Multipartite Dense Coding vs. Quantum Correlation: Noise Inverts Relative Capability of Information Transfer
aa r X i v : . [ qu a n t - ph ] A p r Multipartite Dense Coding vs. Quantum Correlation:Noise Inverts Relative Capability of Information Transfer
Tamoghna Das, R. Prabhu, Aditi Sen(De), and Ujjwal Sen
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
A highly entangled bipartite quantum state is more advantageous for the quantum dense codingprotocol than states with low entanglement. Such a correspondence, however, does not exist even forpure quantum states in the multipartite domain. We establish a connection between the multipartycapacity of classical information transmission in quantum dense coding and several multipartitequantum correlation measures of the shared state, used in the dense coding protocol. In partic-ular, we show that for the noiseless channel, if multipartite quantum correlations of an arbitrarymultipartite state of arbitrary number of qubits is the same as that of the corresponding general-ized Greenberger-Horne-Zeilinger state, then the multipartite dense coding capability of former isthe same or better than that of the generalized Greenberger-Horne-Zeilinger state. Interestingly,in a noisy channel scenario, where we consider both uncorrelated and correlated noise models, therelative abilities of the quantum channels to transfer classical information can get inverted by ad-ministering a sufficient amount of noise. When the shared state is an arbitrary multipartite mixedstate, we also establish a link between the classical capacity for the noiseless case and multipartitequantum correlation measures.
I. INTRODUCTION
In recent times, a lot of interest has been created tocharacterize and quantify quantum correlations in mul-tipartite quantum systems [1–4]. This is due to the factthat the preparation of multiparticle states with quantumcoherence enables us to realize several quantum informa-tion protocols like quantum dense coding [5], quantumteleportation [6], secure quantum cryptography [7], andone way quantum computation [8], in a way that is betterthan their classical counterparts. This increasing interestis further boosted by the latest advances in experimentsto realize multipartite states in different physical systemsincluding photons, ion traps, optical lattices, and nuclearmagnetic resonances [9–14].It has been shown, both theoretically as well as ex-perimentally, that bipartite entanglement is an essentialingredient for a vast majority of known quantum com-munication schemes, involving two parties. Specifically,it has been established that in the case of pure bipartitestates, the capacity of classical information transmissionvia quantum states increases with the increase of anyquantum correlation measure. Such a simple scenario isfacilitated by the fact that quantum correlation in a bi-partite pure quantum state, in the asymptotic domain,can be uniquely quantified by the local von Neumann en-tropy [1]. The situation is far richer in the multipartycase (see Refs. [15–24]). Regarding the entanglementcontent, in a multiparty quantum state, which rangesfrom being bipartite to genuine multipartite, even multi-partite pure states can be classified in several ways [25]and therefore, there is no unique measure that can de-termine quantum correlations, present in the system.Unlike point to point communication, where amongtwo parties, one acts as a sender and the other as a re-ceiver, communication protocols involving multiple par-ties can have various complexities. One possible scenario involves several senders and a single receiver. Suitableexamples for such multipartite communication protocolsinclude, when several news reporters from different loca-tions send various news articles to the newspaper edito-rial office or when several weather observers from differ-ent places communicate their respective weather reportsto the regional meteorological office.In this paper, we connect multipartite communicationprotocols with multiparty quantum correlation measures.In particular, we establish a relation between the capacityof multipartite dense coding and multipartite quantumcorrelation measures of arbitrary multiqubit states. Thiscorrespondence is illustrated both in the case of noiselessand noisy channels for arbitrary shared states. Specifi-cally, we show that in the case of the noiseless quantumchannel, if the capacity of classical information transmis-sion using the generalized Greenberger-Horne-Zeilinger( g GHZ) state [26] is the same as that of any multiqubitpure state, then the multipartite quantum correlation ofthe g GHZ state is either the same or higher than thatof the latter states. The result is generic in the sensethat it does not rely on the choice of the quantum cor-relation measure, and is independent of the number ofparties. Three computable measures, the generalized ge-ometric measure [24, 27, 28], the tangle [29] and discordmonogamy score [30, 31], are considered for obtaining theresults. It is to be noted that while the first multipartymeasure is based on the geometry of the space of mul-tiparty quantum states, the latter two are based on theconcept of monogamy of quantum correlations.The noisy case can be considered at several levelsof complexity. Here we consider the following two sit-uations. First, we consider the case when the noiseis present before the encoding of classical information,while the quantum channel that transmits the encodedstate is noiseless. Secondly, we consider the situationwhen the encoding is performed on a pure shared state,while the post-encoded state is sent via a noisy quan-
FIG. 1. Schematic diagram of the change of status of the g GHZ state in comparison to other multipartite states withrespect to multipartite DC capacity in the presence of noise.The comparison has been made with the states which possessame amount of multipartite quantum correlations as the g GHZ state. The results obtained in this paper shows thatthe g GHZ state is more robust against noise as compared toarbitrary states for the dense coding protocol. This is inde-pendent of the fact whether the noise in the system is fromthe source or in the channel after the encoding. tum channel. In the first case, we begin by obtaininga sufficient condition for dense codeability for arbitrary,possibly mixed, multipartite quantum states. We per-form numerical simulations by generating Haar uniformrank-2 three-qubit states, and find that a great majorityof them are better carriers of classical information thanthe g GHZ state with the same multiparty quantum corre-lation content. Numerical analysis of higher-rank statesare also considered.Going over to the second case, when the quantum chan-nel is noisy, we show that the presence of noise can in-vert the relative capability of information transfer fortwo states with the same multiparty quantum correla-tion content. In particular, we find that the effect of bothcorrelated and uncorrelated covariant noise with respectto the capability of classical information transfer is lesspronounced on the g GHZ state as compared to a genericpure multiparty state. This relative suppression of theeffect of noise for the g GHZ state is what results in theinversion of the relative capabilities of information trans-fer between a g GHZ state and a generic pure multipartyquantum state. A schematic diagram elucidating the sit-uation is presented in Fig. 1.The paper is organized as follows. In Sec. II, we de-scribe the quantum correlation measures that are usedlater in the paper. In the subsequent section (Sec. III),we discuss the capacities of the quantum dense codingprotocol, with and without noise. In Sec. IV, we estab-lish the connection between the capacity and the quan-tum correlation measures for the noiseless channel, whilewe deal with the noisy channels in Sec. V. In partic-ular, we deal with the fully correlated Pauli channel inSec. V A and with the uncorrelated Pauli channel in Sec.V B. We present a conclusion in Sec. VI.
II. QUANTUM CORRELATION MEASURES
Quantum correlations present in bipartite quantumsystems can broadly be classified into two paradigms – (a)entanglement-separability and (b) information-theoretic.Measures of the former paradigm always vanish for sep-arable states. Examples include entanglement of for-mation [32], concurrence [33], distillable entanglement[32], and logarithmic negativity [34]. On the otherhand, information-theoretic quantum correlation mea-sures are independent of entanglement, and examples in-clude quantum discord [35] and quantum work deficit[36]. In the present section, we define concurrence andquantum discord, and then discuss monogamy relationsbased on these two measures. Finally we define a multi-partite entanglement measure, the generalized geometricmeasure, based on the concept of the Fubini-study metric[24, 27] (cf. [28]).
A. Concurrence
Concurrence is a quantum correlation measure for two-qubit systems, which is a monotonic function of entangle-ment of formation [33]. For an arbitrary two-qubit state, ρ AB , it is given by C ( ρ AB ) = max { , λ − λ − λ − λ } , (1)where the λ i ’s are the square roots of the eigenval-ues of ρ AB ˜ ρ AB in decreasing order and ˜ ρ AB = ( σ y ⊗ σ y ) ρ ∗ AB ( σ y ⊗ σ y ), with the complex conjugate being takenin the computational basis. Concurrence vanishes for allseparable states while it is maximal for any maximallyentangled state. B. Quantum Discord
Quantum discord is an information-theoretic quantumcorrelation measure, which is obtained by taking the dif-ference between two inequivalent forms of quantum mu-tual information. Mutual information quantifies the cor-relation between two systems. Classically, it can be de-fined in two equivalent ways. For two variables, X and Y , it is defined as I ( X, Y ) = H ( X ) + H ( Y ) − H ( X, Y ) , (2)where H ( X ) = − P x p x log p x is the Shannon entropy,with p x being the probability of x occurring as a valuefor the classical variable X , and similarly for H ( Y ). H ( X, Y ) denotes the Shannon entropy of the joint prob-ability distribution of X and Y . Using Bayes’ rule, onecan rewrite the mutual information in terms of condi-tional entropy, H ( X | Y ), to obtain I ( X, Y ) = H ( X ) − H ( X | Y ) . (3)In the quantum domain, these two classically equiv-alent definitions of mutual information become unequaland their difference has been proposed to be a measure ofquantum correlation, called the quantum discord [35, 37].For any composite system, ρ AB , quantizing the first def-inition of the mutual information, one obtains I ( ρ AB ) = S ( ρ A ) + S ( ρ B ) − S ( ρ AB ) , (4)where S ( ̺ ) = − Tr( ̺ log ̺ ) is the von Neumann entropyof ̺ . This quantity has been argued to be total corre-lation in the bipartite state [35]. Quantizing the seconddefinition is not straightforward, since the quantity ob-tained by replacing the Shannon entropies by the vonNeumann ones can be negative for some quantum states[38]. To overcome this drawback, one can make a mea-surement on one of the subsystems, say subsystem B , of ρ AB , and the measured conditional entropy of ρ AB canbe obtained as S ( ρ A | B ) = min { B i } X i p i S ( ρ A | i ) , (5)where the rank-1 projection-valued measurement { B i } is performed on the B -part of the system. Here ρ A | i = (1 /p i )(Tr B [( A ⊗ B i ) ρ AB ( A ⊗ B i )]), with p i =Tr AB [( A ⊗ B i ) ρ AB ( A ⊗ B i )], and A being the iden-tity operator on the Hilbert space of subsystem A . Usingthis quantity, one then quantizes the second definition ofmutual information as J ( ρ AB ) = S ( ρ A ) − S ( ρ A | B ) , (6)which has been argued to be a measure of classical cor-relation of the bipartite state [35]. Finally, the quantumdiscord is defined as D ( ρ AB ) = I ( ρ AB ) − J ( ρ AB ) . (7) C. Monogamy score: Tangle and DiscordMonogamy Score
Monogamy of quantum correlations quantifies thesharability of the same in multipartite systems [29]. Foran arbitrary ( N + 1)-party quantum state, ρ A A ...A N B ,let Q A i B ( i = 2 , . . . , N ) be the amount of a certainquantum correlation shared between the pair A i B ( i =2 , . . . , N ), and Q A A ...A N : B represent the same between B and rest of the parties. Here B acts as a “nodal” ob-server. The state ρ A A ...A N B is said to be monogamousfor the quantum correlation measure, Q , if [29] N X i =1 Q A i B ≤ Q A A ...A N : B . (8)Using the terms of the above relation, we define themonogamy score for the quantum correlation measure, Q , as δ Q = Q A A ...A N : B − N X i =1 Q A i B , (9) for B as the nodal observer. Therefore, Q is monoga-mous for a given state when δ Q is positive for that state.Otherwise, the measure is said to be non-monogamousfor that state. The advantage of such a multiparty quan-tum correlation measure is that it can be expressed interms of bipartite quantum correlation measures.In Eq. (9), if Q is chosen to be the square of the concur-rence, then we obtain the tangle [29], which is known tobe monogamous for all multiqubit states [29, 39]. Choos-ing Q to be quantum discord, we obtain the discordmonogamy score [31], which can be negative even forsome three qubit pure states [30]. D. Generalized Geometric Measure
Let us now define a genuine multipartite entanglementmeasure. An N -party pure state is said to be genuinelymultiparty entangled if it is non-separable with respectto every bipartition. The generalized geometric measure(GGM) is obtained by considering the minimal distancefrom the set of all multiparty states that are not gen-uinely multiparty entangled [24, 27] (cf. [28]). Morespecifically, the GGM of an N -party pure quantum state | φ N i is defined as E ( | φ N i ) = 1 − Λ ( | φ N i ) , (10)where Λ max ( | φ N i ) = max |h χ | φ N i| , with the maximiza-tion being taken over all pure states | χ i that are not gen-uinely N -party entangled. It was shown in Ref. [24, 27]that E ( | φ N i ) reduces to E ( | φ N i ) = 1 − max { λ A : B |A∪B = { , , . . . , N } , A∩B = ∅} , (11)where λ A : B is the maximal Schmidt coefficients in the A : B bipartite split of | φ N i . III. QUANTUM DENSE CODING
In this section, we define the capacities of dense coding,when an arbitrary multipartite state is used as a channel,shared between several senders and a single receiver. Wefirst consider the scenario of a noiseless channel and thenderive the capacity for a noisy covariant channel.
A. Quantum dense coding via noiseless channel
The capacity of the quantum dense coding protocolquantifies the amount of classical information that can betransferred via a quantum state used as a channel, whenan additional quantum channel, which may be noiselessor noisy, is also available. We begin with the noiselesscase. For an arbitrary two-party state, ρ SR , shared be-tween the sender, S , and the receiver, R , the capacity ofdense coding (DC) of ρ SR is given by [21, 40, 41] C ( ρ SR ) = 1log d SR max { log d S , log d S + S ( ρ R ) − S ( ρ SR ) } , (12)where d S is the dimension of the Hilbert space on whichthe senders’ part of the state ρ SR is defined, and where d SR is that on which the entire state ρ SR is defined. Here, ρ R is local density matrix of the receiver’s part. Thedenominator, log d SR , is incorporated to make the ca-pacity dimensionless. Note that we are considering thecase of unitary encoding (for non-unitary encoding, see[42, 43]), and where the additional quantum channel isnoiseless. The noisy case is considered below. When S ( ρ R ) − S ( ρ SR ) >
0, a shared quantum state is betterfor classical information transmission than any classicalprotocol with the same resources. It immediately impliesthat any bipartite pure entangled state is a good quan-tum channel for dense coding.In the multipartite regime, let us consider the scenariowhere there are N senders, S , . . . , S N and a single re-ceiver, R . For the quantum state, ρ S ...S N R , shared be-tween the N + 1 parties, the capacity of DC with unitaryencoding and for noiseless additional quantum channel isgiven by [21, 41] C ( ρ S ...S N R ) = 1log d S ...S N R max { log d S ...S N , log d S ...S N + S ( ρ R ) − S ( ρ S ...S N R ) } , (13)where d S ...S N = d S . . . d S N , with d S , . . . , d S N beingthe dimensions of the Hilbert spaces corresponding to theindividual senders, and d S ...S N R is the total dimension ofthe Hilbert space on which the entire multiparty state isdefined. The state is therefore dense codeable if S ( ρ R ) − S ( ρ S ...S N R ) > B. Capacity of quantum dense coding throughnoisy channel
Consider now the situation of classical informationtransmission when the additional quantum channel isnoisy. We assume that after local unitary encoding, theparticles are sent through a noisy quantum channel. Weconsider here a particular class of channels, known asthe covariant channels, denoted by Λ. Such a channelis a completely positive map with the property that itcommutes with a complete set of orthogonal unitary op-erators, { W i } , i.e.,Λ( W i ρW † i ) = W i Λ( ρ ) W † i , ∀ i. (14) The multipartite dense coding capacity has already beencalculated for this channel [43] and is given by C noisy ( ρ S ...S N R ) = 1log d S ...S N R max { log d S ...S N , log d S ...S N + S ( ρ R ) − S (˜ ρ ) } , (15)where˜ ρ = Λ (cid:16) ( U minS S ...S N ⊗ I R ) ρ S ...S N R ( U min † S S ...S N ⊗ I R ) (cid:17) . Here, U minS S ...S N denotes the unitary operator on thesenders’ side, which minimizes the von Neumann entropyof ( U S ...S N ⊗ I R ) ρ S ...S N R ( U † S ...S N ⊗ I R ) over the set ofunitaries { U S S ...S N } , that can be global as well as local.It is reasonable from a practical point of view to assumethat the senders perform local encoding. Then U minS S ...S N will have the form given by U minS S ...S N = U minS ⊗ U minS ⊗ ... ⊗ U minS N . (16)Since only the particles of the senders are sent throughthe noisy channel (after the unitary encoding), the en-tropy of the receiver’s side remains unchanged. Depend-ing on the structure of Λ, the channel can be either cor-related or uncorrelated.In Eqs. (12), (13) and (15), for convenience, we callthe second terms within the maximum in the numerators,divided by the denominators, as the corresponding “raw”capacities. We use the same notation for the raw capacityas the corresponding original capacity, but the contextwill always make the choice clear. Pauli Channel
The Pauli channel is an example of a covariant chan-nel. When an arbitrary two-dimensional quantum stateis sent through the Pauli channel, the state is rotatedby any one of the Pauli matrices, σ x , σ y , σ z , or left un-changed. If the σ x , σ y , and σ z act respectively with theprobabilities λ x , λ y , and λ z , then the transformed stateis ρ ′ = λ x σ x ρσ x + λ y σ y ρσ y + λ z σ z ρσ z + (1 − λ x − λ y − λ z ) ρ. (17)For λ x = λ y = λ z = p/
3, the channel represents thedepolarizing channel [44].
IV. RELATIONSHIP BETWEEN MULTIPARTYQUANTUM CORRELATIONS AND DENSECODING CAPACITY: NOISELESS CHANNEL
In this section, we establish a generic relation betweenthe capacity of dense coding and various quantum cor-relation measures, defined in Sec. II. The analytical re-sults obtained in this section are for quantum systemswith ( N + 1)-qubits, consisting of N senders and a singlereceiver. The numerical simulations that are performedto visualize the results are for three-qubit pure as well asmixed states. Throughout this section, we consider thecase when the additional quantum channel, that is usedpost-encoding, is noiseless. A. Connection between Capacity and MultipartiteQuantum Correlations for Pure States
In a bipartite scenario, all the pure states with thesame amount of entanglement have equal capacity ofdense coding. The entanglement in this case is uniquelyclassified by the von Neumann entropy of the local den-sity matrices and the capacity is maximal for the maxi-mally entangled states.We will see that this simple situation is no more truein the multiparty regime. However, it is still possibleto obtain a generic relation between capacity and en-tanglement. In a multipartite scenario, quantification ofquantum correlations is not unique even for pure statesand hence each measure, in principle, identifies its owndistinct state with maximal quantum correlation. Nev-ertheless, the Greenberger-Horne-Zeilinger (GHZ) state[26] has been found to possess a high amount of multipar-tite quantum correlation, according to violation of certainBell inequalities [45], as well as according to several mul-tipartite entanglement measures [24, 27, 28]. In view ofthese results, we compare the properties of an arbitrary( N + 1)-qubit pure state with that of the ( N + 1)-qubitgeneralized GHZ state ( g GHZ), which is given by | gGHZ i S S ...S N R = √ α | S . . . S N i| R i + √ − αe iφ | S . . . S N i| R i , (18)where α is the real number in (0,1) and φ ∈ [0 , π ). Wefind that if the capacity of dense coding of an arbitrary( N + 1)-party state, | ψ i , and the g GHZ state are thesame, then the quantum correlations of these two statesmay not be the same. However, they follow an ordering,which we establish in the following two theorems. Thefeature is generic in the sense that it holds for drasticallydifferent choices of the quantum correlation measures.Here on, we skip all the subscripts in the notation of thestates, for simplicity.
Theorem 1:
Of all the multiqubit pure states with anarbitrary but fixed multiparty dense coding capacity, thegeneralized GHZ state has the highest GGM.
Proof.
Scanning over α in Eq. (18), one can obtainan arbitrary value of the GGM. Therefore to prove thetheorem, one needs to show that if the multiparty densecoding capacity of an arbitrary ( N + 1)-qubit pure stateis the same as that of an ( N + 1)-party g GHZ state, thenthe genuine multipartite entanglement, as quantified bythe GGM, of that arbitrary pure state is bounded aboveby that of the g GHZ state, i.e., E ( | ψ i ) ≤ E ( | gGHZ i ) . (19) The multipartite dense coding capacities of the ( N +1)-party g GHZ state and the arbitrary pure state, | ψ i , canbe obtained by using Eq. (13), and are given respectivelyby C ( | gGHZ i ) = NN + 1 − α log α + (1 − α ) log (1 − α ) N + 1and C ( | ψ i ) = NN + 1 − λ R log λ R + (1 − λ R ) log (1 − λ R ) N + 1 , where λ R is the maximum eigenvalue of the marginaldensity matrix, ρ R , of the receiver’s part of the state | ψ i .The GGMs for the g GHZ state and the | ψ i are obtainedrespectively by E ( | gGHZ i ) = 1 − α, (20) E ( | ψ i ) = 1 − max[ { l A } ] , (21)where we assume that α ≥ . The set { l A } contains themaximum eigenvalues of the reduced density matrices ofall possible bipartitions of | ψ i . Equating the multipartydense coding capacities for these two states, we obtain α = λ R . (22)Note that λ R ∈ { l A } . Let us now consider the two fol-lowing cases: (1) the maximum in GGM is attained by λ R , and (2) the maximum is attained by an eigenvaluewhich is different from λ R . Case 1:
Suppose λ R = max[ { l A } ]. Then E ( | ψ i ) = 1 − λ R = 1 − α = E ( | gGHZ i ) , (23)by using Eq. (22). Case 2:
Suppose λ R = max[ { l A } ]. Let λ R ≤ λ =max[ { l A } ]. Therefore, we obtain E ( | ψ i ) = 1 − λ ≤ − λ R = 1 − α = E ( | gGHZ i )Hence the proof. (cid:4) We randomly generate 10 arbitrary three-qubit purestates by using the uniform Haar measure on this spaceand plot the behavior of the GGM versus the DC capacityfor these states. As proven in Theorem 1, the scatterdiagram populates only a region outside the paraboliccurve of the g GHZ states. See Fig. 2. Interestingly,therefore, in the plane of the dense coding capacity andthe GGM, there exists a forbidden region which cannotbe accessed by any three-qubit pure state. With respectto dense coding in the noiseless case, therefore, the g GHZstate is the least useful state among all states having anequal amount of the multiparty entanglement.We now show that the result is potentially indepen-dent of the choice of the multipartite quantum correla-tion measure. Towards this aim, let us now consider thetangle and the discord monogamy score, as multipartyquantum correlation measures. The relations between
FIG. 2. (Color online.) GGM vs. multipartite DC capac-ity. GGM is plotted as the ordinate while multipartite DCcapacity is plotted as the abscissa for 10 randomly chosenthree-qubit pure states, according to the uniform Haar mea-sure over the corresponding space (blue dots). The red linerepresents the generalized GHZ states. There is a set of statesfor which, if the capacity matches with a g GHZ state, thentheir GGMs are also equal. For the remaining states, if thecapacity is equal to a g GHZ state, its GGM is bounded aboveby that of the g GHZ state. Note that the range of the horizon-tal axis is considered only when the states are dense codeable.The quantities represented on both the axes are dimension-less. We are considering the case where the post-encodedstates are sent through noiseless channels. these two quantum correlation measures and the capac-ity of DC are established in the following theorem.
Theorem 2:
Of all multiqubit pure states with an arbi-trary but fixed multiparty dense coding capacity, the gen-eralized GHZ state has the highest tangle as well as thehighest discord monogamy score.Note:
The tangle and discord monogamy score are de-fined here by using the receiver of the DC protocol asthe nodal observer.
Proof.
The equality of the multipartite dense coding ca-pacities of the ( N +1)-party g GHZ state and the arbitrarypure state, | ψ i , implies that α = λ R ≥ /
2. Notationsare the same as in the proof of Theorem 1. Note thatthe tangle of the g GHZ state is 4 α (1 − α ). Therefore, wehave δ C ( | ψ i ) = 4 λ R (1 − λ R ) − X i C ( ρ S i R ) ≤ λ R (1 − λ R ) = 4 α (1 − α ) = δ C ( | gGHZ i ) . (24)The inequality in the second line is due to the fact that C ( ρ S i R ) are non-negative, where ρ S i R is the density ma-trix of the sender S i and the receiver R corresponding to FIG. 3. (Color online.) Left: Tangle (vertical axis) vs.multiparty DC capacity (horizontal axis) for randomly gen-erated three-qubit pure states (blue dots). Right: Discordmonogamy score (vertical axis) vs. DC capacity (horizontalaxis) for the same states. In both the cases, the g GHZ statesgive the boundary (red line). The capacity is dimensionless,while the tangle and discord monogamy score are measuredin ebits and bits, respectively. All other considerations arethe same as in Fig. 2. the state | ψ i . Similarly, for the discord score, we have δ D ( | ψ i ) = S ( ρ R ) − X i D ( ρ S i R ) ≤ S ( ρ R ) = S ( α ) = δ D ( | gGHZ i ) , (25)since 0 ≤ D ( ρ S i R ) ≤
1. Here, S ( α ) denotes the vonNeumann entropy of the single-side density matrix of the g GHZ state. Hence the proof. (cid:4)
To visualize the above theorem, we randomly generate10 pure three-qubit states, by using the uniform Haarmeasure in the corresponding space, and prepare scatterdiagrams for tangle versus the multiparty DC capacity(Fig. 3 (left)) and for the discord monogamy score ver-sus the same capacity (Fig. 3 (right)). The simulationsare clearly in agreement with Theorem 2. In particular,and just like for GGM versus the capacity, the planesof ( C, δ C ) and ( C, δ D ) can not be fully accessed by thethree-qubit pure states. B. Capacity vs. Multipartite QuantumCorrelations for Shared Mixed States
We now investigate the relation between DC capac-ity and multipartite quantum correlation measures, whenthe shared state is an arbitrary ( N +1)-party mixed state.In this case, to establish such connection, the main dif-ficulty is that there are only a few quantum correlationmeasures available which can be computed. In this case,therefore, we consider the discord monogamy score as themultipartite quantum correlation measure, since quan-tum discord can be numerically calculated for arbitrarybipartite systems, and investigate its connection with theDC capacity.In Fig. 4, we randomly generate 10 mixed states ofrank-2 in the space of three-qubit states and plot thediscord monogamy score with respect to the DC capac-ity. The random generation is with respect to the uni-form Haar measure induced from that in the appropriatehigher-dimensional pure state space. The numerical sim-ulation reveals that Theorem 2 does not hold for rank-2(mixed) three-qubit states. In particular, we find thatif a g GHZ state and a rank-2 three-qubit mixed statehave the same discord monogamy score, then sharing the g GHZ state is usually more beneficial than the mixedstate, for performing the multiparty DC protocol. Moreprecisely, among randomly generated 10 rank-2 states,there are only 1 .
85% states which satisfy Theorem 2. Wewill later show that a similar picture is true for the noisychannel. This implies that in the presence of noisy en-vironments, irrespective of whether the noise is afflictedbefore or after encoding, it is typically better to share a g GHZ state among states with a given discord monogamyscore, from the perspective of DC capacity. Before pre-senting the results obtained by using numerical simula-tions for higher-rank mixed states, let us discuss the be-havior of the DC capacity, as enunciated in the followingproposition. We will find that it can be used to intu-itively understand the numerical results for higher-rankstates presented below.
Proposition 1:
An arbitrary ( N + 1) -qubit (pure ormixed) state is dense codeable if the maximum eigenvalueof the ( N +1) -party state is strictly greater than the maxi-mum eigenvalue of its reduced state at the receiver’s side. Proof:
An ( N + 1)-qubit (pure or mixed) state, ρ S S ...S N R , is multiparty dense codeable with N senders, S , S , . . . , S N , and a single receiver, R, if and only if thevon Neumann entropy of the reduced state at the re-ceiver’s side is greater than that of the state ρ S S ...S N R ,i.e., S ( ρ R ) > S ( ρ S S ...S N R ) . (26)Let the eigenvalues, in descending order, of the state ρ R ,be given by λ R = { λ ≥ , − λ } . Let the eigenvalues ofthe state ρ S S ...S N R be λ S S ...S N R = { µ i } ri =1 , where r isthe rank of the matrix, and where the µ i ’s are arrangedin descending order. Specifically, µ gives the largesteigenvalue of ρ S S ...S N R . Now, the ordering between thehighest eigenvalues of ρ R and ρ S S ...S N R , i.e., between λ and µ , can have three possibilities, i.e., λ > µ , orthey are equal, or λ < µ .Let us assume that λ ≥ µ . Then, invoking the con-dition of majorization [46], we have λ R ≻ λ S S ...S N R , which implies S ( ρ R ) ≤ S ( ρ S S ...S N R ) . (27)It immediately implies that the state is not densecodeable. Therefore, to obtain dense codeability of ρ S S ...S N R , we must have λ < µ .Hence the proof. (cid:4) Although the above proposition has been presented forqubit systems, it is also valid for an arbitrary (pure ormixed) ( N + 1)-party quantum state in arbitrary dimen-sions, provided ρ R is of rank 2. FIG. 4. (Color online.) Discord monogamy score vs. mul-tipartite DC capacity for Haar uniformly generated rank-2three-qubit states. The red line represents the g GHZ states.About 1.85% of the randomly generated states lie below thered line, and are represented by blue dots. The remaining,represented by green dots, lie above the red line. The hori-zontal axis is dimensionless while the vertical one is measuredin bits.
Let us now move to mixed states with higher-rank. Nu-merically, to obtain high-rank three-qubit mixed states,one possibility is to generate pure states with more thanthree parties. For example, to obtain arbitrary rank-4states of three qubits, 5-qubit pure states can be createdrandomly, and then two parties traced out. However, nu-merical searches become inefficient with the increase ofnumber of parties [47]. To overcome this problem, wecreate mixed states, ρ , of full rank, given by ρ = (1 − p ) ρ + p I , (28)by choosing ρ as arbitrary rank-2 three qubit states, gen-erated randomly from the three-qubit pure states, andwhere I is the identity matrix on the three-qubit Hilbertspace. Moreover, we consider those set of states, ρ , whichare dense codeable. In that case, we find that its DC ca-pacity remains nonclassical only for very small values ofthe mixing parameter p . In Fig. 5, we specifically con-sider the full rank state, ρ , with ρ given by ρ = q | GHZ ih GHZ | + (1 − q ) | GHZ ′ ih GHZ ′ | , (29)where | GHZ ′ i = √ ( | i − | i ). We now plot, inFig. 5, the discord monogamy score and the raw DCcapacity with respect to the mixing parameter p . For q = 1 or q = 0, and p = 0, the capacity is maximumand δ D also gives a maximum. Fig. 5 shows that thereis a small region in which the state remains dense code-able, only when δ D is very high. It is plausible that thecapacity of dense coding for mixed states decreases withthe increase of rank of the state. This is intuitively un-derstandable from the condition in Proposition 1, since
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 δ D q = 0.0q = 0.04q = 0.08q = 0.12q = 0.16q = 0.20q = 0.24 C p δ D FIG. 5. (Color online.) Discord monogamy score and theraw DC capacity are plotted against the mixing parame-ter p , for the rank-8 state, ρ = (1 − p ) ρ + p I . Here ρ = q | GHZ ih GHZ | + (1 − q ) | GHZ ′ ih GHZ ′ | . Each value of q provides a curve, and we present several exemplary curvesin the figure. All the quantities plotted are dimensionless,except δ D , which is measured in bits. the typical high-rank state can have eigenvalues moredistributed than the typical low-rank state. Therefore,the maximal eigenvalue of a shared state typically givesa lower value than that of the receiver’s side, and thecondition in Proposition 1 is thereby satisfied for a verysmall set of states. V. RELATION BETWEEN CAPACITY OFDENSE CODING AND QUANTUMCORRELATIONS FOR NOISY CHANNELS
In this section, we consider the DC capacity of thenoisy channels, for both correlated and uncorrelatednoise, for the ( N + 1)-party state, ρ S S ...S N R , shared be-tween N senders and a single receiver. Here we assumethat the N senders individually apply local unitary op-erations on their parts of the shared state and send theirencoded parts through a covariant noisy channel (see Sec.III B). We now address the extent to which the relation,established in Sec. IV, between multiparty DC capacityfor the noiseless channel and multipartite quantum cor-relation measures, in the case of pure shared quantumstates, still remains valid for the noisy channel scenario.To this end, we consider two extreme scenarios, one inwhich the noise between the different sender qubits arefully correlated, and another in which the same are un-correlated. A. Fully correlated Pauli channel
An ( N + 1)-qubit state, ρ S S ...S N R , after being actedon by the fully correlated Pauli channel, is given byΛ P ( ρ S S ...S N R ) = P i =0 q m ( σ mS ⊗ · · · ⊗ σ mS N ⊗ I R ) ρ S S ...S N R ( σ mS ⊗ · · · ⊗ σ mS N ⊗ I R ) , (30)where P m =0 q m = 1, and q m ≥
0, and where we de-note, for simplicity, σ x = σ , σ y = σ , σ z = σ , and theidentity matrix as σ for the sender qubits. The receiverqubit is acted on only by the identity operator, which wedenote by I R .We now establish the parallel of the ordering in Theo-rem 1 for the fully correlated Pauli channel. Theorem 3:
If the multiparty dense coding capacity ofan arbitrary three-qubit pure state, | ψ i , is the same asthat of the g GHZ state in the presence of the fully cor-related Pauli channel, then the genuine multipartite en-tanglement, GGM, of that arbitrary pure state is boundedbelow by that of the g GHZ state, i.e., E ( | ψ i ) ≥ E ( | gGHZ i ) , (31) provided the following two conditions hold: (i) the largesteigenvalue of the noisy | ψ i state is bounded above by max { q + q , − q − q } , and (ii) the receiver’s sidegives the maximum eigenvalue for the GGM of | ψ i . Proof:
The capacities of multiparty dense coding of the g GHZ state and the three-qubit pure state, | ψ i , afterbeing acted on by the correlated noisy channel, can beobtained from Eq. (15), and are given respectively by C noisyc ( | gGHZ i ) = 23 + H ( α ) − S (˜ ρ gGHZ )3 (32)and C noisyc ( | ψ i ) = 23 + H ( λ R ) − S (˜ ρ ψ )3 (33)where ˜ ρ gGHZ = Λ P (( U minS S ⊗ I R ) | gGHZ ih gGHZ | ( U min † S S ⊗ I R )) with U minS S being the unitary operator atthe senders’ part that minimizes the relevant von Neu-mann entropy (see Sec. III B). Here, we are consideringonly those cases for which the (noisy) capacities of boththe g GHZ state as well as of the | ψ i are non-classical, i.e.,the corresponding noisy states are dense codeable. Re-placing | gGHZ i by | ψ i in ˜ ρ gGHZ , one obtains ˜ ρ ψ . The U minS S is of course a function of the input state. Here , λ R ≥ denotes the maximum eigenvalue of the reduceddensity matrix ρ R of | ψ i . For 0 ≤ x ≤ H ( x ) denotesthe binary entropy function − x log x − (1 − x ) log (1 − x ).For the g GHZ state, the von Neumann entropy of the re-sulting state after sending through the fully correlatedPauli channel is S (˜ ρ gGHZ ) = H ( q + q ), which is inde-pendent of the choice of the local unitary operators.Equating Eqs. (32) and (33), we have, H ( α ) = H ( λ R ) + [ H ( q + q ) − S (˜ ρ ψ )]= H ( λ R ) + [ H ( q + q ) − H ( { λ i } )] , (34)where { λ i } i =1 are the eigenvalues of ˜ ρ ψ in descendingorder. Here H ( { λ i } ) = − P i λ i log λ i . If we assumethat λ ≤ max { q + q , − q − q } , we have { λ i } ≺{ q + q , − q − q } . The relation between majorizationand Shannon entropy [46] then implies that H ( q + q ) ≤ H ( { λ i } ). Therefore, from Eq. (34), we have H ( α ) ≤ H ( λ R ) ⇒ α ≥ λ R , (35)where we assume α ≥ .The GGM for the g GHZ state and the three-qubitstate, | ψ i , are respectively given by E ( | gGHZ i ) = 1 − α and E ( | ψ i ) = 1 − λ max , where λ max is the maximumeigenvalue among the eigenvalues of all the local densitymatrices of | ψ i . If we assume that the eigenvalue from thereceiver’s side attains the maximum, i.e., if λ R = λ max ,using Eq. (35), we obtain E ( | gGHZ i ) = 1 − α ≤ − λ R = E ( | ψ i ) . (36)Hence the proof. (cid:4) The above theorem ekes out a subset of the pure three-qubit state space, for which the g GHZ state is more ro-bust with respect to multiparty DC capacity, againstfully correlated Pauli noise, as compared to any mem-ber of the said subset, provided the g GHZ and the saidmember have equal multiparty entanglement, as quanti-fied by their GGMs. This specific subset of states arethose which satisfy both the conditions ( i ) and ( ii ). Thesituation, at least for this specific subset, has thereforeexactly reversed with respect to the noiseless scenario, asenunciated in Theorem 1. For a given amount of multi-party entanglement content, as quantified by the GGM,the g GHZ state can now be better than other pure states,with respect to the multiport classical capacity. Thenoisy quantum channel can therefore reverse the rela-tive capabilities of classical information transfer of dif-ferent states in multiparty quantum systems. The resultis much more general than what is contained in Theorem3. First of all, the result in Theorem 3 holds even if wereplace the GGM as the multiparty quantum correlationmeasure by the tangle or the discord monogamy score,provided we consider the set of three-qubit pure statesfor which the two-party concurrences or quantum dis-cords respectively vanish, and the receiver is used as thenodal observer. Comparing now with Theorem 2, we seethat the phenomenon of the inversion of the relative capa-bilities for classical information transfer is generic in thissense: it applies irrespective of whether the GGM, or thetangle, or the discord monogamy score is used to measurethe multiparty quantum correlation content. Secondly,we will show below that the phenomenon of reversal ofinformation carrying capacity with the addition of noiseactually holds for a much larger class of states than theones covered by the conditions ( i ) and ( ii ) in Theorem3. We resort to numerical searches by generating Haaruniform three-qubit pure states for this purpose. Thefollowing picture is therefore emerging. Given a three-qubit pure state, | ψ i , and a g GHZ state with the samemultiparty quantum correlation content, the multiparty
FIG. 6. (Color.) GGM (vertical axis) vs. the raw DC capac-ity (horizontal axis) under the fully correlated Pauli channel,when the shared state is an arbitrary three-qubit pure state(orange, green, and blue dots) or the g GHZ state (red line).For all the states, the noise in the channel is fixed to 0.19.We choose q = q = 0 . q = q = 0 .
015 as noiseparameters for the arbitrary as well as the g GHZ state. Thiscorresponds to the Case 1 of Sec. V A. See the discussionthere for further details. Both the axes are dimensionless.The vertical line at the C noisyc = 2 / DC capacity of the g GHZ state is much less affected bynoise than a large class of | ψ i , and in many cases, theordering of the capacities can get reversed in the noisycase as compared to the order in the noiseless case.To perform the numerical searches, we first observethat the C noisyc ( | gGHZ i ) depends on the sum of the twoparameters q and q (or q + q ). By fixing q + q = c (or q + q = 1 − c ), one can set the noise parameter forthe g GHZ state. However, the situation for an arbitrarystate, | ψ i , is more involved, for which the capacity ofdense coding, C noisyc ( | ψ i ), depends individually on all the { q i } . To quantify the randomness of { q i } , and indeed thenoise in the channel, we consider the Shannon entropy, H ( { q i } ). We now consider two extreme cases: one forwhich H ( { q i } ) is maximum and the other in which thesame is a minimum, both subject to the constraint q + q = c , where 0 ≤ c ≤
1. The maximum of H ( { q i } ) isattained when q = q = c/ q = q = (1 − c ) / q and q and any one of q and q are zero. It is alsoevident from Eq. (32), that one should deal with a verylow or very high values of c , for the state to remain densecodeable.We now randomly generate 5 × three-qubit purestates with a uniform Haar measure over that space, andinvestigate the two extreme cases mentioned above, forfixed H ( q + q ) = 0 .
19. We choose the two sets of val-ues for the q i ’s as follows – Case 1: q = q = 0 . q = q = 0 .
015 (see Fig. 6), and Case 2: q =0 . , q = 0 . , q = 0 . , q = 0 .
04 (see Fig. 7). For0
FIG. 7. (Color.) GGM (vertical axis) vs. DC capacity (hori-zontal axis) under the fully correlated Pauli channel. In thiscase, we choose { q i } as q = 0 . , q = 0 . , q = 0 . , q =0 .
04. This corresponds to the Case 2 of Sec. V A. We ran-domly (Haar uniformly) generate 5 × three-qubit purestates. See text for further details. Both axes represent di-mensionless quantities. The vertical line at C noisyc = 2 / fixed H ( q + q ) = 0 .
19, Case 1 is an example for highnoise, and corresponds to the case when H ( { q i } ) is amaximum subject to the constraint H ( q + q ) = 0 . q + q = 0 .
03. Case 2is an example of low noise, and corresponds to a situationthat is close to the case when H ( { q i } ) is a minimum sub-ject to the constraint H ( q + q ) = 0 .
19. We present thelow noise case, when the configuration is slightly awayfrom the analytical minimum to provide a more non-trivial example.Case 1 (Fig. 6): In presence of high noise, we ob-serve that almost all the randomly generated states haveshifted to above the g GHZ state (red line) in the planeof GGM and the raw capacity, C noisyc . As expected, one-third of the randomly generated states satisfy condition( ii ) of Theorem 3. A significantly large fraction (98.6%)of them further satisfies condition ( i ). They are repre-sented by blue dots in Fig. 6 and lie above the g GHZline. The remaining 1.4% are represented by green dots,and may lie below or above the g GHZ curve. The furtherstates are represented by orange dots. Note that we haveplotted the raw capacity in Fig. 6.Case 2 (Fig. 7): For low noise, the randomly generatedstates may fall below or above the red line of the g GHZstates. Again, one-third of the generated states satisfycondition ( ii ). 45 .
6% of them satisfy condition ( i ), arerepresented by blue dots, and fall above the red line. Theremaining 54 .
4% of them are represented by green dots,and can be below or above the g GHZ line. The othertwo-thirds are represented by orange dots, and can againbe either below or above the g GHZ line.The occurrence of the randomly generated states bothbelow and above the curve for the g GHZ states on theplane of the GGM and the capacity is expected from con-tinuity arguments, for low noise. However, if one makes a C noisyc H({q i }) α = 0.02 α = 0.06 α = 0.12 α = 0.20 α = 0.30 α = 0.50 C noisyuc α = 0.02 α = 0.06 α = 0.12 α = 0.20 α = 0.30 α = 0.50 FIG. 8. (Color online.) DC capacity vs. noise for variouschoices of α in the g GHZ state. In the top panel, the ca-pacity of DC is plotted against the noise of the depolarizingchannel, while in the bottom one, the DC capacity is plottedwith respect to the noise in the fully correlated Pauli channel,for the g GHZ state. Different curves correspond to differentvalues of α . The vertical axis starts from 2/3, below whichthe states are not dense codeable. The states remain densecodeable in the presence of moderate to high Pauli noise whilethis is not the case for the uncorrelated depolarizing channel.The horizontal axes are measured in bits. All other quantitiesare dimensionless. comparison between Figs. 2 and 6, it is revealed that ar-bitrary three-qubit pure states require higher amount ofmultipartite entanglement than the g GHZ states to keepthemselves dense codeable in the presence of moderatenoise.We have also numerically analyzed the randomly gen-erated states by replacing the GGM with the tangle andwith the discord monogamy score. We find the behaviorof the DC capacity with these multiparty quantum cor-relation measures to be similar to that between the DCcapacity and the GGM. However, the GGM is more sen-sitive to noise than tangle or discord monogamy score, inthe sense that in the presence of small values of noise pa-rameters, the percentages of states which are below the g GHZ state is much higher in the case of the monogamyscore measures than for the GGM.Therefore, Theorem 3 and the numerical simulationsstrongly suggest that in the presence of fully correlatedPauli noise, the ratio of multipartite entanglement to theDC capacity of the g GHZ state increases at a slower ratethan that of the arbitrary three-qubit pure states, irre-spective of the choice of the multiparty quantum corre-lation measure.
B. Uncorrelated Pauli channel
Consider now a Pauli channel in which the unitary op-erators acting on different subsystems are not correlatedto each other. More specifically, we suppose that eachqubit is acted on by a depolarizing channel with noiseparameter p . Before analyzing the relation between the1 FIG. 9. (Color.) GGM vs. the raw DC capacity, C noisyuc , inthe presence of the uncorrelated noise. Se text for furtherdetails. both axes represent dimensionless quantities. Thevertical line at C noisyuc = 2 / multiparty DC capacity and quantum correlation mea-sures, we compare the multiport dense coding capacitiesfor the correlated channels with those of the uncorrelatedones. A three-qubit state, ρ S S R , after the post-encodedqubits pass through independent (uncorrelated) depolar-izing channels, of equal strength, p , takes the form D ( ρ S S R ) = (1 − p ) ρ S S R + (1 − p ) p X i =1 ( I S ⊗ σ iS ⊗ I R ) ρ S S R ( I S ⊗ σ iS ⊗ I R )+ (1 − p ) p X i =1 ( σ iS ⊗ I S ⊗ I R ) ρ S S R ( σ iS ⊗ I S ⊗ I R )+ p X i = j =1 ( σ iS ⊗ σ jS ⊗ I R ) ρ S S R ( σ iS ⊗ σ jS ⊗ I R ) . In the top panel of Fig. 8, the capacity of DC is plottedagainst the total noise, 2 H ( p ), of the uncorrelated chan-nel, for various choices of α in the g GHZ state. The bot-tom panel represents the DC capacity in the case of thefully correlated Pauli channel with respect to the noise, H ( { q i } ), in this case, for the same g GHZ states. Theamount of correlated Pauli noise that can keep the g GHZstate dense codeable, is therefore higher than that of theuncorrelated noise.To analyze the relation between the DC capacity andquantum correlation, we plot, in Fig. 9, the GGM against C noisyuc , the DC capacity for two senders and a singlereceiver, with the post-encoded quantum systems beingsent to the receiver via uncorrelated depolarizing chan- nels, for arbitrary pure three-qubit states, which are nu-merically generated by choosing 5 × random states.We choose the noise parameter, p , as 0 .
04 for the pur-pose of the figure (Fig. 9). Fig. 8 shows that for smallvalues of p , the g GHZ state remains dense codeable evenfor small values of α . In Fig. 9, The blue dots are theones which satisfy condition ( ii ) of Theorem 3. Note thatcondition ( i ) is not well-defined in the current (uncorre-lated) scenario. Most of them lie above the red curve ofthe g GHZ states. The remaining states are representedby orange dots.
VI. CONCLUSION
For transmission of classical information over noise-less and memory-less quantum channels, the capacity inthe case of a single sender and a single receiver is well-studied. However, point-to-point communication is oflimited commercial use and the exploration of quantumnetworks with multiple senders and receivers is thereforeof far greater interest. Moreover, creation of multipar-tite systems with quantum coherence, the essential in-gredient for several quantum communication as well ascomputational tasks, is currently being actively pursuedin laboratories around the globe. Establishment of con-nections between multipartite quantum correlation andcapacities are usually hindered by the unavailability ofa unique multiparty quantum correlation measure evenfor pure states, and the plethora of possibilities for mul-tiparty communication protocols.For a communication scenario involving several sendersand a single receiver, we establish the relation be-tween capacities of classical information transmission andmultipartite computable quantum correlation measures,both for the noiseless as well as noisy channels. We showthat there are hierarchies among multipartite states ac-cording to the capacities of the dense coding protocol andhence obtain a tool to classify quantum states accordingto their usefulness in quantum dense coding. The resultscan be an important step forward in building up commu-nication networks using multipartite quantum correlatedstates in realizable systems.
ACKNOWLEDGMENTS
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