Quantum state tomography including measurement duration, imperfections and decoherence

Quantum state tomography including measurement duration,

imperfections and decoherence

PRACQSYS, 20-25 July 2015, Sydney.

Pierre Rouchon

QUANTIC Research Team (INRIA/ENS/Mines) Mines ParisTech, PSL Research University

1 / 19

Outline

Filtering versus tomography

Filtering and Stochastic Master Equation (SME) Discrete-time case

Continuous-time (diffusive) case

State tomography with decoherence and imperfections Computation of the likelihood function via the adjoint state Qubit tomography based on fluorescence experimental data

Concluding remarks

2 / 19

Quantum state tomography based on POVM

jπj =I

I Tomography ofρviaN independent measurementsY

associated to POVM: probability Tr(ρπ_j)of each measurement outcomejgiven byπ_j; forN_j the number ofj outcomes,

Y ≡(N_j)withP

jN_j =N, the number of measurements.

I Several estimation methods:

MaxEnt: ρ_ME maximizes−Tr(ρlog(ρ))under the constraints|Tr(ρπ_j)−N_j/N| ≤(Bužek et al, Ann. Phys. 1996).

Compress Sensing: ρ_CSminimizes Tr(ρ)under the constraints

|Tr(ρπ_j)−Nj/N| ≤(Gross et al PRL2010) MaxLike: ρ_MLmaximizes the likelihood function,

ρ7→P(Y |ρ) =Q

j Tr(ρπ_j)Nj

(see, e.g., Lvovsky/Raymer RMP 2009)

Bayesian Mean: ρ_BM∝R

ρP(Y |ρ)P0(ρ)dρwhereP0is some prior distributionP0(ρ)dρ(see, e.g., Blume-Kohout NJP2010).

Low rank, high dimensional systems: see, e.g, key contributions ofRobert Kosutand also ofMadalin Guta.

3 / 19

Quantum filtering versus tomography based on quantum trajectories

the measurement trajectory[0,t[3τ 7→yτ and the initial stateρ₀; seeBelavkinsemilar contributions (links with Monte-Carlo quantum-trajectories).

State tomography: estimation of the initial stateρ=ρ₀from a collection ofNmeasurement trajectories:Y =

y_t⁽ⁿ⁾ withn∈ {1, . . . ,N}andt ∈[0,T].

Process tomography: estimation of a parameterpfrom a known initial stateρ₀and a collection ofNmeasurement trajectoriesY.

Focus on quantum state tomography: decoherence, exp.

imperfections during the measurement durationT can be included via the adjoint stateE already introduced in quantum smoothing¹ Talk contribution:how to compute the likelihood functionP Y/ρ₀ from the stochastic master equation governing filtering.

1Tsang PRL 2009, Gammelmark/Julsgaard/Mølmer PRL 2013, Guevara/Wiseman 2015. . .

4 / 19

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. Randomness, irreversibility and dissipation induced by the measurementof observables withdegenerate spectra.

4. Entanglement and tensor product for composite systems.

VDiscrete-time models²

Take a set of operatorsMµsatisfyingP

µM^†_µMµ=I and a left stochastic matrices(η_yt,µ). Consider the followingMarkov processof stateρ(density op.) and measured outputy:

ρ_t+1 = ^Kyt(ρt)

Tr(^Kyt(ρt)), with proba.Pyt(ρ_t) = Tr(Kyt(ρ_t)) withKy(ρ) =Pm

µ=1η_y,µMµρM^†_µ. It is associated to theKraus map (ensemble average, quantum channel)

E

y

K_y(ρ_t) =X

µ

Mµρ_tM^†_µ.

2see, e.g., the book of Haroche/Raimond and the publications around the LKB photon box.

5 / 19

Continuous/discrete-time Stochastic Master Equation (SME)

Tr(K_yt(ρ_t)), with K_yt(ρ_t) =Pm

µ=1η_yt,µMµρ_tM^†_µ, and proba.Pyt(ρ_t) = Tr(K_yt(ρ_t)).

Ensemble averages correspond toKraus linear maps

E

y

K_y(ρ_t) =X

µ

Mµρ_tM^†_µ with X

µ

M^†_µMµ=I Continuous-time models:stochastic differential systems(see, e.g., Barchielli/Gregoratti, 2009)

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

dW_ν,t driven byWiener processesdWν,t, with measurementsdy_ν,t, dy_ν,t =√

ην Tr

(Lν+L^†_ν)ρ_t

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

d

dtρ=−ⁱ

~[H, ρ] +X

ν

LνρL^†_ν−¹

2(L^†_νLνρ+ρL^†_νLν)

6 / 19

Continuous/discrete-time diffusive SME

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

withdWν,t =dyν,t −√ η_ν Tr

(Lν+L^†_ν)ρ_t

dtgiven by the measurement signaldyν,t, is always a stable filtering process.³ UsingIto rules, it can be written as a "discrete-time" Markov model¯ ⁴

ρ_t+dt=K_dyt(ρ_t)/Tr(K_dyt(ρ_t)) with "partial Kraus maps"

Kdyt(ρ_t) =Mdytρ_tM^†_dyt +P

ν(1−ην)Lνρ_tL^†_νdt M_dyt =I+

−ⁱ

~H−¹₂ P

νL^†_νLν

dt+P

ν

√ηνdy_ν,tL where the probability of outcomedyt = (dyν,t)reads:

P

dy_t ∈Q

ν[ξν, ξν+dξν]. ρ_t

= Tr(Kξ(ρ_t)) Q

νe^{−ξ2ν/2dt√dξν}

2πdt 3H. Amini et al., Russian J. of Math. Physics, 2014, 21, 297-315.

4PR, J. Ralph PRA2015; see also PR Int. Congress of Mathematicians at Seoul 2014, and PhD of Ph. Campagne-Ibracq at ENS-Paris, 2015. 7 / 19

Computation of the likelihood function via the adjoint state (1)

I Denote byPn(ρ)the probability of getting measurement trajectoryn,(y_t⁽ⁿ⁾)_t=0,...,T, knowing the initial stateρ⁽ⁿ⁾₀ =ρ.

I Sinceρ⁽ⁿ⁾_t+1=

Ky(n) t

ρ⁽ⁿ⁾_t Tr

Ky(n)

t

ρ⁽ⁿ⁾_t with Tr K_y(n)

t

ρ⁽ⁿ⁾_t

the probability of having detectedy_t⁽ⁿ⁾knowingρ⁽ⁿ⁾_t , a direct use of Bayes law yieldsPn(ρ) =QT

t=0 Tr Ky_t⁽ⁿ⁾

ρ⁽ⁿ⁾_t

. Some elementary computations show that:

Pn(ρ) = Tr K_y(n)

T

◦. . .◦K_y(n) 0

(ρ) .

I Theadjoint mapK^∗_y ofKyis defined by Tr(AKy(B))≡ Tr K^∗_y(A)B

for all Hermitian operatorsAandB.

Thus Pn(ρ) = Tr

Ky_T⁽ⁿ⁾◦. . .◦K

y₀⁽ⁿ⁾(ρ) I

= Tr ρK^∗

y₀⁽ⁿ⁾◦. . .◦K^∗

y_T⁽ⁿ⁾(I) .

8 / 19

Computation of the likelihood function via the adjoint state (2)

I The normalizedadjoint quantum filter,E_t⁽ⁿ⁾=

K^∗

y(n) t

E_t+1⁽ⁿ⁾

Tr K^∗

y(n) t

E_t+1⁽ⁿ⁾

!

withE_T⁽ⁿ⁾₊₁=I/Tr(I), defines a family of Hermitian and non-negative operators(E_t⁽ⁿ⁾)on unit trace depending only on the measurement dataY.

I We haveK^∗

y₀⁽ⁿ⁾◦. . .◦K^∗

y_T⁽ⁿ⁾(I) =gn(Y)E₀⁽ⁿ⁾with

1

gn(Y) = Tr K^∗

y₀⁽ⁿ⁾◦. . .◦K^∗

y_T⁽ⁿ⁾(I)

independent ofρ.

I ThusPn(ρ) =gn(Y)Tr ρE₀⁽ⁿ⁾

and

P(Y/ρ) =g(Y)

N

Y

n=1

Tr ρE₀⁽ⁿ⁾

whereg(Y) =QN

n=1gn(Y).

9 / 19

MaxLike quantum-state tomography revisited

I MaxLike tomography based on POVMπ_j:ρ_MLmaximizes P(Y |ρ) =Y

j

Tr(ρπ_j)Nj

=Y

n

Tr ρπ_jn

withY ≡(Nj)derived fromjn, the measurement outcome numbern=1, . . . ,N.

I MaxLike tomography based on the adjoint states:ρ_MLmaximizes P(Y |ρ) =g(Y)Y

n

Tr ρE⁽ⁿ⁾

whereE⁽ⁿ⁾=E₀⁽ⁿ⁾ is the adjoint state att =0 associated to measurement trajectory(y_t⁽ⁿ⁾)numbern.

Convex optimization problem: the setDof density operators is convex; the log-likelihood functionf :D 3ρ7→log P(Y |ρ)

is concave (see Robert Kosut talk . . . )

10 / 19

Gradient and Hessian of the log-likelihood function

f(ρ),log(P(Y |ρ)) =log(g(Y)) +

N

X

n=1

log Tr

ρE⁽ⁿ⁾ .

The gradient off,

∇fρ=

N

X

n=1

E⁽ⁿ⁾ Tr E⁽ⁿ⁾ρ. and its Hessian∇²f (self-adjoint super-operator)

ξ7→ ∇²fρ(ξ) =−

N

X

n=1

Tr E⁽ⁿ⁾ξ Tr² E⁽ⁿ⁾ρE⁽ⁿ⁾. result from the following second order expansion:

f(ρ+δρ)−f(ρ)

=

N

X

n=1

Tr E⁽ⁿ⁾δρ

Tr E⁽ⁿ⁾ρ −¹₂ Tr² E⁽ⁿ⁾δρ Tr² E⁽ⁿ⁾ρ

!

+o Tr δρ²

.

11 / 19

Circuit QED: the LPA qubit with fluorescence measurements

No driveH=0,ν∈ {1,2,3}

Two fluorescence measurementsL1=q

1

2T1σ_-andL2=iL1with T₁=4µsand efficienciesη₁=η₂≈1/4.

Dephasing channelL₃=q

1

2Tφσ_z withTφ=35µs (η3=0).

5Ph. Campagne-Ibracq et al., Phys. Rev. Lett., 2014, 112, 180402.

12 / 19

Two quantum filtering trajectories (experimental data)

13 / 19

MaxLike tomography from

I N =

3000 trajectories of length

, with

I

Two measurements of efficiency

:

dy₁= r η

2T

Tr

2T

Tr

For each trajectory, the data corresponds to 2

50 real values (dt

200

I

The resulting

x_ML=

0.0134,

0.0213,

0.9997 is pure since it satisfies

1.

I

Gradient and Hessian of the log-likelihood function

∇f_ρML =





0.12 0.20 9.22



, ∇²f_ρML =





−241.1

3.6 0.4 3.6

1.4 0.4 1.4





14 / 19

Experimental likelihood function corresponding to

N=3000 fluorescence trajectories of length 2T₁. Cross section passing through the center of Bloch sphere zML-axis aligned withρ_ML, close toz-axis,xML-axis close tox-axis.

15 / 19

MaxLike tomography from

I N=3000 trajectories of lengthT =2T₁, withdt = ₂₀¹T₁.

I Two measurements of efficiencyη=¹₄: dy1=

r η

2T₁ Tr(ρσ_x) +dW1, dy2= r η

2T₁ Tr(ρσ_y) +dW2

For each trajectory, the data corresponds to 2×40 real values (dt =200ns).

I The resultingρ_ML

x_ML=−0.0465, y_ML=0.0625, z_ML=0.6787 is mixed since it satisfiesx_ML² +y_ML² +z_ML² <1.

I Gradient and Hessian of the log-likelihood functionf:

∇fρ_ML =0, ∇²fρ_ML =





−231.6 −2.6 1.7

−2.6 −227.5 −0.4 1.7 −0.4 −21.6





16 / 19

Experimental likelihood function corresponding to

N=3000 fluorescence trajectories of length ³₂T₁. Cross section passing through the center of Bloch sphere zML-axis aligned withρ_ML, close toz-axis,xML-axis close tox-axis.

17 / 19

Bayesian mean tomography for

I Another possible estimation is given by ρ_BM∝R

ρP(Y |ρ)P0(ρ)dρwith some prior distributionP0(ρ)dρ.

I With Bloch variables(x,y,z)andP0(ρ)dρ∝dxdydz we have, xBM=

RRR

x²+y²+z²≤1xe^f(x,y,z)dxdydz RRR

x²+y²+z²≤1e^f(x,y,z)dxdydz , yBM=. . .

I With the normalizationf = ¯N¯f withN¯ >0 large , we have approximation ofx_BMvia the asymptotics

RRR

x²+y²+z²≤1g(x,y,z)e^{N¯¯f(x,y,z)}dxdydz = ^eN¯¯f(xML_¯^,yML,zML)

N²

c0+^c_¯¹

N + ^c_¯²

N² +. . . wherec₀,c₁,c₂. . . depend on the derivatives of¯f andgat

(x_ML,y_ML,z_ML)⁶

6For an elementary theory see Bleistein, N. Handelsman, R. : Asymptotic Expansions of Integrals. Dover, 1986.

For the general recent theory called Singular Learning see Watanabe S., Algebraic Geometry and Statistical Learning Theory, Cambridge University Press, 2009.

See also Shaowei Lin, Algebraic Methods for Evaluating Integrals Bayesian Statistics, PhD thesis, University of California, Berkeley, 2011.

18 / 19

Concluding remarks

1. Another validation on theexperimental data of the LKB photon boxunderlying Rybarczyk et al: Past quantum state analysis of the photon number evolution in a cavity, to appear in PRA.

Compensation of photon life-time 1/κcomparable with the time during the QND measurement of photons.

2. In the near future: application toWigner tomography of a cavity fieldbased on the measurement protocol used, e.g., in Leghtas et al.: Confining the state of light to a quantum manifold by engineered two-photon loss; Science, 2015, 347, 853-857.

Compensation for initial thermal state of the probe qubit, qubit measurement errors, cavity field damping and several nonlinear Kerr effects.

3. Extension toparameter estimation(quantum process tomography) where the adjoint state simplifies the gradient computation of the log-likelihood function.

4. Correction to low-rankρ_MLviaρ_BM ∝R

ρP(Y |ρ)P0(ρ)dρand its approximate computation via asymptotics techniques developed formultidimensional integrals of Laplace type.

19 / 19