Compatibility of subsystem states and convex geometry

0.22 MB·English

by  William Hall

Compatibilityofsubsystem states andconvex geometry
William Hall
Department of Mathematics, University of York, Heslington, York YO10 5DD, U.K.∗
Thesubsystemcompatibilityproblem,whichconcernsthequestionofwhetherasetofsubsystemstatesare
compatiblewithastateoftheentiresystem,hasreceivedmuchstudy. Hereweattacktheproblemfromanew
angle, utilisingtheideasofconvexitythathavebeensuccessfullyemployedagainsttheseparabilityproblem.
Analogouslytoanentanglementwitness,weintroducetheideaofacompatibilitywitness,andproveanumber
ofpropertiesabouttheseobjects.Weshowthatthesubsystemcompatibilityproblemcanbesolvednumerically
andefficientlyusingsemidefiniteprogramming, andthatthenumericalresultsfromthissolutioncanbeused
toextractexact analyticresults, anideawhichweusetodisproveaconjecture about thesubsystemproblem
madebyButterleyetal.[Found. Phys. 3683(2006)]. Finally,weconsiderhowtheideascanbeusedtotackle
someimportant variants of thecompatibilityproblem; inparticular, thecaseof identical particles(known as
N-representabilityinthecaseoffermions)isconsidered.
7
0
0
2 I. INTRODUCTION whenasetofsubsystemstatesisnotcompatiblewithastate
forthe entiresystem, and we use these to formthe firstnec-
n
essaryandsufficientconditionforthecompatibilityproblem.
a The fact that quantum mechanical particles can be entan-
However, much as is the case with entanglement witnesses,
J gledwitheachotherleadstoanotionofcompatibilityofquan-
without a full characterisation of compatibility witnesses, in
4 tumstates. Givenafinitesystemofquantummechanicalpar-
practicethistheoremisnotofanyoperationaluse. Nonethe-
ticles, the reduced states of a subset of these particles must
2 less, we do prove a number of interesting results about the
arisefromthepartialtraceofsomedensitystateoftheentire
v natureofcompatibilitywitnesses.
system,andasaresultthesubsystemstatescannotbearbitrary
1 We will show that the ideas of semidefinite programming
positiveoperatorsoftraceone.Asimpleexamplecomesfrom
3
[5, 6] can be used to numerically solve the compatibility
0 the‘monogamyofentanglement’:IftwoparticlesAandBare
problem. Theinspirationforthisliesbehindsimilarapplica-
0 inamaximallyentangledstate,anyotherparticleCcannotbe
tionsofsemidefiniteprogrammingtotheseparabilityproblem
1 entangledatallwitheitherAorB[1,2].
6 [7, 8, 9, 10, 11]. We will give a brief overviewof semidefi-
Therearemanyvariantsofthisproblem,buttheformwhich
0 nite programmingand illustrate how it can be applied to the
wewillbemostconcernedwithinthispaperisthefollowing
/ compatibilityproblem. Furthermore,wewillshowhowana-
h [3]: Given states of all proper subsystems of a multipartite
lyticresultscanbeobtainedfromthenumericalresultsofour
p
quantumsystem,whatarethenecessaryandsufficientcondi-
- solution,andweuseittodisproveaconjectureofButterleyet
t tionsforthesesubsystemstatestobecompatiblewithasingle
n al. [3] concerningthe necessaryandsufficientconditionsfor
state ofthe entire system?. A numberof techniquesanddif-
a thecompatibilityprobleminvolvingthreequbits(thesimplest
u ferentapproacheshavebeenusedtoattackthisproblem,and non-trivialcase).
q manypartialresultshavebeenobtainedinrecentyears.How-
There are other variants of the problem that we will con-
: ever,muchofthecurrentworksontheproblemhaveconcen-
v sider after the main problem. An important situation is
trated on less general forms of the problem, and also on the
i onewhere we onlyhave partialknowledgeof the subsystem
X spectralpropertiesoftheinvolvedstates.
states. Anotheristhecasewherewehavenidenticalparticles
ar Another problem that has been the topic of even greater (bosonsorfermions);thisproblemisofparticularimportance
study in quantum mechanics is the separability problem, of in quantum chemistry in determining the ground state ener-
trying to decide when a multipartite quantum state is entan- giesofasystemofidenticalparticles[12]. Wewillbeableto
gledorseparable(notentangled).Oneapproachtothisprob- givesolutionstobothofthesevariantsusingamodificationof
lemistousethefactthatthesetofseparablestatesisaconvex thesemidefiniteprogramusedtosolvethemainproblem.
set,andtouseallofthegeometricfactsthatfollowfromcon-
vexity. Thisturnsouttobea veryfruitfulapproach,andhas
ledtothedevelopmentofmanypowerfulmathematicaltools,
II. THECOMPATIBILITYPROBLEMANDPREVIOUS
mostnotablytheideaofanentanglementwitness,forattack-
RESULTS
ingtheseparabilityproblem(see[4]forareview).
What we aim to do here is to utilise the idea of convexity
A. Statementoftheproblem
toattackthecompatibilityproblem. Wewillshowthatinthe
correctmathematicalframeworkthesetofcompatiblesubsys-
temstatescanbeviewedasaconvexset. We willdefinethe Let us first establish some nomenclature. For any Hilbert
notionofacompatibilitywitness,whichcanbeusedtoshow space ,let ( )bethesetofboundedoperatorson ,and
K B K K
let ( )bethespaceofdensitystates(i.e.positiveoperators
D K
oftrace1).
Let (i = 1,...,n) be Hilbert spaces with finite di-
i
H
∗Electronicaddress:[email protected] mension di. Define = 1 2 ... n to be the
H H ⊗ H ⊗ ⊗ H
2
Hilbert space for an n-partite quantum system, and define compatiblewithaspecificstateρ ( ),andcheckingthese
∈B H
N = 1,...,n . A set of states on everypropersubsystem conditionsforallpossiblestatesisnotoperationallyfeasible.
{ }
then correspondsto a set of operatorsρ ( ) for There do however exist many results in more constrained
A i∈A i
∈ B ⊗ H
everyA N. versionsoftheproblem. Onequestionthathasbeenparticu-
⊂
The compatibility problem asks the following question: larlystudiedisthefollowing: Whataretheconditionsonthe
givenρ forallA N,whatarethenecessaryandsufficient spectraoftheone-particlereducedstatesforthefullstate to
A
⊂
conditionsforthese statestocompatiblewith astate ρ , haveagivenspectrum?Forexample,apurestatehasapartic-
∈ H
i.e. ularspectrum,andthenecessaryandsufficientconditionson
thespectraoftheone-particlestatesforthemtobethereduced
ρ =Tr ρ (1) statesofapurestatearederivedforanynumberofqubitsby
A N\A
Higuchietal. in[15],forthreequtritsbyHiguchiin[16],and
whereTrX representsthe partialtraceon systems i forall fortwoqubitsanda4-dimensionalsystembyBravyiin[17].
H
i X. Wesaythesetofsubsystemstatesareincompatibleif Asetofnecessaryconditionswereprovedforanynumberof
∈
nosuchstatecanbefound. arbitrarysystemsbyHanetal. in[18]. Finally,themostgen-
Someobviouscompatibilityconditionsarise fromthe fact eralformoftheproblemwassolvedbyKlyachko[19],using
thattwostatesdescribingoverlappingsystemsmustyieldthe methodsofsymplecticgeometrysimilartothoseusedbyKly-
samereducedstateswhentheoverlapistracedout. Formally, achkotosolvealong-standingconjecturebyHorn,whocon-
wecanstatethisasfollows: ifA,B,C N,A B = A jecturedasetofinequalitiesansweringthequestion‘Whatare
⊂ ∩ ∩
C = ,then thepossiblespectraofasumofHermitianmatriceswithgiven
∅
spectra?’. RelatedtothisistheworkofDaftuarandHayden
TrB(ρA∪B)=TrC(ρA∪C). (2) [20],inwhichthepossiblespectraofaone-partyreducedstate
ofabipartitestatearedetermined.
Eveninthesimplestcaseofthreequbits,thereisnoknown As mentioned already, most of this work concentrates on
solution to this problem, and there is no analytic way of de- establishingrelationshipsbetweenthe spectraofthereduced
terminingwhetheranarbitrarysetofsubsystemstatesiscom- and (where one can exist) the total states. In this work we
patiblewithafulldescriptionofthesystem(ornot). intendtomoveawayfromthisfocusandconcentratemoreon
thegeometricaspectsoftheproblemathand.
Other worksthatestablish analytic resultsaboutthe prob-
B. Summaryofcurrentresults lem include the work of Christandl and Mitichison [21], in
whichaconnectionbetweenthisproblemandtherepresenta-
There are very few known necessary criteria for the gen- tiontheoryofthesymmetricgroupisobtained,andthework
eral form of the compatibilityproblem as stated above. The ofJonesandLinden[22],inwhichasystemofqubitsiscon-
followingtheoremisanexampleofsuchacondition: sidered, and it is shown that if we are given all the reduced
statesofmorethanhalfthenumberofqubits,thereisalmost
Theorem 1. Let ρA = TrN\Aρ (where A N = always at most one pure state compatiblewith these subsys-
⊂
1,...,n )bethereducedstatesofan-partystateρ. Then, temstates.
{ }
defining Another important variant of the problem is the case of
identical particles. This extra condition places more con-
∆= ( 1)|A|ρ (3) straintsonthedensitystatefortheentiresystem,aspurestates
A
−
A⊂N ofbosons(fermions)aresymmetric(antisymmetric)underex-
X
changeofanytwoparticles. Acompatibilityproblemthathas
(whereρ hereispaddedoutwithidentitiesonthetracedout
A received particular attention is one that has become known
systems), then if n is odd, ∆ 0. If all of the systems are
asthe N-representabilityproblem- thisis thecase whenwe
≥
qubits,then∆ 1also.
have N identical fermions. The necessary conditionson the
≤
one-partyreducedstateswassolvedbyColeman:
Thiswasprovedinthecaseofthreequbitsin[3]. andthe
resultinitsfullgeneralitycanbeinferredfrom[13]. Further- Theorem2(Coleman[23],Corollary4.3A). Adensitymatrix
more, througha study of the classical analogof the compat-
ρisthereducedone-partystate ofasystem ofN fermionsif
ibility problem, it was conjectured in [3] that in the case of andonlyallofitseigenvaluesareboundedaboveby1/n.
three qubits, the condition 0 ∆ 1 is not only neces-
≤ ≤
saryforsubsystemstatecompatibility,butalsosufficient.The The necessary and sufficient conditions on the two-party
threepartycaseisalsostudiedindetailin[14],whereanum- reducedstate howeverare still not known, and have been of
berof necessaryconditionsonthe spectra ofanythree party greatinterestinthefieldofquantumchemistry,sinceknowing
stateanditstwo-partyreducedstatesareproduced. theconditionsonthetwo-partystateswouldallowthecalcula-
Thefactthatweonlyhavenecessaryconditionsreflectsthe tionofthegroundstateenergyofanN-fermionHamiltonian
difficulty in the problem. A set of subsystem states can be involving2-bodyinteractionsonlyfromthe2-partystateand
provedtobe incompatibleif theyviolateanyoftheseneces- aneffective2-bodyHamiltonian.Thisisstillatopicofactive
sary conditions. However, the only sufficient conditions we research(see[12]andreferencestherein).
have amount to checking whether the subsystem states are Finally, there has been some research on the complexity
3
ofthecompatibilityproblem. Specifically,ithasbeenshown andsinceαρ+(1 α)σ ( ),αρ+(1 α)σ C.
− ∈D H − ∈
that when we have partial knowledge of the reduced states NowwewillproveC iscompact. DefinethefunctionR :
(specifically,anumberofreducedstatesthatispolynomialin ( ) n ( (i))by
B H →⊕i=1B H
the number of quantum systems, each describing a number
ofsystemslessthansomeconstant),thecorrespondingcom- R(ρ)=(Tr1ρ,...,Trnρ). (8)
patibilityproblemisQMA-complete[32]andhencealsoNP-
hard. Itwasalsoshownveryrecentlyin[24]thatthespecific Bydefinition,R( ( ))=C. Sincethetraceisacontinuous
D H
N-representability problem where we are given the 2-party function, R is a continuousfunction. Furthermore, ( ) is
D H
reducedstateisalsointhecomplexityclassQMA-complete. aclosedandboundedset,andsobytheHeine-Boreltheorem
In this paper we will not be concerned with detailed analy- [25],itiscompact.Finally,thecontinuousimageofacompact
siswithrespecttocomplexityclasses,butwewilldiscussthe subsetisitselfcompact,andhenceR( ( ))=Ciscompact.
D H
computationalefficiencyandcomplexityofthealgorithmsin-
volved.
B. Compatibilitywitnesses
III. CONVEXITYANDTHECOMPATIBILITYPROBLEM
Theidea of a entanglementwitness arises fromthe Hahn-
In this section we will use the ideas of convexity on the Banach theorem. An important geometric fact that follows
compatibilityproblem,andintroducethenotionofacompat- fromthistheoremisthefollowing[26,27]:
ibilitywitness, a toolthatwillallowustodetectsetsofsub-
Theorem 3. Let C ,C be disjoint convex closed sets in a
systemstatesthatarenotcompatiblewithanoverallstate. 1 2
realBanachspace,andletC becompact. Thenthereexists
1
a continuous functional f, and a R such that, for every
∈
c C ,c C ,f(c )<a f(c ).
A. Compatiblestatesasaconvexset 1 1 2 2 1 2
∈ ∈ ≤
Wecanusethistheoremtoprovearesultaboutelementsof
Letusdefine (k) = j∈N,j6=k j (i.e. systemkremoved not in C analogousto the result about entangledstates in
fromthetensorpHroduct),⊗andletusHdefine n ( (i)) O[27]:
O ⊂⊕i=1B H
by
Proposition 2. Let σ , but σ / C. Then there ex-
= O =(O ,...,O ) O ( (i)), O† =O . (4) ists W = (W ,...,W∈) O such t∈hat W,σ < 0, and
O { 1 n | i ∈B H i i} 1 n ∈ O h i
W,ρ 0forallρ=(ρ ,...,ρ ) C.
Wedefineaninnerproductonthisspaceby h i≥ 1 n ∈
Proof. The one-element set S(σ) = σ and C are con-
n
{ }
A,B = Tr(A B ). (5) vexcompact(andhenceclosed)subspacesoftherealBanach
i i
h i space . SincealsoS(σ) C = ,wecanapplytheabove
Xi=1 theoreOm. Hence there exis∩ts a line∅ar functionalg : R
ThisspaceisanaturalspacetodefinethesetC by O →
⊂O anda Rsuchthat
∈
C = ρ=(ρ ,...,ρ ) ρ =Tr ρforsomeρ ( ) .
1 n j j
{ | ∈D H } g(σ)<a g(ρ) (9)
(6)
≤
This is the set of compatible subsystem states, or more pre-
forallρ C. BytheRiesz-Fre´chetrepresentationtheorem,
cisely,thesetofvectorsofthenstatesofn 1systemsob- ∈
−
tained from the partialtrace of a full state ρ (ρi corresponds n
to the state on (i)). Thestates of a smaller numberof sys- g(X)= Tr(V X ) (10)
H i i
temscanthenbeobtainedfrompartiallytracinganappropri-
i=1
X
ate(n 1)-systemstate.
The−compatibilityproblemcannowbecastinthefollowing where V = (V1,...,Vn) . Now, defining W = V
∈ O −
form:Letσ =(σ1,...,σn)beavectorofthenstatesofn 1 aI/n,whereI =(11/d1,...,1n/dn),weobtainthat
−
systems. Wearethentryingtoascertainwhetherσ liesinthe
n n
setC.Thisisrathersimilartotheseparabilityproblem-given
Tr(W σ )<0 Tr(W ρ ) (11)
astate ρ, we aretryingto ascertainwhetheritbelongstothe i i ≤ i i
i=1 i=1
set of separable states. This set is convex and compact, and X X
wewillprovethesamefactaboutC. whichistherequiredresult.
Proposition1. Cisaconvex,compactsubsetof .
O Thisresultimmediatelyleadstothefollowingtheorem:
Proof. We willfirstproveC isconvex. Letρ,σ C. Then
∈ Theorem4. Letσ . Thenthe(n 1)-partystatesrepre-
ρi =Triρ,σi =Triσforsomequantumstatesρ,σ ( ). ∈O −
∈B H sentedbyσarecompatiblewithastateofthefullsystem(i.e.
Letα [0,1]. Then
∈ σ C)ifandonlyif W,σ 0forallW suchthat
∈ h i ≥ ∈ O
αρ +(1 α)σ =Tr (αρ+(1 α)σ) (7) W,ρ 0forallρ C.
i i i
− − h i≥ ∈
4
WecallanyW satisfying W,ρ 0forallρ Cacom- establishingtheresult.
h i≥ ∈
patibilitywitness,andwewilldenotethesetofcompatibility
This can be reduced to a finite number of conditions by
witnesses by . If W,ρ < 0, the subsystem states ρ are
W h i taking a basis for the subspace of traceless matrices in each
incompatible,andwesayW detectstheincompatibilityinρ.
Hilbertspace ,andthenformingthetensorproductofthese
Thisisthefirstexampleofa necessaryandsufficientcon- i
bases(thisyieHlds n (d2 1)conditions).
ditionforthecompatibilityproblem,but(ratherlikethesimi- i=1 i −
Wecanalsoconsiderthisprobleminreverse: givenapos-
lartheoremforentanglementwitnesses)itisnotimmediately
Q
itiveoperatorZ ( ), we maywantto ascertainwhether
operationallyuseful,becauseitrequiresacompletecharacter-
∈ B H
thisoperatorisoftheformZ = p(W). Thisisallencapsu-
isationoftheentiresetofcompatibilitywitnesses.
latedinthefollowingproposition:
Proposition 5. Let Z ( ) be positive, such that
C. Propertiesofcompatibilitywitnesses ∈ B H
Tr[Z(T ... T )] = 0ifTr(T ) = 0. ThenZ = p(W)
1 n i
⊗ ⊗
forsomecompatibilitywitnessW.
Tomaketheintroductiontocompatibilitywitnesseseasier,
wewillproveafewresultsaboutsomepropertiesofcompat- Proof. Let Bk,m (m = 1,...,d2k) be a basis of Hermitian
ibility witnesses. The first shows us how to jump between operatorsfor ( k)satisfying
B H
acompatibilitywitnessandapositiveoperatorcharacterising
1 1
thiswitness. Inallthatfollowsthispoint,ifA ( (i)),and B = k; Tr(B )=δ ; Tr(B B )= δ
∈B H k,m d k,m 1m k,m k,n d mn
B ( ), then the operator A B should be interpreted k k
asb∈einBgHami emberof ( )rather⊗than ( (i)) ( )(for (16)
B H B H ⊗B Hi anddefine
example,ifn=3,andA=A A ( (2)),thenA B
1 3
⊗ ∈B H ⊗
shouldbeinterpretedasA1⊗B⊗A3 ∈B(H)). Bm =⊗nk=1Bk,mk. (17)
Proposition3. W isacompatibilitywitnessifandonlyifthe
Define D = dim = d . The set B m =
operatorp(W) ( )definedby H i i { m | i
∈B H 1,...,d2; i = 1,...,n formsa basisfor ( ), andso we
n canwritie } Q B H
p(W)= W 1 , (12)
i i
⊗
Xi=1 Z = zmBm. (18)
where1 istheidentityoperatoron ,ispositive. m∈I
i i X
H
Proof. Letρ C becompatiblewithastateρ. Wenotethat whereI = m m = 1,...,d2; i = 1,...,n , and(from
∈ { | i i }
theorthogonalityconditions(16)),z = DTr(ZB ). Now
n n m m
Tr(W ρ )=Tr (W 1 )ρ =Tr(p(W)ρ) (13) definetheset
i i i i
" ⊗ #
Xi=1 Xi=1 I = m m =1,...,d2, i=1,...,n;
andsotheresultfollows(i.e. ifonesideisgreaterthanzero, C | i i
m =1forsomek=1,...,n . (19)
soistheother). (cid:8) k }
For Z satisfying the assumptions in the statement of the
However, constructing non-trivial entanglement witnesses
proposition, we have that z = Tr(ZB ) = 0 for any
is not as simple as taking W to be positive for all i = m m
i m / I ,asthesebasiselementsareatensorproductoftrace-
1,...,n,asthenTr(Wiσi)≥0foranypositiveσiandsothe less∈opCerators.HenceZ canbewrittenintheform
compatibilitywitnessW neverdetectsanyincompatibilityof
subsystemstates.
Z = z B . (20)
Weshouldnoteherethatthefunctionpisnotaone-to-one m m
function: thereexistO (notnecessarilya compatibility mX∈IC
witness) such that p(O)∈=O0, and hence W′ = W +O is Whenm I ,thebasiselementB isoftheformB 1
a compatibilitywitness such that p(W′) = p(W). We will forsomei∈=C1,...,n, andsoZ isomftheform Wi⊗1i.
returntothispointlater. HenceZ =p(W)forsomeW ,andsinceZ iispois⊗itivei,
Thestructureofp(W)alsoimpliesthefollowingresult: byProposition3,W isacompat∈ibOilitywitness.P
Proposition4. LetW (notnecessarilyacompatibility
∈ O Includedintheaboveproofisamethodofextractingasuit-
witness),andletT ( ),withTr(T )=0.Then,defining
T =T1⊗...⊗Tin,∈TBr(pH(Wi )T)=0. i afobrlemW I∈CWin(s2u0c)hcathnabteZus=edpto(Wco)n.stTruhcettohpeeorapteorrastozrmsBWmi,
Proof. DefineT(i) = T . Weseethat where∈i is determinedby the componentof m thatequals1.
k6=i k
⊗
Where more than one componentequals 1, the operator can
n
contributetoanyoftheappropriateW . Thisfreedomrepre-
Tr(p(W)T) = Tr (W 1 )T (14) i
i i
⊗ ! sents the fact that there is not a unique W satisfying
Xi=1 Z =p(W),butthereareinfactmanysuchw∈itnWesses.
n
= Tr(W T(i))Tr(T )=0, (15) AnothermethodoffindinganappropriateW fromZ
i i ∈W
isthroughthefollowingproposition:
i=1
X
5
Proposition6. LetZ =p(W)forsomeW . Then W′ is finer than W if W,ρ < 0 W′,ρ < 0
∈W • (i.e. W′ detectsatleasthasmainyinco⇒mphatiblesuibsys-
( 1)|A| temstatesasW).
Z = − Z 1 (21)
− d N\A⊗ A
A
A⊆XN,A6=∅ W is an optimal compatibility witness if, apart from
•
W itself,nocompatibilitywitnessisfinerthanW.
whered = d ,and1 = 1 .
A i∈A i A ⊗i∈A i
Thenormalisationconstraintmeansthatwearefairlycom-
Proof. UsingQtheprincipleofinclusion/exclusion,wecanex-
paringwitnesses(undertheabovedefinitionαW isfinerthan
press(20)asfollows:
W whenα = 0,eventhoughtheyclealydetectthesameset
6
ofincompatiblesubsystemstates;infact,withoutthisnormal-
n isation,optimalcompatibilitywitnesseswouldnotexist).
Z = ( 1)k+1 z B (22)
 m m Thefirstobservationofnotehereisthatanoptimalwitness
−
Xk=1 A⊆NX,|A|=kml=1mXw∈heInl∈A  isalsotangential:if
 
= (−1)|A| Tr (Z) 1 (23) min W,ρ =λmin >0 (28)
− d A ⊗ A ρ∈Ch i
A
A⊆XN,A6=∅( 1)|A| thenthewitnessW′ definedbyWi′ =Wi−α1i/nisafiner
= − Z 1 (24) compatibilitywitness,andisalsotangentialtoC atthepoint
− d N\A⊗ A
A⊆XN,A6=∅ A ρ∈C thatminimisestheaboveexpression.However,since
whichistherequiredresult. W,ρ =Tr(p(W)ρ) (29)
h i
Equation(21)allowsustoexpressZintheform ni=1Zi⊗ (whereρisastatecompatiblewithρ),thenλminissimplythe
1i,andsofindasuitableW. Forexample,inthecasen=3, minimaleigenvalueofp(W). Hencewehavethefollowing:
givenZ =p(W),asuitableW isgivenby P
Lemma 1. W is tangentialto C if andonly if p(W) has a
W = 1 11⊗Z2⊗13 + 11⊗Z23; (25) zero eigenvalue, and W is optimalonly if p(W) has a zero
1
3d d d − d d d eigenvalue.
1 2 3 1 3 1
1 1 1 Z 1 Z
W = 1⊗ 2⊗ 3 + 2⊗ 13; (26) Givenanon-optimalcompatibilitywitnessW,wemayask
2
3d1d2d3 − d1d2 d2 howtoproduceafinerwitnessW′ fromit. Supposethatthe
W3 = 1 Z1⊗12⊗13 + Z12⊗13. (27) minimum eigenvalue of p(W) is λWmin > 0, and let P be a
3d1d2d3 − d2d3 d3 compatibilitywitnesssuchthat P,ρ 0forallρ (not
h i≥ ∈O
justthe compatiblesubsystem states), with maximumeigen-
Weagainseethefreedomofchoicein here,assomeofthe
W valueλP . Then,byProposition5,theoperator
termscouldappearinmorethanoneoftheoperatorsW ,but max
i
we havechosen this symmetricexpression. This methodfor
λW
establishingW fromp(W)willbeimportantwhenwecome Z′ =p(W) minp(P) (30)
− λP
totalkaboutusingsemidefiniteprogrammingonthecompat- max
ibilityproblem. defines a compatibility witness W′ which is by definition
finer than W, since W′,ρ = W,ρ λWmin P,ρ <
h i h i − λPmaxh i
D. Optimalcompatibilitywitnesses W,ρ .
h i
Therestofthissectionisdedicatedtoprovingtheconverse
GiventwocompatibilitywitnessesW,W′, ifW′ detects ofthis,i.e.:
all the incompatible subsystem states that W does, we can Theorem 5. A compatibility witness W′ is finer than W if
discard W and use W′ instead. More generally, we need a
andonlyifthereexistsP suchthat P,ρ 0forallρ
notion of an optimal compatibility witness, a witness which h i≥ ∈ O
and
cannotbereplacedbyanotherwitnessthatdetectsasleastas
manyincompatiblesubsystemstates. ′
W =(1 ǫ)W +ǫP (31)
In[28],thenotionofanoptimalentanglementwitnessisin- −
troduced. Asaresult,anumberoftheresultsandproofshere whereǫ (0,1).
∈
are similar to those given in [28], but with some differences
duetothedifferentstructureofthisproblem. Beforeweareinapositiontoprovethistheorem,weneed
Let us take compatibilitywitnesses W satisfying the nor- thefollowinglemma,whichweproveherefirst.
malisation constraint Tr(p(W)) = 1. We start with some
Lemma2. Letρ ,/ C. LetI =1/D(d 1 ,...,d 1 ),
definitions: ∈O ∈ 1 1 n n
the reducedstates of the maximally mixed state on . Then
H
A compatibility witness W is tangentialto C if there there exists x (0,1) such that ρ(x) = (1 x)I +xρ is
• existsρ C suchthat W,ρ =0. compatiblewith∈aquantumstateρ(x) ( )−.
∈ h i ∈B H
6
Proof. For all compatibility witnesses satisfying However,byLemma2,foranyρ ,thereexistsx (0,1)
∈O ∈
Tr(p(W))=1,wehave W,I =1. Define such that ρ(x) = (1 x)I +xρ is compatible with a state
h i ρ(x). Hence W,ρ(x−) = W′,ρ(x) ,andhence W,ρ =
c=min W,ρ (32) W′,ρ i.e. Wh =W′i. h i h i
W h i h i
wheretheminimisationisoverallcompatibilitywitnessesW ProofofTheorem5. First suppose that W = (1 ǫ)W′ +
satisfying Tr(p(W)) = 1. Since ρ / C, c < 0. Also, ǫP, with P,ρ 0 for all ρ . Then −W′,ρ =
min W,ρ(x) = (1∈ x) + xc, and be- 1 ( W,hρ ǫi P≥,ρ ), and henc∈e iOf W,ρ h< 0, ithen
causWeo∈fWP,rTorp(po(sWiti)o)n=14h,forx 1i/(1 c)−,ρ(x)iscompatible 1W−ǫ′,hρ <0i−alsoh. i h i
≤ − h i
withanoverallstate. Conversely, take λ defined as above. Then Lemma 3(iv)
tellsusthatλ 1. Ifλ = 1thenW = W′ i.e. thetheorem
The followinglemma is very similar to Lemma 1 in [28], ≥
statementissatisfiedwithP =0. Ifλ>1,thendefine
and hence the proof is almost identical, with some small
1 1
changes.Westatetheproofherehoweverforcompleteness. P = (λW W′); ǫ=1 . (37)
1 λ − − λ
Lemma3. ForacompatibilitywitnessW,letD = ρ −
O|hW,ρi<0},andfortwowitnessesW,W′,Wdefine{ ∈ TanhdenbybydeLfienmitimona,3W(i)-=(iii()1aboǫv)eW, hP′+,ρǫiP≥as0refqouriarleldρ. ∈ O,
−
W′,σ
λ= min h i . (33) AnexampleofawitnessP withtheabovepropertyisone
σ∈DW (cid:12)hW,σi(cid:12) where each Pi is positive. But there also exist witnesses P′
SupposeW′ isfinerthanW,(cid:12)(cid:12)Thenthef(cid:12)(cid:12)ollowingstatements wherePi′ (cid:3)0(sincee.g.wecanaddα1toPiandsubtractα1
(cid:12) (cid:12) fromP ,j =itoformP′suchthat P′,ρ = P,ρ 0for
aretrue: j 6 h i h i≥
(i) W,ρ =0 W′,ρ 0. allρ∈C. Asaresult,checkingtheconditionofTheorem5is
(ii)hW,ρi <0⇒hW,ρi≤ W′,ρ . difficult,butsuchwitnessesP caneasilybeusedto‘optimise’
(iii)hW,ρi >0⇒hλ W,iρ≥h W′,iρ . compatibilitywitnessesasoutlinedabove.
(iv)λh 1,iwithe⇒qualihtyifanid≥onhlyifWi=W′. Toconcludethissection,wenotethattherearemanymore
≥ results on optimal entanglement witnesses in [28] than we
Proof. (i) Suppose W,ρ = 0, and assume W′,ρ > 0. have presented here. However, the differencein structure in
h i h i
Take σ D , and define ρ(x) = xσ +(1 x)ρ, so that compatibilityandentanglementwitnessesmakesitdifficultto
W
∈ −
ρ(x) D forx (0,1). Forsmallenoughx (0,1),we imitatetheideasin[28].
W
have ∈W′,ρ(x) >∈0,contradictingW′ beingfin∈erthanW.
h i
(ii)Suppose W,ρ < 0. Defineσ = ρ+ W,ρ I, so
that W,σ =h0. (i)thienimpliestheresult. |h i| IV. SEMIDEFINITEPROGRAMMINGANDTHE
(iihi) Suppiose W,ρ > 0, and take σ D . De- COMPATIBILITYPROBLEM
W
h i ∈
fine µ = W,ρ σ + W,σ ρ, so that W,µ = 0.
Then(i) imphliesthiat W|′h,µ <i|0 i.e. W,hσ Wi′,ρ In this section we outline how the compatibility problem
W,ρ W′,σ . Dihvidingbiothsidesb|yh W,iσ|h > 0ian≤d can be cast in the form of a semidefinite program, which
|h i|h i |h i|
W,ρ >0yields makesitamenabletosolutionbynumericalmeans.Semidefi-
h i
niteprogramminghasbeenusedinanumberofwaystoattack
W′,ρ W′,σ theseparabilityproblem[7,8,9,10,11],andtheideasutilised
h i h i . (34)
W,ρ ≤ W,σ inthosepapers(particularlythosein[10])actasaninspiration
h i (cid:12)h i(cid:12)
(cid:12) (cid:12) fortheapplicationhere.
Takingtheinfimumwithrespec(cid:12)ttoσ D(cid:12) ontherighthand
(cid:12) ∈ (cid:12)W We will start by giving a brief and by no means compre-
sideofthisexpressionthenyieldstherequiredresult.
hensiveintroductiontosemidefiniteprogramming. Interested
(iv)λ 1followsimmediatelyfrom(ii). ClearlyifW =
W′ than≥λ=1. Conversely,ifλ=1,than(i)and(iii)imply readerscanreferto[5,6]formoredetails. Wewillthenout-
linehowsemidefiniteprogrammingcanbeusedtoattackthe
that W,ρ W′,ρ forallρ C. Henceif ρ is a state
comphatibleiw≥ithhρ, thein Tr[(p(W∈) p(W′))ρ] 0. Let compatibilityproblem, and we also illustrate how it be used
ρ = ψ ψ fori = 1,...,D,with−theset ψ ≥D being to obtain some interesting analyticalresults, most notablyto
ani ort|hoiniohrmi|albasisfor . SinceTr(p(W)){|=iTi}r(i=p(1W′)), disprovetheconjecturein[3].
H
and
D A. Asummaryofsemidefiniteprogramming
′ ′
0=Tr((p(W) p(W ))= Tr[(p(W) p(W ))ρ ] 0,
i
− − ≥
i=1 A semidefiniteprogram (SDP) is a minimisation problem,
X (35)
whichinvolvestheminimisationofalinearfunctionofavari-
wemusthaveTr[(p(W) p(W′))ρi] = 0foralli. Butthe ablex Rm subjecttoalinearmatrixinequality. Formally,
−
orthonormalbasis was arbitrary, and this implies that for all ∈
theproblemis
ρ C compatiblewithastateρ,wehave
∈ Minimise c.x
′ ′ (38)
W,ρ W ,ρ =Tr[(p(W) p(W ))ρ]=0. (36) subjectto F(x) 0
h i−h i − ≥
7
where this: F(x) will representthe potential compatible state, and
theinequalityF(x) 0willcorrespondtothenecessarypos-
m ≥
itivity of any state compatible with the (n 1)-party states.
F(x)=F0+ xkFk, (39) Within the definition of F(x), the matrix−F will represent
0
k=1
X the part of any compatible state that is fixed by the reduced
and F ,F ,...,F are fixed Hermitian matrices. The min- states. The other matrices represent the freedom in the po-
0 1 m
imisationisperformedoverx Rm. Thesetofxforwhich tentialchoice of anycompatiblestate, and as a result do not
the inequality F(x) 0 hold∈s is known as the feasible re- changethepartialtracesofthepotentialstate.
≥
gion. The SDP is termed feasible if this feasible region is LetB bethebasiselementsfor ( )asdefinedin(17).
m
non-empty. Supposeρ ( ) is a quantumstaBte,Hwith Tr (ρ) = ρ(k).
k
∈ B H
A semidefinite program is an example of a convex opti- Anyρcanbewrittenintheform
misation problem, since the feasible region is convex i.e. if
F(x),F(y) 0,thenforλ [0,1], ρ= ρ B . (43)
≥ ∈ m m
m
F(λx+(1 λ)y)=λF(x)+(1 λ)F(y) 0. (40) X
− − ≥
However,definingI = m m =1 ,and,form I ,
k k k
Itisthisconvexitythathasallowedfastalgorithmstobedevel- { | } ∈
opedtosolvesemidefiniteprogramsnumerically,anddetails B(k) = B (44)
aboutthesealgorithmscanbefoundin[5]. m ⊗i6=k i,mi
An important structural feature of semidefinite program- thenwecanobservethat
mingisafeatureknownasduality.Associatedwithaproblem
oftheform(38),usuallyknownastheprimalproblem,isaso Tr (ρ)= ρ B(k) (45)
k m m
calleddualproblem:
mX∈Ik
Maximise Tr(F Z)
subjectto TZ−r=(FZZ†0),Z=≥c .0, (41) i∪.enk.=t1hIekc=onIdCiti(osneeT(r1k9()ρ)),t=heρn(kp)afirtxiaelstρramcefocronmdit∈ioInks.fiSxinρcme
i i form I [33]. Withthisinmind,wedefine
C
∈
Thisdualproblemisasemidefiniteprogramasitcanbecast
B = ρ B (46)
in the form of (38) [5]. There are a numberof relationships 0 m m
betweenthesetwoproblems. mX∈IC
Let x,Z be feasible solutions to the primal and dual wherethecoefficientsρm arefixedbytheappropriatepartial
• problemrespectively.Then traceconditions.
Theremainingbasiselements(B ,wherem I I )area
m C
∈ \
c.x+Tr(F Z)=Tr(F(x)Z) 0 (42) tensorproductoftracelessmatrices(sinceifm / I ,m =1
0 C k
≥ ∈ 6
forallk,andTr(B ) = 0whenm = 1),andsoyieldzero
k,m
and so Tr(F0Z) c.x i.e. any feasible x can be undertracingofanysystem. Asaresu6lt,theoperator
− ≤
used to compute an upper bound on Tr(F Z), and
0
−
any feasible Z can be used to computea lower bound B(x)=B + x B (47)
0 m m
onc.x.
m∈XI/IC
We say the primal problem is strictly feasible if there
• exits x such that F(x) > 0 (i.e. a strict inequality is (where x R(|I|−|IC|)) also satisfies the appropriate par-
∈
tial trace conditions. Therefore, the compatibility problem
satisfied). Similarly, the dualproblemis strictly feasi-
has now been reduced to the problem of finding x such that
ble if there exists Z > 0 satisfying Tr(F Z) = c . If
i i
eitherofthetwoproblemsarestrictlyfeasible,thenthe B(x) 0.
≥
optimiumvaluesforbothproblemsareequal[5]. AnequivalentproblemistoconsidertheSDPgivenby
Finally,ifx,Z achievetheoptimumvaluesforthepri- Minimise t
• (48)
mal and dual problems respectively, then (42) implies subjectto B + x B +t1 0.
0 m∈I/IC m m ≥
that Tr(F(x)Z) = 0. However, since F(x) 0 and
Z 0,F(x)Z = 0. Thisisknownasthecom≥plemen- TheeffectofthisproblePmistofindanoperatorwiththeappro-
≥
taryslacknesscondition. priate reduced states with the maximum minimal eigenvalue
(givenby t). Ift > 0,thenB(x)canneverbepositive,and
−
the (n 1)-party states are not compatible with a quantum
B. FormulatingthecompatibilityproblemasanSDP stateof−theentiresystem.
Beforewecontinue,wenotethatthereisasecondmethod
Wewillnowdiscusshowtoexpressthecompatibilityprob- ofcalculatingthefixedoperatorB : wenotethattheexpres-
0
lem as a semidefinite programof the form (38). The idea is sionforB givenin(46)issimilartothatgivenin(20),andso
0
8
thetechniquesusedintheproofofProposition6canbeused thenD = dn, leadingtoexponentialcomplexityinn. How-
todeducethat ever, the complexity of the algorithm is still polynomial in
theproblemsize,sincespecifyingthereducedstatewhichex-
( 1)|A| cludessystemkinvolvesspecifyingaD/d D/d Hermi-
B0 =− −d ρN\A⊗1A, (49) tian matrix i.e. (D/d )2 parameters. So ikn×total wke have a
A k
A⊆XN,A6=∅ problemsizeof
where ρ (B N) is the subsystem state for the systems
B ⊂ n
labelled by the elements of B obtained by partial traces of s=O D2 d−2 =O(D2) (54)
appropriate(n 1)-partystates. k !
− kX=1
hencethetimecomplexityisboundedabovebyapolynomial
C. Thedualproblemandcompatibilitywitnesses ins. Hencetheproblemliesinthe complexityclassP. The
exponentialcomplexity in n merely reflects the fact that we
mustsupplyanexponentiallyincreasingamountofinitialdata
Thedualproblemforthecompatibilityproblemisgivenby
intheformofthereducedstates.
Maximise Tr(B Z) Note that this does not contradict the results of [29] dis-
0
subjectto Z−=Z†,Z 0, cussedearlier, asthecompatibilityproblemconsideredthere
Tr(Z)=1,≥ (50) involveshavingonlyanumberofreducedstatesthatispoly-
Tr(B Z)=0forallm I I . nomialinthenumberofquantumsystemsn,andeachreduced
m C
∈ \ state describesthe state of a numberof systems below some
However,Z = p(W) forsomecompatibilitywitnessW: fixedconstant. Hencetheproblemsizehereisonlyapolyno-
Since I I = m m = 2,...,d2,k = 1,...,n , and mialinn,leadingtoverydifferentconclusionsregardingthe
B ,...\,BC f{orms| abkasisfortheskpaceoftraceless}oper- complexityoftheproblem.Wewillreturntothisissuelater.
k,2 k,d2
k
atorson ,Proposition5impliesthatZ = p(W)forsome
k
H
W .
∈W E. NewinsightsfromthecompatibilitySDP
Theupshotofthisis thatiftheoptimumvalueofthedual
problem is positive, the Z which realises this maximum is
suchthatZ =p(W),and We have produced computer code for the compatibility
SDP using MATLAB and a freely available SDP numerical
n solver known as SEDUMI [30]. In much the same way as
0>Tr(B0Z) = Tr B0 Wi 1i (51) wasachievedwhensemidefiniteprogrammingwasappliedto
" k=1 ⊗ !# the separability problem, analytical results can be gained by
X
n inspectionofnumericalresultsduetotheaccuracynumerical
= Tr(ρiWi) (52) SDPsolversworkto.
k=1 A simple exampleofthiscomesfromconsideringthefol-
X
= W,ρ (53) lowingtwo-qubitsubsystemstatesofathreepartysystem:
h i
i.e. W is a compatibility witness that detects the fact that 1
ρ(1) =ρ(2) =ρ(3) = Ψ Ψ ; Ψ = (00 + 11 )
the(n 1)-partyreducedstatesρi arenotcompatiblewitha | +ih +| | +i √2 | i | i
−
quantumstateoftheoverallstate[34]. (55)
which are not compatible with an overallstate (if qubits 1,2
are in a pure entangled state, then qubit 3 must be in a pure
D. Complexity state, which is not the case here). The primalSDP confirms
this incompatibility, and the dual SDP can be used to deter-
The SDP (38) involves optimising over the space Rm, mineanumericalcompatibilitywitness. Fromthisresult, an
analyticcompatibilitywitnesscanbederivedthatdetectsthis
and dealing with the positivity of N N Hermitian ma-
× incompatibility.Let
trices. Numerical SDP solvers are known to have com-
plexity O(m2N5/2) (O(N1/2) iterations, each taking time
1
O(m2N2)[5])[35]. Z = (ψ1 ψ1 + ψ2 ψ2 ) (56)
2 | ih | | ih |
Inourcase,ifwedefineD = d ,then
i i
where
The number of variables iQs equal to I I +1 =
C
• n (d2 1)+1=O(D2); | |−| | 1
i=1 i − ψ1 = (000 011 101 110 ), (57)
| i 2 | i−| i−| i−| i
TQhedimensionofthematricesequalsD. 1
• ψ = (001 + 010 + 100 111 ). (58)
2
| i 2 | i | i | i−| i
andsothecomplexityofthecompatibilitySDPisO(D13/2).
Thisis polynomialin the dimensionof everyindividualsys- This can easily be checked to be a compatibility witness by
tem,butisexponentialinthenumberofpartiesi.e. ifd =d, usingProposition6. DuetothesymmetryofZ,allthethree
i
9
two-partyreducedstatesofZ areequal: Let σ = 1(1 1 1 1). Let us define the following
4 | ih |⊗ ⊗
two-qubitsubsystemstatesofathreequbitsystem:
1
Z =Z =Z = (Ψ Ψ + Φ Φ ); (59)
12 13 23 − − + +
2 | ih | | ih | ρ(k)(p)=(1 p)Tr σ+pΨ Ψ (63)
k + +
− | ih |
1 1 wherep [0,1]. Thissetofsubsystemstatesisclearlycom-
Ψ = (00 11 ), Φ = (01 + 10 ), ∈
− + patible with an overall state when p = 0, and incompatible
| i √2 | i−| i | i √2 | i | i
(60) withanyoverallstatewhenp=1. Wecancalculate∆(ρ)for
and all the one-party reduced states of Z are equal to 1/2. thesereducedstatestobe
HenceacompatibilitywitnessW correspondingtoZisgiven
by ∆(σ) = (1 p)1/4+3p/2(φ1 φ1 + φ2 φ2 )
− | ih | | ih |
p/2(ψ ψ + ψ ψ ) (64)
1 1 2 2
1 1 − | ih | | ih |
W =W =W = + (Ψ Ψ + Φ Φ )
1 2 3 − − + +
−12 4 | ih | | ih | where
(61)
which leads to 3 Tr(ρ(i)W ) = 1/4, detecting the in-
compatibilityasreiq=u1ired. i − φ1 = 1 (3000 + 011 + 101 + 110 ), (65)
P | i √12 | i | i | i | i
ThecompatibilitySDPcanbeusedtohelpusproduceaset
1
of2-qubitreducedstatesthatdisprovestheconjecturein[3]. φ = (001 + 010 + 100 +3111 ), (66)
2
Forthecaseof3qubits,theconjecturestatesthatρ ,ρ ,ρ | i √12 | i | i | i | i
12 13 23
are the 2-qubit reduced states of a three-qubit state ρ if and
onlyif andhence0 ψ ∆ψ 1whenp 1/3.Iftheconjecture
≤h | | i≤ ≤
is true, then when p 1/3, there should be an overallstate
≤
0 ∆(ρ) 1 ρ ρ ρ +ρ +ρ +ρ 1 (62) compatible with the above subsystem states. However, the
1 2 3 12 13 23
≤ ≡ − − − ≤ primalSDPcanbeusedtoshowthisisnottrue,andproduce
where the reduced states are padded out with identities as anumericalcompatibilitywitnessillustratingthisfact. From
statedinTheorem1,andρisthe3-vectorof2-partyreduced this,ananalyticalcompatibilitywitnesscanbeobtained. For
states. thecasep=1/4,itcomesfromtheoperator
2959 0 0 102 0 1715 1715 0
− − −
0 24865 1005 0 766 0 0 1715
 0 1005 24865 0 766 0 0 −1715
−
1 102 0 0 45033 0 766 766 0
Z = 100000 −0 766 766 0 46 0 0 102  (67)
 1715 0 0 766 0 1006 1005 −0 
 − 
 1715 0 0 766 0 1005 1006 0 
 − 
 0 1715 1715 0 102 0 0 228 
 − − − 
 
which can be verified to be positive, to be of the form Z = i.e.Z isanoptimisationof∆(Z+Z′),andcandetectincom-
p(W)throughProposition6,andtowitnesstheincompatibil- patibilitythatthecondition∆(ρ) 0cannot.
≥
ityoftheabovesubsystemstatesforp = 1/4. Thisexample
hencedisprovestheconjecturegivenin[3]. Thiswitnesscan
alsobeseentobeanexampleoftheoptimisationprocesswe
sawinsectionIIID:Defining
Z′ =10−6diag(0,0,0,0,2913,23859,23859,44805)+α1
(68)
V. SOLUTIONSOFVARIANTSOFTHECOMPATIBILITY
andconsideringthedecompositionZ =(Z +Z′) Z′,then
− PROBLEM
itcanbeshownthat
(a) Z+Z′ =∆(Z +Z′);
Thisnew techniqueof using semidefinite programmingto
(b) Z′ =p(P); numerically solve the compatibility problem can be applied
to important variants of the compatibility problem we have
(c) Forlargeenoughα, P,ρ 0forallρ . considered.Wediscusstheseapplicationsinthissection.
h i≥ ∈O
10
A. Partialknowledgeofreducedstates A density state ρ is then totally symmetric if and only if
P ρP = ρ. However, since ρ is positive and of trace one,
S S
Asimplemodificationoftheproblemistoconsiderasitu- thisrelationissatisfiedifandonlyifTr(PSρPS) = Tr(ρ) =
ationwhereweareonlygivenapartialsubsetofthereduced 1. Finally, since PS2 = PS, and by the cyclic property of
states. Theproblem(48)caneasilybemodifiedtosolvethis thetracefunction,we havethatρ istotallysymmetricifand
problem: The number of basis operators Bm that appear in onlyifTr(ρPS) = 1. NotingthatTr(PSρPS) ≤ Tr(ρ) = 1
thefixedoperatorB is reducedaccordingto whichreduced (becausePS isaprojection),wecanchangetheconditionto
0
states are known, and the coefficientsof the remainingbasis Tr(ρPS) 1. Taking all this into account, we can hence
≥
modify problem (48) to accommodate the totally symmetric
operators are then determined by the SDP solver. The dual
condition:
problemtothisprogramwillstillproduceacompatibilitywit-
nessZ = p(W),butwithsomeextraconditionsoftheform
Minimise t
Tr(ZB ) = 0 for the the basis operatorsthatare no longer
m subjectto B + x B +t1 0
fixedintheoperatorB0. 0 m∈I\IC m m ≥
1+Tr(B P )+ x Tr(B P ) 0
Thecomplexityoftheproblemdependsontheproblemsize − P 0 S m∈I\IC m m S ≥
(71)
i.e. the numberof knownreducedstates, and the numberof
P
(notethattwolinearmatrixinequalitiesB (x) 0,B (x)
systems described by these states. Let us consider the case 1 2
≥ ≥
0canbecombinedintooneoftheformB (x) B (x) 0,
discussedin [29] (toreiterate: wehavea numberof reduced 1 2
⊕ ≥
sothisreallyisasemidefiniteprogram).
states p(n), a polynomialin n, each describing a numberof
WecanproduceasimilarSDPforthefermioniccase. The
systems less than some constant k). In this case, the prob-
space of totally antisymmetric pure states is spanned by an
lem size is O(p(n)), whereas the dimension of the matrices
involvedisD, andthereareO(D2)freevariables(sincefix- orthonormalbasis{|Axi}1≤x1<...<xn≤d,where
ing polynomially many reduced states describing at most k
systems can only fix polynomiallymanycoefficientsin B ). 1
0 A = (ρ)x ... x (72)
HencethecomplexityisagainO(D13/2),butthisisnolonger | xi n! E | ρ(1)i | ρ(n)i
polynomialintheproblemsize. r ρX∈Sn
Thisissimplyonecomplexityscenariowecouldimagine;
where (ρ) denotesthe sign (parity) of the permutationρ
we could place other conditions on the number of reduced E ∈
S . Theprojectionontotheantisymmetricsubspaceishence
n
states we know, and how many states they can describe. As
givenby
complexityis not our main focus here, we leave these ques-
tionsforfurtherinvestigation.
P = A A (73)
A x x
| ih |
x1<X...<xn
B. Thebosonicandfermioniccompatibilityproblems andsimilarlythesemidefiniteprogramforthefermioniccom-
patibilityproblemisgivenby
Ifadensitystateρrepresentsthestateofeithernbosonsor
n fermions, there are some additional constraints that much Minimise t
beplacedonρ. Apurestate|ψi ∈ Hrepresentingnbosons subjectto B0+ m∈I\IC xmBm+t1≥0
mustbetotallysymmetricunderparticleexchanges(orswaps 1+Tr(B P )+ x Tr(B P ) 0
− P 0 A m∈I\IC m m A ≥
betweenHilbertspaces);whereasanypurestateofnfermions (74)
P
shouldbetotallyantisymmetricunderparticleexchanges. We can reduce number of free variables in both of these
Since we are dealing with identical particles, the n single problemsbynoticingthatforboththebosonicandfermionic
particle Hilbert spaces are all identical; let their dimension case, the density matrix of n particles is symmetric under
be d. Also, let Bk,m be equal for all k to Bm. The total anyparticleexchange(intheantisymmetriccase, Ax isto-
| i
(anti)symmetry means that all of the k-party reduced states tally antisymmetric, but any operator A A is symmet-
x y
| ih |
will be identical; as a result we will refer to any k-party re- ric because any factors of 1 cancel out). Hence the den-
−
ducedstateasthek-particlereducedstate. sity matrices only depend on symmetrised basis operators
Thespaceoftotallysymmetricpurestateshasanorthonor- BS ,where
{ m}1≤m1≤...≤mn≤d
malbasis S ,where
{| xi}1≤x1≤...≤xn≤d
n
S = n1(x)!...nd(x)! 1/2 x ... x BmS = Bmρ(k)!. (75)
| xi n! | ρ(1)i | ρ(n)i ρX∈Sn Ok=1
(cid:18) (cid:19) ρX∈Sn
(69) Furthertothis,thetwosemidefiniteprogramscanbemodified
(where n (x) is defined to be the number of instances of k again to deal with cases where we have only partial knowl-
k
withinx). Theprojectiononthissubspaceisgivenby edgeofthe subsystemstates, byadjustingthe inequalitiesin
the same way as outlined in section VA. This would allow
P = S S (70) us for example to deal with the N-representability problem
S x x
| ih | discussedinsectionIIB.
x1≤X...≤xn