Semihierarchical quantum repeaters based on moderate lifetime quantum memories

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by  Xiao Liu

Semihierarchical quantum repeaters based on moderate lifetime quantum memories
Xiao Liu,1,2 Zong-Quan Zhou∗,1,2 Yi-Lin Hua,1,2 Chuan-Feng Li†,1,2 and Guang-Can Guo1,2
1CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China
2Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, 230026, China
(Dated: January 23, 2017)
Theconstructionoflarge-scalequantumnetworksreliesonthedevelopmentofpracticalquantum
repeaters. Manyapproacheshavebeenproposedwiththegoalofoutperformingthedirecttransmis-
sion of photons, but most of them are inefficient or difficult to implement with current technology.
Here, we present a protocol that uses a semi-hierarchical structure to improve the entanglement
distributionratewhile reducingtherequirementof memorytimetoarange oftensofmilliseconds.
7
Thisprotocol can beimplemented with a fixeddistance ofelementary linksand fixedrequirements
1
0 onquantummemories, whichareindependentofthetotaldistance. Thisconfigurationisespecially
2 suitable for scalable applications in large-scale quantumnetworks.

n
PACSnumbers: 03.65.Ud,03.67.Hk,03.67.Mn,42.50.-p
a
J

0 I. INTRODUCTION ated earlier will be stored until the neighboring link is
2 ready. In contrast, the EC and ES will be performed

at the same time for all elementary links in the non-
] The distribution of entangled photon pairs over long
h hierarchical quantum repeater. A highly influential pro-
distances plays an essential role in quantum information
p posalby Duan, Lukin, Cirac,andZoller,which is widely
science. Itisthefundamentalrequirementfortherealiza-
- knownasthe DLCZ protocol[9],is atypicalhierarchical
t tionoflong-distancequantumcommunication[1]andfor
n structure. Various improvements have been put forward
theconstructionofopticalquantumnetworks[2]. Entan-
a overthe pastfew yearsto make this protocolmoreprac-
u glement over long distances could also enable large-scale
ticaltoimplement,suchasloweringthestabilityrequire-
q tests of quantum physics [3]. A simple approach is to
ments for the channels via two-photon detections in the
[ createentangledphotonpairslocallyandsendoneofthe
EC and ES stages [10–16], reducing multiphoton errors
photons to the distant location through an optical fiber.
1 throughtheuseofsingle-photonsources[13,16,17],and
v However, the entanglement distribution rate (EDR) de-
increasing the distribution rate at the elementary level
8 creases exponentially with increasing length of optical
by using temporal or spatial multiplexing [18, 19]. How-
1 links because of inevitable photon loss in the fiber. For
7 example, the rate of photons drops ten orders of mag- ever, there is still an intrinsic problem with its structure
5 that limits its applications. An essential requirement for
nitude for 500-kmtransmissionthrough an optical fiber.
0 DLCZ-like protocols is that one must be able to store
Inclassicaltelecommunicationsthisproblemisovercome
. the createdelementary entanglementuntil entanglement
1 through the use of amplifiers. Unfortunately, because
0 of the no-cloning theorem [4], straightforwardamplifica- has been established in the neighboring link, the result-
7 ing higher-levelentanglementagainmust be stored until
tionisnotapplicableinquantumcommunication. Inthis
1 the neighboring higher-level link has been established,
case,quantumrepeatersarerequired,whichcanimprove
: and so on. Thus, the necessary storage time of quantum
v thescalingofphotonlossfromexponentialtopolynomial
i memoriesmust be comparableto the totalentanglement
X in principle [5, 6].
distribution time, which is highly challenging for practi-
A core idea of many quantum repeater schemes is
r cal quantum memories, especially when the distribution
a to create short-distance entanglement between optical
distance is rather long.
memories and then extend the distance of entanglement
Sinclair et al. proposed a non-hierarchical structure
throughentanglementswapping[5, 7]. The sequence be-
tween quantum entanglement creation (EC) and quan- repeater [20] based on an ideal entangled photon source
andspectralmultiplexing. Thisprotocoldoesnotrequire
tum entanglement swapping (ES) is crucial to enhance
the EDR. Many previously proposed quantum repeaters a hierarchical connection of the elementary links. With
protocols can be classified into two categories: hierar- afixedquantumstoragetime,bothECandESprocesses
chical quantum repeaters and non-hierarchical quantum can proceed simultaneously at each attempt. Thus, the
repeaters [6, 8]. In a hierarchical quantum repeater, the system’sclockspeedis drivennotby the communication
EC and ES in the elementary links will be performed in time acrossthe elementary link but by the bandwidth of
a “repeat until success” scheme. The entanglement cre- thequantummemoryandthenumberofspectralmodes.
Despite the appealing performance of this architecture,
therearestillseveralsignificantlimitations. Thisscheme
has low tolerance for the number of spectral modes and
∗email:[email protected] requires the efficiency of the photon source to be close
†email:cfl[email protected] to unity, which constrains its applications and make it
2
Central
control unit
BSM
(a)
Elementary link Elementary link Elementary link Elementary link
(b)
BSM’
(c)
Quantum memory Entangled pairs of photons Single photon detector
FIG.1: (coloronline) Schematicof thesemihierarchical quantumrepeaterarchitecture. (a)Both endsofeach elementary link
containmultimodequantummemoriesandsourcesofentangledphotonpairs,whereeachpairisemittedintoadifferentoptical
mode. Onemember from each pair isstored in a multimodequantummemory while theotheris sent to thecenterof thelink
overaquantumchannel(orangelines)wheretheymeetphotonsgeneratedbythesourcessituatedattheotherendofthelink.
Photonsoccupyingthesamemodethenundergoamode-resolvingBell-statemeasurement(BSM)whichiscomposedofabeam
splitter (BS) and two single-photon detectors (assuming time-bin qubit encoding and temporal multiplexing is employed). A
heralding signal, which originates from a successful BSM, is sent, via classical channel (green lines), to a central control unit
located atthemidpointofthetotaldistributionlength. Thecentralcontrolunitwaitsuntilentanglement(ofremotequantum
memories) has been established over each elementary link (depicted by the dotted arrows), and then informs all quantum
memories to implement the next step of the protocol. (b) The heralded photons are retrieved from adjacent memories such
thattheyarriveindistinguishablyattheBSM’toperformtheES.(c)Theentanglementisdistributedoverthedesireddistance
on thepremise that ES operations at all nodessucceed. Otherwise, thewhole process must start from scratch.
difficulttoimplementwithcurrenttechnology[8]. Inad- quantummemory,notonlyspectralorspatialmodesbut
dition, there are some non-hierarchical protocols based also temporal modes can be used in this protocol. By
on encoding and errorcorrection[21–24], requiring com- introducing more temporal modes, the protocol can be
plicatedentanglementresourcesandgateoperationsthat morepracticallyimplementedbecausetime-resolvedpho-
also make them difficult to achieve. tonemissionanddetectionarewell-developedtechniques
[25]. Toquantifytheperformanceofourprotocol,wecal-
culate the average time required for a successful distri-
II. SEMIHIERARCHICAL QUANTUM bution of an entangled photon pair. Assuming that the
REPEATER PROTOCOL total distribution length L is divided into n elementary
links, the probability of EC for a single elementary link
is given by
Here we present a semihierarchical quantum repeater
scheme that requires only moderate-lifetime quantum m
memories and practical photon sources. It combines the p=1− 1− 1 ηDρ10−2α0Ln 2 , (1)
2
advantages of the above two approaches, that is, hier- (cid:18) (cid:16) (cid:17) (cid:19)
archicalDLCZ-likeprotocolsandnon-hierarchicalproto- whereρis the emissionprobabilityofthe entangledpho-
cols. Specifically, the required memory times are much tonpairsource,withthenumberofmodesdenotedbym.
shorter than those of hierarchical protocols, whereas ηD is the detection efficiency of single photon detectors.
the requirements for photon sources and the number of The whole process is driven by the basic clock inter-
modesarelowerthanthoseofnon-hierarchicalprotocols. val L /c, i.e., the reinitialization time required for an
0
ThediagramofourprotocolispresentedinFig. 1. By attempt to establish entanglement in elementary links
introducing a central control unit for feedforward con- including classical communication, with L = L/n. Af-
0
trol, the ES operations are performed only after the en- ter an expected time Tec, when all n elementary links
tanglement is successfully established over each of the successfullycreateentanglementthroughmanyattempts,
elementary links. Additionally, owing to the on-demand the central control unit waits until it receives the signal
3
from every elementary link and then informs all quan- (a)
tum memories to retrieve photons for ES, which takes
time Tcc = L/c. A successful entanglement distribution A
requiresESoperationsatallnodestosucceed,theproba- D
bility ofwhichcanbe denotedbyPes. Thus,the average
time we need for an entanglement distribution can be
written as
C
1 1
Ttot =(Tec+Tcc)× ×
Pes ηM2 ηD2 (2) B
L f(n,p) L 2n−1
0
= × + × n.
c p c (η2 η2)
(cid:18) (cid:19) M D
with
∞ n n (b)
f(n,p)=p k× 1−(1−p)k − 1−(1−p)k−1 ,
Xk=1 nh i h i o C
(3)
whereηM isthestorageefficiencyofmultimodequantum
memories. α=0.2dB/km and c =2×105 km/s are the B
attenuation coefficient and speed of telecom wavelength
photons transmitted through an optical fiber, respec-
tively. Note that we must wait until all elementary links
succeedinestablishing entanglement,whichtakeslonger
D
thanestablishing a single link by a factorof f(n,p). See
the Appendix for the detailed derivation.
III. SIMULATED PERFORMANCE AND
COMPARISONS WITH OTHER PROTOCOLS
FIG. 2: (color online) (a) Simulated performance of quan-
tum repeater protocols based on different connection types
We can calculate the average entanglement distribu- of the elementary links. The quantity shown is the average
tion time for any distance L and any number of elemen- timeneededtodistributeasingleentangledpairforthegiven
tary links n by knowing the values of all parameters in distance. A: the time required using direct transmission of
the equations above. Assuming that ηM =ηD =ρ=0.9 10-GHz single photons through fibers. B: the proposed pro-
and m = 100, we are able to plot the distribution time tocol that uses semihierarchical structure. C: the protocol of
Ref. [18]. D: the protocol of Ref. [20]. (b) The required
as a function of total distance. First, let us consider
average memory time of different quantum repeater proto-
the shortest distribution time by choosing the optimal
cols. Thequantityshownistheaveragememorytimeneeded
value of link number n, shown as curve B in Fig. 2(a).
to distribute a single entangled pair for the given distance.
Direct transmission of a photon, sent over optical fiber,
The letters refer to the same protocols as above. For all the
thatoriginatesfromasourceoperatingat10GHzis also curves we have assumed ηM = ηD = 0.9 and m = 100. For
included in Fig. 2(a) (curve A), as a reference. Our curveB and curve D, the emission probability ρ=0.9 . The
protocol outperforms direct transmission at an approxi- shaded area of curve B represents a one standard deviation
mately 500-km fiber length, and the advantages become uncertaintyof requiredmemory time, thedetailed derivation
increasingly obvious as the distance increases. of which is contained in the Appendix. All curves are drawn
We compare the performance of our proposed semi- bychoosing the optimal link number.
hierarchicalquantumrepeaterprotocoltootherprotocols
based on different architectures with multiplexing. The
averagedistributiontimewithrespecttodistributiondis- and spectral multiplexing. The system’s clock speed is
tanceoftheseprotocolsisplottedinFig. 2(a)usingsim- driven by the bandwidth of quantum memory and the
ilar parameters as assumed in curve B. The protocol of number of spectral modes. We assume the bandwidth
Ref. [18] (curve C) uses photon-pair sources and multi- andbandwidthinefficiencytobethesameasinRef. [20].
mode memories to implement a temporally multiplexed One essential requirement for the quantum repeater
version of the DLCZ protocol. Due to multiphoton er- protocol is that the quantum memories in each electu-
rors,theemissionprobabilityofphotonpairsischosento ary link are required to store the photons until the ES
obtain a final fidelity F =0.9, as in Ref. [18]. Curve D, proceeds. In our semihierarchical scheme, we separate
whichcorrespondstothenon-hierarchicaldesignprotocol the EC process from ES process by using a central con-
proposed in Ref. [20], combines two-photon interference trol unit. From Eq. (2), it is easy to find the average
4
(a) (a)
E
B
D
B
C
(b) (b)
B
D
B
E
FIG. 3: (color online) Robustness of different protocols with FIG. 4: (color online) (a) The average time needed to dis-
respect to thereduction of mode numberand emission prob- tributeasingleentangledpairforagivendistanceintwosce-
ability. (a) The average time for the distribution of an en- narios. B: when choosing the best n corresponding to total
tangled pair for a distance L=1000 km as a function of the distance. E:whenfixingthelengthofelementarylinksto125
modenumberm. (b)Theaveragetimeforthedistributionof km. The curve E is drawn in two different styles–solid and
anentangledpairforadistanceL=1000kmasafunctionof dashed–and a change in styles means that the optimal link
the emission probability ρ. Curve C is not included because number increments by one. The bolded dots on the curve E
thepairemissionprobabilityoftheprotocolproposedbyRef. represent nodes of the elementary links. Segments with the
[18] can not be too high due to multiphoton errors. The let- samestyleatbothsidesofonespecificboldeddotmeansthat
ters refer to the same protocols as in Fig. 2. All curves are theentanglementistransmittedviafibersfromthisnode. (b)
drawnbychoosingthebestlinknumbertoobtaintheoptimal Thememorytimeneededtodistributeasingleentangledpair
distribution time. forthegivendistanceintwoscenarios. B:whenchoosingthe
best n corresponding to total distance. E: when fixing the
length of elementary links to 125 km. The shaded areas rep-
required storage time for memories, written as resent the uncertaintyof required memory time.
L f(n,p) L
0
htMi= c × p + c. (4) strongrobustnessforreducedmodenumbermandemis-
sionprobabilityρofphotonsources. Wechangeeitherm
Usingthesameparametersassumedabove,wecanalso or ρ, while the other parametersremainthe same, as as-
plottheaveragememorytimehtMiasafunctionoftotal sumed before, and then plot the total distribution time
distance L. Fig. 2(b) shows htMi as a function of L as a function of m or ρ, respectively. The results are
whenlinknumbernischosenfortheoptimaldistribution shown in Fig. 3. It is clear that the performance of our
time. Owing to our very different structure, the average proposed architecture remains acceptable even when m
memorytimeofthesemihierarchicalrepeaterislessthan drops to 10 or ρ drops to 0.3, which is readily accessible
1 s for a distribution distance of 1600 km, a factor of 27 for practical use considering current techniques [25, 26].
shorter than that of the hierarchical protocol proposed Note thatf(n,p)growsveryslowly aslink number in-
in Ref. [18]. creases and p is determined by the length of elementary
Another advantage of our semihierarchical quantum linkL0 (notonlylinknumber);asaresulthtMiislargely
repeater protocol is that the total distribution time has determined by L . We can consider another scenario:
0
5
setting the length of elementary links to an appropriate atcryogenictemperaturesbyapplyingaspecialmagnetic
constant. Within the realistic range, the smaller L we fieldusedtoinduceamagnetic-fieldinsensitivetransition
0
choose, the smaller htMi we can obtain. Meanwhile, the when combined with dynamical decoupling [34]. More-
total distribution time Ttot relies on the values of L0 as over, the same idea can also be used in praseodymium
well. Thus, we can obtain a fine tradeoff among tM and doped YSO to increase the storage time up to seconds
Ttot bysetting L0 to 125km togetshortermemorytime withstrongpulses[35,36]. AFC-basedmultimode quan-
and reasonable distribution time. The distribution time tum memories currently suffer from low memory effi-
asafunctionoftotaldistanceisdrawnascurveEinFig. ciency. Thismaybeimprovedbyeitherputtingthemem-
4(a), with all other parameters as before. The bolded oryinsideacavity[37,38]orrecallingthephotonsinthe
dotsonthecurverepresentLthatsatisfyL=nL ,which backward direction [27]. The ultimate limit of the these
0
are nodes of the elementary links. For L that is not an materialsis farfrombeing reached. We thereforebelieve
integer multiple of L , we can distribute the entangle- that the memory efficiency and storage time can be fur-
0
ment to one side of its nearest nodes (which side of the ther improved in the near future to fit the requirements
nodes is determined by the total distribution time) and of our proposed protocol.
transmit the entangled photons to the desired distance Notably,anon-demandsingle-photonsourcewith66%
through optical fibers. The curve E in Fig. 4(b) shows extraction efficiency from a resonantly driven quantum
required memory time as a function of L for L = 125 dotinamicropillar[25]andsingle-photondetectorswith
0
km. Compared with curve B, the distribution time of highefficiencyupto95%andextremelylowintrinsicdark
whichis optimized by choosingthe best n corresponding counts have already been demonstrated [39, 40]. The
to different L (i.e., L0 is not a constant), htMi of curve combinationofallofthesedevelopingtechnologiesmakes
E is greatly reducedandhas little dependence onL. For semihierarchical quantum repeaters seem feasible in the
instance,the storagetime neededforL=1600kmisap- near future.
proximately 26.5 ms, which is a nearly 32-fold decrease
comparedwiththe0.84sofcurveB.Moreover,theloca-
tions of repeater stations (nodes) are fixed for different V. CONCLUSIONS
distribution distances if we fix the length of elementary
links, which is an economical way to build a real-word The quantumrepeater protocolweproposedtakesad-
quantum repeater and is suitable for future scalable ap- vantage of a semihierarchical structure, which improves
plications. the overall entanglement distribution time and reduces
the required lifetime of quantum memories to a mod-
erate and accessible range. More intriguingly, the de-
IV. IMPLEMENTATION mands for the emission probability of entangled single-
photon sources and multimode capacity of quantum
memories are also significantly reduced compared with
We will now briefly discuss the implementation of our
non-hierarchical protocols. We discussed two scenarios
proposed scheme. Multiplexed quantum memories with
in the paper. The first one with optimal length of ele-
on-demand read-out can be realized based on rare-earth
mentary links has the shortest distribution time and the
doped crystals since they have excellent coherence prop-
second one with an appropriate fixed length of elemen-
erties at cryogenic temperatures, while also providing
tary links makes the required memory lifetime drop to
stronglight-mattercouplingthroughhighnumberdensi-
tens of milliseconds and nearly independent of the total
ties. Theyalsoexhibitlargestaticinhomogeneousbroad-
distribution distance. This protocol greatly reduces the
ening of the optical transitions, which can be tailored
technicaldifficultyfortherealizationofanefficientquan-
and used as a resource for temporally [26–28] and spec-
tum repeater and is well suited for scalable applications
trally [20] multiplexed quantum memories. At the sin-
in a large-scale quantum network.
gle photon level, on-demand quantum memories have
been demonstrated tens of microsecond storage times in
praseodymium doped yttrium orthosilicate (YSO) crys-
VI. APPENDIX
tal[29]andmillisecondstoragetimes ineuropiumdoped
YSOcrystal[30]usingspin-waveatomicfrequencycombs
(AFCs). The latter experiment also demonstrated tem- A. The derivation of entanglement distribution
poral multimode storage with five modes; this number time
was further increased to 50 without sacrificing storage
time [31]. Except for temporal modes, other forms of Herewepresentthedetailedanalyticalderivationlead-
modes suchas spatialmodes [32,33] andspectralmodes ingtothequantitativeformulasinthepaper. Thecalcu-
can also be used together to fulfill multimode multiplex- lationoftheaveragetimeforthesuccessfulcreationofan
ing. For example, 100 modes can be realized by two entangled state across the overall quantum channel can
spectralandfivespatialmodesinadditiontotentempo- be divided into two parts: The first part is the average
ralmodes. Itisworthmentioningthateuropiumnuclear time neededbeforethe ESstepinvolvingthe BSM’mea-
spinsinYSOhaveshowedquantumlifetimesofupto6h surements,includingtheexpectedtimeoftheECprocess
6
acrossallelementarylinksandtheclassiccommunication where
time needed for feedforward control. The second part is
k−1 k−1
the probability of the ES process and final retrieval. P (k)=p (1−p)k−1 =1−(1−p)k. (A6)
We will first derive the probability of EC for a sin-
gle elementary link. The total distance L is split into Xk=1 Xk=1
n elementary links, so the length for each elementary The ith term of Eq. (A5) represents that i of n ele-
link is L0 = L/n. All sources are probabilistic single- mentary links create their entanglement successfully at
pair entanglement sources with emission probability ρ the kth attempt, whereas the other (n−i) elementary
and mode number m. To deterministically create entan- links succeed before the kth attempt. Using the bino-
glement between the ends of each elementary link, each mial theorem
photonmusttraveladistanceofL /2andbedetectedby
0
∞
amseiansgulree-pmheontton(BdSeMte)c.tor(SPD)forasuccessfulBell-state (a+b)n = Cniaibn−i, (A7)
i=0
Considering a single elementary link, using only one X
optical mode, the probability of a successful BSM at the one finds
center station is given by
n n
k−1 k−1
p1m = 21 × ηDρ10−2α0Ln 2, (A1) Pn(k)=(P (k)+"k=1P (k)#) −"k=1P (k)#
(cid:16) (cid:17) Xn X n
where ηD is the detection efficiency of SPDs, the pre- = 1−(1−p)k − 1−(1−p)k−1 .
factor1/2isthemaximumsuccessfulprobabilityofBSM
h i h i (A8)
with linear optics, and 10−2α0Ln is the transmission effi- The corresponding expectation value becomes
ciency corresponding to a distance of L /2 in telecom
0
wavelength optical fibers. f(n,p) ∼ ∞
If each source emits m different optical modes at one = k = kPn(k), (A9)
p
attempt, the probability that at least one mode results D E Xk=1
inasuccessfulBSM,whichcreatestheelementaryentan-
giving
glement, is
m ∞ n n
p=1−(1−p1m)m =1− 1− 1 ηDρ10−2α0Ln 2 . f(n,p)=p k× 1−(1−p)k − 1−(1−p)k−1 .
(cid:18) 2(cid:16) (cid:17) (cid:19)(A2) Xk=1 nh i h (A10i)o
For each attempt, one elementary link must wait the Thus, the average time to create entanglement across
time required for the photons to propagate from the all elementary links is given by
sources to the central station and for the information
about the result to propagate back to the memories. L0 f(n,p)
Tec = × . (A11)
Thusthebasicclockintervaloftheelementarycreationis c p
L /c,wherecisthespeedoftelecomwavelengthphotons
0
After each elementary link successfully creates entan-
in an optical fiber.
glement, a signal is sent to a feedforward control mod-
One has an exponential distribution of attempt times
ule located at the center of the total distance through
k for a success probability p, written as
a classical channel. The central control unit waits until
P (k)=(1−p)k−1p, (A3) entanglements are successfully established in all elemen-
tarylinksandtheninformsallquantummemoriestoim-
which gives an expectation value of plement the next step of the protocol, which takes time
∞ Tcc =L/c.
1
hki= kP(k)= . (A4) Onceallproceduresmentionedabovearecomplete,ES
p betweenneighboringlinks isattempted. Thisprocessin-
k=1
X cludesrecallingphotonsfromthememorieswithanover-
Thus,theexpectedtimeofECforoneelementarylink allefficiencyofηM andaBSM’.Iftherearenelementary
is (L0/c)/p. We wait until n elementary links all suc- links, there are (n−1) such ES operations, giving
cessfully create their entanglement. Assume that the ex-
pected attempt times is increased to f(n,p)/p. We will 1 n−1
denote the distribution for this combined attempt times Pes = ηM2 ηD2 . (A12)
2
∼ (cid:18) (cid:19)
by Pn(k) and its expectation value by k . One has
The entanglement is distributed over the desired dis-
D E
n k−1 n−i tance on the premise that ES operations at all nodes
Pn(k)= Cni(P (k))i P (k) , (A5) succeed. Otherwise, the whole process must start from
i=1 "k=1 # scratch. Finally, the entanglementis useful only if it can
X X
7
be retrieved from the memories at the ends of the quan- saryto calculate the standarddeviationof memorystor-
tumchannelanddetected,sotheprobabilityη2 η2 must age time. Consider the distribution of one specific en-
M D
be considered. tangled photon pairs. Assume the attempt times of this
Therefore, the average time we need for an entangle- clock cycle is k. We can write the actual storage time of
ment distribution can be given as the whole distribution process as
1 1 L L
Ttot =(Tec+Tcc)× Pes × ηM2 ηD2 tM(k)= c0 ×k+ c. (A15)
(A13)
L f(n,p) L 2n−1
0
= × + × n. TheprobabilitydistributionofattempttimesisPn(k)
c p c (η2 η2)
(cid:18) (cid:19) M D as calculated above. Therefore, the variance of memory
time is
B. The standard deviation of memory time ∞
D(tM)= (tM(k)−htMi)2×Pn(k) , (A16)
From the calculations above, it is easy to find the av- Xk=1h i
erage storage time required for memories, written as
which gives the standard deviation
L f(n,p) L
0
htMi= c × p + c. (A14) ∞ L f(n,p) 2
0
σ(tM)= k− ×Pn(k) .
v c p
Ifwewanttofullyaccesstherequirementforthemem- uk=1((cid:20) (cid:18) (cid:19)(cid:21) )
uX
ories of the proposed repeater protocol, it is also neces- t (A17)
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