Optimal super dense coding over memory channels

Optimal super dense coding over memory channels

Z. Shadman shadman@thphy.uni-duesseldorf.de    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 INFM-Unit di Pavia, Via Bassi 6, 27100, Pavia, Italy
July 6, 2019
Abstract

We study the super dense coding capacity in the presence of quantum channels with correlated noise. We investigate both the cases of unitary and non-unitary encoding. Pauli channels for arbitrary dimensions are treated explicitly. The super dense coding capacity for some special channels and resource states is derived for unitary encoding. We also provide an example of a memory channel where non-unitary encoding leads to an improvement in the super dense coding capacity.

PACS numbers

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

I Introduction

Super dense coding is one of the notable areas in which quantum entanglement plays a crucial role. By this protocol, due to the nonlocal properties of quantum entanglement, it is possible to communicate two bits of classical information by sending one qubit only Bennett (). The first attention, after proposing the super dense coding protocol, was given to various scenarios over noiseless channels and unitary encoding hiroshima (); ourPRL (); Dagmar (). In this case one starts from a -dimensional bipartite shared state between the sender Alice and the receiver Bob. Alice performs with probability a local unitary operation on her subsystem to encode classical information through the state Subsequently, she sends her subsystem to Bob (ideally via a noiseless channel). The ensemble that Bob receives is . The maximal amount of classical information that can reliably be transmitted in this process is known as super dense coding capacity. It has been shown that for noiseless channels and unitary encoding, the capacity is given by ziman (); hiroshima (). Here, is Bob’s reduced density operator with , and is the von Neumann entropy. Without the additional resource of entangled states, a -dimensional quantum state can be used to transmit the information . Hence, quantum states for which , i.e., those which are more mixed locally than globally, are the useful states for super dense coding.

A realistic quantum system usually suffers from unwanted interactions with the outside world. Optical fibers and an unmodulated spin chain Bose-channel () are examples of such quantum channels which are suitable for long- and short-distance quantum communication, respectively. Super dense coding in the situation when the quantum states experience noise in the transmission channels was studied in zahra-paper (). In zahra-paper () uncorrelated noise (i.e., memoryless channels) was discussed. For those cases (channels and states) where the von Neumann entropy fulfills a specific condition, the super dense coding capacity was derived. Explicitly, for the two-dimensional uncorrelated depolarizing channel, it was shown that Alice and Bob do not win by sending classical information via a super dense coding protocol with unitary encoding if there is too much noise.

In this paper, memory effects along the transmission channel are taken into account. In this scenario the noisy channel acting on two subsystems cannot be expressed as a product of two independent channels acting on each subsystem separately. In particular, we investigate the bipartite super dense coding scenario for a correlated Pauli channel and unitary or non-unitary encoding. Such kinds of channels were originally analyzed from the point of view of optimization of the classical information transmission mp (); memory-quasi-chiara (); cerf ().

The paper is organized as follows. In Sec. II we review the Holevo bound as a key concept in finding the super dense coding capacity. We discuss the mathematical definition of the Holevo quantity in the presence of an arbitrary channel . Section III is devoted to the super dense coding capacity in the presence of a correlated Pauli channel and unitary encoding. We give examples of correlated channels and initial states for which the capacity is explicitly determined. Section IV is dedicated to the correlated Pauli channel and non-unitary encoding. We compare the capacities related to both unitary and non-unitary encoding and also discuss a case where non-unitary encoding has an advantage over unitary encoding. Finally, in Sec. V, we summarize the main results.

Ii capacity of super dense coding

The performance of a given composite state for super dense coding is usually quantified by the Holevo quantity, maximized over all possible encodings on Alice’s side. A theorem stated by Gordon Gordon () and Levitin Levitin (), and proved by Holevo Holevo-chi-quantity (), states that the amount of accessible classical information () contained in an ensemble is upper bounded by the so-called -quantity , often also referred to as the Holevo quantity. This upper bound holds for any measurement that can be performed on the system, and is given by

(1)

where is the average ensemble state and is the von Neumann entropy of . From the concavity of the von Neumann entropy it follows that the Holevo quantity is non-negative. The Holevo bound (1) is achievable in the asymptotic limit Schumacher-Westmoreland (); Holevo-capacity ().

ii.1 Holevo quantity in the presence of noise

A quantum channel is a communication channel which can transmit quantum information. Physically, a noisy quantum channel is a communication channel that is affected by interaction with the environment. Mathematically, a noisy quantum channel can be described as a completely positive trace preserving (CPTP) map acting on the quantum state that is transmitted. We consider to be a CPTP map that acts on the encoded state . For being the ensemble that Bob receives, the Holevo quantity is given by

(2)

where is the average state after transmission through the noisy channel and is the relative entropy. 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 unitary operators , chosen with the probabilities , namely

(3)

In the following, we will concentrate on the optimization of the Holevo quantity in order to find the super dense coding capacity.

Iii Super dense coding via correlated Pauli channels

We will now consider quantum channels with memory, where noise in consecutive uses of the channel is correlated. We specifically consider correlated Pauli channels mp (); memory-quasi-chiara (); cerf (), modelled as follows. Consider first a single Pauli channel, whose action on a -dimensional density operator is given by

(4)

where are probabilities (i.e., and ). The unitary displacement operators are defined as

(5)

The above operators satisfy and , and commute up to a phase,

(6)

As the operators in Eq. (4) are unitary, the Pauli channel is unital, i.e., it preserves the identity. Now, let and be two -dimensional Pauli channels (4) with the elements and , respectively. Based on the elements of these two channels, a model of a correlated Pauli channel is defined as

where the probability is given by , and the parameter () quantifies the correlation degree. For the two channels and are uncorrelated and act independently on Alice’s and Bob’s subsystems, respectively. For , the global channel (LABEL:k-Pauli-channels) is called fully correlated and for other values of , different from zero and one, the global channel is partially correlated.

For a single sender, a single receiver, and a correlated Pauli channel as well as unitary and non-unitary encoding, we derive two explicit expressions for the super dense coding capacity. We show that both unitary and non-unitary encoding problems reduce to the problem of finding a single CPTP map (in the case of unitary encoding this is a specific unitary transformation) that minimizes the output von Neumann entropy after its application and the action of the channel on the input state . For the case of unitary encoding we find examples for the optimal unitary operator.

iii.1 Unitary encoding

This subsection treats the optimization of the Holevo quantity for a correlated Pauli channel and unitary encoding. We first introduce an upper bound on the Holevo quantity and we then show that this upper bound is reachable and thus coincides with the super dense coding capacity. This procedure is phrased in the following Lemma.
Lemma 1. Let

(8)

be the Holevo quantity with , the average state and the correlated Pauli channel defined via Eq. (LABEL:k-Pauli-channels). Let 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

(9)

where and is the dimensional Pauli channel (4) on Bob’s subsystem. The subscript “un”  refers to unitary encoding.
Proof: We start by introducing an upper bound on the Holevo quantity (8). Since is a unitary operator that leads to the minimum of the output von Neumann entropy, for we have

The von Neumann entropy is subadditive and the maximum entropy of a -dimensional system is . Therefore we can write

(10)

where we have used that . This statement can be proved as follows. By using the definition of a correlated Pauli channel eq. (LABEL:k-Pauli-channels) and of the average state, and noting that acts on Alice’s side, we have

(11)

which completes this part of the proof. To show that the upper bound (10) is achievable, we consider the ensemble with being the displacement operators of Eq. (5). The Holevo quantity for this ensemble is denoted by and is given by

(12)

In hiroshima (), for an arbitrary bipartite state , it was shown that . By using this property, and noting that acts only on Alice’s side, we find that the argument in the first term on the RHS of (12) is given by

(13)

Furthermore, the second term on the RHS of Eq. (12) can be expressed in terms of the unitary operator and the channel. By inserting the action of the correlated Pauli channel, using Eq. (6), from which follows that and commute up to a phase, and since the von Neumann entropy is invariant under unitary transformation, we can write

(14)

Inserting Eqs. (13) and (14) into Eq. (12), one finds that the Holevo quantity is equal to the upper bound given in Eq. (10) and consequently, this is the super dense coding capacity.

By Lemma 1, we proved that, in order to determine the super dense coding capacity, it is enough to find an optimal that minimizes the channel output von Neumann entropy . In the next two sections we give examples of channels and initial states for which can explicitly be determined.

iii.2 Correlated quasi-classical channel

A dimensional quasi-classical depolarizing channel (or simply quasi-classical channel) is a particular form of a dimensional Pauli channel memory-quasi-chiara (); cerf (), as given in Eq. (4). For this channel, the probabilities of the displacement operators are equal for and any phase shift labeled by , and they differ from the rest of the probabilities which are also equal, i.e.,

(15)

The quasi-classical channel is characterized by a single probability parameter . With probability , a displacement occurs and with probability , no displacement occurs to the quantum signal. Like in the classical case, can also be seen as the amount of noise in a channel.

In the following we will consider as a resource state a Werner state with . In the presence of a correlated quasi-classical channel, as defined via Eqs. (LABEL:k-Pauli-channels) and (15), we find . Thus the dimension is . For two-dimensional systems the displacement operators, defined in Eq. (5), are either the identity or the Pauli operators, i.e.,

(16)

The correlated quasi-classical channel is in this case

(17)

where with and . In order to find , we start with the most general unitary operator

(18)

where and are complex variables which satisfy . The output of a correlated quasi-classical channel for an arbitrary input state is invariant under the unitary transformation memory-quasi-chiara (), i.e.,

(19)

By the above property (19), the channel outputs for the input states and are equal and therefore we can conveniently use the average of both states instead of only . This average is given by

with and being

(21a)
(21b)

After applying the quasi-classical channel, using Eqs. (19) and (LABEL:matrix-werner), and the concavity of the von Neumann entropy, we find the following lower bound

(22)

In the last line we have used that both are lower bounded by . The proof for this statement is as follows. We can rewrite (a similar argument holds for ) up to a global phase as

(23)

After applying the correlated quasi-classical channel to the state , we arrive at

(24)

The von Neumann entropy of a quantum state is defined via its eigenvalues. The eigenvalues of Eq. (24) are

(25)

To minimize the von Neumann entropy , the eigenvalues should diverge as much as possible with respect to the parameter . The eigenvalues are independent of . Thus, the von Neumann entropy is minimized when we maximize while we minimize . This is the case for and it leads to be the Bell state which proves the above statement, namely

(26)

The lower bound on the von Neumann entropy (22) is reachable: It is not difficult to see that the variables and , which correspond to being the identity operator. We have thus found a unitary operator which minimizes the output entropy. Therefore, the super dense coding capacity for a Werner state in a correlated quasi-classical channel, according to Eq. (9), is given by

(27)

For , the Werner state reduces to a Bell state . Therefore, the super dense coding capacity, according to Eq. (27), for a Bell state and in the presence of a correlated quasi-classical channel, is given by

(28)

In Fig. 1 and Fig. 2, we visualize the super dense coding capacity for the correlated quasi-classical channel as a function of the parameters , and [Eq.(27)]. In Fig. 1, we consider a Bell state, i.e., , as a function of the noise parameter and the correlation degree . In Fig. 2, the noise parameter is fixed to and we vary and the parameter characterising the Werner state.

Fig. 1. (Color online) The super dense coding capacity for a correlated quasi-classical channel and a Bell state ( ), as a function of the noise parameter and the correlation degree .

Fig. 2. (Color online) The super dense coding capacity for a correlated quasi-classical channel and a Werner state, as a function of the correlation degree and the parameter . The noise parameter is .

iii.3 Fully correlated Pauli channel

In this section we give another example for which we determined . That is the case of a fully correlated Pauli channel and a Werner state. As mentioned above, a fully correlated Pauli channel is a special form of a correlated Pauli channel (LABEL:k-Pauli-channels) when . For , it is given by

(29)

where and are either the identity or the Pauli operators.

We again consider the Werner states as resource states. For a fully correlated Pauli channel (29) we determine the operation . To do so, we derive a lower bound on where is an arbitrary unitary operator. By using the concavity of the von Neumann entropy and also by using the invariance of the von Neumann entropy under unitary transformations, the lower bound on takes the form

(30)

where we use the notation .

By using the invariance of a Bell state under the action of a fully correlated Pauli channel, i.e., , it follows that the lower bound (30) is reachable by the identity operator. Then and the super dense coding capacity, according to (9), is given by

(31)

The Werner state reduces to a Bell state for . Since the Bell state is invariant under the action of a fully correlated Pauli channel, its von Neumann entropy is zero. Therefore, using Eq. (31), the super dense coding capacity for a shared Bell state and a fully correlated Pauli channel (29), is two bits. It is the maximum information transfer for . This shows that no information at all is lost to the environment and this class of channels behaves like a noiseless one. This behavior corresponds to the results of mp (); memory-quasi-chiara ().

Iv Non-unitary encoding

So far, we have assumed that the encoding in the super dense coding protocol is unitary. The super dense coding protocol with non-unitary encoding for noiseless channels has been discussed by M. Horodecki and Piani CPTP (), M. Horodecki et al. unitaryoptimal1 (), and Winter unitaryoptimal2 (). In this section we consider the possibility of performing non-unitary encoding in the presence of a correlated Pauli channel. Let us consider to be a completely positive trace preserving (CPTP) map. Alice applies the map on her side of the shared state , thereby encoding as . The rest of the scheme is similar to the case of unitary encoding. Alice sends the encoded state with the probability to Bob through the correlated Pauli channel . Now, the question is: which ensemble of CPTP maps achieves the super dense coding capacity? In other words, what is the optimum Holevo quantity with respect to the encoding and ? To answer this question, first we give the definition for the super dense coding capacity with a correlated Pauli channel and non-unitary encoding:

(32)

where is defined via (LABEL:k-Pauli-channels). Similar to the unitary encoding case in section III.1, we find an upper bound on the Holevo quantity (32) and then we show that this upper bound is reachable by a pre-processing before unitary encoding. The above statement will be expressed in the following Lemma.

Lemma 2. Let be the Holevo quantity (32), and let be the map that minimizes the von Neumann entropy after application of this map and the correlated Pauli channel to the initial state , i.e., minimizes the expression . Then the super dense coding capacity is given by

where and is the dimensional Pauli channel (4) on Bob’s subsystem.

Proof: The von Neumann entropy is sub-additive and the maximum entropy of a -dimensional system is , and since is a map that leads to the minimum of the entropy after applying it and the channel to the initial state , we have the upper bound

By using , we find the upper bound

(34)

We now show that the ensemble with and where is defined in (5), reaches the upper bound (34). In other words, the optimal encoding consists of a fixed pre-processing with and a subsequent unitary encoding. This is analogous to the case of noiseless channels and uncorrelated Pauli channels CPTP (); zahra-paper (). Below we prove the above claim.

The Holevo quantity of the ensemble is

With an argument similar to the case of unitary encoding, the first term on the RHS of (LABEL:optiensemble) is given by

(36)

Furthermore, for the second term on the RHS of (LABEL:optiensemble) we have

(37)

Inserting Eqs. (36) and (37) into Eq. (LABEL:optiensemble), one finds that the Holevo quantity is equal to the upper bound given in Eq. (34). Consequently, the super dense coding capacity with non-unitary encoding is determined by Eq. (LABEL:capacity-non-unitary).

A comparison of Eqs. (LABEL:capacity-non-unitary) and (9) shows that applying an appropriate pre-processing on the initial state before the unitary encoding may increase the super dense coding capacity, with respect to only using unitary encoding for the case of a correlated Pauli channel. However, for some examples, no better encoding than unitary encoding is possible. For instance, since two bits is the highest super dense coding capacity for , our results derived in Sec. III.3 for fully correlated Pauli channel and the Bell state provide an example where no pre-processing can improve the capacity. However, examples exist for which non-unitary pre-processing is useful to increase the super dense coding capacity. In the next section we provide an explicit example.

iv.1 Pre-processing can improve capacity

Here, we show that for a two-dimensional Bell state in the presence of a correlated quasi-classical channel, a non-unitary pre-processing , which is not necessarily , can improve the super dense coding capacity. To show this claim, consider the completely positive trace preserving pre-processing , with the Kraus operators and . Alice applies on her side of the Bell state and transforms the Bell state to . Therefore, according to Eq. (LABEL:capacity-non-unitary), for a correlated quasi-classical channel, a Bell state, and a pre-processing , the amount of information that is transmitted by this process is given by

(38)

where is the noise parameter for a quasi-classical channel (15). Since is not necessarily the optimal pre-processing, is not also necessarily the capacity. We name (38) the transferred information. We now compare the transferred information (38) with the capacity (28) which is achieved by applying only unitary encoding. In the range of we find that the capacity is always higher than the transferred information , i.e., . Therefore, in this range, the chosen pre-processing does not improve the capacity. In the range of , the capacity with unitary encoding (28) and the transferred information with the pre-processing (38) coincide for , shown as the dashed red curve in Fig. 3. Note that corresponds to the . The function is invariant under the simultaneous exchange