PHYSICAL REVIEW A 88, 042329 (2013)

Genuine multiparty quantum entanglement suppresses multiport classical information transmission

R. Prabhu, Aditi Sen(De), and Ujjwal SenHarish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India

(Received 9 October 2012; published 22 October 2013)

We establish a universal complementarity relation between the capacity of classical information transmissionby employing a multiparty quantum state as a multiport quantum channel, and the corresponding genuinemultipartite entanglement. The classical information transfer is from a sender to several receivers by usingthe quantum dense coding protocol with the multiparty quantum state shared between the sender and severalreceivers. The relation is derived when the multiparty entanglement is the relative entropy of entanglement aswell as when it is the generalized geometric measure. The relation holds for arbitrary pure or mixed quantumstates of an arbitrary number of parties in arbitrary dimensions.

DOI: 10.1103/PhysRevA.88.042329 PACS number(s): 03.67.Hk, 03.67.Mn

The discoveries of the quantum communication strategies,beginning with the protocols of quantum dense coding [1],quantum teleportation [2], and quantum key distribution [3],have revolutionized the way we think about modern commu-nication schemes. The importance of such protocols lies in thefact that they can efficiently transmit classical or quantuminformation in a way that is better than what is possibleby using classical protocols. Such quantum communicationschemes have already been realized experimentally and, in par-ticular, experimental quantum dense coding has been reportedin several systems including photonic states, ion traps, andnuclear magnetic resonance [4]. The protocols were initiallyintroduced for communication between two separated parties.This is true for the theoretical discussions of these protocols aswell as for the experimental demonstrations of the same. It hasalready been established that, in the case of such bipartitecommunication schemes between a sender and a receiver,shared quantum correlations play a key role.

Commercialization of these protocols demand the imple-mentations of these protocols in a multipartite scenario [5].Classical information transmission, in the form of quantumdense coding, has already been introduced in multiparticlesystems [6]. Further work in this direction includes Ref. [7].

In this paper, we consider the multiport quantum channelfor transmitting classical information where a sender wishesto send classical information individually to several receiversby employing the quantum dense coding protocol with amultiparty shared quantum state between all the partners. Wefind that the quantum advantage in this quantum dense codingscheme is suppressed by the genuine multipartite entanglementof the shared multiparty quantum state.

Let us stress here that the quantity, coherent information,which quantifies the quantum advantage in the dense codingprotocol can also be used to study quantum error correc-tion [8], “partial” quantum information [9], and distillableentanglement [10]. The complementarity is first demonstratedby using the relative entropy of entanglement [11] and holdsfor arbitrary (pure or mixed) quantum states of an arbitrarynumber of parties and in arbitrary dimensions. To showthat the complementarity is potentially generic, we go on todemonstrate the complementarity for an independent measureof genuine multisite entanglement, the generalized geometricmeasure [12], in which case the relation again holds for pure aswell as mixed multisite quantum states of an arbitrary number

of parties in arbitrary dimensions. Due to the monogamy ofquantum correlations [13], one intuitively expects that a highmultipartite entanglement of a quantum state will suppressthe reduced bipartite entanglement and hence will reducethe capacity of multiport dense coding. Since a quantitativestatement of the constraint on the multiparty entanglement dueto the monogamy of bipartite entanglement is as yet missing,it is not straightforward to relate the monogamy of bipartitequantum correlations with a quantum advantage of multiportchannel capacities. The complementarity relation derived inthis paper can shed light towards a quantitative understandingin this direction.

Quantum dense coding capacity and the quantumadvantage. Quantum dense coding (DC) is a quantum com-munication protocol by which one can transmit classicalinformation encoded in a quantum system from a sender toa receiver [1]. The available resources for the transmissionare a shared quantum state and a noiseless quantum channelto transmit the sender’s part of the shared quantum stateto the receiver’s end. If the sender, called Alice, and thereceiver, called Bob, share a bipartite quantum state �AB ,then the amount of classical information (in bits) that thesender can send to the receiver is given by [6,14,15] C(�AB) =max{log2 dA, log2 dA + S(�B) − S(�AB)}, where dA is thedimension of Alice’s Hilbert space and �B is the local-densitymatrix of Bob’s subsystem. S(·) denotes the von Neumannentropy of its argument and is defined as S(σ ) = −tr(σ log2 σ ),for an arbitrary quantum state σ . The capacity C(�AB) reachesits maximum when Alice and Bob share a maximally entangledstate, which is a pure state with completely mixed local-densitymatrices. Without using the shared quantum state but using thenoiseless quantum channel, Alice will be able to send log2 dA

bits. This process of sending classical information withoutusing the shared quantum state is referred to as the “classicalprotocol.” Using the shared quantum state is therefore advan-tageous if S(�B) − S(�AB) > 0. A bipartite quantum state issaid to be dense codeable in that case. Correspondingly, thequantity Cadv(�AB) = max{S(�B) − S(�AB),0} is identified asthe “quantum advantage” in a dense coding protocol fromAlice to Bob. Note that C(�AB) = log2 dA + Cadv(�AB).

Let us now move on to a multiport situation and supposenow that there are N + 1 parties who share a quantum state�AB1B2...BN

. Moreover, we assume that among the N + 1parties, A is the sender and the others, i.e., B1,B2, . . . ,BN are

042329-11050-2947/2013/88(4)/042329(4) ©2013 American Physical Society

R. PRABHU, ADITI SEN(DE), AND UJJWAL SEN PHYSICAL REVIEW A 88, 042329 (2013)

the receivers. Let us consider a situation where A wants to send,individually, classical information to the Bis (i = 1,2, . . . ,N ).In this multiport case, the quantum advantage is naturallydefined as

Cmaxadv (�AB1B2...BN

) = max{Cadv(�ABi)|i = 1,2, . . . ,N}, (1)

where �ABiis the local-density matrix of A and Bi (i =

1,2, . . . ,N). This “multiport dense coding quantum advan-tage” quantifies the amount of classical information that canbe sent from the sender A to the N receivers by using thequantum dense coding protocol with the shared quantum state�AB1B2...BN

, over and above the amount of classical informationthat can be sent by using a classical protocol. It is to be notedthat the multiport dense coding protocol under considerationrequires a multiparty state, and the successful implementationof the protocol will take place between the sender and aparticular receiver, with the choice of the receiver (from amongthe many available) being dependent upon the multipartyshared state that is considered as the channel for the saidprotocol.

Genuine multipartite entanglement measures. We nowintroduce two multipartite entanglement measures which canquantify genuine multiparticle entanglement of an arbitraryquantum state �A1A2...An

shared between n parties.Relative entropy of entanglement. The relative entropy of

entanglement was proposed as a measure of entanglement foran arbitrary multipartite state, �A1A2...An

[11] as

ER(�A1A2...An) = min

σ∈n-genS(�A1A2...An

||σ ),

where ‘‘n-gen” is the set of all n-party multipartite quantum(pure or mixed) states which are not genuinely multipartiteentangled, and the relative entropy, S(�||σ ), between � andσ , is defined as S(�||σ ) = tr(� log2 � − � log2 σ ). An n-partyquantum state is said to be genuinely multiparty entangled ifit cannot be written as a probabilistic mixture of multipartyquantum states which are separable across at least one biparti-tion of the n parties. As an example, for three-party quantumsystems between A1, A2, and A3, a probabilistic mixture oftwo quantum states which are respectively separable acrossA1 : A2A3 and A2 : A1A3 is not genuinely multisite entangled.ER is the “relative entropy distance” of the correspondingmultisite state from the convex set of all multipartite stateswhich are not genuinely multiparty entangled.

Generalized geometric measure. The generalized geometricmeasure (GGM) [12] was first proposed to be a measureof genuine multipartite entanglement of a pure multipartyquantum state by using a distance function of the given purestate from all pure multisite states which are not genuinelymultipartite entangled and is defined as

E(|ψ〉A1A2...An

) = min(1 − |〈φ|ψ〉|2),

where the minimization is over all |φ〉A1A2...Anthat are not

genuinely multipartite entangled.Using the convex roof approach, the generalized geometric

measure for an arbitrary mixed quantum state can be definedas E(�A1A2...An

) = min∑

i piE(|ψi〉A1A2...An), where the mini-

mization is performed over all pure-state decompositions of�A1A2...An

= ∑i pi(|ψi〉〈ψi |)A1A2...An

.

Complementarity. We now establish complementarity re-lations between the amount of classical information thatcan be sent through the multisite quantum state �AB1B2...BN

,as quantified by the multiport dense coding quantumadvantage (Cmax

adv ), with genuine multipartite entanglementmeasures—relative entropy of entanglement and generalizedgeometric measure.

Multiport dense coding advantage vs relative entropy ofentanglement. Let A, B1, B2, ..., BN be N + 1 observerssharing an arbitrary (N + 1)-party (pure or mixed) quantumstate, �AB1B2...BN

, of arbitrary dimensions. In the multiportdense coding protocol that we consider, we assume that A

is the sender and the Bis (i = 1,2, . . . ,N ) are the receivers.We now prove the following complementarity between thequantum advantage in this multiport scenario with the relativeentropy of entanglement:

Theorem 1. For the arbitrary multipartite pure or mixedquantum state �AB1B2...BN

in arbitrary dimensions, the relativeentropy of entanglement and the multiport dense codingquantum advantage satisfy

Cmaxadv + ER � log2 d, (2)

where d is the maximal dimension of the Hilbert spaces ofthe Bis.

Proof. One can obtain an upper bound on the multipartyrelative entropy as follows:

ER(�AB1B2...BN) = min

σ∈(N+1)-genS(�||σ )

� minσ ′∈sep1

S(�||σ ′) ≡ EAB1:restR (�AB1:B2...BN

)

� EAB1:restf (�AB1:B2...BN

) � S(�AB1 ). (3)

Here the set “sep1” is the set of quantum states ofA,B1,B2, . . . ,BN which are separable across AB1 : B2 . . . BN .E

AB1:restR (�AB1:B2...BN

) and EAB1:restf (�AB1:B2...BN

) respectivelydenote the relative entropy of entanglement and the entan-glement of formation [16] of the state �AB1B2...BN

in the samebipartition. The second inequality is obtained due to the factthat the relative entropy of entanglement is bounded aboveby the entanglement of formation [17], while the third followsfrom the fact that the von Neumann entropy of the local-densitymatrix is an upper bound of the entanglement of formation forbipartite states [18].

For the multipartite state �AB1B2...BN, the dense coding

advantage can be written as Cmaxadv (�AB1B2...BN

) = max{SB1 −SAB1 ,SB2 − SAB2 , . . . ,SBN

− SABN,0}, where SBi

= S(�Bi)

and SABi= S(�ABi

) are the single-site and two-site vonNeumann entropies of �AB1B2...BN

, respectively. Considerthe instance when SB1 − SAB1 attains the maximum. ThenCmax

adv (�AB1B2...BN) = SB1 − SAB1 . Adding this relation with

Eq. (3), we obtain Cmaxadv (�AB1B2...BN

) + ER(�AB1B2...BN) �

SB1 � log2 dB1 . If the maximum of Cmaxadv is obtained for the

pair, say ABi , we have to choose the corresponding bipartitionin the relative entropy of entanglement in Eq. (3) and, in thatcase, the complementarity relation will be bounded above bythe logarithm of the dimension of subsystem Bi . Therefore,in general, we have max{log2 dB1 , log2 dB2 , . . . , log2 dBN

} =log2 d, as the upper bound for the sum Cmax

adv + ER . Hence theproof. �

042329-2

GENUINE MULTIPARTY QUANTUM ENTANGLEMENT . . . PHYSICAL REVIEW A 88, 042329 (2013)

The complementarity relation which is established aboveclearly indicates that a high genuine multipartite entanglementwill lower the advantage of the same state for transmittingclassical information. The result is universal in the sense thatit holds for an arbitrary pure or mixed quantum state of anarbitrary number of parties in arbitrary dimensions.

Relation between dense coding advantage and GGM.Towards showing that the obtained complementarity relation isgeneric, we consider another genuine multiparty entanglementmeasure: the generalized geometric measure. The relation isfirst proven for pure multiparty quantum states in arbitrarydimensions. For simplicity, we assume that the state liesin (Cd )⊗N+1, with arbitrary dimension d. Subsequently weshow that the relation also holds for arbitrary mixed states,ρAB1B2...BN

.Consider an (N + 1)-party pure state, |ψ〉AB1B2...BN

, whichis employed by the sender A to perform dense coding with thereceivers Bi (i = 1,2, . . . ,N ).

Theorem 2. The sum of the advantage in dense codingand the generalized geometric measure, both appropriatelynormalized, for the arbitrary pure state |ψ〉AB1B2...BN

is boundedabove by unity, i.e.,

1

log2 dCmax

adv + d

d − 1E � 1. (4)

Here the factors 1log2 d

and dd−1 , which stand for dense coding

advantage and GGM, respectively, are normalizations thatmake the individual terms have a maximal value of unity.

Proof. Let us assume, without loss of generality, that themaximum in the multiport quantum advantage in dense codingis attained for SB1 − SAB1 , i.e., Cmax

adv (|ψ〉AB1B2...BN) = SB1 −

SAB1 . We now note that the GGM for the state |ψ〉AB1B2...BN

can be shown to be given by E(|ψAB1B2...BN〉) = 1 − max{�j }

[12], where �j are the maximal eigenvalues of the local-density matrices of all possible bipartite partitions of thestate |ψ〉AB1B2...BN

. Therefore, we have E(|ψ〉AB1B2...BN) �

1 − λAB1 , with λAB1 being the maximum eigenvalue of the two-party reduced-density matrix �AB1 of |ψ〉AB1B2...BN

. Adding therelations for Cmax

adv (|ψ〉AB1B2...BN) and GGM, we obtain

Cmaxadv

(|ψ〉AB1B2...BN

)

log2 d+ d

d − 1E(|ψ〉AB1B2...BN

)

� SB1

log2 d− SAB1

log2 d+ d

(1 − λAB1

)

d − 1.

To complete the proof, we have to show that the sum of the lasttwo terms is nonpositive. The sum of these two terms of theabove equation is a convex function. Therefore, its maximumis attained at the extremal points, i.e., for λAB1 = 1

d2 , 1. WhenλAB1 = 1, the sum vanishes and at λAB1 = 1

d2 , it is negative.The proof follows from the fact that SB1 � log2 d. �

We now consider quantum states of N + 1 parties which arepossibly mixed and are in arbitrary dimensions. For simplicity,we assume that the state is defined on (Cd )⊗N+1, with arbitrarydimension d.

Theorem 3. For the arbitrary (possibly mixed) quantumstate �AB1B2...BN

, the sum of the normalized quantum advantagein dense coding and the normalized generalized geometricmeasure is bounded above by unity.

0.0 0.5 1.0 1.5 2.0 2.5 3.0�1.0

�0.8

�0.6

�0.4

�0.2

0.0

Θ

ΔC

FIG. 1. (Color online) Left panel shows the saturation of the com-plementarity quantity. The complementarity quantity δC = Cmax

adv +E − 1 is plotted against the state parameter θ of the generalizedGHZ state. When δC = 0, saturation of complementarity is achieved.The right panel presents the projection of the δC surface, for thegeneralized W state, on the plane of the state parameters θ ′ and φ′.The dark violet regions show the area where δC vanishes, so that thecomplementarity relation is saturated. The δC axis is dimensionlessand all other axes are in radians.

Proof. The genuine multiparty entanglement mea-sure, GGM, of �AB1B2...BN

is given by E(�AB1B2...BN) =∑

k pkE(|ψk〉〈ψk|), where the ensemble {pk,|ψk〉} forms theoptimal pure-state decomposition of the state �AB1B2...BN

forobtaining the minimum in the convex roof of the GGM.Suppose that λk

AB1is the maximum eigenvalue of the two-party

reduced-density matrix, of the parties A and B1, of the state|ψk〉. Then we have E(�AB1B2...BN

) �∑

k pk(1 − λkAB1

). Onecan show that

d

d − 1E(ρAB1B2...BN

) �∑

k

pk

S(trB2...BN

|ψk〉〈ψk|)

log2 d

� S(�AB1 )

log2 d.

We have used concavity of the von-Neumann entropy to getboth inequalities. Now, along with the quantum advantage indense coding and using the above inequality, we obtain

Cmaxadv (�AB1B2...BN

)

log2 d+ d

d − 1E(�AB1B2...BN

)

� SB1 − SAB1

log2 d+ SAB1

log2 d� 1. (5)

Here we have assumed, without loss of generality, that themaximum of Cmax

adv is attained by the AB1 pair. �We now illustrate that the obtained complementary relation

between the multiport dense coding quantum advantage andgenuine multipartite entanglement is tight. For this inves-tigation, let us first consider the generalized Greenberger-Horne-Zeilinger (GHZ) state [19], |ψGHZ〉 = cos θ |000〉 +sin θ |111〉, where θ ∈ (0,π ). We plot the complementarityquantity δC = 1

log2 dCmax

adv + dd−1E − 1 with respect to θ [see

left panel of Fig. 1. We see that the saturation of δC occurs atθ = π

4 and 3π4 . Note here that the complementarity relation

of Theorem 2 implies that δC is always nonpositive, andthat the vanishing of δC implies that the bound is tight.Similarly, we consider the generalized W states [20], |ψW 〉 =sin θ ′ cos φ′|011〉 + sin θ ′ sin φ′|101〉 + cos θ ′|110〉, with θ ′ ∈(0,π ) and φ′ ∈ (0,2π ), which are known to be inequivalent

042329-3

R. PRABHU, ADITI SEN(DE), AND UJJWAL SEN PHYSICAL REVIEW A 88, 042329 (2013)

to the generalized GHZ states by stochastic local operationsand classical communication [20]. In the right panel of Fig. 1,we depict the projection of δC on the (θ ′,φ′) plane for thegeneralized W state. It clearly indicates that there are regionswhere δC is saturated.

Summarizing, we have established a complementary re-lationship between the quantum advantage of the multiportclassical capacity of a multiparty quantum state used as aquantum channel and the genuine multipartite entanglement of

the same state. The relation is demonstrated for two genuinemultipartite entanglement measures—the relative entropy ofentanglement and the generalized geometric measure. Therelation holds for pure or mixed quantum states of arbitrarydimensions and of an arbitrary number of parties.

R.P. acknowledges support from the DST, Government ofIndia, in the form of an INSPIRE faculty scheme at the Harish-Chandra Research Institute, India.

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