases, the one-way deficit increases for small h and
e e
D D decreases for large h, see Fig. 2 (c) for γ = 0.5. When
0 0 γ →1, maximum of the one-way deficit is attained near
0 0.5 1 1.5 0 0.5 1 1.5
h h h=1, see Fig. 2 (d).
(c) 0.5 (d) 1 In Fig. 2 (a) for the case that γ → 0, our model re-
γ=0.5 Maximal
duces to XX model. We find that the one-way deficit
deficit
ficit0.25 γ 0.5 is nonzero in the domain h ∈ [0,1) and then suddenly
De becomes zero when h≥1. As the XX model undergoes
a first order transition at the critical point h = 1 from
0 0
0 0.5 1 1.5 0 0.5 1 1.5 fully polarized to a critical phase with quasi-long-range
h h
order, we conclude that one-way deficit can be used to
detectquantumphaseoftheXX model. Theconclusion
FIG. 2: (Color online) One-way deficit of two adjacent spins
in the bulk for the XY model: (a-c) one-way deficit of XY is in consistent with the result obtained in [27].
model for γ → 0 (XX model), γ = 1 (transverse field Ising InFig.2(b)forthecasethatγ =1,themodelreduces
model)andγ =0.5respectively;(d)Maximalone-waydeficit to transverse field Ising model. We find that one-way
of the XY model. deficit of Ising model increases for small h and decreases
for large h. When one-way deficit achieve the maximum
nearly ath=1,transversefield Ising model undergoes a
The eigenvalues of p ρ + p ρ are the same with the
0 0 1 1 first order transition. We infer that one-way deficit can
eigenvalues of the states (I ⊗V†)(p ρ +p ρ )(I ⊗V),
0 0 1 1 also be used to detect quantum phase of the transverse
and
field Ising model.
(I⊗V†)(p ρ +p ρ )(I ⊗V)
0 0 1 1
1
= (I +rσ )⊗I III. CONCLUSION
3
4
1
+ (sz3I+c1z1σ1+c2z2σ2+c3z3σ3)⊗σ3.(21) We have given a method to analytically evaluate the
4
one-waydeficitofthe thermalgroundstatesoftwoadja-
The eigenvalues of the states in the equation (21) are cent spins in the bulk for the XY model in the thermo-
dynamic limit. We have drawn the diagram of one-way
1
w = 1−sz ± (r−c z )2+c2z2+c2z2 (22) deficit of the XY model. We find that we can use one-
1,2 4(cid:18) 3 q 3 3 1 1 2 2(cid:19) way deficit to detect quantum phase of the XX model
1 and transverse field Ising model. We find the sudden
w = 1+sz ± (r+c z )2+c2z2+c2z2 .(23)
3,4 4(cid:18) 3 q 3 3 1 1 2 2(cid:19) deathoftheone-waydeficitinXX modelasthestrength
of transverse field increases to h = 1. For the trans-
The entropy of Π ρabΠ is S( Π ρabΠ ) = verse field Ising model, the one-way deficit of the ther-
k k k k k k
4 malgroundstateofnearbytwospinsarrivesitsmaximal
− w logw . WPhen γ,h are fixedP, r,s,c ,c ,c is
i=1 i i 1 2 3
conPstant. By using z12 + z22 + z32 = 1, it converts the value at the critical point h = 1. On one hand, our re-
problem about min S( Π ρabΠ ) to the problem sults may shedlights onthe study of properties of quan-
{Πk} k k k tum correlations in different quantum phases of many
about the function of threePvariables z1,z2,z3 for mini-
body systems. On the other hand, our investigations
mum, that is
will also benefit many applications including quantum
minS( Π ρabΠ )= min S( Π ρabΠ )(.24) phase transition detection and the capacity of quantum
k k k k
{Πk} Xk {z12+z22+z32=1} Xk computation evaluation in critical systems. Also numer-
ical techniques such as DMRG, MPS and exact diago-
By Eqs. (1), (6), (24), the one-way deficit of the X nalization methods deserve to be used to investigate ex-
states in Eq. (3) is given by tensive problems of quantum phase transitions from the
perspective of quantum information and quantum cor-
∆→(ρab)
relations, including finite-temperature phase transitions,
4 4 phase transitions of other quantum many-body models
= min (− w logw )+ λ logλ .(25)
i i i i or even topological phase transitions.
{z12+z22+z32=1} Xi=1 Xi=1
Next, we use the equation above to calculate the one-
way deficit of two adjacent spins in the bulk for the XY Acknowledgments
model and analyze the in different phases..
The main results are shown in Fig. 1, in which we WewouldliketothankJin-JunChenforusefuldiscus-
plotthe one-waydeficitoftwoadjacentspins inthe bulk sions. ThisworkwassupportedbytheScienceandTech-
4
nology Research Plan Project of the Department of Ed- ucation of Jilin Province in the Twelfth Five-Year Plan.
[1] J. Oppenheim, M. Horodecki, P. Horodecki, and R. [14] T. J. Osborne and M. A. Nielsen, Phys. Rev. A 66,
Horodecki, Phys.Rev. Lett.89, 180402 (2002). 032110 (2002).
[2] M.Horodecki,K.Horodecki,P.Horodecki,R.Horodecki, [15] R. Dillenschneider, Phys. Rev.B 78, 224413 (2008).
J.Oppenheim,A.Sen(De),andU.Sen,Phys.Rev.Lett. [16] J. Cui, J. P. Cao, and H. Fan, Phys. Rev. A 85, 022338
90, 100402 (2003). (2012)
[3] K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. [17] J. Cui, M. Gu, L. C. Kwek, M. F. Santos, H. Fan, and
Vedral,Rev. Mod. Phys. 84. 1655 (2012). V. Vedral, Nat Commun 3, 812 (2012).
[4] M. Horodecki, P. Horodecki, R. Horodecki, J. Oppen- [18] J. Cui, L. Amico, H. Fan, M. Gu, A. Hamma, and V.
heim, A. Sen(De), U. Sen, and B. Synak, Phys. Rev. A Vedral, Phys. Rev.B 88, 125117 (2013).
71, 062307 (2005). [19] F.Franchini,J.Cui, L.Amico,H.Fan,M.Gu,V.Kore-
[5] M. Horodecki, P. Horodecki, and J. Oppenheim, Phys. pin,L. C.Kwek,and V.Vedral,Phys.Rev.X4, 041028
Rev.A 67, 062104 (2003). (2014).
[6] A.Streltsov, H. Kampermann, and D. Bruß, Phys. Rev. [20] S.L.Liu,Q.Quan,J.J.Chen,Y.R.Zhang,W.L.Yang,
Lett.108, 250501 (2012) and H.Fan, Sci. Rep.6, 29175 (2016).
[7] T. K. Chuan, J. Maillard, K. Modi, T. Paterek, M. Pa- [21] A. Streltsov, H.Kampermann, and D. Bruß, Phys. Rev.
ternostro, and M. Piani, Phys. Rev. Lett. 109, 070501 Lett. 106, 160401 (2011).
(2012). [22] V.Mukherjee,U.Divakaran,A.Dutta,andD.Sen,Phys.
[8] T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Rev.B76,174303(2007);S.Garnerone,N.T.Jacobson,
Monroe, and J. L. OBrien, Nature (London) 464, 45 S. Haas, and P. Zanardi, Phys. Rev. Lett. 102, 057205
(2010). (2009); F. Franchini, A. R. Its, and V. E. Korepin, J.
[9] L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Phys. A 41, 025302 (2008).
Mod. Phys. 80, 517 (2008). [23] S. Y. Liu, Y. R. Zhang, W. L. Yang, and H. Fan, Ann.
[10] A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature Phys 362, 805. (2015).
(London) 416, 608 (2002). [24] P. Pfeuty, Ann.Phys. 57, 79 (1970).
[11] L. A. Wu, M. S. Sarandy, and D. A. Lidar, Phys. Rev. [25] W.Son,L.Amico,F.PlastinaandV.Vedral,Phys.Rev.
Lett.93, 250404 (2004). A 79, 022302 (2009).
[12] R.Oru´sandT.C.Wei,Phys.Rev.B82,155120 (2010). [26] S. Luo, Phys. Rev.A 77, 042303 (2008).
[13] G.Vidal,J.I.Latorre,E.RicoandA.Kitaev,Phys.Rev. [27] Y. K. Wang, arXiv:1601.02750.
Lett.90, 227902 (2003).