Supplementary information:Autonomous Maxwell’s demon in a cavity QED system

Baldo-Luis Najera-Santos,1 Patrice A. Camati,2 Valentin Metillon,1 Michel

Brune,1 Jean-Michel Raimond,1 Alexia Auffeves,2 and Igor Dotsenko1

1Laboratoire Kastler Brossel, College de France, CNRS, ENS-Universite PSL,Sorbonne Universite, 11 place Marcelin Berthelot, F-75231 Paris, France

2Universite Grenoble Alpes, CNRS, Grenoble INP, Institut Neel, 38000 Grenoble, France(Dated: June 26, 2020)

Section I presents the derivation of the entropy conservation in a qubit-demon-cavity system,given by Eq. (2) of the main paper. In Section II we describe the basic experimental protocol interms of quantum logic circuit. In Sections III we give the definition of a cavity thermal field, explainits experimental preparation and control, and present the maximum-likelihood reconstruction of itsphoton-number distribution. Section IV presents a thermal state of a two-level atom and then showthe measured populations of three atomic states, used for coding a qubit and a demon, dependingon the initial state of the atom. Finally, in Section V we present a theoretical model describing ourexperimental protocol taking into account the main experimental imperfections, such as read-outefficiency, atomic relaxation, and detection errors.

I. CONSERVATION OF ENTROPY

In order to obtain Eq. (2) of the main paper, we startfrom the following identity for the relative entropy (di-vergence):

D(ρX||GβX

)= βX [U (ρX)−Feq

X ]− S (ρX) , (1)

where ρX is an arbitrary state for the system, UX ≡U (ρX) = Tr

[HXρX

]is the internal energy for some

Hamiltonian HX, GβX= e−βX(HX−Feq

X ) is a Gibbs state,

and FeqX = − (βX)

−1ln Tr

[e−βXHX

]is the equilibrium free

energy. Applying this identity for two different timeswith the same reference Gibbs state (∆Feq

X = 0) andtaking their difference one obtains

∆SX = βX∆UX −∆DX, (2)

after rearranging the terms. Considering the two instantsof time to be before and after the feedback step and forthe systems X ∈ Q,C, we substitute ∆SX from Eq. (2)into ∆SQC = ∆SQ + ∆SC − ∆IQ:C, where IQ:C denotesthe mutual information between two systems. Aftersuch a substitution and recognizing the following identity

D(ρXY||GβX

⊗ GβY

)= D

(ρX||GβX

)+D

(ρY||GβY

)+ IX:Y

one finally arrives at

∆SQC = βQ∆UQ + βC∆UC −∆DQC. (3)

This is one way to write the entropy change of the ther-modynamic system QC during the feedback step. Theonly assumption behind Eq. (3) was that there is no in-teraction Hamiltonian before and after the feedback step.This is naturally accomplished in our setup since the Ry-dberg atom passes through the cavity, hence effectivelyturning the interaction on, while inside the cavity, andthen off, after leaving the cavity.

Now, we write the same entropy change in a differentform, relating it to the demon D. This is easily achievedby rewriting the definition of mutual information IQC:D

for the partition QC:D of our system, obtaining

∆SQC = ∆SQDC −∆SD + ∆IQC:D. (4)

Before we put these equations together, we state someproperties of the feedback step of our protocol that can bechecked to be true and that are necessary for the deriva-tion:

(i) since there is no driving on any of the subsystemsduring the feedback step, the first law of thermody-namics allows us to identify ∆UX = QX as the heatabsorbed by the system X;

(ii) the total energy change is conserved (isolated uni-tary evolution), i.e., ∆UQDC =∆UQ+∆UD+∆UC =0;

(iii) the total entropy is conserved (unitary process), i.e.,∆SQDC = 0;

(iv) the reduced demon state ρD does not change duringthe feedback step and hence its energy and entropyare conserved, i.e., ∆UD = 0 and ∆SD = 0;

(v) initial state of the qubit and the cavity before thefeedback step are the Gibbs state, hence ∆DQC =DQC, i.e., the change is given by the final divergenceonly.

Property (iv) is true because the demon state afterthe read-out step is diagonal in the computational ba-sis which is also the basis for the controlled unitary inFig. S1. From properties (i) and (iv) one can see that noenergy is transferred to the demon. Together with (ii) itresults in QC = −QQ, meaning that the heat absorbed

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by the qubit is the same as the heat given off by the cav-ity and vice versa. With properties (i), (ii), (iv), and (v),Eq. (3) becomes

∆SQC = δβQC −DQC, (5)

where δβ = βC − βQ. With properties (iii) and (iv),Eq. (4) becomes

∆SQC = ∆IQC:D. (6)

Together, Eqs. (5) and (6) give Eq. (2) of the main paper.

II. LOGICAL STATES AND EFFECTIVEQUANTUM CIRCUIT

In order to independently describe the qubit and de-mon states, we map the three atomic levels |e〉, |g〉, |f〉into a subspace of a two-qubit Hilbert space of Q and D as

|e〉 = |1Q〉 ⊗ |0D〉,|g〉 = |0Q〉 ⊗ |0D〉, (7)

|f〉 = |0Q〉 ⊗ |1D〉.

We refer to this “logical basis” as qubit-demon, or QDbasis for short. Since the density operator describing the

initial state of the atom is ρ(0)A = |g〉〈g|, the initial QD

state is ρ(0)QD = |0Q0D〉〈0Q0D|.

Figure S1 shows an effective quantum circuit describ-ing the autonomous operation of our Maxwell’s demon.Initial states of C and Q are thermal, while the demon Dis initially in its ground state. The read-out process is re-alized by a controlled-NOT gate between Q and D. Thefeedback is modelled by a controlled unitary operationUfb between C and Q conditioned on D. Note that bothcontrolled gates are conditioned on a control qubit in |0〉state. In Section V we describe in detail each element ofthis circuit.

FIG. S1. Effective quantum circuit describing the au-tonomous Maxwell’s demon operation. The operations in adashed box are unitary. The final measurement is used toreconstruct the system’s state and then to compute all ther-modynamic quantities.

Note also that all states in the current work have no co-herences and are thus diagonal in energy basis. However,

for the sake of generality, we continue to use density ma-trices (ρ) instead of state populations (ρii) when describ-ing state evolutions and calculating entropic quantities.Finally, the fourth state of the QD system, |1Q〉 ⊗ |1D〉,is never populated and, thus, do not need to be codedonto the atom. Since both physical and logical states ofthis mapping are orthogonal in their respective Hilbertspaces, these two sets are isomorphic and therefore theirmapping is bijective with no loss of information or changein entropy.

III. CAVITY THERMAL FIELD

Definition

The concept of temperature in quantum thermody-namics is very similar to its counterpart in statisticalphysics. According to Boltzmann, the probability of find-ing the system, that is in thermal equilibrium at temper-ature T , in level |m〉 decreases exponentially with thelevel energy Em:

P (m) = Z−1 exp

(− EmkBT

), (8)

where the normalization parameter Z is the partitionfunction defined as

Z =

∞∑m=0

exp

(− EmkBT

). (9)

In the following, we use an inverse temperature definedas β = (kBT )−1.

For a cavity C at temperature βC and with resonantfrequency ωC/2π = 51 GHz, we get a thermal photon-number distribution

PCβC

(n) = (1− e−~ωCβC) e−n~ωCβC =nth

n

(1 + nth)n+1, (10)

with the thermal (mean) photon number

nth =1

e~ωCβC − 1. (11)

The cavity thermal field and the corresponding inversetemperature, respectively, read

GCβC=

∞∑n=0

PCβC

(n) |n〉〈n|, (12)

βC = (~ωC)−1 ln

(1 + nth

nth

). (13)

Preparation

Each experimental sequence starts by erasing the resid-ual microwave field in the cavity C with a sequence of

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several tens of atoms prepared in the ground state |g〉and resonantly absorbing all photons from C. To preparea thermal field, we inject into the empty C a series ofcoherent microwave pulses generated by the microwavesource CC. All injections have equal absolute amplitudeα and duration τ , but have different and random phases.During these injections the cavity field undergoes a ran-dom walk in phase space, starting from the space origin.The average field amplitude after Ninj injections, i.e., af-ter a total injection time tinj = Ninjτ , is given by

αinj = α√Ninj = α

√tinj

τ. (14)

Consequently, the mean photon number 〈n〉 = α2inj is

not expected to grow quadratically with time, as it doesfor the coherent pumping, but rather linearly. After alarge number of injections, the photon-number distribu-tion P (n) converges to the Boltzmann distribution witha thermal photon number nth equal to the mean pho-ton number: nth ≈ 〈n〉. We have checked numerically,that after only Ninj = 10 injections with an amplitudeα < 0.1 photons, the built-up distribution is close to theBoltzmann one. A more reliable verification of the suc-cessful thermal field preparation is done experimentally,see below.

Calibration

The thermal field injection is experimentally pre-calibrated using the Ramsey interferometer R1-R2 [1].The method is similar to the one presented in Ref. [2,Appendix A.3]. Each photon number state in the cavityshifts the Ramsey fringe in a well-defined way dependingon the interferometer settings (i.e., dephasing per pho-ton). The Ramsey signal, i.e., the population transfer ofthe QND atoms from their initial state |g〉 to |e〉, dependson the photon number n as

pe(n, φr) = y0 +c

2cos (nφ0 + φr) . (15)

The offset y0 and the contrast c of the Ramsey fringescan be calibrated independently on the vacuum field. Forthe cavity field with a photon-number distribution P (n)the Ramsey signal is a sum of individual photon numbercontributions:

pe(φr) =

∞∑n=0

pe(n, φr)P (n). (16)

In the case of a coherent field of amplitude β (i.e., con-stant phase injections), P (n) reads

P (n) = e−〈n〉〈n〉n

n!(17)

FIG. S2. Calibration of the injection amplitude with Ramseyfringes. Points are experimental, lines are fits. Blue (red) isfor the Ramsey phase φr1 = 0 (φr2 = π/2). Full points andsolid lines are for the thermal field. Open points and dashedlines are for the coherent field. The fitted parameters areγcoh = 2.1 ms−1 and γth = 0.39 ms−1.

with 〈n〉 = |β|2. The total Ramsey signal is then

pcohe = y0 +

c

2e(cosφ0−1)〈n〉 cos

(φr + 〈n〉 sinφ0

). (18)

Assuming that 〈n〉 = γcoh t2inj, we get the theoretical de-

pendence of pcohe on tinj.

For the thermal field, P (n) is given by (10) and theRamsey signal is

pthe = y0 +

c

2

(1 + nth) cosφr − nth cos(φr − φ0)

(1 + nth(1− cosφ0))2 + (nth sinφ0)2. (19)

If we assume that nth = γth tinj, we obtain the depen-dence of pth

e on tinj.We use here the same interferometer settings as those

used for the cavity state reconstruction in the main pa-per. Namely, the dephasing per photon is φ0 = π/2 andwe use two interleaved ensembles of QND atoms withRamsey phases φr1 = 0 and φr2 = π/2 aligned to max-imize the atom population pe in state |e〉 for the vac-uum state and for the one-photon state, respectively. Werecord the dependence of the Ramsey signal (i.e., atomicpopulation) on the injection duration. By comparing itto the theoretical variation for the Poisson (i.e., for a co-herent field) and Boltzmann (i.e., for a thermal field)photon-number distribution, we obtain the calibratedmean photon numbers.

Figure S2 shows pthe (tinj) and pcoh

e (tinj), measured andfitted, for two Ramsey phases (blue and red for φr1 andφr2, respectively). Open points and dashed lines are forthe coherent field, while full points and solid lines arefor the thermal field. The points are the measured pop-ulation transfer and the lines are the corresponding fits

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FIG. S3. Reconstructed cavity thermal state P CβC

(n). Bluepoints are reconstructed populations. Solid line with opencircles is a fit with a Boltzmann distribution (10), given nth =0.63± 0.04 photons.

with (18) and (19). For the thermal field, the injectionphase is randomly changed every 0.1 ms. From this quickmeasurement, which requires no sophisticated state re-construction, we get a value of γth = 0.39 ms−1 and thusnth = 0.39 photons after 1 ms injection with the mi-crowave source power used in Fig. S2.

Reconstruction

We reconstruct the photon-number distribution ofthe thermal field with our quantum state tomographymethod [3]. Figure S3 shows the reconstructed P (n) af-ter the incoherent pumping (i.e., with random phases) ofC with 10 weak injections. The data fit to the thermalfield distribution results in nth = 0.63 ± 0.04 photonsand, according to (13), in βC = (0.95±0.04)(~ωC)−1 andTC = 2.6 ± 0.1 K. This value is used in all experimentalsequences presented in the paper. The residual noise onP (n) for higher photon numbers (n ≥ 4) results in themean photon number 〈n〉 = 0.68 slightly higher than thefitted nth.

We repeat the same analysis for different injectiontimes tinj, i.e., after different number of injections Ninj

of 0.1 ms duration. Figure S4 presents the dependenceof the reconstructed nth on tinj. Note that the power ofthe microwave source SC in this test is set higher thanthat used in the main experiment. Starting from about 4injections, nth grows linearly, as expected from the ran-dom walk in the phase space.

For the experiments presented in the paper, we applyNinj = 10 weak injections of τ = 0.1 ms duration, result-ing in tinj = 1 ms. The power of the microwave sourceSC is set to have nth = 0.63 photons.

FIG. S4. Mean photon number versus injection duration. Theinjection phase is randomly changed every 0.1 ms. Startingfrom Ninj ≤ 3 injections, the growth of 〈n〉 is close to linear.In the experiment, we use Ninj = 10 injections to prepare athermal field. For thermal fields with 〈n〉 > 1.5 the currentreconstruction with the Hilbert space dimension of 7 photonsstarts to produce inaccurate state estimation.

IV. ATOMIC STATE MANIPULATIONS

Thermal state

We consider a qubit with states |0〉 and |1〉. The pop-ulation of its excited state |1〉 at temperature βQ reads

nQ ≡ PQβQ

(1) =1

exp(~ωQβQ) + 1, (20)

where nQ is the qubit occupancy. The qubit thermalstate is then a statistical mixture of two pure states withthe corresponding probabilities:

GQβQ= PQ

βQ(0) |0〉〈0|+ PQ

βQ(1) |1〉〈1|, (21)

where the ground state population is PQβQ

(0) = 1−PQβQ

(1).

Using (20), we get for the inverse temperature

βQ = (~ωQ)−1 ln

(1− nQnQ

). (22)

Initial state

We set the initial inverse temperature βQ of our atomicqubit Q (states |g〉 and |e〉) with a resonant microwavepulse applied to Q in state |g〉 by microwave source Seg.The excited state population oscillates with the pulse du-ration tQ as

nQ ≡ P (e) = π0 −cQ2

cos (ΩtQ) . (23)

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We omit in the following the subscripts and superscriptsβQ and Q for the atomic populations where obvious. Theoffset π0 and the contrast cQ account for the imperfectionof these Rabi oscillations (spacial dispersion of atoms inan atomic sample, microwave field inhomogeneity, detec-tion errors, etc). The Rabi frequency Ω depends on themicrowave power. Each P (e) corresponds to a uniquequbit temperature βQ given by (22). Thus, we controlβQ by choosing the pulse duration tQ.

Figure S5(a) presents the probability P (a) to detectthe atom in one of its three states (a ∈ f, g, e) de-pending on the pulse duration tQ. The observed Rabifrequency is Ω/2π = 77 kHz. For tQ = 0 some atoms aredetected in |e〉, because of the state discrimination errorη of our nonideal detector. We also observe some atomsin |f〉, because of the combination of η and the atomicspontaneous relaxation from |g〉 to |f〉 between the statepreparation and detection. As expected, P (f) decreasestogether with P (g).

Qubit-cavity energy exchange

Figure S5(b) shows the atomic populations P (a) afterthe adiabatic passage transfer between Q and C. At a halfRabi period (tQ ≈ 6.5µs), P (g) and P (e) are opposite tothose in Fig. S5(a), since Q has emitted a photon into C,passing from |e〉 to |g〉. The probability for Q in |g〉 toabsorb a photon equals the probability to have at least 1photon in C. Therefore, for tQ = 0 and Q prepared mainlyin |g〉, PQ (e|tQ = 0) = 1 − PC(0) and PQ (g|tQ = 0) =PC(0), where PC(0) is the probability of the vacuum fieldin C. Here, βC = 0.90(~ωC)−1, resulting in PC(0) = 0.59,which is close to the observed value of PQ (e|tQ = 0) =0.52 ± 0.04 if considering the reduced contrast c of theRabi oscillations.

Demon read-out

Now, we consider the effect of the demon read-out ontothe energy exchange between Q and C, at temperaturesβQ(tQ) and βC(tinj). Unlike the previous case, the atom istransferred from |g〉 to |f〉 with a demon read-out pulse,before the Q-C energy exchange. Figure S5(c) shows thestate of the atom, after read-out and before the interac-tion. As expected, the levels |g〉 and |f〉 have exchangedtheir populations, compared to Fig. S5(a). The remain-ing population P (g) for tinj = 0 reveals the limited demonpulse efficiency, ηD ≈ 0.95, due to the inhomogeneity ofthe radiating microwave field.

Figure S5(d) displays the final state of the atoms afterthe feedback (i.e., adiabatic passage between Q and C).The population P (f) remains the same as in Fig. S5(c),since the |g〉 → |f〉 transition is far detuned from thecavity frequency. Moreover, we observe an almost perfect

transfer from |e〉 to |g〉, showing the high efficiency of theadiabatic passage.

The current readout process is implemented bytransferring the atomic population between two non-degenerate levels. However, since this transfer is inducedby a strong classical coherent field, the atom (and, thus,the system in total) does not get entangled with the ex-ternal field. Consequently, there is no information (en-tropy) flow outside of the system and all entropy analysisis valid. On the other hand, the energy of the field modedoes increase on average. In principle, it could be used atthe very end of our protocol to reset the demon D backto its initial state.

V. THEORETICAL MODEL

In this chapter we present a theoretical model describ-ing the evolution of our QDC system during the exper-imental sequence and taking into account the main ex-perimental imperfections which affect the measurementresults. This model is used for all theoretical curves inthe main paper. Note that for the bijective state mappingbetween physical and logical states implemented here, allQDC operators are defined in a unique way.

Thermal state preparation

We model the qubit preparation in the thermal stateusing the following unitary transformation Uth:

Uth(θ) =√

1− nQ IQ − i√nQ σY

=

(√1− nQ −√nQ√nQ

√1− nQ

), (24)

where nQ is given by (23) and IQ = ID = I2 is the identityoperator in a two-dimensional Hilbert space. The QDdensity matrix transforms according to

ρthQD =

(Uth ⊗ ID

)ρ

(0)QD

(U†th ⊗ ID

)(25)

= nQ|1Q0D〉〈1Q0D|+ (1− nQ) |0Q0D〉〈0Q0D|−√nQ(1−nQ)

|1Q0D〉〈0Q0D|+ |0Q0D〉〈1Q0D|

.

Although the qubit has non-vanishing coherences, we arenot able to track them all along the experiment, andthey are averaged out over several protocol repetitions.Neglecting the coherence, expression (25) simplifies to

ρthQD = nQ |1Q0D〉〈1Q0D|+ (1− nQ) |0Q0D〉〈0Q0D|. (26)

The initial state of the cavity C is a thermal state GCβC

with temperature βC given by (12). Thus, the initial jointQDC density matrix reads

ρthQDC = ρth

QD ⊗ GCβC. (27)

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FIG. S5. Population of the atomic states, with and without demon read-out, versus the duration of the |g〉 → |e〉 microwavepulse before and after feedback (energy exchange). The atom is initially in state |g〉. The points represent experimentalprobabilities of atomic detection in |e〉 (red), |g〉 (blue), and |f〉 (green). The lines are sine fits.

Demon read-out

The read-out step is implemented by a unitary trans-formation Urd:

Urd = |1Q〉〈1Q| ⊗ ID − i |0Q〉〈0Q| ⊗ σY (28)

The limited efficiency ηD of our imperfect read-out istaken into account by mixing two possible evolutions:successful Urd with probability ηD and unsuccessful IQ ⊗ID with probability (1 − ηD). The QD density matrix isthen transformed according to

ρrdQD = ηD ( Urd ρ

thQD U

†rd ) + (1− ηD)ρth

QD

= nQ |1Q0D〉〈1Q0D|+(1− ηD)(1− nQ) |0Q0D〉〈0Q0D|+ηD(1− nQ) |0Q1D〉〈0Q1D| (29)

and the QDC density matrix is

ρrdQDC = ρrd

QD ⊗ GCβC. (30)

Feedback operation

The feedback operation controlling the heat exchangeis realized with the adiabatic population transfer betweenQ and C. The calibrated efficiency is closed to unity. Thisoperation transforms the QDC joint state as

ρfbQDC = Ufb ρ

rdQDC U

†fb. (31)

where the unitary transformation Ufb is modelled by

Ufb = σ− ⊗ |0D〉〈0D| ⊗ b+ + σ+ ⊗ |0D〉〈0D| ⊗ b−+|0Q〉〈0Q| ⊗ |0D〉〈0D| ⊗ |0C〉〈0C|+IQ ⊗ |1D〉〈1D| ⊗ IC. (32)

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Here, we have defined a set of cavity and qubit operators:

b+|n〉 = |n+ 1〉, b− |n〉 = |n− 1〉, b+ = b†−;

σ+|0〉 = |1〉, σ− |1〉 = |0〉, σ+ = σ†−. (33)

The first and second terms in (32) describe the energytransfer from Q to C and from C to Q, respectively. Thethird term accounts for the case when both Q and C arein their ground states and no heat exchange is thus possi-ble. Finally, the fourth term corresponds to the situationwhen D actively prevents the heat exchange.

The joint QC population probabilities (i.e., the diag-onal elements of the joint QC density matrix) after thefeedback are given by

P∅(n, 1Q) = (1− nQ)PC(n+ 1) (34)

P∅(n, 0Q) = nQPC(n− 1)

P∅(0, 0Q) = (1− nQ)PC(0)

P d(n, 1Q) = (1− ηD) (1− nQ )PC(n+ 1)

P d(n, 0Q) = nQPC(n− 1) + ηD (1− nQ)PC(n)

P d(0, 0Q) = (1− nQ)PC(0)

where superscripts ∅ and d indicate the absence or pres-ence of the demon read-out, respectively. The effect ofD is essentially decreasing the heat loss of C by a factorof (1 − ηD), that is the probability for D to ignore Q instate |0Q〉.

Atomic relaxation

The atomic levels are subjected to spontaneous relax-ation to lower lying states. This relaxation is efficientlydescribed by the set of rate equations:

dρee/dt = −Γρee,

dρgg/dt = −Γρgg + Γρee, (35)

dρff/dt = −Γρff + Γρgg.

The relaxation times for the atomic levels e, g and fare about 33 ms, 30 ms and 27 ms, respectively. Forthe sake of simplicity, we consider the relaxation rateΓ ≈ (30 ms)−1 to be the same for these levels and weneglect the possible thermal excitation of atoms due tothermal background radiation in our experiment. Thesolution to (35) is

ρee = ρee,0 e−Γt,

ρgg = (ρee,0Γt+ ρgg,0) e−Γt, (36)

ρff =

(1

2ρee,0(Γt)2 + ρgg,0Γt+ ρff,0

)e−Γt,

where subscript 0 refers to the initial atomic populations.The normalization condition (ρee+ρgg +ρff = 1) has to

be taken into account by dividing (36) by the sum of itsright-hand side terms.

The relaxation process occurs during all experimen-tal sequence. The atomic velocity (250 m/s) and thefixed geometry of the experimental components (Ramseyzones, cavities, detector, etc) define the time intervals be-tween the thermal state preparation, read-out, feedbackand detection. We apply the atomic state transforma-tion (36) to each of these intervals.

Imperfect state detection

The imperfect state resolution of our detector leads tothe effective mixing of the detected atomic states. Wemodel it with the following Kraus map:

ρ→∑

a,b∈e,g,f

Ma→b ρM†a→b, (37)

where the erroneous detection of an atomic state |a〉 asstate |b〉 with probability εa→b is described by

Ma→b =√εa→b |b〉〈a|. (38)

The closure relation implies that

εa→a = 1−∑b 6=a

εa→b. (39)

We have independently calibrated the detection errorsand obtained the following values for εa→b:

εe→b = 0.02 δb,g + 0.98 δb,e,

εg→b = 0.07 δb,e + 0.02 δb,f + 0.91 δb,g, (40)

εf→b = 0.07 δb,g + 0.035 δb,e + 0.90 δb,f ,

where the Kronecker δb,a = 1 iff a = b and δb,a = 0otherwise.

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