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Enhancement of Transmission Rates in Quantum Memory Channels with Damping

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We consider the transfer of quantum information down a single-mode quantum transmission line. Such quantum channel is modeled as a damped harmonic oscillator, the interaction between the information carriers -a train of N qubits- and the oscillator
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    a  r   X   i  v  :   0   9   0   3 .   1   4   2   4  v   1   [  q  u  a  n   t  -  p   h   ]   8   M  a  r   2   0   0   9 Enhancement of transmission rates in quantum memory channels with damping Giuliano Benenti, 1,2 Antonio D’Arrigo, 3 and Giuseppe Falci 3 1 CNISM, CNR-INFM & Center for Nonlinear and Complex Systems,Universit`a degli Studi dell’Insubria, Via Valleggio 11, 22100 Como, Italy  2  Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano, Italy  3  MATIS CNR-INFM, Consiglio Nazionale delle Ricerche and Dipartimento di Metodologie Fisiche e Chimiche per l’Ingegneria,Universit`a di Catania, Viale Andrea Doria 6, 95125 Catania, Italy  (Dated: November 22, 2013)We consider the transfer of quantum information down a single-mode quantum transmission line.Such quantum channel is modeled as a damped harmonic oscillator, the interaction between theinformation carriers -a train of   N   qubits- and the oscillator being of the Jaynes-Cummings kind.Memory effects appear if the state of the oscillator is not reset after each channel use. We show thatthe setup without resetting is convenient in order to increase the transmission rates, both for thetransfer of quantum and classical private information. Our results can be applied to the micromaser. PACS numbers: 03.67.Hk, 03.67.-a, 03.65.Yz Quantum communication channels [1, 2] use quantum systems as carriers for information. One can employthem to transfer classical information, by encoding clas-sical bits by means of quantum states. Furthermore thereare some peculiar issues strictly related to the quantumcomputation: to transfer an (unknown) quantum statebetween different subunits of a quantum computer, tohold in memory a quantum state waiting to process itlater, to distribute entanglement among different parties.A key problem in quantum information is the determi-nation of the classical and quantum  capacities   of noisyquantum channels, defined as the maximum number of bits/qubits that can be reliably transmitted per use of the channel. These quantities characterize the channel,giving an upper bound to the channel efficiency per use.In any realistic implementation, errors occur due tothe unavoidable coupling of the transmitted quantumsystems with an uncontrollable environment. Noise canhave significant low frequency components, which tra-duce themselves in memory effects, leading to relevantcorrelations in the errors affecting successive transmis-sions. Important examples in this context are photonstraveling across fibers with birefringence fluctuating withcharacteristic time scales longer than the separation be-tween consecutive light pulses [3] or low-frequency im-purity noise in solid state implementations of quantumhardware [4]. Memory effects become unavoidably rele-vant when trying to increase the  transmission rate   [5],that is, to reduce the time interval that separates twoconsecutive channel uses.Quantum channels with memory attracted increasingattention in the last years, see [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] and references therein. Coding theo- rems have been proved for classes of quantum memorychannels [6, 7]. Memory effects have been modeled by Markov chains [8, 9, 10, 11, 12] and the quantum ca- pacity has been exactly computed for a Markov chaindephasing channel [10, 11, 12]. Various kind of memory channels are been studied, for example: purely dephas-ing channels [11, 12, 13], lossy bosonic channels [14]; also spin chains have been studied as models for the chan-nel itself  [15]. Hamiltonian models of memory chan-nels [11, 12, 13, 15] aim at a description directly referring to physical systems and enlight another important exam-ple of a noisy quantum channel, namely the memory of a quantum computer [16].In this work the quantum channel is modeled as adamped harmonic oscillator, and we consider transfer of quantum information through it. A train of   N   qubits issent down the channel (initially prepared in its groundstate) and interacts with it during the transit time. If thestate of the oscillator is not reset after each channel use,then the action of the channel on the  k -th qubit dependson the previous  k − 1 channel uses. The oscillator acts asa  local   “unconventional environment” [5, 6, 9, 17], cou- pled to a memoryless reservoir damping both its phasesand populations, which mimics any cooling process re-setting the oscillator to its ground state. The model isvisualized by a qubit-micromaser [18] system, the qubittrain being a stream of two-level Rydberg atoms injectedat low rate into the cavity. Unconventional environmentscapture essentials features of solid state circuit-quantumelectrodynamics (QED) [19] devices and in this contextthe model may describe the architecture of a quantummemory. The low injection rate is required in order toavoid collective effects such as superradiance. Atoms in-teract with the photon field inside the cavity and mem-ory effects are relevant if the lifetime of photons is longerthan the time interval between two consecutive channeluses. In what follows we will show that it is convenientto use the channel without resetting in order to increasethe transmission rates, both for the transfer of quantumand classical private information. The quantum capacity.–   N   channel uses correspond toa  N  -qubit input state  ρ , which may be chosen with prob-ability  {  p i }  from a given ensemble  { σ i }  of the  N  -qubit  2Liouville space ( ρ  =   i  p i σ i ). Due to the coupling touncontrollable degrees of freedom, the transmission is ingeneral not fully reliable. The output is therefore de-scribed by a linear, completely positive, trace preserving(CPT) map for  N   uses, E  N  ( ρ ). For memoryless channels E  N   =  E  ⊗ N  1  , where  E  1  indicates the single use, and thequantum capacity  Q  can be computed as [16, 20, 21] Q  = lim N  →∞ Q N  N  , Q N   = max ρ I  c ( E  N  ,ρ ) ,  (1) I  c ( E  N  ,ρ ) =  S  [ E  N  ( ρ )] − S  e ( E  N  ,ρ ) .  (2)Here  S  ( ρ ) =  − Tr[ ρ log 2 ρ ] is the von Neumann en-tropy,  S  e ( E  N  ,ρ ) is the  entropy exchange   [16], defined as S  e ( E  N  ,ρ ) =  S  [(  I ⊗ E  N  )( | ψ  ψ | )], where  | ψ  ψ |  is anypurification of   ρ . That is, we consider the system  S , de-scribed by the density matrix  ρ , as a part of a largerquantum system  RS ;  ρ  = Tr R | ψ  ψ |  and the referencesystem  R  evolves trivially, according to the identity su-peroperator  I  . The quantity  I  c ( E  N  ,ρ ) is called  coherent information   [16] and must be maximized over over allinput states  ρ . In general  I  c  is not subadditive [16], i.e. Q N  /N   ≥  Q 1 . When memory effects are taken into ac-count the channel does not act on each carrier indepen-dently,  E  N    =  E  ⊗ N  1  , and Eq. (1) in general only providesan upper bound on the channel capacity. However, forthe so-called  forgetful channels   [6], for which memory ef-fects decay exponentially with time, a quantum codingtheorem exists showing that the upper bound can be sat-urated [6]. The model.–   The overall Hamiltonian governing thedynamics of the system ( N   qubits), a local environment(harmonic oscillator) and a reservoir is defined as (    = 1) H ( t ) = H 0  + V   + δ  H ,  H 0  =  ν   a † a + 12  +  ω 2 N   k =1 σ ( k ) z  ,V   =  λ N   k =1 f  k ( t )  a † σ ( k ) −  + aσ ( k )+  .  (3)The qubits-oscillator interaction  V   is of the Jaynes-Cummings kind, and we take  λ  real and positive. Cou-pling is switchable:  f  k ( t ) = 1 when qubit  k  is inside thechannel (transit time  τ   p ),  f  k ( t ) = 0 otherwise. The term δ  H  describes both the reservoir’s Hamiltonian and thelocal environment-reservoirinteraction and causes damp-ing of the oscillator (the cavity mode in the micromaser),that is, relaxation and dephasing with time scales  τ  d  and τ  φ , respectively. Two consecutive qubits entering thechannel are separated by the time interval  τ  .One can argue that the resonant regime  ν   ∼  ω  is themost significant when describing the coupling to modesinducing damping. We work in the interaction picture,where the effective Hamiltonian at resonance is given by˜ H  =  e i H 0 t ( V   +  δ  H ) e − i H 0 t (we will omit the tilde fromnow on).We assume  τ   p  ≪  τ,τ  φ ,τ  d , so that non-unitary effectsin the evolution of the system and the oscillator can beignored during the crossing time  τ   p . Between two succes-sive pulses the oscillator evolves according to the stan-dard master equation (obtained after tracing over thereservoir)˙ ρ c  = Γ  aρ c a † −  12 a † aρ c −  12 ρ c a † a  .  (4)The asymptotic decay (channel reset) to the ground state | 0   takes place with rate Γ, so that  τ  d  = 1 / Γ. We intro-duce the memory parameter  µ ≡ τ  d / ( τ   + τ  d ): fast decay τ  d  ≪ τ   yields the memoryless limit  µ ≪ 1, whereas  µ  1when memory effects come into play. The memoryless limit –   In this limit damping actsas a built-in reset for the oscillator to its ground state ρ c (0) =  | 0  0 |  after each channel use. We consider ageneric single-qubit input state, ρ 1 (0) = (1 −  p ) | g  g | + r | g  e | + r ⋆ | e  g | +  p | e  e | ,  (5)with  {| g  , | e }  orthogonal basis for the qubit,  p  real and | r | ≤    p (1 −  p ). Given the initial, separable qubit-oscillator state  ρ 1 (0) ⊗ ρ c (0), we have E  1 [ ρ 1 (0)] = Tr c { U  ( τ   p )[ ρ 1 (0) ⊗ ρ c (0)] U  † ( τ   p ) } ,  (6)with  U  ( τ   p ) unitary time-evolution operator determinedby the undamped Jaynes-Cummings Hamiltonian. It iseasy to obtain [23], in the  {| g  , | e }  basis, E  1 [ ρ 1 (0)] =  1 −  p cos 2 ( λτ   p )  r cos( λτ   p ) r ⋆ cos( λτ   p )  p cos 2 ( λτ   p )  ,  (7)with  λ  frequency of the Rabi oscillations between levels | e, 0   and  | g, 1  . Eq. (7) corresponds to an amplitude-damping channel:  E  1 [ ρ 1 (0)] =   1 k =0 E  k ρ 1 (0) E  † k , wherethe Kraus operators [1, 2]  E  0  = | g  g | + √  η | e  e | ,  E  1  = √  1 − η | g  e | , with  η  = cos 2 ( λτ   p )  ∈  [0 , 1]. This channelis degradable [24] and therefore to compute its quantumcapacity it is sufficient to maximize the coherent infor-mation over single uses of the channel. Maximizationis achieved by classical states ( r  = 0) and one obtains Q  = max  p ∈ [0 , 1] { H  2 ( ηp ) − H  2 [(1 − η )  p ] }  if   η >  1   2 , where H  2 ( x ) =  − x log 2 x  −  (1  −  p )log 2 (1  −  x ) is the binaryShannon entropy, while  Q  = 0 when  η ≤  12  [24]. Memory channels: validity of Eq. ( 1) –   Memory ap-pears in our model when  τ   is finite. To show that theregularized coherent information still represents the truequantum capacity we follow the arguments made for for-getful channels in Ref. [6]. The key point is the use of adouble-blocking strategy mapping, with a negligible er-ror, the memory channel into a memoryless one. Weconsider blocks of   N   + L  uses of the channel and do theactual coding and decoding for the first  N   uses, ignoringthe remaining  L  idle uses. We call ¯ E  N  + L  the resulting  3CPT map. If we consider  M   uses of such blocks, the cor-responding CPT map ¯ E  M  ( N  + L )  can be approximated bythe memoryless setting (¯ E  ( N  + L ) ) ⊗ M  . One can use Eq. (1)to compute the quantum capacity if  [6]  ¯ E  M  ( N  + L ) ( ρ s ) − (¯ E  ( N  + L ) ) ⊗ M  ( ρ s )  1  ≤ h ( M  − 1) c − L ,  (8)where  ρ s  is a  M  ( N   +  L ) input state,  h >  0,  c >  1 areconstant and   ρ  1  = Tr   ρ † ρ  is the trace norm [1] (notethat  c  and  h  are independent of the input state  ρ s ). Onecan prove [25] that, due to the exponentially fast chan-nel (cavity) reset to the ground state, inequality (8) isfulfilled. Therefore, quantum capacity can be computedfrom the maximization (1) of coherent information. Lower bound for the quantum capacity.–   For the modelHamiltonian (3) computation of the coherent informationfor a large number  N   of channel uses is a difficult task,both for analytical and numerical investigations, even forseparable input states,  ρ  =  ρ 1 (0) ⊗ N  . Indeed, interac-tion with the oscillator entangles initially independentqubits. Nevertheless, a lower bound to the quantum ca-pacity can be computed if   τ  φ  ≪  τ  . This happens whenadditional mechanisms of pure dephasing (without re-laxation) not explicitly included in Eq. (4) dominate theshort time dynamics, the typical situation, e.g., in thesolid state. The net effect is that the qubit-oscillatorphase correlations accumulated during their mutual in-teraction are lost before a new qubit enters the channel.We account for additional dephasing by tracing over eachqubit after it crossedthe channel. This enables to addressthe problem for a very large number of channel uses, atleast numerically. We have  I  c ( E  N  ,ρ ) =  N k =1 I  ( k ) c  , where I  ( k ) c  ≡ I  c [ E  ( k )1  ,ρ 1 (0)] and the CPT map  E  ( k )1  depends on k  due to memory effects in the populations of the oscil-lator. Steady state –   In the above strongly dephased regimethe oscillator state  ρ ( k ) c  after  k  channel uses is diagonaland determined by the populations  { w ( k ) n  } . The buildup of the map that governs the populations dynamicsrequires the computation of the intermediate populations { ˜ w ( k ) n  } , obtained after the Jaynes-Cummings interactionof the  k -th qubit with the oscillator:  ˜ w ( k )0  =  w ( k − 1)0  [1 −  pS  21 ] + w ( k − 1)1  (1 −  p ) S  21 , ˜ w ( k ) n  =  w ( k − 1) n − 1  pS  2 n  + w ( k − 1) n  [(1 −  p ) C  2 n +  pC  2 n +1 ] + w ( k − 1) n +1  (1 −  p ) S  2 n +1 , n ≥ 1 , (9)where we have used the shorthand notation  S  n  =sin(Ω n τ   p ) and  C  n  = cos(Ω n τ   p ), with Ω n  =  λ √  n . Thenthe mapping from { ˜ w ( k ) n  }  to  { w ( k ) n  }  is obtained after an-alytically solving the master equation (4) for the popu-lations [26]. The overall mapping  { w ( k − 1) n  } → { w ( k ) n  }  isthen numerically iterated. As shown in Fig. 1 (top) asteady state distribution is eventually reached.Following Eq. (2) we compute  I  ( k ) c  =  S  ( ρ ( k )1  ) − S  ( ρ ( k )1 R  ),with  ρ ( k )1  and  ρ ( k )1 R  output states for the  k -th qubit and 1102030405060 k  0.20.30.40.5     I   c   (   k   ) τ =10 τ p , µ  ~ 0.81 τ =20 τ p , µ  ~ 0.68 τ=40 τ p , µ  ~ 0.52 τ =10 2 τ p , µ  ~ 0.30 τ =10 3 τ p , µ  ~ 0.04 012345 n 10 -4 10 -3 10 -2 10 -1 10 0   w   n 0204060 k  00.10.20.30.4    <  a    +   a   > FIG. 1: Top:   a † a   as a function of the number  k  of channeluses (left) and steady-state populations  w n  (right, numer-ically computed at  k  = 200). Bottom: coherent informa-tion  I  ( k ) c  as a function of   k . Parameter values:  η  = 0 . 8 (i.e., λτ  p  ≈ 0 . 46) and  λτ  d  = 20. for the  k -th qubit plus its reference system, respectively.State  ρ ( k )1  is obtained as in Eq. (6), but with initial stateof the oscillator  ρ ( k − 1) c  instead of the ground state  ρ c (0).We obtain ρ ( k )1  =  ∞ n =0 w ( k − 1) n ×  (1 −  p ) C  2 n  +  pS  2 n +1  rC  n C  n +1 r ⋆ C  n C  n +1  (1 −  p ) S  2 n  +  pC  2 n +1  . (10)State  ρ ( k )1 R  can be conveniently computed by choosing thepurification of   ρ 1 (0) as in Ref. [24] and again consid-ering the oscillator initially in the state  ρ ( k − 1) c  . Since ρ ( k ) c  reaches a steady state, the same must happen for I  ( k ) c  . This expectation is confirmed by the numerical datashown in Fig. 1 (bottom). The optimization of the reg-ularized coherent information (1) over separable inputstates is then simply obtained by maximizing the sta-tionary value of the coherent information over  ρ 1 (0). Theobtained  I  c -value provides a lower bound to the quantumcapacity of the channel. Transmission rates.–   The (numerical) optimization isachievedwhen r  = 0 (we havechecked it for severalvaluesof   η  and Γ) and  p  =  p opt  in Eq. (5). Note that  p opt  maystrongly depend on the time separation  τ   between con-secutive channel uses [see Fig. 2 (top right)], namely onthe degree of memory of the channel. As shown in Fig. 2(top left) the coherent information, optimized over sepa-rable input states, turns out to be a growing function of  τ  , that is, a decreasing function of the degree of memoryof the channel. The memoryless setting  τ   ≫  τ  d  mightappear to be the optimal choice. However, long waitingtimes  τ   ≫  τ  d  are required to reset the quantum chan-nel (cool the harmonic oscillator/ cavity) to its ground  4 020406080100 λτ 10 -2 10 -1     I   c    /   (        λ     τ       ) 020406080100 λτ 0.10.30.60.9    I   c 020406080100 λτ 0.10.20.30.40.5   p   o  p   t 0.20.40.60.8 µ 10 -2 10 -1     I   c    /   (        λ     τ       ) FIG. 2: Top left: steady state coherent information  I  c  (opti-mized over separable input states) as a function of the di-mensionless time separation  λτ   between consecutive chan-nel uses. Top right: optimal input state parameter  p opt  vs. λτ  . Bottom left: same data as in the top left panel, but forthe transmission rates  I  c / ( λτ  ). Bottom right: transmissionrates as a function of the memory parameter  µ . Parame-ter values:  η  = 0 . 95 ( λτ  p  ≈  0 . 22) (black curves),  η  = 0 . 7( λτ  p  ≈  0 . 58) (gray curves),  λτ  d  = 20. Dotted curves cor-respond to  p  =  p opt ( τ   → ∞ ), and show, in the case withlower performances of the channel ( η  = 0 . 7), the importanceof optimization. state after each channel use, thus reducing the transmis-sion rate. It appears preferable to consider the quantumtransmission rate  R Q  ≡  Q/τ  , defined as the maximumnumber of qubits that can be reliably transmitted perunit of time [5]. Fig. 2 (bottom) shows that in order to enhance  R Q  it is convenient to choose  τ   ≪  τ  d , namelymemory factors  µ  close to 1. In other words, by takinginto account memory effects, one can make more efficientthe use of the available transmitting resource.These results are relevant also for the  secure   trans-mission of classical information, then for cryptographicpurposes. The reference quantity is, for this case, the private classical capacity   C   p , defined as the capacity fortransmitting classical information protected against aneavesdropper [21]. It was recently shown [27] that for degradable channels, as it is the case of our model in thememoryless limit,  C   p  =  Q . Since the private classicalcapacity is always lower bounded by the coherent infor-mation [28], our results also show that the setup withoutresetting is convenient to increase the transmission rate R  p  ≡ C   p /τ   of private classical information. Discussion –   Eq. (3) models for instance dephasing in amicromaser emerging from fluctuations in the laser field.In the solid state scenario it may describe communica-tion by electrons or chiral quasiparticles [29] sent downa mesoscopic channel where they interact with opticalphonons. As an effective model Eq. (3) has a broadrange of applications since the unconventional environ-ment [17] describes the most relevant part of the inter-action with a bunch of phonon modes producing qubitradiative decay. In such solid-state systems the phonondephasing time  τ  φ  is expected to be much shorter thanthe phonon decay time scale  τ  d . In these cases we haveshown that a setup without memory resetting is con-venient in order to increase the rate of transmission of quantum information and private classical information.The noisy quantum channel Eq. (3) also describes thedynamics of a quantum memory [16], which may be im-plemented by coupling  N   superconducting qubits to amicrostrip cavity, in a circuit-QED [19] architecture. Inthis case, the use of cavities with moderate quality fac-tor [30] might be a good trade-off between reducing deco-herence and avoidingcross-talksgeneratingentanglementbetween the qubits crossing the channels. Our resultsshow that in such situation it is convenient to use thechannel without resetting to increase the rate of sequen-tial processing of each qubit. [1] M. A. Nielsen and I. L. 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