Genuine Multiparty Quantum Entanglement Suppresses Multiport ...

Genuine Multiparty Quantum Entanglement Suppresses Multiport Classical Information Transmission R. Prabhu, Aditi Sen(De), and Ujjwal Sen

arXiv:1208.6535v2 [quant-ph] 13 Aug 2013

Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India We establish a universal complementarity relation between the capacity of classical information transmission by employing a multiparty quantum state as a multiport quantum channel, and the corresponding genuine multipartite entanglement. The classical information transfer is from a sender to several receivers by using the quantum dense coding protocol with the multiparty quantum state shared between the sender and several receivers. The relation is derived when the multiparty entanglement is the relative entropy of entanglement as well as when it is the generalized geometic measure. The relation holds for arbitrary pure or mixed quantum states of an arbitrary number of parties in arbitrary dimensions.

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 communication schemes. The importance of such protocols lies in the fact that they can efficiently transmit classical or quantum information in a way that is better than what is possible by using classical protocols. Such quantum communication schemes have already been realized experimentally, and in particular, experimental quantum dense coding have been reported in several systems including photonic states, ion traps, and nuclear magnetic resonance [4]. The protocols were initially introduced for communication between two separated parties. This is true for the theoretical discussions of these protocols as well as for the experimental demonstrations of the same. It has already been established that in the case of such bipartite communication schemes between a sender and a receiver, shared quantum correlations play a key role.

arbitrary (pure or mixed) quantum states of an arbitrary number of parties and in arbitrary dimensions. To show that the complementarity is potentially generic, we go on to demonstrate the complementarity for an independent measure of genuine multisite entanglement, the generalized geometric measure [12], in which case the relation again holds for pure as well as mixed multisite quantum states of an arbitrary number of parties in arbitrary dimensions. Due to the monogamy of quantum correlations [13], one intuitively expects that a high multipartite entanglement of a quantum state will suppress the reduced bipartite entanglements, and hence will reduce the capacity of multiport dense coding. Since a quantitative statement of the constraint on the multiparty entanglement due to the monogamy of bipartite entanglement is as yet missing, it is not straightforward to relate the monogamy of bipartite quantum correlations with a quantum advantage of multiport channel capacities. The complementarity relation derived in this paper can shed light towards a quantitative understanding in this direction.

Commercialization of these protocols demand the implementations of these protocols in a multipartite scenario [5]. Classical information transmission, in the form of quantum dense coding, has already been introduced in multiparticle systems [6]. Further work in this direction include Refs. [7].

Quantum dense coding capacity and the quantum advantage. Quantum dense coding (DC) is a quantum communication protocol by which one can transmit classical information encoded in a quantum system from a sender to a receiver [1]. The available resources for the transmission are a shared quantum state and a noiseless quantum channel to transmit the sender’s part of the shared quantum state to the receiver’s end. If the sender, called Alice, and the receiver, called Bob, share a bipartite quantum state ̺AB , then the amount of classical information (in bits) that the sender 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 the dimension of Alice’s Hilbert space and ̺B is the local density matrix of Bob’s subsystem. S(·) denotes the von Neumann entropy of its argument and is defined as S(σ) = −tr(σ log2 σ), for an arbitrary quantum state σ. The capacity C(̺AB ) reaches its maximum when Alice and Bob share a maximally entangled state, which is a pure state with completely mixed local density matrices. Without using the shared quantum state but using the noiseless quantum channel, Alice

In this paper, we consider the multiport quantum channel for transmitting classical information where a sender wishes to send classical information individually to several receivers by employing the quantum dense coding protocol with a multiparty shared quantum state between all the partners. We find that the quantum advantage in this quantum dense coding scheme is suppressed by the genuine multipartite entanglement of the shared multiparty quantum state. Let us stress here that the quantity, coherent information, which quantifies quantum advantage in the dense coding protocol can also be used to study quantum error correction [8], “partial” quantum information [9], and distillable entanglement [10]. The complementarity is first demonstrated by using the relative entropy of entanglement [11], and holds for

2 will be able to send log2 dA bits. This process of sending classical information without using the shared quantum state is referred to as the “classical protocol”. Using the shared quantum state is therefore advantageous if S(̺B ) − S(̺AB ) > 0. A bipartite quantum state is said to be dense-codeable in that case. Correspondingly, the quantity Cadv (̺AB ) = max{S(̺B ) − S(̺AB ), 0} is identified as the “quantum advantage” in a dense coding protocol from Alice to Bob. Note that C(̺AB ) = log2 dA + Cadv (̺AB ). Let us now move on to a multiport situation and suppose now that there are N + 1 parties who share a quantum state ̺AB1 B2 ...BN . Moreover, we assume that among the N + 1 parties, A is the sender and the others, i.e B1 , B2 , . . . , BN are the receivers. Let us consider a situation where A wants to send, individually, classical information to the Bi ’s (i = 1, 2, . . . , N ). In this multiport case, the quantum advantage is naturally defined as max (̺AB1 B2 ...BN ) = max{Cadv (̺ABi )|i = 1, 2, . . . , N }, Cadv (1) where ̺ABi is the local density matrix of A and Bi (i = 1, 2, . . . , N ). This “multiport dense coding quantum advantage” quantifies the amount of classical information that can be sent from the sender A to the N receivers by using the quantum dense coding protocol with the shared quantum state ̺AB1 B2 ...BN , over and above the amount of classical information that can be sent by using a classical protocol. It is to be noted that the multiport dense coding protocol under consideration requires a multiparty state, and the successful implementation of the protocol will take place between the sender and a particular receiver, with the choice of the receiver (from among the many available) being dependent upon the multiparty shared state that is considered as the channel for the said protocol. Genuine multipartite entanglement measures. We now introduce two multipartite entanglement measures which can quantify genuine multiparticle entanglement of an arbitrary quantum state ̺A1 A2 ...An shared between n parties. Relative entropy of entanglement – The relative entropy of entanglement was proposed as a measure of entanglement for an arbitrary multipartite state, ̺A1 A2 ...An [11] as

between A1 , A2 , and A3 , a probabilistic mixture of two quantum states which are respectively separable across A1 : A2 A3 and A2 : A1 A3 is not genuinely multisite entangled. ER is the “relative entropy distance” of the corresponding multisite state from the convex set of all multipartite states which are not genuinely multiparty entangled. Generalized geometric measure (GGM) – The generalized geometric measure [12] was first proposed to be a measure of genuine multipartite entanglement of a pure multiparty quantum state, by using a distance function of the given pure state from all pure multisite states which are not genuinely multipartite entangled, and is defined as E(|ψiA1 A2 ...An ) = min(1 − |hφ|ψi|2 ), where the minimization is over all |φiA1 A2 ...An that are not genuinely multipartite entangled. Using the convex roof approach, the generalized geometric measure for an arbitrary mixed P quantum state can be defined as E(̺A1 A2 ...An ) = min i pi E(|ψi iA1 A2 ...An ), where the minimization is performed over all pure state P decompositions of ̺A1 A2 ...An = i pi (|ψi ihψi |)A1 A2 ...An . Complementarity. We now establish complementarity relations between the amount of classical information that can be sent through the multisite quantum state ̺AB1 B2 ...BN , as quantified by the multiport dense coding max quantum advantage (Cadv ), with genuine multipartite entanglement measures – relative entropy of entanglement and generalized geometric measure. Multiport dense coding advantage vs relative entropy of entanglement – Let A, B1 , B2 ,..., BN be N + 1 observers sharing an arbitrary (N + 1)-party (pure or mixed) quantum state, ̺AB1 B2 ...BN , of arbitrary dimensions. In the multiport dense coding protocol that we consider, we assume that A is the sender and Bi ’s (i = 1, 2, . . . , N ) are the receivers. We now prove the following complementarity between the quantum advantage in this multiport scenario with the relative entropy of entanglement. Theorem 1: For the arbitrary multipartite pure or mixed quantum state ̺AB1 B2 ...BN in arbitrary dimensions, the relative entropy of entanglement and the multiport dense coding quantum advantage satisfy max Cadv + ER ≤ log2 d,

ER (̺A1 A2 ...An ) = min S(̺A1 A2 ...An ||σ),

(2)

σ∈n-gen

where “n-gen” is the set of all n-party multipartite quantum (pure or mixed) states which are not genuinely multipartite entangled, and the relative entropy, S(̺||σ), between ̺ and σ, is defined as S(̺||σ) = tr(̺ log2 ̺ − ̺ log2 σ). An n-party quantum state is said to be genuinely multiparty entangled if it cannot be written as a probabilistic mixture of multiparty quantum states which are separable across at least one bipartition of the n parties. As an example, for three-party quantum systems

where d is the maximal dimension of the Hilbert spaces of the Bi ’s. Proof. One can obtain an upper bound on the multiparty relative entropy as follows ER (̺AB1 B2 ...BN ) =

min

σ∈(N+1)-gen

S(̺||σ)

AB1 :rest ≤ min S(̺||σ ′ ) ≡ ER (̺AB1 :B2 ...BN ) σ∈sep1

≤ EfAB1 :rest (̺AB1 :B2 ...BN ) ≤ S(̺AB1 ).

(3)

3 Here the set “sep1 ” is the set of quantum states of A, B1 , B2 , . . . , BN which are separable across AB1 :rest AB1 : B2 . . . BN . ER (̺AB1 :B2 ...BN ) and AB1 :rest Ef (̺AB1 :B2 ...BN ) respectively denote the relative entropy of entanglement and the entanglement of formation [16] of the state ̺AB1 B2 ...BN in the same bipartition. The second inequality is obtained due to the fact that the relative entropy of entanglement is bounded above by the entanglement of formation [17], while the third follows from the fact that the von Neumann entropy of the local density matrix is an upper bound of the entanglement of formation for bipartite states [18]. For the multipartite state ̺AB1 B2 ...BN , the dense codmax ing advantage can be written as Cadv (̺AB1 B2 ...BN ) = max{SB1 −SAB1 , SB2 −SAB2 , . . . , SBN −SABN , 0}, where SBi = S(̺Bi ) and SABi = S(̺ABi ) are the single-site and two-site von Neumann entropies of ̺AB1 B2 ...BN respectively. Consider the instance when SB1 − SAB1 atmax tains the maximum. Then Cadv (̺AB1 B2 ...BN ) = SB1 − SAB1 . Adding this relation with Eq. (3), we obmax tain Cadv (̺AB1 B2 ...BN ) + ER (̺AB1 B2 ...BN ) ≤ SB1 ≤ max is obtained for the log2 dB1 . If the maximum of Cadv pair, say ABi , we have to choose the corresponding bipartition in the relative entropy of entanglement in Eq. (3), and in that case, the complementarity relation will be bounded above by the logarithm of the dimension of subsystem Bi . Therefore, in general, we have max{log2 dB1 , log2 dB2 , . . . , log2 dBN } = log2 d, as max the upper bound for the sum Cadv + ER . Hence the proof.  The complementarity relation which is established above clearly indicates that a high genuine multipartite entanglement will lower the advantage of the same state for transmitting classical information. The result is universal in the sense that it holds for an arbitrary pure or mixed quantum state of an arbitrary number of parties in arbitrary dimensions. Relation between dense coding advantage and GGM Towards showing that the obtained complementarity relation is generic, we consider another genuine multiparty entanglement measure, the generalized geometric measure. The relation is first proven for pure multiparty quantum states in arbitrary dimensions. For simplicity,
⊗N +1 we assume that the state lies in Cd , with arbitrary dimension d. Subsequently we show that the relation also holds for arbitrary mixed states, ρAB1 B2 ...BN . Consider an (N + 1)-party pure state, |ψiAB1 B2 ...BN , which is employed by the sender A to perform dense coding with the receivers Bi ’s (i = 1, 2, . . . , N ). Theorem 2: The sum of the advantage in dense coding and the generalized geometric measure for the arbitrary pure state |ψiAB1 B2 ...BN is bounded above by unity, i.e., d 1 C max + E ≤ 1. log2 d adv d−1

(4)

d , respectively for dense Here the factors log1 d and d−1 2 coding advantage and GGM, are normalizations that make the individual terms to have maximal value as unity. Proof. Let us assume, without loss of generality, that the maximum in the multiport quantum advantage in dense coding is attained for SB1 − SAB1 , i.e. max Cadv (|ψiAB1 B2 ...BN ) = SB1 − SAB1 . We now note that the GGM for the state |ψiAB1 B2 ...BN can be shown to be given by [12] E(|ψAB1 B2 ...BN i) = 1 − max{Λj }, where Λj ’s are the maximal eigenvalues of the local density matrices of all possible bipartite partitions of the state |ψiAB1 B2 ...BN . Therefore, we have E(|ψiAB1 B2 ...BN ) ≤ 1−λAB1 , with λAB1 being the maximum eigenvalue of the two-party reduced density matrix ̺AB1 of |ψiAB1 B2 ...BN . max Adding the relations for Cadv (|ψiAB1 B2 ...BN ) and GGM, we obtain max d Cadv (|ψiAB1 B2 ...BN ) + E(|ψiAB1 B2 ...BN ) log2 d d−1 SAB1 d(1 − λAB1 ) SB1 − + . ≤ log2 d log2 d d−1

To complete the proof, we have to show that the sum of the last two terms is non-positive. The sum of these two terms of the above equation is a convex function. Therefore, its maximum is attained at the extremal points, i.e. for λAB1 = d12 , 1. When λAB1 = 1, sum vanishes and at λAB1 = d12 , it is negative. The proof follows from the  fact that SB1 ≤ log2 d. We now consider quantum states of N + 1 parties which are possibly mixed, and are in arbitrary dimensions. For simplicity, we assume that the state is defined
⊗N +1 on Cd , with arbitrary dimension d. Theorem 3: For the arbitrary (possibly mixed) quantum state ̺AB1 B2 ...BN , the sum of the quantum advantage in dense coding and the generalized geometric measure is bounded above by unity. Proof. The genuine multiparty entanglement measure, GGM,P of ̺AB1 B2 ...BN is given by E(̺AB1 B2 ...BN ) = k pk E(|ψk ihψk |), where the ensemble {pk , |ψk i} forms the optimal pure state decomposition of the state ̺AB1 B2 ...BN for obtaining the minimum in the convex roof of the GGM. Suppose that λkAB1 is the maximum eigenvalue of the two-party reduced density matrix, of the parties A and PB1 , of the state |ψk i. Then we have E(̺AB1 B2 ...BN ) ≤ k pk (1 − λkAB1 ). One can show that P S(trB2 ...BN |ψk ihψk |) ) S(̺ d ≤ logABd1 . k pk d−1 E(ρAB1 B2 ...BN ) ≤ log2 d 2 We have used concavity of the von-Neumann entropy to get both the inequalities. Now, along with the quantum advantage in dense coding, and using the above inequality, we obtain max d Cadv (̺AB1 B2 ...BN ) + E(̺AB1 B2 ...BN ) log2 d d−1 SAB1 SB1 − SAB1 + ≤ 1. (5) ≤ log2 d log2 d

4 the Harish-Chandra Research Institute, India. 0.0 -0.2 -0.4 ∆C

-0.6 -0.8 -1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Θ

FIG. 1. (Color online.) Saturation of the complementarity max quantity. The complementarity quantity, δC = Cadv +E −1 is plotted against the state parameter, θ, of the generalized GHZ state (Left). When δC = 0, saturation of complementarity is achieved. The right figure presents the projection of the δC surface, for the generalized W state, on the plane of the state parameters θ′ and φ′ . The dark violet regions show the area where δC vanishes, so that the complementarity relation is saturated. The δC axis is dimensionless and all other axes are in radians.

Here we have assumed, without loss of generality, that max the maximum of Cadv is attained by the AB1 pair.  We now illustrate that the obtained complementary relation between the multiport dense coding quantum advantage and genuine multipartite entanglement is tight. For this investigation, let us first consider the generalized GHZ state [19], |ψGHZ i = cos θ|000i + sin θ|111i, where θ ∈ (0, π). We plot the complementarity quantity d max δC = log1 d Cadv + d−1 E − 1 with respect to θ (see Fig. 2 1(left)). We see that the saturation of δC occurs at θ = π4 and 3π 4 . Note here that the complementarity relation of Theorem 2 implies that δC is always non-positive, and that the vanishing of δC implies that the bound is tight. Similarly, we consider the generalized W states [20], |ψW i = sin θ′ cos φ′ |011i + sin θ′ sin φ′ |101i + cos θ′ |110i, with θ′ ∈ (0, π) and φ′ ∈ (0, 2π), which are known to be inequivalent to the generalized GHZ states by stochastic local operations and classical communication [20]. In Fig. 1 (right), we depict the projection of δC on the (θ′ , φ′ )plane for the generalized W state. It clearly indicates that there are regions where δC is saturated. Summarizing, we have established a complementary relationship between the quantum advantage of the multiport classical capacity of a multiparty quantum state used as a quantum channel and the genuine multipartite entanglement of the same state. The relation is demonstrated for two genuine multipartite entanglement measures – the relative entropy of entanglement and the generalized geometric measure. The relation holds for pure or mixed quantum states of arbitrary dimensions and of an arbitrary number of parties. R.P. acknowledges support from the DST, Government of India, in the form of an INSPIRE faculty scheme at

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