One-shot conclusive multiport quantum dense coding capacities
Chirag Srivastava Anindita Bera Aditi Sen De Ujjwal Sen







Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India



Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India
Department of Applied Mathematics, University of Calcutta, 92 A.P.C. Road, Kolkata 700 009, India
Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem
91 904, Givat Ram, Israel


Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India



Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India
    




We introduce a probabilistic version of the one-shot quantum dense coding protocol
in both two- and multiport scenarios, and refer to it as conclusive quantum 
dense coding. Specifically, we analyze the corresponding capacities of shared states between two, three, and more qubits, and two qutrits. We identify 
cases where Pauli and generalized Pauli operators are not sufficient as encoders to attain the optimal one-shot conclusive quantum dense coding capacities. We find that there is a rich connection between the capacities, and the bipartite and multipartite entanglements of the shared state.
















One-shot conclusive multiport quantum dense coding capacities
    Ujjwal Sen
    {kuny, yifanchen, roumposg, ccolby, janders}@google.com
=============================================================



§ INTRODUCTION








Any communication protocol typically involves two parties, a sender and a receiver. A sender 


encodes the information in a physical system, sends it through a physical channel, and then the other party – the receiver – performs a decoding. In this paper,
we are interested in a class of communication schemes that deal with transfer of classical information through quantum states shared between distant parties, broadly known as quantum dense coding (DC). The first instance of a DC protocol was provided by Bennett and Wiesner <cit.> in 1992,
in which they proposed to use a single copy of a certain shared state between two parties, say, Alice (sender) and Bob (receiver), and a noiseless quantum channel between them, as resources. Utilizing these resources, Alice is able to transfer an amount of classical information that is higher than the corresponding “classical limit" <cit.>, for certain shared states.

It turns out that such an advantage over classical protocols is obtained due to the quantumness present in a shared quantum state, named as entanglement 


<cit.>. Over the years, 
it has also been realized that entangled states are not only useful for sending classical information but are also beneficial for different quantum information processing tasks <cit.>, and such tasks have been performed with different physical systems like


 
 photons <cit.> and ions <cit.>  (see also 
 
 <cit.>).








In 1996, Mattle et al. performed an experiment to demonstrate the quantum DC protocol by using the polarization degree of freedom of 

photons <cit.>. See also <cit.>.

There is a significant body of work on generalizations of the original DC protocol
in which the unitary encoding at the sender's node is on a single copy of the state while the decoding process of the receiver involves multiple copies, and refer to them as the generalized DC protocols. See, e.g., <cit.>. See also <cit.>.

In a different direction,

Mozes et al. <cit.> have introduced a DC protocol, called 

“deterministic dense coding" (DDC) protocol, where both encoding and decoding are performed at the single-copy level of the shared quantum state between Alice and Bob, and Alice is able to send classical data to Bob in an error-free mode, by using non-maximally entangled pure states. Several interesting questions including generalization of a DDC scheme to a multiparty scenario have been addressed in recent years

<cit.>.

In this paper, we introduce 

the concept of conclusive dense coding (CDC), in which Alice uses a single copy of a pre-shared non-maximally entangled pure (quantum) state with Bob to send complete classical messages unambiguously. The protocol, which, like the DDC scheme, is a single-shot strategy,  allows for some error (because of its probabilistic nature), but the users know when an error has been committed, and for a fixed shared state, 
the aim 

is to devise a quantum strategy (encoding and decoding) to minimize the error. And in case an error is not committed, Alice is successful in sending the complete classical message what she intended to send. 


To clarify the meaning of sending a complete classical message, suppose that Alice intended to send information about the color and the softness of the ball. And if a protocol, sometimes, allows to send information about its color only, then it is not complete information. In the CDC protocol, we equate receipt of such partial information with receipt of no information. The situation is similar to e.g. certain online transactions where both card verification value (“CVV") and one-time password (“OTP") are needed, and possessing any one is useless for the task. 
Previous works on probabilistic versions of the dense coding protocol can be found in <cit.>, although none of the 
schemes include the 

stipulation
of complete information transfer.


 An important feature of utilizing CDC instead of the related DDC protocol for a shared state of two d-dimensional quantum systems is that the latter is not useful for sending a (classical) message that has more than d^2-2 values if the shared state is not maximally entangled <cit.>, while the former is. Probabilistic versions of the DC protocols and their capacities have often been calculated in previous works by using generalized Pauli operators as encoders. Such encoding may not provide the optimal capacity, as is the case for several of the scenarios considered here. 
 
 We also comment on the relation of the CDC capacity with the entanglement in the shared state. 
 
 
 We subsequently move over to the case of multiport conclusive quantum dense coding
with 

two senders and a single receiver sharing a genuine pure tripartite entangled state. Recently, it was shown that generalized Greenberger-Horne-Zeilinger (GHZ) <cit.> states are not useful for DDC except at the GHZ point <cit.>. Here we reveal that quantum advantage of CDC for every pure state, entangled in the senders : receiver bipartition. We also report that there is a marked difference between the generalized GHZ and the generalized W <cit.> states, in terms of the relation between their multiparty conclusive quantum dense coding capacities and (bipartite and multipartite) entanglement contents. 

We arrange the paper in the following way. In Sec. <ref>, we begin by recounting the original DC protocol and its generalization to non-maximally entangled states. We then describe the one-shot conclusive DC scheme, including the motivation to consider the same.

Next, in Sec. <ref>, we frame the capacity of the CDC protocol 
for arbitrary shared pure states.

We illustrate the CDC capacities for arbitrary two-qubit and two-qutrit pure states 

in Sec. <ref>. In Sec. <ref>, we extend our protocol to a multiple senders and single receiver picture, who are situated at different locations, followed by concluding remarks in Sec. <ref>.







§ CONCLUSIVE QUANTUM DENSE CODING




We begin by providing a brief description of the 
quantum dense coding protocol, discovered by Bennett and Wiesner <cit.>. Two parties, named as Alice and Bob, share a two-qubit system in the singlet state, |ψ^-|=⟩(|01|-⟩|10|)⟩/√(2), with |0|$⟩ and|1|$⟩ being orthogonal eigenvectors of the Pauli-σ_z operator. Alice applies the set of local unitaries {𝕀,σ_x,σ_y,σ_z}, where 𝕀 is the identity operator on the qubit space and σ_i, i=x, y, z, are the Pauli spin matrices, with equal probabilities, on her part of |ψ^-⟩. She then sends her part to Bob using a quantum channel that can noiselessly carry a qubit, thereby creating an ensemble of the states  {|ψ^±⟩,|ϕ^±⟩} 
<cit.>, with equal probabilities, at Bob's laboratory.






As these states are orthogonal, Bob can, in principle, distinguish between them with certainty, and hence

Alice in this case can send log_2 4 =2 bits of classical information, or equivalently, a 4-valued classical message, to Bob. When the shared state is not entangled, Alice can send only 1 bit, i.e., a two-valued classical message, to Bob, and is the “classical limit" in this case. 

Instead of possessing a singlet, or a state local unitarily connected to it, which provides the highest capacity for dense coding, 

a non-maximally entangled state can be shared between Alice and Bob, and may turn out to be the natural vehicle for transferring classical data at a certain stage of a quantum machine. Suppose, therefore, that Alice and Bob share an arbitrary quantum state ρ_AB on ℂ^d⊗ℂ^d, and  Alice wants to send the classical information  i to Bob. Depending on the classical message i, Alice performs the unitary operation U_i on her part of the shared state. We now suppose that Alice and Bob have at their disposal, a quantum channel that can noiselessly transfer a d-dimensional quantum state. Using this channel, Alice sends her part of ρ_AB, after the action of U_i, to Bob. Bob therefore possesses a quantum ensemble, {p_i, ρ_AB^i}, and his task is to find as much information as possible by performing a quantum measurement on the ensemble. The information gathered after performing a measurement by Bob can be quantified by the mutual information <cit.> between the measurement results and the index i. This mutual information is dependent upon the measurement strategy of Bob and the encoding parameters of Alice, and a maximization over these strategies and parameters will provide us the amount of classical information that can be sent from Alice to Bob. This maximum is bounded above by  <cit.>

    C_a(ρ_AB)=log_2d+{S(ρ_B)-S(ρ_AB),0}

bits, where S(σ)=-tr(σlog_2 σ) is the von Neumann entropy <cit.> of σ and ρ_B=tr_A(ρ_AB) is the reduced density matrix of Bob's subsystem. This bound can be asymptotically attained, assuming an encoding that is product over different uses of the quantum channel <cit.>. C_a is usually referred to as the DC capacity of ρ_AB. It is measured in bits, being a consequence of the choice of 2 as the base of the logarithms appearing in Eq. (<ref>). We will continue to use the same base of logarithms appearing in this paper, so that the other capacities and mutual information are also measured in bits, even though the unit will often be kept silent in the definitions. It is important to note here that the first term in Eq. (<ref>), log_2d bits (i.e., a d-valued classical message), provides the “classical limit", being defined as the maximum number of bits that can be transferred through a quantum channel that noiselessly transfers a d-level quantum system, without the additional resource of any pre-shared entangled state between the laboratories at the ends of the channel. The term {S(ρ_B)-S(ρ_AB),0} is the “quantum advantage" in DC with the state ρ_AB, being non-zero for states with S(ρ_B)>S(ρ_AB), and being equal to entropy of entanglement <cit.> for pure shared states. The quantum DC protocol has been generalized in several directions. In particular, Braunstein et al. <cit.> and Zhang et al. <cit.> have proposed DC protocols for continuous variables and the authors in Refs. <cit.> have generalized the DC protocol to the multipartite case.
 









 





 
 At this point, it is important to stress that the DC capacity, C_a, given in Eq. (<ref>), is the maximization of the mutual information between the message sent (the index i of the ensemble {p_i,ρ_i}) and the message received (the outcome of measurement by Bob). While this is an important quantity in theory and practice of information transmission, its value does not ensure that the retrieved 
 information is the same as that sent, unless the mutual information touches  2log_2d bits. If the maximal mutual information touches the 2log_2d bits limit, it is possible to send a d^2-valued classical message without any error. However, if the maximal mutual information is lower than 
  2log_2 d, one cannot ensure an error-free transmission of a classical message that has more than d values (but less than d^2). A d-level classical message can always be sent exactly, as discussed before, since we have a noiseless qudit channel at our disposal. Also the DC capacity, C_a, is achieved, in general, only when an infinite number of copies of the initial state, required for decoding, is shared.
  

 
It is therefore interesting to consider another approach of DC which deals with conclusive transfer of classical information through a single copy of a pure shared quantum state. 

 The protocol is conclusive in the sense that when an N-valued classical message, encoded in an N-element ensemble on ℂ^d⊗ℂ^d, is obtained by Bob, with N>d, it is possible to unambiguously distinguish between the N options with a non-zero probability. Note that we must have N⩽ d^2. The distinguishing protocol is “unambiguous” in the sense that while there may be errors in discrimination between the N elements of the ensemble, we are able to know when an error has been committed. In case the discrimination succeeds with unit probability, the protocol is termed as 
 
 “deterministic dense coding"  <cit.>. It is clear therefore that deterministic dense coding is a special, and important, instance of conclusive dense coding. 

 
In a different approach to probabilistically send classical information over quantum channels, as considered by Hao et al. <cit.> (see also <cit.>), 

partial transfer of information was allowed from Alice to Bob.

To describe it, let us consider an example where the information about whether a ball 
is either hard or soft, and is either black or blue, is to be sent from Alice to Bob, implying that there are two bits of information that are to be sent. Suppose now that the sender is able (or decides) to communicate the information only about the color of the ball. In the scenario considered by Hao et al., this constitutes sending one bit of information. This is different from the approach in conclusive quantum dense coding, as followed in this paper, since we consider this incidence of sending partial information (when only the color of the ball is sent), as equivalent to not sending any information. There exist situations in which, until and unless one has the full information about the system, it is effectively a “failure". For example, consider the situation where a thief has stolen a person's debit card and wants to do an online transaction with a merchant. But certain cards have an additional level of security, e.g. an one-time password sent to a pre-decided mobile phone, in absence of which, the stolen card is, happily, useless for any transaction.





§ CAPACITY OF CONCLUSIVE QUANTUM DENSE CODING


  
  Let us describe the conclusive dense coding protocol in a little more detail. Suppose that two parties, Alice and Bob, share the state |Ψ|$⟩ ofℂ^d⊗ℂ^d, and also
  
  possess a quantum channel that can noiselessly transfer ad-dimensional quantum system. With this doublet of resources, the task is to transfer anN-level classical message unambiguously, for which they adopt the following strategy. Alice applies anarbitraryensemble of unitaries,U_i, defined onℂ^d, with respective probabilities,p_i, creating the ensemble{p_i,|Ψ_i|}⟩, where|Ψ_i|=⟩U_i⊗𝕀|Ψ|$⟩, i=1,2,…,N. Here, 𝕀 denotes the identity operator on ℂ^d. Without loss of generality, we can choose U_1=𝕀, as this fixes the local basis on Alice's side. Alice subsequently uses the noiseless quantum channel to send her part of the shared state to Bob, so that the two-system ensemble {p_i,|Ψ_i|}⟩ is entirely in Bob's laboratory.   
  
  If  |Ψ_i|$⟩ are mutually orthogonal, they can, in principle, be distinguished exactly by Bob, and the entireN-level classical message will be obtained by Bob, with unit probability. To attain orthogonality, we need to find the corresponding unitaries that lead to this orthogonality, and this is the program of deterministic dense coding. If, however, the|Ψ_i|$⟩ are not mutually orthogonal, they can still be unambiguously distinguished with non-unit probability, which is non-zero if the |Ψ_i|$⟩ are linearly independent <cit.>. More precisely, Ref. <cit.> shows that the set of linearly independent states|Ψ_1|,⟩ |Ψ_2|,⟩ …, |Ψ_N|$⟩ can be recognized with probabilities   γ_1, γ_2, …, γ_N, respectively, if and only if

    X^(1)-Γ⩾ 0,
 
 where X^(1)=⟨Ψ|_i|Ψ_j|$⟩ andΓ=γ_iδ_ij. Due to the fact that the states|Ψ_i|$⟩ appear, in the ensemble in possession of Bob after Alice's use of the noiseless quantum channel, with probabilities p_i, the total probability of successfully recognizing the states is given by ∑_i=1^Np_iγ_i. 

 

In order to obtain the capacity of conclusive dense coding, let us first define the classical mutual information, Ĩ(i:m), between the message, i, which can take values, say, 1,2,…,N, sent with probabilities p_i, and the outcome m of a measurement,


having 
values, say, 1',2',…,M', received with the probabilities q_m, as <cit.>

    Ĩ(i:m) = H({p_i}) - ∑_m=1'^M' q_mH({p_i|m}_i),
 
where p_i|m is the conditional probability of the classical message i for the outcome 

m, and 

H({ζ_j})=-∑_jζ_jlog_2 ζ_j is the Shannon entropy of the probability distribution ζ_j.


However, this standard definition and conceptualization of classical mutual information between two classical random variables is applicable to the case at hand, i.e. in the setting of conclusive dense coding, only after certain crucial modifications. First of all, we will only consider those strategies as valid conclusive dense coding protocols, for which every measurement outcome m unambiguously corresponds to only one classical message i or does not provide any information. And furthermore, we only consider strategies for which every i corresponds to at least one m that occurs with a non-zero probability in the measurement. Consider, for example, a strategy in which we have i=1, 2, 3 and m=1, 2, 3, such that m=1 implies either i=1 or i=2, m=2 implies i=3, and m=3 implies that nothing can be deduced about i. This strategy is discarded because the outcome m=1 does not unambiguously imply a particular i. Consider another strategy where we have i=1, 2 and m=1, 2, 3 such that m=1 implies i=1, m=2 implies i=2, and m=3 implies that nothing can be deduced about i, and moreover, q_2=0. This strategy is also disregarded because i=2 is never implied by any outcome. So, for an arbitrary but fixed m, the conditional probability p_i|m has the following properties: 


  * Either p_i|m=1 for a particular i, and =0 for the rest. Or p_i|m=0 for all i.

  * For every i, there exists at least one m=m', with q_m≠ 0, such that p_i|m'=1.

Therefore, the outcomes can be divided into two categories - ones for which there is no remaining disorder (uncertainty) about the ensemble, and ones for which there is no or incomplete depletion of disorder about the same. Let us denote these sets by 𝒮 and ℱ respectively. Let the corresponding mutual information be denoted by I(i:m). For cases when m ∈ℱ, we assume that no information has been transmitted, so that the remnant disorder is the full disorder, viz. H({p_i}). Therefore,   

    I(i:m)   =   H({p_i})-∑_m∈𝒮 q_m × 0 - ∑_m∈ℱ q_m H({p_i})	
       =   (1-∑_m∈ℱ q_m)H({p_i}).

Now, 1-∑_m∈ℱ q_m is exactly equal to ∑_j=1^N  p_jγ_j, that has already appeared earlier. Therefore, we get

    I(i:m)= ∑_j=1^N  p_jγ_j H({p_i}).
   

 The free parameters of the function I(i:m), for a given shared state and a given quantum channel, are the p_i and U_i, with γ_i being functions of U_i. The conclusive dense coding capacity, C_N,  for sending an N-level classical message, is defined as the maximum of I(i:m) over the space of free parameters. We further assume that the classical messages appear with equal probabilities, i.e., p_i=1/N. Hence, the conclusive dense coding capacity for sending an equiprobable N-valued classical message, given a shared pure quantum state in ℂ^d⊗ℂ^d and a noiseless d-dimensional quantum channel, is given by
 
    C_N (|Ψ⟩)= max_U_i∑_i=1^N γ_i/Nlog_2N,

 where the γ_i are constrained by the relations in (<ref>).



 For an N-level classical message with N>d, one cannot send the message conclusively, using an unentangled shared pure state. This is because, any set of N states that the sender can form (using local unitaries), will become linearly dependent, if the parties share an unentangled pure state. So the “classical limit" is zero for N-level classical messages with N>d, and we say that there is a “quantum advantage" when C_N is strictly higher than zero. We define the CDC 
 capacity for a given state |Ψ⟩ as 

    C(|Ψ⟩)=max_N C_N,
 
where the maximization is performed over all N>d. For N≤ d, one 
can ignore the shared state and reach a capacity of log_2N

for all |Ψ⟩.


 There are several works on quantum dense coding, e.g. in  Refs. <cit.>, in which a “standard encoding scheme” utilizing generalized Pauli operators have been used. Generalized Pauli operators are given by U_mn=X^m Z^n, where X and Z are defined by their actions on the elements |l|$⟩ of an orthonormal basis of the Hilbert spaceℂ^d:

    X|l|   ⟩=    |l⊕ 1|,⟩
    
    Z|l|   ⟩=    e^2 π i l/d|l|.⟩

Here⊕denotes addition modulod, andm,n=1, 2,…, d. 

The conclusive dense coding capacity obtained 
by
using the standard encoding scheme, 

for sending anN-valued classical message, is denoted as C_N^P, and the maximum of these capacities overNasC^P. 





 
 
 
 
 
 
















































§ PROPERTIES OF BIPARTITE  CAPACITY
   
In this section,  we investigate the behavior of 

the CDC capacities for arbitrary pure states inℂ^2 ⊗ℂ^2, and then 

inℂ^3 ⊗ℂ^3, with respect to state parameters.






 §.§ CDC protocol in ℂ^2 ⊗ℂ^2

Let us consider that an arbitrary pure state|Ψ^(2)|=⟩α|00|+⟩√(1-α^2)|11|$⟩ in ℂ^2⊗ℂ^2, shared between Alice and Bob.  Without loss of generality, we assume that α is real and that 0⩽α⩽1/√(2). The absence of a relative phase in |Ψ^(2)|$⟩ does not affect the generality of the considerations. We now discuss about the conclusive dense coding capacity of this state. 



 In Fig. <ref>, we depict the capacityC_N, forN = 3and4, with respect to the state parameterα. Clearly, all such states, whenever entangled, provide an advantage over the case when there is no entanglement for being used as a resource in conclusive quantum dense coding, as their CDC capacities lie beyond the classical limit. The constrained optimizations here, and further on in the paper, are performed by non-linear optimization procedures <cit.>.
 

 For the maximally entangled states, i.e., for α=1/√(2),C_N = log_2 N, because all of the efficiencies of identification (γ_i) become unity. Clearly, in this scenario, DDC can be executed. But for non-maximally pure entangled states, this situation does not arise and hence DDC is not possible in this case for bothN=3and 4.  So, we are able to reproduce the result of Mozes et al. <cit.> in which they showed that the entire family of pure states inℂ^2 ⊗ℂ^2except the maximally entangled state is not useful for DDC. However, the CDC capacity remains higher than the classical limit, whenever the states are entangled.    
 
We also compute the CDC capacity,C_N^P, using Pauli operators. ForN=3and 4, they are given by the following analytical form (see Appendix):

    C_N^P=4-N+2α^2(2N-4)/Nlog_2N.

An interesting fact is that the CDC capacities obtained using non-linear optimization overarbitraryunitaries matches with the ones obtained using Pauli operators:

    C_N=C_N^P.


Note thatC_4is continuous forα∈[0,1/√(2)], andC_3is continuous forα∈(0,1/√(2)]. However,lim_α→0^+C_3(α)is not the same asC_3(α=0).C_3(α=0)is clearly vanishing, as forα=0, the shared state is the product state|11|$⟩, which, when rotated by Alice can produce at most two linearly independent states that can never be used to conclusively transfer a 3-valued classical message. But, if α≠ 0, 
although

close to zero, |Ψ^(2)|=⟩α|00|+⟩√(1-α^2)|11|$⟩ can be transformed by Alice to the ensemble consisting of|Ψ^(2)|$⟩, σ_x ⊗𝕀|Ψ^(2)|$⟩, andσ_y ⊗𝕀|Ψ^(2)|$⟩, 
which can be conclusively recognized with probabilities γ_1=1, γ_2=γ_3=2α^2, respectively. This leads to a conclusive transfer of  1+4α^2/3log_23 bits, which is actually the optimal one, as shown in 
Eqs. (<ref>) & 

(<ref>). And for α→ 0^+, this optimal transfer of information tends to a non-zero value.

Let us now consider the CDC capacity maximized over N>2. From Fig. <ref>, it is clear that the CDC capacity (C)

is C_3 for a certain range, viz. 0<α≲ 0.53, and C_4 for the rest,
viz.

0.53≲α⩽1/√(2). 
Note that α≈ 0.53 (accurate up to the second decimal point) 
is the point where C_3 and C_4 intersect.


 The 
 
 capacity, C_a=1 + S(ρ_B), is also plotted with respect to α in Fig. <ref>. Clearly, C_a is an upper bound of all single-shot capacities, viz. C_3 and C_4. Note thatC_adiffers from entanglement entropy (i.e., the von Neumann entropy of the local density matrix of the shared pure state) just by an additive scalar in the case of pure bipartite shared states that we are considering, and is a strictly increasing function of the state parameterα. 
 Interestingly, we observe thatC_3andC_4, too, are strictly increasing functions of the state parameter, thereby implying that they are 
 
 strictly increasing functions of the  entanglement of the shared state.
 
 While entanglement entropy is the asymptotic entanglement measure for pure two-party quantum states <cit.>,α^2andαare also entanglement measures (or monotones) for single-copy transformations between pure two-party quantum states under local quantum operations and classical communication <cit.>. In this sense,C_3andC_4are manifestly increasing functions of entanglement monotones.  








 §.§ CDC protocol in ℂ^3 ⊗ℂ^3


Let us now move to an arbitrary pure bipartite state in inℂ^3⊗ℂ^3, given by|Ψ^(3)|=⟩α|00|+⟩β|11|+⟩√(1-α^2-β^2)|22|$⟩,

where {|0|,⟩ |1|,⟩ |2|}⟩ forms an orthonormal basis in ℂ^3. Without loss of generality, we choose the coefficients of |00|,⟩ |11|,⟩ |22|$⟩ in|Ψ^(3)|$⟩ as real and positive,   α⩽β, and α^2+β^2 ⩽2/3. The conclusive dense coding capacities C_N are plotted in Fig. <ref> for N = 4, 5,…, 9 with respect to state parameters α and β. We see that for N=8,9, C_N does not reach log_2N except for the maximally entangled state (α=β=1/√(3)), so that DDC is not implementable in these cases, confirming the results obtained in <cit.>. However, for 4 ⩽ N ⩽ 7,

C_N does reach log_2N even for non-maximally entangled states (the bright yellow regions in the panels in Fig. (<ref>)).

We also find that the area  in parameter space  for which the corresponding states render themselves for DDC decreases with  increasing N.

For fixed N, the computation of C_N and C reveals that their optimal values are not obtained from generalized Pauli operators, but from other unitaries, showing that C^P_N ≠ C_N, which is in contrast to the two-qubit case.

To visualize the gap, we plot the quantity C-C^P against α and β
in Fig. <ref>.


Another observation is that C^P/log_2N does not reach unity for non-maximally entangled states, implying that generalized Pauli operators are not  enough to implement DDC in non-maximally entangled states.

In ℂ^3 ⊗ℂ^3, we moreover find that the connection between entanglement and the capacities of CDC is far richer than the same for 

states in ℂ^2 ⊗ℂ^2. First note that the entanglement monotones for single-copy transformations for the state |Ψ^(3)|$⟩ areα^2+β^2andα^2. For asymptotic transformations, the entanglement can be quantified by the local von Neumann entropy. A look at the panels  forN=4, N=5, andN=6in Fig. <ref> tells us thatC_4, C_5,andC_6broadly behave as functions ofα^2+β^2. On the other hand,C_7, C_8,andC_9(see the bottom row of panels in Fig. <ref>), andC(middle panel in Fig. <ref>) broadly behave as local von Neumann entropy, plotted in the right panel in Fig. <ref>. In case of the latter instances, the projected capacities are convex aroundα=β=0until a certainα^2+β^2, after which they concave outwards. 



 



§ MULTIPORT CONCLUSIVE DENSE CODING


Until now, we have 

been discussing the protocol of conclusive dense coding with a single sender and a single receiver. Point to point communication is however not the reasonable one in all instances, and in this section, we consider the case of performing the CDC protocol with two senders and a single receiver, each situated at a separate location. We suppose that the three parties share a three-qubit pure state, with one qubit at each party. We study the CDC capacity,C_N, of such a state, with which the senders aim to send anN-level classical message to the receiver. Additionally, there are two noiseless qubit channels from the two senders  to the receiver. If the parties do not share an entangled state in the senders : receiver bipartition, then the senders can send at most a 4-level message, even if the senders are allowed global unitaries. We will however be interested in the scenario where only local unitaries are possible to be utilized by the senders for encoding theN-level classical message. Let us call the senders as Alice 1 and Alice 2, denoting them respectively asA_1andA_2, while the receiver is still called Bob (B). In case the senders wish to send a classical message ofN>4levels, application ofN>4local unitaries,U_i^A_1A_2=U̅_i^A_1⊗U̅̅̅_i^A_2, withi=1,2,…,N, is necessary, with Alice 1 applying theU̅_i^A_1and Alice 2 theU̅̅̅_i^A_2. Suppose now that the shared state is|Ψ|_⟩A_1 A_2 B. We again deal with the case of equiprobable classical messages, so that after the action of the local unitaries by two Alices, the resulting ensemble consists of the states|Ψ_i|_⟩A_1 A_2 B = U̅_i^A_1⊗U̅̅̅_i^A_2⊗𝕀^B |Ψ|_⟩A_1 A_2 B, with equal probabilities. In case|Ψ|$⟩ is a product state in the A_1 A_2:B bipartition, and N>4, the |Ψ_i|$⟩ will be linearly dependent for any choice of the unitaries by the Alices (even global ones), and a conclusive transfer of information will not be possible, as a conclusive distinguishing of linearly dependant states is forbidden in quantum mechanics <cit.>.
 We are, therefore, interested in studying the CDC capacity,C_N, for shared state which is entangled in the senders : receiver bipartition, and where5⩽ N ⩽ 8. Furthermore, we choose to work with genuinely tripartite pure entangled states, where the adjective “genuinely" is used to imply that the state is entangled across all bipartitions.  




 
 
The family of three-qubit genuinely entangled pure states is a disjoint union of the paradigmatic classes 
of GHZ (Greeenberger-Horne-Zeilinger) and W. These classes are characterized by the fact that a single copy of any state within any class can be transformed into any other state within the same class by stochastic local quantum operations and classical communication (SLOCC) between the three observers, while it cannot be transformed into any state of the other class <cit.>. Important sets of states within these two classes are respectively the generalized GHZ and generalized W states. Previously it has been proved that deterministic dense coding of generalized GHZ states for the non-trivial case (transfer of a more than 4-level classical message) is not possible unless the shared state is a GHZ state. However, with some specific parameter values, the DDC capacities of three-qubit generalized W states can be non-classical <cit.>. 


It is also interesting to study the behavior of CDC capacity with the entanglement of the initial shared pure state. In the case of two senders and a single receiver, there are two kinds of entanglement one can watch out for. One is the genuine multipartite entanglement present in the state and other is the entanglement in the senders : receiver bipartition of the state. As a measure of genuine multiparty entanglement, we employ the generalized geometric measure (GGM) <cit.>, while in the latter case, we evaluate entanglement entropy in the senders : receiver bipartition. In this section, we first present a definition of GGM, and then study the CDC capacity of generalized GHZ and W states. Finally, we relate their behavior with GGM and entanglement entropy in the senders : receiver bipartition.