Mutual information and spontaneous symmetry breaking

0.74 MB·English

by  A. Hamma

Mutualinformationandspontaneoussymmetrybreaking
A. Hamma,1,∗ S. M. Giampaolo,2 and F. Illuminati2
1CenterforQuantumInformation,InstituteforInterdisciplinaryInformationSciences,TsinghuaUniversity,Beijing100084,P.R.China
2DipartimentodiIngegneriaIndustriale,Universita` degliStudidiSalerno,I-84084Fisciano(SA),Italy
(Dated:October29,2015)
Weshowthatthemetastable,symmetry-breakinggroundstatesofquantummany-bodyHamiltonianshave
vanishingquantummutualinformationbetweenmacroscopicallyseparatedregions,andarethusthemostclassi-
calonesamongallpossiblequantumgroundstates.Thisstatementisobviousonlywhenthesymmetry-breaking
groundstatesaresimpleproductstates,e.g. atthefactorizationpoint. Ontheotherhand,symmetry-breaking
statesareingeneralentangledalongtheentireorderedphase,andtoshowthattheyactuallyfeaturetheleast
macroscopiccorrelationscomparedtotheirsymmetricsuperpositionsishighlynontrivial.Weprovethisresult
ingeneral, byconsideringthequantummutualinformationbasedonthe2−Re´nyientanglemententropyand
6
usingalocalityresultstemmingfromquasi-adiabaticcontinuation. Moreover,intheparadigmaticcaseofthe
1
0 exactlysolvableone-dimensionalquantumXY model,wefurtherverifythegeneralresultbyconsideringalso
2 thequantummutualinformationbasedonthevonNeumannentanglemententropy.

n
PACSnumbers:03.67.Mn,05.30.Rt,75.10.Pq
a
J

6 I. INTRODUCTION

]
l The emergence of a macroscopic classical behavior from
e
- a microscopic quantum world can be explained in terms of
r
t decoherencetotheenvironmentthatquicklydestroystheco-
t.s chaetrse)nt[1s]u.peTrphoesisteiolencsteodfpmoianctreorscstoapteics ombujestcttshe(nScbhero¨fdaicntgoer-r l A d(A,B) B
a
m ized states with respect to a tensor product structure that is
localinrealspace[2,3]. Similarly,superselectioninducedby
-
d decoherence due to weak interactions with the environment
n playsakeyrolealsointhephenomenonofspontaneoussym- E
o metry breaking, where different ordered sectors with broken
c
symmetryaredynamicallydisconnectedandarethustheonly
[
states that are metastable [4], whence the notion of sponta-
2 neoussymmetrybreaking[5]. FIG.1: Amany-bodyquantumsystemispartitionedinthreedistin-
v guishablesubsystems,A,B,andtheremainderE,sothatthetotal
In the paradigmatic case of the quantum Ising model, the
1 HilbertspaceacquiresthetensorproductstructureH=HA⊗HB⊗
ground space of the ferromagnetic phase at zero transverse
6 HE. Thequantityd(A,B)isthedistancebetweenthetworegions
9 field h is spanned by two orthogonal product states |0(cid:105)⊗N A,B,andlisthedistancedefiningtheneweffectivesupportafteran
6 and |1(cid:105)⊗N which are in the same class of pointer states of adiabaticdeformationofoperatorswithinitialsupportonHA,HB.
0 thetypical√decoherenceargument,whilethesymmetricstates
. Ψ = 1/ 2(|0(cid:105)⊗N ±|1(cid:105)⊗N) realize macroscopic coherent
1 ±
0 superpositions that are not stable under decoherence [1, 4].
5 Therefore,atzerotransversefieldh,thesituationisveryclear: tems possess in general nonvanishing entanglement [9–11],
1 theonlystablestatesarethosethatmaximallybreakthesym- aswellasquantum[12–14]andclassicalcorrelations[6]. In-
: metryoftheHamiltonian,andatthesametime,thosethatfea- deed,thesymmetry-breakinggroundstatescanbe,“locally”,
v
i ture vanishing macroscopic total correlations, including en- moreentangledthansomenearbysymmetricstates[15]. On
X tanglement,betweenspatiallyseparatedregions. theotherhand,itisalwaysimplicitlyassumedthatsuchstates
r Asweturnontheexternalfieldh, wehaveawholerange arenotmacroscopicallycorrelated,whiletheirsymmetricsu-
a
of values where, before a critical value h = h is reached, perpositions are, in complete analogy with the case h = 0.
c
thereisamagneticorderassociatedtospontaneoussymmetry Although this is a very plausible picture, a rigorous proof
breaking[6],andthedecoherenceargumentapplieswithinthe has never been provided, due to the mathematical difficul-
entire ordered phase. This means that, again, the only stable ties in dealing with measures of entanglement and correla-
statesarethosethatmaximallybreaktheHamiltoniansymme- tions based on the von Neumann entropy; see, e.g., the dif-
try[7,8].However,nowthesymmetry-breakingstatesareen- ficultiesinprovingtheboundary(area)lawingenericgapped
tangled,andtheirmixed-statereductionsonarbitrarysubsys- systems[16,17],orinprovingthestabilityoftopologicalen-
tanglement entropy in topologically ordered states[18]. The
symmetry-breaking states obey the boundary law for entan-
glement[19–22],whilethemacroscopiccorrelationsfeatured
∗Correspondingauthor:[email protected] bythesuperpositionoftwodifferentsymmetrybrokensectors
2
areoforderone. formations of the ground state and we will study the behav-
Thequestionisthenaboutwhichquantityoneshouldlook iorofthemacroscopicmutualinformation. Thedeformation
at in order to distinguish the presence of macroscopic corre- corresponds to the adiabatic continuation of the ground state
lations,amongallpossiblesourceofentanglementandcorre- obtainedbyswitchingonthetransversemagneticfieldh. We
lations? Historically,thekeyconceptsthathavebeenconsid- willprovethat,intheentiresymmetry-breakingphase,theto-
eredaretheoffdiagonallongrangeorder(ODLRO)[23]and, tal macroscopic correlations and, a fortiori, the macroscopic
morerecently,thetwo-siteconcurrence(entanglementoffor- entanglement,asmeasuredbythemutualinformation,vanish
mation)atlargedistance[24,25].IfthereiseitherODLROor in the maximally symmetry-breaking ground states at large
nonvanishingconcurrencebetweentwositesindifferentclus- spatial separations between arbitrarily selected macroscopic
ters A and B, then also the two clusters must be entangled, subsystems. On the other hand, we will also prove that, as
since the total state of the global system is pure. This is an longasthedeformationissufficientlysmall,themacroscopic
important point, because the reduced subsystems being in a mutual information remains finite in symmetric states. Later
separablestatedoesnotimplythattheremustbenoentangle- on, we will apply the results of the general analysis to spe-
mentbetweenthetwoclustersinapurestate. Evenifallthe cificspinmodels,showingthattheresultholdingforslightly
remaining correlations are classical, they are due to the fact deformed symmetric states is in fact valid for all symmetric
thattheoverallstateisapureentangledstate. states in the entire symmetry-breaking (ordered) phase, until
On the other hand, the reverse argument needs not ap- aquantumphasetransitionpointisreached.
ply: it is possible that macroscopic clusters are entangled Proving the general result analytically would be quite a
even if measures of two-point correlations, like the concur- daunting task if one were to consider the von Neumann en-
rence, are vanishing. For example, this can happen in the tropy S. Rather, we will resort to a related quantity, the
2D toric code [26] or in the one dimensional cluster mod- 2−Re´nyientropy,S ,andthecorresponding2−Re´nyimutual
2
els [27–29] where all two-site concurrences are identically information.Forspecific,computable,examples,wewillthen
zero and yet the macroscopic block entanglement entropy is showthatthegeneralconclusionsreachedusingthe2−Re´nyi
finite.Alsostatesthatpossessvolumelawforthebipartiteen- entropyholdaswellusingthevonNeumannentropy.
tanglementtypicallyhavenotwo-siteentanglement, because TheRe´nyientropiesofindexαaredefinedas
of monogamy. Moreover, concurrence and ODLRO have to
becomputedforeveryspecificmodelandcannotprovideuni- Sα(A)=(1−α)−1log2TrραA , (2)
versalclassifications.
Inthepresentwork,wewillshowthatmacroscopiccorrela- and they are all bona fide entanglement monotones [36]. In
tionsaregenerallynonvanishingingenericsymmetricstates, particular, the 2−Re´nyi entropy S2 is an experimentally ac-
and vanishing in symmetry-breaking states. To this end, we cessiblequantity,sinceitistheexpectationvalueoftheswap
consider the total correlations (classical plus quantum) be- operatorontwocopiesofaquantumstate[37],whilethevon
tween two generic subsystems A and B of the total system, NeumannentropyS isnotanexperimentallyfriendlyobserv-
asmeasuredbythemutualinformation[30]: able,becauseitrequirescompletestatetomography,whichis
essentially impossible on a many-body state. Concrete pro-
I(A|B)=S(A)+S(B)−S(AB), (1) posals for measuring S resort on quantum switches [38], or
2
onmultiparticleinterferometry[39].
whereS(X)isthevonNeumannentropyofthedensityma- Wethusconsiderthe2−Re´nyimutualinformationdefined
trixpertainingtosubsystemX,asshownpictoriallyinFig.1. as
Themutualinformationisindeedabonafidemeasureoftotal
correlations(classicalplusquantum)[31]. Iffortwoarbitrary I (A|B):=S (A)+S (B)−S (AB). (3)
2 2 2 2
subsystems A,B spatially separated by arbitrarily large dis-
tancesd(A,B)themutualinformationI(A|B)isvanishing, Unlike the quantum mutual information defined in Eq. (1),
weareassuredthattherearenomacroscopiccorrelationsand, the2−Re´nyimutualinformationmaynotbepositivedefined
in particular, no macroscopic entanglement and no macro- foreverystateinHilbertspace. Inordertoensurepositivity,
scopicquantumcorrelations(allcorrelationandentanglement one can regularize this definition in terms of 2−Re´nyi rela-
measures are non-negative defined). Otherwise, taking into tive entropies, as shown in Refs. [40, 41]. However, on the
accountthatthetotalsystemisinaglobalpurestate,thetwo class of states of interest for the present investigation, non-
subsystemsmustbemacroscopicallyentangledandquantum regularized 2−Re´nyi mutual information is always positive
correlated. ItisimmediatetoverifythatI(A|B)vanisheson definite. Moreover, as already mentioned, in the paradig-
symmetry-breaking states that are product states of the form matic case of the exactly solvable one-dimensional quantum
|0(cid:105)⊗N and|1(cid:105)⊗N ,whileforsymmetricsuperpositionsΨ it XY model,wewillalsomakedirectcomparisonwiththevon
±
isalwaysoforderone, irrespectiveoftheactualvalueofthe Neumann-basedmutualinformation,findingcompleteagree-
distanced(A,B). ment. Finally,theclusteringpropertyandothergeneralprop-
We will show that the above result is in fact valid in gen- ertiesofthe2−Re´nyimutualinformationthatareatthecore
eral. In order to prove such statement, our strategy is the of our investigation and that we will prove below, can be
following. Starting from a factorization point, i.e. a point provenwithouttoomuchefforttoholdvalidalsoforthereg-
in which the system admits a fully separable pointer state as ularized versions. Therefore, in the following, also in order
global ground state [32–35], we will consider adiabatic de- toavoidunnecessaryformalmathematicalcomplications,we
3
willconsideronlythenon-regularizedversionofthe2−Re´nyi fiedatthefactorizationpoint.
mutualinformation. As the Hamiltonian parameters are changed adiabatically
By adopting such strategy, we will obtain the following andmoveawayfromthefactorizationpoint,ground-stateen-
mainresult:intheorderedphasecorrespondingtothesponta- tanglementdoesnotcomesolelyfromthemacroscopicsuper-
neousbreakingofsomesymmetryofamany-bodyHamilto- position of disconnected sectors, but also from the fact that
nianwithnonvanishinglocalorderparameter, themutualin- fully factorized states are no longer ground states. They can
formationbetweentwoarbitrarilyselectedsubsystemsAand berepresentedasU(s)|0(cid:105)⊗N,U(s)|1(cid:105)⊗N,whereU(s)isthe
B separated by a distance d(A,B) reaches its maximum in unitaryoperatorthatconnectstheinstantaneousgroundstates
thesymmetry-preservinggroundstatesandisupperbounded of the Hamiltonian H(s). The operator U(s) is an entan-
by exp(−O(d(A,B))) in the maximally symmetry-breaking gling operator, showing that the symmetry breaking ground
ground states, i.e. the ground states that maximize the order states are now entangled. This raises the problem whether
parameter. theyarestillonlylocallyentangledoriftheyhavedeveloped
Therefore,weestablishrigorouslythat-atleastforsomefi- some macroscopic entanglement. Similarly, for the symmet-
niterangeofvalueswithinthephase-spontaneoussymmetry ric states will be U(s)Ψ , it is not immediately obvious to
±
breaking corresponds to the suppression of macroscopic co- what extent their entanglement is macroscopic or not for a
herent superpositions, and symmetry-breaking ground states genericvalueofs. Inthefollowing,weshowthattheentan-
are the ones selected in real world by environmental deco- glementandthecorrelationsduetotheadiabaticcontinuation
herence, in complete analogy with pointer states. In the fol- U(s)arelocal,i.e. theyvanishinthelimitofarbitrarilylarge
lowing, wespecializetotheclassofglobalZ symmetry. In spatialseparationsd(A,B).
2
particular,wewillconsiderthespecific,butparadigmatic,ex- In order to proceed, we need to discriminate clearly be-
ampleoftheofquantumXY models,andwewillshowthat tween the macroscopic and the local contributions to I. We
macroscopic entanglement survives until a quantum phase first recall that by S (A) we mean the quantity S (A) =
2 2
transitionoccurs. However,thecentralelementsofthisresult −log Q where Q is the purity defined as Q = Tr[ρ2],
2 A A A A
holdingeneralandtheremainingonescanbeeasilyadapted and ρ is the reduced density matrix from the ground state
A
to every instance of spontaneous symmetry breaking for dif- ofthetotalsystemtothesubsystemcontainedinthefinitere-
ferentHamiltoniansanddifferentclassesofsymmetrygroups. gion A. To compute the Re´nyi entropy of order 2 we will
use the identity Q := Trρ2 = Tr(S ρ⊗2), where ρ⊗2
A A A
represents two copies of the original full state on the dou-
II. CLUSTERINGOF2-RE´NYIENTROPYAND bled Hilbert space H⊗H(cid:48). The operator SA is the permu-
LONG-RANGEORDER tation operator (swap operator) of order two with support on
HAA(cid:48) only: SA = S˜A ⊗IA¯ where S˜A = ⊗a∈AS˜a and Sa
Macroscopic long-range total correlations are revealed by is the permutation operator on the a-th spin of the system,
a nonvanishing mutual information I between two arbitrary i.e.Sa|i1,...ia,...,in(cid:105)⊗|j1,...ja,...,jn(cid:105)=|i1,...ja,...,in(cid:105)⊗
regions A and B separated by arbitrarily large distances. If |j1,...ia,...,jn(cid:105).
I vanishes when A and B are separated by a distance larger Next, we exploit a locality result, stemming essentially
thanwhatdefinesthemacroscopicinteractionscale, thenwe from the Lieb-Robinson bound [43]. Indeed, for the mutual
areassuredthatthereisnomacroscopicentanglement. informationbasedonthevonNeumannentropy,arecentsem-
Let us first consider the case of fully factorized ground inal contribution has shown that I(A|B) is an upper bound
states. These states are realized at a precise and unique set for the two-point correlation functions and a lower bound to
ofvaluesoftheHamiltonianparametersinanorderedphase, exponentially decreasing functions of the ratio between the
the so-called factorization point or, in spin systems, the fac- d(A,B)andthecorrelationlength[44].
torizingfield,firstdiscoveredinRef.[32]. Thegeneraltheory FollowingHastingsandWen[45],letusconsideramany-
(cid:80)
of ground-state factorization in terms of the response to lo- bodyHamiltoniansumoflocalterms,H(s)= h (s)with
i i
cal unitary perturbations was fully developed much later, in a finite gap ∆E above the low energy sector for some fi-
Refs.[33–35]. Factorizedgroundstatescanonlyoccurinan niteintervalofvaluesoftheHamiltonianparameterss(then,
orderedphaseofstronglyinteractingmany-bodysystems. In- out of this interval a quantum phase transition may occur).
deed, they are always maximally symmetry-breaking ground Moreover, the local operators are assumed to be bounded:
stateswithdegeneracyequaltothedimensionofthesymme- (cid:107)h (s)(cid:107) < ∞. If the ground state of H(s) is known for
i
trygroupand,beingfullyproductstates,theyhaveatrivially a particular, fixed set of values of the Hamiltonian parame-
vanishing mutual information I. Remarkably, at the factor- ters, say s , we may obtain it for any other generic set s by
0
izationpoint,allothergroundstatescanalwaysbeexpressed the quasi-adiabatic continuation U(s) induced by a continu-
as coherent linear superpositions of the fully factorized and ous deformation of H(s). A local operator O with support
A
maximally symmetry-breaking ground states [42]. Then, by onAtransformsasO (s) = U†(s)O U(s). Thenewoper-
A A
construction,symmetricgroundstatesfeatureanonvanishing ator O (s) has support on the whole Hilbert space. Never-
A
I, no matter how large the distance d(A,B) between the A theless, the locality result implies that we can arbitrarily ap-
andB regions. Moreover,therearenofurthertypesofentan- proximate it with an operator O (s) that has support only
A(cid:48)
glement and quantum correlations involved. Therefore, the over the Hilbert space associated to a region with diameter
classicalityofsymmetry-breakingstatesisimmediatelyveri- diam(A(cid:48))=diam(A)+l,aslongaslislargerthanthecor-
4
relationlengthξ inducedbythegap∆E,andbythismaking Since for s = 0 the expectation values of (cid:104)a(cid:48)b(cid:48)|S (0)|ab(cid:105)
A
anerrorboundedinthisway: (cid:107)O (s)−O (s)(cid:107)≤Ke−l/ξ. are positive definite, by continuity they are still positive for
A A(cid:48)
TheconstantK growslikelD whereD isthespatialdimen- a small enough value of s. So we see that, for small s,
sion(e.g. thelatticedimensionforlocalizedspins). the second term in Eq. 8, is negative but that the I (A|B)
2
Letnowρbethegroundstateofthesystemfors=s .The staysstrictlypositive,forarbitraryd(A,B). Thisisalsotrue
0
purity of the restriction of the evolved state ρ(s) to a spatial for any non trivial superposition of the symmetry breaking
regionC reads sectors given by the amplitudes α , so that only the maxi-
a
mally symmetric breaking ones have exactly zero I (A|B)
QC(s) = Tr(cid:2)U(s)⊗2ρ⊗2(U†(s))⊗2SC(cid:3) (asd(A,B)→∞). 2
= Tr(cid:2)ρ⊗2(U†(s))⊗2S U(s)⊗2(cid:3) Inthefollowing,wedeterminetheexactvalueofI (A|B)
C 2
(cid:39) Tr(cid:2)ρ⊗2S (s)(cid:3) . (4) formacroscopiccoherentsuperpositionswitharbitrarilylarge
C+l
d(A,B) in the entire ordered phase, beyond the perturbative
HereS denotesthepermutationoperatorwithsupporton case of small s in the case in which A and B are made by a
spins thCa+tlare at most at distance l from C, and the (cid:39) sign singlespin. WeprovethisresultexplicitlyformodelswithZ2
meansthattheerrorisexponentiallysmallinl/ξ. Ifthesub- symmetry, but the central elements of the proof are valid for
system C is the union of disjoint and macroscopically sepa- arbitrarysymmetrygroupsandarbitrarydimensionofAand
ratedsubsets,C =A∪B,wehave B. In fact, our proof implies that for a maximally symme-
try broken ground state all the correlations function between
Q = Tr(cid:2)ρ⊗2(U†(s))⊗2S S U(s)⊗2(cid:3) twoveryfarsubsystemsfactorizeintheproductoftheexpec-
C A B
= Tr(cid:2)ρ⊗2S (s)S (s)(cid:3) . (5) tation value of the local operators. And this implies that the
A+l B+l mutualinformationbetweenAandB mustvanish. Whenwe
turn to consider symmetric ground states, some of these lo-
IfthedistanceseparatingAandB ismuchlargerthanl, i.e.
calexpectationvaluesmustvanishbecausethelocaloperator
d(A,B)(cid:29)l,wecanwrite
doesnotcommutewithatleastoneoftheparityoperatorsthat
Q (cid:39) Tr(cid:2)ρ⊗2S (s)⊗S (s)⊗I (cid:3) , (6) definethesymmetrygroupoftheHamiltonian. Asresultthe
C A+l B+l E mutual information is expected to be different from zero. A
comprehensiveanalysisonthedependenceonthesizeofthe
where E is the complement to A and B together, see Fig. 1.
subsystemsand/orthesymmetrygroupoftheHamiltonianis
Assumefirstthattheinitialstateats = s isoneofthecom-
0
inprogress[46].
pletely factorized ground states (which, we recall, are also
maximally symmetry-breaking ground states). In this case,
ρ(s )=ρ ⊗ρ ⊗ρ andweobtain
0 A B E
Q (cid:39) Tr(cid:2)ρ⊗2S (s)(cid:3)Tr(cid:2)ρ⊗2S (s)(cid:3) . (7)
C A+l B+l
III. LONG-RANGEMUTUALINFORMATIONIN
Therefore,thepurityintheregionC =A∪Bistheproductof MODELSWITHZ SYMMETRY
2
thepuritiesinthetwoseparatedregionsAandB fromwhich
itfollowsimmediatelythatI (A|B)(cid:39)0.
2 In the following, we focus on spin-1/2 systems with a
Letusnextconsidertheoppositecaseinwhichtheground
global Z symmetry, thus described by Hamiltonians that
state for s = s is a macroscopic coherent superposition 2
0 commute with the parity operator along a fixed spin direc-
(Schro¨dinger cat) of the fully factorized ground states. We
tion, i.e. P =⊗ σµ. In such systems spontaneous symme-
show that any such superposition leads to a non vanish- µ i i
try breaking is associated to the presence of a two-fold de-
ing I (A|B. Since we need to consider two copies of the
2 generate ground state and an off diagonal long-range order
ground state, we will label the factorized states by |a(cid:105) and
along a spin direction σν that is orthogonal to σµ. We show
|b(cid:105), respectively, on each copy. So, we can write |a(cid:105) = i i
that throughout the entire ordered phase the long-range mu-
|a(cid:105) ⊗|a(cid:105) ⊗|a(cid:105) andsimilarlyfor|b(cid:105). Therefore,|ρ(cid:105)⊗2 =
(cid:80)A α αB|a(cid:105) |a(cid:105)E|a(cid:105) ⊗|b(cid:105) |b(cid:105) |b(cid:105) ,where(cid:80) |α |2 = tualinformationvanishesidenticallyonstatesthatmaximally
a,b a b A B E A B E a a breakthesymmetry, whileitremainsstrictlypositiveonany
(cid:80) |α |2 = 1. For the sake of an explicit evaluation, con-
b b macroscopic coherent superposition of the two broken sym-
sider the case of a doubly degenerate ground state manifold
metrysectors. Forthesakeofsimplicitywefocusonthecase
and subsystems A and B with equal size (number of spins).
inwhichsubsystemsAandB areeachonemadebyasingle
Forsymmetricsuperpositionswethenobtain:
spinbuttheresultscanbeextendedstraightforwardlytomore
(cid:88) generalchoices.
I (A|B)(cid:39)log 4−2log (cid:104)a(cid:48)b(cid:48)|S (s)|ab(cid:105)
2 2 2 A+l
Thetwoorthogonalsymmetricgroundstates|e(cid:105)and|o(cid:105)(re-
aba(cid:48)b(cid:48)
(cid:88) spectively,theeven-symmetricandtheodd-symmetricground
+log (cid:104)a(cid:48)b(cid:48)|S (s)|ab(cid:105)2
2 A+l states) form a convenient basis that allows to write all other
aba(cid:48)b(cid:48) groundstates|g(cid:105)intheorderedphaseastheirlinearsuperpo-
=log 4+log (cid:80)aba(cid:48)b(cid:48)(cid:104)a(cid:48)b(cid:48)|SA+l(s)|ab(cid:105)2 . (8) sitions: |g(u,v)(cid:105) = u|e(cid:105)+v|o(cid:105). Thereduceddensitymatrix
2 2 ((cid:80)aba(cid:48)b(cid:48)(cid:104)a(cid:48)b(cid:48)|SA+l(s)|ab(cid:105))2 ρC of the two-spin block C = A ∪ B can be expressed in
5
termsofthe2-pointcorrelationfunctionsasfollows[9]: 1.0
ρC(u,v)=41 (cid:88) GiA,iB(u,v)σAiAσBiB , (9) 0.8 1.0
iA,iB 0.8
0.6
where the expectations GiA,iB(u,v)=(cid:104)g(u,v)|σiAσiB|g(u,v)(cid:105) h 0.6
A B
areonproductsofPaulimatricesσiA andσiB. Asshownin 0.4
A B 0.4
Ref.[8],allcorrelationfunctionscanbeassociatedtospinop-
erators that either commute or anti-commute with the parity 0.2 0.2
operator Pµ. Therefore, the reduceddensity matrix ρC(u,v) 0
can be expressed as the sum of a symmetric part that coin- 0.0
cides with the density matrix of the symmetric ground state, 0.0 0.2 0.4 0.6 0.8 1.0
ρ(s)(u,v), and an antisymmetric part, that is a traceless ma- Γ
C
trix,ρ(a)(u,v). Takingintoaccountthefactthatthetwosym-
C FIG. 2: (Color Online) Behavior of the mutual information
metricgroundstatesfallintwoorthogonaleigenspacesofthe (I(s)(∞)) between two spins at infinite distance in the symmet-
Parity,itisstraightforwardtoverifythatρ(s)(u,v)isindepen- ric2groundstateoftheone-dimensionalXY model(thermodynamic
C
dent of the superposition amplitudes and hence ρ(s)(u,v) ≡ limit)asafunctionoftheanisotropyγ andtransversefieldhinthe
C ferromagnetic phase 0 ≤ h ≤ h = 1. Within the ferromagnetic
ρ(CsT).he reduced density matrix of a symmetric ground state phase,itisalwaysmν > 0,andhcenceI2(s)(∞) > 0. Ontheother
hand,m = 0eitherath = 1,oratγ = 0. Onlyatthesepoints
ν c
thusreads I(s)(∞)=0.
2
1
ρ(s) = (11+m (σµ +σµ)+m2σµσµ +m2σνσν). (10)
C 4 µ A B µ A B ν A B
thetransversefield. Forh<h =1thesystemisferromagnet-
c
In Eq. (10) m is the expectation value of the local oper- icallyorderedandischaracterizedbyatwofoldground-state
µ
ator that commutes with the parity while mν is the local or- degeneracysuchthattheZ2paritysymmetryunderinversions
derparameter. ExploitingEq.(10)onecanderivethemutual alongthespin-z directionisbrokenbysomeelementsofthe
information I(s) and evaluate its asymptotic expression for groundspace. Usingtheanalyticalsolution,inFig.2wehave
d(A,B)→∞2: plottedthebehaviorofI2(s)(∞)intheferromagneticphase.
Giventhetwosymmetricgroundstates,theso-calledeven
(cid:20) m4 (cid:21) |e(cid:105) and odd |o(cid:105) states belonging to the two orthogonal sub-
I(s)(∞)=log 1+ ν . (11)
2 2 (1+m2)2 spaces associated to the two possible distinct eigenvalues of
µ
theparityoperator, anysymmetry-breakinglinearsuperposi-
Asm (cid:54)= 0throughouttheentireorderedphase,theabove tionoftheform
ν
relationshowsthepresenceofmacroscopicentanglementand
|g(u,v)(cid:105)=u|e(cid:105)+v|o(cid:105) (13)
correlationsinthesymmetric,coherentsuperpositionground
statesthroughouttheentirephase. Uptothispoint,thisresult is also an admissible ground state, with the complex super-
is valid for any model with Z2 symmetry. The actual val- positionamplitudesuandvconstrainedbythenormalization
uesofmν ofcoursedependonthespecificmodelconsidered. condition|u|2+|v|2=1.Takingintoaccountthattheevenand
Here,weanalyticallycomputetheresultforthequantumXY oddgroundstatesareorthogonal,theexpectationvaluesofop-
model. eratorsthatcommutewiththeparityoperatorareindependent
The one-dimensional spin-1/2 quantum XY Hamiltonian of the superposition amplitudes u and v. On the other hand,
with ferromagnetic nearest-neighbor interactions in a trans- spinoperatorsthatdonotcommutewiththeparitymayhave
versefieldwithperiodicboundaryconditionsreads: nonvanishingexpectationvaluesonsuchlinearcombinations
andhencebreakthesymmetryoftheHamiltonianEq.(12).
N(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
(cid:88) 1+γ 1−γ In the asymptotic macroscopic regime d(A,B) → ∞,
H=− σxσx + σyσy +hσz ,
2 i i+1 2 i i+1 i the general two-spin reduced density matrix for an arbitrary
i=1 groundstatereads
(12)
wonhesriteeσiiµ,,γµis=thxe,ya,nzis,oatrreopthyepPaaruamliespteirn-i1n/t2heopxeyraptolarnsea,chtinigs ρC(u,v)=ρ(Cs)+ 41(uv∗+vu∗)[mν(σAν +σBν)
the transverse magnetic field along the z direction, and the + m m (σµσν +σνσν)] . (14)
periodicboundaryconditionsσµ ≡σµensureinvarianceof µ ν A B A B
N+1 1
themodelHamiltonianunderspatialtranslations. The corresponding expression for the mutual information
Such model can be solved analytically [6, 47, 48] and, I (∞)reads
2
hence, the phase diagram can be determined exactly and in
greatdetail. Inthethermodynamiclimit,foranyγ∈(0,1],a I (∞)=log(cid:20)1+ m4ν(1−(uv∗+vu∗)4) (cid:21) . (15)
quantumphasetransitionoccursatthecriticalvalueh =1of 2 2 (1+m2 +(uv∗+vu∗)2m2)2
c µ ν
6
0.8æà æà æà æà æàæàæàæàæàæà 0.8ìæà ìæà ìæà ìæà ìæàìæàìæàìæàìæàìæà 0.8 0.8
ì ì ì ì ìììììì
ABÈL 0.4 ABÈL 0.4ò ò ò ò òòòòòò ABÈL 0.4 ABÈL 0.4
IH2 ò ò ò ò òòòòòòIH IH2 IH
ô
ô ô ô ôôôôôô
0çô çô çô çô çôçôçôçôçôçô 0ç ç ç ç çççççç 0 0
0 1 2 0 1 2 0 Π Π 0 Π Π
4 2 4 2
LnHrL LnHrL Θ Θ
FIG. 3: (Color online) Behavior of the mutual information based FIG. 4: (Color online) Behavior of the mutual information based
on the 2-Re´nyi entropy, I2(A|B) (left), and the von Neumann en- on the 2-Re´nyi entropy, I2(A|B) (left), and the von Neumann en-
tropy,I(A|B)(right),asfunctionsofthelogarithmoftheinter-spin tropy,I(A|B)(right),asfunctionsoftheground-statesuperposition
distancerfordifferentsuperpositionsintheferromagneticphaseof parameter θ, with the parametrization u = cosθ, v = sinθ, for
theone-dimensionalXY modelat(γ = h = 0.5). Blackcircles: two extreme values of the distance r between A and B. Dashed
u = 1, v = 0 (symmetric state); red squares: u = cos(0.05π), black curve: r = 1. olid red curve: r = ∞. The absolute mini-
v = sin(0.05π);greendiamonds: u = cos(0.1π),v = sin(0.1π); mumisalwaysrealizedatθ=π/4,correspondingtothemaximally
brownup-triangles: u = cos(0.15π),v = sin(0.15π);bluedown- symmetry-breaking ground state. At this point, both measures of
triangles: u = cos(0.2π), v = sin(0.2π); violet empty circles: mutualinformationvanishexactlyforr=∞.
u = cos(0.25π), v = sin(0.25π)(maximallysymmetry-breaking
ground state). The two definitions feature the same qualitative be-
havior;inparticular,theybothvanishinandonlyinthemaximally
asthedifferencebetweenmutualinformation-whichaccounts
symmetry-breakinggroundstates.
forallcorrelations,bothclassicalandquantum-andtheopti-
mal classical correlations between A and B, by maximizing
over all the measurement on B: C = max [S(ρˆ )−
Due to the normalization constraint, |u|2 +|v|2 = 1, the AB {Bˆk} A
S (ρˆ |{Bˆ })]. It is therefore sensible to verify whether
fractioninEq.(15)issemi-positivedefinedandvanishesonly C AB k
it may be a good quantifier of macroscopic quantum coher-
eitheratm =0,i.e.inthedisordered,classicalparamagnetic
ν
phase, or when (uv∗ +vu∗) = 1. Therefore, in the ordered ence. Thelong-rangepairwisequantumdiscordbetweentwo
spins in the ground state of the one-dimensional XY chain
phase the only ground states with vanishing long-range mu-
hasbeenrecentlyinvestigatedinRefs.[8,12–14]. Itturnsout
tualinformation,andhencevanishingmacroscopicentangle-
thatsuchquantityfeaturesalong-rangebehaviorquiteanalo-
mentandcorrelations,arethemaximallysymmetry-breaking
goustothatofthemutualinformation,withthecrucialdiffer-
ground states. At the other end of the spectrum, it is easy to
ence that it vanishes identically in all possible ground states
verifythatthemaximumofI ,forafixedvalueoftheparam-
2
as m → 0. From a mathematical point of view this can
etersm andm ,isalwaysachievedinthetotallysymmetric µ
µ ν
beeasilyexplainedconsideringthat, insuchacase, thetwo-
(even)andantisymmetric(odd)states,theabsolutemaximum
spin reduced density matrix in the symmetric ground states
being obtained for m = 0 and m = 1. Finally, since the
µ ν
atasymptoticallylargeinter-spindistanceisindistinguishable
one-dimensionalXY modelallowsfortheexactevaluationof
fromtheoneobtainedbythesymmetry-breakingGibbsstates
allentropiesintheRe´nyihierarchy,inFig.3wecomparethe
atzerotemperature.
mutual information based on the 2-Re´nyi and the one based
Thelocalizableentanglementinamany-bodypurestateis
onthevonNeumannentropy,findingcompletequalitativeand
definedasthemaximalamountofpairwiseentanglementbe-
quantitativeagreement. Inparticular,theybothvanishin,and
tweentwospinsi,jatarbitrarydistancethatcanbeachieved,
onlyin,themaximallysymmetry-breakinggroundstates. Fi-
nally,usingtheparametrizationu=cosθ,v =sinθ,inFig.4 on average, by performing generalized measurements on all
other spins [50, 51]. This naturally defines an entanglement
wealsoreportandcompareforcompletenessthebehaviorof
the2-Re´nyibasedandthevonNeumannbasedmutualinfor- length χ that diverges in symmetric states. Numerics show
mation as functions of the superposition parameter θ for the thatinthemaximallysymmetry-breakinggroundstatesofthe
two extreme cases of nearest-neighbor distance r = 1 and XY chainthelocalizableentanglementbehaveslikethecon-
asymptoticdistancer =∞,findingagainperfectagreement. nected spin-spin correlation functions Qxx that are bound to
decay exponentially at long distance. Therefore long-range
pairwise entanglement defined via the localizable entangle-
ment features a behavior quite similar to that of the mutual
IV. COMPARISONWITHOTHERINDICATORSOF information. However,thisisjustpairwiseentanglement(al-
MACROSCOPICCOHERENTSUPERPOSITIONS
thoughatlongdistance),soitdoesnotquitecapturethenotion
ofamacroscopicsuperposition. Moreover, themutualinfor-
Quantum discord [31, 49] is a measure of quantum cor- mation is more readily generalized to thermal mixed states,
relations more general than entanglement that may exist in theGibbsstatesatfinitetemperature,forwhichtheevaluation
mixedquantumstates,includingseparableones. Itisdefined ofthemutualinformationpresentsnoparticulardifficulty.
7
V. CONCLUSIONSANDOUTLOOK metry breaking below a critical temperature. At thermal
equilibrium the system will be described by the Gibbs state
We investigated macroscopic entanglement [52] through ρeq =Z−1(cid:80)i,ae−βEi|Ei(cid:105)(cid:104)Ei|a, where with a we have ex-
thebehaviorofthequantummutualinformationbetweentwo plicitlylabeleddifferentsectors. Belowacriticaltemperature
macroscopically separated blocks of dynamical variables in Tc,ifalabelsthesectorswithbrokensymmetry,spontaneous
thegroundstateofmany-bodysystemsfeaturingspontaneous symmetrybreakingmeansthatineverysinglerealizationthe
symmetry breaking. This quantity detects macroscopic to- labelawillbefixedbytheinitialconditions. Therefore, our
tal correlations, including entanglement. The main result of statement is that Gibbs states obtained by fixing a feature
this paper is that in the entire phase with broken symmetry, theleastlong-rangemutualinformationcomparedtoallother
the symmetry-breaking states have vanishing long-distance non-maximallysymmetry-breakingGibbsstates[46].
mutual information, while the latter remains finite for any
nonmaximallysymmetry-breakingsuperposition,attaininga Fromadifferentperspective,weremarkthatmany-bodylo-
maximum for the totally symmetric states. This fact is easy calization has recently become a subject of great interest for
toprovewhenconsideringsymmetry-breakingstatesthatare thecondensedmattercommunity. Systemswithstrongdisor-
completely unentangled (fully factorized), whose symmetric derfeaturingmany-bodylocalizationfailtothermalizeandto
superpositions are GHZ states. In order to prove this feature obey the eigenstate thermalization hypothesis [55], because
in the entire ordered phase, a much more challenging task, their eigenstates are more weakly entangled than in typical
wefollowedastrategybasedontwoingredients: (i)adopting non-integrable systems. It would be interesting to see how
measures of mutual information based on the 2−Re´nyi and the techniques developed in the present work might help in
onthevonNeumannentropies,and(ii)exploitinglocalityre- providing rigorous results regarding the clustering of mutual
sultsaboutquasiadiabaticcontinuationofquantumstatesde- information in such systems. In fact, our techniques can be
rivedbyusingtheLieb-Robinsonbounds[43,53,54]. Inthis used to investigate also systems that involve long-range en-
waywewereabletoprovethatspontaneoussymmetrybreak- tanglement and correlations, such as topologically ordered
ingselectsthemany-bodystateswithvanishinglong-distance states [56, 57] and their resilience in the presence of pertur-
mutualinformation,andthusmacroscopicallyleastentangled, bationsoratfinitetemperature[26,58,59].
andthereforemostclassical. Acknowledgments.—Weacknowledgeusefulconversations
In perspective, we are concerned with the investigation withF.Verstraete. Wealsoacknowledgeusefulcommentsby
of several open problems. In particular, it would be in- G.AdessoandP.Zanardionanearlierdraftofthemanuscript.
teresting to extend our analysis to the case of subsys- AH thanks the National Basic Research Program of China
tems of arbitrarily variable size, in order to observe pos- Grant2011CBA00300,2011CBA00301theNationalNatural
sible threshold effects, and to generic classes of symmetry Science Foundation of China Grant No. 61361136003 and
groups [46]. Moreover, we are interested in studying the No. 11574176. S.M.G.andF.I.acknowledgetheFP7Coop-
case of globally mixed states [24] and, in particular, equi- erationSTREPProjectEQuaM-EmulatorsofQuantumFrus-
libriumthermalstatesofmodelsfeaturingspontaneoussym- tratedMagnetism(GANo. 323714).
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