Quantum discord in spin systems with dipole-dipole interaction

0.34 MB·English

by  E. I. Kuznetsova

Quantum discord in spin systems with dipole-dipole interaction
Elena I.Kuznetsova∗ and M. A. Yurishchev
Institute of Problems of Chemical Physics of Russian Academy of Sciences,
Chernogolovka, 142432, Moscow Region, RUSSIA
(Dated:)
Thebehavioroftotalpurelyquantumcorrelation(discord)indimersconsistingofdipolar-coupled
spins 1/2 is studied. We found that the discord Q = 0 at the absolute zero temperature. With
increasing the temperature T, at first the quantum correlations in the system increase, smoothly
reach themaximum, and then turn again into zero according to theasymptotic law T−2. It is also
shownthatinabsenceofexternalmagneticfieldB,theclassicalcorrelationCatT →0is,viceversa,
3
1 maximal. Our calculations predict that in crystalline gypsum CaSO4·2H2O the value of natural
(B =0) quantum discord between nuclear spins of hydrogen atoms is maximal at the temperature
0
2 of 0.644 µK, and for 1,2-dichloroethane H2ClC−CH2Cl the discord achieves the largest value at
T =0.517µK.Inbothcases,thediscordequalsQ≈0.083bit/dimerwhatis8.3%ofitsupperlimit
n intwo-qubitsystems. WeestimatealsothatforgypsumatroomtemperatureQ∼10−18 bit/dimer,
a and for 1,2-dichloroethane at T =90 K thediscord is Q∼10−17 bit per a dimer.
J

3 PACSnumbers: 03.65.Ud,03.67.Mn,75.10.Jm,75.50.Xx
2

] I. INTRODUCTION the new kind of correlations. One discovered
h
alsothatdiscordcandetectthequantumphase
p
- The notion quantum discord has been intro- transitions13,14. Moreover, it has been shown
t
n duced in the quantum information theory by that in contrast to entanglement and thermo-
a Zurek1. Discord was regarded as “a measure dynamicalquantities,the discordmakesitpos-
u of a violation of classicality of a joint state of sible to catch the approach of quantum phase
q two quantum subsystems”1. However later its transitionsevenatfinitetemperatures15. Other
[
initial definition undergone some changes. important features of discord have been also

1 In 2001, Henderson and Vedral and then in- noted. By this, a surprising fact turned out:
v “Almost all quantum states have nonclassical
dependent of them Ollivier and Zurek in their
4 papers2,3 (seealso4,5)performedanalysisofev- correlations”16. Achieved up to now results on
9
thetheoryandapplicationsofquantumdiscord
erypossiblecorrelationsI inabipartitesystem
3
are given in the recent reviews17,18.
5 and suggested the ways to extract from them,
. on the one hand, the purely classical part C The goal of this paper is to study the be-
1
and,onthe otherhand,only the quantumcon- havior of discord in spin systems with dipo-
0
3 tribution Q. The quantum excess of correla- lar couplings. There is a large number of sub-
1 tions, Q=I C, has been called “discord” — stancesmagneticinteractionsinwhichhavethe
−
: in the modern understanding of this term3. dipole-dipolecharacter,andexchangeandindi-
v
The authorsofabovepapersestablishedalso rect ones are weak enough. By this, the spins
i
X that the quantum correlation can be non-zero both electrons and nuclears can serve as ele-
r even in separable (but mixed) states. In other mentarymagneticmoments. Theclassofdipo-
a
words,quantumcorrelationsarenotexhausted larmagnetswithelectronspinsincludes,forex-
by entanglement (E). Entanglement, which ample, the Tutton salts, alums19, and numer-
can relate the different parts of a system even ous salts of rare earth elements20. The typi-
when there are no interactions between these cal temperatures at which the effects of their
parts (the Einstein-Podolsky-Rosen effect), is dipole interactions show themselves lie in the
only a special kind of quantum correlations. millikelvin region. However the spin-lattice in-
Since the 80–90s of past century the entangle- teractions in electron paramagnets are strong
mentwasconsideredasafundamentalresource that leads to short relaxation times too. Nu-
for quantum informationprocessing,teleporta- clearspins,ofcourse,haveanindubitableasset:
tion, cryptography, metrology, and other tasks the spin-lattice relaxation times for them can
in quantum technology6–10. achieveminutes andhours. Using the available
It is remarkable that quantum discord can NMR data for the two classical examples, gyp-
also lead to a speedup over classical compu- sum CaSO 2H O21 and 1,2-dichloroethane
4 2
·
tation and lead even without containing much H ClC CH Cl22 (see also, e.g.,23,24) which
2 2
entanglement11,12. This important property contain−thesufficientlyisolatedpairsofdipolar-
of discord evoked extremely great interest to coupled nuclear hydrogen spins, we estimate
2
the discord between these spins. Taking into account the Bayes rule one can
In following sections of the paper we give rewrite the right part of Eq. (2) in Shannon’s
definitions forthe differentcorrelations,formu- nonsymmetric form
latethemodel,calculatetheclassicalandquan-
tuminformationcorrelationsinit,estimatethe I′(X :Y)=H(X) H(X Y), (4)
− |
discord for materials with spin-nuclear dimers,
where
and, lastly, briefly summarize the results ob-
tained.
H(X Y)=H(X,Y) H(Y) (5)
| −
is the condition entropy. In the classical case,
II. CLASSICAL AND QUANTUM I′ I.
CORRELATIONS I≡n the quantum information theory6,29, the
equation (2) is replaced by the new definition
In statistical theory, a degree of relationship
I(A:B)=S(ρ )+S(ρ ) S(ρ ), (6)
(correlation) between two random variables x A B AB
−
and y with the joint probability distribution
which serves as a measure of quantum mutual
function p(x,y) is often measured by covaria-
informationbetweenthetwosubsystemsAand
tions or by Pearson’s correlation coefficient
B composing together the joint system AB =
A B. In Eq.(6), ρ is the density matrix of
(x x)(y y) ∪ AB
R= − − . (1) thejointsystemAB,ρAandρB arethereduced
√D1√D2 density matrices, respectively, for subsystems
A and B, and S(ρ) (ρ = ρ ,ρ ,ρ ) repre-
Herethebardenotestheaverageovertheprob- { A B AB}
sents the von Neumann entropy
ability distribution and D = (x x)2 and
1
−
D2 =(y y)2 aredispersionsrespectivelyforx S(ρ)= Trρlogρ. (7)
− −
and y. Notice that in the mathematical statis-
tics other types of correlation coefficients are It is important that I =0 is the necessary and
used also. One should emphasize that the con- sufficient condition for the factorizationρAB =
dition R = 0 does not imply, generally speak- ρA ρB, what means, of course, the absolute
⊗
ing,theindependenceofrandomvariables,i.e., independence(non-correlativety)ofAandB in
that p(x,y)=p (x)p (y)25,26. the product state. Therefore, in the quantum
1 2
In the classical information theory informationtheory,onetakesthemutualinfor-
(e. g.,27,28), one uses the notion of mutual mationtomeasurethetotal(bothclassicaland
information quantum)correlationsbetweenthe twosubsys-
tems of bipartite quantum system.
I(X :Y)=H(X)+H(Y) H(X,Y) 0 (2) Ontheotherhand,measurementsperformed
− ≥
on one system, in general, influence on the
between twoobjects X Y. Here H(X), H(Y), quantum state of another system (see, for
and H(X,Y) are the Shannon entropies instance,30). Postulating that the total classi-
cal part of correlations is the maximal amount
H(X)= p (x)logp (x), of information about one subsystem, say A,
1 1
−
thatcanbeextractedbyperformingameasure-
x
X
mentontheothersubsystemB,Hendersonand
H(Y)= p (y)logp (y),
− 2 2 Vedral2 suggested to take as a measure of clas-
y
X sical correlation the quantity
H(X,Y)= p(x,y)logp(x,y), (3)
−Xx,y C(ρAB)=m{Baix}{S(ρA)− piS(ρiA)}. (8)
i
X
where p (x) = p(x,y), p (y) = p(x,y).
1 y 2 x
Here B is a complete set of measurements
(A choose of logarithm base defines the infor- i
{ }
P P on the subsystem B,
mation measure unities: bits, nats, dits, hart-
ley.) It is remarkable that now the equality
ρi =Tr (B ρ B+)/Tr (B ρ B+) (9)
I = 0 is a necessary and sufficient condition A B i AB i AB i AB i
for the independence of X and Y. This prop- is the remaining state of A after obtaining the
erty allows to use the mutual information as a outcome i on B, and
measureofinformationcorrelationbetweenthe
systems X and Y25. p =Tr (B ρ B+) (10)
i AB i AB i
3
is the probability to detect the result i. Inthese equations, µ is the magneticconstant
0
OllivierandZurek3,onthe contrary,focused (magnetic permeability of free space), γ is the
their attention on an extraction of quantum gyromagneticratio,σ arethevectorsofPauli
1,2
correlations. Further analysis of the measure- matrices at the sites 1 and 2, r is the distance
mentsledtothegeneralizationofexpression(4) betweenthe spins ina dimer, n is the unitvec-
for the quantum case, tor in the direction from one spin to the other,
andBisthevectorofmagneticfieldinduction.
I′(A:B)=S(ρ ) p S(ρi ). (11) The dipole-dipole interaction reflects the exact
A − i A
i law (in that sense that it does not contain the
X
fitting parameters), the interaction is sharply
(It is obvious that the righthand of this equal-
anisotropicandlong-acting(incontrast,say,to
ity is a non-optimized classical correlation of
the exchange interaction).
HendersonandVedral.) Inthepaper3,themin- In the spherical coordinates when B =
imal difference I −I′ ≡ Q has been identified (0,0,B) and n = (sinθ,0,cosθ), the Hamilto-
withanamountofquantumcorrelationandhas
nian (13) – (15) takes the form
been called the quantum discord — a measure
of the quantum excess of correlations, a mea-
µ γ2~2
sinutroeaocfctohuenqtutahnattumnessofcorrelations. Taking H= 4π0 4r3 [σ1σ2−3(σ1xsinθ+σ1zsinθ)
1
(σxsinθ+σysinθ)] γ~B(σz +σz).
I =C+Q, (12) × 2 2 − 2 1 2
(16)
we see the equivalentness of results of
Henderson-Vedral and Ollivier-Zurek. It has been shown31 that when the polar angle
Quantumdiscorddisplaysanumber ofprop- θ = π (the externalfieldisappliedperpendicu-
erties (see, e.g., the reviews17,18). We note be- 2
larlyto the directionofdimerlongitudinalaxis
tweenthemthefollowingones. Forapurestate,
theentanglementinthesystemismaximal. On
discord coincides with the entanglement E. In
the contrary,whenθ =0orπ (the longitudinal
mixedstates,thequantumcorrelation(discord)
dimer axis is parallel to the external field) the
can present even in that case when the entan-
entanglementbetweenspinsisabsent(belowwe
glement is absent. Quantum discord Q 0.
prove this strongly/exactly).
≥
Discordislimitedfromabovebythe entropyof
Becausethespecialinterestistodiscoverand
one subsystem, Q S(ρ ). In particular, if
≤ A(B) investigate the quantum correlations without
the systemis two-qubitandthe logarithmbase
entanglement, we will consider from this point
for entropy equals two then Q 1.
≤ the case θ = 0. By this, the Hamiltonian (16)
takes the form
III. HAMILTONIAN AND DENSITY
1 1
MATRIX = D(σxσx+σyσy+∆σzσz) h(σz+σz),
H 2 1 2 1 2 1 2 −2 1 2
(17)
Considerasystemconsistingoftwoidentical where the dipolar coupling constant equals
particles with the spins 1/2 which couple be-
tweenthemselvesbythemagneticdipole-dipole µ γ2~2
0
interaction. Let moreover, the external homo- D = (18)
4π 2r3
geneous magnetic field with induction B was
applied to the system. Then the Hamiltonian and the normalized external field is
of a model can be written as (see, e. g.,6)
h=γ~B. (19)
= + , (13)
dd Z
H H H
where the dipolar part is In dipole-dipole coupled dimer, the anisotropy
parameter is ∆= 2. However, below we will,
Hdd = 4µπ0 γ42r~32[σ1σ2−3(n·σ1)(n·σ2)] (14) itnhesovmaeluceasseosff∆or.t−hBeusatkaellofggreanpehriaclailtya,nedxtennud-
merical material in our paper is presented for
and the Zeeman energy equals
∆= 2.
−
1 The Hamiltonian (17) corresponds to the
Z = γ~(σ1+σ2)B. (15) XXZ model in Z field. In the matrix forn, it
H −2
4
is given as and partition function is
∆D h Z =2(chβD+e−βD∆chβh)eβD∆/2. (29)
2 −
∆D D
= −2 . Expressions (28) satisfy the condition
H D ∆D
−2
 ∆D+h a+d+2b=1, (30)
 2 
 (20)
which provides the normalization Trρ=1.
The 2 2 subblock presented here is a cen-
× Expandingthedensitymatrix(27)intopow-
trosymmetric matrix which under the orthog-
ers of Pauli matrices we obtain it in the (nor-
onal transformation
mal) Bloch form
1 1 1
O = (21) 1
√2 1 1 ρ= [1+(a d)(σz +σz)+2v(σxσx+σyσy)
(cid:18) − (cid:19) 4 − 1 2 1 2 1 2
undegoes in diagonal form. +(1 4b)σzσz]. (31)
− 1 2
TheenergyspectrumoftheHamiltonian(20)
Expansioncoefficientsaretheunaryandbinary
withcondition∆= 2consistsoftwoindepen-
− correlators,
dent on external field levels
2
E =2D, E =0 (22) m σz = σz =a d= e−βD∆/2shβh,
1 2 ≡h 1i h 1i − Z
(32)
and two levels
4
E3,4 =−D±h. (23) Gk ≡hσ1zσ2zi=1−4b=1− Z eβD∆/2chβD,
(33)
Because D >0, the ground state energy is
2
E0 =−D+|h|. (24) G⊥ ≡hσ1xσ2xi=hσ1yσ2yi=2v =−Z eβD∆/2shβD.
(34)
In absence of externalfield, the groundstate is
Brackets denote the trace operation for the
two-fold degenerate.
expression in brackets with density operator,
We will consider the dimer in a termal equi-
= Tr(ρ). The coefficient a d = m in the
librium state. In this case, its density matrix h·i · −
expansion (31) is equal to the z components
ρ ρ has the Gibbs form
≡ AB (projections) of the Bloch vectors for the re-
1 duceddensitymatricesofsubsystemsAandB.
ρ= exp( β ), (25) Moreover,thequantitiesm,G ,andG havea
Z − H k ⊥
physical sense, correspondingly,as the normal-
whereβ =1/kBT,kB isBoltzmann’sconstant, izedmagnetization,longitudinalandtransverse
and Z is the partition function, components of correlation matrix. Relations
Z =Trexp( β ). (26) 1 1
− H a= 4(1+2m+Gk), b= 4(1−Gk),
Performing necessary calculations we find that 1 1
v = G , d= (1 2m+G ) (35)
in the original (standard) basis 00 , 01 , 10 2 ⊥ 4 − k
| i | i | i
11 the density matrix has the structure
| i give the expressions for the matrix elements of
a the density operator (27) through the system
b v correlators.
ρ= , (27)
 v b  FromEq.(32)–(34)wefindthe hightemper-
d ature behavior for the magnetization and cor-
 
  relation functions,
where
1 h ∆ h D 2
a= 1 exp[ β(∆D h)], m(T,h)= +O(1/T3),
Z − 2 − 2 kBT − 4 D (cid:18)kBT(cid:19)
(36)
d= 1 exp[ β(∆D+h)],
Z − 2
(28) ∆ D 1 h 2 D 2
b= Z1 exp(β∆2D)chβD, Gk(T,h)=−2 kBT − 4"1−(cid:18)D(cid:19) #(cid:18)kBT(cid:19)
v = 1 exp(β∆D)shβD, +O(1/T3), (37)
−Z 2
5
1 D ∆ D 2 the equation34
G (T,h)=
⊥
−2 k T − 4 k T
B (cid:18) B (cid:19) C˜ =2max w √u u ,0 (45)
1 1 h 2 D 3 1 {| |− 1 2 }
+ + +O .
8 3 D k T T4 serves for calculation of concurrence. The
h (cid:18) (cid:19) i(cid:18) B (cid:19) (cid:18) (cid:19) expression (45) is a particular case of Hill-
(38)
Woottersformula32,33 whichallowstocalculate
Thus, when T , the main contributions the pairwise concurrence in general two-qubit
→ ∞
to the correlatorstend to zeroaccordingto the system.
law 1/T. By this, a weak external field does Our density matrix (27) has the form of
not exercise an influence on the G and G . Eq. (44). Using Eqs. (28) we find that
k ⊥
Moreover,the leading term in the expansionof
the transverse correlator G does not depend w √u u = v √ad 0, (46)
⊥ | |− 1 2 − − ≤
on the anisotropy ∆.
At lower temperature, the correlation func- if ∆ 1. This inequality is valid for arbi-
≤ −
tions forthe dipolardimer (D >0,∆= 2)in trary external field B. Hence, the concurrence
absence of external field behave as − (45)andtogetherwithittheentanglement(43)
are equal identically to zero. So, in the dipole-
D
G 1 exp( ), (39) dipole (∆ = 2) dimer under question, the
k|T→0 ≈ − −kBT quantum entan−glement is absent for all tem-
peratures and arbitrary longitudinal fields.
1 D
G exp( ). (40)
⊥ T→0
| ≈−2 −k T
B V. INFORMATION CORRELATIONS
If the field h = 0, the additional statistical
6
weightsattheexponentsarise(duetoachange In this section, we give a calculation of in-
of ground state of the system; see below), formationcorrelationsin dipolardimer both in
and out magnetic field.
D
G 1 2exp( ), (41)
k|T→0 ≈ − −k T
B
A. Arbitrary external field
D
G exp( ). (42)
⊥ T→0 Discord in a system with the density matrix
| ≈− −k T
B having the structure (27) equals35,36
Thus, the lower temperature behavior of cor-
relators is described by a function of the form Q=min Q1,Q2 , (47)
{ }
e−1/x.
where
a
IV. QUANTUM ENTANGLEMENT Q =S S alog
1 A− AB − 2 a+b
(cid:18) (cid:19)
b b
The entanglement through the relation
blog blog
− 2 a+b − 2 b+d
(cid:18) (cid:19) (cid:18) (cid:19)
1+ 1 C˜2 1+ 1 C˜2 d
E = − log − dlog , (48)
− p2 2 p2 ! − 2 b+d
(cid:18) (cid:19)
1 1 C˜2 1 1 C˜2
− − log − − (43)
− p2 2 p2 ! Q2 =SA−SAB −δ1log2δ1−δ2log2δ2, (49)
is expressed via the concurrence C˜32,33. and
In the case of density matrix having the 1
δ = [1 ((a d)2+4v2)1/2]. (50)
block-diagonalform 1,2 2 ± −
u1 In equations (48) and (49),
x w
ρ= 1 , (44)
 w∗ x  S = (a+b)log (a+b) (b+d)log (b+d)
2 A − 2 − 2
u (51)
 2 
 
6
isthevonNeumannentropyofreduceddensity
matrix ρ and
A
S = alog a dlog d (b+v)log (b+v)
AB − 2 − 2 − 2
(b v)log (b v) (52)
− − 2 −
equalsthevonNeumannentropyforthedensity
matrix ρ of full system.
The total correlation is I =2S S , and
A AB
−
classical one C = I Q. These relations and
−
also(47)–(52)together withEqs.(28) and(29)
define the quantum discord, classical, and to-
talcorrelationsasfunctions ofthetemperature
and externalmagnetic field by arbitraryvalues
Figure 1: Quantum discord in dipolar system
of parameters D and ∆.
(∆ = −2) as a function of temperature and ex-
At high temperatures, ternal longitudinal magnetic field.
1 D 2 ∆ D 3
Q = + +O(1/T4),
1
4ln2 k T 8ln2 k T
(cid:18) B (cid:19) (cid:18) B (cid:19) field, a = d and S = S = 1. Besides, using
(53) A B
expressions (33) and (34), it is not difficult to
1 D 2 establish that by ∆ < 1 one has Gk G⊥
Q2 = (1+∆2) +O(1/T3). (54) (belowthiswillbeseeni−ngraphics). The≥re|fore|,
8ln2 k T
(cid:18) B (cid:19) in absence of magnetic field, the classical part
Thus, the main terms of these expressions do of mutual correlations is
not depend on the external magnetic field. 1
From the relations (53) and (54), one can see C = 2[(1+Gk|)log2(1+Gk|)+(1−Gk|)log2(1−Gk|),
that by ∆ > 1 the discord, according to (57)
| |
Eq. (47), is defined by the branch Q1. Hence, and the quantum discord equals
when ∆ > 1 or ∆ < 1, the quantum discord
− 1
by high temperatures behaves as Q= [(1+2G G )log (1+2G G )
4 ⊥− k 2 ⊥− k
1 1 2(1 G )log (1 G )+(1 2G G )
Q|T→∞ ≈ 4ln2(k T/D)2. (55) − − k 2 − k − ⊥− k ·
B log (1 2G G )]. (58)
2 − ⊥− k
Thus,itdoesnotdependonbothexternalmag-
How can one see from Eq. (57), the classical
netic field or interaction anisotropy ∆. With
correlationiscompletelydeterminedbythelon-
increasing the temperature, the quantum cor-
gitudinal correlator G . Using expressions for
relations decrease according to the law 1/T2, k
the correlation functions (33), (34) and setting
i. e., essentially rapidly than ordinary statisti-
h=0in them, wegetthe followingformulafor
cal correlations which, how it can was empha-
the discord,
sized above, tend to zero according to the law
1/T. D sh D ch D ln[ch D ]
From equations presented above, we estab- Q(T)= k(cBhT( DkB)T+−exp(kBTD∆))lknB(2T).
lishalsothatbyhightemperaturestheclassical kBT −kBT
correlation is (59)
∆2 1 Fora dipolardimer (D >0,∆= 2)inzero
C (56) −
|T→∞ ≈ 8ln2(k T/D)2 field and at lower temperatures, we have
B
1
and does not depend on external field. Q exp( D/k T). (60)
T→0 B
| ≈ 2 −
Arbitrary order derivatives with respect to the
B. Dimer in absence of external field temperature for the function in the right hand
ofthis equationis zeroatT =0. Therefore,by
In important particular case of zero external small deviation of temperature from absolute
field,theinformationcorrelationscanbecalcu- zero, Q 0. This is connected with existence
lated via the simpler formulas37. In absence of of a gap≈in the energy spectrum of the system.
7
0.1
∆= 2
−
0.08 h/D=0
0.06 1
Q
0.04 2
0.02
Figure 2: Isolines of quantum discord in dipolar 0
system (∆=−2). 0 0.5 1 1.5 2 2.5 3
kBT/D
Figure 3: Discord isoterms by different values of
external field.
VI. DISCUSSION
A behaviour of quantum discord in the
dipole-dipole system under question (∆ = 2)
solution of transcendental equation
−
is shown in Fig. 1. We see the smooth hill-like
surfacestretchedinthetemperatureaxisdirec-
tion. Along a straight line h = 0, two ridges x(e∆·x+chx+∆ shx)=(shx+∆ chx)ln(chx),
go; one ridge goes in the direction of tempera- · ·
(63)
ture decreasing and the other in the direction
where x=D/k T. This equation follows from
B
of its increasing. The surface is symmetrical
thecondition∂Q/∂T =0inwhichthefunction
underreflectioninverticalplanegoingthrough
Q(T) is given by expression (59).
the hill top and the straight line h = 0 in the
temperature-fieldplane. Bygiventemperature, OnecanseefromFig.3that,withincreasing
the discord is maximal in absence of external the field, the discordmaximumismovedinthe
magnetic field, that is, the field leads only to a directionofhighertemperatures. Thisisagood
suppressionofquantumcorrelation. Atthe ab- feature. Howeverbythisthevalueofdiscordat
solutezerotemperature,thediscordindimeris the maximum is less than its value atthe same
identically equal to zero. In the high tempera- temperature in absence of external field.
turelimit,quantumcorrelationisalsovanished.
We see also that with the help of external
In Fig. 2, the cross sections (profiles) of a dis-
field and temperature (these parameters are in
cordsurfaceareshownbydifferentvaluesofQ.
ourhands),onecancontrolthediscordvarying
It is seen that the isolines form a set of non-
its value from zero to 0.083 bit per a dimer.
crossing ovals.
Tostudyindetailthebehaviorofdiscord,we Let us consider the case when the dipolar
represented in VI its temperature dependence dimer is in absence of external field. The tem-
bydifferentvaluesofexternalfield. Thediscord peraturedependencesofordinaryspin-spincor-
reaches the largest value at h=0 and relationfunctionsG and G (G 0,there-
k ⊥ ⊥
| | ≤
foreitsabsolutevalueistaken),aswellastotal
k T /D =0.881297... . (61) classical correlation C and discord Q are pre-
B m
sented in Fig. 4. At zero temperature, both
longitudinal and total classical correlation are
The discord in this point equals
equal to their maximal magnitudes. The tem-
peratureincreasingleadsonlytotheirdecrease.
Q =0.083061... . (62)
m Both the value of transverse correlation G
⊥
| |
and quantum discord are zero at T =0.
what is 8.3% of maximal value which is possi-
bleinanytwo-qubitsystem. Thevalue(61),we To clarify the situation in correlationbehav-
havefound both by a numericalsearchof max- ioratzero,weconsiderthelimitofdensityma-
imum for the function Q of k T/D and from trix (27) when T 0. Using the expressions
B
→
8
CORRELATIONS CORRELATIONS
1 1
∆= 2 ∆= 2
− −
0.8 Gk h/D=0 0.8 Gk h/D=0.1
0.6 0.6
C
C
0.4 0.4
0.2 0.2
|G⊥| |G⊥|
Q Q
0 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3
kBT/D kBT/D
Figure4: Temperaturebehaviorofstatistical (G Figure5: ThesameasinFig.4butinpresenceof
k
and|G |),classical (C),andquantum(Q)correla- a field.
⊥
tions in absence of magnetic field.
(the value G = 0 witnesses this). With in-
⊥
for its matrix elements (28) one finds creasing the temperature these degrees of free-
dom“revive”andthe systemfromclassicalbe-
1/2
comes a quantum one. But when the temper-
0
ρ = . (64) ature fluctuations begin to exceed the system
|T=0  0  energygap,the temperatureproducesits usual
1/2
  destroying actionand the correlationsgo down
 
to non.
This matrix has the expansion
From Fig. 4 and asymptotical behavior of
1 correlations,it is not difficult to notice a corre-
ρ= (0 0 0 0 +1 1 1 1 , (65)
2 | ih |A⊗| ih |B | ih |A⊗| ih |B spondence in qualitative behavior, on the one
hand, C andG2 and,onthe otherhand,Qand
i. e., it is a sum of the direct products of sepa- k
rateparticles. Suchstatebelongstotheclassof G2. The latter correspondence is not an ac-
⊥
purelyclassicalones38 andtherefore,according cident. Indeed, if the quantum correlation is
to the criteria39, all quantum correlations in it measuredby geometricaldiscordQg, i. e., by a
are entirely absent. This explains that we ob- distance(insenseoftheHilbert-Schmidtnorm)
tainedQ =0. Onthe otherhand,classical from the given state ρ to the nearest classical
T=0
correlatio|ns are available and equal the total one,thegeneralformula39,40appliedtotheden-
correlations C =I. sity matrix (31) by an absence of external field
So the behavior of system at T =0 is purely yields
classical. However, with increasing the tem-
Q =G2. (67)
perature, the behavior acquires the quantum g ⊥
features, i. e., the temperature leads to the
Consider now the situation in a presence of
generation of quantum correlations. Such un-
externalmagnetic field. In Fig. 5, the behavior
usual phenomenon (it is much ordinary when
of different correlations is again shown but for
the temperature destroys the quantum states)
h/D =0.1. ItisseenthatcorrelationsG , G ,
k ⊥
one can explain by following. In the Hamil-
and Q were changed weakly. But the behavior
tonian (17) with ∆ < 1, the purely classical
− ofclassicalcorrelationC undergoesanessential
ferromagnetic contribution
change—ittendsnowtozerowhenT 0. At
→
H =∆Dσzσz (66) the point of absolute zero temperature, there
zz 1 2
are no any correlations in the system at all.
is dominate at lower temperatures. The trans- To understand a happening, we turn out
versecomponentsofspinsareactually“frozen” again to the density matrix. The matrix (27)
9
Thus, any statistical relation between spins is
1
also absent.
∆= 2
−
Fig. 6 shows the behavior of classical corre-
0.8
lation C upon the external field. The curves
kBT/D=0.25 havetheformofbell-likesplashes. Theirlargest
0.6 value is arrived at in the point when the field
C vanishes. Withdecreasingthetemperature,the
0.1 maximumsbecomemorenarrowandtheirvalue
0.4
tends to the value C = 1. At T = 0, a splash
becomes infinitely thin,
0.2
1, h=0
C = . (72)
0
-0.4 -0.2 0 0.2 0.4 (0, h=0
h/D 6
Figure 6: Classical correlation vesus the field by The external field sweeping will on a moment
diffrrent values of temperature. lead to a spasmodic jump appearance of classi-
calcorrelationandatthesametimetoadisap-
pearance of this correlation by going the field
of the zero point. By this, the quantum corre-
in the limit T 0 is equal now to both
→ lation in the system does not arise.
1
Let’sturntotheavailableexperimentaldata.
0
ρ = , (68) Measurements performed by the nuclear mag-
|T=0  0  netic resonance (NMR) at room temperature
0
  show that in gypsum crystals CaSO4 2H2O
  ·
the distance between protons in each water
if h>0 or
molecule is r = 0.158 nm21 (see also, e. g.,
0 the book23). For protons, the gyromagnetic
0 ratio, as it is known24, equals γ = 2.675
ρ|T=0 = 0 , (69) 108 rad/(cT), therefore the dipole-dipole cou-·
·
1 pling constant (18) in gypsum is D/k =
  B
  0.73 µK (in temperature units). Consequently,
if h < 0. These states correspond to the com- inaccordwith(61),themaximumdiscordvalue
pletely orderings of spins along the field. Both Q = 0.083 will arrive at the temperature
max
matrices (68) and (69) are factorized into a di- 0.64 µK. At room temperature (T = 300),
rect product of two density matrices 2 2, the discordin gypsum, accordingto (55), must
×
be equal to Q 2 10−18. In spite of ex-
1 ∼ ·
tremely small value of quantum correlations in
0 1 0 1 0
= , spin-nuclear systems at room temperatures, at
 0  0 0 ⊗ 0 0
(cid:18) (cid:19) (cid:18) (cid:19) present the attempts are undertaken to detect
0
  their by NMR methods41–44.
 
0
As an another example, let us consider 1,2-
0 0 0 0 0
= (.70) dichloroethane ClH C CH Cl. In this com-
 0  0 1 ⊗ 0 1 2 − 2
(cid:18) (cid:19) (cid:18) (cid:19) pound, two protons at each carbon atoms are
1
  coupled much stronger between themselves by
 
This means that the states (68) and (69) are dipole-dipole interactionthan with protonsbe-
completely uncorrelatedand therefore any cor- long to an other carbon atom. NMR mea-
relations are absent in them. surementsperformedonsoliddichloroethaneat
Emphasize that G = 0 at T = 0 does not the temperature 90 K have shown that here
k
contradict to the said a6 bove. In the factorized r = 0.17(2) nm22 (see also23,24). Using again
statemustbezeroonlythecenteredcorrelator. the relations (18) and (61), we estimate the
By h = 0, the spins at the point of absolute temperature for the discord maximum in this
zero te6mperature are ordered and therefore substance as Tm = 0.517 µK. At the temper-
ature 90 K, the value of quantum correlations
(σz σz )(σz σz ) =0. (71) must equal Q 1.5 10−17.
h 1 −h 1i 2 −h 2i i ∼ ·
10
VII. CONCLUSIONS It was shown also that the classical correla-
tions have a sharp maximum in the point of
In the paper, a study of information correla- zero external magnetic field when T 0.
→
tionsindipolardimersbothinabsenceofexter- A qualitative analogy between the quantum
nalmagneticfieldandinthefielddirectedalong discordandthesquaredcorrelatorG⊥ hasbeen
the longitudinal axis of a dimer has been per- found. In supporting this observation, it was
formed. It bas been shown that the quantum shownthatthegeometricaldiscordequalsQg =
correlations are completely absent at the ab- G2⊥.
solute zero temperature and arbitrarystrength In the paper, the estimates for the thermal
ofmagnetic field but they arise withincreasing quantumdiscordbetweenspinsofhydrogennu-
the temperature. We proposed an interpreta- cleous 1H in gypsum and 1,2-dichloroethane
tion such of phenomenon. have been done.
In the high temperature region, the discord Our investigation can be extended to many-
obeys the law nuclear clusters if to perform a density matrix
reduction for all spins except any two. Exper-
Q G2 1/(k T/D)2. imental NMR data are available for clusters in
∼ ⊥ ∼ B
aform,forexample,triangles,tetrahedron,lin-
This, in particular, means that at room, ni- ear magnetic structures, etc.,23,45,46.
trogen, or helium temperatures, the non-zero
quantitiesofspin-spincorrelationsbetweenthe
xxoryyspincomponentscanserveasawitness ACKNOWLEDGMENT
ofquantumcorrelations. Thespin-spincorrela-
tions go down with increasing the temperature This researchwas supported by the program
according to the essentially slow law T−1 and No. 8 of the Presidium of RAS.
one candirectly measurethem, for example, in
scattering experiments.
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