Design and Stability of Quantum Filters with Measurement Imperfections: discrete-time and continuous-time cases

Design and Stability of Quantum Filters with Measurement Imperfections:

discrete-time and continuous-time cases

Pierre Rouchon

Mines-ParisTech,

Centre Automatique et Systèmes Mathématiques et Systèmes

pierre.rouchon@mines-paristech.fr

Yale, April 26, 2013.

Based on collaborations with

Hadis Amini, Michel Brune, Igor Dotsenko, Serge Haroche, Zaki Leghtas, Mazyar Mirrahimi, Jean-Michel Raimond,

Clément Sayrin and Ram Somaraju

Quantum filtering: some references . . .

I Quantum stochastic equations: R. L. Hudson and K. R.

Parthasarathy. Quantum Itô’s formula and stochastic evolutions.

Commun. Math. Phys., 93:301-323, 1984.

I Infinite dimensional analysis, noncommutative probability, quantum information: V. Belavkin. Quantum stochastic calculus and quantum nonlinear filtering. Journal of Multivariate Analysis, 1992, 42, 171-201.

I Control theory: Bouten, L.; R. van Handel & James, M. R. An introduction to quantum filtering. SIAM J. Control Optim., 2007, 46, 2199-224.

I Discrete-time approximation: Bouten, L. & Van Handel, R.

Discrete approximation of quantum stochastic models. J. Math.

Phys., AIP, 2008, 49, 102109-19.

I Quantum Monte Carlo trajectories:Dalibard, J.; Castin, Y. &

Mølmer, K. Wave-function approach to dissipative processes in quantum optics. Phys. Rev. Lett., 1992, 68, 580-583.

The first experimental realization of a quantum

state feedback²

1

The LKB photon box: sampling time (∼100µs) long enough to estimate in real-time the quantum-stateρand to compute the control u=Ae^ıΦas a function ofρ(quantum state feedback).

1Courtesy of Igor Dotsenko

2C. Sayrin et al., Nature, 1-September 2011

Experimental data

Anopen-loop trajectory starting from coherent state with an average of 3 photons relaxes towards vacuum (decoherence due to finite photon life time around 70 ms)

Detection efficiency 40%

Detection error rate 10%

Delay 4 sampling periods

The quantum filter takes into account cavity decoherence, measure imperfections and delays (Bayes law).

Truncation to 9 photons

Stabilization around 3-photon state

Several "quantum states":

|ψ_ki,ρbk

and

ρb^est_k

.

QuantumFilter_Controller PhotonBox

u

y y

Detector Coherent

pulse

u

est est

Thestate estimationρb^est_k used in the feedback law takes into account, measure imperfections, delays and cavity decoherence:

I Derived from Bayes law: depends on past detector outcomes between 0 andk; computed recursively from an initial valueρb^est₀ ;

I Stable and tends to converge towardsρb_k,the expectation value ofρ_k =|ψ_kihψ_k|knowing its initial valueρ₀=ρb₀and the past detector outcomes between 0 andk.

Outline

Quantum filter of the LKB Photon Box Markov chain in the ideal case

The Markov chain with detection errors The Markov chain with cavity decoherence Structure of the complete quantum filter Discrete-time quantum systems

Markov chains in the ideal case

Markov chains with imperfections and decoherence Stability and convergence issues

Bayesian parameter estimations Continuous-time quantum filters

From discrete-time to continuous-time models

SDE driven by Poisson and Wiener processes

SDE with imperfections and decoherence

Conclusion

Markov chain in the ideal case (1)

I System

S corresponds to a quantized cavity mode:

H_S

=

( _∞

X

n=0

ψⁿ|ni |

(ψ

ⁿ

)

^∞_n=0∈

l

²

(

C

)

)

,

where

|ni

represents the Fock state associated to exactly n photons inside the cavity

I Meter

M is associated to atoms :

H_M

=

C²

, each atom admits two energy levels and is described by a wave function c

_g|gi

+ c

_e|ei

with

|c_g|²

+

|c_e|²

= 1; atoms leaving B are all in state

|gi

I

When an atom comes out B, the state

|Ψi_B ∈ H_S⊗ H_M

of the composite system atom/field is

separable

|Ψi_B

=

|ψi ⊗ |gi.

Markov chain in the ideal case (2) C

B

D R

₁

R

₂

I When an atom comes outB: |Ψi_B =|ψi ⊗ |gi.

I Just before the measurement inD, the state is in general entangled(not separable):

|Ψi_R2 =U_SM |ψi ⊗ |gi

= M_g|ψi

⊗ |gi+ M_e|ψi

⊗ |ei whereU_SMis the total unitary transformation (Schrödinger propagator) defining the linear measurement operatorsMgand M_eonH_S. SinceU_SMis unitary,M_g^†M_g+M_e^†M_e=I.

Markov chain in the ideal case (3)

Just before the measurement inD, the atom/field state is:

M_g|ψi ⊗ |gi+M_e|ψi ⊗ |ei

Denote byµ∈ {g,e}the measurement outcome in detectorD: with probabilitypµ=hψ|M_µ^†Mµ|ψiwe getµ. Just after the measurement outcomeµ,the state becomes separable:

|Ψi_D= ^√1_p

µ (Mµ|ψi)⊗ |µi= (Mµ|ψi)⊗ |µi q

hψ|Mµ^†Mµ|ψi .

Markov process(density matrix formulationρ∼ |ψihψ|)

ρ+=







Mg(ρ) = ^MgρM

† g

Tr(MgρM_g^†), with probabilitypg = Tr

MgρM_g^†

; Me(ρ) = ^MeρMe†

Tr(MeρMe^†), with probabilityp_e= Tr

M_eρM_e^† .

Kraus map:

E

^{(ρ+/ρ) =K(ρ) =MgρMg†+MeρMe†.}

Markov chain with detection errors (1)

I ρ₊= ¹

Tr(MµρMµ^†)MµρM_µ^† when the atom collapses inµ=g,e.

This happens with probability Tr MµρM_µ^† .

I Detection error rates: P(y =e/µ=g) =η_g∈[0,1]the probability of erroneous assignation toewhen the atom collapses ing;P(y =g/µ=e) =η_e∈[0,1](given by the contrast of the Ramsey fringes).

Bayes law gives the probability that the atom collapses inµ=g knowing the detector outcomey =g:

P(µ=g/y =g) =

(1−η_g)Tr

MgρM_g^† (1−η_g)Tr

MgρM_g^†

+η_eTr

MeρM_e^†

sinceP(y =g/µ=g) = (1−η_g)andP(y =g/µ=e) =η_e.

Markov chain with detection errors (2)

The expectation valueρb₊ofρ₊knowingρand the imperfect detection y =g is given by

ρb+=P(µ=g/y =g) ^MgρM

† g

Tr(^MgρM^†_g)+P(µ=e/y =g) ^MeρMe†

Tr(^MeρM^†_e) Since

P(µ=g/y =g) = ^(1−ηg)Tr(MgρM_g^†)

(1−ηg)Tr(MgρM^†g)^+ηeTr(MeρMe^†)

andP(µ=e/y =g) =1−P(µ=g/y =g), this expectation value ρb+is given by

ρb+= ¹

Tr((1−ηg)MgρM_g^†+η_eMeρM_e^†)

(1−η_g)M_gρM_g^†+η_eM_eρM_e^†

Similarly wheny =e, the expectation valueρb₊is given by

ρb+= ¹

Tr((1−ηe)MeρM_e^†+η_gMgρM_g^†)

(1−η_e)MeρM_e^†+η_gMgρM_g^†

Markov chain with detection errors (3)

We get

ρb+=











(1−ηg)MgρM^†_g+ηeMeρM_e^†

Tr((1−ηg)MgρM_g^†+η_eMeρM_e^†), with prob. Tr

(1−ηg)MgρM_g^†+ηeMeρM_e^†

;

ηgMgρM_g^†+(1−ηe)MeρM_e^†

Tr(^ηgMgρM^†_g+(1−ηe)MeρM_e^†) with prob. Tr

ηgMgρM_g^†+ (1−ηe)MeρM_e^† .

Key point:

Tr

(1−η_g)M_gρM_g^†+η_eM_eρM_e^†

and Tr

η_gM_gρM_g^†+ (1−η_e)M_eρM_e^†

are the probabilities to detecty =g ande, knowingρ.

Withη_µ⁰_,µbeing the probability to detecty =µ⁰knowing that the atom collapses inµ, we have

ρb+= P

µηµ⁰,µMµρM_µ^† Tr

P

µηµ⁰,µMµρMµ^†

when we detecty =µ⁰.

The probability to detecty =µ⁰knowingρis Tr P

µηµ⁰,µMµρM_µ^† .

The Markov chain with cavity decoherence

When the sampling time∆T is much smaller than the photon life time T_cav, cavity decoherence (at zero temperature) can be described approximatively by the Kraus map

ρ7→M0ρM₀^†+M−ρM₋^† withM₀= (1−_2T^∆T

cav)I−_2T^∆T

cavNandM−= q∆T

Tcava

M0andM−can be seen as "measurement" operators corresponding to information catched by the "environment", information unknown in the real life but known in "Matlab/Simulink world":

I M₀corresponds to no photon destruction during the sampling interval∆T; probability Tr

M₀ρM₀^† .

I M−corresponds to one photon destruction during the sampling interval∆T; probability Tr

M−ρM₋^† .

The fact that we do not have access to this information can be interpreted as a detection error of 50%forM0andM−. We get

ρb+=M0ρM₀^†+M−ρM₋^†.

Photon-box quantum filter parameterized by left stochastic matrix

η_µ⁰_,µ ³ ρb^est_k+1

=

¹

Tr P

µη_µ0,µMµρb^est_k Mµ^†

P

µη_µ⁰_,µ

M

_µρb^est_k

M

µ^†

with

I

we have a total of m = 3

×

7 = 21 Kraus operators M

_µ

. The "jumps" are labeled by

µ

= (µ

^a, µ^c

) with

µ^a∈ {no,

g, e, gg, ge, eg, ee} labeling atom related jumps and

µ^c ∈ {o,+,−}cavity decoherence jumps.

I

we have only m

⁰

= 6 real detection possibilities

µ⁰ ∈ {no,

g, e, gg, ge, ee} corresponding respectively to no detection, a single detection in g, a single detection in e, a double detection both in g, a double detection one in g and the other in e, and a double detection both in e.

µ⁰\µ (no, µ^c) (g, µ^c) (e, µ^c) (gg, µ^c) (ee, µ^c) (ge, µ^c)or(eg, µ^c)

no 1 1−_d 1−_d (1−_d)² (1−_d)² (1−_d)²

g 0 _d(1−ηg) _dηe 2_d(1−_d)(1−ηg) 2_d(1−_d)ηe _d(1−_d)(1−ηg+ηe) e 0 _dηg _d(1−ηe) 2_d(1−_d)ηg 2_d(1−_d)(1−ηe) _d(1−_d)(1−ηe+ηg)

gg 0 0 0 ²_d(1−ηg)^{2 2}_dη²_e ²_dηe(1−ηg)

ge 0 0 0 2²_dηg(1−ηg) 2²_dηe(1−ηe) ²_d((1−ηg)(1−ηe) +ηgηe)

ee 0 0 0 ²_dη²_g ²_d(1−ηe)^{2 2}_dηg(1−ηe)

3Somaraju, A.; Dotsenko, I.; Sayrin, C. & PR. Design and Stability of Discrete-Time Quantum Filters with Measurement Imperfections. American Control Conference, 2012, 5084-5089.

Markov chain in ideal life (e.g. Matlab/Simulink world): pure state

ρk

ρk+1

=

Mµk

(ρ

k

) =: M

_µkρk

M

_µ^†_k

Tr

M

_µkρk

M

_µ^†_k

I To each measurement outcomeµis attached the Kraus operator Mµdepending onµand also on time (not explicitly recalled here, Mµ=Mµ,k could depend onk).

I µ_k is a random variable taking valuesµin{1,· · ·,m}with probabilitypµ,ρk = Tr Mµρ_kM_µ^†

. Conservation of probability (P

µpµ,ρ=1 for allρ) is guarantied byPm

µ=1M_µ^†Mµ=I.

I The Kraus mapK(ρ) =Pm

µ=1MµρM_µ^† provides

E

^{(ρk+1/ρk) =K(ρk)}

The Markov chain in real life: mixed states,

ρbk

and

ρb^est_k

(1)

⁴ Takeρ_k+1= ¹

Tr(MµkρkMµ^†k)

Mµkρ_kM_µ^†_k

with measure imperfections and decoherence described by theleft stochastic matrixη:

ηµ⁰,µ∈ |0,1]is the probability of having the imperfect outcome µ⁰∈ {1, . . . ,m⁰}knowing that the perfect one isµ∈ {1, . . . ,m}.

I ρb_k =

E

^{ρk|ρ0, µ0}0, . . . , µ⁰_k−1

can be computed efficiently via the following recurrence

ρb_k+1= ¹

TrPm µ=1η_µ0

k,µMµρb_kMµ^†





m

X

µ=1

η_µ⁰

k,µMµρb_kM_µ^†





where the detector outcomeµ⁰_k takes valuesµ⁰ in{1,· · ·,m⁰} with probabilitypµ⁰,ρbk = Tr

Pm µ=1η_µ⁰

k,µMµρb_kM_µ^† .

I Thus

E

^{(ρbk+1|ρbk) =K(ρbk) =Pm}µ=1Mµρb_kM_µ^†.

4Somaraju, A.; Dotsenko, I.; Sayrin, C. & PR. Design and Stability of Discrete-Time Quantum Filters with Measurement Imperfections. American Control Conference, 2012, 5084-5089.

The Markov chain in real life: mixed states,

ρbk

and

ρb^est_k

(2)

ρb_k =

E

^{ρk|ρ0, µ0}0, . . . , µ⁰_k−1

is given by

ρb_k+1= ¹

TrPm µ=1η_µ0

k,µMµρb_kMµ^†





m

X

µ=1

η_µ⁰

k,µMµρb_kM_µ^†





with theperfect initialization:ρb₀=ρ₀. ρb^est_k+1= ¹

Tr Pm

µ=1η_µ0

k,µMµρb^est_k Mµ^†

Pm µ=1η_µ⁰

k,µMµρb^est_k M_µ^†

but with imperfect initializationρb^est₀ 6=ρ₀.

This filtering process is stable⁵: the fidelityF(ρb_k,ρb^est_k )is a sub-martingale for anyηandMµ:

E

^{F(ρbk+1,ρbest}k+1)/ρb_k

≥F(ρb_k,ρb^est_k )

Convergence ofρb^est_k towardsρb_k whenk 7→+∞is an open problem.⁶

5PR. Fidelity is a Sub-Martingale for Discrete-Time Quantum Filters IEEE Transactions on Automatic Control, 2011, 56, 2743-2747 .

6A partial result (continuous-time): R. van Handel. The stability of quantum Markov filters. Infin. Dimens. Anal. Quantum Probab. Relat. Top. , 2009, 12, 153-172.

Bayesian parameter estimations

Consider detector outcomesµ⁰_k corresponding to a parameter valuep¯ poorly known. Assume to simplify that eitherp¯=aor¯p=b, with a6=b. We can discriminate betweenaandband recover¯pvia the following Bayesian scheme using information contained in theµ⁰_k’s:

ρb^est_a,k+1=

P

µη^a_µ0

k,µM_µ^aρb^est_a,kM_µ^a† Tr

P

p

P

µη^p

µ0

k,µMµ^pρb^est_p,kMµ^p

†, ρb^est_b,k+1=

P

µη^b_µ0

k,µM^b_µρb^est_b,kM^b_µ^† Tr

P

p

P

µη^p

µ0

k,µMµ^pρb^est_p,kMµ^p

†

with initializationρb^est_a,k+1=ρb^est_b,k+1=ρb^est₀ /2 whereρb^est₀ is some guess ofρb₀assuming initial probability of ¹₂ to havep¯=aand¯p=b.

This estimation/filtering process is also stable:

•F(ρb_k,ρb^est_a,k) +F(ρb_k,ρb^est_b,k)is a sub-martingale

• Tr ρb^est_a,k

, Tr ρb^est_b,k

) estim. of proba. to havep¯=a,¯p=b.

Direct generalization to a continuumof choices forp¯∈[p_min,p^max] (see⁷for a first experimental use)

7Brakhane, S.; Alt, W.; Kampschulte, T.; Martinez-Dorantes, M.; Reimann, R.; Yoon, S.; Widera, A. & Meschede, D. Bayesian Feedback Control of a Two-Atom Spin-State in an Atom-Cavity System. Phys. Rev. Lett., 2012, 109, 173601-

Dynamical models with a precise structure

Discrete-time modelsareMarkov chains

ρ_k+1= 1

pµ(ρ_k)Mµρ_kM_µ^† with proba. pµ(ρ_k) = Tr Mµρ_kM_µ^† associated toKraus maps(ensemble average, open quantum channels)

E

^{(ρk+1/ρk) =K(ρk) =X}

µ

Mµρ_kM_µ^† with X

µ

M_µ^†Mµ=I

Continuous-time modelsarestochastic differential systems dρ=

−i[H, ρ] +LρL^†−¹

2(L^†Lρ+ρL^†L) dt +

Lρ+ρL^†− Tr (L+L^†)ρ ρ

dw driven byWiener processes⁸dw =dy− Tr (L+L^†)ρ

dt with measurey and associated toLindbaldmaster equations:

d dtρ=−ⁱ

~[H, ρ] +LρL^†−¹

2(L^†Lρ+ρL^†L)

8Another possibility: SDE driven by Poisson processes.

From discrete-time to continuous-time: heuristic connection For Monte-Carlo simulations of

d

ρ

=

−i[H, ρ] +

LρL

^†−¹

2

(L

^†

Lρ +

ρL^†

L)

dt +

Lρ +

ρL^†−

Tr

(L + L

^†

)ρ

ρ

dw

take a small sampling time dt, generate a random number

dwt

according to a Gaussian law of

standard deviation√

dt, and set ρ_t+dt

=

Mdyt

(ρ

t

) where the jump operator

Mdyt

is labelled by the measurement outcome

dyt

= Tr (L + L

^†

)

ρ_t

dt +

dwt

:

Mdyt

(ρ

t

) =

^I+(−iH−

1

2L^†L)dt+dytL

ρt I+(iH−¹₂L^†L)dt+dytL^†

Tr

I+(−iH−¹₂L^†L)dt+dytL

ρt I+(iH−¹₂L^†L)dt+dytL^†.

Then

ρ_t+dt

remains always a density operator and using the

Itô rules

(dw of order

√

dt and

dw²≡

dt ) we get the good

dρ =

ρ_t+dt −ρt

up to O((dt)

^3/2

) terms.

From discrete-time to continuous-time: heuristic connection (end)

For the Lindblad equation d

dt

ρ

=

−ⁱ

~

[H, ρ] + LρL

^†−¹

2

(L

^†

Lρ +

ρL^†

L) take a small sampling time

dt

and set

ρt+dt

= I +

dt

(−iH

−¹₂

L

^†

L)

ρt

I +

dt

(iH

−¹₂

L

^†

L)

+

dt

Lρ

_t

L

^†

Tr I +

dt

(−iH

− ¹₂

L

^†

L)

ρt

I +

dt

(iH

−¹₂

L

^†

L)

+

dt

Lρ

t

L

^†.

Then

ρ_t+dt

remains always a density operator and

d

dtρ

= (ρ

_t+dt−ρ_t

)/dt up to O(dt ) terms.

SDE driven by Poisson and/or Wiener processes

dρ

t

=

L(ρ_t

) dt+

mw

X

ν=1

Λ

_ν

(ρ

t

) dw

_t^ν

+

mP

X

µ=1

Υ

_µ

(ρ

t

)

dN

_t^µ−

Tr

C

_µρt

C

_µ^†

dt

,

where

I L(ρ_t

) :=

−i

[H, ρ

_t

] +

PmP

µ=1L^P_µ

(ρ

t

) +

Pmw

ν=1L^w_ν

(ρ

t

),

L^P_µ

(ρ) :=

−¹₂{C_µ^†

C

_µ, ρ}

+ C

_µρC_µ^†, L^w_ν

(ρ) :=

−¹₂{L^†_ν

L

_ν, ρ}

+ L

_νρL^†_ν

; Υ

_µ

(ρ) :=

^CµρC

† µ

Tr

CµρC_µ^†−ρ,

Λ

_ν

(ρ) := L

_νρ

+

ρL^†ν −

Tr

(L

_ν

+ L

^†ν

)ρ

ρ

I

Detector click no

µ

is related to the Poisson process dN

_t^µ

= N

^µ

(t + dt )

−

N

^µ

(t) = 1 and happens with probability Tr

C

µρt

C

µ†

dt;

I

Continuous detector y

_t^ν

is related to the Wiener process dw

_t^ν

by dy

_t^ν

= dw

_t^ν

+ Tr

(L

ν

+ L

^†ν

)ρ

t

dt.

Quantum filter in the ideal case

dρt =L(ρ_t)dt+

mw

X

ν=1

Λν(ρ_t)dw_t^ν+

mP

X

µ=1

Υµ(ρ_t) dN_t^µ− Tr Cµρ_tC_µ^† dt

,

and the associated quantum filter

dρb^est_t =L(ρb^est_t )dt+

mw

X

ν=1

Λν(ρb^est_t ) dy_t^ν− Tr (Lν+L^†_ν)ρb^est_t dt

+

mP

X

µ=1

Υµ(ρb^est_t ) dN_t^µ− Tr Cµρb^est_t C_µ^† dt

.

It can be rewritten as follows

dρb^est_t =L(ρb^est_t )dt+

mw

X

ν=1

Λν(ρb^est_t ) Tr (Lν+L^†_ν)ρ_t

−Tr (Lν+L^†_ν)ρb^est_t dt

+

mw

X

ν=1

Λ_ν(ρb^est_t )dw_t^ν+

mP

X

µ=1

Υ_µ(ρb^est_t ) dN_t^µ− Tr Cµρb^est_t C_µ^† dt

.

Quantum filters with imperfections and decoherence

⁹

(1)

I

Imperfection model for the Poisson processes dN

_t^µ

:

I real outcomesµ⁰∈ {0,1, . . . ,m⁰_P}

I ideal outcomesµ∈ {0,1, . . . ,m_P}.

I (m_P⁰ +1)×m_Pleft stochastic matrix η^P = (η^P_µ0,µ)_0≤µ0≤m⁰_P,1≤µ≤mP

I positive vectorη¯^P= (¯η_µ^P0)_1≤µ⁰_≤m⁰

P inR^m

0 P

+ .

I

Imperfection model for the diffusion processes dw

_t^ν

:

I m_w⁰ real continuous signalsy_t^ν0 withν⁰∈ {1, . . . ,m_w⁰ },

I mw ideal continuous signalsy_t^νwithν∈ {1, . . . ,mw}

I correlationm_w⁰ ×mw matrixη^w= (η_ν^w0,ν)_1≤ν⁰_≤m⁰_{w,1≤ν≤mw}, with 0≤η_ν^w0,ν≤1 andPmrw

ν⁰=1η_ν^w0,ν≤1.

9see last chapter of H. Amini. Stabilization of discrete-time quantum systems and stability of continuous-time quantum filters. PhD thesis, Mines-ParisTech, September 2012.

Quantum filters with imperfections and decoherence (2)

dρb_t = L(ρb_t)dt+

m⁰_w

X

ν⁰=1

pη¯^w_ν0Λb_ν⁰(ρb_t)dwb_t^ν0

+

m⁰_P

X

µ⁰=1

Υb_µ⁰(ρb_t)

dNb_t^µ0−η¯_µ^P0dt−

mP

X

µ=1

η^P_µ0,µTr Cµρb_tC_µ^† dt

I η¯^w_ν0 =Pmw

ν=1η^w_ν0,ν ,Υb_µ⁰(ρ) := ^η¯

P µ0ρ+P_mP

µ=1η_µ^P_0,µCµρC^†_µ

¯ η^P

µ0+P_mP

µ=1η^P

µ0,µTr(CµρCµ^†)−ρ, bΛν⁰(ρ) =bLν⁰ρ+ρbL^†_ν0− Tr

(bLν⁰+bL^†_ν0)ρ

ρ, bLν⁰ := (Pmw

ν=1η_ν^w0,νLν)/η¯_ν^w0;

I the jump detectorµ⁰corresponds toNb^µ0(t):

dNb_t^µ0 =Nb^µ0(t+dt)−Nb^µ0(t) =1 happens with probability Pbµ⁰(ρb_t) = ¯η^P_µ0dt+PmP

µ=1η_µ^P0,µTr Cµρb_tC_µ^† dt;

I the continuous detectorν⁰refers toby_t^ν0 anddwb_t^ν0: dby_t^ν0 =dwb_t^ν0+p

¯ η_ν^w0 Tr

(bLν⁰+bL^†_ν0)ρb_t dt.

Quantum filters with imperfections and decoherence (3)

dρb_t = L(ρb_t)dt+

m⁰_w

X

ν⁰=1

pη¯^w_ν0Λbν⁰(ρb_t)dwb_t^ν0

+

m⁰_P

X

µ⁰=1

Υbµ⁰(ρb_t)

dNb_t^µ0−η¯_µ^P0dt−

mP

X

µ=1

η^P_µ0,µTr Cµρb_tC_µ^† dt

and the associated quantum filter

dρb^est_t = L(ρb^est_t )dt+

m⁰_w

X

ν⁰=1

pη¯_ν^w0bΛν⁰ρb^est_t )

dyb_t^ν0−p

¯ η^w_ν0 Tr

(bLν⁰+bL^†_ν0)ρb^est_t dt

+

m_P⁰

X

µ⁰=1

Υbµ⁰(ρb^est_t )

dNb_t^µ0−η¯^P_µ0dt −

mP

X

µ=1

η_µ^P0,µTr Cµρb^est_t C_µ^† dt

Quantum filtering combines the following key points

1. Bayes law: P(µ⁰/µ) =P(µ/µ⁰)P(µ⁰)/ P

ν⁰P(µ/ν⁰)P(ν⁰) . 2. Schrödinger equationsdefining unitary transformations.

3. Partial collapse of the wave packet:irreversibility and

convergence are induced by the measure of observablesOwith degenerate spectra,O=P

µλµPµ:

I measure outcomeλ_µ with proba.pµ=hψ|Pµ|ψi= Tr(ρPµ) depending|ψi,ρjust before the measurement

I measure back-action if outcomeµ:

|ψi 7→ |ψi+ = Pµ|ψi

phψ|Pµ|ψi, ρ7→ρ+ = PµρPµ

Tr(ρPµ) 4. Tensor product for the description of composite systems(S,M):

I Hilbert spaceH=H_S⊗ H_M

I HamiltonianH=HS⊗IM+Hint+IS⊗HM I observable on sub-systemM only:O=IS⊗ O_M.