Multipartite correlations in mutually unbiased bases

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by  David Sauerwein

Multipartite correlations in mutually unbiased bases
David Sauerwein,1 Chiara Macchiavello,2 Lorenzo Maccone,2 and Barbara Kraus1
1Institute for Theoretical Physics, University of Innsbruck, Technikerstraße 21a, A-6020 Innsbruck, Austria
2Dip. Fisica and INFN Sez. Pavia, University of Pavia, via Bassi 6, I-27100 Pavia, Italy
We introduce new measures of multipartite quantum correlations based on classical correlations
in mutually unbiased bases. These classical correlations are measured in terms of the classical
mutualinformation, which hasa clear operational meaning. Wepresentsufficient conditions under
which these measures are maximized. States that have previously been shown to be particularily
important in the context of LOCC transformations fulfill these conditions and therefore maximize
the correlation measures. In addition, we show how the new measures can be used to detect high-
dimensional tripartite entanglement by using only a few local measurements.
7
1
0
2 I. INTRODUCTION find new truly multipartite applications of the quantum
correlations they exhibit.
n
a The fact that quantum systems can exhibit correla- To this end, different approaches are being pursued.
J tionsthathavenoclassicalanalogis atthe coreofquan- One approach is to qualify and quantify quantum corre-

5 tum theory. In the last decades it has been realizedthat lations in terms of their transformation properties. As
2 these quantum correlations are not only of fundamental entanglement is a resource under LOCC, it is natural
interest; they can moreover be manipulated to achieve to study LOCC transformations of multipartite quan-
]
h tasks that are impossible with classical devices. These tum states [3]. Moreover, more general transformations
p applications include quantum communication, quantum have been considered, such as separable transformations
- computation and quantum simulation [1]. Hence, a lot [2,10]orǫ-nonentanglingoperations[11],despitethefact
t
n of effort has been devoted to reach a better understand- that they lack a clear physical meaning. Another ap-
a ing of quantum correlations and to find new ways to de- proach is to find measures to quantify the quantum cor-
u
tect, quantify and manipulate them. While the theory relations contained in multipartite states. These include
q
[ of bipartite quantum correlations is well developed, the approaches based on, e.g. quantum entropic quantities,
multipartite case still holds many open problems [2, 3]. concurrences and polynomial invariants [2, 3].
1
The drastic difference between the bipartite and mul- In recent work [12, 13], the concept of complementar-
v
ity, another distinctive feature of quantum theory, has
2 tipartite case is particularly clear in the case of quan-
been used to quantify quantum correlations of bipartite
1 tum entanglement, which is fundamental to many ap-
4 plications in quantum information theory. For bipartite quantum systems. In the present work, we generalize
7 quantum systems there exists (up to local unitary oper- these results to the multipartite setting. Two observ-
0 ables are called complementary if certainty about the
ations) one single most entangled state, the maximally
1. entangled state. Its name is justified by the fact that measurementoutcomeofoneobservableimpliescomplete
0 spatially separated parties can deterministically obtain uncertainty about the outcome of the other observable.
7 Forcomplementaryobservablesthe absolutevalueofthe
any other bipartite state from the maximally entangled
1 overlap between any eigenstate of the first observable
state by manipulating it via local operations assisted by
:
v classicalcommunication(LOCC)[4]. Themaximallyen- with any eigenstate of the second observable is constant
i tangledstateishencetheoptimalbipartiteentanglement [14] and their eigenbases are called mutually unbiased.
X
Mutuallyunbiasedbases(MUBs)havemanyapplications
resource. Itmaximizesanymeasureofentanglementand
r rangingfromquantumstate tomography,quantumerror
a serves as a standard with which the resourcefulness of
correction, and quantum cryptography to the detection
other entangled bipartite quantum states can be com-
of entanglement [14].
pared. This situation changes drastically if one enters
the multipartite realm. Multipartite quantum systems In this work, we use the notion of complementary
can be entangledin many inequivalentways [5] and gen- observables to study quantum correlations. The cen-
erallytheredoes notexistaunique maximallyentangled tral idea is that only quantum correlated states can ex-
state thatcanserveasa standardto assessthe resource- hibit strong classical correlations in the measurement
fulness of other quantum states [6]. In the last years, outcomes of local complementary observables. We use
several classes of multipartite entangled quantum states mutually unbiased bases to define a novel set of mea-
withinterestingpropertieshavebeenidentifiedandsome sures of quantum correlations, . has a clear
N N
{C } C
of these have motivated novel multipartite applications, operational meaning as it measures multipartite quan-
such as one-way quantum computing [7], secret sharing tum correlations in terms of the maximal classical mu-
[8] or applications in quantum metrology [9]. But the tual information that a single party can share with the
correlations these states contain are nevertheless often other parties by measuring the state in N mutually un-
only partially understood. A better understanding of biased bases. This generalizes the bipartite measure of
their properties seems however crucial if one wants to “complementary correlations” introduced in [12] to the
2
multipartite case. We investigate which states are max- lations. Moreover,wereviewconceptsfromentanglement
imally correlated with respect to and present some theory that we use in this work.
N
C
necessary and some sufficient conditions for this to be
the case, thereby providingnew physicalinsight into the
quantumcorrelationstheycontain. Moreover,wepresent
severalotherapplicationsof N forpureandmixedquan- A. Mutual information
C
tum states. One application is entanglement detection.
While the effort required to certify entanglement gener-
We denote by ( ) the set of density operators on
ally increases rapidly with the system size [15], we show
D H
uhionweCtNripcaarntibteeeunsteadntgoledmeteenctteemvepnlohyiignhg-doinmlyenasiofenwallgoecna-l tishemHeailsbuerretdsopnaceρ H. I(fCIad)b,atshisisByi=eld{s|ba(i)sip}edic=−i0fi1cofouCItd-
∈ D
measurementsettings. Thestructureoftheremainderof come i 0,...,d 1 , with probability p(i ;ρ). The
∈ { − } |B
this paper is the following. In Sec. II we introduce our uncertainty with respect to the outcome of the mea-
notation and recall important definitions and results on surement is quantified by the Shannon entropy of this
MUBs and entropic uncertainty relations. We moreover probability distribution, which we denote by H( ρ) =
B|
reviewsomeconceptsfromentanglementtheorythatare p(i ;ρ)log(p(i ;ρ)). Here and in the following
− i |B |B
used in this work. After that we present the following thePlogarithm is taken to base two. As we are only in-
results. terested in the entropic properties of measurement out-
comes, and not in the corresponding outcomes, we often
(i) New correlation measures (Sec. III): We identify anoperatorX (no degeneracy)with its eigenba-
present a set of new correlation measures, N , sis and write e.g. H(X ρ) H( ρ) and analogously
and discuss its basic properties. {C } forBotherquantities. Moreo|ver≡,foraBs|tateρofacompos-
itesystemofnsubsystemswemeasurethemutualdepen-
(ii) States maximizing (Sec. IV): In Lemma 1
CN dence of the outcomes of measurements on different sets
we first present a generalized version of a result
ofsubsystemsviatheirmutualinformation. Morespecif-
from[12]onbipartitestatesthatmaximize (see
CN ically, we consider sets of subsystems A,B 1,...,n
also[16]). InLemma3wepresentnecessaryandin ⊂ { }
that form a partition of 1,...,n , i.e. A B = and
Theorem5sufficientconditionsfora(multipartite) A B = 1,...,n , and{measurem}ents in t∩he basi∅s (l)
statetoyieldthemaximalquantumcorrelationsas ∪ { } B
on subsystem l. The mutual information between the
measured by .
CN measurement outcomes in A and B is then
(iii) Examples of pure states maximizing (Sec.
N
C
V): We use Theorem 5 to show that several previ- I( (l) : (l) ρ)
l∈A l∈B
ously studied few-body quantum states maximize {B } {B } |
=H( (l) ρ) H( (l) (l) ;ρ). (1)
,discusspropertiesof(n>3)-partitepurestates l∈A l∈A l∈B
N {B } | − {B } |{B }
C
that maximize and present examples thereof.
N
C
Moreover,weinvestigatehow changesunderlo- Here, we used the conditional entropy of the outcomes
N
C
cal operations assisted by classical communication obtained at A conditioned on the outcomes of B,
(LOCC).
H( (l) (l) ;ρ)
(iv) Detection of mixed state entanglement (Sec. l∈A l∈B
{B } |{B }
tVrIip):aWrtietesheonwtahnogwlemCNenctan(Lbeemumsead6toanddeteLcetmgmenaui7n)e. =H({B(l)}l∈{1,...,n}|ρ)−H({B(l)}l∈B|ρ).
In particular, we demonstrate that even for high-
Note here that (l) and (l′) are allowed to be different
dimensionalsystemsandusingonly twolocalmea-
B B
forl =l′. Iftheglobalsystemispartitionedintoonesub-
surement settings, can be employed to detect
CN syste6m, j, and the rest, 1,...,n / j ¯j, we often use
genuine tripartite entanglement in the vicinity of
the generalized GHZ state. the notation I( (j) : ({¯j) ρ) for}th{e }sa≡ke of readability
B B |
and analogous abbreviations for other quantities. That
(v) Generalization to mutually unbiased mea- is, I( (j) : (¯j) ρ) denotes the mutual information be-
surements (Sec. VII): We discuss how the defini- tweenBsubsysBtem|jandtherestwhenthebases (l) are
l
tionof N canbegeneralizedfrommeasurementsin measured. Let us denote the dimensions of th{eBsu}bsys-
C
MUBs to, e.g., include measurements in mutually tems A and B with D = d for S A,B . Then
unbiased measurements (MUMs) [17]. the mutual informatioSn inQElq∈.S(1l) is upp∈e{r boun}ded by
min log(D ),log(D ) . The bound is reached iff the
A A B
{ }
outcomes of the measurements by one subset of parties
II. PRELIMINARIES AND NOTATION S A,B withD =min D ,D arecompletelyun-
S A B
∈{ } { }
certain, but as soon as the measurement results of the
Inthissubsectionweintroduceournotationandrecall other subsystems are known, one can predict them with
important results on MUBs and entropic uncertainty re- certainty.
3
B. Mutually unbiased bases theothersisexpressedinthewell-knownMaassen-Uffink-
inequality[21]. Fortwoarbitrarybases ′ = b′(i) d−1
B1 {| 1 i}i=0
Letusnowreviewsomedefinitionsandresultsconcern- and ′ = b′(i) d−1 it states that
B2 {| 2 i}i=0
ingmutuallyunbiasedbases(see[14]forarecentreview).
Forad-dimensionalsystemasetofbases{Bk}Nk=1,where H(B1′|ρ)+H(B2′|ρ)≥−log(c), ∀ρ∈D(H), (3)
= b (i) d−1, is called a set of mutually unbiased
Bbfoakrseasll({iM,|jUkBsi)0},iif=..|0h.b,kd(i)|1bk.′(Fj)rio|m2 =th1e/ddeffionritailolnki6=tisk′claenadr w{Bhke}reNk=c1 t=hismyaiexlid,js|htbh′1e(ie)n|bt′2r(ojp)iic|2u.ncFerotraiantsyetreloaftiMonUBs
tphraotpearntyyob∈fats{hiseisnysatesm−et’so}sftaMteUtBhsa,t{iBskc}oNkm=p1,ledmeesnctraibreystoa H(Bk|ρ)+H(Bk′|ρ)≥log(d), ∀k 6=k′. (4)
the others. That is, knowing the outcome of a measure- This relation shows that there is a tradeoff between the
ment in basis that projects the quantum state onto prior knowledge one can attain about the outcomes of
k
B
bk(i) impliesthatonehasnopriorknowledgeaboutthe measurements in basis k and k′.
| i B B
outcomeofameasurementinacomplementarybasis,i.e. By simply applying inequality Eq. (4) to always two
H( k′ bk(i) bk(i))=log(d), for all k′ =k. different MUBs it is trivial to find the uncertainty rela-
B || ih | 6
There exist at most N = d+1 MUBs inCId [18] and tion
a set of two MUBs always exists [14]. If the maximal
N
N
number of d + 1 MUBs is reached, the corresponding H( ρ) log(d). (5)
k
set is called complete. A complete set of MUBs can be Xk=1 B | ≥ 2
constructed if d is prime or a power of a prime (see e.g.
[19, 20]). For the smallest dimension which is not of this In [22] this relation was improved to
kind, i.e. for d = 6, it is not known whether a complete
N
set of MUBs exists. N +d 1
H( ρ) Nlog − . (6)
k
In this work we often use that a particular complete B | ≥− (cid:18) dN (cid:19)
Xk=1
set of MUBs is given by the eigenbases of some of the
generalizedPaulioperatorsifdisprime. Theirdefinition As mentioned above,a complete set of N =d+1 MUBs
is as follows. For every k = (k ,k ) 0,...,d 1 2 a isknowntoexistifdisapoweroftwo. Thelowerbound
1 2
∈ { − }
generalized Pauli operator is defined as can then be further improved to [23]
S =Xk1Zk2, (2) d+1 d d d d
d,k d d H( ρ) log + +1 log +1 . (7)
k
B | ≥ 2 2 (cid:18)2 (cid:19) (cid:18)2 (cid:19)
where Xk=1
d−1
D. Concepts from entanglement theory
X = k+1 mod d k ,
d
| ih |
kX=0
Let us now review some concepts from entanglement
d−1
Z = ωk k k , theory that we use in this work. Two states are called
d Xk=0 d| ih | local-unitarily (LU)-equivalent if they can be converted
into each other by applying local unitaries. Moreover,
and ω = exp(2πi/d). For d = 2 the operators in Eq. we denote the maximally entangled bipartite state by
d
(2) are proportional to the Pauli operators. The eigen- φ+ d−1 i i . ThegeneralizedGHZstateofnsub-
| i∝ i=0 | i| i
basesofthesetofgeneralizedPaulioperators Sd,(0,1) systemsPwithlocaldimensiondisdenotedby GHZd,n
{ }∪ | i∝
S d−1 constitute a complete set of MUBs for d d−1 i ⊗n.
{ d,(1,m)}m=0 i=0 | i
prime [24]. The eigenvectors of S , i d−1, fulfill PThe first concept that we briefly review here are
S i = ωi i . Notice also thatdt,hke {ei|gkein}bi=as0is of Z stochastic LOCC (SLOCC) classes. Two states ψ and
isd,tkh|ekicompudt|aktiional basis and the mutually unbiasedd φ are in the same SLOCC class if ψ can be| tirans-
| i | i
Fourier basis is the eigenbasis of X , i.e. i = i formed into φ and vice versa via LOCC with some fi-
and i =1/√d d−1 ωmi m fodr i 0(cid:12),(.0.,1.),(cid:11)d 1| i. nite probabil|itiy of success. The states are then called
(1,0) m=0 d | i ∈{(cid:12) − } SLOCC equivalent. The states that are SLOCC equiv-
More(cid:12)over,(cid:11)theeigenbPasesofZd,Xd constituteasetoftwo
(cid:12) alent to ψ constitute its SLOCC class. It is known
MUBs for any d [14]. | i
that there are only two different SLOCC classes of gen-
uinely tripartite entangled three-qubit states [5]. The
first class is represented by the GHZ state, GHZ ,
2,3
C. Entropic uncertainty relations | i
and the second class is represented by the W state,
W 100 + 010 + 001 . There are, however, in-
| i ∝ | i | i | i
The fact that each basis in a set of MUBs describes a finitely many SLOCC classes of genuinely multipartite
property of the system’s state that is complementary to entangled three-qutrit or four-qubit states [25, 26].
4
The second concept that we review here is the one of Here we used the notation (l)
{Bk }k,l ≡
the maximally entangled set (MES) introduced in [6]. (l) . In Eq. (12) we optimize
It is the set of all truely multipartite entangled states {Bk }k∈{1,...,N},l∈{1,...,n}
over all such sets of MUBs. Let us mention here again
in a Hilbert space, H, that cannot be obtained from any that (l) and (l′) are allowed to differ for l = l′. That
otherLU-inequivalentstateviaLOCC.Thatis,theMES Bk Bk 6
is,the basesonthe differentsubsystemscanbedifferent.
is the minimalsetfromwhichanyother truelymultipar-
Moreover, all bases on any subsystem are mutually
tite entangled state can be generated deterministically
unbiased.
viaLOCC.TheMESisthusageneralizationofthemax-
Let us first comment on some general properties of
imallyentangledbipartite stateto multipartitequantum
. Notice first that is defined for any multipartite
systems. Note that, while the MES of bipartite systems CN C2
system as there always exist two MUBs on each subsys-
contains only φ+ , the MES of three-qubits contains al-
| i tem. IfasetofL>2MUBsexistsoneachsubsystemwe
ready infinitely many states [6, 27, 28].
obtain a whole class of correlation measures, L .
Finally, we review the definitions of the source and {CN}N=2
As the maximal number of MUBs on subsystem l is up-
accessible entanglement, which were introduced in [29].
per bounded by d +1 the number of functions in this
For a quantum state ρ ( ) one can define the sets l
∈D H class is L minldl + 1. In order to make the func-
≤
Ms(ρ)={σ ∈D(H) s.t. σ −L−O−C−C→ρ}, (8) tizioendssiunch{CtNha}tctohmepyacraanblreetaocheaacthmotohsetr1t/hneyanrenloogrm(d˜a)l-,
Ma(ρ)={σ′ ∈D(H) s.t. ρ−L−O−C−C→σ′}, (9) where d˜i = min{di, l∈¯idl}, which is indepPeni=de1nt ofiN
and saturated iff allQof the mutual informations in Eq.
i.e. the set of states from (to) which ρ can be obtained
(11) are maximized. In this work we consider only sys-
(transformed) via LOCC, respectively. For any mea-
temsforwhichd˜ =d foralli 1,...,n suchthatthe
sure µ on the state space one can measure their volumes i i ∈{ } n
maximalvalueofthefunctions is1/n log(d ).
aVnkd(ρa)c≡cesµs(ibMleke(nρt)a)nfgolremke∈nt{ass,a} and define the source Moreover, note that for N > N{′CeNv}ery statePmi=a1ximizinig
N also maximizes N′.
C C
Es(ρ)≡1− supVs(Vρ)(σ) and Ea(ρ)≡ supVaV(ρ)(σ), (10) witFhortbhiepaernttirtoepsiycstmemeassuC2recooifnc“icdoemspulpemtoenntoarrmyacliozrarteiloan-
σ s σ a
tions” introduced in [12], where it has been shown that
respectively. The former quantifies how easy it is to
onlyentangledbipartitestatescanexhibitcorrelationsin
generate ρ from other states via LOCC, while the latter
mutuallyunbiasedbasesthatarestrongenoughtoexceed
quantifies the potentiality of ρ to be converted to other
a certain value of . In Sec. IV we review these results
2
states via LOCC. Due to their operational character, C
and show how they can be extended to the multipartite
it is easy to show that they are indeed entanglement
case. In the subsequent sections we investigate the set
measures [29]. The source and accessible entanglement
of correlation measures in detail and use them to
N
and generalizations thereof have been calculated and {C }
studycorrelationsofmultipartite pureandmixedstates.
usedtostudyandcharacterizefew-bodyentanglementin
Weidentifynecessarypropertiesofstatestomaximize
N
[29, 30]. Here, we compare these entanglement measures C
and present also conditions that are sufficient for max-
for some states with the new correlation functions we
imization. Moreover, we use to study the entan-
N
introduce in the next section. {C }
glement properties of multipartite quantum states. We
show, for example, how it can be used to detect even
high-dimensional genuine tripartite entanglement using
only few measurement settings.
III. NEW CORRELATION MEASURES
In this subsection we introduce a set of new correla- IV. STATES MAXIMIZING CN
tion measures for multipartite quantum systems based
on measurements in mutually unbiased bases. We con-
In this section we determine some properties a state
sider a n-partite quantum system with Hilbert space ρ ( n CIdi) necessarily has to fulfill in order to
eHxi=stsNonni=e1CaIcdhi fsourbwsyhsitcehmalse∈to{f1N,..≥.,n2}M.UWBes,th{eBnk(l)d}eNkfi=n1e, roevae∈crh,DwtheNepuirp=ep1seenrtbocounndditCiNon(sρ)th=a1t/anrePsniu=ffi1lcoiegn(dti)t.oMreoarceh-
the function
this bound. We first consider bipartite states and then
N n proceed with a study of the multipartite case.
(l) 1 (l) (¯l)
C (ρ, )= I( : ρ), (11)
N {Bk }k,l nN Bk Bk |
Xk=1Xl=1
A. Bipartite states
and we introduce the correlationfunction
CN(ρ)={Bmk(l)a}xk,lCN(ρ,{Bk(l)}k,l). (12) entIanntghliesdsustbasteecstiaornewtheeexotnelnydstthaetersesinultt(hCIadtmCaIxdi)mtahlalyt
D ⊗
5
maximize (see [12]) to a more general situation (see where q˜ are probabilities, V are isometries mapping
2 k k
also [16]).C In order to do so, we use Holevo’s Theo- CId toCI{d′ a}nd where Vk and Vk′ have orthogonal images,
rem[31],whichwereviewhereforthesakeofreadability. i.e. Vk†Vk′ =0, for k 6=k′.
It is easy to see that the state ρ can be reversibly
Let us consider the following bipartite scenario with AB
transformedinto the maximally entangledstate φ+ via
parties Alice (A) and Bob (B) (see also [1]). Suppose | i
local operations of Alice. ρ hence contains the same
Alice encodes the random variable X 1,...,m us- AB
ing the ensemble {pX(i),ρX(i)}mi=A1 o∈f q{uantumAs}tates entFaonrgtlhemeseankteaosf|rφe+adi.ability,wegivehereonlyanoutline
and Bob performs a measurement corresponding to
of the proof, which is presented in Appendix A in full
the positive-operator valued measure (POVM) elements
Q (i) mB on this ensemble to obtain the random vari- detail. It follows directly from Holevo’s Theorem that
{ Y }i=1 I(X :Y )+I(Z :Y )=2log(d) can be achieved only if
able Y 1,...,m . Then Holevo’s theorem states X Z
B
∈ { }
that for any such measurement Bob may perform, S ρ(B) =log(d) (16)
(cid:16) (cid:17)
I(X :Y) S(ρ) pX(i)S(ρX(i)), (13) and
≤ −
Xi S ρ(B)(i) =0, i and for R X,Z (17)
(cid:16) R (cid:17) ∀ ∈{ }
where S(ρ) tr(ρlog(ρ)) is the von Neumann entropy
and ρ= i≡pX−(i)ρX(i). where ρ(B) = trA(ρAB) = ipR(i)ρ(RB)(i), for R ∈
P X,Z . Eqs. (16-17)statethPatI(X :YX)+I(Z :YZ)=
Here, we consider the case where Alice and Bob ini- {2log(d}B) only if ρ(B) is completely mixed and Alice can
tially share a quantum state ρ (CId′ CId), with preparepurestatesonBob’ssystemusinganyoneofthe
AB
d′ d. Alice can measure her∈syDstem ⊗in order to measurements. Moreover,usingEq. (14)itiseasytosee
≥ that the pure state ensembles created on Bob’s system
create an ensemble on Bob’s system, which encodes a
have to correspond to MUBs. In Appendix A we show
random variable as described above. In order to en-
that these conditions can only be fulfilled if ρ can be
code the random variable X she performs the measure- AB
expressed as described in Eq. (15). It is straightforward
mentcorrespondingtothePOVMelements P (j) and
X
thereby creates the ensemble p (i),ρ(B)(i{) on B}ob’s toseethattheoperationsneededtotransformρAB tothe
{ X X } maximallyentangledstate canalwaysbe included in the
system[46]. In orderto encode Z she performs P (j′)
{ Z } measurements of Alice. Hence, it is enough to show the
and thereby prepares {pZ(i′),ρ(ZB)(i′)} on Bob’s system. “if”-partforthestate φ+ ,whichhasalsobeenshownin
Moreover, Alice prepares the ensembles in such a way | i
[12, 16]. The maximal correlationis reachedif Alice and
that they fulfill the following additional condition. For
Bob measure either both in the computational basis, i.e.
themeasurements Q (i) and Q (i′) onBob’ssys-
{ YX } { YZ } the eigenbasis of Zd, or the mutually unbiased Fourier
temthatallowhimtomaximizeI(X :Y )andI(Z :Y ),
X Z basis, i.e. the eigenbasis of X . A straightforwardcalcu-
d
respectively, it holds that
lationshowsthatthesemeasurementsandthe ensembles
that Alice creates indeed fulfill Eq. (14).
tr(QYX(i)ρZ(B)(j′))=tr(QYZ(i′)ρ(XB)(j))=1/d (14) Wecannowlookatthemorespecialsituationinwhich
the two measurements by Alice are restricted to MUBs.
for all i,j,i′,j′. That is, the measurement that allows
The measurement setting described before Lemma 1 is
Bob to extract maximal information on X (Z), i.e. to
thenidenticaltothescenarioconsideredinthedefinition
maximize I(X : YX) (I(Z : YZ)), cannot be used to of . For d′ = d it is easy to see that no mixed state
2
extract any information about the other random vari- C
of the form given in Eq. (15) exists. Any state that
able, Z (X), respectively, as then all measurement out- maximizes thus has to be LU-equivalent to φ+ . We
2
comes are equally likely. However, if Alice informs Bob C | i
therefore obtain the following corollary of Lemma 1 (see
about which one of the two random variables she has
also [12, 16]).
used for the encoding, e.g. by sending one classical
bit, Bob can extract the maximal information about Corollary 2. A bipartite state ρ (CId CId) maxi-
∈ D ⊗
the corresponding random variable. Note, however,that mizes 2 iff it is LU-equivalent to the maximally entan-
I(X : Y )+I(Z : Y ) 2log(d). Using Holevo’s Theo- gled staCte, φ+ .
X Z ≤ | i
rem we prove the following lemma (see also [12, 16]). Since, as mentioned before, any bipartite state that
maximizes maximizes as well, we have that any
Ld′emmd,ath1e.reFoerxiastbmipaearstiutreesmtaetnetsρAaBs d∈esDcr(CIibde′d⊗bCIedfo)r,ewfiothr btoipbaertpituerestaaCntNdeLmUa-xeiqmuiizvianlgeCnC2tNtofoφr+d′.=Hdowneevceers,saitriilsyehasays
wh≥ich I(X : YX)+I(Z : YZ) = 2log(d) iff ρAB admits to find examples of mixed states| ofisystems with d′ > d
the decomposition thatmaximize usingEq. (15). Inthefollowingsection
2
C
we derive some necessary and some sufficient conditions
ρAB = q˜k(Vk⊗1l) φ+ φ+ (Vk†⊗1l), (15) a multipartite state has to fulfill in order to maximize
Xk (cid:12)(cid:12) (cid:11)(cid:10) (cid:12)(cid:12) CN.
6
B. Multipartite states ψ n CIdi as
| i∈ i=1
N
n
Let us first derive some necessary conditions a state S = S GL(CI,d ) s.t. S ψ = ψ , (19)
ψ i
ρ ∈ D( niCIdi) has to fulnfill in order to reach the up- { ∈Oi=1 | i | i}
per bouNnd CN(ρ)=1/n i=1log(di). Before that, recall i.e. as the set of all local symmetries of ψ . Using this
that in order to determiPne N all systems are measured | i
C notation, we state the following theorem (recallthe defi-
in N different MUBs and for each of these measurement
nition of S in Eq. (2)).
settings the mutual information between the measure- d,k
ment results obtained for any single system and the rest Theorem 5. Let ψ (CId)⊗n be a pure state with the
are considered (see Eq. (11 - 12)). Using Lemma 1 it is | i ∈
following properties.
easy to show the following lemma.
ψ has completely mixed single-subsystem reduced
• | i
Lemma 3. A multipartite state maximizes CN only if it states, i.e. tr¯l(|ψihψ|)∝1l,∀l ∈{1,...,n}, and
admits a decomposition as described in Eq. (15) in the
n Smk,l S , where m = 0 NI ,
bipartition of any single subsystem and the rest. • { l=1 d,k }k∈K ⊂ ψ { k,l 6 }k,l ⊂
anNd 0,...,d 1 2, = N, is such that
K ⊆ { − } |K|
Proof. Note that ρ maximizes only if it maximizes the corresponding set of generalized Pauli opera-
N
C
also C2. From the definition of C2 we have that ρ tors, {Sd,k}k∈K, is mutually unbiased.
maximizes this correlation function only if it maximizes
Then every φ ψ maximizes .
LU N
the corresponding function in each bipartite splitting | i≃ | i C
2
C
ofsubsysteml 1,...,n withthe rest. Itfollowsfrom Proof. It is clear that φ ψ maximizes iff ψ
∈{ } | i ≃LU | i CN | i
the definition of 2 and Lemma 1 that this is possible does. Hence,inordertoproofthetheoremitissufficient
C
only if ρ admits a decomposition as described in Eq. to show that it holds for ψ itself.
| i
(15) in the bipartition of any single subsystem and the We expand ψ on the first n 1 subsystems in the
| i −
rest. This proves the statement. eigenbasis of S , i d−1, where k , as
d,k {| ki}i=0 ∈K
We have stated in Corollary 2 that for bipartite sys- ψ = ~i φ (~i) . (20)
k k
tems with equal local dimensions, only pure states can | i ~i∈{0,..X.,d−1}n−1(cid:12)(cid:12) E(cid:12)(cid:12) E
maximize C . In contrastto that, we havethe following (cid:12) (cid:12)
N
observation for multipartite systems. Here, we used the notation ~i = (i(1),...,i(n−1)) and
~i i(1) ... i(n−1) is a state ofthe firstn 1
Otdhimabtseemnrsavioxaintm.ioizne C4N. ,Tehveerneiefxaislltsmubisxyedstemmuslthipaavretittheesstaamtees es(cid:12)(cid:12)(cid:12)uqkbuEsayt≡isot(cid:12)(cid:12)nemsk.n(cid:11)⊗ASsmNk⊗,lnl=(cid:12)(cid:12)ψ1Sdm=,kkk,l(cid:11)ψ∈hSaψs,towehoklndo.wUtshinagt E−thqe.
l=1 d,k | i | i
(20) and tNhat S i =ωi i we obtain
An example is the mixed three-qutrit state ρ = d,k| ki d| ki
ABC
n
trR(|ΩiRABChΩ|), where Smk,l ψ = ~i ωPnj=−11mk,ji(j)Smk,n φ (~i)
2 Ol=1 d,k | i X~i (cid:12)(cid:12) kE d d,k (cid:12)(cid:12) k E
1 (cid:12) (cid:12)
|ΩiRABC = 3 |iiR|jiA|i+j mod 3iB|i+2j mod 3iC = ~ik φk(~i) = ψ . (21)
iX,j=0 X~i (cid:12)(cid:12) E(cid:12)(cid:12) E | i
(18) (cid:12) (cid:12)
This equation is fulfilled iff
is an absolutely maximally entangled state (AMES)
presented in [32]. AMES are states of N-subsystems for Smk,n φ (~i) =ω−Pnj=−11mk,ji(j) φ (~i) , ~i. (22)
which any N/2 -subsystem reduced state is completely d,k (cid:12) k E d (cid:12) k E ∀
⌊ ⌋ (cid:12) (cid:12)
mixed (see e.g. [32]), where ⌊·⌋ denotes the floor of Hence, φk(cid:12)(~i) is an eigenvector of S(cid:12)d,k with eigenvalue
a number. One can show that N(ρABC) = log(3), (cid:12) E
for N ∈ {2,3,4}, which is the mCaximal value for all ωq(~i), w(cid:12)(cid:12)here q(~i) = −1/mk,n jn=−11mk,jij mod d, i.e.
N 2,3,4 . Thesevaluesareachievedifthe individual iff P
∈{ }
subsystems are all measured in the basis of the same
φ (~i) =c (~i) q(~i) (23)
three-qutrit generalized Pauli operators. k k k
(cid:12) E (cid:12) E
(cid:12) (cid:12)
The next theorem provides a condition that is suffi- for some complex(cid:12) number ck(~i).(cid:12) Reinserting Eq. (23)
cient to show that a state maximizes . Before we into Eq. (20) we see that
N
C
statethistheorem,weintroducethefollowingdefinitions.
We denote by GL(CI,d) the setof invertible operatorson ψ = ck(~i) ~ik q(~i)k . (24)
CId and define the stabilizer of a pure multipartite state | i X~i (cid:12)(cid:12) E(cid:12)(cid:12) E
(cid:12) (cid:12)
7
Hence, ψ isasuperpositionoftensorproductsofeigen- V. EXAMPLES OF PURE STATES
| i
states of Sd,k with the following property. The state of MAXIMIZING CN
anyn 1subsystemsdeterminesthestateoftheremain-
−
ing subsystem. This implies that the outcomes of mea- In this section we use the theorems and the lemmata
surementsof{|iik}onanyn−1subsystemsdeterminethe proven in the previous section to study the entangle-
outcome of a measurement performed in the same basis ment of multipartite pure states. We provide examples
onthe lastsubsystem. Atthe sametime,the outcomeof of states that maximze and study the set in
N N
themeasurementononesubsystemiscompletelyrandom the contextofLOCCtranCsformations. Astate ψ{Cm}eet-
if one is unaware of the outcomes of the other measure- ing the prerequisits of Theorem 5 is interesting|initerms
ments, as the single-subsystem reduced states of ψ are of state transformations via LOCC as it has nontrivial
| i
completely mixed. Considering both of these facts, we local unitary symmetries and completely mixed single-
obtain subsystem reduced states. It then follows from the re-
sults in [34, 35] that ψ is convertible, i.e. it can be
I(S(l) :S(¯l) ψ ψ )=log(d), (25) | i
d,k d,k|| ih | transformed deterministically via LOCC to some other
(non-LU-equivalent)stateinitsSLOCCclass. Inthefol-
for all l 1,...,n . As this holds for all k and lowingwestudytheentanglementofsomeofthesestates
∈ { } ∈ K
the eigenbases of the corresponding N = generalized and related ones and compare it with .
Pauli operators are mutually unbiased, th|iKs|shows that {CN}
(ψ)=log(d), (26)
N
C A. Three-qubit states
which is the maximal possible value.
The GHZ state, GHZ 000 + 111 , is
2,3
| i ∝ | i | i
Note that it is straightforwardto generalize this theo- (up to LUs) the unique genuinely tripartite entan-
reminsuchawaythatitalsoincludesotherlocalunitary gled pure three-qubit state with completely mixed
symmetries U = U1 ... Un for which the spectrum single-subsystem reduced states. Hence, it is the
of the local unitaries⊗Ui is⊗{ωdj}dj=−01 for all i∈{1,...,n}. only pure state that can potentially maximize CN,
Recall also that any state that maximizes N also maxi- according to Lemma 3. Indeed, the maximal value
mizes CN′ for N′ ≤N. C of C2(GHZ) = 1 can be reached if either σx or σz is
There are many interesting states that meet the pre- measured on each of the subsystems. Note however,
requisits of Theorem 5, as we show in the next section. that is not maximized if measurements of σ are
3 y
C
In the proof of Theorem 5 it became evident that hav- included, i.e. if all Pauli operators are measured.
ing a symmetry nl=1Sdm,kk,l ∈ Sψ and having all single- In the following we call the scenario where all Pauli
subsystem reduceNd states completely mixed implies Eq. operators are measured the Pauli setting. As the
(25),i.e. measurementsonn 1subsystemsinthecorre- y-measurementdoes notgive riseto any correlations,we
−
spondingbasisdeterminetheoutcomeofthelastsubsys- haveC (GHZ , σ(l) )=2/3. Numer-
tem. It might be tempting to believe that the reverse is 3 2,3 { j }j∈{x,y,z},l∈{1,2,3}
ical calculations suggest that in fact (GHZ ) = 2/3
3 2,3
true aswellforastate ψ with completelymixedsingle- C
and that there is, moreover, no three-qubit pure state
| i
subsystem reduced states. That is, one might think that
that exceeds this value. Hence, there appears to be no
the existence ofsuchcorrelationsinabasis b d−1 en-
{| ji}j=0 three-qubit pure state that exhibits correlations in three
tails the existence of a unitary U = dj=−01eiφj|bjihbj| mutually unbiased bases that are strong enough to yield
such that P 3 >2/3.
C
n
Uml ψ = ψ , (27)
| i | i
Ol=1
1. States in the maximally entangled set
for some set of integers m = 0 NI . That this is not
l
{ 6 } ⊂
the case can be seen by the example of the four-qutrit
appearstoverywellcapturethe multipartite corre-
state ψ 0120 + 1201 + 2012 . This state is C2
| i ∝ | i | i | i lations of states in the GHZ class, while its value for the
maximally correlated in the computational basis in the
W state, W 100 + 010 + 001 , is comparatively
waydescribedabove. Itis, however,easyto showthatit | i ∝ | i | i | i
low ( (W) = 0.685). Due to that reason, we examine
does not have a nontriviallocal symmetry U that fulfills C2
states in the intersection of the MES (see Sec. II) with
Eq. (27) and is diagonal in the computational basis.
the GHZ class.
Every state in the GHZ class is LU-equivalent to a
In what follows, we use the insights gainedin this sec-
state of the form [6]
tiontostudythecorrelationpropertiesofsomemultipar-
tite quantum states via the set of correlation functions
ψ (~g;z) g g g P GHZ , (28)
N and present applications of these measures. | GHZ i∝ x1 ⊗ x2 ⊗ x3 z| 2,3i
{C }
8
where ~g = (x ,x ,x ) RI3 , z CI, z 1. More-
1 2 3 ∈ ≥0 ∈ | | ≤
over, gxj are invertible operators such that gx†jgxj = 1
1/21l+x σ , for all j 1,2,3 , and P = diag(z,1/z).
j x z
∈ { }
Note that we choose here gxj = 1/21l+xjσx. It has 0.8
been shown in [6] that the set of sptates in the GHZ class
that are also in the MES is given by the set of states
0.6
with z = 1. The GHZ state obviously corresponds to
ψ (~0;1) .
GHZ 0.4
(cid:12) E
(cid:12) Here, we focus on MES states ψ ((x,x,x);1)
(cid:12) | GHZ i
that are symmetric under exchange of subsystems.
0.2
decreases monotonically with increasing x > 0 as
2
C
can be seen in Fig. 1. This shows that the quantum
correlations contained in the GHZ state, as measured
0 0.1 0.2 0.3 0.4 0.5
by , are particularily strong also in comparison with
2
C
other states in the MES. Let us stress here again that
of a generic state does not need to be optimized for FIG. 1: The quantum correlations of the three-qubit state
N
C
a measurement setting for which all parties measure the |ψGHZ((x,x,x);1)i,whichisanelementoftheMES,asmea-
samebases. However,for ψ ((x,x,x);1) wefindnu- suredbyC2. Theaverageofthecorrelationsobtainedbymea-
GHZ
| i suring in the x- and z-basis (solid red line) are equal to the
merically that (x)=1/2(Q (x)+Q (x)) holds, where
2 x z
Q =1/3 3 CI(σ(l) :σ(¯l))arethecorrelationsobtained correlationsobtainedifweoptimizeoverallMUBs(reddots)
i l=1 i i which shows that the optimal choice of MUBs is in fact the
if σi is mPeasured on each subsystem, for i x,y,z . former setting. C2 monotonically decreases with x. The cor-
∈ { }
The correlations Qx(x) decrease only slowly, while relationsinthex-basis,Qx,(uppermostdashedline;red)are
Qz(x) declines more rapidly. This is because the local larger than the correlations Qz (middle dashed line; green).
operators that are applied to the GHZ state in order Qy (lowestdashedline;blue)reachesamaximumatx=1/4.
to obtain ψ ((x,x,x);1) have, in the Pauli basis, Thethree-tangle,τ3 (thingreyline;dashed),declinessimilar-
GHZ
only a com|ponent along σ iand hence correlations in ily toC2.
x
the x-basis are favoured. Interestingly, the decline of
Q (x),Q (x) from their maximal value with increasing
x z
x > 0 is accompanied by the appearance of correlations close to 1 (see Fig. 2). Then the transformation con-
in the y-basis; Q (x) increases from zero to a maximal tinuestoLOCC-reachablestateswithlargercomponents
y
value at x = 1/4. Clearly all correlations disappear xj for j 1,2,3 ,of~g (see Eq. (28)), which also deter-
∈{ }
as x approaches the value 1/2 and ψ ((x,x,x);1) mine their z-parameter. Interestingly, the z-parameter
GHZ
approaches a separable state. In Fig.|1 we also comparei deviates a lot from z = 1 even if ~g is only changed
| |
with the three-tangle, τ [37]. Both measures decline slightly [47]. As a result, the correlations in the x- and
2 3
Cin a similar manner with increasing x. However, the z-basis (the Pauli setting) differ significantly from 2 in
C
three-tangle declines more rapidly. thecourseoftheprotocol. However,astheLOCCproto-
col proceeds, even the optimal setting yields only small
correlations. This shows that the quantum correlations
measuredby canbeuseduprapidlyintheprocessofa
2
C
LOCC transformation, while the source and the accessi-
2. States in the accessible set of the GHZ state
ble entanglement decline much slower. Interestingly, the
correlationsmeasuredbythe three-tangle,τ ,and are
The GHZ state is the three-qubit state which is con- 3 C2
almost identical for the states considered in this LOCC
vertible to the most other states via LOCC. That is, it
transformation.
has the largestaccessibleentanglementofallthree-qubit
states[29]. Itisinterestingtoinvestigatehow changes
2
C
under LOCC operations. In Fig. 2 we show how these
B. Three-qutrit states
changescomparetothe onesofthe sourceandthe acces-
sible entanglement (see Sec. II) for a LOCC transforma-
tioninwhichastatewithveryhigh istransformedinto We haveseenabovethatthere exists(up toLUs) only
2
C
other states along a specific path in Hilbert space. The one pure three-qubit state, the GHZ state, that can po-
conditions that the states on such a path have to fulfill tentially exhibit perfect correlations in MUBs as mea-
have been determined in [29, 36]. Note that, in order to sured by the functions . However, we have numeri-
N
{C }
be very precise, we do not start at the GHZ state itself, calevidence thateventhe GHZ state ofthree qubits can
asithasanaccessibleentanglementwhichisnotdirectly maximize only , but not . In contrast to that, we
2 3
C C
comparabletotheoneofotherstates(see[29]). Westart show now that, by increasing the local dimension, one
atthestate ψ ((x ,x ,x );z ) ,withx =0.001and can find infinitely many three-qutrit states that exhibit
GHZ 0 0 0 0 0
| i
z = 0.99999 0.00099i with a value of that is very theseperfectcorrelationsinfourMUBs. Weconsiderthe
0 2
− C
9
1 where ~g = (g(1),g(2),g(3)) CI3, k 0,1,2 2 and
∈ ∈ { }
g(j) span 1l,S ,S such that g(j)†g(j) = 1/31l+
0.8 gk(j)S∈3,k + g{(j)∗e3−,kiν−k3S,−3k,−}k. Here, thke phkases νk are
such that S3†,k = eiνkS3,−k. Moreover, gk(j) 6∝ 1l, ex-
0.6 cept for Ψ (a,b,c) itself. We consider states with
3,3
| i
~g =(x,x,x) RI3. Recall that Ψ (a,b,c) exhibits, in
∈ + | 3,3 i
0.4 contrast to the three-qubit GHZ state, the same strong
correlations for measurements in all of the generalized
Pauli bases. We will therefore not see any qualitative
0.2
differences in the investigation of (ψ(~g;k;a,b,c)) for
N
C
differentvaluesofkandhencechoosek =(1,0). Inorder
tosimplifythecomparisonwiththequbitcase,wemore-
0 0.01 0.02 0.03
over choose a = 1,b = c = 0 and thereby states in the
SLOCC class of the generalized GHZ state, GHZ .
3,3
FIG. 2: The initial state close to the GHZ state (see main | i
In Fig. 3 we see, similar to the three-qubit case, that
text) is successively transformed via LOCC into states with
(ψ((x,x,x);(1,0);1,0,0)) decreases with increasing x.
parameters ~g = (x,x,x),z(x), where x > 0 and z(x) is C4
We call the bases of S ,S ,S and S the Z-
determined by x (see Eq. (28)). The uppermost line (0,1) (1,0) (1,1) (1,2)
, X-, XZ- and XZZ-basis, respectively, and denote the
(black,crosses)depictsthesourceentanglementandtheother
straight line (black, asterics) the accessible entanglement of correlations in the basis W by QW = 1/3 3l=1I(W(l) :
the states. The upper curved line (red, circles) represents C2 W(¯l)). The correlation among the measurPements in the
and the lower curved line (blue, squares) the correlations C2 X-basis remains high even for large values of g, while
measured in the x- and z-basis (the Pauli setting). Notice thecorrelationsinthe othermutuallyunbiasedbasesde-
that we investigated more intermediate states in the range
crease faster. In contrast to the three-qubit case, all of
x∈[0.001,0.002] to better resolve the decline of the correla-
these correlation functions are strictly monotonic func-
tionsinthePaulisetting. Theblacklinewithtrianglesshows
tions of x.
thevalues of τ3.
states
1.6
Ψ (a,b,c) =a(000 + 111 + 222 ) (29)
3,3
| i | i | i | i
+b(012 + 201 + 120 )
| i | i | i
+c(021 + 210 + 102 ),
1.2
| i | i | i
where a,b,c CI. It is straightforwardto show that [28]
∈
S⊗3 S . (30) 0.8
{ 3,k}k∈{0,1,2}2 ⊂ Ψ3,3(a,b,c)
Note that the seed states of generic three-qutrit SLOCC
classes correspond to a subset of these states for which
0.4
S S [28]. These states are the
{ 3,k}k∈{0,1,2}2 ≡ Ψ3,3(a,b,c)
representatives of the SLOCC classes that are dense in
the set of three-qutrit pure states [26].
According to Theorem 5, the fact that Ψ3,3(a,b,c) 0 0.1 0.2 0.3
| i
have completely mixed single-subsystem reduced states
and the existence of the local symmetries in Eq. (30)
FIG. 3: Correlations of the three-qubit state
imply that (Ψ (a,b,c)) = log(3) for N 2,3,4 .
CN 3,3 ∈ { } |ψ((x,x,x);(1,0);1,0,0)i, which is an element of the
In particular, generic three-qutrit seed states maximize
MES. The correlations obtained by measuring in the Z-, X-,
all N. Analogously to the three-qubit case, we can XZ- and XZZ-basis (solid red line) are in good agreement
C
again consider the correlations of states in the intersec- with the correlations obtained if we optimize over the mutu-
tion of the MES and the SLOCC class represented by ally unbiased bases (red dots). C4 monotonically decreases
Ψ3,3(a,b,c) . In contrast to the three-qubit case, most with x. The correlations in the X-basis, QX, decline slower
s|tates in thie three-qutrit MES are not convertible via (uppermost dashed line; red) compared with thecorrelations
LOCC[28]. Theconvertiblestates,however,aretheonly QZ (middle dashed line; green). The correlations QXZ and
interesting ones in terms of LOCC transformations. For QXZZ are identical and the weakest (lowest dashed line;
blue).
three qutrits they can be expressed as [28]
ψ(~g;k;a,b,c) =g(1) g(2) g(3) Ψ (a,b,c) , (31)
| i k ⊗ k ⊗ k | 3,3 i
10
C. States of more than three subsystems VI. DETECTION OF MIXED STATE
ENTANGLEMENT
There are also genuinely multipartite entangled states
In this subsection we show how can be used
ψ (CId)⊗n, with n > 3, that reach the maximal value {CN}
| i ∈ to detect entanglement of multipartite mixed states
of (ψ) = log(d) for some N. Note that, for exam-
CN of systems with arbitrary dimensions. We focus here
ple, all generic four-qubit seed states have completely
on the systems with local dimension d. However,
mixed single-qubit reduced states and their stabilizer is
the results can be easily generalized to systems with
σ⊗4 , where σ 1l (see e.g. [6]). Therefore,
{ i }i∈{0,x,y,z} 0 ≡ different local dimensions. We first show that any state
they maximize and according to Theorem 5. The
same holds for Ca2class Cof32m-qubit states (m 2) with of a n-partite system exceeding a certain value of CN
stabilizer σ⊗2m presented in [35]. ≥Note that has to be entangled, which generalizes the results on
{ i }i∈{0,x,y,z} bipartite entanglement detection presented in [12, 13].
alsomanygraphstates[38]maximize . Forthesestates
C3 For tripartite systems with Hilbert space = (CId)⊗3
one can construct Pauli symmetries from their stabilizer H
we moreover show that this entanglement has to be
that fulfill the requirements of Theorem 5. An example
tripartite in nature if of the corresponding state
is the n-qubit GHZ state for n even. Further examples CN
exceeds another (higher) threshold value. Hence,
are the states correspondingto the binary tree graphsof {CN}
can be used to detect genuine tripartite entanglement
eight or of 16 vertices for which all leaf nodes have the
using a number of local measurements in MUBs that
same depth. Another example is the four-qutrit AME
scales at most linearly with the local dimension. This is
state in Eq. (18), which maximizes if the generalized
C2 in contrastto the fact that the number of measurements
Pauli operators Z and X are measured on each of the
3 3 required for entanglement detection generally grows
subsystems. Going to higher-dimensional systems, it is
rapidly with the size of the system [15]. In [40, 41]
moreovereasytoseethatS⊗n GHZ = GHZ ,for
d,k | d,ni | d,ni related approaches to the detection of multipartite
all k 0,...,d 1 2, if n = m d. Using this, and the
entanglement via MUBs that are not based on classical
∈ { − } ·
factthatthegeneralizedPaulioperatorsformacomplete
information measures were proposed. We compare our
set ofMUBs for prime d, it followsfrom Theorem5 that
results to those and show that, even if only two local
GHZ maximizes for all N 2,...,d+1 if d
d,m·d N measurement settings are used, i.e. if a lower bound on
| i C ∈{ }
isaprimenumber. Thecorrespondingmeasurementsare
is measured, our method is comparably useful for
2
given by the N mutually unbiased generalized Pauli op- C
detecting also high-dimensional tripartite entanglement.
erators. Another interesting state is the Aharanov state
Another advantage of the detection method presented
(see e.g. [39]), which is defined as
here is that one does not need to know the phase re-
lationbetweenthedifferentmeasurements(seee.g. [41]).
d−1
1
= ǫ i ...i (CId)⊗d, (32) A state of a system composed of n different d-level
|Sdi √d!i1,.X..,id=0 i1,...,id| 1 di∈ sρyste=ms ispcaρl(l1e)d fuρl(l2y) se.p.a.rabρle(n)i,fwitheisreeρx(pi)reisssaibsletaates
sep j j j ⊗ j ⊗ ⊗ j j
where ǫi1,...,id denotes the generalized Levi-Civita` sym- oafsswyestwePmillie,xppjla≥in0inantdhePfojlplojw=in1g., tIhtactan be easily seen,
bol. It is easy to see that maximizes the mutual
d
|S i
information between any subsystem and the rest, if N
each of the systems is measured in the computational I ( (l) : (¯l)) Nlog(d) f(N,d), (33)
ρsep Bk Bk ≤ −
basis. Moreover, this also holds if any other basis is Xk=1
measured, as the Aharonov state fulfills U⊗d =
|Sdi |Sdi for all parties l 1,...,n , where f(N,d) is such that
for any unitary U. This implies that |Sdi maximizes the entropic unc∈er{tainty rel}ation
whenever N MUBs exist onCId. In particular,
N d
C |S i
maximizes for any d and it maximizes whenever N
2 d+1
d is the powCer of a prime number. C H( k ρ) f(N,d), ρ (CId), (34)
B | ≥ ∀ ∈D
Xk=1
Thestatesmentionedsofarareallmultipartiteentan- holds. NotefirstthatthestatementinEq. (33)hasbeen
gled. Thereare,however,biseparablestatesofmorethan proven in [12] for separable states of bipartite systems
threesubsystemsthatmaximize . Itisstraightforward in the special case of N = 2 and f(2,d) = log(d).
N
to see that for a system with HilCbert space =(CId)⊗n, Note further that any fully separable state is of course
H
where n is even, any n/2 pairs of maximally entangled separable with respect to any bipartition of the parties.
states maximize . Note, however, that such a bisepa- It is then easy to see that the proof in [12] can be
2
C
rable pure state ψ canbe easily distinguished from generalized to the statement above.
bisep
| i
genuinelymultipartitepurestatesastheyobviouslyfulfill
S(ρ ) = 0 for some genuine subsystem A ( 1,...,n , ConsideringthesumoverallpartiesofEq. (33),weob-
A
{ }
where ρ =tr (ψ ψ ). tain that all fully separable states cannot exceed a value
A {1,...,n}\A bisep bisep
| ih |