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, CNRINFM & 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 CNRINFM, 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 singlemode 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 JaynesCummings 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 classical 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 determination 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 traduce themselves in memory eﬀects, leading to relevantcorrelations in the errors aﬀecting successive transmissions. Important examples in this context are photonstraveling across ﬁbers with birefringence ﬂuctuating withcharacteristic time scales longer than the separation between consecutive light pulses [3] or lowfrequency impurity noise in solid state implementations of quantumhardware [4]. Memory eﬀects become unavoidably relevant 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 dephasing channels [11, 12, 13], lossy bosonic channels [14]; also
spin chains have been studied as models for the channel itself [15]. Hamiltonian models of memory channels [11, 12, 13, 15] aim at a description directly referring
to physical systems and enlight another important example 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 resetting the oscillator to its ground state. The model isvisualized by a qubitmicromaser [18] system, the qubittrain being a stream of twolevel Rydberg atoms injectedat low rate into the cavity. Unconventional environmentscapture essentials features of solid state circuitquantumelectrodynamics (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 interact with the photon ﬁeld inside the cavity and memory 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 probability
{
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 described 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 entropy,
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
, described 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 superoperator
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 account the channel does not act on each carrier independently,
E
N
=
E
⊗
N
1
, and Eq. (1) in general only providesan upper bound on the channel capacity. However, forthe socalled
forgetful channels
[6], for which memory effects decay exponentially with time, a quantum codingtheorem exists showing that the upper bound can be saturated [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 qubitsoscillator interaction
V
is of the JaynesCummings kind, and we take
λ
real and positive. Coupling 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 environmentreservoirinteraction and causes damping 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 nonunitary eﬀectsin the evolution of the system and the oscillator can beignored during the crossing time
τ
p
. Between two successive pulses the oscillator evolves according to the standard 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 introduce 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 builtin reset for the oscillator to its ground state
ρ
c
(0) =

0
0

after each channel use. We consider ageneric singlequbit 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 qubitoscillator 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 timeevolution operator determinedby the undamped JaynesCummings 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 amplitudedamping 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 information 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 appears in our model when
τ
is ﬁnite. To show that theregularized coherent information still represents the truequantum capacity we follow the arguments made for forgetful channels in Ref. [6]. The key point is the use of adoubleblocking strategy mapping, with a negligible error, 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 corresponding 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 channel (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, interaction with the oscillator entangles initially independentqubits. Nevertheless, a lower bound to the quantum capacity can be computed if
τ
φ
≪
τ
. This happens whenadditional mechanisms of pure dephasing (without relaxation) not explicitly included in Eq. (4) dominate theshort time dynamics, the typical situation, e.g., in thesolid state. The net eﬀect is that the qubitoscillatorphase correlations accumulated during their mutual interaction 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 oscillator.
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 JaynesCummings 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 analytically solving the master equation (4) for the populations [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 steadystate populations
w
n
(right, numerically computed at
k
= 200). Bottom: coherent information
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 considering 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 regularized coherent information (1) over separable inputstates is then simply obtained by maximizing the stationary 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 consecutive 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 separable 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 channel (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
(optimized over separable input states) as a function of the dimensionless time separation
λτ
between consecutive channel 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
. Parameter values:
η
= 0
.
95 (
λτ
p
≈
0
.
22) (black curves),
η
= 0
.
7(
λτ
p
≈
0
.
58) (gray curves),
λτ
d
= 20. Dotted curves correspond 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 transmission 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
transmission 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 information [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 communication 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 environment [17] describes the most relevant part of the interaction with a bunch of phonon modes producing qubitradiative decay. In such solidstate 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 convenient 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 implemented by coupling
N
superconducting qubits to amicrostrip cavity, in a circuitQED [19] architecture. Inthis case, the use of cavities with moderate quality factor [30] might be a good tradeoﬀ between reducing decoherence and avoidingcrosstalksgeneratingentanglementbetween the qubits crossing the channels. Our resultsshow that in such situation it is convenient to use thechannel without resetting to increase the rate of sequential processing of each qubit.
[1] M. A. Nielsen and I. L. Chuang,
Quantum computation and quantum information
(Cambridge University Press,Cambridge, 2000).[2] G. Benenti
et al.
,
Principles of quantum computation and information
, vol. II (World Scientiﬁc, Singapore, 2007).[3] K. Banaszek
et al.
, Phys. Rev. Lett.
92
, 257901 (2004).[4] Y. Makhlin
et al.
, Rev. Mod. Phys.
73
, 357 (2001); E.Paladino
et al.
, Phys. Rev. Lett.
88
, 228304 (2002); G.Falci
et al.
, Phys. Rev. Lett.
94
, 167002 (2005); G. Ithier
et al.
, Phys. Rev. B
72
, 134519 (2005).[5] V. Giovannetti, J. Phys. A
38
, 10989 (2005).[6] D. Kretschmann and R. F. Werner, Phys. Rev. A
72
,062323 (2005).[7] N. Datta and T. C. Dorlas, J. Phys. A
40
, 8147 (2007).[8] C. Macchiavello and G. M. Palma, Phys. Rev. A
65
,050301(R) (2002).[9] G. Bowen and S. Mancini, Phys. Rev. A
69
, 012306(2004).[10] H. Hamada, J. Math. Phys.
43
4382 (2002)[11] A. D’Arrigo
et al.
, New J. Phys.
9
, 310 (2007).[12] M. B. Plenio and S. Virmani, Phys. Rev. Lett.
99
, 120504(2007); New J. Phys.
10
, 043032 (2008).[13] D. Rossini
et al.
New J. Phys.
10
115009 (2008)[14] O. V. Pilyavets
et al.
, Phys. Rev. A
77
, 052324 (2008).[15] A. Bayat
et al.
, Phys. Rev. A
77
, 050306(R) (2008).[16] H. Barnum
et al.
, Phys. Rev. A
57
, 4153 (1998).[17]
Focus on Quantum Dissipation in Unconventional Environments
, M. Grifoni and E. Paladino Eds., New J. Phys.
10
(2008); F. Cavaliere et al., New J. Phys.
10
, 115004(2008); A. Garg, Jour. Chem. Phys.
83
, 4491 (1985); F.Plastina and G. Falci, Phys. Rev.
B 67
, 224514 (1993).[18] P. Meystre and M. Sargent III,
Elements of quantum optics
(4th Ed.) (Springer–Verlag, Berlin, 2007).[19] A. Wallraﬀ et al., Nature
431
, 162 (2004); J.M. Fink etal., Nature
454
, 315 (2008).[20] S. Lloyd, Phys. Rev. A
55
, 1613 (1997).[21] I. Devetak, IEEE Trans. Inf. Theory
51
, 44 (2005).
5
[22] I. Devetak and P.W. Shor, Comm. Math. Phys.
256
, 287(2005).[23] X.y. Chen, preprint arXiv:0802.2327 [quantph].[24] V. Giovannetti and R. Fazio, Phys. Rev. A
71
, 032314(2005).[25] A. D’Arrigo
et al.
, in preparation.[26] G. G. Carlo
et al.
, Phys. Rev. A
69
, 062317 (2004).[27] G. Smith, Phys. Rev. A
78
, 022306 (2008).[28] B. W. Schumacher and M. D. Westmoreland, Phys. Rev.Lett.
80
, 5695 (1998).[29] A. Mitra, I. Aleiner, and A.J. Millis, Phys. Rev. B 69,245302 (2004); A. Naik
et al.
, Nature
443
, 193 (2006);K.S. Novoselov
et al.
, Nature
438
, 197 (2005).[30] A. A. Houck
et al.
, Nature
449
, 328 (2007).