Strong subadditivity for log-determinant of covariance matrices and its applications

0.35 MB·

by  Gerardo Adesso

LETTERTOTHEEDITOR
Strong subadditivity for log-determinant of covariance
matrices and its applications
6
1
0
GerardoAdesso
2
SchoolofMathematicalSciences,TheUniversityofNottingham,
ul UniversityPark,NottinghamNG72RD,UnitedKingdom
Email:[email protected]
J

2
1 R.Simon

Optics&QuantumInformationGroup,TheInstituteofMathematicalSciences,
]
h C.I.T.Campus,Tharamani,Chennai600113,India
p Email:[email protected]
-
t
n Abstract. We prove that the log-determinant of the covariance matrix obeys the strong
a subadditivity inequality for arbitrary tripartite states of multimode continuous variable
u quantum systems. This establishes general limitations on the distribution of information
q encodedinthesecondmomentsofcanonicallyconjugateoperators.Theinequalityisshownto
[ bestrongerthantheconventionalstrongsubadditivityinequalityforvonNeumannentropyina

classofpuretripartiteGaussianstates.Wefinallyshowthatsuchaninequalityimpliesastrict
2 monogamy-typeconstraintforjointEinstein-Podolsky-Rosensteerabilityofsinglemodesby
v Gaussianmeasurementsperformedonmultiplegroupsofmodes.
6
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2
3 PACSnumbers:03.67.Mn,03.65.Ta,03.65.Ud,42.50.Dv
0
.
1
0
6
1. Introduction
1
:
v The formulation of classical information theory, thanks primarily to the seminal work by
Xi Shannon[1],ledtoaremarkablybroadspectrumofconcreteapplicationsinthelastcentury,
encompassing in particular systems theory, signal processing, communication and control,
r
a complexity and cybernetics. The more recent and still ongoing developments in quantum
information theory [2] have opened the way for even more exciting and unprecedented
scenarios in the processing of information, with quantum technologies well in the course
of revolutionising industrial sectors such as data storage, encryption, sensing, learning, and
computing[3].
Whileclassicalandquantumtheoryradicallydifferinthebasicsetofrulesdetermining
the possible and the impossible for the manipulation of information, the two theories rest
on some common formal pillars with far-reaching physical implications. Crucial in both
cases is in fact the concept of entropy H as quantifier of information (or, more precisely,
of uncertainty), respectively formalised as Shannonentropy H = −(cid:80) P(x)logP(x) fora
X i i i
Strongsubadditivityforlog-determinantofcovariancematricesanditsapplications 2
classicalrandomvariable X takingvalues{x}withprobabilitydistribution P(x),andasvon
i i
NeumannentropyH =−tr(ρlogρ)foraquantumstateρ.‡
ρ
A fundamental limitation for the distribution of entropy in a composite system is then
establishedbythestrongsubadditivity(SSA)inequality[4],whichimpliesthenonnegativity
of the mutual information as a measure of total correlations, and guarantees that the latter
quantity never increases upon discarding subsystems. For a tripartite classical or quantum
systemABC,thiscanbeformallyexpressedas
H +H −H −H ≥0. (1)
AB BC A C
The SSA inequality is straightforward to prove in the classical case, but far less trivial to
establishinthequantumcase[5,6].
In this Letter we prove that an alternative quantifier of information that can be defined
inthequantumcase,namelythelog-determinantofthecovariancematrixofaquantumstate,
also obeys a SSA inequality formally analogous to Eq. (1). We prove that this alternative
SSAinequalityisstrongerthanandimpliesthetraditionalSSAforthevonNeumannentropy,
in a class of pure tripartite Gaussian states. We then show that the SSA inequality for log-
determinanthasimportantimplicationsforlimitingtheEinstein-Podolsky-Rosen(EPR)joint
steerability[7]ofquantumstatesinamultipartitesetting.
2. Continuousvariablesystemsandlog-determinantofcovariancematrices
We focus on continuous variable composite quantum systems described by infinite-
dimensional Hilbert spaces [8], as exemplified by a set of n quantum harmonic oscillators
(modes). To describe the most general state ρ of such systems, one requires in principle
an infinite hierarchy of moments of the canonically conjugate quadrature operators {q ,p }
j j
defined on each mode j = 1,...,n. However, in many practical situations, one can extract
already valuable information by considering the first and second moments of the state only.
Of these, the first moments play no role in determining any informational quantity, as they
canbefreelyadjustedbylocalphasespacedisplacements;weshallhenceassumevanishing
first moments in all the states considered in the following with no loss of generality. What
remainscentralisthenthecovariancematrix(CM)V ,whoseelementsaredefinedas[9]
ρ
(V ) =tr[ρ(R R +R R )], (2)
ρ jk j k k j
whereR = {R ,...,R } = {q ,p ,q ,p ,...,q ,p }isthevectorofthecanonicaloperators,
1 2n 1 1 2 2 n n
andwehaveadoptedthenaturalunitconventionsuchthatV =Iifρistheground(vacuum)
ρ
stateofeachoscillator. Anypositivedefinite,real,symmetric2n×2nmatrixV isavalidCM
ofaphysicalstateiffitobeysthebonafideconditionstemmingfromtheuncertaintyprinciple
[10]:
(cid:32) (cid:33)
0 1
V+iσ⊕n ≥0, withσ= . (3)
−1 0
The CM of an arbitrary state can be reconstructed efficiently by homodyne detections
[11, 12]. Confining the description of a state ρ to its CM V is analogous to implementing
ρ
a ‘small oscillations’ approximation for classical oscillators. In the quantum case, for
any ρ (with vanishing first moments), one can always define a reference state χ uniquely
ρ
specifiedbytheCMV : thestateχ willbelongtothewell-studiedclassofGaussianstates
ρ ρ
[13, 9], which are central resources in continuous variable optical and atomic technologies
‡ Logarithms are usually assumed in base 2 for finite-dimensional systems and in natural base for infinite-
dimensionalsystems;however,theanalysisofthispaperdoesnotdependonanyspecificchoice.
Strongsubadditivityforlog-determinantofcovariancematricesanditsapplications 3
includingnetworkedcommunication,phaseestimationand(ifsupplementedbynon-Gaussian
detections) one-way quantum computation [14, 13]. Useful sufficient criteria to detect
nonclassical correlations such as inseparability [15, 16], steerability [17, 7] and nonlocality
[18,19]ofanarbitrarystateρcanbeformulatedandaccesseddirectlyatthelevelofitsCM,
althoughtheywilltypicallybenecessaryonlyforGaussianstates. Infact, onecanquantify
the‘error’inapproximatingastateρbyitsCMintermsofthenon-Gaussianityofρ,which
canbeinturnmeasuredbythedifferenceinentropy[20,21]betweenthereferenceGaussian
stateχ andtheoriginalρ.
ρ
Quantifying the degree of information (or uncertainty) in a CM V can thus provide
ρ
important indications regarding the corresponding properties of any state ρ compatible with
such CM, which will be the more accurate the less ρ deviates from Gaussianity. In this
respect, notice that most non-Gaussian resources considered in current protocols (such as
photon-subtractedandphoton-addedstates)[22]areconstructedasdeviationsfromGaussian
reference states, which means that precious quantitative indications on their degrees of
information and (for composite systems) correlations can be gained from the CM alone
[23,24]. Fromnowon,weshallthenspeakdirectlyofCMsandmeasuresappliedtothem.
Wedefinethelog-determinantofaCMV as
M =log(detV). (4)
V
TheideathatM mayberegardedasanindicatorofinformationakinto(butdifferentfrom)
V
conventional entropy can be understood as follows. For a Gaussian state ρ, the purity is
givenbytrρ2 =(detV )−1/2,henceM isamonotonicallydecreasingfunctionofthepurity.
ρ Vρ
Precisely, 1M is equal to the Re´nyi entropy of order 2 of a Gaussian state with CM V ,
2 Vρ ρ
which is in turn equal to the Shannon entropy of its Wigner quasi-probability distribution
(moduloanadditiveconstant)[25]. Forageneralnon-GaussianstateρwithCMV ,wecan
ρ
theninterpretM asaquantifierofuncertaintyinitssecondmoments,expressedby(twice)
Vρ
theRe´nyientropyoforder2ofthereferenceGaussianstateχ withthesameCMV .
ρ ρ
3. Strongsubadditivityforlog-determinantandrelatedinequalities
Givenanarbitraryn-modecontinuousvariablesystempartitionedintothreegroupsofmodes
forming subsystems ABC, with n + n + n = n, we denote by V and M the CM of
A B C α α
(sub)system α and its log-determinant, respectively. In the following, we shall establish the
centralresultofthisLetter,announcedbythenextTheorem.
Theorem 1 (SSA inequality for log-determinant of CMs). For any tripartite CM V , the
ABC
followinginequalityholds,
M +M −M −M ≥0, (5)
AB BC A C
which by comparison with Eq. (1) will be referred to as the SSA inequality for the log-
determinantoftheCM.
Before moving to the proof of the main Theorem, it is instructive to give a simple
demonstrationofthefactthatordinary(weak)subadditivityholdsforlog-determinantofCMs.
Proposition 1 (Subadditivity for log-determinant of CMs). For any bipartite CM V , the
AB
followinginequalityholds,
M ≤M +M . (6)
AB A B
Strongsubadditivityforlog-determinantofcovariancematricesanditsapplications 4
Proof. Let V be the CM of a bipartite state, with reduced subsystem CMs V and V . In
AB A B
blockform,wecanwrite
(cid:32) (cid:33) (cid:32) (cid:33)
V = VA Voff = LT VA 0 L, (7)
AB VT V 0 V¯
off B AB\A
where
V¯AB\A =VB−VoTffVA−1Voff (8)
(cid:32) (cid:33)
is the Schur complement of VA in VAB, and L = 0I VA−1IVoff . Since VoTffVA−1Voff ≥ 0,
we have that detV¯AB\A ≤ detVB, with equality holding iff Voff = 0. It follows then that
detV ≤detV detV ,whichupontakinglogarithmsimpliestheclaim. (cid:3)
AB A B
Theinequality(6)isanalogoustotheordinarysubadditivityofentropy,H ≤H +H .
AB A B
AGaussianstatesaturatestheinequality(6)iffitisaproductstate, butnon-Gaussianstates
can saturate it even if they are not product states, provided their CM takes the direct sum
form √VAB = VA ⊕ VB. One such example is the non-Gaussian entangled state |ψAB(cid:105) =
|00(cid:105)/ 2+(|02(cid:105)+|20(cid:105))/2,whosecorrelationsareallinhigherordermoments[23].
The validity of ordinary subadditivity for log-determinant prompts us to proceed and
tackletheproofoftheSSAinequality(5)announcedinTheorem1. Tothisend,wemakeuse
oftwomathematicalingredients.
Lemma1. Thelog-determinantisconcaveoverthesetofallpositivedefinitematrices.
Proof. This follows from a well known result of classical information theory [26]. Let
V ,V ,...,V bem×mpositivedefinitematrices;andletλ ,λ ,...,λ beasetofprobabilities,
λ1j ≥20,(cid:80)jλjl = 1. Then,det(cid:16)(cid:80)lj=1λjVj(cid:17) ≥ (cid:81)lj=1(detVj)λ1j. T2akingllogarithmsweobtainthe
desiredconcavityproperty,logdet(cid:16)(cid:80) λ V (cid:17)≥(cid:80) λ logdetV . (cid:3)
j j j j j j
Lemma1establishesthatconcavity,theprimarypropertyofentropy,holdsforthelog-
determinantofCMs. Thenextauxiliaryresultweneedisasfollows.
Lemma2. ThedifferenceM −M isconcaveoverthesetofallbipartiteCMsV .
AB A AB
Proof. LetV beanm×mpositivedefinitematrix,andletV bethe(m−l)×(m−l)matrix
(m) (m−l)
obtainedbydeletinginV asetoflchosenrowsandthecorrespondingcolumns. Without
(m)
loss of generality, V can be taken as the leading (m−l)-dimensional diagonal block of
(m−l)
V . Wehavethen[27]thatlogdetV −logdetV isconcaveoverthesetofallm×m
(m) (m) (m−l)
positivedefinitematrices.ChoosingnowV =V andV =V ,theclaimisproven. (cid:3)
(m) AB (m−l) A
Notice that this is true even though the difference in Lemma 2 can be negative, as it
doeshappenformostcasesofinterestinquantuminformationtheory(e.g.bipartiteentangled
states); analogousresultsholdforthecorrespondingvonNeumannentropicquantityH −
AB
H ,whosenegativityhasbeeninterpretedasaresourceforquantumstatemerging[28].
A
Equippedwiththeseresults,theproofofTheorem1cannowbecompleted.
Proof(ofTheorem1). Lemma2readilyimpliesthat
(M −M )+(M −M ) (9)
AB A BC C
is concave over all tripartite CMs V . Since {V } form a convex set, concavity implies
ABC ABC
that the quantity in Eq. (9) achieves its minimum value at one of the extreme points of this
set, i.e., on aCM V with det(V ) = 1. Anysuch CM describesa pureGaussian state,
ABC ABC
Strongsubadditivityforlog-determinantofcovariancematricesanditsapplications 5
forwhichwehavedetV = detV anddetV = detV ,whichmeansthatthequantityin
AB C BC A
Eq.(9)evaluatestozeroinanysuchcase. ThisconcludestheproofoftheSSAinequalityfor
thelog-determinantofCMsasanticipatedinEq.(5). (cid:3)
WenowproceedwithsomeremarksonTheorem1. First,Proposition1triviallyfollows
as a corollary of Theorem 1. Next, we notice that the SSA inequality (5) is saturated not
onlybypuretripartiteGaussianstates, butalsobyallstates(Gaussianornot)forwhichV
B
issymplectic(i.e.suchthatVBσ⊕nBVBT = σ⊕nB),thatisbystatesρABC = ρAC ⊗ρB whereρAC
is arbitrary and ρ is a pure Gaussian state; the CM of these states can be written in block
B
diagonalform,V =V ⊕V .
ABC AC B
Ifwefocusinsteadonthecaseofadifferentlypartitionedproductstateρ =ρ ⊗ρ
ABC AB C
whereρ isnowapureGaussianstate, andweconsiderboththeSSAinequality(5)andits
C
variantwhenAandBareswapped,weobtaintheinequality
M ≥|M −M |, (10)
AB A B
whichisformallyanalogoustotheAraki-LiebtriangleinequalityforShannon/vonNeumann
entropies[4,29].
Finally,letusrecallthat,givenanyCMV ,itcanbe‘purified’toasymplectic(positive
ABC
definite)CMV withdetV =1,sothatforglobalbipartitionsofthisfour-partiteCM
ABCD ABCD
one has detV = detV , detV = detV , and so on. Then, the inequality (5) can be
AB CD A BCD
recastas(finallyrelabellingDasCforaestheticconvenience)
M +M ≥M +M , (11)
AB AC A ABC
reminiscentofthecelebratedcounterpartofEq.(1)forShannon/vonNeumannentropies,
H +H ≥H +H , (12)
AB AC A ABC
whichisalsotypicallyreferredtoasSSAinequalityininformationtheoryliterature. Notice
that if V is assumed to be the CM of a tripartite Gaussian state, the inequality (11)
ABC
reproducestheonedemonstratedfortheRe´nyientropyoforder2in[25](seealso[30]). We
remarkthatin[25]anassumptionofGaussianityofstateswasmade, whilehereanexplicit
(and particularly didactic) proof of the SSA for the log-determinant of CMs of arbitrary
Gaussianornon-Gaussianstateshasbeenpresented.
4. Comparisonsbetweenlog-determinantandvonNeumannentropy
OnemightwonderwhetherthereexistsahierarchicalrelationbetweentheSSAinequalities
for the log-determinant M , Eq. (5), and for the von Neumann entropy H , Eq. (1), in
Vρ ρ
arbitrarytripartitestatesρ. However,thetwoinequalitiesareprimafacieincomparable,asit
canbeseenthattheyaresaturatedfordifferentclassesofstatesingeneral[27]. Furthermore,
whileM canbecomputedeasilyforanystatebasedonsecondmoments,H doesnotadmit
Vρ ρ
amanageableexpressioninarbitrarycontinuousvariablestates,whichrendersthecomparison
evenmoredifficulttoundertake. Nevertheless,ifwefocusourattentionontoGaussianstates,
somepartialanswerscanbeobtained.
RecallthatthevonNeumannentropyofanarbitraryn-modeGaussianstatewithCMV
canbecomputedinclosedformviatheexpression[31,32]
(cid:88)n ν +1 (cid:32)ν +1(cid:33) ν −1 (cid:32)ν −1(cid:33)
H = j log j − j log j , (13)
V
2 2 2 2
j=1
where {ν }n are the symplectic eigenvalues of V, obeying ν ≥ 1 ∀j as a consequence
j j=1 j
of the bona fide condition (3). The latter quantities can be evaluated by noting that the
Strongsubadditivityforlog-determinantofcovariancematricesanditsapplications 6
spectrum of the matrix (−Vσ⊕nVσ⊕n) is of the form {ν2,ν2,...,ν2,ν2}, i.e., contains the
1 1 n n
squaredsymplecticeigenvaluesofV withdoubledegeneracies.
Exploitingacomparisonbetweenvariousentropiesperformedin[33],wehavethatthe
von Neumann entropy H admits tight lower and upper bounds as a function of the log-
V
determinantM ,foranyn-modeGuassianstatewithCMV,givenby
V
f (M )≤H ≤ f (M ), (14)
1 V V n V
with
fn(m)= 2n(cid:34)log(cid:32)emn4−1(cid:33)+e2mn log(cid:18)coth4mn(cid:19)(cid:35) . (15)
For any real m ≥ 0 and integer n ≥ 1, the function f (m) is monotonically increasing with
n
bothmandn,andisconcaveinm;furthermore,sincelimm→0+ fn(m)=0,itfollowsthat fn(m)
isalsosubadditiveinm. Therefore,foranyx,y≥0,thefollowingholds,
f (x)+ f (y)≥ f (x+y)≥(cid:2)f (2x)+ f (2y)(cid:3)/2.
n n n n n
Noticethatif wesetn = 1inEq. (14), theupperandlower boundscoincide, meaning
that the von Neumann entropy is a simple monotonic, concave, and subadditive function of
the log-determinant of the CM for all single-mode Gaussian states, while for n > 1 we can
onlysaythatH isconstrainedbetweentwomonotonic,concave,andsubadditivefunctions
V
ofM ,withtheupperboundarybecominglooserwithincreasingnumbernofmodes.
V
WecannowshowthattheSSAinequalityforthelog-determinantisinfactstrongerthan
theconventionalSSAinequalityforthevonNeumannentropyinarelevantinstance.
Theorem2(SSAhierarchyforpureGaussianstates). LetV withdetV =1denotethe
ABC ABC
CMofa(n +n +n )-modepureGaussianstatesuchthatthereducedCMV hassymplectic
A B C A
spectrum{1,...,1,ν }. Then,theSSAforthelog-determinant,Eq.(11),impliestheSSAfor
nA
thevonNeumannentropy,Eq.(12).
Proof. Forapurestate,M = 0andEq.(11)rewritesasM ≤ M +M . Ifsubsystem
ABC A B C
A is in a Gaussian state whose CM V has (n − 1) symplectic eigenvalues equal to 1
A A
(correspondington −1vacuainitsnormalmodedecomposition),thenitsentropicproperties
A
areequivalenttothoseofasinglemodewithsymplecticeigenvalueν ,meaninginparticular
nA
thatthevonNeumannentropyofAsaturatesthelowerboundinEq.(14)[33]. Wehavethen
thefollowingchainofinequalities:
H = f (M )≤ f (M +M )≤ f (M )+ f (M )≤H +H , (16)
A 1 A 1 B C 1 B 1 C B C
wherewehaveusedrespectivelythemonotonicityof f (m)andtheSSAforlog-determinant
1
inthefirstinequality,thesubadditivityof f (m)inthesecondinequality,andthelowerbound
1
ofEq.(14)inthethirdinequality. Eq.(16)yieldsH ≤H +H ,concludingtheproof. (cid:3)
A B C
WeremarkthatTheorem2holdsinparticularforallpuretripartiteGaussianstateswith
n = 1 and n ,n arbitrary. In order to provide a simple illustration of the Theorem, let
A B C
us consider the instance of pure three-mode Gaussian states (n = n = n = 1). Up to
A B C
localunitaries,theirCMV isfullyspecifiedbythreesymplecticinvariants,whichcanbe
ABC √ √
identifiedwiththedeterminantsofthethreereducedCMs,thatis,a= detV ,b= detV ,
√ A B
andc = detV [34,35]. Inthiscase,consideringallpermutationsofthethreemodes,the
C
SSAconstraintstaketheformofatriangleinequality
|S −S |≤S ≤S +S , (17)
A B C A B
Strongsubadditivityforlog-determinantofcovariancematricesanditsapplications 7
ℋ
ℳ

Figure1. (Colouronline)HierarchyofSSAinequalitiesasformalisedbyTheorem2. The
plotshowsacross-sectionoftheregionsdefinedbythetriangleinequality(17)forpurethree-
modeGaussianstateswithlocalsymplecticinvariants{a,b,c}atfixedc=2,withtheentropy
functionScorresponding,inorderofincreasingstrength,to: (i)thevonNeumannentropy
H (blueoutermostregionwithdottedboundary);(ii)thelog-determinantoftheCMM(red
intermediateregionwithdashedboundary);and(iii)thesqrt-determinantoftheCMD(green
innermostregionwithsolidboundary),definedinthetext. Thelatterinequalitydelimitsthe
physicalparameterspaceofallpurethree-modeGaussianstates[34,35].
with S ≡ H for the von Neumann entropy, and S ≡ M for the log-determinant. In Fig.1
wecomparetheregionsdefinedbytheseinequalitiesinthespaceofparameters{a,b,c}. The
figureshows,asproveninTheorem2,thatthelog-determinantSSAdefinesasmallerregion
and is thus stronger than the von Neumann SSA. However, the set of physical three-mode
pureGaussianstatesisdelimitedbyanevenstrongertriangleine√quality,obtainedbysetting
S ≡ D in Eq. (17), with the sqrt-determinant function D = detV −1 [34]. The latter
V
inequality,whichcanbeseenasasolutiontotheGaussianmarginalproblemforn = 3[36],
furtherincorporatestherequirementthattheCMV mustobeythebonafidecondition(3),
ABC
whiletheSSAinequalityintheform(11)forthelog-determinantonlyreliesonthepositivity
oftheCM,V >0(seealso[25,30,27]),whichisweakerthanEq.(3).
ABC
5. ApplicationstoEPRsteeringofmultimodestates
In the remaining part of the Letter, we investigate applications of the SSA inequality (5)
for log-determinant of CMs to characterising possibilities and limitations of EPR steering
in continuous variable systems. Let us briefly introduce the necessary concepts. Steering,
intended in a bipartite setting as the possibility for Alice to remotely prepare Bob’s system
in different states depending on her own local measurements, is a genuine manifestation
of quantum correlations that embodies the crux of the original EPR paradox [37], and was
recognised by Schro¨dinger as evidence of the “amazing knowledge” allowed by quantum
mechanics[38,39].
Letρ beabipartitestate,andletaandbbemeasurementoperatorsonsubsystems A
AB
(operatedbyAlice)andB(operatedbyBob),withrespectiveoutcomesαandβ.Bydefinition
Strongsubadditivityforlog-determinantofcovariancematricesanditsapplications 8
[7],thestateρ isA→ Bsteerableiff,forallpairsaandb,themeasurementstatisticsobeys
AB
p(α,β|a,b;ρ )(cid:44)(cid:80) p p(α|a,λ)p(β|b,τλ), (18)
AB λ λ B
that is, it cannot be interpreted as arising from correlations between a random local hidden
variable (λ) for Alice and a random local hidden state (τλ) measured by Bob. Here p is a
B λ
probability distribution and p(α,β|a,b;ρ ) = tr[(Πa ⊗Πb)ρ ], where Πa is the projector
AB A B AB A
satisfyingaΠa =αΠa.
A A
Foratwo-modecontinuousvariablesystem, asufficientconditiontodetectsteerability
[17] can be expressed in terms of the violation of Heisenberg-type uncertainty relations for
theconditionalvariancescorrespondingtomeasurementsofcanonicallyconjugateoperators.
Let q and p be quadrature operators on Bob’s mode, satisfying [q ,p ] = i, and define
B B B B
thevariancesthatAlicededucesforBobbylinearinferencebasedonherownmeasurements
ofapairofuncharacterisedoperatorsq and p ,e.g.∆ q = (cid:104)[q −qest(q )]2(cid:105) ,where
A A inf,A B B B A ρAB
qest(q ) = g q for some optimised value of the linear gain coefficient g (and similarly for
B A q A q
p ). Onehasthenthatthestateρ isA→ Bsteerableif[17]
B AB
E (ρ )=∆ q ∆ p <1 (19)
B|A AB inf,A B inf,A B
The criterion in Eq. (19) can detect steerability due to quadrature (Gaussian) measurements
which act on second moments. If one optimises it over all possible choices of canonically
conjugate pairs (i.e. over local phase space symplectic transformations for Alice and Bob),
thentheminimumofE canbeexpressedonlyintermsoftheCMV ofρ [40,41,42]:
A|B AB AB
(cid:16) (cid:17) detV
min E (U ⊗U )ρ (U ⊗U )† =detV¯ = AB , (20)
UA⊗UB B|A A B AB A B AB\A detVA
where V¯ denotes the Schur complement of V in V as in Eq. (8). In this form, the
AB\A A AB
criterion can be extended to an arbitrary number of modes: given a bipartite state ρ with
AB
CMV ,iftheSchurcomplementisnotitselfabonafideCMinthesenseofEq.(3),i.e.if
AB
V¯AB\A+iσ⊕nB (cid:3)0, (21)
then ρ is A → B steerable. Steerable states are useful resources for one-sided device-
AB
independent quantum key distribution [43], subchannel discrimination [44], and secure
continuousvariableteleportation[45].
InthespecialcaseofbipartiteGaussianstatesρ , Eq.(21)isnecessaryandsufficient
AB
for steerability by Gaussian measurements [7, 41]. Accordingly, a quantitative measure of
Gaussian steerability has been proposed for a (n +n )-mode bipartite Gaussian state [40],
A B
definedas

GA→B(VAB)= 0−,(cid:80)j:ν¯AB\A<1ν¯AljoBg\A(cid:16)ν¯≥AjB1\A∀(cid:17)j,=o1t,h.e.r.w,nisBe;, (22)
j
where{ν¯AB\A}denotethesymplecticeigenvaluesofV¯ .InthespecialcaseofBcomprising
j AB\A
one mode only (n = 1), the Schur complement V¯ has only one symplectic eigenvalue
B AB\A
(cid:113)
ν¯AB\A = detV¯ ,hencetheaboveexpressionsimplifiesto
AB\A
(cid:40) (cid:41)
(cid:12) 1
GA→B(VAB)(cid:12)(cid:12)nB=1 =max 0, 2(MA−MAB) , (23)
wherewehaveadoptedtheexpressioninEq.(4)forthelog-determinant.Thelog-determinant
is therefore useful to capture the quantitative degree of steerability of mode B by Gaussian
measurements performed on the multimode subsystem A, as detectable at the level of CMs.
Notice however that, very recently, examples of Gaussian states unsteerable by Gaussian
Strongsubadditivityforlog-determinantofcovariancematricesanditsapplications 9
A B C
Figure 2. (Colour online) Monogamy of EPR steering by Gaussian measurements as
formalisedbyTheorem3. ForanystateρABC ofamultimodesystempartitionedintothree
subsystemsABC,whereAandCarecomposedofanarbitrarynumberofmodeswhileBis
composedofasinglemode,ifAcansteerBbyGaussianmeasurements,thenCcannotsteerB
byGaussianmeasurements,andviceversa.ThisisadirectconsequenceoftheSSAinequality
forthelog-determinantoftheCMofρABCpresentedinTheorem1.
measurements (i.e. with GA→B = 0) have been found, which are nonetheless steerable,
accordingtodefinition(18),bymeansofsuitablenon-Gaussianmeasurements[46,47].
Consider now a tripartite setting. Quite interestingly, if only quadrature (Gaussian)
measurementsandsecondmomentsareconsideredforsteeringdetection,thenaverystrong
limitation occurs: for an arbitrary (Gaussian or non-Gaussian) three-mode state ρ , the
ABC
monogamyconstraint
E (ρ )E (ρ )≥1, (24)
B|A ABC B|C ABC
holds[48,49,50]. Thismeansthatitisimpossibletodetectanysimultaneoussteeringofthe
singlemodeBbythesinglemodesAandCifusingsecondmomentcriteria.
WecannowgeneralisethisresulttothecaseofpartiesAandC comprisinganarbitrary
numberofmodes,whilethesteeredpartyBremainsformedbyasinglemode(seeFig.2).
Theorem 3 (No-joint steerability of one mode by multimode Gaussian measurements). Let
ρ be an arbitrary (n +n +n )-mode quantum state with n ,n arbitrary and n = 1.
ABC A B C A C B
Thenitisimpossibleforρ tobesimultaneouslyA→ BandC → BsteerablebyGaussian
ABC
measurements.
Proof. The claim follows by combining the SSA inequality (5) for log-determinant of CMs
with the CM-based steering criterion (21). Denoting by V the CM of the composite
ABC
tripartite system (with nB = 1), we have in fact that V¯AB\A + iσ⊕nB (cid:3) 0 is equivalent
to detV¯AB\A = detVAB/detVA < 1, and similarly V¯CB\C + iσ⊕nB (cid:3) 0 is equivalent to
detV¯ =detV /detV <1.ToaccomplishsimultaneoussteeringofmodeBbygroupsA
CB\C BC C
andCbasedonsecondmoments,onewouldthusneed(detV /detV )(detV /detV )<1,
AB A BC C
orequivalently,takinglogarithms,M +M −M −M <1. Butthisisimpossibleasit
AB BC A C
contradictsEq.(5),henceconcludingtheproof. (cid:3)
The SSA inequality for the log-determinant implies therefore a limitation for joint
steerability based solely on CMs in arbitrary continuous variable states. Specifically,
subsystems A andC cannot simultaneously steer the single-mode subsystem Bby Gaussian
measurements, although this no-go may be circumvented by non-Gaussian measurements
evenontripartiteGaussianstatesρ [47].
ABC
However,itiseasytoseethat,assoonas Bismadeofatleasttwomodes,suchastrict
monogamyislifted,andBcanbesteeredbypartiesAandCsimultaneouslyalreadyinanall-
Gaussiansetting.Asanexample,consideraGaussianstateoffourmodes1,2,3,4,andgroup
themsuchthatsubsystemAisassignedmode1,subsystemBisassignedmodes2and3,and
subsystemC is assigned mode 4. For illustration, we can focus on the family of four-mode
purestatesintroducedin[51](seeFigure1therein),whoseCMtakestheform
V ≡V =S (a)S (a)S (s)ST (s)ST (a)ST (a), (25)
ABC 1234 3,4 1,2 2,3 2,3 1,2 3,4
Strongsubadditivityforlog-determinantofcovariancematricesanditsapplications 10
whereS (r)denotesatwo-modesqueezingsymplectictransformationactingonmodesiand
i,j
j with real squeezing degree r [9]. These states are symmetric under swapping of modes
1 ↔ 4 and 2 ↔ 3. Therefore, their 1 → (2,3) and 4 → (2,3) steerability properties are
the same; with respect to our grouping, modes A and C are thus able in principle to steer
simultaneouslythetwo-modegroupBbythesameamount. Toseewhetheranysuchsteering
ispossibleatall,wecancalculatetheGaussiansteerabilitymeasure(22)[40]intherelevant
settings. We find that as soon as a,s > 0, i.e., as soon as the state is not a product state,
then it is both A → B andC → B steerable by Gaussian measurements, with its Gaussian
steerabilityGA→B(V ) = GC→B(V )beingamonotonicallyincreasingfunctionofaand
ABC ABC
s,notreportedhere.ThisdoesnotcontradictthegeneralSSAinequality(5),whichholdswith
equalityonthisexampleasdetV =1. Theexplanationisthat,whenthesteeredpartyhas
ABC
morethanonemode,thesymplecticspectrumenteringEq.(22)doesnotdependonlyonthe
determinantoftheSchurcomplement,hencesteerabilitycannotbedecidedsolelyintermsof
abalanceoflog-determinants.
6. Conclusions
In this Letter we demonstrated that the log-determinant, a simple informational quantity
definedonthecovariancematrixofanycontinuousvariablestate,behavesasafullyfledged
entropy,obeyingthefundamentalstrongsubadditivityinequality. Inaparticularclassofpure
tripartiteGaussianstatesofanarbitrarynumberofmodes,weshowedthatsuchaconstraint
isstrongerthantheconventionalstrongsubadditivityinequalityforvonNeumannentropy. It
wouldbeveryinterestingasafuturedirectiontoinvestigatewhetherthishierarchybetween
strongsubadditivitiesholdstrueingeneral,ormaybereversedonotherclassesofstates.
Our result implies a strict limitation on the joint steerability of one quantum harmonic
oscillator by two other groups of oscillators, within a steering detection setting based on
second moments. This is in turn relevant for practical applications, e.g. in the context of
securequantumcommunication[13].Inatypicalquantumopticslaboratorywhereoperations
(includingmaliciousattacks)arelimitedtotheGaussiantoolbox,itisimpossibleforasingle
modeinBob’spossessiontobesteeredbymorethanonepartneratonce. Suchamonogamy
ensures that Bob’s exclusive pairing with Alice (who can operate on multiple modes), for
thepurposesofentanglementverification[7]andone-sideddevice-independentquantumkey
distribution [43], cannot be disrupted by the attempts of an eavesdropper Claire. It will
be interesting to investigate other applications of this no-go result in the context of secure
teleportationandtelecloningprotocolsinvolvingthreeormoreparties[45].
Moreextensionsandadditionalstrenghteningsofthestrongsubadditivityinequalityfor
log-determinant of covariance matrices, taking into account physical requirements such as
theuncertaintyprinciple,andinspiredbyseminalormoremoderndevelopmentsinclassical
and quantum information theory, are certainly worthy of further investigation [30, 52]. In
particular, weanticipatethatitispossibletodefinearemaindertermforEq.(11)bymeans
of a Gaussian recovery map [52], in analogy to the latest advances obtained for the strong
subadditivity of von Neumann entropy by Fawzi and Renner [53]. A more comprehensive
characterisation of (Gaussian and non-Gaussian) states saturating the strong subadditivity
inequality for log-determinat also deserves a separate study. We finally notice that other
monogamy-typeconstraintsoncontinuousvariablesteeringwithinmultipartitenetworkshave
beenrecentlyexploredandreportedelsewhere[54].