Models and feedback stabilization of open quantum systems

Models and feedback stabilization of open quantum systems

International Congress of Mathematicians (ICM2014) August 13 - 21, Seoul , Korea.

Pierre Rouchon

Centre Automatique et Systèmes Mines ParisTech

PSL Research University

Quantum feedback: the back-action of the measurement.

A typical stabilizing feedback-loop for a classical system

system

controller

Two kinds of stabilizing feedbacks for quantum systems 1. Measurement-based feedback: controller is classical;

measurement back-action on the systemS is stochastic (collapse of the wave-packet); the measured outputy is a classical signal; the control inputuis a classical variable appearing in some controlled Schrödinger equation;u(t) depends on the past measurementsy(τ),τ≤t.

2. Coherent/autonomous feedback and reservoir engineering:the systemSis coupled tothe controller, another quantum system; the composite system,HS⊗Hcontroller, is an

open-quantum system relaxing to some target (separable) state.

2 / 33

Controlling quantum degrees of freedom

Several applications:

I Nuclear Magnetic Resonance (NMR) applications;

I Quantum chemical synthesis;

I High resolution measurement devices (e.g. atomic/optic clocks);

I Quantum information processing:quantum computation and quantum communication.

Physics Nobel prize 2012:

Serge Haroche David J. Wineland

Nobel prize: ground-breaking experimental methods that enablemeasuring and manipulation of individual quantum systems.

Outline

The LKB photon box

First experimental realization of a quantum-state feedback (2011) Why density operatorρinstead of wave function|ψi

Stabilization of "Schrödinger cats" by reservoir engineering

Model structure of open quantum systems

Conclusion: some open issues

4 / 33

The first experimental realization of a

The photon box of the Laboratoire Kastler-Brossel (LKB):

group of S.Haroche (Nobel Prize 2012), J.M.Raimond and M. Brune.

u y

1

Stabilization of a quantum state with exactlyn=0,1,2,3, . . .photon(s).

Experiment:C. Sayrin et. al., Nature 477, 73-77, September 2011.

Theory:I. Dotsenko et al., Physical Review A, 80: 013805-013813, 2009.

R. Somaraju et al., Rev. Math. Phys., 25, 1350001, 2013.

H. Amini et. al., Automatica, 49 (9): 2683-2692, 2013.

1

Three quantum features emphasized by the LKB photon box

d

dt|ψi=−_~ⁱH|ψi, d

dtρ=−_~ⁱ[H, ρ], H=H0+uH1

2. Origin of dissipation: collapse of the wave packetinduced by the measurement of observableOwith spectral decomp.P

µλµPµ:

I measurement outcomeµwith proba.

Pµ=hψ|Pµ|ψi= Tr(ρPµ)depending on|ψi,ρjust before the measurement

I measurement back-action if outcomeµ=y:

|ψi 7→ |ψi⁺= P_y|ψi

phψ|P_y|ψi, ρ7→ρ+= P_yρP_y Tr(ρPy) 3. Tensor product for the description of composite systems(S,M):

I Hilbert spaceH=H^S⊗ H^M

I HamiltonianH=H_S⊗I_M+H_int+I_S⊗H_M

I observable on sub-systemM only:O=I_S⊗O_M.

2S. Haroche and J.M. Raimond.Exploring the Quantum: Atoms, Cavities and Photons.Oxford Graduate Texts, 2006.

6 / 33

Composite system built with an harmonic oscillator and a qubit.

I SystemS

corresponds to a quantized harmonic oscillator:

HS = ( _∞

X

n=0

ψ_n|ni

(ψ_n)^∞_n=0∈l²(C) )

,

where

represents the Fock state associated to exactly

photons inside the cavity

I MeterM

is a qu-bit, a 2-level system (idem 1/2 spin system) :

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

with

|c_g|²+|c_e|²=

1; atoms leaving

are all in state

I State of the full system|Ψi ∈ HS⊗ HM

:

|Ψi=

+∞

X

n=0

Ψ_ng|ni ⊗ |gi+ Ψ_ne|ni ⊗ |ei, Ψ_ne,Ψ_ng ∈C.

Ortho-normal basis:

.

S: quantum harmonic oscillator (spring system)

I Hilbert space:

HS =n P

n≥0ψ_n|ni, (ψ_n)_n≥0∈l²(C)o

≡L²(R,C)

I Quantum state space:

D={ρ∈ L(HS), ρ^† =ρ, Tr(ρ) =1, ρ≥0}.

I Operators and commutations:

a|ni=√

n|n-1i,a^†|ni=√

n+1|n+1i; N=a^†a,N|ni=n|ni;

[a,a^†] =I,af(N) =f(N+I)a;

Dα=e^{αa†−α†a}. a=X +iP= ^√1

2 x+_∂x^∂

, [X,P] =ıI/2.

I Hamiltonian:H_S/~=ω_ca^†a+uc(a+a^†).

(associated classical dynamics:

dx

dt =ω_cp, ^dp_dt =−ω_cx−√ 2uc).

I Classical pure state≡coherent state|αi α∈C: |αi=P

n≥0

e^{−|α|2/2√αn}

n!

|ni;|αi ≡π^1/41 e^ı

√2x=αe^−(x−

√ 2<α)2 2

a|αi=α|αi,Dα|0i=|αi.

|0

|1

|2 ω

| n

ω

u

... ...

8 / 33

M: 2-level system, i.e. a qubit (half-spin system)

I Hilbert space:

H^M =C²=n

cg|gi+ce|ei, cg,ce∈C o

.

I Quantum state space:

D={ρ∈ L(H^M), ρ^† =ρ, Tr(ρ) =1, ρ≥0}.

I Operators and commutations:

σ_-=|gihe|,σ₊=σ_-^†=|eihg| σ_x=σ_-+σ₊=|gihe|+|eihg|; σ_y=iσ_-−iσ₊=i|gihe| −i|eihg|; σ_z =σ₊σ_-−σ_-σ₊=|eihe| − |gihg|; σ_x²=I,σ_xσ_y =iσ_z,[σ_x,σ_y] =2iσ_z, . . .

I Hamiltonian:H_M/~=ω_qσ_z/2+uqσ_x.

I Bloch sphere representation:

D=n

1

2 I+xσx+yσ_y+zσ_z (x,y,z)∈R³, x²+y²+z²≤1o

| g

| e ω

u

The Markov model (1)

C

B

D R

R

B R

I When atom comes outB,|ΨiB of the full system isseparable

|ΨiB=|ψi ⊗ |gi.

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

|ΨiR2 =U_SM |ψi ⊗ |gi

= M_g|ψi

⊗ |gi+ M_e|ψi

⊗ |ei whereUSM is a unitary transformation (Schrödinger propagator) defining the linear measurement operatorsM_gandM_eonH^S. SinceU_SMis unitary,M^†_gMg+M^†_eMe=I.

10 / 33

The Markov model (2)

C

B

D R

R

H _{S M}

The unitary propagatorU_SM is derived from Jaynes-Cummings HamiltonianH_SMin the interaction frame.

Two kind of qubit/cavity Halmitonians:

resonant,H_SM/~=i Ω(vt)/2

a^†⊗σ_-−a⊗σ₊ , dispersive,HSM/~= Ω²(vt)/(2δ)

N⊗σ_z, whereΩ(x) = Ω₀e^−x

2

w2,x =vt withv atom velocity,Ω₀vacuum Rabi pulsation,wradial mode-width and whereδ=ω_q−ω_c is the detuning between qubit pulsationω_qand cavity pulsationω_c (|δ| Ω₀).

The Markov model (3)

Just beforeD, the field/atom state isentangled:

Mg|ψi ⊗ |gi+Me|ψi ⊗ |ei

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

|ΨiD=√¹

Py

(M_y|ψi)⊗ |yi= q ^My

hψ|M^†_yMy|ψi|ψi

!

⊗ |yi. Markov process(density matrix formulationρ∼ |ψihψ|)

ρ₊=







MgρM^†_g

Tr(^MgρM^†_g), with probabilityPg= Tr

MgρM^†_g

;

MeρM^†_e

Tr(MeρM^†_e), with probabilityPe= Tr

M_eρM^†_e . Kraus map:

E

12 / 33

A controlled Markov process (input

Stateρ: the density operator of the photon(s) trapped in the cavity.

Outputy: quantum projective measurement of the probe atom.

Theideal modelreads

ρ_k+1=



















DukMgρ_kM^†_gD^†_uk Tr

M_gρ_kM^†_g yk =gwith probabilityPg,k = Tr

M_gρ_kM^†_g

DukMeρ_kM^†_eD^†_uk Tr

Meρ_kM^†_e y_k =ewith probabilityPe,k = Tr

M_eρ_kM^†_e

I Displacement unitary operator(u∈R):D_u=e^ua†−uawith a=upper diag(√

1,√

2, . . .)the photon annihilation operator.

I Measurement Kraus operators in the linear dispersive case M_g=cos_φ

0N+φR

2

andM_e=sin_φ

0N+φR

2

:M^†_gM_g+M^†_eM_e=I withN=a^†a=diag(0,1,2, . . .)the photon number operator.

u =

0: Quantum Non Demolition (QND) measurement of photons.

ρ_k+1=



















cos

φ₀N+φR

2

ρ_kcos

φ₀N+φR

2

Tr

cos²

φ₀N+φR

2

ρ_k with prob. Tr cos²_φ

0N+φ_R 2

ρk

sin

φ₀N+φR

2

ρ_ksin

φ₀N+φR

2

Tr

sin²

φ₀N+φR

2

ρ_k with prob. Tr sin²_φ

0N+φ_R 2

ρk

Steady state:any Fock stateρ=|n¯ihn¯|(n¯∈N) is a steady-state (no other steady state when(φ_R, φ₀, π)areQ-independent)

Martingales:for any real functiong,Vg(ρ) = Tr(g(N)ρ)is a martingale:

E

Convergence to a Fock statewhen(φ_R, φ₀, π)areQ-independent:

V(ρ) =−¹2

P

nhn|ρ|ni²is a super-martingale with

E

For a realization starting fromρ₀, the probability to converge towards the Fock state|¯nihn¯|is equal to Tr(|n¯ihn¯|ρ₀) =h¯n|ρ₀|n¯i. 14 / 33

Structure of the stabilizing quantum-state feedback scheme

I Goal: stabilization of the steady-state|n¯ihn¯|(controller set-point).

I At each time stepk:

1. readyk the measurement outcome for probe atomk. 2. update the quantum state estimationρ_k−1toρ_k fromyk

3. computeuk as a function ofρ_k(state feedback).

4. apply the micro-wave pulse of amplitudeuk.

Observer/controllerexploiting thequantum separation principle³: 1. real-time state estimationbased on asymptotic observer: here

quantum filteringtechniques;

2. state feedbackstabilization towards a stationary regime: here control Lyapunovtechniques constructed with open-loop martingales Tr(g(N)ρ)and inversion of a Laplacian matrix.

3L. Bouten and R. van Handel: On the separation principle of quantum control. InQuantum Stochastics and Information: Statistics, Filtering and Control, V. P Belavkin and M. I. Guta (Eds.) World Scientific, 2008.

Experimental closed-loop data

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

16 / 33

Outline

The LKB photon box

First experimental realization of a quantum-state feedback (2011) Why density operatorρinstead of wave function|ψi

Stabilization of "Schrödinger cats" by reservoir engineering

Model structure of open quantum systems

Conclusion: some open issues

LKB photon-box: Markov process with detection errors (1)

I With pure stateρ=|ψihψ|, we have ρ₊=|ψ₊ihψ₊|= 1

Tr

MµρM^†_µMµρM^†_µ

when the atom collapses inµ=g,ewith proba. 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: expectationρ+of|ψ+ihψ+|knowingρand the imperfect detectiony.

ρ₊=











(1−ηg)MgρM^†_g+η_eMeρM^†_e

Tr(^(1−ηg)MgρM^†g+ηeMeρM^†e)^ify=g, prob. Tr

(1−ηg)MgρM^†_g+ηeMeρM^†_e

;

η_gMgρM^†_g+(1−ηe)MeρM^†_e

Tr(^ηgMgρM^†_g+(1−ηe)MeρM^†_e)^ify=e, prob. Tr

ηgMgρM^†_g+ (1−ηe)MeρM^†_e .

ρ+does not remain pure: the quantum stateρ+becomes a mixed

state;|ψ+ibecomes physically irrelevant (not numerically). _{18 / 33}

LKB photon-box: Markov process with detection errors (2)

ρ+=











(1−ηg)MgρM^†_g+η_eMeρM^†_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.

Generalizationby merging aKraus mapK(ρ) =P

µMµρM^†_µwhere P

µM^†_µMµ=I with aleft stochastic matrix(ηµ⁰,µ):

ρ+= P

µη_y,µMµρM^†_µ Tr

P

µη_y,µMµρM^†_µ when we detecty =µ⁰. The probability to detecty =µ⁰knowingρis Tr

P

µη_µ⁰_,µMµρM^†_µ .

Photon-box quantum filter: 6

21 left stochastic matrix

Tr(^Pµη_yk,µMµρkM^†µ) P

µη_yk,µMµρ_kM^†_µ where

I we have a total ofm=3×7=21 Kraus operatorsMµ. The

"jumps" are labeled byµ= (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 onlym⁰=6 real detection possibilities

y =µ⁰∈ {no,g,e,gg,ge,ee}corresponding respectively to no detection, a single detection ing, a single detection ine, a double detection both ing, a double detection one ingand the other ine, and a double detection both ine.

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

no 1 1−d 1−d (1−d)² (1−d)² (1−d)²

g 0 d(1−ηg) dηe 2d(1−d)(1−ηg) 2d(1−d)ηe d(1−d)(1−ηg+ηe) e 0 dηg d(1−ηe) 2d(1−d)ηg 2d(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) 20 / 33

Outline

The LKB photon box

First experimental realization of a quantum-state feedback (2011) Why density operatorρinstead of wave function|ψi

Stabilization of "Schrödinger cats" by reservoir engineering

Model structure of open quantum systems

Conclusion: some open issues

Wigner functions of some quantum states for an harmonic oscillator

n≥0

e^{−|α|2/2√αn}

n!

|ni;

Phase-cat states:N |αi+| −αi .

Wigner functionW^ρassociatedρ:W^ρ:C3ξ→ π² Tr e^iπND−ξρDξ

Re(ξ)

Im(ξ)

−0.6

−0.4

−0.2 0 0.2 0.4 Fock state |n=0> Fock state |n=3> Coherent state |α=1.8> 0.6

Coherent state |-α> Statistical mixture of

|-α> and |α> Cat state |-α>+|α>

-3-3

0 0

3 3

22 / 33

Stabilizing "Schrödinger cats"

-α

2.

Cavity mode (system) atom (controller)

R1

R2

Box of atoms

Aim:

engineer atom-mode interaction, to stabilize |-α +|α

DC field:

(controls atom frequency) ENS experiment

Jaynes-Cumming Hamiltionian

H(t)/~=ω_ca^†a⊗IM+ω_q(t)IS⊗σ_z/2+iΩ(t) a^†⊗σ_-−a⊗σ₊ /2 with the open-loop controlt7→ω_q(t)combiningdispersiveω_q6=ω_c andresonantω_q=ω_cinteractions.

Key issues:convergenceofρ_k+1=K(ρ_k) =M_gρ_kM^†_g+M_eρ_kM^†_e

4A. Sarlette et al: Stabilization of Nonclassical States of the Radiation Field in a Cavity by Reservoir Engineering. Physical Review Letters, Volume 107, Issue 1, 2011.

Convergence of

iterates towards

-α

2

ρ^Kerr_k+1=K^Kerr(ρ^Kerr_k ) =M^Kerr_g ρ^Kerr_k (M^Kerr_g )^†+M^Kerr_e ρ^Kerr_k (M^Kerr_e )^†. withM^Kerr_g =cos(^u₂)cos(θN/2) +sin(^u₂)^{sin(θ√N/2)}

N a^† and

M^Kerr_e =sin(^u₂)cos(θN+1/2)−cos(^u₂)a^{sin(θ√N/2)}

N .

Assume|u| ≤π/2,θ₀=0,θ_n∈]0, π[forn>0 and limn7→+∞θ_n=π/2, then (Zaki Leghtas, PhD thesis (2012))

I exists aunique common eigen-state|ψ^KerriofM^Kerr_g andM^Kerr_e : ρ^Kerr_∞ =|ψ^Kerrihψ^Kerr|fixed point ofK^Kerr.

I if, moreovern7→θ_nis increasing, limk7→+∞ρ^Kerr_k =ρ^Kerr_∞. For well chosen experimental parameters,ρ^Kerr_∞ ≈ |α_∞ihα_∞|and h^Kerr_N ≈πN²/2. Sincee^−iπ2N2|α_∞i= ^e−iπ/4√

2 |α_∞i+i|-α∞i :

k7→+∞lim ρ_k = ¹₂

|α_∞i+i|-α∞i

hα_∞|+ih-α∞|

6

= ¹₂|α_∞ihα_∞|+¹₂|-α∞ih-α∞|.

24 / 33

Outline

The LKB photon box

First experimental realization of a quantum-state feedback (2011)

Why density operatorρinstead of wave function|ψi

Stabilization of "Schrödinger cats" by reservoir engineering

Model structure of open quantum systems

Conclusion: some open issues

Discrete-time models of open quantum systems

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 dissipation are induced by the measurement of observables with

degeneratespectra.

4. Tensor product for the description of composite systems.

VDiscrete-time models:Markov processesof stateρ, (density op.):

ρ_k+1=

Pm

µ=1η_µ0,µMµρ_kM^†_µ

Tr(^Pmµ=1η_µ0,µMµρkM^†µ), with proba. Pµ⁰(ρ_k) =Pm

µ=1η_µ⁰_,µTr

Mµρ_kM^†_µ associated toKraus maps(ensemble average, quantum channel)

E

µ

Mµρ_kM^†_µ with X

µ

M^†_µMµ=I

and left stochastic matrices (imperfections, decoherences)(ηµ⁰,µ).

26 / 33

Continuous/discrete-time Stochastic Master Equation (SME)

ρ_k+1=

Pm

µ=1η_µ0,µMµρ_kM^†_µ

Tr(^Pmµ=1η_µ0,µMµρ_kM^†µ), with proba. Pµ⁰(ρ_k) =Pm

µ=1ηµ⁰,µTr

Mµρ_kM^†_µ with ensemble averages corresponding toKraus linear maps

E

µ

Mµρ_kM^†_µ with X

µ

M^†_µMµ=I

Continuous-time models:stochastic differential systems

dρ_t = −_~ⁱ[H, ρt] +X

ν

Lνρ_tL^†_ν−¹₂(L^†_νLνρ_t+ρ_tL^†_νLν)

! dt

+X

ν

√ην

Lνρ_t+ρ_tL^†_ν− Tr

(Lν+L^†_ν)ρ_t ρ_t

dWν,t

driven byWiener processesdW_ν,t, with measurementsy_ν,t, dy_ν,t =√ην Tr

(Lν+L^†_ν)ρ_t

dt+dW_ν,t, detection efficiencies ην ∈[0,1]andLindblad-Kossakowskimaster equations (ην ≡0):

d

dtρ=−_~ⁱ[H, ρ] +X

ν

LνρL^†_ν−¹₂(L^†_νLνρ+ρL^†_νLν)

Continuous/discrete-time diffusive SME

(L+L^†)ρ_t

dt+dW_t and detection efficiencyη ∈[0,1], the quantum stateρ_t is usually mixed and obeys to

dρ_t =

−_~ⁱ[H, ρ_t] +Lρ_tL^†−¹₂(L^†Lρ_t+ρ_tL^†L) dt +√η

Lρt+ρ_tL^†− Tr

(L+L^†)ρ_t ρ_t

dWt

driven by the Wiener processdWt

WithIt¯o rules, it can be written as the following "discrete-time" Markov model

ρ_t+dt= Mdytρ_tM^†_dyt + (1−η)LρtL^†dt Tr

Mdytρ_tM^†_dy

t + (1−η)Lρ_tL^†dt

withM_dyt =I+

−_~ⁱH−¹2

L^†L

dt+√ηdy_tL.

28 / 33

Continuous/discrete-time jump SME

θ+ηTr Vρ_tV^† dt,and detection imperfections modeled byθ≥0 andη∈[0,1], the quantum stateρ_t is usually mixed and obeys to

dρ_t =

−i[H, ρ_t] +Vρ_tV^†−¹₂(V^†Vρ_t+ρ_tV^†V) dt +

θρ_t+ηVρ_tV^† θ+ηTr(Vρ_tV^†)−ρ_t

dN(t)−

θ+ηTr Vρ_tV^† dt

FordN(t) =0we have

ρ_t+dt= M0ρ_tM₀^†+ (1−η)Vρ_tV^†dt Tr

M0ρ_tM₀^†+ (1−η)Vρ_tV^†dt withM0=I− iH+¹₂V^†V

dt.

ForN(t+dt)−N(t) =1we have a similar transition rule ρ_t+dt= _Tr(•)^• whereρ_tis replaced byρ˜_t = θρ_t+ηVρ_tV^†

θ+ηTr(Vρ_tV^†).

Continuous/discrete-time diffusive-jump SME

The quantum stateρtis usually mixed and obeys to dρt =

−i[H, ρt] +LρtL^†−¹

2(L^†Lρt +ρtL^†L) +VρtV^†−¹

2(V^†Vρt+ρtV^†V)

dt +√

η

Lρt+ρtL^†− Tr

(L+L^†)ρt

ρt

dWt

+

θρt+ηVρtV^† θ+ηTr(VρtV^†)−ρt

dN(t)−

θ+ηTr

VρtV^† dt

FordN(t) =0we have

ρt+dt = Mdy_tρtM_dy^†

t+ (1−η)LρtL^†dt+ (1−η)VρtV^†dt

Tr

MdytρtM_dy^†

t + (1−η)LρtL^†dt+ (1−η)VρtV^†dt

withMdy_t =I− iH+¹₂L^†L+¹₂V^†V dt+√

ηdytL.

ForN(t+dt)−N(t) =1we have a similar transitionρt+dt= _Tr(•)^• whereρt

is replaced byρ˜t = θρt+ηVρtV^† θ+ηTr(VρtV^†).

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Continuous/discrete-time general diffusive-jump SME

The quantum stateρ_tis usually mixed and obeys to

dρ_t= −i[H, ρ_t] +X ν

Lνρ_tL^†_ν−¹₂(L^†_νLνρ_t+ρ_tL^†_νLν) +Vµρ_tV_µ^†−¹₂(V_µ^†Vµρ_t+ρ_tV_µ^†Vµ)

! dt

+X ν

√η_ν

L_νρ_t+ρ_tL^†_ν−Tr

(L_ν+L^†_ν)ρ_t ρ_t

dW_ν,t

+X µ







θµρ_t+P

µ0η_µ,µ0V_µ0ρ_tV^† µ0 θµ+P

µ0η_µ,µ0Tr V_µ0ρ_tV^†

µ0 −ρ_t









dN_µ(t)− θ_µ+X

µ0 η_µ,µ0Tr

V_µ0ρ_tV^† µ0 dt





whereη_ν∈[0,1],θ_µ, η_µ,µ0≥0 withη_µ0=P

µη_µ,µ0≤1 are parameters modelling measurements imperfections.

When∀µ,dN_µ(t) =0, we have

ρ_t+dt=

M_dytρ_tM^†

dyt+P

ν(1−η_ν)L_νρ_tL^†_νdt+P

µ(1−η_µ)V_µρ_tV_µ^†dt Tr

M_dytρ_tM^†

dyt+P

ν(1−ην)Lνρ_tL^†_νdt+P

µ(1−η_µ)Vµρ_tV_µ^†dt

withM_dyt =I− iH+¹₂P

νL^†_νL_ν+¹₂P µ V_µ^†V_µ

dt+P ν

√ηνdy_νtL_νand where dy_ν,t=√

η_νTr

(L_ν+L^†_ν)ρ_t

dt+dW_ν,t.

If, for someµ,Nµ(t+dt)−Nµ(t) =1, we have a similar transition ruleρ_t+dt= _Tr(•)^• but whereρ_tis replaced

byρ˜_t=

θµρ_t+P

µ0η_µ,µ0V_µ0ρ_tV^† µ0 θµ+P

µ0η_µ,µ0Tr V_µ0ρ_tV^†

µ0 .

Useful for positiveness-preserving numerical schemes

Conclusion: some open issues

I Few available convergence results in the low rank case:

most of available results are for full rank density operators either for Kraus maps (quantum channels)

ρ_k+1=K(ρ_k) =P

µMµρ_kM^†_µ, or for Lindblad-Kossakowski master equations :

d

dtρ=−_~ⁱ[H, ρ] +P

νLνρL^†_ν−¹2(L^†_νLνρ+ρ_tL^†_νLν).

I Continuous-time models with quantum input signal ?

Stochastic master equations driven by Wiener processes valid for classical (coherent) input signals of amplitudeu(see, e.g., the(S,L,H)-theory of quantum networks, J. Gough and M.

James, IEEE Trans. AC 2009); modelling issues for quantum input signals such as|ui+|-ui.

I The curse of dimensionality:composite quantum systems rely on tensor products whereas composite classical systems rely on Cartesian products . . . .

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Acknowledgments

I Former PhDs and PostDocs:Hadis Amini, Hector Bessa Silveira, Zaki Leghtas, Ram Somaraju.

I LKB Physicists: Michel Brune, Igor Dotsenko, Sébastien Gleyzes, Serge Haroche, Jean-Michel Raimond,Bruno Peaudecerf, Clément Sayrin, Xingxing Zhou.

I Quantic project (INRIA/ENS/MINES):Benjamin Huard, François Mallet, Mazyar Mirrahimi, Alain Sarlette,Rémi Azouit, Landry Bretheau, Philippe Campagne, Joachim Cohen, Emmanuel Flurin, Ananda Roy, Pierre Six.

I Mathematicians: Karine Beauchard, Jean-Michel Coron, Thomas Chambrion, Bernard Bonnard, Ugo Boscain, Sylvain Ervedoza, Stéphane Gaubert, Andrea Grigoriu, Claude Le Bris, Yvon Maday, Vahagn Nersesyan, Clément Pellégrini, Paulo Sergio Pereira da Silva, Jean-Pierre Puel, Lionel Rosier, Julien Salomon, Rodolphe Sepulchre, Mario Sigalotti, Gabriel Turinici.

I Mines Paris-Tech: Brigitte d’Andrea-Novel, Silvère Bonnabel, François Chaplais, Florent Di Méglio, Jean Lévine, Philippe Martin, Nicolas Petit, Laurent Praly.

I And also: Lectures at Collège de France, ANR projects CQUID