Distributed quantum dense coding with two receivers in noisy environments
Abstract
We investigate the effect of noisy channels in a classical information transfer through a multipartite state which acts as a substrate for the distributed quantum dense coding protocol between several senders and two receivers. The situation is qualitatively different from the case with one or more senders and a single receiver. We obtain an upper bound on the multipartite capacity which is tightened in case of the covariant noisy channel. We also establish a relation between the genuine multipartite entanglement of the shared state and the capacity of distributed dense coding using that state, both in the noiseless and the noisy scenarios. Specifically, we find that in the case of multiple senders and two receivers, the corresponding generalized GreenbergerHorneZeilinger states possess higher dense coding capacities as compared to a significant fraction of pure states having the same multipartite entanglement.
I Introduction
Quantum entanglement is one of the essential ingredients in quantum information processing tasks which include superdense coding BennetDC (), teleportation TeleRef (), quantum error correction Quantum_error (), quantum secret sharing QCryp (); qsecretsharing (), and one way quantum computation onewayQC (). It was shown that such protocols can provide advantage over the corresponding classical protocols amaderreview (). Moreover, classical information as well as quantum state transfer via quantum channels have been successfully realized in the laboratory over reasonably large distances DCexp (); teleexp ().
Any communication protocol involves three major steps – (1) encoding of the information in a physical system, (2) sending the physical system through a physical channel, and (3) decoding the information. In this paper, we are interested in the communication scheme which deals with the transfer of classical information encoded in a quantum state shared between distant parties, and is known as quantum dense coding (DC) BennetDC (); dcgeneral (). Capacity of the dense coding protocol have been evaluated in several scenarios involving a single receiver. These include the cases of a single sender as well as multiple senders and in both noiseless and noisy scenarios BennetDC (); dcgeneral (); DistriDC (); Dense_multi (); Shadmansob (). An important tool here is the Holevo bound on the accessible information for ensembles of quantum states HolevoQuantity1 (); HolevoQuantity2 (). The situations when there are multiple senders and/or multiple receivers are involved, have been termed as distributed quantum dense coding DistriDC (); Dense_multi (). The capacity in the noiseless case for two receivers has been estimated in DistriDC (); Dense_multi (), where the Holevolike upper bound on locally accessible information for ensembles of quantum states of bipartite systems was used AccLOCC ().
In this paper, we estimate the capacity of distributed quantum dense coding for two receivers in the noisy case. The receivers are allowed to perform local (quantum) operations and classical communication (LOCC), and we term the communication protocol as the LOCCDC protocol and the corresponding capacity as the LOCCDC capacity. We begin by finding an upper bound on the capacity for arbitrary noisy channels between the senders and the receivers. A tighter bound in closed form is obtained for the case of covariant channels covariancenoise (). When the shared state is a GreenbergerHorneZeilinger (GHZ) state GHZ () and when the noisy channels are among the amplitude damping, phase damping, or the Pauli channels, the upper bounds on the LOCCDC capacities are explicitly evaluated. Furthermore, we relate the LOCCDC capacity with the multiparty entanglement in the shared state, in both noiseless and noisy cases. We had recently observed in the case of several senders and a single receiver that noise in the channel inverts relative capability of information transfer in dense coding between generic multiparty pure quantum states and the corresponding generalized GHZ (gGHZ) states Noiseinvert (). Here we find that such inversion does not take place in the case of two receivers (and several senders): The gGHZ state provides better classical informationcarrying capacity for both noiseless and noisy cases in comparison to a significantly high fraction of pure states in the corresponding Hilbert space.
The paper is organized as follows. In Sec. II, we discuss the multiparty DC capacity for more than one receiver, with the decoding operations being restricted to LOCC. In the case of multiple senders and two receivers, we establish an upper bound on the DC capacity for noisy quantum channels. A tighter upper bound on the LOCCDC capacity in presence of covariant noise is obtained in Sec. II.1.1. In Sec. III, we evaluate closed forms of LOCCDC capacity for some specific noisy quantum channels, when a fourqubit GHZ state is shared. In Sec. IV, we briefly introduce the generalized geometric measure (GGM), a genuine multiparty entanglement measure. We establish connections between the entanglement measure with the upper bound on information transfer in Sec. V. Finally, we present a conclusion in Sec VI.
Ii Quantum dense coding for more than one Receiver
We consider the quantum dense coding protocol with an arbitrary number of senders and two receivers. Let an party quantum state, , be shared between senders, , and two receivers, and . And among them, some of the senders send their encoded quantum state to the first receiver while the rest will send to the second receiver, through noiseless or noisy quantum channels.
The amount of classical information that the senders can send to the receivers depends on four factors – encoding procedures used by the senders, the probability of the sampling rate of different encodings, properties of channels by which the encoded states have to be sent, and the measurement strategies used by the receivers to decode the message. Let us first concentrate on the case when the decoding procedures which the receivers are allowed to make are global operations. The capacity of dense coding, in this case, reduces to optimization of the Holevo quantity over unitary encodings (for different encodings, see HoroPiani ()) and probabilities. The multiparty DC capacity for an arbitrary multiparty state , with N senders, and , and two receivers, and , who are in this case together and denoted by , is given by dcgeneral (); DistriDC (); Dense_multi ()
where is the dimension of the Hilbert space of all the senders, and is the von Neumann entropy of the density matrix, . In this paper all logarithms will be with base 2, and there by all capacities will be measured in bits. Such a situation can arise, e.g., if teleports his Female () quantum systems to after obtaining the postencoded systems from the senders.
The case when the receivers are at distant locations and when teleportation or global operations are not allowed, has a further two possibilities. When the receivers are not allowed to communicate between themselves, the corresponding DC capacity is additive with respect to the receivers, and is known as the LODC protocol DistriDC (); Dense_multi (). When the receivers are allowed to perform LOCC for decoding, the protocol is called LOCCDC. It is the second case which is considered in this paper, and we will now describe it in some detail. Consider again a multiparty state, , shared between N senders and two receivers, and . To send the classical information, , which occurs with probability , some of the senders, say, , perform either local or global unitary operations, denoted by with probabilities , on their parts of the shared state and send it to the receiver . The rest of the senders, , also perform unitary operations, , with probabilities on their parts and send it to (see Fig. 1). Finally, the receivers, and possess an ensemble of state , where , and , with and being the identity operators in the receiver Hilbert spaces. The receivers, and , now apply an LOCC protocol in the bipartition to decode the information that the senders have sent.
The LOCCDC protocol can be considered for the noiseless channel DistriDC (); Dense_multi (), or when the channels from the senders and the receivers are noisy. We first deal with the general noisy channel and then consider the covariant channel.
ii.1 Capacity of Dense Coding for Many Senders and Two Receivers – Noisy Channels
In this section, our aim is to estimate the capacity when multiple senders send their encoded parts of the shared quantum state to the two receivers by using a general noisy quantum channel. In a realistic situation, the transmission channel can not be kept completely isolated from the environment, and hence noise almost certainly acts on the encoded parts of the senders’ side while sending their parts through the shared channels.
Mathematically, noise in the transmission channel is a completely positive trace preserving map (CPTP), , acting on the state space of the senders’ part of the transmitted state. Therefore, the receivers, and , after the transmission, possess the distorted ensemble, , in the bipartition, where . To estimate the capacity, the party quantum state, , can be expanded as
(1)  
where , , and are respectively bases in the Hilbert space , of all the senders, and of the receiver .
After the action of the CPTP map, , on the encoded state, we get
(2) 
where is collectively or individually acting only on the senders’ subsystems.
The amount of classical information that can be extracted from the ensemble, , by LOCC, is given by AccLOCC ()
(3) 
where , and . The Holevo bound HolevoQuantity1 () on accessible information is asymptotically achievable HolevoQuantity2 (). However, for two receivers AccLOCC (), such asymptotic achievability has not yet been proven. Therefore, unlike the cases when a single receiver is involved BennetDC (); dcgeneral (); DistriDC (); Dense_multi (); Shadmansob (), the LOCCDC capacity can only be estimated with an upper bound DistriDC (); Dense_multi ().
To obtain the capacity of LOCCDC in the noiseless scenario, one has to maximize the righthandside (R.H.S.) of inequality (3) over unitary encodings and probabilities with and we obtain DistriDC (); Dense_multi ()
(4) 
where with , , and , .
Like in the noiseless case, to obtain the capacity of LOCCDC in a noisy scenario, one has to maximize the R.H.S. of (3) over unitaries and probabilities. The ensemble in the noisy scenario, involve the CPTP map .
(5)  
If we apply the subadditivity of von Neumann entropy in the and bipartitions for the first two terms, we have
(6) 
where and , with , . The second inequality is due to the fact that the maximum von Neumann entropy of a dimensional quantum state is .
To deal with the third term in the R.H.S. of (5), let us assume that and are two unitary operators acting on subsystems and of respectively. Let us suppose that after passing through the noisy transmission channel , those unitaries give the minimum von Neumann entropy among all the von Neumann entropies of , of the ensemble. Consider , and the corresponding reduced density matrices
(7) 
(8) 
Since entropy is concave, one should expect that the set, , of real numbers, which depend on the unitary operators or must have a minimum value, denoted by , which can be achieved by the unitary operators and . Hence we have
(9) 
which implies
(10) 
One should note here that and independently minimize and , respectively. For example, to minimize the von Neumann entropy, of , we already traced out the other partition of and and hence the minimization procedure in depends only on . Similar argument holds for also. Thus we have the following theorem.
Theorem 1: For arbitrary noisy channels between multiple senders and the two receivers, the LOCC dense coding capacity, involving two receivers, is bounded above by the quantity
(11) 
Here and are respectively given in Eqs. (7) and (8). The question remains whether there exists any noisy channel for which the upper bound can be made tighter than the one given in Eq. (II.1). We will address the question below.
ii.1.1 Covariant Noisy Channel
We will now deal with a class of noisy channels called the covariant channels. For an arbitrary quantum state in dimensions, the CPTP map, , is said to be covariant, if one can find a complete set of orthogonal unitary operators, , acting on the state space of , such that
(12) 
satisfies the orthogonality condition, given by
(13) 
and the completeness relation
(14) 
where is any operator in the same Hilbert space as . After encoding at the senders’ side, we assume that the senders’ part are sent through the noisy covariant channel, . After passing through the channel, the resulting state is given by
(15) 
where we use the form of an arbitrary party quantum state given in Eq. (1), and, , is a covariant noise acting on the state space of , satisfying Eq. (12), with the complete set of orthogonal unitary operators belonging to the linear operator space . We are going to show that in this case, the maximization involved in the upper bound on the capacity depends on the fixed unitary operator and the Kraus operator of the channel .
Let , with probabilities , and with probabilities , be two complete sets of orthogonal unitary operators satisfying Eq. (13), respectively acting on the Hilbert spaces of the senders , and . Without loss of generality, we assume that the first bunch of senders send their encoded parts to the receiver , while the rest sends to the receiver . Let
(16)  
One can always write and , where acting on the senders state space, satisfying Eqs. (13) and (14), commutes with the covariant map, , while and are fixed unitary operators. Therefore, after the encodings and passing through the covariant channel, the ensemble states of the DC protocol are
where we have used the covariant condition, given in Eq. (12), on . Let us denote as . The reduced density matrix shared between and is given by
(18)  
where we have used the fact that any bipartite state, , satisfy
(19) 
The HilbertSchmidt decomposition of on in the bipartition is given by
(20) 
where , and respectively are the generators of and , and where and , are real numbers. Using this form, the reduced density matrix of the output state is given by
(21) 
where the second equality comes from the fact that . Since neither the CPTP map nor the unitary operators are acting on the part of the shared state in the receiver’s side, , we have . Finally, we have
(22) 
and similarly
(23) 
Note that in the case of arbitrary noise, the above equalities were inequalities as given in (6).
Let us now consider the third term in the R.H.S. of (5). For example, if , we have
(24) 
where we use Eq. (18) and the fact that unitary operations do not change the spectrum of any density matrix.
Interestingly, does not depend on and . It only depends on the fixed unitary operators and the covariant channel, . The remaining task is to minimize , by varying the ’s. Note that we have already shown that the first two terms in the R.H.S. of (5) are independent of maximizations. We now suppose that the minimum value of is which will be achieved by setting . Similarly, for , we have that the optimal is , for the optimal unitary . Both the above inequalities can be achieved by using orthogonal unitary operators applied with equal probabilities. We have therefore proved the following proposition.
Proposition 1: For any covariant noisy channel between an arbitrary number of senders and two receivers in a multiparty DC protocol, the capacity of LOCCDC is bounded above by
where are given by
(26) 
and
(27) 
Here
Depending on the specific covariant channels, the minimum unitaries, and can be obtained. We find minimum unitaries for certain specific channels in the next section, where both covariant as well as noncovariant channels will be considered. In Theorem 1, we proved that for an arbitrary noisy channel, the upper bound on the LOCCDC capacity as given in inequality (5) is further bounded above by the expression given in Eq. (II.1). Proposition 1 shows that for covariant noisy channels, the two upper bounds are equal.
Iii Some examples of noisy quantum channels
In this section, we consider the shared state as the fourqubit GHZ state GHZ (), and consider different types of noisy channels. Undoubtedly, the GHZ state is one of the most important multiparty states, having maximal multiparty entanglement GM (); GGM () as well as maximal violations of certain Bell inequalities GHZbellineq (). Moreover, it has been successfully realized in laboratories by using several physical systems, including photons and ions GHZexp (). Our aim is to find the minimum unitary operators involved in and for different channels for this state, when the latter is used for LOCCDC.
A fourqubit GHZ state shared between two senders, and two receivers, is given by
(28) 
We are now going to find out the that minimizes , where
(29) 
and
(30) 
Here . Note that acts only on the senders’ subsystems. We also denote as .
To find the form of and , let us consider an arbitrary unitary matrix, acting on a sender’s subsystem, given by
(31) 
for , where and . To find , we require only to manipulate the , since is not involved in . A similar statement is true for .
Let us now consider three classes of noisy channels, viz.

the amplitude damping,

phase damping, and

Pauli channels.
Note that only the Pauli channel is a covariant one. In all the examples considered in this section, we consider that there are local channels which act on the individual channels running from the two senders to the two receivers. Note that from the perspective of the actual realizations, this is the more reasonable scenario.
These channels play important roles in the problem of decoherence Preskil (). The amplitude damping channel has been used to model the spontaneous decay of a photon from an excited atomic state to its ground state, while the phase damping one can correspond to scattering events. Pauli channels include a reasonably large class of quantum channels like the bit flip, and depolarizing channels, and also play an important role in the problem of decoherence.
iii.1 Amplitude Damping Channel
A qubit in the state , after passing through the amplitude damping channel is given by
(32) 
where the Kraus operators, , are given by
satisfying the condition
(33) 
with .
In the dense coding scenario, the senders, and , send their parts of the fourqubit GHZ state through local amplitude damping channels, after encoding, and the corresponding output state is given by