Optimal super dense coding over noisy quantum channels
Zahra Shadman, Hermann Kampermann, Chiara Macchiavello, Dagmar Bruss
OOptimal super dense coding over noisy quantumchannels
Z Shadman ∗ , H Kampermann , C Macchiavello and D Bruß Institute f¨ur Theoretische Physik III, Heinrich-Heine-Universit¨at D¨usseldorf,D-40225 D¨usseldorf, Germany Dipartimento di Fisica “A. Volta” and INFM-Unit´ a di Pavia, Via Bassi 6, 27100,Pavia, ItalyE-mail: ∗ [email protected] Abstract.
We investigate super dense coding in the presence of noise, i.e., thesubsystems of the entangled resource state have to pass a noisy unital quantumchannel between the sender and the receiver. We discuss explicitly the case of Paulichannels in arbitrary dimension and derive the super dense coding capacity (i.e. theoptimal information transfer) for some given resource states. For the qubit depolarizingchannel, we also optimize the super dense coding capacity with respect to the inputstate. We show that below a threshold value of the noise parameter the super densecoding protocol is optimized by a maximally entangled initial state, while above thethreshold it is optimized by a product state. Finally, we provide an example of a noisychannel where non-unitary pre-processing increases the super dense coding capacity,as compared to only unitary encoding.PACS numbers: 03.67.-a, 03.67.Hk, 03.65.Ud a r X i v : . [ qu a n t - ph ] O c t ptimal super dense coding over noisy quantum channels
1. Introduction
In quantum information processing, entanglement can be used as a resource for superdense coding, as introduced by Bennett et al. [1]. Essential to this communicationprotocol 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 intoanother state via a local operation, taken from some set of operations. The sender’ssubsystem is then transmitted to the receiver (ideally via a noiseless channel), whoidentifies the global state in an optimal way. The super dense coding capacity is definedto be the maximal amount of classical information that can be reliably transmitted tothe receiver for a given initial state. In the last years attention has been given to variousscenarios of super dense coding over noiseless channels [2, 3, 4]. It has been proved thatfor noiseless channels and for unitary encoding, the super dense coding capacity is givenby [2] C = log d + S ( ρ b ) − S ( ρ ) , (1)where ρ is the initial resource state shared between the sender (Alice) and the receiver(Bob). Here, d is the dimension of Alice’s system, ρ b is Bob’s reduced density operatorand S ( ρ ) = − tr( ρ log ρ ) is the von Neumann entropy. Without the additional resource ofentangled states, a d -dimensional quantum state can be used to transmit the informationlog d . Hence, quantum states for which S ( ρ b ) − S ( ρ ) > S ( ρ b ) − S ( ρ ) > et al [7].In a realistic scenario however noise is unavoidably present. The central themeof this paper is the question: how does noise in the transmission channel affect thesuperdense coding capacity? Here, we focus on the case of a single sender and asingle receiver, assuming unitary encoding at first, and then generalizing to non-unitary encoding. Physically, noise is a process that arises through interaction withthe environment. Mathematically, a noisy quantum channel can be described as acompletely positive trace preserving linear map Λ, acting on the quantum state. Inthis paper we will study two different scenarios of noisy channels: first, we will assumethat the sender Alice and the receiver Bob share already a bipartite quantum state ρ (it could e.g. have been distributed to them by a third party). After Alice’s localencoding operation, she sends her part of the quantum state to Bob via the noisy channel,described by the map Λ a , see Figure 1. We call this the case of a one-sided channel. ptimal super dense coding over noisy quantum channels ρ and sends onepart of it via a noisy channel, described by the map Λ b , to Bob, thus establishing theshared resource state for super dense coding. When the two parties want to use thisresource, Alice does the local encoding and then sends her part of the state via thechannel Λ a to Bob, see Figure 2. We call this case a two-sided channel. Λ a Bob ρ Alice {W , p } i i
Figure 1.
One-sided noise: Bipartite super dense coding with an initially entangled state ρ , shared betweenAlice and Bob. Alice applies the unitary operator W i , taken from a set { W i } with probability { p i } , on her part of theentangled state ρ . She sends the encoded state with probability p i over a noisy channel, described by the map Λ a , toBob. In the first approach we assume that Λ a just affects Alice’s subsystem, but that there is no noise on Bob’s side. Λ a BobAlice ( ρ ) b Λ {W , p } i i Figure 2.
Two-sided noise: Bipartite super dense coding with an initially entangled state ρ , shared betweenAlice and Bob. In the second approach, the noisy channel Λ a influences Alice’s subsystem after encoding while the noisychannel Λ b has already affected Bob’s side in the distribution step of the initial state ρ . The paper is organized as follows: in Section II we discuss the definition of theHolevo quantity for an ensemble of states in the presence of a noisy channel. Weintroduce a certain condition on the von Neumann entropy and we derive the superdense coding capacity for those cases where this condition is fulfilled. In Sections III andIV, we give examples of initial states and channels for which this condition on the vonNeumann entropy is satisfied, and calculate their optimal super dense coding capacityexplicitly. Section V provides a comparison between the super dense coding capacitiesin the presence of a one-sided or two-sided ptimal super dense coding over noisy quantum channels
2. Super dense coding capacity
In the super dense coding protocol Alice and Bob share a bipartite entangled quantumstate ρ . Alice performs local unitary operations W i with probability p i (where (cid:80) i p i = 1)on ρ to encode classical information through the state ρ i , i.e. ρ i = ( W i ⊗ ) ρ ( W i † ⊗ ) . (2)We consider Λ : ρ i → Λ( ρ i ) to be any completely positive map that acts on the sharedstate ρ i . (Below Λ will describe the noise acting on the ensemble states.) The ensemblethat Bob(s) receives is { Λ( p i , ρ i ) } . The amount of classical information transmitted viaa quantum channel is measured by the Holevo quantity or χ -quantity. This quantity forthe ensemble { Λ( p i , ρ i ) } is given by χ = S (cid:16) Λ( ρ ) (cid:17) − (cid:88) i p i S (Λ( ρ i )) = (cid:88) i p i S (cid:16) Λ( ρ i ) (cid:107) Λ( ρ ) (cid:17) , (3)where Λ( ρ ) = (cid:80) i p i Λ( ρ i ) is the average state and S ( η ) is the von Neumann entropy of η .The symbol S ( σ (cid:107) ρ ) denotes the relative entropy, defined as S ( σ (cid:107) ρ ) = tr( σ log σ − σ log ρ ).Note that χ is a function of the resource state ρ , the encoding { p i , W i } and the channelΛ. For brevity of notation we will not write explicitly these arguments of χ .The super dense coding capacity C for a given resource state ρ is defined to be themaximum of the Holevo quantity χ with respect to { p i , W i } , that is C = max { p i ,W i } ( χ ) . (4)In this paper we consider bipartite systems, where each subsystem has finitedimension d . A general density matrix on C d ⊗ C d in the Hilbert-Schmidt representationcan be conveniently decomposed as ρ = ⊗ ρ b d + 1 d (cid:32) d − (cid:88) i =1 r i λ i ⊗ + d − (cid:88) i,k =1 t ik λ i ⊗ λ k (cid:33) , (5)where ρ b = tr a ρ represents Bob’s reduced density operator and λ i are the generatorsof the SU( d ) algebra with tr λ i = 0. The parameters r i , s i , t ik are real numbers. Weintroduce the set of unitary operators { V i } , defined as V i =( m,n ) | j (cid:105) = exp( 2 π i njd ) | j + m (mod d ) (cid:105) . (6)These operators satisfy the condition d − tr( V i V † j ) = δ ij . Integers m and n run from 0to d − d unitary operators V i . We will consider in the followingthe case of unital noisy channels acting on Alice’s and Bob’s systems, namely channelsdescribed by the completely positive mapΛ( ρ ) = (cid:88) m K m ρK † m , (cid:88) m K † m K m = , (cid:88) m K m K † m = , (7) ptimal super dense coding over noisy quantum channels K m are Kraus operators. Here, the first condition on the Kraus operatorscorresponds to trace preservation, and the second condition guarantees the unitalproperty Λ( ) = . We will show in this section that for unital memoryless noisyquantum channels and certain initial resource states, the set of unitary operators { V i } with equal probabilities is the optimum encoding and leads to the maximum of theHolevo quantity.We will first prove in Lemma 1 some properties that hold for the specific encoding { V i } . In the following the symbol τ i will denote the resource state after encoding with V i , whereas τ will denote the resource state after encoding with an arbitrary unitaryoperation U . The ensemble average after the specific encoding with { V i } , the probabilitydistribution p i = 1 /d and after action of the channel will be denoted as ˜ ρ . - For similarmethods in the case of noiseless channels see also [2]. Lemma 1.
Let Λ a ( σ a ) = (cid:80) m A m σ a A † m and Λ b ( σ b ) = (cid:80) ˜ m B ˜ m σ b B † ˜ m be any twounital channels which act on Alice’s and Bob’s side, respectively. For an initial resourcestate ρ shared between Alice and Bob, the global channel Λ ab then acts asΛ ab ( ρ ) = (cid:88) m, ˜ m ( A m ⊗ B ˜ m ) ρ (cid:16) A † m ⊗ B † ˜ m (cid:17) . (8)Then, the following statements hold: For τ i = ( V i ⊗ ) ρ ( V i † ⊗ ), with V i being defined in (6), the average ˜ ρ of theensemble { p i = d , Λ ab ( τ i ) } d − i =0 takes the form ˜ ρ = ⊗ Λ b ( ρ b d ). For τ = ( U ⊗ ) ρ (cid:0) U † ⊗ (cid:1) with U being any unitary operator acting on Alice’ssystem, tr (Λ ab ( τ ) log ˜ ρ ) = − S ( ˜ ρ ). The relative entropy between Λ ab ( τ ) and ˜ ρ can be expressed as S (Λ ab ( τ ) (cid:107) ˜ ρ ) = S ( ˜ ρ ) − S (Λ ab ( τ )). Proof 1-a).
In [2] it was shown that the average of the ensemble { p i = d , τ i } d − i =0 is (cid:88) i d τ i = ⊗ ρ b d . (9)By using (9), the linearity of the channel and its unital property, the average of theensemble { p i = d , Λ ab ( τ i ) } d − i =0 is˜ ρ = (cid:88) i d Λ ab ( τ i ) = Λ ab ( d ⊗ ρ b ) = ⊗ Λ b ( ρ b d ) . (10) Proof 1-b).
In Lemma (1-a) we showed that ˜ ρ = ⊗ Λ b ( ρ b d ) and hence, log ˜ ρ = ⊗ log Λ b ( ρ b d ). Therefore: ptimal super dense coding over noisy quantum channels ab ( τ ) log ˜ ρ ) = tr (cid:34)(cid:32)(cid:88) m A m U U † A † m (cid:33) ⊗ (cid:32)(cid:88) ˜ m B ˜ m ρ b d B † ˜ m log Λ b ( ρ b d ) (cid:33) + 1 d (cid:32) d − (cid:88) i =1 r i (cid:88) m A m U λ i U † A † m (cid:33) ⊗ (cid:32)(cid:88) ˜ m B ˜ m B † ˜ m log Λ b ( ρ b d ) (cid:33) + 1 d d − (cid:88) i,k =1 t ik (cid:32)(cid:88) m A m U λ i U † A † m (cid:33) ⊗ (cid:32)(cid:88) ˜ m B ˜ m λ k B † ˜ m log Λ b ( ρ b d ) (cid:33)(cid:35) . (11)By using the linearity of the trace and the relationstr[ (cid:88) m A m U U † A † m ] = tr[ (cid:88) m A m A † m ] = tr[ ] , (12)tr[ (cid:88) m A m U λ i U † A † m ] = tr[ U λ i U † (cid:88) m A † m A m ]= tr[ U λ i U † ] = tr[ λ i ] = 0 (13)we can writetr (Λ ab ( τ ) log ˜ ρ ) = tr a tr b (cid:34)(cid:88) m, ˜ m ⊗ (cid:16) B ˜ m ρ b d B † ˜ m log Λ b ( ρ b d ) (cid:17)(cid:35) = tr b (cid:104) Λ b ( ρ b ) log Λ b ( ρ b d ) (cid:105) = − S ( ˜ ρ ) . (14) Proof 1-c).
Using the definition of the relative entropy S ( σ (cid:107) ρ ) = tr( σ log σ − σ log ρ )and the result of Lemma (1-b) we can write S (Λ ab ( τ ) (cid:107) ˜ ρ ) = tr(Λ ab ( τ ) log Λ ab ( τ ) − Λ ab ( τ ) log ˜ ρ )= S ( ˜ ρ ) − S (Λ ab ( τ )) . (15) (cid:50) We now show that for resource states with a certain symmetry property, namely forthose states where the von Neumann entropy after the channel action is independent ofthe unitary encoding, the encoding with the equally probable operators { V i } , as givenin (6), is optimal. Our proof follows the line of argument developed in [2]. Lemma 2.
Let τ i denote the resource state after encoding with V i , given in (6).Let ˜ χ = S ( ˜ ρ ) − d d − (cid:88) i S (Λ ab ( τ i )) (16) ptimal super dense coding over noisy quantum channels { p i = d , Λ ab ( τ i ) } , where ˜ ρ is the average stateof this ensemble and Λ ab ( · ) is defined in (8). For all the channels Λ ab and all initialstates ρ for which S (Λ ab ( τ )) = 1 d d − (cid:88) i S (Λ ab ( τ i )) (17)holds, ˜ χ is the super dense coding capacity. Here τ = ( U ⊗ ) ρ (cid:0) U † ⊗ (cid:1) , as we definedalready above, with U being any unitary operator. Proof.
Let us consider an arbitrary encoding, leading to an ensemble { p i , Λ ab ( ρ i ) } . Wewill show that its Holevo quantity χ cannot be higher than ˜ χ in (16), if the condition(17) is fulfilled.If S (Λ ab ( τ )) = d (cid:80) d − i S (Λ ab ( τ i )), then from (16) and Lemma (1-c),˜ χ = S (Λ ab ( τ ) (cid:107) ˜ ρ ) . (18)Since this equation holds for any τ that fulfills (17), it specially holds for ρ i , i.e.˜ χ = S (Λ ab ( ρ i ) (cid:107) ˜ ρ ) = (cid:88) i p i S (Λ ab ( ρ i ) (cid:107) ˜ ρ ) . (19)Using Donald’s identity, see [8], the right hand side of the above equation can bedecomposed as (cid:88) i p i S (Λ ab ( ρ i ) (cid:107) ˜ ρ ) = (cid:88) i p i S (Λ ab ( ρ i ) (cid:107) Λ ab ( ρ )) + S (Λ ab ( ρ ) (cid:107) ˜ ρ ) (20)with Λ ab ( ρ ) = (cid:80) i p i Λ ab ( ρ i ). The first term on the right hand side is the Holevo quantityfor any arbitrary ensemble { p i , Λ ab ( ρ i ) } . Hence,˜ χ = χ + S (Λ ab ( ρ ) (cid:107) ˜ ρ ) . (21)Since the relative entropy S (Λ ab ( ρ ) (cid:107) ˜ ρ ) is always positive or zero we can say that ˜ χ isalways bigger or equal than χ and hence, ˜ χ is the super dense coding capacity. (cid:50) From Lemma 2 we find that˜ χ = S ( ˜ ρ ) − S (Λ ab ( τ )) . (22)Since the above equation holds for τ = ( U ⊗ ) ρ (cid:0) U † ⊗ (cid:1) with any unitary U , itespecially holds for τ = ρ . Hence, whenever the condition (17) is true, the super densecoding capacity is given by C = ˜ χ = S ( ˜ ρ ) − S (Λ ab ( ρ )) , (23)where ˜ ρ is the average of the ensemble after encoding with the specific (and equallyprobable) unitaries { V i } and after the channel action, as introduced in Lemma 1. Asan interpretation of this formula, note that the action of a noisy channel typicallywill increase the entropy of a given state, and therefore will decrease the dense codingcapacity of the original resource state.In the next two sections we will study examples of channels and bipartite statessatisfying the condition (17), and evaluate explicitly the corresponding super densecoding capacities. ptimal super dense coding over noisy quantum channels One-sided d -dimensional Pauli channel A d -dimensional Pauli channel [9] that acts just on Alice’s side is defined byΛ Pa ( ρ i ) = d − (cid:88) m,n =0 q mn ( V mn ⊗ ) ρ i ( V † mn ⊗ ) , (24)where q mn are probabilities (i.e. q mn ≥ (cid:80) mn q mn = 1). The operators V mn , definedin (6) with a slightly different notation for the indices, can be expressed as V mn = d − (cid:88) k =0 exp (cid:18) iπknd (cid:19) | k (cid:105)(cid:104) k + m (mod d ) | . (25)They satisfy tr V mn = dδ m δ n and V mn V † mn = , and have the properties V mn V ˜ m ˜ n = exp (cid:18) iπ ˜ nmd (cid:19) V m + ˜ m ( mod d ) ,n +˜ n ( mod d ) , (26)tr[ V mn V † ˜ m ˜ n ] = dδ m ˜ m δ n ˜ n , (27) V mn V ˜ m ˜ n = exp (cid:18) iπ (˜ nm − n ˜ m ) d (cid:19) V ˜ m ˜ n V mn . (28)As the Kraus operators of one-sided Pauli channel (24) are unitary it is a unital channel. A Bell state in d × d dimensions is defined as | ψ (cid:105) = √ d (cid:80) d − j =0 | j (cid:105) ⊗ | j (cid:105) . The set of theother maximally entangled Bell states is then denoted by | ψ mn (cid:105) = ( V mn ⊗ ) | ψ (cid:105) , for m, n = 0 , , ..., d −
1. We will show that for a Bell state shared between Alice and Bob,and with a one-sided d -dimensional Pauli channel, the condition (17) is fulfilled. Wewill first prove the following Lemma. Lemma 3.
Let us define π mn := ( V mn U ⊗ ) ρ ( U † V † mn ⊗ ), where U is a unitaryoperator, ρ = | ψ (cid:105)(cid:104) ψ | and V mn is defined in (25). For m (cid:54) = ˜ m , n (cid:54) = ˜ n , π mn π ˜ m ˜ n = 0 (29)holds. Proof.
In Appendix B we show that ρ ( U † V † mn V ˜ m ˜ n U ⊗ ) ρ = 0 for m (cid:54) = ˜ m , n (cid:54) = ˜ n . Hence, π mn π ˜ m ˜ n = ( V mn U ⊗ ) ρ ( U † V † mn V ˜ m ˜ n U ⊗ ) ρ (cid:124) (cid:123)(cid:122) (cid:125) ( U † V † ˜ m ˜ n ⊗ ) = 0 (cid:50) By using the orthogonality property (29) and the purity of the density operators π mn ,we can write ptimal super dense coding over noisy quantum channels S (Λ Pa ( τ )) = S (cid:0) Λ Pa (cid:0) ( U ⊗ ) ρ ( U † ⊗ ) (cid:1)(cid:1) = S d − (cid:88) m,n =0 q mn ( V mn U ⊗ ) ρ ( U † V † mn ⊗ ) (cid:124) (cid:123)(cid:122) (cid:125) := π mn = H ( { q mn } ) , (30)where H ( { q mn } ) = − (cid:80) m,n q mn log q mn is the Shannon entropy. We note that the vonNeumann entropy S (Λ Pa ( τ )) is independent of the unitary encoding U . Consequently,for a one-sided d -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-sidedPauli channel in d dimensions takes the form C one − sided P d Bell = S ( d ⊗ ρ b ) − H ( { q mn } ) = log d − H ( { q mn } ) (31)for m, n = 0 , , ..., d −
1. Using (1) we notice that the super dense coding capacity of a d × d -dimensional Bell state in the noiseless case is given by log d . Thus, in the presenceof a one-sided Pauli channel the super dense coding capacity is reduced by the amount H ( { q mn } ) with respect to the noiseless case - i.e. the channel noise is simply subtractedfrom 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 | ψ (cid:105) = U a ⊗ U b | ψ (cid:105) . Actually, Lemma 3 still holds in this case and thereforealso the derivation of the capacity (31). We will now evaluate the super dense coding capacity for an input Werner state ρ W = − ηd + ηρ with 0 ≤ η ≤
1. The Werner state ρ W in the presence of a one-sided d -dimensional Pauli channel provides another example of states and channels thatsatisfy (17).Using (30), { q mn } is the set of eigenvalues of Λ Pa (cid:2) ( U ⊗ ) ρ ( U † ⊗ ) (cid:3) . The Paulichannel is a linear and unital map. Expressing the identity matrix in a suitablebasis, we arrive at S (cid:0) Λ Pa (cid:0) ( U ⊗ ) ρ W ( U † ⊗ ) (cid:1)(cid:1) = S (cid:18) η Λ Pa (cid:2) ( U ⊗ ) ρ ( U † ⊗ ) (cid:3) + 1 − ηd (cid:19) = S (cid:18) diag (cid:18) ηq + 1 − ηd , ..., ηq d − ,d − + 1 − ηd (cid:19)(cid:19) = H (cid:18) { ηq mn + 1 − ηd } (cid:19) . (32)From (32) it is apparent that the output channel entropy is independent of the unitaryencoding. Consequently, the super dense coding capacity, according to (23), is given by ptimal super dense coding over noisy quantum channels C one − sided P d Werner = log d − H ( { − ηd + ηq mn } ) . (33)The above capacity is also achieved by any other state with the form U a ⊗ U b ρ W U † a ⊗ U † b .
4. Two-sided d -dimensional depolarizing channel. In (24) we introduced the concept of a one-sided d -dimensional Pauli channel. A two-sided d -dimensional Pauli channel is then defined byΛ P ab ( ρ i ) = d − (cid:88) m,n, ˜ m, ˜ n =0 q mn q ˜ m ˜ n ( V mn ⊗ V ˜ m ˜ n ) ρ i ( V † mn ⊗ V † ˜ m ˜ n ) . (34)The d -dimensional depolarizing channel is a special case of a d -dimensional Paulichannel, with probability parameters q mn = − p + pd , m = n = 0 pd , otherwise . (35)for the noise parameter p , with 0 ≤ p ≤
1, and m, n = 0 , ..., d − Lemma 4.
Let Λ dep ab denote a two-sided d -dimensional depolarizing channel. Fora state ρ and bilateral unitary operator U a ⊗ U b , we have S (cid:16) Λ dep ab (cid:16) ( U a ⊗ U b ) ρ ( U † a ⊗ U † b ) (cid:17)(cid:17) = S (Λ dep ab ( ρ )) . (36) Proof : Considering Λ dep a and Λ dep b to be the d -dimensional depolarizing channelsthat act on Alice’s and Bob’s system, respectively, it is straightforward to verify thatΛ dep a ( λ i ) = (1 − p ) λ i , (37)(where λ i are as before the generators of SU ( d )), and analogously for Bob’s system.Using the decomposition (5) for ρ and the following relation (proved in theAppendix A): Λ dep a ( U a λ i U † a ) = (1 − p ) U a λ i U † a , (38)it is then easy to prove the following covariance property of the channel:Λ dep ab (cid:16) ( U a ⊗ U b ) ρ ( U † a ⊗ U † b ) (cid:17) = ( U a ⊗ U b ) (cid:104) Λ dep ab ( ρ ) (cid:105) ( U † a ⊗ U † b ) . (39) ptimal super dense coding over noisy quantum channels (cid:50) As a consequence of Lemma 4 we can conclude that for a two-sided d -dimensionaldepolarizing channel the entropy for a given initial state ρ is independent of the unitaryencoding, namely S (cid:16) Λ dep ab (cid:0) ( U ⊗ ) ρ (cid:0) U † ⊗ (cid:1)(cid:1)(cid:17) = S (cid:16) Λ dep ab ( ρ ) (cid:17) . (40)Therefore, (17) holds and, according to (23), the super dense coding capacity for a givengeneral resource state ρ , with a two-sided d -dimensional depolarizing channel is givenby C two − sided dep d ( ρ ) = S (cid:18) d ⊗ Λ dep b ( ρ b ) (cid:19) − S (cid:16) Λ dep ab ( ρ ) (cid:17) = log d + S (cid:16) Λ dep b ( ρ b ) (cid:17) − S (cid:16) Λ dep ab ( ρ ) (cid:17) . (41)Notice that since Lemma 4 holds for any local unitary U a ⊗ U b , the capacity (41)depends only on the degree of entanglement of the input state ρ . In other words, allinput states with the same degree of entanglement have the same super dense codingcapacity.Comparing the above expression (41) with the one for the noiseless case, given by C = log d + S ( ρ b ) − S ( ρ ), one realizes that in the case of two-sided noise the channelthat affects Bob’s subsystem enters twice, both in the von Neumann entropies for thelocal and the global density matrix. In (41) we obtained the super dense coding capacity of an arbitrary given initial resourcestate ρ for the two-sided d -dimensional depolarizing channel. In this subsection weperform the optimization of the super dense coding capacity over the initial state oftwo qubits for the two-sided | ϑ α (cid:105) can be written in the Schmidt bases {| u i (cid:105)} , {| v i (cid:105)} as | ϑ α (cid:105) = √ − α | u v (cid:105) + √ α | u v (cid:105) with 0 ≤ α ≤ /
2. Two local unitaries V a and V b convert the computational bases to the Schmidt bases. Therefore, | ϑ α (cid:105) in computationalbases can be written as | ϑ α (cid:105) = V a ⊗ V b ( √ − α | (cid:105) + √ α | (cid:105) ). In (36) we showed that theoutput von Neumann entropy of the two-sided depolarizing channel is invariant underprevious local unitary transformations. Therefore | ϑ α (cid:105) and | ϕ α (cid:105) = √ − α | (cid:105) + √ α | (cid:105) lead to the same dense coding capacity. We can thus parametrize a pure initial state asa function of a single real parameter, namely as the state | ϕ α (cid:105) , and follow the approach ptimal super dense coding over noisy quantum channels α and the noise parameter p is given by C two − sided dep α ( | ϕ α (cid:105)(cid:104) ϕ α | ) = 1 − ξ log ξ − ξ log ξ + γ log γ + γ log γ + 2 γ log γ , (42)where γ i (with i = 1 , , ,
4) are the eigenvalues of Λ dep ab ( | ϕ α (cid:105)(cid:104) ϕ α | ) and ξ s (with s = 1 , dep b ( ρ b,α ), where ρ b,α = tr a ( | ϕ α (cid:105)(cid:104) ϕ α | ). The eigenvalues γ i and ξ s are explicitly given by γ , = 12 (cid:16) − p (1 − p ± (1 − p ) (cid:112) − pα (2 − p )(1 − α ) (cid:17) ,γ = γ = p − p ,ξ = α − pα + p ,ξ = 1 − α + pα − p . (43)We can now maximize expression (42) over the variable α , for a given noiseparameter p , and find interesting results. They are illustrated in Figure 3, where weplot the superdense coding capacity in (42) as a function of the noise parameter p ,for various values α . We find that there is a threshold value p t ≈ . ≤ p ≤ .
345 the value α = 1 / p ≥ . α = 0, i.e. product states are best for densecoding. As shown graphically in the close-up of Figure 3, the curves for intermediatevalues of α are always lower than α = 1 / α = 0. In order to prove this claim,we also evaluated C two − sided dep α =1 / − C two − sided dep α in the range of 0 ≤ p ≤ .
345 and C two − sided dep α =0 − C two − sided dep α in the range of 0 . ≤ p ≤ α and p . We found that these two functions are positive or zero. Thus, for pure initialstates it is always best to either use maximally entangled states or product states,depending on the noise level. ptimal super dense coding over noisy quantum channels Α (cid:61) (cid:144) Α (cid:61)
Α (cid:61)
Α (cid:61)
Figure 3.
The super dense coding capacity for the two-sided depolarizing channel in 2 dimensions, C two − sided dep α , as function of the noise parameter p , for α = 0, α = 0 . α = 0 . α = 1 /
2. For the definition of α see main text. For 0 ≤ p ≤ .
345 a Bell state, i.e. α = 1 /
2, leads to the optimal capacity, while for 0 . ≤ p ≤ α = 0). In the following we call the super dense coding capacity of an initial Bell state | ϕ / (cid:105) in the presence of a two-sided C two − sided dep Bell . Using(42) with α = 1 /
2, this capacity is given by C two − sided dep Bell = 2 + 1 + 3(1 − p ) − p )
4+ 3 1 − (1 − p ) − (1 − p ) . (44)The super dense coding capacity with an initial product state | ϕ (cid:105) in the presence ofa two-sided C ch dep .From (42) with α = 0 it follows that C ch dep = 1 + p p − p − p . (45)Note that (45) is identical to the classical channel capacity of the depolarizing channelfor qubits [11].We now show that using mixed initial states as a resource cannot increase the superdense coding capacity, i.e. | ϕ / (cid:105) and | ϕ (cid:105) are the optimal input states for the range ofnoise parameter 0 ≤ p ≤ .
345 and 0 . ≤ p ≤
1, respectively. To show this claim wefirst write the super dense coding capacity (41) in the form of the relative entropy C two − sided dep d ( ρ ) = S (Λ ab ( ρ ) (cid:107) d ⊗ Λ b ( ρ b )) . (46) ptimal super dense coding over noisy quantum channels ρ k ,i.e. ρ mix = (cid:80) k p k ρ k , and ρ b,mix = tr a ( ρ mix ) = (cid:80) k p k ρ b,k , we can write C ρ mix = S (Λ ab ( ρ mix ) (cid:107) ˜ ρ ) = S (Λ ab ( ρ mix ) (cid:107) d ⊗ Λ b ( ρ b,mix ))= S ( (cid:88) k p k Λ ab ( ρ k ) (cid:107) (cid:88) k p k d ⊗ Λ b ( ρ b,k )) ≤ (cid:88) k p k S (Λ ab ( ρ k ) (cid:107) d ⊗ Λ b ( ρ b,k )) . (47)In the above inequality we have used the subadditivity of the relative entropy, i.e. S ( (cid:80) i p i r i (cid:107) (cid:80) i q i s i ) ≤ (cid:80) i p i S ( r i (cid:107) s i ) + H ( p i (cid:107) q i ), where H ( ·(cid:107)· ) is the Shannon relativeentropy, defined as H ( p i (cid:107) q i ) = (cid:80) i p i log p i q i [13]. We showed before that the super densecoding capacity of a pure state for 0 ≤ p ≤ .
345 is upper bounded by the super densecoding capacity of a Bell state | ϕ / (cid:105) , and for 0 . ≤ p ≤ | ϕ (cid:105) . Remembering that ρ k is pure, and using (46), we find that for0 ≤ p ≤ . C ρ mix ≤ (cid:88) k p k S (Λ ab ( ρ k ) (cid:107) d ⊗ Λ b ( ρ b,k )) ≤ C two − sided dep , (48)and for 0 . ≤ p ≤ C ρ mix ≤ (cid:88) k p k S (Λ ab ( ρ k ) (cid:107) d ⊗ Λ b ( ρ b,k )) ≤ C ch dep , (49)which proves our claim.It is interesting to note that the optimal capacity for the two-sided qubitdepolarizing channel is a non-differentiable function of the noise parameter p , and thatthe optimal states are either maximally entangled or separable. In other words, there isa transition in the entanglement of the optimal input states at the particular thresholdvalue of the noise parameter p t ≈ .
5. Super dense coding capacity versus channel capacity
In this section, we consider the question of whether or not it is reasonable in thepresence of noise to use the super dense coding protocol for the transmission of classicalinformation? To answer this question, we provide a comparison between the classicalcapacity of a 2-dimensional depolarizing channel and the super dense coding capacities ptimal super dense coding over noisy quantum channels one-sided and two-sided one-sided C one − sided dep = 2 + 4 − p − p p p . (50)The super dense coding capacity for a two-sided C ch dep of the 2-dimensional depolarizing channel is achieved by an ensemble of pure states belonging toan orthonormal basis, say {| (cid:105) , | (cid:105)} at the channel input, with equal probability andperforming a complete von Neumann measurement in the same basis over the channeloutput [11]. Its expression is given explicitly in (45).In Figure 4, we plot C one − sided dep , C two − sided dep , C ch dep , and C = 1 in terms of thenoise parameter p . As we expect, the first three capacities C one − sided dep , C two − sided dep and C ch dep decrease as the noise increases. As expected, the super dense coding capacityof a one-sided C one − sided dep is greater than theclassical capacity C ch dep for all values of p , as the additional resource of entanglementis used in dense coding. The comparison between C two − sided dep and C ch dep illustratesthat for 0 . ≤ p ≤ two-sided . ≤ p ≤ p (cid:61) p (cid:61) C one (cid:45) sided dep C two (cid:45) sided dep C ch dep C (cid:61) Figure 4.
The classical capacity C ch dep of the 2-dimensional depolarizing channel and the super densecoding capacities for an initial Bell state in the presence of a one-sided and two-sided C one − sided dep and C two − sided dep , respectively, as functions of the noise parameter p . ptimal super dense coding over noisy quantum channels C one − sided dep corresponds to the entanglement assisted capacityfor the depolarizing channel [15]. According to (50) for p = 0 .
252 the super dense codingcapacity for an initial Bell state via the one-sided one . The maximum information that can be transmitted by two-dimensionalsystems without any source of entangled states is C = 1. That is, for p = 0 .
252 thesuper dense coding capacity reaches the classical limit, as can be seen in Figure 4. Itwas shown in [14] that the classical limit of the quantum teleportation protocol, whenusing a Bell state and distributing one subsystem of it via a depolarizing channel, isreached at p = 1 /
3. In the absence of noise, quantum teleportation and super densecoding are two equivalent protocols [12]. According to our results this is not true inthe presence of noise, as we have shown explicitly for the depolarizing channel: here,the quantum/classical boundary for super dense coding occurs at a different noise valuethan for quantum teleportation.We point out that the expression (31) for the dense coding capacity of a Bell stateprovides a lower bound to the entanglement-assisted capacity of a general Pauli channel.
6. Non-unitary encoding for the d -dimensional Pauli channel So far, we assumed that the encoding in the super dense coding protocol is unitary.In this section we consider the possibility of performing non-unitary encoding on theinitial state and discuss explicitly the case of the depolarizing channel. Let us considerΓ i to be a completely positive trace preserving (CPTP) map. Alice applies the mapΓ i on her side of the shared state ρ , thereby encoding ρ as ρ i = [Γ i ⊗ ]( ρ ) := Γ i ( ρ ).The super dense coding protocol with non-unitary encoding for noiseless channels hasbeen discussed by M. Horodecki et al. [17], M. Horodecki and Piani [5], and Winter[18]. In this section we introduce an upper bound on the Holevo quantity for a two-sided d -dimensional Pauli channel, and show that this upper bound is reachable by apre-processing before unitary encoding. Our arguments follow a similar line as in [5],where non-unitary encoding was studied for the case of noiseless channels. Lemma 5.
Let χ = S (cid:0)(cid:80) i p i Λ Pab ( ρ i ) (cid:1) − (cid:80) i p i S (cid:0) Λ Pab ( ρ i ) (cid:1) be the Holevo quantitywith ρ i = Γ i ( ρ ) and let Λ Pab ( ρ ) be a general two-sided d -dimensional Pauli channel definedvia Λ Pab ( ρ ) = d − (cid:88) m,n, ˜ m, ˜ n =0 q mn ˜ m ˜ n ( V mn ⊗ V ˜ m ˜ n )( ρ )( V † mn ⊗ V † ˜ m ˜ n ) (51)with (cid:80) d − m,n, ˜ m, ˜ n =0 q mn ˜ m ˜ n = 1. Let Γ M ( · ) := [Γ M ⊗ ]( · ) be the map that minimizes thevon Neumann entropy after application of this map and the channel Λ ab to the initialstate ρ , i.e. Γ M minimizes the expression S (cid:0) Λ Pab (Γ M ( ρ )) (cid:1) . Then, the Holevo quantity χ is upper bounded by χ ≤ log d + S (cid:0) Λ Pb ( ρ B ) (cid:1) − S (cid:0) Λ Pab (Γ M ( ρ )) (cid:1) . (52) ptimal super dense coding over noisy quantum channels Proof: Γ M ( · ) is a map that leads to the minimum of the entropy after applying itand the channel to the initial state ρ . Therefore, χ = S (cid:32)(cid:88) i p i Λ Pab ( ρ i ) (cid:33) − (cid:88) i p i S (cid:0) Λ Pab (cid:0) ρ i (cid:1)(cid:1) ≤ S (cid:32)(cid:88) i p i Λ Pab ( ρ i ) (cid:33) − S (cid:0) Λ Pab (Γ M ( ρ )) (cid:1) . Since the von Neumann entropy is subadditive and since the maximum entropy of a d -dimensional system is log d , we have χ ≤ log d + S (cid:32) tr a (cid:32)(cid:88) i p i Λ Pab ( ρ i ) (cid:33)(cid:33) − S (cid:0) Λ Pab (Γ M ( ρ )) (cid:1) . Now, since tr a (cid:80) i p i Λ Pab ( ρ i ) = Λ Pb ( ρ b ) it follows that χ ≤ log d + S (cid:0) Λ Pb ( ρ b ) (cid:1) − S (cid:0) Λ Pab (Γ M ( ρ )) (cid:1) . (cid:50) If the upper bound in (52) is achievable, then it is equal to the super densecoding capacity. We consider the ensemble { ˜ p i , ˜Γ i ( ρ ) } with ˜ p i = d and ˜Γ i ( ρ ) =( V i ⊗ )Γ M ( ρ )( V † i ⊗ ), where V i is defined in (6). We will show in the following thatthis ensemble achieves the upper bound in (52). In other words, the optimal encodingconsists of a fixed pre-processing with Γ M and a subsequent unitary encoding. This isanalogous to the case of noiseless channels, for which the same statement was shown in[5]. The Holevo quantity of the ensemble { ˜ p i , ˜Γ i ( ρ ) } is˜ χ = S (cid:32)(cid:88) i d Λ Pab (cid:16) ˜Γ i ( ρ ) (cid:17)(cid:33) − (cid:88) i d S (cid:104) Λ Pab (cid:16) ˜Γ i ( ρ ) (cid:17)(cid:105) . (53)By using (9) and noting that Γ M acts only on Alice’s side, and by using Lemma1-a), we find that the average of Λ Pab (cid:16) ˜Γ i ( ρ ) (cid:17) , i.e. the argument in the first term on theRHS of (53), is given by (cid:88) i d Λ Pab (cid:16) ˜Γ i ( ρ ) (cid:17) = d ⊗ Λ Pb ( ρ b ) . (54)Furthermore, the second term on the RHS of (53) is given by (cid:88) i d S (cid:16) Λ Pab (cid:16) ˜Γ i ( ρ ) (cid:17)(cid:17) = (cid:88) i d S (cid:16) Λ Pab (cid:16) ( V i ⊗ ) Γ M ( ρ ) (cid:16) V † i ⊗ (cid:17)(cid:17)(cid:17) = 1 d (cid:88) i S (cid:32) ( V i ⊗ ) (cid:34) d − (cid:88) m,n, ˜ m, ˜ n =0 q mn ˜ m ˜ n ( V mn ⊗ V ˜ m ˜ n ) Γ M ( ρ ) (cid:16) V † mn ⊗ V † ˜ m ˜ n (cid:17)(cid:35) · ptimal super dense coding over noisy quantum channels · ( V † i ⊗ ) (cid:17) = 1 d (cid:88) i S (cid:2) Λ Pab (Γ M ( ρ )) (cid:3) = S (cid:2) Λ Pab (Γ M ( ρ )) (cid:3) (55)where in the second line of the above equations we have inserted the action of thechannel, defined in (51), and we have used (28), from which it follows that V i and V mn commute up to a phase.Inserting (54) and (55) into (53), one finds that the Holevo quantity ˜ χ is equal tothe upper bound given in (52). Consequently, the super dense coding capacity withnon-unitary encoding is C = log d + S (cid:0) Λ Pb ( ρ b ) (cid:1) − S (cid:2) Λ Pab (Γ M ( ρ )) (cid:3) . (56)Thus, we have shown above for the case of a d -dimensional Pauli channel that applyingthe appropriate pre-processing Γ M on the initial state ρ before the unitary encoding { V i } may increase the super dense coding capacity, with respect to only using unitaryencoding. Our results derived in section 4 provide an example where pre-processingindeed leads to an improvement: Consider the case of a two-sided . ≤ p ≤
1, see Figure 3. To reach the optimal super dense coding capacity inthis case, Alice applies a measurement as a pre-processing, projecting the Bell stateonto | (cid:105) or | (cid:105) ; afterwards she applies the unitary encoding. As we showed above, thesuper dense coding capacity for product states is equal to the capacity of the depolarizingchannel, given in (45). Thus, in this case we reach a higher super dense coding capacitythan without pre-processing. The effect of pre-processing is illustrated in Figure 5,which is an excerpt of Figure 4. p=0.3450.2 0.4 0.6 0.8 1.0 p0.51.01.52.0C Figure 5.
The solid curve is the optimal super dense coding capacity with a Bell state in the presence of a two-sided . ≤ p ≤ ptimal super dense coding over noisy quantum channels
7. Conclusions
In conclusion, we investigated the bipartite super dense coding protocol in the presenceof a unital noisy channel, which acts either only on Alice’s subsystem after encoding( one-sided channel) or both on Alice’s and Bob’s subsystems ( two-sided channel). Forthose cases where the von Neumann entropy fulfills a specific condition, we derived thesuper dense coding capacity. We showed that a one-sided d -dimensional Pauli channelfor the resource of Bell and Werner states fulfills the above mentioned condition on thevon Neumann entropy. Our condition on the von Neumann entropy is also satisfiedfor a two-sided d -dimensional depolarizing channel. For these examples, we derived theexplicit optimal super dense coding capacity, as a function of the initial resource state.When the initial state can be chosen, we found for the case of a two-sided one-sided and two-sided two-sided d -dimensional Pauli channel. We showed that the optimal strategy is toapply a pre-processing before the unitary encoding. We gave an example of super densecoding for an initial Bell state and a two-sided Acknowledgments
We are grateful for discussions with Alexander Holevo, Barbara Kraus and ColinWilmott. This work was partially supported by the EU Integrated Project SCALA,the European Project CORNER and Deutsche Forschungsgemeinschaft (DFG). ptimal super dense coding over noisy quantum channels Appendix A.
We give here a proof for (38). We expand
U λ i U † in terms of { V mn } . By using the factthat λ i is traceless, we haveΛ depa ( U λ i U † ) = Λ depa d − (cid:88) m,n (cid:54) =(0 , γ mn V mn . Here, Λ depa ( · ) is a linear map that is given by Λ depa ( · ) = (cid:80) d − m, ˜ n =0 q ˜ m ˜ n V ˜ m ˜ n ( · ) V † ˜ m ˜ n . Then wecan write Λ depa ( U λ i U † ) = d − (cid:88) m,n (cid:54) =(0 , γ mn Λ depa ( V mn )= d − (cid:88) ˜ m, ˜ n =0 d − (cid:88) m,n (cid:54) =(0 , γ mn q ˜ m ˜ n V ˜ m ˜ n V mn V † ˜ m ˜ n . By using (28) and unitarity of V mn , we haveΛ depa ( U λ i U † ) = d − (cid:88) ˜ m, ˜ n =0 d − (cid:88) m,n (cid:54) =(0 , γ mn q ˜ m ˜ n exp (cid:18) iπ ( n ˜ m − ˜ nm ) d (cid:19) V mn = d − (cid:88) m,n (cid:54) =(0 , γ mn V mn d − (cid:88) ˜ m, ˜ n =0 q ˜ m ˜ n exp (cid:18) iπ ( n ˜ m − ˜ nm ) d (cid:19) . For q ˜ m ˜ n we replace the expression of (34) and we then haveΛ depa ( U λ i U † ) = d − (cid:88) m,n (cid:54) =(0 , γ mn V mn − p + pd d − (cid:88) ˜ m, ˜ n =0 exp (cid:18) iπ ( n ˜ m − ˜ nm ) d (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) δ ,m δ ,n = (1 − p ) d − (cid:88) m,n (cid:54) =(0 , γ mn V mn = (1 − p ) U λ i U † Therefore, (38) is proved.
Appendix B.
In Lemma 3 we need to prove that ρ ( U † V † mn V ˜ m ˜ n U ⊗ ) ρ = 0. We here show that (cid:104) ψ | ( U † V † mn V ˜ m ˜ n U ⊗ ) | ψ (cid:105) = 0, from which the previous statement follows. Due to (27)for m (cid:54) = ˜ m and n (cid:54) = ˜ n the expression V † mn V ˜ m ˜ n is traceless and { V jk } d − j,k =0 form a complete ptimal super dense coding over noisy quantum channels V † mn V ˜ m ˜ n = (cid:80) ( j,k ) (cid:54) =(0 , β jk V jk with expansion coefficients β jk .Therefore, (cid:104) ψ | ( U † V † mn V ˜ m ˜ n U ⊗ ) | ψ (cid:105) = (cid:88) ( j,k ) (cid:54) =(0 , β jk (cid:104) ψ | ( U † V jk U ⊗ ) | ψ (cid:105) = 1 d (cid:88) ( j,k ) (cid:54) =(0 , d − (cid:88) m,n =0 β jk (cid:104) mm | ( U † V jk U ⊗ ) | nn (cid:105) = 1 d (cid:88) ( j,k ) (cid:54) =(0 , d − (cid:88) m,n =0 β jk (cid:104) m | U † V jk U | n (cid:105)(cid:104) m | n (cid:105) = 1 d (cid:88) ( j,k ) (cid:54) =(0 , β jk tr[ U † V jk U ] = 1 d (cid:88) ( j,k ) (cid:54) =(0 , β jk tr[ V jk ] = 0 . Since ρ = | ψ (cid:105)(cid:104) ψ | , we arrive at ρ ( U † V † mn V ˜ m ˜ n U ⊗ ) ρ = 0 , (B.1)which completes the proof. ptimal super dense coding over noisy quantum channels References [1] Bennett C H and Wiesner S J 1992
Phys. Rev.
Lett. J. Phys. A Math. Gen. Phys. Rev.
Lett. Int. J.Quant. Inform. Phys. Rev.
Lett. Phys. Rev.
Lett. Math. Proc. Camb. Phil. Soc.
Phys. Rev.
Lett. J. Mod. Optics IEEE Transactions onInformation Theory J. Phys. A Math. Gen. Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, United Kingdom)[14] Horodecki M, Horodecki P and Horodecki R 1999
Phys. Rev. A et al Phys. Rev.
Lett. Phys. Rev. A Quantum Inf. Comput. J. Math. Phys.43