Canonical and micro-canonical typical entanglement of continuous variable systems

0.29 MB·English

by  A. Serafini

Canonical and micro-canonical typical entanglement of
continuous variable systems
7 A.Serafini,O.C.O.Dahlsten,D.Gross,andM.B.Plenio
0 InstituteforMathematicalSciences,53Prince’sGate,ImperialCollegeLondon,London
0 SW72PG,UKandQOLS,BlackettLaboratory,ImperialCollegeLondon,LondonSW7
2 2BW,UK

n
a Abstract. We present a framework, compliant with the general canonical principle of
J statistical mechanics, to define measures on the set of pure Gaussian states of continuous
0 variable systems. Within such a framework, we define two specific measures, referred to
1 as‘micro-canonical’ and‘canonical’, andapply them tostudysystematically thestatistical
propertiesofthebipartiteentanglementofn-modepureGaussianstates(asquantifiedbythe

entropyofasubsystem). Werigorouslyprovethe“concentrationofmeasure”aroundafinite
2
average,occurringfortheentanglementinthethermodynamicallimitinboththecanonicaland
v
themicro-canonicalapproach.Forfiniten,wedetermineanalyticallytheaverageandstandard
1
deviationoftheentanglement(asquantifiedbythereducedpurity)betweenonemodeandall
5
the other modes. Furthermore, wenumerically investigate more general situations, clearly
0
showingthattheonsetoftheconcentrationofmeasurealreadyoccursatrelativelysmalln.
1
0
7
0
/
h
p
-
t
n
a
u
q
:
v
i
X
r
a
Canonicalandmicro-canonicaltypicalentanglementofcontinuousvariablesystems 2
1. Typicalentanglementinquantuminformationtheory
DuetotheexponentiallyincreasingcomplexityoftheHilbertspacesofmultipleconstituents,
a complete theoretical characterisation of the entanglement of general quantum systems of
many particles turns out to be a daunting task [1]. A viable approach towards such a
characterisation consists in focusing on the “typical”, statistical properties of the quantum
correlations of multipartite systems, when the states of the system are assumed to be
distributed accordingto a particular‘measure’. This strategy is firstly aimed at simplifying
theproblemathandbyrestrictingattentiononthetypical(andthus,inasensetobeprecisely
specifiedinthefollowing,“overwhelminglylikely”)featuresoftheentanglementofasystem
whosestateisapttobedescribedbythechosenmeasures. Furthermore,thiskindofanalysis
isabletoshedlightonthegeneralpropertiesoftheentanglementofphysicalsystems.
For finite dimensional quantum systems, a natural, ‘uniform’ measure on pure states
emergesfromthe“Haar”measureoftheunitarygroup(i.e.,fromtheleft-andright-invariant
measure underapplicationof anyunitary transformation),whose elementsallow to retrieve
anystatewhenappliedtoanothergivenstartingpurestate. Onsuchgrounds,awelldefined
typicalentanglementoffinitedimensionalsystemscanbeaddressedandanalysed. Original
studiesinthisdirectionwereundertakenwellbeforethedevelopmentoftheformaltheoryof
entanglement developed in quantum information science [1]: In 1978, Lubkin considered
the expected entropy of a subsystem when picking pure quantum states at random from
the uniform measure[2]. Let us recall that the von Neumann entropy S = trρlnρ of a
−
subsystem in state ρ properlyquantifies, for globallypurestates, the entanglementbetween
thesubsystemandtheremainderofthesystem. Lubkinshowedthatoneexpectsthisquantity
to be nearly maximal. Pagels and Lloyd arrived later at the same qualitative conclusions,
following an independent line of thought [3]. Their work was expanded by Page, who
conjecturedanexactformulafortheaverageentropyofasubsystemS [4],reading
m,n
mn
1 m 1
S = − , (1)
m,n
k − 2n
k=n+1
X
foraquantumsystemofHilbertspacedimensionmninarandompurestate,andasubsystem
ofdimensionm n. ThisrelationwaslaterprovenbyFoongandKanno[5].
≤
This general line of enquiry was revisited and considerably extended in the setting
of quantum information theory by Hayden, Leung and Winter in Ref. [6], where they
extensively studied the ‘concentration of measure’ around the average of the entanglement
probabilitydistributionwithincreasingn. Theyalsopointedoutthatthisstudymayprovide
a way of simplifying the theory of entanglement which contains a plethora of locally
inequivalent classes. In other words, as already mentioned, restricting statements to the
“typicalentanglement”allowsonetoignoreseveralunessentialcomplications.Recentresults
on the physical interpretation of Page’s conjecture can be found in Refs. [7, 8], where it
is proven that a circuit of elementary quantum gates on a quantum circuit is expected to
maximallyentanglethestatetoafixedarbitraryaccuracy,withinanumberofgatesthatgrows
onlypolynomiallyinthenumberofqubitsoftheregister.
A further simplification in the analysis of the entanglement can be achieved by
considering the typical entanglement of ‘particularly relevant’ (according to the specific
problem at hand) subsets of states. For example, it has been found that ‘stabilizer states’
(a countable set of states playing a central role in quantum error correcting codes) are also
typically maximallyentangled [9, 10], similarly to the set of all states. Such investigations
are interesting per se, as they unveilthe potentialand limitations hidden in the adoption of
Canonicalandmicro-canonicaltypicalentanglementofcontinuousvariablesystems 3
restrictedclassesofstatesand,furthermore,provideuswithamoredetailedunderstandingof
theentanglementpropertiesofthetotalstatespace.
Hereweexpandtheseconsiderationsintotherealmofcontinuousvariables,
i.e. ofquantumsystemsdescribedbypairsofcanonicallyconjugatedobservableswith
continuousspectra. Suchsystems,rangingfrommotionaldegreesoffreedom
ofparticlesinfirstquantisationtobosonicfieldsinsecondquantisation,areubiquitous
toallareasofquantumphysics,beingprominentinquantumoptics(astheyembodythelight
fieldinsecondquantisation),atomicphysics(notably,inthedescriptionofatomicensembles),
quantumfieldtheory(astheyencompassanybosonicfield),inadditiontotheircrucialrolein
molecularandatomicphysics.
Quantum systems described by operators with continuous spectra live in infinite
dimensionalHilbertspaces. Therefore,a firstnaivetryinthisdirectioncouldbetotakethe
infinitedimensionallimitofEq.(1). PageshowedthatEq.(1)impliesthatS lnm m
m,n ≃ −2n
for1 m n. Underthatrestriction, andnotingthatthemaximalentanglementis lnm,
≪ ≤
wecanthenmaketheobservationthattheratiooftheentanglementaveragetothemaximum
tends to unity as n , but that the two quantities both diverge logarithmically. From
−→ ∞
this perspective, the entanglement could be said to be typically infinite in the continuous
variablesetting. Eventhoughmathematicallyreasonable(inthelimit’ssense),thisstatement
is definitely questionable, physicallyand practically. In fact, in any practical situation, one
willdealwithafinitetotalenergyorwithfinite“temperatures”(bothquantitieswillbedefined
preciselyinourtreatment),whereasinfinitelyentangledstatesrequireaninfiniteenergytobe
created. In this paper, we shall restrict our attention on pure Gaussian states whose typical
entanglement we shall study under two different measures, both inspired by arguments of
thermodynamicalnatures,butapttodescribedifferentsituations. Thepreviouslymentioned
divergences,stillpotentiallyemergingintheGaussiansetting, willbetamedbyintroducing
properprescriptionswhichwillgenerallyaffecttheenergyofthesystem,byimposingeither
asharpupperboundoranexponentiallydecayingdistributionofenergies(inthisrespect,see
also Ref.[11]foradetaileddiscussionoftheimpactofsimilarconstraintsonentanglement
measures). We will show that such prescriptions induce the occurrence of a finite typical
entanglementinthelimitofaninfinitenumberoftotalconstituents.Evenforafinitenumber
oftotaldegreesoffreedomandfiniteupperboundtotheenergy–entailingtheexistenceofa
finitemaximalentaglement–thetypicalentanglementconcentratesaroundavaluewhichis
welldistantfromtheallowedmaximum.
Notice thatGaussian states are certainlythe most prominentclass of states notonlyin
quantum informationwith continuousvariables, but also, more broadly, in quantum optics,
astheycanbegeneratedandmanipulatedwithrelativeease(evenashighlyentangledstates
[12,13,14]),canbeusedfortheimplementationofquantumcommunicationandinformation
protocols[15] and serve as a powerfultesting groundfor the theoreticalcharacterisation of
entanglementproperties[16].
The micro-canonical measure, which we will apply here to the study of the typical
entanglement, has been already employed in the analysis of the quantum teleportation of
Gaussianstateswithgenericsecondmoments[17].Inparticular,themicro-canonicalaverage
quantum fidelity and a corresponding “classical threshold”, have been evaluated for the
teleportation of states with null first moments and arbitrary second moments under the
standard continuous variable teleportation protocol (see, e.g., [15] for a description of the
scheme).
Letusfinallymentionthatadefinitionofmicro-canonicalaverageentanglementhasbeen
veryrecentlyaddressedforfinitedimensionalsystemsaswell[18],withamajoremphasison
thepossibilityofreducingtime-averagestoensemble-averages.
Canonicalandmicro-canonicaltypicalentanglementofcontinuousvariablesystems 4
This paper is organised as follows. In Section 2 we review some preliminary facts
about Gaussian states and set the notation. In Section 3 we review the definition of the
micro-canonical measure on pure Gaussian states, already introduced in [17], completing
our previous analysis with the inclusion of comments and mathematical details previously
omitted, and we extend the existing framework to encompass a ‘canonical’ measure as
well. Section4 containsa rigorousproofofthe ‘concentrationofmeasure’,commontothe
two measures introducedhere, i.e. of the fact that the entanglementprobabilitydistribution
concentratesinthethermodynamicallimit, aroundafinite‘thermal’average,awayfromthe
allowed maximum. Even though this result had been anticipated in Ref. [17], this proof is
originalandaddsfurtherinsightintothematter. Section5presentsadetailedstudyaboutthe
typicalentanglementofpureGaussianstateswithfinitenumberofdegreesoffreedom,where
bothanalyticalfindingsandnumericalevidencesarereported. Conclusionsandoutlookare
found in Section 6. Three appendices complement the work, one of which (AppendixB)
contains the most technical steps needed to prove the concentration of measure, while in
AppendixA a specific Baker-Campbell-Hausdorffrelation is derived, and in AppendixC a
derivationoftheexpressionofthemaximalentanglementforgivenenergyispresented.
2. Preliminaries
We consider bosonic continuous variable (CV) quantum mechanical systems described
by n pairs of canonically conjugated operators xˆ ,pˆ with continuous spectra, like
j j
{ }
motional degrees of freedom of particles in first quantisation or bosonic field operators
in second quantisation. Grouping the canonical operators together in the vector Rˆ =
(xˆ ,...,xˆ ,pˆ ,...,pˆ )T allows to express the canonical commutation relations (CCR) as
1 n 1 n
[Rˆ ,Rˆ ]=2iΩ ,wherethe‘symplecticform’Ωisdefinedas
j k jk
0
Ω= n 1n ,
0
n n
(cid:18) −1 (cid:19)
0 and standingforthenullandidentitymatrixindimensionn.
n n
1
Anystateofann-modeCVsystemisdescribedbyapositive,trace-classoperator̺. For
anystate̺,letusdefinethe2n-dimensionalvectoroftheexpectationvalues(“firstmoments”)
ofthecanonicaloperatorsR(withentriesR )as
j
R Tr[Rˆ ̺]
j j
≡
andthe2n 2nmatrixofsecondmoments,or“covariancematrix”,σ(withentriesσ )as
i,j
×
σ Tr[ Rˆ ,Rˆ ̺]/2 Tr[Rˆ ̺]Tr[Rˆ ̺].
i,j i j i j
≡ { } −
Also,throughoutthepaper,wewillrefertothe‘energy’ofastate̺astotheexpectation
valueoftheoperatorHˆ = n (xˆ2 +pˆ2). Thisdefinitioncorrespondstotheenergyofa
0 j=1 j j
freeelectromagneticfieldintheopticalscenario(andtodecoupledoscillatorsinthegeneral
P
case). Inourconvention,asdeterminedbythefactor2appearingintheCCR,thevacuumofa
singlemodehascovariancematrixequaltotheidentity(thussimplifyingsignificantlyseveral
expressions),withenergy2(theadoptedenergyunitis~ω/4foramodeoffrequencyω). The
energyisdeterminedbyfirstandsecondmomentsaccordingto
Tr(̺Hˆ )=Tr(σ)+ R 2 ,
0
k k
where R istheusualeuclideannormofthevectorR.
k k
Canonicalandmicro-canonicaltypicalentanglementofcontinuousvariablesystems 5
GaussianstatesaredefinedasthestateswithGaussiancharacteristicfunctionsandquasi-
probabilitydistributions,definedoveraphasespaceanalogoustothatofclassicalHamiltonian
dynamics. Aswellknown,apurestate ψ isGaussianifandonlyifitcanbeobtainedby
G
| i
transforming the vacuum 0 under an operation generated by a polynomial of the second
| i
orderinthecanonicaloperators.Informulae(uptoanegligibleglobalphasefactor):
ψ =Gˆ 0 ei(RˆTARˆ+RˆTb) 0 , (2)
G A,b
| i | i≡ | i
where A and b are, respectively, a real 2n 2n matrix and a real 2n-dimensional vector.
Because of the CCR and of the unitarity ×of Gˆ ei(RˆTARˆ+RˆTb), A can be chosen
A,b
≡
symmetric, without loss of generality, while b is a generic real vector. A and b determine
theCMandthesecondmomentsoftheGaussianstate,thuscompletelydeterminingit.
Theunitaryoperatorcanalwaysberewrittenas(seeAppendixA)
Gˆ = ei(RˆTARˆ)ei(RˆTMb), (3)
A,b
for some matrix M (as shown in AppendixA, M = ΩA−1( e4AΩ)/4 for invertible
2n
1 −
A’s). Firstandsecondorderoperationscanthusbegenerally‘decoupled’.
First order operations correspond to local displacements in phase space. While such
operationsdonotaffectlocalentropies(andthustheentanglement)ofmultipartitestates,they
doaffectthe energyofthestates, whichwillplayacentralrolein whatfollows. Moreover,
let us notice that the group of these transformations is non-compact, being isomorphic to
the abelian 2n under the addition composition rule. In the following, we will show how
R
the first moments can be consistently incorporated in the presented framework. However,
becausetheirinclusionin thestudyof thestatistical propertiesofthe entanglementis justa
technicality (addingno significantinsight), we will set them to zero in the investigationsto
come.
As for second order transformations, determined by the matrix A, they can be
convenientlymappedintothegroupSp ofrealsymplectictransformations,actinglinearly
2n,
in phase space (as second order transforRmations acting on the Hilbert space make up the
multi-valuedmetaplectic representationof the symplectic group[19]). Recall that a matrix
S belongstothesymplecticgroupSp ifandonlyifitpreservestheantisymmetricform
2n,
Ω: S SL(2n, ) : S Sp RSTΩS = Ω. Let us also recall that a symplectic
2n,
transfo∈rmation SRacts by co∈ngruencRe o⇔n a covariance matrix σ: σ STσS. Of course,
7→
symplectic transformationscan in generalaffect both the entanglementand the energyof a
state. Thealgebraofgeneratorsofthesymplecticgroupiscomprisedofallthematricesthat
can be written as ΩJ, where J is some 2n 2n symmetric matrix [20] (in this notation,
×
generatorsare not complexified, so that S = eΩJ). Such generatorsdo not have a definite
symmetry (i.e., they are not necessarily symmetric or antisymmetric). Choosing a basis
of the algebra such that each generator of the basis is either symmetric or antisymmetric,
allowsonetodistinguishbetweenacompactsubgroupK(n)=Sp SO(2n)(spawned
2n,
by antisymmetric generators) and a non-compact subgroup (arisingRfr∩om skew-symmetric
generators). Notice also that, since compacttransformationsareindeedorthogonal,theydo
notaffecttheenergyofthestatestheyactupon(explicitly,Trσ and R 2arebothinvariant
k k
underphasespace“rotations”).
Remarkably, the subgroup K(n) is isomorphic to U(n). Because this fact will be
exploited throughout the whole work, we shall sketch its proof here. Let us define the
transformationOby
X Y
O = ,
W Z
(cid:18) (cid:19)
Canonicalandmicro-canonicaltypicalentanglementofcontinuousvariablesystems 6
where X, Y, W and Z are n n real matrices. It is straightforward to show that this
transformationissymplecticand×orthogonalifandonlyifZ =X,W = Y,XTX+YTY =
andXTY YTX =0,sothat −
1 −
X Y
O = . (4)
Y X
(cid:18) − (cid:19)
Now, let U = X +iY be a matrix with real part X and imaginary part Y. The unitarity
condition on U corresponds exactly to the previous two conditions on X and Y, thus
demonstrating the existence of a bijective mapping from U(n) to K(n). The preservation
ofthecompositionrulecanbestraightforwardlycheckedout. Incidentally,thisisomorphism
impliesthatK(n)hasn2independentparameters.
Let us rephrase Eq. (2) to give a transparent parametrisation of pure Gaussian states
in phase space terms, by considering the action of first and second order operations on the
covariancematrixandonthefirstmoments:
σ =STS, with S Sp , R 2n (5)
2n,
∈ R ∈R
(recall that, in our units, the covariance matrix of the vacuum is the identity). Indeed,
becauseofthepeculiarnatureoftheircharacteristicfunctions,Gaussianstatesarecompletely
determinedbyfirstandsecondmomentsofthecanonicaloperators.
More generally,let usalso recallthatthe CM Σ of any, pureor mixed, Gaussian state
canbewrittenas
Σ=STνS , (6)
whereS Sp andν = diag(ν ,...,ν ,ν ,...,ν )isadiagonalmatrixwithdouble-
2n, 1 n 1 n
valued ei∈genvalueRs called the “Williamson normal form” of Σ [21, 22] (corresponding to
the normal-modes decomposition of positive definite quadratic Hamiltonians). The real
quantities ν are referred to as the ‘symplectic eigenvalues’ of Σ and can be computed
j
astheeige{nva}luesofthematrix iΩΣ. Thesymplecticeigenvaluesholdalltheinformation
| |
about the entropic quantities of the Gaussian state in question. In particular, the ‘purity’
µ Tr̺2oftheGaussianstate̺withCMΣisdeterminedas
≡
n
µ=1/ ν =1/√DetΣ,, (7)
j
j=1
Y
whilethevonNeumannentropyS Tr(̺ln̺)reads
≡−
n
S = h(ν ), (8)
j
j=1
X
withthe‘entropicfunction’h(x)givenby
x+1 x+1 x 1 x 1
h(x)= log ( ) − log ( − ). (9)
2 2 2 − 2 2 2
Beingthe eigenvaluesof iΩΣ, thesymplecticeigenvaluesare continuouslydeterminedby
| |
‘symplecticinvariants’(i.e. byquantitiesdependingontheentriesoftheCMinvariantunder
symplectic transformations), defined as the coefficients of the characteristic polynomial of
suchamatrix[24,25]).Thisobservationwillbeusefullateron.
Canonicalandmicro-canonicaltypicalentanglementofcontinuousvariablesystems 7
3. MeasuresonthesetofpureGaussianstates
The present section is devoted to the definition of consistent measures on the set
of pure Gaussian states, introducing a broad framework motivated by fundamental
statistical arguments. We will review the construction of the ‘micro-canonical measure’,
already introduced in Ref. [17], complementing such earlier studies with discussions and
mathematical details. Furthermore, within the same general framework, we will present a
novel,“canonical”measure,thusextendingourprevioustreatment.
A ‘natural’ measure to pick would be one invariantunder the action of the operations
whichgeneratethesetofstateswearefocusingon.Intheprevioussection,wehaveanalysed
such a set of operations for pure Gaussian states, showing that it amounts to symplectic
operations and displacements. One would thus be tempted to adopt the left- and right-
invariant measure (i.e., the Haar measure) over such groups. Unfortunately, because the
symplecticgroupisnoncompact,theexistenceofaHaarmeasureonthewholegroup[from
which a measure for pure Gaussian states could be derived via Eq. (5)] is not guaranteed.
Notably, even if such a measure could be constructed, it would not be normalisable, giving
rise to distributions with unboundedstatistical moments. Moreover, some prescription has
obviously to be introduced also to handle the first moments which are, in general, free to
varyinthenon-compactR2n (noticethattheEuclideanvolumeisobviouslyinvariantunder
left translations but is not a proper measure in the space of first moments because it is not
normalisable,duetothenon-compactnessofR2n).
To copewith suchdifficultieswe willintroduce,in analogywith statistical mechanical
treatments,assumptionsontheenergyofthestatesunderexamination,whichwillconstitute
our‘privileged’physicalobservable(inasensetobeelucidatedinthefollowing). Aproper
structure to introducea measure is inspired by a well knowndecompositionof an arbitrary
symplectictransformationS:
S =O′ZO, (10)
where O,O′ K(n) = Sp(2n, ) SO(2n) are orthogonalsymplectic transformations,
∈ R ∩
while
Z =Z′ Z′−1 , (11)
⊕
where Z′ is a diagonal matrix with eigenvalues z 1 j. The set of such Z’s forms a
j
≥ ∀
non-compactsubgroupofSp (correspondingtolocalsqueezings),whichwillbedenoted
2n,
byZ(n). Thevirtueofsucha dRecomposition,knownas“Euler”(or“Bloch-Messiah”[23])
decomposition,isimmediatelyapparent,asitallowsonetodistinguishbetweenthedegreesof
freedomofthecompactsubgroup(essentially‘angles’,rangingfrom0to2π,whichmoreover
do not affect the energy) and the degrees of freedom z ’s with non-compact domain. In
j
particular,applyingEulerdecompositiontoEq.(5)leadsto
σ =OTZ2O. (12)
Duetotherotationalinvarianceofthevacuuminphasespace,thenumberoffreeparameters
ofapureGaussianstateofann-modesystemisthusn2+3n(takingthe2nindependentfirst
momentsintoaccount).
Quite naturally, we shall assume the n2 parameters of the transformation O to be
distributedaccordingtotheHaarmeasureofthecompactgroupK(n),carriedoverfromU(n)
throughtheisomorphismdescribedbyEq.(4). Thesetofsuchparameterswillbecompactly
referredtoasϑ,whilethecorrespondingHaarmeasurewillbedenotedbydµ (ϑ).
H
WehavethusidentifiedthevariableswhichparametriseanarbitrarypureGaussianstate
and imposed a distribution on a subset of them. A ‘natural’ measure has yet to emerge
Canonicalandmicro-canonicaltypicalentanglementofcontinuousvariablesystems 8
for the non-compact variables z and R . In order to further constrain the choice of
j j
{ } { }
measuresonsuchvariablesweshallinvokenowafundamentalstatisticalargument. Intheir
kinematicalapproachto statistical mechanics[26], Popescu, Short and Winter introduceda
generalprinciple,whichtheyrefertoasgeneralcanonicalprinciple,statingthat
“Givenasufficientlysmallsubsystemoftheuniverse,almosteverypurestateofthe
universeissuchthatthesubsystemisapproximatelyinthe‘canonicalstate’̺ .”
c
The ‘canonical state’ ̺ is, in our case, the local reduction of the global state picked
c
from a distribution of states with maximal entropy under the constraint of a maximal total
energyE. That is, quite simply, a “thermalstate”, which is a Gaussian state with nullfirst
moments and CM σ = (1 + T/2) . Here the ‘temperature’ T is defined by passage to
c
1
the“thermodynamicallimit”,thatisforn andE ,(E 2n)/n T (assuming
→∞ →∞ − →
k =1fortheBoltzmannconstant).Foreaseofnotation,inthefollowing,thesymbol will
B
≃
imply thatthe equalityholdsin the thermodynamicallimit, e.g.:(E 2n)/n T. Notice
− ≃
that we have required here the introduction of a maximum energy E or, alternatively, of a
temperatureT. Inpointoffact, suchrequirementsarenecessarytohandlethenon-compact
partofthesymplecticgroup.Notealsothat,inprinciple,twooptionsareopeninthisrespect,
asonecanintroduceeitheranupperboundtotheenergyoratemperature: essentially,these
twodistinctoptionscharacterisethetwodistinctapproaches(micro-canonicalandcanonical)
whichwewilldetailinthenextsection.
Because the canonical state ̺ is Gaussian with vanishing first moments, the general
c
canonical principle can be fully incorporated into our restricted (Gaussian) setting. As we
haveshowninRef.[17],thecompliancewiththegeneralcanonicalprincipleenforcesarather
stringentrestrictiononthedistributionofthenon-compactvariables.Inparticular,thegeneral
canonical principle is always satisfied if, in the thermodynamical limit, such variables are
independentandidenticallydistributed(i.i.d)[27]. Tokeepourexpositionlighterandmore
readable,wewillnotrepeatherethetechnicalderivationofthisimplication. Actually,itwill
be entirely subsumed, a posteriori, by the derivation of the “concentration of measure” in
Sec.4.
Before movingon with the definition ofspecific measures, let us commentonthe first
moments R . Thegeneralcanonicalprincipleimposesthat, in thethermodynamicallimit,
{ }
theirdensityofprobabilityp′(R)tendtoaδ-distributioncenteredin0(infact,thecanonical
statehasvanishingfirstmoments). Asuitableexampleisp′(R) = (nλ/π)nexp( nλ R 2)
− k k
forsomeconstantλ(noticethat R 2 isthefirstmoments’contributiontotheenergy). Let
k k
usremarkthatthisclassofdistributionsencompassestheonesusuallyadoptedforcoherent
states,inthecomputationofclassicalteleportationthresholds[28,29].Fromnowon,wewill
justsetthefirstmomentstozero:theycanbecoherentlyincorporatedintoourgeneralpicture
followingtherecipegivenabove.
Letusnowturnto secondmomentsandsumupourlineofthoughtsofar: inspiredby
mathematicalconsiderationsandguidedbyphysicalarguments,wehavedefinedadistribution
for the “compact” degrees of freedom ϑ (essentially, their Haar measure) and specified a
prescriptionfor the non-compactparametersz ’s. Severalchoicesare then possible, within
j
thisprescription,todealwiththevariablesz ’s. We willdescribenowindetailtwoofsuch
j
choices,whichwewillthenapplytostudythetypicalentanglement.
3.1. Micro-canonicalmeasure
As a first approach, we will introduce a micro-canonical measure on the class of n-mode
pure Gaussian states with an energy upper boundedby E. Notice that such a restriction is
Canonicalandmicro-canonicaltypicalentanglementofcontinuousvariablesystems 9
essentiallyequivalenttofixingthetotalenergyof(n+1)-modestatestoE+2(infact,inthe
latterinstance, theenergyoftheadditionalmodeisnotindependentandmerely‘makesup’
toreachthefixedtotalamount).Notably,thetwoapproachesareobviouslyindistinguishable
inthethermodynamicallimit.
Noticethattheparametersz ,whosedistributionislefttodefine,determinetheenergy
j
E pertaining to each decoupled mode j of the Euler decomposition, according to (see
j
Eq.(12))
1
E =z2+ . (13)
j j z2
j
Here, we will assume a Lebesgue (‘flat’) measure for the local energies E ’s (uniquely
j
determining the squeezings z ’s, as z 1), inside the region Γ = E : E E
j j E
≥ { | | ≤ }
boundedbythelinearhypersurfaceoftotalenergyE (here,E = (E ,...,E )denotesthe
1 n
vectorofenergies,withallpositiveentries,while E = n E ). Moreexplicitly,denoting
| | j=1 j
bydp (E)theprobabilityoftheoccurrenceoftheenergiesE,onehas
mc
P
dp (E)= dnE dE ...dE if E Γ ,
mc 1 n E
N ≡N ∈
dp (E)=0 otherwise, (14)
mc
where isanormalisationconstantequaltotheinverseofthevolumeofΓ .Noticethatsuch
E
N
aflatdistributionistheonemaximisingtheentropyintheknowledgeofthelocalenergiesof
thedecoupledmodes.Inthisspecificsensesuchvariableshavebeenprivileged,onthebasisof
bothmathematical(theEulerdecomposition)andphysical(analogywiththemicro-canonical
ensemble) grounds. Let us also mention that, as will become apparent in the next section,
employing the variables E leads to a remarkable simplification of the expression of the
j
{ }
averagesoverthe Haar measure of the compactsubgroup. While purelyformal, thisaspect
yieldssomesignificantinsightintotheprivilegedroleofsuchvariablesincharacterisingthe
statisticalpropertiesofphysicalquantities.
The micro-canonicalaverageQ (E) overpureGaussian states at maximalenergyE
mc
ofthequantityQ(E,ϑ)determinedbythesecondmomentsalonewillthusbedefinedas
Q (E)= dµ (ϑ) dEQ(E,ϑ), (15)
mc H
N
Z ZΓE
where the integrationoverthe Haar measureis understoodto be carried outoverthe whole
compactdomainofthevariablesϑ. Moreexplicitly,theintegralovertheenergiesE canbe
recastas
E−2(n−1) E−Pnj=−11Ej
dE = dE ... dE (16)
1 n
N
ZΓE Z2 Z2
(each energyis lower boundedby the vacuumenergy,equalto 2 in the conventionadopted
here),determiningthenormalisationas =n!/(E 2n)nandleadingtoamarginaldensity
N −
ofprobabilityP (E ,E)foreachoftheenergiesE givenby
n j j
n E 2 n−1
P (E ,E)= 1 j − . (17)
n j
E 2n − E 2n
− (cid:18) − (cid:19)
Clearly, the energies E are not i.i.d. for finite n. However, as is apparent from Eqs. (16)
j
and (17), in the thermodynamical limit the upper integration extremum diverges for each
Ej while, for the marginal probability distribution, one has Pn(Ej,E) e−EjT−2/T. in
≃
the thermodynamicallimit, the decoupledenergiesare distributed accordingto independent
Boltzmann distributions, with the parameter T playing the role of a temperature, in
Canonicalandmicro-canonicaltypicalentanglementofcontinuousvariablesystems 10
compliance with the equipartition theorem and equivalence of statistical ensembles of
classicalthermodynamics.Thisargumentshowsthatthemicro-canonicalmeasurefulfillsthe
generalcanonicalprinciple.Also,itnaturallybringsustointroducea‘canonical’measureon
thesetofpureGaussianstates.
3.2. Canonicalmeasure
In the ‘canonical’ approach, we will assume for the energies E a probability distribution
dp (E)reading
c
e−(|E|−2n)/T n e−(Ej−2)/T
dp (E)= dE = dE , (18)
c Tn T j
j=1(cid:18) (cid:19)
Y
introducinga‘temperature’T. Thisdistributionmaximisestheentropyontheknowledgeof
the continuousvariablesE ’s forgivenaveragetotalenergyE , suchthatnT = E (the
j av av
latterrelationiseasilyderivedbyapplyingLagrangemultipliers).
The ‘canonical’ average Q (T) over pure Gaussian states at temperature T of the
c
quantityQ(E,ϑ)determinedbythesecondmomentsalonewillthusbedefinedas
e−(|E|−2n)/T
Q (T)= dµ (ϑ) dEQ(E,ϑ), (19)
c H Tn
Z Z
wheretheintegrationovertheenergiesisunderstoodtobecarriedoutoverthewholeallowed
domain(E 2 j).
j
≥ ∀
As already elucidated, the micro-canonical and canonical approaches coincide in the
thermodynamicallimit, as one should expectin analogywith the indistinguishabilityof the
classicalstatisticalensemblesinthethermodynamicallimit.
4. Concentrationofmeasure
In the present section, we will study the statistical properties of the entanglement of pure
Gaussianstatesinthethermodynamicallimitunderthemeasuresintroducedintheprevious
section.Morespecifically,weshallfocusonthebehaviouroftheentanglementofasubsystem
of m modes (as quantified by the von Neumannentropy of the reduction describingsuch a
subsystem),keepingmfixedandlettingthetotalnumberofmodesn . Aswehaveseen,
→∞
thetwomeasurescoincideinthislimit(when,inthemicro-canonicaltreatment,theenergy–
notablytheotherextensivequantityinplay–divergesaswell). Itwillthussufficetoconsider
the canonicalmeasure, and the micro-canonicalaverageswill be retrieved uponidentifying
T (E 2n)/n. Inthissection,theshorthandnotationxwillstandfortheaverageofthe
≡ −
quantityxwithrespecttothecanonicalstate-spacemeasure.Underthepreviousassumptions,
wewilldeterminetheaverageasymptoticentanglementandrigorouslyprovethatthevariance
of the entanglementtends to zero in the thermodynamicallimit. The latter property, which
hadbeenpreviewedinRef.[17],willbereferredtoas“concentrationofmeasure”.
LetusfirstrecallthatthevonNeumannentropySofthem-modereductionisdetermined
by the local symplectic eigenvalues ν , for1 j m of the reduced m-mode CM γ
j
{ ≤ ≤ }
accordingto:
m
S = h(ν ), (20)
j
j=1
X