Quantum channels with a finite memory

0.16 MB·English

by  Garry Bowen

Quantum channels with a finite memory
Garry Bowen1,∗ and Stefano Mancini2,†
1Centre for Quantum Computation, Clarendon Laboratory,
University of Oxford, Oxford OX1 3PU, United Kingdom.
2Dipartimento di Fisica, Universit`a di Camerino, I-62032 Camerino, Italy.
(Dated: February 1, 2008)
Inthispaperwestudyquantumcommunicationchannelswithcorrelatednoiseeffects,i.e., quan-
tumchannelswithmemory. Wederiveamodelforcorrelatednoisechannelsthatincludesachannel
memory state. We examine the case where the memory is finite, and derive bounds on the clas-
sical and quantum capacities. For the entanglement-assisted and unassisted classical capacities it
is shown that these bounds are attainable for certain classes of channel. Also, we show that the
structure of any finite memory state is unimportant in the asymptotic limit, and specifically, for a
4 perfect finite-memorychannelwherenoinformation islosttotheenvironment,achievingtheupper
0 boundimplies that the channelis asymptotically noiseless.
0
2 PACSnumbers: 03.65.Ud,03.67.Hk,89.70.+c

n
a
J I. INTRODUCTION the model used to describe the correlated noise of the
5 channel, and is not necessarily a physical consideration.
1 Quantum communication through noisy channels has, Thecorrelationsbetweenerrorsmaybeconsideredeither
temporally overeachuse of a single channel, or spatially
to date, mainly concentrated on quantum channels that
2 between uses of many parallel channels. The dimension
are memoryless. A memoryless channel is defined as
v of the memory is determined by the number of Kraus
0 a noisy channel where the noise acts independently on
operators in the single channel expansion and the corre-
1 each symbol transmitted through the channel. In classi-
lationlengthofthechannel,whichmaybedefinedasthe
0 cal information theory the discrete memoryless channel
5 (DMC) is well understood in terms of the capacity of maximum number of channel uses for which the noise is
0 the channel, and the capacity of such channels is invari- notconditionallyindependent. Anychannelwithafinite
3 correlation length may be generated by a channel with
ant under inclusion of feedback or shared random cor-
0 a finite memory, according to this model. Although a
relations [1]. The existing capacity theorems for quan-
/
h tum channels have also concentrated on the memoryless physical interpretation of the model is not necessary to
p achievethegoalofdeterminingcapacitytheorems,itmay
case [2, 3, 4, 5]. The calculation of capacities for classi-
- give an understanding of the physical motivation. Over
t cal channels with correlated noise, or memory channels,
n short times the environment with which the transmitted
has had much more limited success. One type of classi-
a stateactsmaybeassumedtobearbitrarilylarge,within-
cal memory channel for which the capacity is known is
u
teractions between components of the vast environment
q the channel with Markov correlated noise. In this pa-
essentially making the recovery of the information im-
: per, we examine the quantum extension of the channel
v possible. The memory, however, may be interpreted as
with Markov correlated noise. In particular, we exam-
i a subspace of the environment which does not “decay”
X ine a model of a correlated noise channel which utilizes
overthe timescale of separate uses of the channel, andis
r unitary operations between the transmitted states, an
a environment,andasharedmemorystate,anddetermine thereforedependentonthepreviousstateofthechannel.
some of the characteristics of such a class of quantum
channels.
The fact that entangled alphabet states may increase
the classical capacity of a particular correlated noise
channelhasbeenshown[6]. Forthecorrespondingmem-
orylesschannelthecapacityisknowntobeadditive,and
hence entangled input states cannot increase the classi- A physical example of a memory channel is the recent
calcapacityofthememorylesschanneloverproductstate proposalbyBose[7],whichusesunmodulatedspinchains
encoding. totransmitquantuminformation. Inthiscasethe initial
The phrase finite memory is used to describe one as- state introduced to the chain by Alice acts as both the
pect of the model, a memory state of finite dimension. input state and a part of the memory state for further
The finiteness of the memory is defined only in terms of uses of the channel, as it is assumed Alice replaces each
transmitted state with a new spin state after each use
ofthe channel,whilstthe remainingelements ofthe spin
chain constitute both the physical channel, the memory
∗Electronicaddress: [email protected] of the channel, and the output state, which may be re-
†Electronicaddress: [email protected] moved from the chain for future decoding by Bob.
2
II. A MODEL FOR QUANTUM MEMORY whereρ andρ′ aretheinputandtheoutputstateofthe
Q Q
CHANNELS channel while the trace over the environment is over all
environment states. If the unitaries factor into indepen-
TheKrausrepresentationtheorem[8]isanelegantand dent unitaries acting on the memory and the combined
powerful method of representing quantum dynamics in state and environment, that is, Un,MEn = Un,EnUM,
twodifferentways,asasumoveroperatorsactingonthe then the memory traces out and we have a memoryless
state, or alternatively as a unitary evolution of a state channel. If the unitaries reduce to Un,M, we can callit a
and environment. The unitary interaction model pro- perfect memory channel, as no information is lost to the
vides an intuitive understanding of open quantum sys- environment.
tems, as well as providing a method of calculation. In Thequestionremainsastowhatchannelscanbemod-
deriving our model of a quantum memory channel, we eledbyEq. (4)? FromtheKrausrepresentationtheorem
try to preserve the useful aspects that such a unitary [8], we know that for any block of length n, then any
representation provides. channel acting on the n states may be modeled with a
memory plus environmentof dimension at most d2n, for
dthedimensionofthechannel. However,theunitaryop-
A. Unitary Representation of Memoryless
erationmaynotbefactorableintoaproductofoperators
Channels
acting in the form of Eq. (4).
Aquantumchannelisdefinedasacompletelypositive,
tracepreservingmapfromthesetofdensityoperatorsto
itself. Any such map may be represented as a unitary
operation between the system state and an environment C. Examples of Finite-Memory Channels
with a known initial state. For a single channel use the
output state is given by,
A naive “memory” channel can be considered by the
ρ′ =Tr U ρ 0 0 U† (1) two qubit channel given by the Kraus operators, A0 =
Q Eh QE(cid:0) Q⊗| Eih E|(cid:1) QEi 21(I⊗I), and, A1 = 21(σZ ⊗σZ), and can be modeled by
using a memory state that is initially in the 0 0 state,
with ρ the input state, and ρ′ the output state. For a | ih |
Q Q andisthetargetofaCNOToperationbyonlytwoqubits
sequence of transmissions through the channel,
beforebeingresettotheinitialstate. However,thischan-
nelis essentiallyjust a memorylesschannelinthe higher
ρ′ =Tr U ...U ρ 0 ...0 0 ...0
Q Eh n,En 1,E1(cid:0) Q⊗| E1 Enih E1 En|(cid:1) danimdecnasnionnotspthaecree,ftorraenbsmeictotninsgidqeruedditussoeffudlimasenasmioondfeoluorf,
U† ...U† (2)
× 1,E1 n,Eni a memory channel. This channel also does not fit into
the model of Eq. (4) as the memory is erased separately
= Λ ... Λ ρ (3)
n 1 Q
⊗ ⊗ after every two qubits. All such channels which may be
(cid:0) (cid:1)
wherethestateρ nowrepresentsa(possiblyentangled) factored into memoryless channels for some finite num-
Q
input state across the n channel uses, and the environ- ber ofuses maythereforebe describedusing the existing
ment state is a product state 0 ...0 = 0 ... properties known for memoryless channels.
| E1 Eni | E1i⊗ ⊗
|0Eni. Asimpleexampleofaperfectmemorychannelisanex-
tension of the qubit dephasing channel. For this channel
CNOT gates operate between the qubits going through
B. A Unitary Model for Memory Channels
the channel and a target memory state, initially given
as 0 0 , which replaces the environment. The out-
M M
| ih |
One model of a quantum memory channel is where put states have the same reduced density matrices as if
eachstate goingthroughthe channelacts witha unitary they had passed through a memoryless dephasing chan-
interaction on the same channel memory state, as well nel,butthestatesarealsocorrelatedacrosschanneluses,
as an independent environment. The backaction of the that is, a product input state does not necessarily give a
channelstateonthemessagestatethereforegivesamem- productoutputstate. Wecallthischannelthecorrelated
ory to the channel. The general model thus includes a dephasing channel.
channel memory M, and the independent environments
UsingtheunitarySWAPgatetomodelachannelsim-
for each qubit E . Hence,
i ply acts as a shift by a single state. Since for this “shift
channel” the SWAP gate the unitaries act to increment
ρ′ =Tr U ...U ρ M M
Q MEh n,MEn 1,ME1(cid:0) Q⊗| ih | tbhloecikndoefxnfoinrptuhtespoonsliytitohneolfasttheinipnuptutstsattaeteiss,ntohtenreocnova-
0 ...0 0 ...0 U† ...U†
⊗| E1 Enih E1 En| 1,ME1 n,MEni erable. Hence the transmission rate for intact states for
(cid:1)
blocks of size n is simply 1 1/n, which approaches a
=TrMhΛn,M...Λ1,M(cid:0)ρQ⊗|MihM|(cid:1)i (4) noiseless channel in the limit−n→∞.
3
D. Channels with Markovian Correlated Noise [p ,p ,...,p ], for a family of m operators, by the re-
0 1 m
lationship,
An important class of channels that may be repre-
sented by the memory channel model are channels with α~ =Γ−1~p (9)
Markovian correlated noise. A Markovian correlated
noise channel of length n, is of the form,
for Γ the transition matrix with entries p , and α~ the
j|i
Λ(n)ρ= p p ...p p squares of the amplitudes for the initial memory state
in|in−1 in−1|in−2 i1|i0 i0
i0X,...,in |Mi = j√αj|jMi. This is, of course, provided that
A ... A ρ A† ... A† (5) the tranPsition matrix is not singular. For a singular ma-
× in ⊗ ⊗ i0 in ⊗ ⊗ i0 trix we may utilize a different unitary operation V on
(cid:0) (cid:1) (cid:0) (cid:1) the initial use of the channel, which will not change the
where the set A are Kraus operators for single uses of
ik asymptotic behavior of the channel. We may also utilize
the channel on state k [6]. The motivation for looking
a mixed initial memory state ρ = α j j , in-
at Markovian channels is that the properties of typical M j j| Mih M|
steadofthe pure state, without affectPing the behaviorof
sequencesgeneratedfromMarkoviansourcesarewellun-
the channel.
derstood,andthetypicalsequencesoferrorsgeneratedin
Eq. (5)willbedirectlyrelatedtothesetypicalsequences. The derivation of the specific model for the correlated
dephasing channel in Eqs. (6) and (7) differs from the
Thecorrelateddephasingchannelmaybedescribedus-
ing the memoryto correlatethe dephasingerrorforeach prescription given in Eq. (8), in that it does not re-
quire the extra environment. The unitary operation on
qubit, that is, the probability of the kth qubit under-
going a phase error is determined exactly by whether an the initial states produces orthogonal outputs, whereas
in the general case the states for each j in Eq. (8)
erroroccurredonthepreviousqubit. Thus,forthecorre- M
| i
lated dephasing channel with error operators A =I(n) may not necessarily be orthogonal without the environ-
0n
andA1n =σZ(n), acting onthe nthqubit, the conditional omretnhto.goInfatlhteooaulltpouthtesrtoautetpfuortsagegniveernat|ejdMbiyidniffEeqr.en(t8i)niis-
probabilities are given by p = δ , with an initial
kn|jn−1 jk tial memory states, then the final environment state for
probabilityoferrorgivenbyp =p =1/2. Thischannel
0 1 this particular output can “overlap” and does not need
may be generated using the unitary operation,
to be orthogonal to the other environment states. This
U φ(i) 0 = φ(i) 0 (6) occurs in the correlated dephasing channel, and results
i,M M M
| i| i | i| i inthe channelrequiringnoenvironmentatall. However,
Ui,M|φ(i)i|1Mi=σZ(i)|φ(i)i|1Mi (7) itshallbe shownthat the behaviorofthese twodifferent
channel constructions is identical, as the actual size of
withaninitialmemorystate M =1/√2(0 + 1 ). The the memory becomes irrelevant in the asymptotic limit,
| i | i | i
equivalence of the controlled phase gate in Eqs. (6) and provided it is finite.
(7) to the use of a CNOT with a memory initially in
The noisy channel described by Macchiavello and
the 0 0 state is obtained by noting U =
M M CPHASE Palma [6] may be described in the context of Eq. (8),
(I |H)UihCNO|T(I H), for H a qubit Hadamard rota- withthe erroroperatorsgivenby the identityA =I(n)
⊗ ⊗ 0n
tiononthememorystate. Thechannelisasymptotically (n) (n)
andthe PaulispinmatricesA =σ ,A =σ ,and
noiseless, as all states with an even number of 1 ’s are 1n X 2n Y
invariant, and therefore this subspace may be |miapped A3n = σZ(n), and the transition matrix elements defined
as p = (1 µ)p +µδ , where µ is a correlation pa-
onto by simply adding a single ancilla qubit. The en- k|j k jk
−
coding map from ⊗n to ⊗(n+1), may then transform rameter. Thesteadystateprobabilitiesforthistransition
states with even nHumbers Hof 1 ’s to the same state ten- matrix are given by the uniform distribution.
| i
soredwith 0 ,andthosestateswithoddnumbersof 1 ’s
| i | i
to these states tensored with 1 . The coded subspace
of ⊗(n+1) is then noiseless. |Tihe rate of transmission
H
through n+1 uses of the channel is therefore n/(n+1),
III. CAPACITIES FOR FINITE-MEMORY
which approaches unity in the limit n .
→∞ CHANNELS
ForageneralchannelwithMarkoviancorrelatednoise,
that is p = p for all i < j, the channel
j|j−1 j|(j−1)(j−2)...i
maybegeneratedusingthemodelgiveninEq. (4),where Having generated a model for the channel, we must
the unitary operator is given by, address whether such a model is instructive in obtain-
ing capacity theorems for the channels the model rep-
Ui,MEi|φ(i)i|jMi|0Eii= √pk|jA(ki)|φ(i)i|kMi|jEii. resents. The existence of a unitary representation of an
Xk interactionwithanenvironmentdoesallowtheextension
(8) ofresultsfrommemorylesschannelswhichrelyonsimilar
The initial memory state determined by the values of arguments, such as the coherent information bound and
the initial probability vector for the error operators the quantum Fano inequality [9].
4
A. Results for Classical Capacities with A = A A , and ρ a possibly entan-
in,in−1 in ⊗ in−1
gled input state across the two transmissions through
Anupperboundontheclassicalinformationthatmay the channel. This construction may be extended for ar-
be sent through the memory channel is given by the bitrary lengths n. In the case that the initial distribu-
Holevo bound [10]. The maximum mutual information tionpi0 isequaltothe steadystatedistributionpi0 =p˜i,
generated between sender and receiver, per channel use, the approximations in Eqs. (11) and (12) become exact.
for n channels is then given by, This is true for all lengths n, with diagρM = diagρ˜M
always, where diagρ is the density matrix formed from
1 the diagonalelements ofρ. Thereforethe achievable rate
S(n) = max S p Tr Λ(n) ρi ρ
max {pi,ρi}n(cid:20) (cid:16)Xi i Mh M (cid:0) Q⊗ M(cid:1)i(cid:17) iWseosbttmaionreedlanimdm(HedSiWat)eltyhefororemmt[h3e, 4H].olevo–Schumacher–
p S Tr Λ(n) ρi ρ (10)
−Xi i (cid:16) Mh M (cid:0) Q⊗ M(cid:1)i(cid:17)(cid:21)
where for each n, Λ(n) = Λ ...Λ , is a channel,
M n,M 1,M The correlateddephasing channel gives an easy exam-
and the asymptotic limit is achieved by taking n .
The ensemble of states ρi = ρi is a set of states→ge∞n- pleoftheachievablilityofthecapacity,asforthischannel
Q A any initial distribution is a steady state probability. A
erated by the sender, Alice, for unassisted communica-
rateequaltothe unassistedclassicalcapacityisachieved
tion, or ρi = ρi is a set of shared entangled states
Q AB using the orthogonal states 0 , 1 with a priori prob-
between sender, Alice, and receiver, Bob, for entangle- {| i | i}
ability of p = 1/2 for this channel, and hence the limit
mentassistedcommunication,withtherequirementthat i
is achieved in this case when n = 1. The entanglement
ρi =ρ . To reduce the number ofsubscripts, the use of
B B assisted capacity for this channel C = 2 is, however,
the notation ρi ρi for the signal states shall be used E
≡ Q only achieved in the asymptotic limit as the block size
for the rest of this section.
n .
Theargumentforachievingthisupperbounddoesnot →∞
extend easily to the memory channel case. The problem
lies in the fact that the coding for the channelcannotbe
brokenupintoblocksofnuses,asthememorystatemay
be entangled with the previous block and thus may not
In the case that the initial error probabilities differ
be identical for each block.
from the steady state, much of the derivation above is
The bound in Eq. (10) is achievable for a class of reg-
still applicable. From the convergence properties of reg-
ular Markovian correlated noise channels. The channels
ular Markovian sequences, we know that diagρ ρ˜
M M
are assumed to be representable by unitary Kraus op- →
as n becomes large, where ρ˜ is the diagonal density
M
erators (and are therefore unital), and have initial error
matrix with eigenvalues equivalent to the steady state
probability distributions equal to the steady state prob-
probabilities. Similarly, for any δ > 0 there exists an n
abilities. The asymptotic use of the channel may be seg-
for which the total probability of the atypical sequences
mented into approximate channels of length n. That is,
of Kraus operators is less than δ. This follows from the
bytracingoutallotherstatesforeachlengthnsegment,
behavior of regular Markovian sources in the Shannon
whaevoebΛt(ali)n aΛc(hna)nne..l.whΛe(rne).foFrroamtotthaeltlheneogrtyholf≫Manrkwove theory [1]. The contributionto the state Λ(n)ρ when the
≈ ⊗ ⊗ initialprobabilities are notthe steady state probabilities
chains,weknowthattheapproximatechannelforaprod-
may therefore be small enough such that the bounds on
uct state input is given for a single use by,
the total probability of error may be made arbitrarily
small asymptotically, although at present this remains
Λ(1)ρ p˜ A ρA†
≈ in in in an open question.
Xin
=Tr Λ(1) ρ ρ˜ (11)
M M ⊗ M
(cid:0) (cid:1)
where p˜ = p˜ are the steady state probabilities, ρ˜ is
in i M
the memory density matrix with the p˜ on the diagonal, Foranychannelwithafinitememorywherethecapac-
i
and n is taken to be suitably large. The derivations re- ityequalstheupperbounditmaybeseenthatthe exact
quiredforthisapproximationareshownintheAppendix. nature of the memory has little effect on the asymptotic
Similarly, with n large,two uses the channelare approx- behaviorof the channel. The correlateddephasing chan-
imated by, nel, where two possible constructions exist each with a
different sized memory state, is an example. To analyze
Λ(2)ρ p p˜ A ρA† thebehaviorweassumethatBobhasaccesstothemem-
≈inX,in−1 in|in−1 in−1 in,in−1 in,in−1 ory after the block is sent, and as such he can measure
the information in M as well, then reset the memory to
=Tr Λ(2) ρ ρ˜ (12)
M M ⊗ M a given initial state before the next block. This gives an
(cid:0) (cid:1)
5
achievable rate, the use of quantum dense coding and quantum telepor-
tation, giving the equality C = 2Q [14]. The actual
E E
1
R lim max S p Λ(n) ρi ρ nature of the channel does not affect this relationship.
i M
≡n→∞{pi,ρi}n(cid:20) (cid:16)Xi (cid:0) ⊗ (cid:1)(cid:17)
piS Λ(n) ρi ρM (13) C. The Quantum Fano Inequality
−Xi (cid:16) (cid:0) ⊗ (cid:1)(cid:17)(cid:21)
1
lim max S p Tr Λ(n) ρi ρ ThequantumFanoinequality[9]isusedto giveacon-
i M M
≤n→∞{pi,ρi}n(cid:20) (cid:16)Xi h (cid:0) ⊗ (cid:1)i(cid:17) verse to any quantum coding theorems. The inequality
describesthelossinfidelityofthetransmittedstatesthat
−Xi piS(cid:16)TrMhΛ(n)(cid:0)ρi⊗ρM(cid:1)i(cid:17)+2log2dM(cid:21) odcucruinrgsdtruaentsomtihsesieoxncthharnoguegohftehnetrcohpayntnoelt.heenvironment
(14) Takingastateρ witha purificationintermsofa ref-
Q
2 erence system R, such that, ρ = Tr ψ ψ , we
=Sm(na)x+ nlog2dM (15) define the entanglementfidelityQasF =Rh|ψQQRR|iρh′QRQ|Rψ|QRi,
where ρ′ is the total output state following the trans-
where Eq. (14) follows from Eq. (13) by strong subad- QR
mission of ρ through the noisy channel. The quantum
ditivity and the factor 2log d is an upper bound on Q
2 M Fano inequality may be applied to the finite memory
entropy of the memory state living in a space of dimen-
channel by simply noting that the entropy exchange to
siond . TheboundfortherateRofachannelgenerated
M the environment E may be rewritten as,
from tracing both the environment and the memory, is
nthSe(nn)sa,nwdhwiicchhewdobuyldthaeptperromasc,hntShm(ena)xc+ha2nlnoegl2cdaMpa≥citnyRfo≥r S(ρ′E)=S(ρ′MQR)≤S(ρ′M)+S(ρ′QR). (16)
max
the channel including access to the memory, as n , It is assumed here that the memory state is initially
→ ∞
for any finite memory channel. The channel capacity is pure, as it does not affect the derivation compared to
thus only affected by the loss of information to the en- a mixed memory state. This is because any finite mem-
vironment, and the loss of information into the memory ory state may be purified with another finite reference
state may be seen to vanish in the asymptotic limit. For system. This is also equivalent to applying the Fano
aperfectmemorychannelthechannelwillbeasymptoti- inequality using S(ρ′ ) as the environment, and then
ME
callynoiseless,aswasshownfortheexamplesoftheshift utilizing the Araki–Lieb inequality to obtain S(ρ′ )
QR ≥
channel and the correlateddephasing channel. S(ρ′ ) S(ρ′ ).
E − M
This leads to a Fano inequality for channels with a
finite memory,
B. Results for Quantum Capacities
S log d +h(F)+(1 F)log (d2 1) (17)
E ≤ 2 M − 2 −
The quantum capacities are determined by the max-
imum asymptotic rates at which intact quantum states forS theentropyexchangewiththeenvironment,F the
E
may be transmitted through a noisy quantum channel. entanglementfidelity,h(F)= F log F (1 F)log (1
− 2 − − 2 −
The coherent information bound [9, 11] on the quantum F) the binary entropyof the entanglementfidelity, d the
capacityappliesdirectlytothecaseofmemorychannels. dimension of , and d the dimension of the memory.
Q M
H
The role the memory plays in the coherent information For a single channel use, this inequality may be weak,
bound may be seen by examining the converse to the butinthecaseofmultipleusestheinequalitycanbecome
bound, the quantum Fano inequality, which is shown in stronger. Thisisduetotheaverageentropyexchangefor
the next section. a large number of channel uses N being given by,
Thereexistanumberofquantumcapacitiesdependent
on available additional resources. Primarily there is the 1 S 1 log d +h(F)+(1 F)log (d2N 1)
unassisted quantum capacity Q, the capacities assisted N E ≤ N 2 M − 2 −
(cid:2) (cid:3)
byclassicalsidechannelsQ ,QFB,Q ,denotingforward, 2(1 F)log d (18)
1 2 ≈ − 2
backward (feedback), and two way classical communica-
tion respectively, and, the entanglement assisted quan- where the first two terms in the sum on the right hand
tum capacity Q , achievable when sender and receiver side may be made arbitrarily small, given large enough
E
share unlimited amounts of entanglement prior to com- N. Thismaybeinterpretedasthefactthatahighentan-
munication taking place. For memoryless channels the glementfidelityovermanyusesofthechannelnecessarily
situationisslightlysimplifiedbytheequivalenceQ =Q impliesalowaverageentropyexchangewiththeenviron-
1
[12, 13], whether this holds for channels with memory is ment. Intheasymptoticlimittheparticularchannelcon-
not yet known. structionused,andtheexactnatureofanyfinitememory
Theentanglementassistedquantumcapacityissimply state,areboth“irrelevant”intermsoftheboundsonthe
relatedtotheentanglementassistedclassicalcapacityby channel capacity.
6
IV. CONCLUSION The error operators are determined by the diagonal el-
ements of the memory only, the off-diagonal matrix ele-
A model for a class of quantum channels with mem- ments have no effect on the channel.
ory has been proposed. The class of channels that may 2. Convergence of the Diagonal Elements of the
Memory State
be described by this model include the set of channels
with Markovian correlated quantum noise. For these
types of channels it has been shown that the memory
The exact nature of the memory state itself depends
state required to generate the channel is finite. These fi-
on the states transmitted through the channel. Per-
nite memory channels have similar asymptotic behavior
haps surprisingly, however, the diagonal elements of the
to the quantum memoryless channels, in that they may
state are independent of the transmitted states. To
be essentially described by the loss of information to an
show this we note for a memory initially in the state
initial product state environment after a unitary inter-
ρ = λ j l , the new memory state after one
action with the states transmitted through the channel. M jl jl| MihM|
iteratioPn of the channel is,
The size of the memory state is finite, and so the effect
on loss from the channel is vanishing in the asymptotic
limit. The simplest demonstration is the case of perfect
λ Tr U φ j 0 0 l φ U†
memory channels where no information at all is lost to jl QE QME| Qi| Mi| Eih E|hM|h Q| QME
the environmentandso achievementofthe upper bound Xjl (cid:2) (cid:3)
ocanlltyhegicvaepaacniotyisefolerssthqisuacnlatsusmofchchaannnnele.ls will asymptoti- = λjlTrQ(cid:20) √pk|jpm|lδjlAk|φQi|kMihmM|hφQ|A†m(cid:21)
Xjl kXmn
It has also been demonstrated that Holevo–
Schumacher–Westmoreland coding can achieve the = λjlδjl √pk|jpm|lhn|Ak|φi|kMihmM|hφ|A†m|ni
capacity bound for channels with Markov correlated Xjl kXmn
nthoeisei,niwtihaelreertrhoer Kprroabuasboiplietireastoarrseareequuanlittaory,thperosvtiedaidnyg = λjj √pk|jpm|jhφ|A†mAk|φi|kMihmM|
Xj Xkm
state probabilities for the regular Markov chain.
The unitary representation of the channel also allows = λ p φA†A φ k k
for derivations ofbounds onthe quantumcapacity using jj(cid:20) k|jh | k k| i| Mih M|
Xj Xk
thecoherentinformation,andapplicationofthequantum
Fano inequality to finite memory channels. + √pk|jpm|jhφ|A†mAk|φi|kMihmM|(cid:21)
kX6=m
APPENDIX: EVOLUTION OF THE CHANNEL
AND MEMORY STATE ifm=k then the unitariesgiveA† A =I, thereforethe
m k
diagonalelementsofρ undergotheprocessofaMarkov
M
1. Derivation of the Channel from the Unitary chain. The off-diagonalelements do not necessarily van-
Construction ish, but they do not affect the error operators acting on
the transmitted states. Only the diagonal elements of
Here it is shown that the diagonal elements of the the memory state affect the behavior of the channel.
memory state determine the error operators for the
next transmitted state. For the memory state ρ =
M
λ j l , the channel for the next transmitted
jl jl| MihM|
Pstate is given by,
λ Tr U φ j 0 0 l φ U† ACKNOWLEDGMENTS
jl ME QME| Qi| Mi| Eih E|hM|h Q| QME
Xjl (cid:2) (cid:3)
= λjlTrM(cid:20) √pk|jpm|lδjlAk|φQi|kMihmM|hφQ|A†m(cid:21) byWthheileOtxhfoisrdw-AorukstwraalsiauTndruerstta,ktehne,HGaBrmwsawsorstuhppTorrutsetd,
Xjl Xkm
and Universities UK.
= λjlδjl √pk|jpm|lAk|φihn|kMihmM|nihφ|A†m
Xjl kXmn
= λ p A φ φ A†
jj k|j k| Qih Q| k
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