# Testing sequential quantum measurements: how can maximal knowledge be extracted?

**ABSTRACT** The extraction of information from a quantum system unavoidably implies a modification of the measured system itself. In this framework partial measurements can be carried out in order to extract only a portion of the information encoded in a quantum system, at the cost of inducing a limited amount of disturbance. Here we analyze experimentally the dynamics of sequential partial measurements carried out on a quantum system, focusing on the trade-off between the maximal information extractable and the disturbance. In particular we implement two sequential measurements observing that, by exploiting an adaptive strategy, is possible to find an optimal trade-off between the two quantities.

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Page 1

Testing sequential quantum

measurements: how can maximal

knowledge be extracted?

Eleonora Nagali1, Simone Felicetti1, Pierre-Louis de Assis2,3, Vincenzo D’Ambrosio1, Radim Filip4

& Fabio Sciarrino1

1Dipartimento di Fisica, Sapienza Universita ` di Roma, Roma 00185, Italy,2Institut Nel, CNRS et Universit Joseph Fourier, BP 166,

F-38042 Grenoble Cedex 9, France,3Departamento de Fisica, Universidade Federal de Minas Gerais, Caixa Postal 702,

30123-980, Belo Horizonte, Brazil,4Department of Optics, Palacky ´ University, 17. listopadu 12, 771 46 Olomouc, Czech Republic.

The extraction of information from aquantum systemunavoidably implies amodification of the measured

systemitself.Inthisframeworkpartialmeasurementscanbecarriedoutinordertoextractonlyaportionof

theinformationencodedinaquantumsystem,atthecostofinducingalimitedamountofdisturbance.Here

we analyze experimentally the dynamics of sequential partial measurements carried out on a quantum

system, focusing on the trade-off between the maximal information extractable and the disturbance. In

particularweimplementtwosequentialmeasurementsobservingthat,byexploitinganadaptivestrategy,is

possible to find an optimal trade-off between the two quantities.

T

experiment3.Thedualitybetweentheinformationavailableonanunknownquantumsystemandthedisturbance

inducedbyameasurementprocessisofutmostrelevancewheninvestigatingthequantumworld4–6andliesatthe

basis of the security of quantum cryptographic protocols7. In this framework, a partial measurement approach

can be adopted to extract only a limited amount of information from the quantum system at the cost of limited

induced disturbance8–12. Such partial measurement technique allows to perform consecutive observations (i.e.

sequential measurements) on the same quantum system in order to investigate its properties without destroying

it13–21. In such context a question arises whether it is possible to extract an optimal amount of information from

sequential measurements, compared to the degree of disturbance induced on the system. The aim of this paper is

to investigate experimentally the trade-off between information gained and disturbance induced by partial

sequential measurements on a quantum system22. Conceptually our experiment is similar to a double-slit experi-

mentFig.1-awherethewhich-wayinformationis acquiredviasuccessive measurements onthesameparticle. In

particularweimplementtwosequentialmeasurementsperformedonthesamequantumsystemandobservethat

the optimal trade-off that characterizes the single measurement can be retrieved by adopting a proper adaptive

strategy. Such result, observed for N 5 2 sequential measurements, can be extended for any value of N.

The experimental analysis is carried out through the interaction between a quantum state and an ancillary

qubit (the ‘meter’) on which projective measurements have been performed11. Such quantum states have been

encoded in two different degrees of freedom of a single photon. The performed analysis on the trade-off between

knowledgeanddisturbancehasalsoallowedustoobserveaZeno-likebehavior23ofthemeasurementdynamicsas

a function of the strength of the interaction between the system and the meter.

Inordertoquantifytheparametersinvolvedintheexperimentherepresented,werefertotwocomplementary

figures of merit that lie at the basis of the quantum properties of a system. Let us refer to the schematic

representation of the well-known double-slit experiment, shown in Fig. 1-a. According to quantum mechanics

theory, the presence of a photon passing through the two slits is manifested by fringes with a defined visibility.

Performing a partial measurement corresponds to an observer who tries to distinguish if the photon passes

through path 0 or 1. Such which-path information quantifies the knowledge K extracted from the system, and

affects the visibility of the fringes24. Indeed, once it is perfectly known where the photon passes, no fringes are

observed. According to these observations, in this experiment we adopt the following figures of merit:

he measurement process represents one of the most distinctive aspects of quantum mechanics with respect

toclassicalphysics1,2.Themainresultofquantummeasurementtheoryistheunavoidabledisturbanceofthe

quantum state by the measuring process, as epitomized by the early Heisenberg x-ray microscope thought

SUBJECT AREAS:

QUANTUM PHYSICS

QUANTUM OPTICS

GENERAL PHYSICS

FIBRE OPTICS AND OPTICAL

COMMUNICATIONS

Received

27 March 2012

Accepted

18 May 2012

Published

19 June 2012

Correspondence and

requests for materials

should be addressed to

F.S. (fabio.sciarrino@

uniroma1.it)

SCIENTIFIC REPORTS | 2 : 443 | DOI: 10.1038/srep00443

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i)The knowledge K [ 0,1

rectly discriminating the quantum states belonging to the com-

putational basis {j0æ, j1æ}, analogously to the capability of

discriminating between path 0 or 1 in the double-slit

experiment. Mathematically K can be defined as K~

P

is jjæ. K 5 1 corresponds to maximal knowledge and is achieved

by projective measurements {j0æÆ0j, j1æÆ1j}, while K , 1 can be

achievedbyameasurementwithreducedinteractionstrengthto

which we refer as partial measurement. Partial measurements

are interactions between two systems, a meter and a target, that

leave the meter in one of a set of non-orthogonal states, as

opposed to strong measurements that leave the meter in one

of a set of perfectly distinguishable orthogonal states. Indeed,

the strength of a measurement can be related to the distinguish-

ability between elements of the set of output states of the meter.

When K R 0, we perform a weak measurement25,26.

ii) In order to quantify the overall disturbance related to the mea-

surement process we need to define a figure of merit analogous

to the fringes visibility in the double slit experiment. Here we

choose to consider as quantum system a qubit belonging to

bipartite entangled state: see Fig. 1-b. Indeed in this case any

irreversible disturbance effect on a single subsystem B could be

revealed by estimating the degree of correlation in the whole

quantum state rAB. The adoption of an entangled state allows

to exploit quantum state tomography to estimate the disturb-

ance of the channel via concurrence C[ 0,1

is complementary to the knowledge, and it gets lower (C , 1) as

the information extracted from the system increases.

½? corresponds to the capability of cor-

ip i,i

ð Þ{P

i=jp i,j

ð Þ

??? ??? where p(i, j) is the probability that

the state is identified as jiæ by the measurement, when the input

½?27,28.Such parameter

Results

As first step, we analyze the trade-off in a single measurement strat-

egy, represented by the quantum circuit in Fig. 1-b: box MK1. We

consider the singlet state y{

j

parties A and B. The measurement strategy gains information on

qubit B by entangling it with the meter M through a interaction of

iAB~1ffiffiffi

2

p

10

jiAB{ 01

jiAB

??shared by

variablestrength,parametrized byy,andthenperforming aproject-

ive measurement on M. Specifically, the qubit B interacts with M,

initialized as j0æM, through the following unitary transformation:

^U y

ð Þ i j iB0 j iM~ i j iBai

Both ja0æ and ja1æ can be expressed in terms of y: ja0æM 5

cos yj0æM1 sin yj1æM; ja1æM5 cos yj0æM2 sin yj1æM. In general

thedisturbanceduetothemeasurementprocesscaninducedifferent

quantum channels on the system, however, as predicted by theory22,

the unbiased nondestructive measurement described above disturbs

the state only by introducing a phase-damping channel. Hence the

concurrence of the final state rABdepends on the strength of the

coupling with the meter, and reads C 5 jÆa0ja1æj 5 jcos 2yj, equal to

zero when ja0æMand ja1æMare orthogonal. The optimal value of

extractable knowledge can be expressed in terms of y, and reads

K 5 jsin 2yj. This expression holds when we perform a projective

measurement on the meter in the diagonal basis

0 j iM+ 1 j iM

ffiffiffi

residual entanglement is represented by the curve C~

so that when all the information available is extracted (K 5 1), the

initial entanglement is completely lost (C 5 0).

The former relations have been experimentally implemented

encoding the initial singlet state in the polarization of two photons

Aand B,

2

and vertical polarization, respectively. The experimental setup is

reported in Fig. 2 and we provide all details in the Methods section.

ThemeterqubitMhasbeenencodedinadifferentdegreeoffreedom

of the photon B, the linear momentum. In this way the extracted

information corresponds to the correlations between the polariza-

tion of the photon A and the path of the photon B. When all the

information is extracted, and thus the strength of the coupling is

maximum, the correlation is perfect.

In order to control the interaction between the qubit B and the

meter, we exploit a Sagnac interferometer with a polarizing beam

splitter (PBS), that interfaces the polarization to the transmitted and

reflected spatial modes29–33: Fig. 2-b. We note that such interfacing

between the two degrees of freedom could be achieved also in a

j iMwhere i 5 0, 1, and jaiæ is a pure state.

+

jiM~

2

p

, which is found out to be the one that maximizes the

parameter K. Thus the optimal trade-off between knowledge and

ffiffiffiffiffiffiffiffiffiffiffiffiffi

1{K2

p

28,

1ffiffiffip

HV

jiAB{ VH

jiAB

??,where(H, V)are linearhorizontal

Figure 1 |(a)Schemeofthetwo-slitsexperimentwiththeinterferencefringes(ontheright),whosevisibilitydependsontheinformationextractedonthe

path followed by the single photon. (b) Circuital scheme of the sequential measurements scenario. The qubit B is coupled with the ancillary qubit |0æM.

ThemeasurementresultaffectsthenextsequentialonebyadaptingthemeasurementbasisthrougharotationRiinthenextmeasurementprocess(MK2).

(c) Scheme for the implementation of two sequential measurements strategy on single photon states. An individual photon passing through the whole

measurement apparatus is detected only in one of the four output ports, which indicates which combination of results for MK1and MK2is obtained.

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SCIENTIFIC REPORTS | 2 : 443 | DOI: 10.1038/srep00443

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Mach-Zehnder34, however the Sagnac interferometer provides a

higher stability and thus is more suitable for experimental purposes.

The interferometer, here and after denoted as the measurement kit

MK, has been aligned in a non-degenerate configuration, where the

two internal modes are spatially separated and propagate clockwise

(mode a) and anticlockwise (mode b). Such configuration allows an

independentmanipulationofthepolarizationonaandbbytwohalf-

waveplates rotated at angles haand hb, respectively, providing a

controlled modification of the coupling strength y associated to

the information extraction. Indeed y is fixed by the waveplates in

the interferometer by the relation y~p

have been expressed in terms of the physical angle hbas in order to

perform a projective measurement on the meter qubit in the basis

j6æM, corresponding to the output modes of the interferometer, the

two waveplates of the Sagnac have a fixed shift relation, equal to

hb5 ha1 p/4. The projective measurement on the meter qubit

corresponds to the selection of one of the output modes 0 and 1 of

the PBS. According to definition in (i), in the single measurement

4{2hb. The latter relations

case

p 0,H

ð

that an input photon with polarization j emerges on the mode cor-

responding to the outcome i. On the other hand the concurrence of

the state rABafter the measurement process acting on qubit B has

been estimated from the density matrix reconstructed via quantum

state tomography27. In order to estimate probabilities p(i, j) and the

elements of the density matrix, we have recorded the coincidence

counts between the single photon detector DAand detectors D0and

D1on the output modes of the first measurement kit, measuring

around 600 events per second. Experimental results are reported in

Fig. 3, and compared to theoretical expectations evaluated taking

into account imperfections due to the PBS and to the source of

entangled states. We observe that the single measurement process

saturates the optimal trade-off between information extracted and

disturbance, allowed by quantum mechanics22.

Let us now address the following questions: how can knowledge be

extracted from sequential measurements? Is it still possible to achieve

the same optimal trade-off in such scenario? We consider twosequen-

tial measurement processes of equal strength. We note that if the first

observer does not extract all the information on the state, the second

one can still extract some information from the system. Such mea-

surement process is represented by the whole scheme in Fig. 1-b,c.

Each measurement process introduces a specific amount of decoher-

ence, reflected by a lowering of the concurrence after each step. The

degree of entanglement of the state after the two sequential measure-

mentsgivesanindicationofthetotaldisturbanceinducedinthewhole

process. The analysis on sequential measurements has been carried

out considering two separate cases: the first one concerningindepend-

ent measurements, the second adaptive ones. In both cases in order to

experimentally estimate the knowledge and the concurrence of the

state, we recorded coincidence counts between detectors [DA, D00],

[DA, D01], [DA, D10], and [DA, D11]: see Fig. 2-a.

Firstly we consider two independent sequential measurements, that

is, the second projection on the state is performed independently of the

outcome of the first one. In this case the concurrence shows a depend-

ence from the whole amount of knowledge extracted Ktot as

Cind~1{ Kind

tot

able from the system does not achieve the optimal trade-off with the

decoherence induced. In Fig. 4-a we report the experimental behavior

of the concurrence for this measurement strategy (black squares),

where the total knowledge Ktothas been evaluated as in the single

the

Þzp 1,V

knowledge

Þ{p 1,H

2

K hasbeenmeasured asK~

ððÞ{p 0,V

ðÞ

, where p(i, j) is the probability

??2, thus the maximum amount of information extract-

Figure 2 | (a) Experimental setup adopted for the sequential measurement strategy for N 5 2 measurement processes. The entangled state has been

generatedviaspontaneousparametricdownconversion.(b)SchemeofthemeasurementkitMKi:thekitisbasedonaSagnacinterferometerwhichallows

to separately manipulate the horizontal and vertical polarizations adopting two half-waveplates (HWP) oriented at angles haand hb, related by a shift of

p/4 for the optimal configuration.

Figure 3 | Theoretical expectations (red dashed line) compared to

experimental data (black squares) for concurrence as function of the

knowledge for the single measurement case. Continuous line reports

theoretical expectations rescaled to experimental imperfections as the

parameters of the PBS (tH5 rV5 0.992, rH5 tV5 0.008) and the

concurrence of the initial state (Cin5 (0.95 6 0.01)).

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SCIENTIFIC REPORTS | 2 : 443 | DOI: 10.1038/srep00443

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measurement procedure, combining outcomes 00 with 01 and 10 with

11. This strategy is related to the scenario in which a series of inde-

pendent observers estimate an unknown state of a quantum system by

performing consecutive measurements over the very same system35.

Assecondbenchmarkwehaveconsideredwhetheritispossibleto

achieve the optimal trade-off exploiting sequential measurements.

Thus we have analyzed the case of two sequential adaptive measure-

ments, where a feed-forward between the two measurements allows

to achieve the optimal trade-off Copt~

what observed for the single measurement22. We note that such

strategy is based on the one proposed for discrimination of multiple

copies of quantum states24. As schematically shown in Fig. 1-b, the

resultsfromthe firstmeasurement kitdeterminean adaptation, i.e.a

rotationinthemeterbasis,forthesubsequentmeasurementprocess.

Therefore, classical communication is required between the sequen-

tial measurements and they cannot be treated as independent any-

more. Depending on the outcome 0 or 1 of MK1, two different basis

of analysis, generically indicated as

are applied on the meter qubit in MK0

coslij0æ1sinlij1æ.Bothparameters{l0,l1}aredeterminedinorder

to maximize the extracted knowledge and depend on the decoher-

ence induced by the first measurement process. In Fig. 4-b we report

thenumericaldeterminationofparameters{l0,l1}dependingonthe

measurement strength y of the first kit.

The adaptive strategy has been implemented experimentally by

rotating the waveplates in the Sagnac of MK2, thus modifying the

basis of the meter qubit depending on the outcome of the measure-

ment carried out by MK1. We note that an intrinsic feed-forward

takes place in the adaptation process since two different rotations of

thebasisareperformedinthesecondmeasurementprocess,depend-

ing on which output arm of the first interferometer the photon gets

out, as shown in Fig. 1-b and Fig. 2-a. Different values of parameters

{l0, l1} leads to different values of the physical angles {ha, hb} and

hencethebasisforthemeterprojectionisnotingeneralthediagonal

one, so that the relation hb5 ha1 p/4 does not hold anymore.

In Fig. 4-a we report the theoretical behavior (red line) of concur-

rence as function of the global knowledge Ktotand the experimental

results (red dots) for the adaptive measurement strategy, where the

value of Ktothas been estimated with the same relation adopted for

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1{K2

tot

p

, analogously to

b0

ji, b\

2and MK1

0

??

???and

b1

ji, b\

1

??

???,

2, where jbiæ 5

the single measurement process, where outcomes i refer only to the

second kit. We find a good agreement with theoretical predictions

rescaled to experimental imperfections, thus achieving the predicted

optimal trade-off. We conclude that even performing several weak

transitionsfromquantumtoclassicalworldthroughtheinformation

extraction, is possible to keep the same degree of disturbance of a

single transition.

Let us now analyze the previous results from the point of view of

information extraction within the dynamics of sequential measure-

ments, observing how knowledge accumulates after N 5 2 adaptive

measurements as a function of K, the knowledge that would be

extracted from each measurement if it were the only one performed:

see Fig. 4-c, red squares and dashed-dot line. We stress that we

consider two sequential measurements of equal strength. As a gen-

eralobservation, weexpectthat,astheknowledgeextractedfromthe

first measurement gets closer to 1, a lowering of the extractable

knowledgefromthesecondprocesstakesplace.Hereweexperiment-

ally verified that, according to theoretical predictions in22, the total

knowledge extracted after two equal adaptive sequential measure-

ments does not additively accumulate as Ktot~2?K. Indeed the

adaptation between the two processes introduces additional quad-

ratic terms in the accumulation law, so that the total knowledge is

equal to Ktot~

2?K2{?K4

. Such behavior can be generalized to N

sequential adaptive measurements22.

Finally, Fig. 4-d experimentally demonstrates that the adaptive

concurrence Cadapt

tot

after two identical measurements is a concave

function of?K, as in the single measurement case.This is at variance

with the scenario in which the information is extracted with a clas-

sical strategy and where the concurrence is a convex function of?K,

leading to an exponential decay. The qualitative behavior of the

function curvature that characterizes strategies allowing an optimal

trade-off between knowledge and decoherence, is responsible for a

Zeno-like effect22. Indeed the quantum Zeno-effect refers to a situ-

ation in which a quantum system, if observed frequently by project-

ive measurements, varies slower than the exponential decay law.

Here we verified that for the case N 5 2 the concurrence scales with

?K as Cadapt

tot

?K

ð Þ<1{N

weakly affected when a small amount of information is acquired in

each single measurement.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p

?K2

2. Moreover the amount of entanglement is

Figure 4 | (a)Concurrence Casfunction ofthe knowledge forN52sequentialmeasurements adopting anindependent strategy(black squares)andthe

adaptiveone(redcircles).Blackandredlinesrepresenttheoreticalexpectations(dashedlinesfortheidealcase,continuousonesrescaledbyexperimental

imperfections) for the two approaches. (b) Numerical determination of adaptive basis depending on the measurement carried out in the first kit,

expressed by the parameter y. (c) Experimental knowledge after N 5 2 sequential adaptive measurements (red squares) compared to theoretical

predictions for classical (black line), adaptive extraction (dashed-dot red line), and after N 5 1 measurement (dashed blue line). (d) Experimental and

theoretical behavior of concurrence as function of?K. Black squares and line refer to experimental N 5 2 adaptive measurements and theoretical

expectations, respectively. Analogously red dots and line refers to the experimental and theoretical results for the single quantum measurements.

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Discussion

Insummary,wehavereportedtheexperimentalanalysisofthetrade-

off between acquired knowledge from a quantum state and the

detrimental effect on the system itself, by performing sequential

measurements.Wehaveexperimentallyinvestigatedhowknowledge

can be accumulated from two sequential measurements, verifying

that an optimal trade-off can be achieved when an adaptive strategy

is adopted. Finally we have observed that in sequential adaptive

measurements, the knowledge accumulation rule can lead to a

Zeno-like behavior of the entanglement dynamic of the system here

considered. Future steps might be the study of the extension for

multiple measurements and the application to continuos measure-

ments for controlled dynamics and others quantum information

protocols36,37. Here we considered sequential measurements of com-

patible observables, future work will focus on different scenarios of

complementary measurements38and non-compatible observables.

Methods

Entangled state source. The initial singlet state encoded in the polarization of single

photonshasbeenimplementedthroughspontaneousparametricdownconversionin

a 1.5 mm thick b-borate of barium crystal (BBO) cut for type-II phase matching,

pumped by 700 mW of the second harmonic of a Ti:Sa mode-locked laser beam with

repetition rate equal to 76 MHz. The photons are generated with wavelength l 5

795 nmandspectralbandwidthDl53 nm,asdeterminedbytwointerferencefilters

(IF). The spatial and temporal walk-off is compensated by inserting a half-waveplate

and a 0.75 mm thick BBO crystal (C) on each output mode kAand kB

Fig. 2-a. The detected photon-pair generation rate of the source is 8 kHz.

ThephotononmodekAinsenttoastandardpolarizationanalysissetup,composed

by a half-waveplate (HWP), a quarter waveplate (QWP) and a polarizing beam

splitter (PBS), and then coupled to a single mode fiber (SM) connected to a single

photon counter module DA. The photon on mode kBis sent through a single mode

fiber to the single or sequential measurement kit MKi.

39, shown in

Sequential adaptive quantum measurements. The adaptive measurement can be

described by the following operators, where the subscripts 1,2 refer to projective

measurements on the first and the second meter qubit, respectively:

P00~ z

j

j

i z

i {

h

h

j16 b0

j16 b1

j

j

i b0

i b1

h

h

j2

j2

;

P01~ z

j

j

i z

i {

h

h

j16 b\

j16 b\

0

??

?

b\

0

b\

1

?

??

2

P10~ {

;

P11~ {

1

??

?

?

??

2

ð1Þ

The states jb0,1æ are defined as follows:

bi

j i ¼ cosli0 j izsinli1 j i

where j6æ is the projection basis adjusted to gain maximal knowledge from first meter

qubit. The elements Pijsatisfy the condition Si,jPij5 1. The overal prediction on the

computationalbasisstatesj0æandj1æofqubitBisperformedonlyrespectivelytothelast

indexof Pij. The optimal parameters l0,1are chosentomaximize the overall knowledge

Ktoton the computation basis. By exploiting the adaptive measurement, maximal

knowledge is reached with optimal collective measurement on both the meters. The

minimal disturbance effect from this unbiased nondemolition measurement is pure

phase damping and this optimal synthesis of the sequential measurements leads to a

single optimal measurement with minimal phase decoherence22.

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Acknowledgements

This work was supported by project FIRB- Futuro in Ricerca (HYTEQ), Finanziamento

Ateneo 2009 of Sapienza Universita‘ di Roma, and European project PHORBITECH of the

FET program (Grant No. 255914). P.-L. de Assis acknowledges the financial support of the

CNPq, CAPES and FAPEMIG Brazilian research funding agencies. R.F. acknowledges

grant P205/12/0577 of GACR.

Author contributions

R.F., S.F., E.N., F.S. Conceived the theory and designed the experiments. V.D., P.D., S.F.,

E.N., F.S. performed the experiments and analyzed the data. All authors discussed the

results and wrote the manuscript.

Additional information

Competing financial interests: The authors declare no competing financial interests.

License: This work is licensed under a Creative Commons

Attribution-NonCommercial-ShareAlike 3.0 Unported License. To view a copy of this

license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/

How to cite this article: Nagali, E. et al. Testing sequential quantum measurements: how

can maximal knowledge be extracted?. Sci. Rep. 2, 443; DOI:10.1038/srep00443 (2012).

www.nature.com/scientificreports

SCIENTIFIC REPORTS | 2 : 443 | DOI: 10.1038/srep00443

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