Quantum Filtering and Dynamical Parameter Estimation

Quantum Filtering and Dynamical Parameter Estimation

[email protected]

Isaac Newton Institute, Cambridge, July 2014.

Based on collaborations with

Hadis Amini, Michel Brune, Igor Dotsenko, Serge Haroche, Zaki Leghtas, Mazyar Mirrahimi, Clément Pellegrini, Jean-Michel Raimond, Clément Sayrin and Ram Somaraju

1 / 27

The first experimental realization of a

The LKB photon Box

Group of Serge Haroche, Jean-Michel Raimond and Michel Brune.

1

Stabilization by ameasurement-based feedback of photon-number states (sampling time 80µs) Experiment:C. Sayrin et. al., Nature 477, 73-77, September 2011.

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

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

1Animation realized by Igor Dotsenko

Experimental closed-loop data

C. Sayrin et. al., Nature 477, 73-77, Sept. 2011.

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,

measurement imperfections and delays (Bayes law).

Truncation to 9 photons

Stabilization around 3-photon state

3 / 27

Ideal model: Markov chain with input

and output

Quantum state :ρthe density operator of the photons . Output:y ∈ {g,e}measurement of the atom.

ρ_k+1=



















DukMgρ_kM_g^†D^†_uk Tr

M_gρ_kM_g^†, yk =g with proba.Pg,k = Tr

Mgρ_kM_g^†

DukMeρ_kM_e^†D_u^†_k Tr

Meρ_kM_e^† , y_k =ewith proba.Pe,k = Tr

Meρ_kM_e^†

QND measurement operators:M_g=cos_φ

0(N+1/2)+φR

2

et Me=sin_φ

0(N+1/2)+φR

2

withN=a^†a=diag(0,1,2, . . .).

Unitary control operator :Du=e^{ua†−u∗a}whereais the photon annihilation operator.

Goal :stabilize state with exactly¯nphoton(s),ρ¯=|¯nih¯n|, that are open-loop stationary state foru=0.

Structure of this quantum-state feedback (ideal case)

Observer-Controller

I Non linear filtering of the measurementsk 7→yk provides an estimateρ^estofρ:

ρ^est_k+1= D_ukM_ykρ^est_k M_y^†_kD^†_uk Tr

Mykρ^est_k M_y^†_k .

Quantum filter in the sense of Belavkin.

I Thestabilizing feedbackuk =f(ρ^est_k )ensuring convergence towardsρ¯is based onLyapunovdesign:

uk =Argmin

u E V(ρ_k+1)|ρ_k =ρ^est_k , u whereV is a well chosensuper-martingaleconstructed with open-loop martingales attached to the QND process.

2The global convergence proof of such observer/controller for the realistic case is given in H. Amini et. al., Automatica, 49 (9): 2683-2692, 2013.

5 / 27

In the realistic case:

and

.

QuantumFilter_Controller PhotonBox

u

y y

Detector Coherent

pulse

u

est est

u

y

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

I Derived from Bayes law: depends on past detector outcomes between 0 andk;

computed recursively from an initial valueρ^est₀ ;

I Stable and tends to converge towardsρk,the expectation valueof|ψkihψk| knowing its initial value|ψ₀ihψ₀|and the past detector outcomes from 0 tok.

Outline

Quantum filtering: discrete-time case

Quantum filtering: continuous-time case

Conclusion

7 / 27

For the photon box, quantum filtering combines

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 measurement of observablesO withdegenerate spectra,O=P

µλµPµ:

I measurement outcomeλµwith proba.

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

I measurement 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.

LKB photon-box: Markov chain in the ideal case (1)

I SystemS

corresponds to a quantized cavity mode:

H_S = ( _∞

X

n=0

ψⁿ|ni |(ψⁿ)^∞_n=0∈l²(C) )

,

where

represents the Fock state associated to exactly

photons inside the cavity

I

Meter

is associated to atoms :

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

with

1; atoms leaving

are all in state

I

When an atom comes out

of the composite system atom/field is

|Ψi_B =|ψi ⊗ |gi.

9 / 27

LKB photon-box: Markov chain in the ideal case (2)

C

B

D R

R

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 =USM |ψi ⊗ |gi

= Mg|ψi

⊗ |gi+ Me|ψi

⊗ |ei whereUSMis 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.

LKB photon-box: Markov chain in the ideal case (3)

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

Pµ

(Mµ|ψi)⊗ |µi= (Mµ|ψi)⊗ |µi q

hψ|Mµ^†Mµ|ψi .

Markov process(density matrix formulationρ∼ |ψihψ|)

ρ+=















M_gρM_g^† Tr

MgρM_g^†, with probabilityPg= Tr

M_gρM_g^†

;

MeρM_e^† Tr

MeρM_e^†, with probabilityPe= Tr

M_eρM_e^† .

Kraus map:

E

11 / 27

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

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

Tr MµρMµ^†

MµρM_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+ηeMeρMe^†)^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

ηgMgρMg^†+ (1−ηe)MeρMe^†

.

ρ+does not remain pure: the quantum stateρ+becomes a mixed state;|ψ+ibecomes physically irrelevant (not numerically).

Photon-box quantum filter: 6

21 left stochastic matrix

Tr(^Pµη_µ0,µMµρ^est_k Mµ^†) P

µηµ⁰,µMµρ^est_k M_µ^† with

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

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}_dη_e^{2 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)

13 / 27

The Markov chain with imperfections:

and

Take|ψ_k+1ihψ_k+1|= ¹

Tr(Mµk|ψkihψk|Mµ^†k)

Mµ_k|ψ_kihψ_k|M_µ^†_k

with measurement imperfections and decoherence described by theleft stochastic matrixη:η_µ⁰_,µ∈ |0,1]is the probability of having the imperfect outcomeµ⁰∈ {1, . . . ,m⁰}knowing that the perfect one is µ∈ {1, . . . ,m}.

The optimal quantum filter: ρ_k =

E

|ψ_kihψ_k|

|ψ₀i, µ⁰₀, . . . , µ⁰_k−1

can be computed efficiently via the following recurrence

ρ_k+1= ¹

TrPm µ=1η_µ0

k,µMµρkMµ^†





m

X

µ=1

η_µ⁰

k,µMµρ_kM_µ^†





where the detector outcomeµ⁰_k takes valuesµ⁰in{1,· · · ,m⁰}with probabilityPµ⁰,ρk = Tr

Pm µ=1η_µ⁰

k,µMµρ_kM_µ^† .

Stability and convergence issues (1)

I The quantum stateρ_k =

E

|ψ_kihψ_k|

|ψ₀i, µ⁰₀, . . . , µ⁰_k−1

is given by the following optimalBelavkin filtering process

ρ_k+1= ¹

Tr Pm

µ=1η_µ0

k,µMµρkMµ^†





m

X

µ=1

η_µ⁰

k,µMµρ_kM_µ^†





with theperfect initialization:ρ₀=|ψ₀ihψ₀|.

I Its estimateρ^estfollows the same recurrence

ρ^est_k+1= ¹

TrPm µ=1η_µ0

k,µMµρ^est_k Mµ^†





m

X

µ=1

η_µ⁰

k,µMµρ^est_k M_µ^†





but withimperfect initializationρ^est₀ 6=|ψ₀ihψ₀|.

A natural question :ρ^est_k 7→ρ_k whenk 7→+∞?

15 / 27

Stability and convergence issues (2)

ρ_k+1=

Pm µ=1η_µ0

k,µMµρ_kM_µ^† Tr

Pm µ=1η_µ0

k,µMµρ_kMµ^†

, ρ^est_k+1=

Pm µ=1η_µ0

k,µMµρ^est_k M_µ^† Tr

Pm µ=1η_µ0

k,µMµρ^est_k M^†µ

Proba. to getµ⁰_kat stepk, Tr Pm

µ=1η_µ⁰

k,µMµρ_kM_µ^†

, depends onρ_k.

I Convergenceofρ^est_k towardsρ_k whenk 7→+∞is an open problem.

A partial result (continuous-time) due to R. van Handel:The stability of quantum Markov filters. Infin. Dimens. Anal.

Quantum Probab. Relat. Top. , 2009, 12, 153-172.

I Stability³: the fidelityF(ρ_k,ρ^est_k ) = Tr² p√

ρ_kρ^est_k √ ρ_k

is a sub-martingale for anyηandMµ:

E

≥F(ρ_k,ρ^est_k ).

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

The key inequality underlying

is sub-martingale

For

I

any set of

matrices

with

µ=1M_µ^†M_µ=

1,

I

any partition of

into

1 sub-sets

,

I

any Hermitian non-negative matrices

and

of trace one, the following inequality holds

ν=p

X

ν=1

Tr



 X

µ∈Pν

M_µρM_µ^†



F

P

µ∈PνMµσMµ^†

Tr P

µ∈PνMµσMµ^†

,

P

µ∈PνMµρMµ^†

Tr P

µ∈PνMµρMµ^†

!

≥F(σ, ρ)

where

Tr

σρ√ σ

.

Proof combines Cauchy-Schwartz inequalities with a lifting procedure based on Ulhmann’s theorem.

4PR. Fidelity is a Sub-Martingale for Discrete-Time Quantum Filters. IEEE Transactions on Automatic Control, 2011, 56, 2743-2747.

17 / 27

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 =ρ₀assuming initial probability of¹₂ to have¯p=aandp¯=b. At stepk,

Pa,k = Tr ρb^est_a,k

,Pb,k = Tr ρb^est_b,k

) are the proba. to havep¯=a,

¯p=b, knowing the initial stateρ₀and the past detection outcomes.

This dynamical parameter estimation process is stable: if the true value of the parameter isathenPa,k is a sub-martingale.

5See Kato, Y. & Yamamoto, N. Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on, 2013, 1904-1909

Discrete-time translation of

Gambetta, J. & Wiseman, H. M., Phys. Rev. A, 2001, 64, 042105

and of Negretti, A. & Mølmer, K. , New Journal of Physics, 2013, 15, 125002.

Discrete-time models of open quantum systems

Four features:

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).

19 / 27

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, ρ_t] +X

ν

Lνρ_tL^†_ν−¹

2(L^†_νLνρ_t+ρ_tL^†_νLν)

! dt

+X

ν

√η_ν

Lνρ_t+ρ_tL^†_ν− Tr

(Lν+L^†_ν)ρ_t ρ_t

dWν,t

driven byWiener processdW_ν,t =dy_ν,t−√ ην Tr

(Lν+L^†_ν)ρ_t dt with measurementsyν,t, detection efficienciesην ∈[0,1]and Lindblad-Kossakowskimaster equations (ην ≡0):

d dtρ=−ⁱ

~[H, ρ] +X

ν

Lνρ_tL^†_ν−¹

2(L^†_νLνρ_t+ρ_tL^†_νLν)

Continuous/discrete-time diffusive SME

With a single imperfect measurement dy_t =√

ηTr

(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^†−¹

2(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+

−ⁱ

~H−¹₂ L^†L

dt+√ ηdy_tL.

21 / 27

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, ρt] +Vρ_tV^†−¹

2(V^†Vρ_t+ρ_tV^†V) dt +

θρ_t+ηVρtV^† θ+ηTr(Vρ_tV^†)−ρ_t

dN(t)−

θ+ηTr Vρ_tV^† dt

ForN(t+dt)−N(t) =1we haveρ_t+dt= θρ_t+ηVρ_tV^† θ+ηTr(Vρ_tV^†). 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+¹₂ η Tr Vρ_tV^†

I−V^†V dt.

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

ForN(t+dt)−N(t) =1we haveρt+dt= θρt+ηVρtV^† θ+ηTr(VρtV^†). FordN(t) =0we have

ρt+dt = Mdy_tρtM_dy^†

t+ (1−η)LρtL^†dt+ (1−η)VρtV^†dt Tr

Mdy_tρtM_dy^†

t + (1−η)LρtL^†dt+ (1−η)VρtV^†dt withMdy_t =I+ −iH−¹₂L^†L+¹₂ ηTr VρtV^†

I−V^†V dt+√

ηdytL.

23 / 27

Continuous/discrete-time general diffusive-jump SME

The quantum stateρ_tis usually mixed and obeys to

dρ_t= −i[H, ρ_t] +X ν

L_νρtL^†_ν−¹₂(L^†_νL_νρt+ρtL^†_νL_ν) +V_µρtV_µ^†−¹₂(V_µ^†V_µρt+ρtV_µ^†V_µ)

! 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







θ_µρ_t+P

µ0η_µ,µ0V_µρ_tV_µ^† θµ+P

µ0η_µ,µ0Tr V_µ0ρtV^†

µ0 −ρ_t









dNµ(t)− θµ+X

µ0 η_µ,µ0Tr

V_µ0ρ_tV^† µ0 dt





whereην∈[0,1],θµ, η_µ,µ0≥0 withη_µ0=P

µη_µ,µ0≤1 are parameters modelling measurements imperfections.

If, for someµ,N_µ(t+dt)−N_µ(t) =1, we haveρ_t+dt=

θµρ_t+P

µ0η_µ,µ0V_µ0ρ_tV^† µ0 θµ+P

µ0η_µ,µ0Tr V_µ0ρtV^†

µ0 . When∀µ,dN_µ(t) =0, we have

ρ_t+dt=

M_dytρ_tM^†

dyt+P

ν(1−ην)Lνρ_tL^†_νdt+P

µ(1−η_µ)Vµρ_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

η_µTr

V_µρ_tV_µ^†

I−V_µ^†V_µ dt+P

ν

√η_νdy_νtL_νand

wheredy_ν,t=√ η_νTr

(L_ν+L^†_ν)ρ_t

dt+dW_ν,t.

Could be used as a numerical integration scheme that preserves the positiveness ofρ.

Continuous-time diffusive SME and quantum filtering

For clarity’sake, take a single measurementy_tassociated to operator Land detection efficiencyη∈[0,1]. Thenρ_tobeys to the following diffusive SME

dρ_t =−i[H, ρt]dt+

LρtL^†−¹

2(L^†Lρt+ρ_tL^†L) dt +√

η Lρt+ρ_tL^†− Tr (L+L^†)ρ_t ρ_t

dWt

driven by the Wiener processesWt, Sincedy_t =√

ηTr (L+L^†)ρ_t

dt+dW_t, the estimateρ^est_t is given by

dρ^est_t =−i[H,ρ^est_t ]dt+

Lρ^est_t L^†−¹

2(L^†Lρ^est_t +ρ^est_t L^†L) dt +√

η Lρ^est_t +ρ^est_t L^†− Tr (L+L^†)ρ^est_t ρ^e_t

dyt−√

ηTr (L+L^†)ρ^est_t dt

. initialized to any density matrixρ^est₀ .

25 / 27

Stability of diffusive quantum filtering

Assume that(ρ,ρ^est)obey to dρt =−i[H, ρt]dt+

LρtL^†−¹

2(L^†Lρt+ρ_tL^†L) dt +√

η Lρt+ρ_tL^†− Tr (L+L^†)ρ_t ρ_t

dWt

dρ^est_t =−i[H,ρ^est_t ]dt+

Lρ^est_t L^†−¹

2(L^†Lρ^est_t +ρ^est_t L^†L) dt +√

η Lρ^est_t +ρ^est_t L^†− Tr (L+L^†)ρ^est_t ρ^est_t

dW_t +η Lρ^est_t +ρ^est_t L^†− Tr (L+L^†)ρ^est_t

ρ^est_t

Tr (L+L^†)(ρ_t −ρ^est_t ) dt

| {z }

correction terms vanishing whenρt=ρ^est_t

.

Then for anyH,Landη∈[0,1],F(ρ_t,ρ^est_t ) = Tr² p√

ρ_tρ^est_t √ ρ_t

is a sub-martingale:

t7→

E

is non-decreasing.

6H. Amini, C. Pellegrini, PR: Stability of continuous-time quantum filters with measurement imperfections.http://arxiv.org/abs/1312.0418

Metric for the distance between

and its estimate

I

1

remains a

and their associated quantum filters when they are driven simultaneously by several Wiener and Poisson processes.

I

Petz has given, via the theory of operator monotone functions, a

:

d

dtρ=−i[H, ρ] +X

ν

L_νρL^†_ν −¹

2(L^†_νL_νρ+ρL^†_νL_ν)

.

I

Could we exploit Petz results to characterize "metrics"

D(ρ,ρ^est)

that are super-martingale for all Belavkin SMEs.

and filters ?

7D. Petz. Monotone metrics on matrix spaces.Linear Algebra and its Applications, 244:81–96, 1996.

27 / 27