Optimal super dense coding over noisy quantum channels

Optimal super dense coding over noisy quantum channels

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

We investigate super dense coding in the presence of noise, i.e., the subsystems of the entangled resource state have to pass a noisy unital quantum channel between the sender and the receiver. We discuss explicitly the case of Pauli channels in arbitrary dimension and derive the super dense coding capacity (i.e. the optimal information transfer) for some given resource states. For the qubit depolarizing channel, we also optimize the super dense coding capacity with respect to the input state. We show that below a threshold value of the noise parameter the super dense coding protocol is optimized by a maximally entangled initial state, while above the threshold it is optimized by a product state. Finally, we provide an example of a noisy channel where non-unitary pre-processing increases the super dense coding capacity, as compared to only unitary encoding.

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

1 Introduction

In quantum information processing, entanglement can be used as a resource for super dense coding, as introduced by Bennett et al. [1]. Essential to this communication protocol is an entangled initial state that is shared between sender(s) and receiver(s), together with the property that an entangled state can be transformed by the sender into another state via a local operation, taken from some set of operations. The sender’s subsystem is then transmitted to the receiver (ideally via a noiseless channel), who identifies the global state in an optimal way. The super dense coding capacity is defined to be the maximal amount of classical information that can be reliably transmitted to the receiver for a given initial state. In the last years attention has been given to various scenarios of super dense coding over noiseless channels [2, 3, 4]. It has been proved that for noiseless channels and for unitary encoding, the super dense coding capacity is given by [2]

 C=logd+S(ρb)−S(ρ), (1)

where is the initial resource state shared between the sender (Alice) and the receiver (Bob). Here, is the dimension of Alice’s system, is Bob’s reduced density operator 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 are the states which are useful for dense coding. The relation cannot hold for quantum states with positive partial transpose [3]. Therefore, states that are useful for dense coding always have a non-positive partial transpose (NPT). However, the converse is not true: There exist states which are NPT but which are not useful for dense coding. One can then classify bipartite states according to their usefulness for super dense coding [4]. Besides the case of a single sender and receiver sharing an initial pure entangled state and using unitary encoding some other scenarios also have been discussed: many senders and either one or two receivers, initially entangled mixed states, non-unitary encoding, etc. [1, 2, 4, 5]. Super dense coding has been realized in optical experiments with polarized photons by Mattle et al. [6], and for continuous variables by Li et al[7].

In a realistic scenario however noise is unavoidably present. The central theme of this paper is the question: how does noise in the transmission channel affect the superdense coding capacity? Here, we focus on the case of a single sender and a single receiver, assuming unitary encoding at first, and then generalizing to non-unitary encoding. Physically, noise is a process that arises through interaction with the environment. Mathematically, a noisy quantum channel can be described as a completely positive trace preserving linear map , acting on the quantum state. In this paper we will study two different scenarios of noisy channels: first, we will assume that the sender Alice and the receiver Bob share already a bipartite quantum state (it could e.g. have been distributed to them by a third party). After Alice’s local encoding operation, she sends her part of the quantum state to Bob via the noisy channel, described by the map , see Figure 1. We call this the case of a one-sided channel. Second, we consider the case where Alice prepares the bipartite state and sends one part of it via a noisy channel, described by the map , to Bob, thus establishing the shared resource state for super dense coding. When the two parties want to use this resource, Alice does the local encoding and then sends her part of the state via the channel to Bob, see Figure 2. We call this case a two-sided channel.

Figure 1. One-sided noise: Bipartite super dense coding with an initially entangled state , shared between Alice and Bob. Alice applies the unitary operator , taken from a set with probability , on her part of the entangled state . She sends the encoded state with probability over a noisy channel, described by the map , to Bob. In the first approach we assume that just affects Alice’s subsystem, but that there is no noise on Bob’s side.

Figure 2. Two-sided noise: Bipartite super dense coding with an initially entangled state , shared between Alice and Bob. In the second approach, the noisy channel influences Alice’s subsystem after encoding while the noisy channel has already affected Bob’s side in the distribution step of the initial state .

The paper is organized as follows: in Section II we discuss the definition of the Holevo quantity for an ensemble of states in the presence of a noisy channel. We introduce a certain condition on the von Neumann entropy and we derive the super dense coding capacity for those cases where this condition is fulfilled. In Sections III and IV, we give examples of initial states and channels for which this condition on the von Neumann entropy is satisfied, and calculate their optimal super dense coding capacity explicitly. Section V provides a comparison between the super dense coding capacities in the presence of a one-sided or two-sided 2-dimensional depolarizing channel, and the classical capacity of a 2-dimensional depolarizing channel. In Section VI we consider the case of non-unitary encoding and show an example where pre-processing is useful to increase the dense coding capacity of the initial resource state in the presence of the noisy channel.

2 Super dense coding capacity

In the super dense coding protocol Alice and Bob share a bipartite entangled quantum state . Alice performs local unitary operations with probability (where ) on to encode classical information through the state , i.e.

 ρi=(Wi⊗\mathbbm1)ρ(Wi†⊗\mathbbm1). (2)

We consider to be any completely positive map that acts on the shared state . (Below will describe the noise acting on the ensemble states.) The ensemble that Bob(s) receives is . The amount of classical information transmitted via a quantum channel is measured by the Holevo quantity or -quantity. This quantity for the ensemble is given by

 χ=S(¯¯¯¯¯¯¯¯¯¯¯Λ(ρ))−∑ipiS(Λ(ρi))=∑ipiS(Λ(ρi)∥¯¯¯¯¯¯¯¯¯¯¯Λ(ρ)), (3)

where is the average state and is the von Neumann entropy of . The symbol denotes the relative entropy, defined as . Note that is a function of the resource state , the encoding and the channel . For brevity of notation we will not write explicitly these arguments of .

The super dense coding capacity for a given resource state is defined to be the maximum of the Holevo quantity with respect to , that is

 C=max{pi,Wi}(χ). (4)

In this paper we consider bipartite systems, where each subsystem has finite dimension . A general density matrix on in the Hilbert-Schmidt representation can be conveniently decomposed as

 ρ=\mathbbm1⊗ρbd+1d2⎛⎝d2−1∑i=1riλi⊗\mathbbm1+d2−1∑i,k=1tikλi⊗λk⎞⎠, (5)

where represents Bob’s reduced density operator and are the generators of the algebra with . The parameters are real numbers. We introduce the set of unitary operators , defined as

 Vi=(m,n)|j⟩=exp(2πinjd)|j+m(modd)⟩. (6)

These operators satisfy the condition . Integers and run from 0 to such that we have unitary operators . We will consider in the following the case of unital noisy channels acting on Alice’s and Bob’s systems, namely channels described by the completely positive map

 Λ(ρ)=∑mKmρK†m,∑mK†mKm=\mathbbm1,∑mKmK†m=\mathbbm1, (7)

where are Kraus operators. Here, the first condition on the Kraus operators corresponds to trace preservation, and the second condition guarantees the unital property . We will show in this section that for unital memoryless noisy quantum channels and certain initial resource states, the set of unitary operators with equal probabilities is the optimum encoding and leads to the maximum of the Holevo quantity.

We will first prove in Lemma 1 some properties that hold for the specific encoding . In the following the symbol will denote the resource state after encoding with , whereas will denote the resource state after encoding with an arbitrary unitary operation . The ensemble average after the specific encoding with , the probability distribution and after action of the channel will be denoted as . - For similar methods in the case of noiseless channels see also [2].

Lemma 1. Let and be any two unital channels which act on Alice’s and Bob’s side, respectively. For an initial resource state shared between Alice and Bob, the global channel then acts as

 Λab(ρ)=∑m,~m(Am⊗B~m)ρ(A†m⊗B†~m). (8)

Then, the following statements hold:
1-a) For , with being defined in (6), the average of the ensemble takes the form .
1-b) For with being any unitary operator acting on Alice’s system, .
1-c) The relative entropy between and can be expressed as .
Proof 1-a). In [2] it was shown that the average of the ensemble
is

 ∑i1d2τi=\mathbbm1⊗ρbd. (9)

By using (9), the linearity of the channel and its unital property, the average of the ensemble is

 ~ρ = ∑i1d2Λab(τi)=Λab(\mathbbm1d⊗ρb)=\mathbbm1⊗Λb(ρbd). (10)

Proof 1-b). In Lemma (1-a) we showed that and hence, . Therefore:

 +1d2⎛⎝d2−1∑i=1ri∑mAmUλiU†A†m⎞⎠⊗(∑~mB~mB†~mlogΛb(ρbd)) +1d2d2−1∑i,k=1tik(∑mAmUλiU†A†m)⊗(∑~mB~mλkB†~mlogΛb(ρbd))⎤⎦. (11)

By using the linearity of the trace and the relations

 \tr[∑mAmUU†A†m]=\tr[∑mAmA†m]=\tr[\mathbbm1], (12)
 \tr[∑mAmUλiU†A†m] = \tr[UλiU†∑mA†mAm] (13) = \tr[UλiU†]=\tr[λi]=0

we can write

 \tr(Λab(τ)log~ρ) = \tra\trb[∑m,~m\mathbbm1⊗(B~mρbdB†~mlogΛb(ρbd))] (14) = \trb[Λb(ρb)logΛb(ρbd)]=−S(~ρ).

Proof 1-c). Using the definition of the relative entropy and the result of Lemma (1-b) we can write

 S(Λab(τ)∥~ρ) = \tr(Λab(τ)logΛab(τ)−Λab(τ)log~ρ) (15) = S(~ρ)−S(Λab(τ)).

We now show that for resource states with a certain symmetry property, namely for those states where the von Neumann entropy after the channel action is independent of the unitary encoding, the encoding with the equally probable operators , as given in (6), is optimal. Our proof follows the line of argument developed in [2].

Lemma 2. Let denote the resource state after encoding with , given in (6). Let

 ~χ=S(~ρ)−1d2d2−1∑iS(Λab(τi)) (16)

be the Holevo quantity for the ensemble , where is the average state of this ensemble and is defined in (8). For all the channels and all initial states for which

 S(Λab(τ))=1d2d2−1∑iS(Λab(τi)) (17)

holds, is the super dense coding capacity. Here , as we defined already above, with being any unitary operator.
Proof. Let us consider an arbitrary encoding, leading to an ensemble . We will show that its Holevo quantity cannot be higher than in (16), if the condition (17) is fulfilled.

If , then from (16) and Lemma (1-c),

 ~χ=S(Λab(τ)∥~ρ). (18)

Since this equation holds for any that fulfills (17), it specially holds for , i.e.

 ~χ=S(Λab(ρi)∥~ρ)=∑ipiS(Λab(ρi)∥~ρ). (19)

Using Donald’s identity, see [8], the right hand side of the above equation can be decomposed as

 ∑ipiS(Λab(ρi)∥~ρ)=∑ipiS(Λab(ρi)∥¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Λab(ρ))+S(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Λab(ρ)∥~ρ) (20)

with . The first term on the right hand side is the Holevo quantity for any arbitrary ensemble . Hence,

 ~χ=χ+S(¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯Λab(ρ)∥~ρ). (21)

Since the relative entropy is always positive or zero we can say that is always bigger or equal than and hence, is the super dense coding capacity.

From Lemma 2 we find that

 ~χ=S(~ρ)−S(Λab(τ)). (22)

Since the above equation holds for with any unitary , it especially holds for . Hence, whenever the condition (17) is true, the super dense coding capacity is given by

 C=~χ=S(~ρ)−S(Λab(ρ)), (23)

where is the average of the ensemble after encoding with the specific (and equally probable) unitaries and after the channel action, as introduced in Lemma 1. As an interpretation of this formula, note that the action of a noisy channel typically will increase the entropy of a given state, and therefore will decrease the dense coding capacity of the original resource state.

In the next two sections we will study examples of channels and bipartite states satisfying the condition (17), and evaluate explicitly the corresponding super dense coding capacities.

3 One-sidedd-dimensional Pauli channel

A -dimensional Pauli channel [9] that acts just on Alice’s side is defined by

 ΛPa(ρi)=d−1∑m,n=0qmn(Vmn⊗\mathbbm1)ρi(V†mn⊗\mathbbm1), (24)

where are probabilities (i.e. and ). The operators , defined in (6) with a slightly different notation for the indices, can be expressed as

 Vmn=d−1∑k=0exp(2iπknd)|k⟩⟨k+m(modd)|. (25)

They satisfy and , and have the properties

 VmnV~m~n=exp(2iπ~nmd)Vm+~m(modd),n+~n(modd), (26)
 \tr[VmnV†~m~n]=dδm~mδn~n, (27)
 VmnV~m~n=exp(2iπ(~nm−n~m)d)V~m~nVmn. (28)

As the Kraus operators of one-sided Pauli channel (24) are unitary it is a unital channel.

3.1 Bell states

A Bell state in dimensions is defined as . The set of the other maximally entangled Bell states is then denoted by , for . We will show that for a Bell state shared between Alice and Bob, and with a one-sided -dimensional Pauli channel, the condition (17) is fulfilled. We will first prove the following Lemma.

Lemma 3. Let us define , where is a unitary operator, and is defined in (25). ï¿½For ,,

 πmnπ~m~n=0 (29)

holds.
Proof.
In Appendix B we show that ï¿½for ,. Hence,

 πmnπ~m~n=(VmnU⊗\mathbbm1)ρ00(U†V†mnV~m~nU⊗\mathbbm1)ρ000�(U†V†~m~n⊗\mathbbm1)=0

By using the orthogonality property (29) and the purity of the density operators , we can write

 S(ΛPa(τ)) = S(ΛPa((U⊗\mathbbm1)ρ00(U†⊗\mathbbm1))) (30) = S⎛⎜⎝d−1∑m,n=0qmn(VmnU⊗\mathbbm1)ρ00(U†V†mn⊗\mathbbm1):=πmn⎞⎟⎠ = H({qmn}),

where is the Shannon entropy. We note that the von Neumann entropy is independent of the unitary encoding . Consequently, for a one-sided -dimensional Pauli channel with an initial Bell state, the condition (17) is satisfied. The super dense coding capacity (23) for an initial Bell state and a one-sided Pauli channel in dimensions takes the form

 Cone−sidedPdBell = S(\mathbbm1d⊗ρb)−H({qmn})=logd2−H({qmn}) (31)

for . Using (1) we notice that the super dense coding capacity of a -dimensional Bell state in the noiseless case is given by . Thus, in the presence of a one-sided Pauli channel the super dense coding capacity is reduced by the amount with respect to the noiseless case - i.e. the channel noise is simply subtracted from the super dense coding capacity with noiseless channels.

Notice that the same capacity is achieved also for any maximally entangled state, i.e. for any . Actually, Lemma 3 still holds in this case and therefore also the derivation of the capacity (31).

3.2 Werner states

We will now evaluate the super dense coding capacity for an input Werner state with . The Werner state in the presence of a one-sided -dimensional Pauli channel provides another example of states and channels that satisfy (17).
Using (30), is the set of eigenvalues of . The Pauli channel is a linear and unital map. Expressing the identity matrix in a suitable basis, we arrive at

 S(ΛPa((U⊗\mathbbm1)ρW(U†⊗\mathbbm1)))=S(ηΛPa[(U⊗\mathbbm1)ρ00(U†⊗\mathbbm1)]+1−ηd2\mathbbm1) =S(diag(ηq00+1−ηd2,...,ηqd−1,d−1+1−ηd2)) =H({ηqmn+1−ηd2}). (32)

From (32) it is apparent that the output channel entropy is independent of the unitary encoding. Consequently, the super dense coding capacity, according to (23), is given by

 Cone−sidedPdWerner=logd2−H({1−ηd2+ηqmn}). (33)

The above capacity is also achieved by any other state with the form .

4 Two-sided d-dimensional depolarizing channel.

In (24) we introduced the concept of a one-sided -dimensional Pauli channel. A two-sided d-dimensional Pauli channel is then defined by

 ΛPab(ρi)=d−1∑m,n,~m,~n=0qmnq~m~n(Vmn⊗V~m~n)ρi(V†mn⊗V†~m~n). (34)

The d-dimensional depolarizing channel is a special case of a d-dimensional Pauli channel, with probability parameters

 qmn=⎧⎨⎩1−p+pd2,m=n=0pd2,otherwise. (35)

for the noise parameter , with , and .

In the following Lemma we make the statement that the von Neumann entropy of a state that was sent through the two-sided depolarizing channel is independent of any local unitary transformations that were performed before the action of the channel.

Lemma 4. Let denote a two-sided d-dimensional depolarizing channel. For a state and bilateral unitary operator , we have

 S(Λdepab((Ua⊗Ub)ρ(U†a⊗U†b)))=S(Λdepab(ρ)). (36)

Proof: Considering and to be the d-dimensional depolarizing channels that act on Alice’s and Bob’s system, respectively, it is straightforward to verify that

 Λdepa(λi)=(1−p)λi, (37)

(where are as before the generators of ), and analogously for Bob’s system.

Using the decomposition (5) for and the following relation (proved in the Appendix A):

 Λdepa(UaλiU†a)=(1−p)UaλiU†a, (38)

it is then easy to prove the following covariance property of the channel:

 Λdepab((Ua⊗Ub)ρ(U†a⊗U†b))=(Ua⊗Ub)[Λdepab(ρ)](U†a⊗U†b). (39)

Since the von Neumann entropy is invariant under unitary transformations, the proof of Lemma 4 is complete.

As a consequence of Lemma 4 we can conclude that for a two-sided -dimensional depolarizing channel the entropy for a given initial state is independent of the unitary encoding, namely

 S(Λdepab((U⊗\mathbbm1)ρ(U†⊗\mathbbm1)))=S(Λdepab(ρ)). (40)

Therefore, (17) holds and, according to (23), the super dense coding capacity for a given general resource state , with a two-sided -dimensional depolarizing channel is given by

 Ctwo−sideddepd(ρ) = S(\mathbbm1d⊗Λdepb(ρb))−S(Λdepab(ρ)) (41) = logd+S(Λdepb(ρb))−S(Λdepab(ρ)).

Notice that since Lemma 4 holds for any local unitary , the capacity (41) depends only on the degree of entanglement of the input state . In other words, all input states with the same degree of entanglement have the same super dense coding capacity.

Comparing the above expression (41) with the one for the noiseless case, given by , one realizes that in the case of two-sided noise the channel that affects Bob’s subsystem enters twice, both in the von Neumann entropies for the local and the global density matrix.

4.1 Super dense coding capacity and optimal initial state

In (41) we obtained the super dense coding capacity of an arbitrary given initial resource state for the two-sided -dimensional depolarizing channel. In this subsection we perform the optimization of the super dense coding capacity over the initial state of two qubits for the two-sided 2-dimensional depolarizing channel. Thus, we derive the optimal value of the super dense coding capacity, if Alice and Bob have a depolarizing channel available for the transfer of 2-dimensional quantum states and can choose the initial resource state.

A pure state of two qubits can be written in the Schmidt bases as with . Two local unitaries and convert the computational bases to the Schmidt bases. Therefore, in computational bases can be written as . In (36) we showed that the output von Neumann entropy of the two-sided depolarizing channel is invariant under previous local unitary transformations. Therefore and lead to the same dense coding capacity. We can thus parametrize a pure initial state as a function of a single real parameter, namely as the state , and follow the approach of Ref. [10]. The super dense coding capacity (41) of a pure state of two qubits as a function of and the noise parameter is given by

 Ctwo−sideddep2α(|φα⟩⟨φα|) = 1−ξ1logξ1−ξ2logξ2 (42) + γ1logγ1+γ2logγ2+2γ3logγ3 ,

where (with ) are the eigenvalues of and (with ) are the eigenvalues of , where . The eigenvalues and are explicitly given by

 γ1,2=12(1−p(1−p2)±(1−p)√1−4pα(2−p)(1−α)) , γ3=γ4=p2(1−p2) , ξ1=α−pα+p2 , ξ2=1−α+pα−p2 . (43)

We can now maximize expression (42) over the variable , for a given noise parameter , and find interesting results. They are illustrated in Figure 3, where we plot the superdense coding capacity in (42) as a function of the noise parameter , for various values . We find that there is a threshold value , where two curves cross each other: for the value leads to the highest super dense coding capacity, i.e. the optimal initial resource state is a Bell state. For , the optimal choice is , i.e. product states are best for dense coding. As shown graphically in the close-up of Figure 3, the curves for intermediate values of are always lower than or . In order to prove this claim, we also evaluated in the range of and in the range of as functions of the parameters and . We found that these two functions are positive or zero. Thus, for pure initial states it is always best to either use maximally entangled states or product states, depending on the noise level.

Figure 3. The super dense coding capacity for the two-sided depolarizing channel in 2 dimensions, , as function of the noise parameter , for , , and . For the definition of see main text. For a Bell state, i.e. , leads to the optimal capacity, while for the optimal initial state is a product state ().

In the following we call the super dense coding capacity of an initial Bell state in the presence of a two-sided 2-dimensional depolarizing channel . Using (42) with , this capacity is given by

 Ctwo−sideddep2Bell = 2+1+3(1−p)24log1+3(1−p)24 (44) + 31−(1−p)24log1−(1−p)24 .

The super dense coding capacity with an initial product state in the presence of a two-sided 2-dimensional depolarizing channel is denoted in the following as . From (42) with it follows that

 Cchdep2=1+p2logp2+2−p2log2−p2. (45)

Note that (45) is identical to the classical channel capacity of the depolarizing channel for qubits [11].

We now show that using mixed initial states as a resource cannot increase the super dense coding capacity, i.e. and are the optimal input states for the range of noise parameter and , respectively. To show this claim we first write the super dense coding capacity (41) in the form of the relative entropy

 Ctwo−sideddepd(ρ)=S(Λab(ρ)∥\mathbbm1d⊗Λb(ρb)). (46)

Since any mixed state can be written as a convex combination of pure states , i.e. , and , we can write

 Cρmix = S(Λab(ρmix)∥~ρ)=S(Λab(ρmix)∥\mathbbm1d⊗Λb(ρb,mix)) (47) = S(∑kpkΛab(ρk)∥∑kpk\mathbbm1d⊗Λb(ρb,k)) ≤ ∑kpkS(Λab(ρk)∥\mathbbm1d⊗Λb(ρb,k)).

In the above inequality we have used the subadditivity of the relative entropy, i.e. , where is the Shannon relative entropy, defined as [13]. We showed before that the super dense coding capacity of a pure state for is upper bounded by the super dense coding capacity of a Bell state , and for it is upper bounded by the product state . Remembering that is pure, and using (46), we find that for

 Cρmix≤∑kpkS(Λab(ρk)∥\mathbbm1d⊗Λb(ρb,k))≤Ctwo−sideddep2 , (48)

and for

 Cρmix≤∑kpkS(Λab(ρk)∥\mathbbm1d⊗Λb(ρb,k))≤Cchdep2 , (49)

which proves our claim.

It is interesting to note that the optimal capacity for the two-sided qubit depolarizing channel is a non-differentiable function of the noise parameter , and that the optimal states are either maximally entangled or separable. In other words, there is a transition in the entanglement of the optimal input states at the particular threshold value of the noise parameter . Notice that a similar transition behavior in the entanglement of the optimal input states for transmission of classical information was found also for the qubit depolarizing channel with correlated noise [16]. It is interesting that in the present context the transition behavior arises in a memoryless channel and is not related to correlations introduced via the noise process.

5 Super dense coding capacity versus channel capacity

In this section, we consider the question of whether or not it is reasonable in the presence of noise to use the super dense coding protocol for the transmission of classical information? To answer this question, we provide a comparison between the classical capacity of a 2-dimensional depolarizing channel and the super dense coding capacities of a one-sided and two-sided 2-dimensional depolarizing channel, for the resource of an initial Bell state. Since the depolarizing channel is a special form of a Pauli channel, according to (31) the super dense coding capacity for a one-sided 2-dimensional depolarizing channel for an initially shared Bell state is

 Cone−sideddep2=2+4−3p4log4−3p4+3p4logp4. (50)

The super dense coding capacity for a two-sided 2-dimensional depolarizing channel with a Bell state as resource is given in (44). The classical capacity of the -dimensional depolarizing channel is achieved by an ensemble of pure states belonging to an orthonormal basis, say at the channel input, with equal probability and performing a complete von Neumann measurement in the same basis over the channel output [11]. Its expression is given explicitly in (45).

In Figure 4, we plot , , , and in terms of the noise parameter . As we expect, the first three capacities ,