Distributed super dense coding over noisy channels

# Distributed super dense coding over noisy channels

Z. Shadman    H. Kampermann    D. Bruß Institute für Theoretische Physik III, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany    C. Macchiavello Dipartimento di Fisica “A. Volta” and INFN-Sezione di Pavia, Via Bassi 6, 27100, Pavia, Italy
July 11, 2019
###### Abstract

We study multipartite super dense coding in the presence of a covariant noisy channel. We investigate the case of many senders and one receiver, considering both unitary and non-unitary encoding. We study the scenarios where the senders apply local encoding or global encoding. We show that, up to some pre-processing on the original state, the senders cannot do better encoding than local, unitary encoding. We then introduce general Pauli channels as a significant example of covariant maps. Considering Pauli channels, we provide examples for which the super dense coding capacity is explicitly determined.

PACS numbers

03.67.-a, 03.67.Hk, 03.65.Ud

## I Introduction

The notion of multipartite super dense coding was introduced by Bose et al. Bose-multi-first () to generalize the Bennett-Wiesner scheme Bennett () of super dense coding to multiparties. In this scheme it was shown that the use of a multipartite entangled state can allow a single receiver to read messages from more than one source through a single measurement. A generalization of this multipartite super dense coding to higher dimensions was given by Liu et al. multi-sdc-Liu-higher-dim (). Distributed super dense coding was also widely discussed in ourPRL (); Dagmar () in which two scenarios of many senders with either one or two receiver(s) were addressed. For a single receiver, the exact super dense coding capacity was determined and it was shown that the senders do not need to apply global unitaries to reach the optimal capacity, but each sender can perform a local encoding on her side. As a result, it was shown that bound entangled states with respect to a bipartite cut between the senders (Alices) and the receiver (Bob) are not “multi” dense-codeable. Furthermore, a general classification of multipartite quantum states according to their dense-codeability was investigated.

The above multipartite scenarios were discussed for noiseless systems. However, in a realistic super dense coding scheme noise is unavoidably present in the system. We assume here that noise is present only in the transmission channels and the other apparatuses involved are perfect. In zahra-paper (); zahra-paper2 (), the bipartite super dense coding for both correlated and uncorrelated channels was discussed. In the present paper we generalize those schemes to the multipartite case in the presence of covariant noise. We investigate the scenario of more than one sender with a single receiver, considering both unitary and non-unitary encoding. We follow two avenues. First, we will consider the case where the senders are far apart and can only apply local operations. Second, we will assume that the senders are allowed to perform global operations. Since the amount of classical information that can be extracted from an ensemble of quantum states can be measured by the Holevo quantity Gordon (); Levitin (); Holevo-chi-quantity (), the super dense coding capacity for a given resource state is defined to be the maximal amount of this quantity with respect to the encoding procedure. In the present paper we focus on the optimization problem of the Holevo quantity in order to find the super dense coding capacity, considering local (non)unitary encoding as well as global (non)unitary encoding.

The paper is organized as follows. In Sec. II we first review the mathematical definition of the Holevo quantity for an ensemble of multipartite states when the parties are connected through a completely positive trace preserving map (a noisy channel). Considering unitary encoding, and a covariant channel, for both scenarios of local and global encoding, we then find an expression for the super dense coding capacity. This expression only involves to find a single unitary operator acting on the resource state. In Sec. III we discuss the Pauli channel as a typical example of a covariant map. We then give examples of Pauli channels and initial states for which the single unitary operator is explicitly determined. In Sec. IV, considering non-unitary encoding, we derive the multipartite super dense coding capacity in the presence of covariant channels up to a pre-processing on the resource state. We investigate both local and global encoding. Furthermore, we discuss the Pauli channel as particular map. In Sec. V we summarize the main results. Finally, in the Appendix, we provide proofs for two Lemmas reported in the paper.

## Ii super dense coding capacity with many senders and one receiver in the presence of noisy channels

A quantum channel is a communication channel which can transmit a quantum system and can be used to carry classical information. If the transfer is undisturbed the channel is noiseless; if the quantum system interacts with some other external systems (environment), a noisy quantum channel results. Mathematically, a quantum channel can be described as a completely positive trace preserving (CPTP) map acting on the quantum state that is transmitted. Considering a noisy transmission channel, the multipartite super dense coding scheme works as follows: a given quantum state is distributed between Alices and a single Bob (in our scenario, Bob’s subsystem experiences noise in this stage). Then, Alices perform with the probability a unitary operation on their side of the state , thus encoding classical information through the state , where is the identity operator on the Bob’s Hilbert space and , is a set of indices for Alices. Subsequently, the Alices send their subsystems of the encoded state through the noisy channel to Bob. We consider to be the CPTP map (quantum channel) that globally acts on the multipartite encoded state . By this process, Bob receives the ensemble . By performing suitable measurements, Bob can extract the accessible information about this ensemble which is given by the Holevo quantity Holevo-chi-quantity ()

 χ\textmdun({ρ{i},p{i}}) = S(∑{i}p{i}Λ\textmda1...\textmdak\textmdb(ρ{i})) (1) − ∑{i}p{i}S(Λ\textmda1...\textmdak\textmdb(ρ{i})),

where is the von Neumann entropy, and the logarithm is taken to base two. The subscript “un”  refers to unitary encoding. The super dense coding capacity for a given resource state and the noisy channel is defined to be the maximum of the Holevo quantity with respect to the encoding , i.e.

 C\textmdun=max{W\textmda1...\textmdak{i},p{i}}χ\textmdun({ρ{i},p{i}}). (2)

For an illustration, see Fig. 1.

Fig. 1. (Color online) Super dense coding with a distributed quantum state between four parties (three Alices and a single Bob). The straight black lines show the entanglement between the parties and the dashed red curves show the transmission channels between Alices and Bob. The Alices encode with the ensemble and in the next step, they send their subsystems of the encoded state through the channel to the receiver, Bob. The ensemble that Bob gets is . In this process, based on optimal encoding by the Alices, the maximal amount of classical information which is defined to be the capacity is transferred (see main text).

### ii.1 Covariant noisy channels

In this section we determine the super dense coding capacity for a special class of channels, up to a single unitary operator acting on the given state . The channels we consider, denoted by , commute with a complete set of orthogonal unitary operators , namely they have the property

 Λ\textmdc\textmda1...\textmdak\textmdb(~V{i}ρ~V†{i})=~V{i}Λ\textmdc\textmda1...\textmdak\textmdb(ρ)~V†{i}, (3)

for the set of unitary operators which satisfy the orthogonality condition . According to hiroshima (), for this set it is guaranteed that where is an arbitrary operator. The property (3) is usually referred to as covariance holevo (). Here we will consider local unitary operators, namely of the form

 ~V{i}=V\textmda1i1⊗...⊗V\textmdakik. (4)

In the following we will first discuss the case that the Alices are far apart and they are restricted to local unitary operations in the presence of a covariant channel with the property (3). We will then investigate the case where the Alices are allowed to perform entangled unitary encoding.

#### ii.1.1 Senders performing local unitary operators

In this scenario, the th Alice applies a local unitary operator with probability on her subsystem of the shared state . The optimization of the Holevo quantity is given in the following lemma.

Lemma 1. Let

 χ\textmdlo\textmdun = S(∑{i}p{i}Λ\textmdc\textmda1...\textmdak\textmdb(ρ{i})) (5) − ∑{i}p{i}S(Λ\textmdc\textmda1...\textmdak\textmdb(ρ{i})),

be the Holevo quantity with

 ρ{i} = (6) (W\textmda1i1⊗W\textmda2†i2⊗...⊗W\textmdakik⊗\mathbbm1\textmdb),

and be a covariant channel with the property (3). The superscript “lo”  refers to local encoding. Let

 U\textmdlo\textmdmin:=U\textmda1\textmdmin⊗...⊗U\textmdak\textmdmin (7)

be the unitary operator that minimizes the von Neumann entropy after application of this unitary operator and the channel to the initial state , i.e. minimizes the expression . Then the super dense coding capacity is given by

 C \textmdlo\textmdun=logD\textmdA+S(Λ\textmd\textmdb(ρ\textmdb))

where is the dimension of the Hilbert space of the Alices, and .
Proof: The von Neumann entropy is subadditive. The maximum entropy of a -dimensional system is . Since is a unitary operator that leads to the minimum of the output von Neumann entropy, an upper bound on Holevo quantity (5) can be given as

 χ\textmdlo\textmdun ≤ S(∑{i}p{i}Λ\textmdc\textmda1...\textmdak\textmdb(ρ{i}))−S(Λ\textmdc\textmda1...\textmdak\textmdb (9) ≤ logD\textmdA+S(Λ\textmd\textmdbρ\textmdb)−S(Λ\textmdc\textmda1...\textmdak\textmdb((U\textmdlo\textmdmin⊗\mathbbm1\textmdb) ρ\textmda1...\textmdak\textmdb(U\textmdlo†\textmdmin⊗\mathbbm1\textmdb))).

In the next step, we show that the upper bound (9) is reachable by the ensemble where was defined in Eqs. (3) and (4).

The Holevo quantity for the ensemble is denoted by and is given by

 ~χ\textmdlo\textmdun = S(∑{i}1D2\textmdAΛ\textmdc\textmda1...\textmdak\textmdb(~U{i}ρ\textmda1...\textmdak\textmdb~U{i}†)) (10) − ∑{i}1D2\textmdAS(Λ\textmdc\textmda1...\textmdak\textmdb(~U{i}ρ\textmda1...\textmdak\textmdb~U{i}†)).

By using the covariance property (3), the argument in the first term on the RHS of (10) is given by

 ∑{i}1D2\textmdAΛ\textmdc\textmda1...\textmdak\textmdb((~U{i}⊗\mathbbm1\textmdb)ρ\textmda1...\textmdak\textmdb(~U{i}†⊗\mathbbm1\textmdb)) (11) = 1D2\textmdA∑{i}(~V{i}⊗\mathbbm1\textmdb)[Λ\textmdc\textmda1...\textmdak\textmdb((U\textmdlo\textmdmin⊗\mathbbm1\textmdb)ρ\textmda1...\textmdak\textmdb (U\textmdlo†\textmdmin⊗\mathbbm1\textmdb))]:=ϱ(~V†{i}⊗\mathbbm1\textmdb)

The density matrix with the bipartite cut between the Alices and Bob, and in the Hilbert-Schmidt representation, can be decomposed as

 ϱ = \mathbbm1\textmda1...\textmdakD\textmdA⊗Λ\textmdb(ρ\textmdb)+∑jrjλ\textmda1...\textmdakj⊗\mathbbm1\textmdb (12) + ∑j,ktjkλ\textmda1...\textmdakj⊗λ\textmdbk,

where the are the generators of the algebra, and are the generators of the algebra with . The parameters and are real numbers. By exploiting the equation , and since each is traceless, we can write

 ∑{i}~V{i}λ\textmda1...\textmdakj~V†{i}=0. (13)

By using this property and the decomposition (12), we find that the argument in the first term on the RHS of (10) is given by

 S(∑{i}1D2\textmdAΛ\textmdc\textmda1...\textmdak\textmdb(~U{i}ρ\textmda1...\textmdak\textmdb~U{i}†)) (14)

Furthermore, the second term on the RHS of Eq. (10) can be expressed in terms of the unitary operator and the channel. By using the covariance property (3), and since the von Neumann entropy is invariant under a unitary transformation, we can write

 ∑{i}1D2\textmdAS(Λ\textmdc\textmda1...\textmdak\textmdb((~U{i}⊗\mathbbm1\textmdb)ρ\textmda1...\textmdak\textmdb(~U{i}†⊗\mathbbm1\textmdb))) = 1D2\textmdA∑{i}S((~V{i}⊗\mathbbm1\textmdb)[Λ\textmdc\textmda1...\textmdak\textmdb((U\textmdlo\textmdmin⊗\mathbbm1\textmdb)ρ\textmda1...\textmdak\textmdb (U\textmdlo†\textmdmin⊗\mathbbm1\textmdb))](~V†{i}⊗\mathbbm1\textmdb)) =

Inserting Eqs. (14) and (LABEL:average-entropy1) into Eq. (10), one finds that the Holevo quantity is equal to the upper bound given in Eq. (9) and therefore this is the super dense coding capacity.

As we can see from the capacity expression (LABEL:local-dc-covariant), all the parameters are known except the single unitary operator . However, for some specific situations like noiseless channels, i.e. for , this unitary operator has already been identified as the identity operator. The capacity for noiseless channels is then given by . We also provide more examples in the next section.

#### ii.1.2 Senders may perform entangled unitaries

We will now investigate the case where the Alices are allowed to apply entangled unitary operators. The question we want to address is: can the Alices increase the information transfer by applying entangled unitaries? To answer this question we follow a strategy similar to the case of local encoding, mentioned in the previous part. The difference is that instead of local unitaries , Alices encode with the global unitary operators with the probabilities . In order to find the optimal encoding and thus the super dense coding capacity, we optimize the Holevo quantity (1). The optimization procedure is similar to Lemma 1. The difference is that we now have a global unitary operator which minimizes the output von Neumann entropy. We can then show that the optimal encoding is given by the ensemble , and the super dense coding capacity for this situation is given by

 C\textmdg\textmdun=logD\textmdA+S(Λ\textmd\textmdb(ρ\textmdb))

The difference between the capacities and is the occurrence of the local and global unitary transformation and , respectively.

## Iii Pauli noise as a model for a covariant quantum channel

In the present section we will consider the explicit case of Pauli channels, namely channels whose action on a -dimensional density operator is given by

 Λ\textmdP(ξ)=d−1∑m,n=0qmnVmnξV†mn, (17)

where are the displacement operators defined as

 Vmn=d−1∑k=0exp(2iπknd)|k⟩⟨k+m(modd)|. (18)

They satisfy , and . They also have the properties

 \textmdtr[VmnV†m′n′]=dδmm′δnn′, (19a) VmnVm′n′=exp(2iπ(n′m−nm′)d)Vm′n′Vmn, (19b) VmnVm′n′=exp(2iπn′md)Vm+m′(\textmdmodd),n+n′(\textmdmodd).

The superscript “P”  in (17) refers to the Pauli channel. Here are probabilities (i.e. and ). Since the operators are unitary, the Pauli channel (17) maps the identity to itself (it is a unital channel).

In the case of parties we can consider a general Pauli channel which globally acts on the Alices’ subsystems (after encoding) and Bob’s subsystem (in the distribution stage) as

 Λ\textmdP\textmda1...\textmdak\textmdb(ξ)=∑{mini}q{mini}(V\textmda1m1n1⊗...⊗V\textmdakmknk⊗ V\textmdbmk+1nk+1)ξ(V\textmda1†m1n1⊗...⊗V\textmdak†mknk⊗V\textmdb†mk+1nk+1), (20)

where the probabilities add to one. Here, the notations stand for Alices and stands for Bob.

Since the displacement operators commute up to a phase, it is straightforward to see that the Pauli channel (20) is a covariant channel. Therefore, the capacities for local and global unitary encoding are a special form of Eqs. (LABEL:local-dc-covariant) and (LABEL:global-dc-covariant), respectively, and are given by

 C\textmdlo,P\textmdun=logD\textmdA+S(Λ\textmdP\textmdb(ρ\textmdb))

and

 C\textmdg,P\textmdun=logD\textmdA+S(Λ\textmdP\textmdb(ρ\textmdb))

where, in both of the above equations, represents Bob’s reduced density operator and is the -dimensional Pauli channel (17) acting on Bob’s subsystem.

This general model of Pauli channels includes both the case of a memoryless channel, where the Pauli noise acts independently on each of the parties and the probabilities are products of the single party probabilities , or more generally the case where the action of noise is not independent on consecutive uses but is correlated. For example, for uses of a Pauli channel we can define a correlated Pauli channel in the multipartite scenario as follows

 q{mini} (23) = (1−μ12)...(1−μk,k+1)qm1n1...qmk+1nk+1 + μ12(1−μ13)...(1−μk,k+1)δm1m2δn1n2qm1n1 qm3n3...qmk+1nk+1 + (1−μ12)μ13...(1−μk,k+1)δm1m3δn1n3qm1n1 qm2n2qm4n4...qmk+1nk+1 . . . + (1−μ12)...(1−μk−1,k+1)μk,k+1δmkmk+1δnknk+1 qm1n1qm3n3...qmk−1nk−1qmk+1nk+1 + μ12μ13(1−μ14)...(1−μk,k+1)δm1m2δn1n2 δm1m3δn1n3qm1n1qm4n4...qmk+1nk+1 . . . + μ12...μk−1,k+1(1−μk,k+1)δm1m2δn1n2...δm1mkδn1nk qm1n1qmk+1nk+1 + μ12...μk,k+1δm1m2δn1n2...δm1mk+1δn1nk+1qm1n1.

Here, between every two individual channels we have defined a correlation degree with which correlates the channel to the channel ( ). Thus, for parties we have correlation degrees . For instance, correlates the channel one and two, correlates the channel k and Bob’s channel, etc. If for all and , then the channels are independent or, in other words, we are in the memoryless (or uncorrelated) case. As mentioned above, this channel can be expressed as a product of independent channels acting seperately on each subsystem. If for all and , we have a fully correlated Pauli channel. For other values of other than zero and one, the channel (20) is partially correlated. For two uses of a Pauli channel, the expression (23) reduces to with a single correlation degree mp (). We considered this situation in zahra-paper (); zahra-paper2 () for the case of bipartite super dense coding.

In the next section, we give examples for which the unitaries and are determined. For these examples, we show that both capacities are the same. Thus the Alices can reach the optimal information transfer via local encoding.

## Iv Examples

In this section, we show examples of multipartite systems for which or/and are determined. One example is a correlated Pauli channel (20) and copies of the Bell state. Noise here acts just on the Alices’ subsystem. Another example is a fully correlated Pauli channel and a GHZ state as well as copies of a Bell diagonal state, both for . The last example will be the depolarizing channel with uncorrelated noise.

### iv.1 k copies of a Bell state and a correlated Pauli channel

In this section we discuss the example that the Alices and Bob share copies of the Bell state. We consider the situation when there is no noise on Bob’s side, and the Alices’ shares of the Bell states globally experience a correlated Pauli channel (see Fig. 2). This example satisfies the situation discussed in Sec. II.1.1. Therefore, the capacity follows from Eqs. (LABEL:local-dc) and (LABEL:global-dc).

A Bell state in dimensions is defined as . The set of the other maximally entangled Bell states is denoted by , for . We prove that the von Neumann entropy is invariant under arbitrary unitary rotation of the state after application of the channel , i.e.

 S(Λ\textmdP\textmda1...\textmdak((U\textmda1...\textmdak⊗\mathbbm1\textmdb1...\textmdbk)(ρ\textmda1\textmdb100⊗...⊗ρ\textmdak\textmdbk00) (24) (U\textmda1...\textmdak†⊗\mathbbm1\textmdb1...\textmdbk))) = S(Λ\textmdP\textmda1...\textmdak(ρ\textmda1\textmdb100⊗...⊗ρ\textmdak\textmdbk00)).

To show this claim, we first prove the following lemma.

Lemma 2. Let

 ρ\textmda1\textmdb100⊗...⊗ρ\textmdak\textmdbk00=|Φ\textmda1\textmdb100...Φ\textmdak\textmdbk00⟩⟨Φ\textmda1\textmdb100...Φ\textmdak\textmdbk00|,

be copies of the Bell states with different dimensions . Let us define

 π{mini}:=(V\textmda1m1n1⊗...⊗V\textmdakmknk⊗\mathbbm1\textmdb1...\textmdbk) (U\textmda1...\textmdak⊗\mathbbm1\textmdb1...\textmdbk)(ρ\textmda1\textmdb100⊗...⊗ρ\textmdak\textmdbk00)(U\textmda1...\textmdak† ⊗\mathbbm1\textmdb1...\textmdbk)(V\textmda1†m1n1⊗...⊗Vak†mknk⊗\mathbbm1b1...bk), (26)

where is an arbitrary unitary operator and are the operators in Eq. (18). For different states ,

 π{mini}π{m′in′i}=0, (27)

holds.

A proof for this Lemma is presented in the Appendix. Using the orthogonality property (27), and the purity of the density operator , the channel output entropy can be written as

 S(Λ\textmdP\textmda1...\textmdak((U\textmda1...\textmdak⊗\mathbbm1\textmdb1...\textmdbk)(ρ\textmda1\textmdb100⊗...⊗ρ\textmdak\textmdbk00) (U\textmda1...\textmdak†⊗\mathbbm1\textmdb1...\textmdbk)))=S⎛⎝∑{mini}q{mini}π{mini}⎞⎠ =H({q{mini}}), (28)

where is the Shannon entropy. Consequently, the channel output entropy is just determined by the channel probabilities and it is invariant under unitary encoding. Therefore, both local encoding and global encoding leads to the same capacity in Eqs. (LABEL:local-dc) and (LABEL:global-dc). That is

 Ck\textmd−copy\textmd\textmdun,B = logd21+logd22+...+logd2k − H({q{mini}}), ≠ kC\textmdone−copy\textmd\textmdun,B. (29b)

The subscript “B”  refers to a Bell state. As we can see from Eq. (29), for a correlated Pauli channel, the capacity of copies of a Bell state is not additive except when for all and , i.e. the case of an uncorrelated Pauli channel with . Then the capacity for copies is times the capacity of a single copy with dimension . That is

 Ck\textmd−copy,unco\textmd\textmdun,B = k(logd2−H({qmn})) (30) = kC\textmdone−copy,unco\textmd\textmdun,B.

If for all and , i.e. the case of a fully correlated Pauli channel with , by using Eq. (29), we have

 Ck\textmd−copy\textmd\textmdun,B,f = logd2+...+logd2−H({qmn}) (31) = k(