Quantum robots perturbed by Levy processes: Stochastic analysis and simulations

Commun Nonlinear Sci Numer Simulat 83 (2020) 105142

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Research paper

Quantum robots perturbed by Levy processes: Stochastic analysis and simulations Rohit Singla a,∗, Harish Parthasarathy b a b

Amity University Uttar Pradesh, India University of Delhi, India

a r t i c l e

i n f o

Article history: Received 21 February 2019 Revised 2 October 2019 Accepted 1 December 2019 Available online 2 December 2019 Keywords: Levy noise Schrodinger equation Quantum mechanical time independent perturbation theory Ito’s formula for poisson process

a b s t r a c t In the present work, the problem of quantization of a robot via the Schrodinger equation with Jerk Levy noise and the wave function plots is analyzed. Quantum average and quantum fluctuation of the trajectory positions are displayed. It is observed that the quantum robots are molecules whose link angles can be varied resulting in changing chemical properties, thus justifying the importance of this work also in physics of biological engineering. we have also made computations on the statistics of the wave function as well as transition probabilities between two stationary states. The quantum analysis has been based on using time independent perturbation theory to evaluate the approximate evolution operator of the unperturbed robot. © 2019 Elsevier B.V. All rights reserved.

1. Introduction In this paper, the motion of a 2-link robot in the presence of gravitational forces and a classical torque process τ (t) applied at the joints using the Schrodinger wave equation is quantized. To initialize the process, the Hamiltonian of the robot in the absence of torques is expressed in terms of the canonical angular position and momentum operators. Using time independent perturbation theory, the approximate eigenfunctions and eigenvalues of the unperturbed 2-link robot (by unperturbed, we mean in the absence of torque) are calculated. Using these approximate eigenfunctions and eigenvalues, the expression for the approximate quantum evolution operator of the unperturbed robot system is then arrived at. The perturbing torque process is introduced thereafter and the approximate evolution operator for the resulting time dependent system using the Dyson series, also known as time dependent perturbation theory is derived. It is assumed that the torque process comprises of a compound Poisson process and using It’s formula for the Poisson process, It is explained how the potential that is modulated by the Poisson component must be chosen so as to ensure unitary evolution which is required for the conservation of probability. Using the truncated Dyson series approximation, the approximate formulae for the characteristic functional of the wave function of the robot over a fixed time duration are then obtained. The Poisson noise can be regarded as a jerk component to the torque caused by irregular vibrations in the motors at the joints or in the external force applied on the robot (as seen in nano-electromechanical machines). The Dyson series approximation is then also used to obtain an approximate expression for the transition probability of the quantum robot from one stationary state to another under the influence of the compound Poisson noise. These theoretical expressions are likely to find applications in estimating the quantum robot parameters from transition probability measurements. Our theoretical analysis ends with a ∗

Corresponding author. E-mail addresses: [email protected], [email protected] (R. Singla).

https://doi.org/10.1016/j.cnsns.2019.105142 1007-5704/© 2019 Elsevier B.V. All rights reserved.

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R. Singla and H. Parthasarathy / Commun Nonlinear Sci Numer Simulat 83 (2020) 105142

discussion of the dynamics of a quantum system interacting with a bath that generates both compound Poisson noise and Brownian motion noise. By averaging over the bath noise, a general quantum Liouville equation for the density operator of the quantum system is derived which is a version of the well known Sudarshan-Lindblad equation also called the quantum Fokker-Planck equation. Finally, simulation results for the wave function evolution as well as the quantum averages and quantum dispersion of the robot angles in the presence of Poisson noise are presented. Owing to the presence of this classical noise, these quantum averages display classical fluctuations. Our simulation studies are based on normalized values of the robot parameters and Planck’s constant. To justify these normalized values, a discussion that shows how by scaling the link masses, link lengths, time, angles and torque amplitude, one can arrive at the normalized quantum model from the actual Planck’s scale model is included in our simulation studies. It is necessary to emphasize that, molecules like CO2 , H2 O, H2 S having two covalent bonds are examples of two link quantum robots and the bond angles can be altered by applying external torques and forces in the form of electromagnetic fields. By altering these link angles, one may hope to change the chemical properties of these molecules thereby enabling manufacture of new drugs in medicinal research. 2. Review of literature Before presenting our problem in detail, we briefly survey below a few articles in the literature related to noise in quantum systems and its relation to our work. In [1] the authors start with the Sudarshan-Lindblad equation that governs the evolution of a quantum system coupled to a noisy bath. The evolution contains a Hamiltonian term [H, ρ t ] and a noisy part

−

1 ∗ (Lk Lk ρt + ρt L∗k Lk − 2Lk ρt L∗k ) 2

(1)

k

This equation can also be obtained by writing the evolution equation for the system plus bath using the noise processes of the Hudson-Parthasarathy [2] theory and then tracing out over the bath variables. The authors then give condition on H, {Lk } an invariant subsystem to exist. Then they derive conditions for an initial state not to evolve under Markovian dynamics asymptotically as a density of an invariant subsystem. Then they design a feedback Hamiltonian that causes the original system state to evolve into a state supported by a given subspace (attractive system). These ideas can be exploited in the quantum robot problem as follows. The desired subspace may be spanned by wave function ψ (q1 , q2 ) supported over a set [q1 −  , q1 +  ] × [q2 −  , q2 +  ] i.e with a large probability, the angular position of the links assume values around given angles. This means that after a long time, the quantum system behaves nearly classically with an error margin of 2 . In [3] the stochastic optimal control problem for cost function that are quadratic in the control input process has been solved when the governing state stochastic differential equation is linear in the control input. The method used is the standard Hamilton-Jacobi equation (HJE). A tensor power series method is then developed for obtaining an approximate solution to this HJE. This has then been applied to quantum optimal control where the wave function evolution equation is naturally linear in the control input that modulates a potential (i.e bilinear in control input and output wave function). Optimal tracking of a reference wave function is studied when apart from the control input, there is White Gaussian noise that modulates another potential. This can readily be generalized to robot SDE’s driven by Levy noise with a control torque input especially in the quantum context. The only difference is that instead of using the generator of a diffusion process, we need to use the generator of a Markov process driven by a Levy input. Our classical robot differential equation is

dq = q˙ dt

(2a)

dq˙ = (−M (q )−1 N (q, q˙ ) + M (q )−1 τ (t ))dt + M (q )−1 dW (t )

(2b)

where W(t) is a Levy process. This dynamics is linear in the control torque input τ (t) and hence the techniques of the authors can be used to minimize



E

T 0



||







q(t ) q (t ) − d q˙ (t ) q˙d (t )

||2 dt

(3)

In the quantum context, the wave function evolves as

dψ (t ) = [−iHodt + V1 dw(t ) + iτ (t )T qdt ]ψ (t )

(4)

where Ho = p = −i∇q and V1 is a function of q, p with w(t) being a Levy process. Again {ψ (t), t ≥ 0} is Markovian and the methods of the paper can be used to minimize 1 T −1 p, 2 p M (q )

 E

T 0

ψ (t ) − ψd (t )2 dt + E



T 0

[< ψ (t ), −iτ (t )T [Ho, q]ψ (t ) >]dt

(5)

the latter term being the expected machine power consumed by the robot. Note that i[Ho, q] = q˙ for the unperturbed system. In [4] a different approach to noise in quantum mechanics has been proposed. The system is modelled as a single harmonic oscillator and the bath as a sum of independent harmonic oscillator. the interaction between the system and bath

R. Singla and H. Parthasarathy / Commun Nonlinear Sci Numer Simulat 83 (2020) 105142

3

is a bilinear combination of the system and bath variables. Then a linear transformation is applied to the system and bath variables resulting in the overall Hamiltonian expressible as a sum of independent harmonic oscillators. In other words, the interaction has been diagonalized. The partition function for the system plus bath is then obtained using expressions for the quantized energy levels of a harmonic oscillator. The quantum characteristic function of the system plus bath is calculated by solving a partial differential equation. The crucial idea of [4] is to introduce stochastic effects into quantum system via statistical mechanics of the Gibbs densities at a given temperature. This approach can also be adopted in quantum robots. We have to first determine the stationary states and energy eigenvalues of the robot Hamiltonian in the absence of external  torque. Then calculate the partition function n e−β En = T r (e−β H ) and determine the position and momentum correlations

T r (qα qβ e−β H (q,p) )/T r (e−β H )

(6)

T r (qα pβ e−β H (q,p) )/T r (e−β H )

(7)

T r ( pα pβ e−β H (q,p) )/T r (e−β H )

(8)

etc. we can then determine the coefficients of the Levy process in our quantum model in the limit t → ∞ so that the corresponding correlations matches those obtained in [4]. 3. Quantum mechanics of a two link robot with external torque Here, we quantize the robot using the Schrodinger wave equation for wave function of the angular variables. The Hamiltonian H0 of the 2-link robot is expressed in terms of the angular vector q and the angular momentum vector p. This Hamiltonian contains a gravitational potential component V(q). By expanding the inverse of the mass moment of inertia matrix M(q), we are able to express H0 as the sum of two bouncing ball Hamiltonians and a perturbation term. Time independent perturbation theory is used to calculate approximately the energy eigenvalues and eigenfunctions of H0 . The Schrodinger evolution equation of this quantum 2-link robot is then perturbed by a compound Poisson process and to ensure unitarity of the evolution, a condition on the potential that is modulated by the Levy process is derived. Then time dependent perturbation theory is applied to derive approximate expressions for the transition probability of the robot system from one stationary state to another. 3.1. Application of time independent perturbation theory for calculating the approximate unperturbed evolution operator Let q1 denote the angle made by the first link with respect to the x-axis and q2 the angle made by the second link with respect to the x-axis. Thus, q2 − q1 is the angle between the two links. While quantizing the motion of a 2-link robot in the presence of a gravitational field described by the potential

V ( q ) = V ( q1 , q2 ) =

m1 gl1 l2 sin(q1 ) + m2 g(l1 sin(q1 ) + sin(q2 )) 2 2

(9)

We take as our unperturbed Hamiltonian

H0 (q, p) =

1 T −1 p M (q ) p + V (q ) 2

(10)

where M (q ) = M0 + cos(q2 )M1 with M0 a constant +ve definite matrix and M1 a constant Hermitian matrix such that M1 ≤ M0 and −M1 ≤ M0 i.e −M0 ≤ M1 ≤ M0 , i.e M0 ± M1 are both +ve definite matrices. This condition is necessary and significant for M(q) to be +ve definite ∀q2 ∈ [0, 2π ]. Then

M−1 (q ) = M0−1/2 (I + cos(q2 − q1 )M0−1/2 M1 M0−1/2 )−1 M0−1/2 = M0−1 +

∞  (cos(q2 − q1 ))n −1/2 −1/2 M0 (M0 M1 M0−1/2 )n M0−1/2 n! n=1

∞  (cos(q2 − q1 ))n −1 = M0−1 + (M0 M1 )n M0−1 n! n=1

∞  (cos(q2 − q1 ))n −1 = M0−1 + M0 (M1 M0−1 )n n!

(11)

n=1

The Hamiltonian is

H0 = +

1 T −1 p M0 p + 2





m1 gl1 m2 gl2 m1 gl1 + m2 gl1 q1 + q2 + ∈ { (sin(q1 ) − q1 ) + m2 g(l1 (sin(q1 ) − q1 ) 2 2 2

l2 (sin(q2 ) − (q2 )))}+ ∈ H1 (q, p) 2

(12)

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R. Singla and H. Parthasarathy / Commun Nonlinear Sci Numer Simulat 83 (2020) 105142

After making an orthogonal canonical transformation, this becomes

H0 =

p21 p2 + 2 + α1 q1 + α2 q2 + ∈ H1 (q, p) 2λ1 2λ2

(13) p2

where ∈ H1 (q, p) is the second term on the rightmost of equation (11). The Hamiltonian 2λ1 + α1 q1 is that of a bouncing 1 ball and its bound states are obtained using the Airy function [5]. Let En (λ1 , α1 ), n = 1, 2, . . . be its energy eigenvalues and un (q1 |λ1 , α1 ), n = 1, 2, . . . its corresponding bound states. Then using perturbation theory, the approximate energy eigenstates of H0 are

unm (q|λ, α ) = un (q1 |λ1 , α1 )um (q2 |λ2 , α2 )+ ∈ δ unm (q1 , q2 |λ, α ), n, m ≥ 1

(14)

with corresponding energy eigenvalues

En (λ1 , α1 ) + Em (λ2 , α2 ) + Enm (λ, α ) = Enm (λ, α ) + Enm (λ, α )

(15)

E nm(λ, α ) = ∈< un (.|λ1 , α1 )  um (.|λ2 , α2 )|H1 | × un (.|λ1 , α1 )  um (.|λ2 , α2 ) >  2π  2π = u¯ n (q1 |λ1 , α1 )u¯ m (q2 |λ2 , α2 )H1 × (un (q1 |λ1 , α1 )um (q2 |λ2 , α2 ))dq1 dq2

(16)

where

0

0

δ unm is determined using H0 δ unm + H1 (un  um ) = Enm (λ, α )δ unm + Enm (un  um )

(17)

so

< ur  us |δ unm >= −

< ur  us |H1 |un  um > Ers (λ, α ) − Enm (λ, α )

(18)

where (r, s) = (n, m). Then the stationary state wave functions of H0 are up to ( ∈ ) given by

un  um (q )+ ∈ δ unm (q )  u (q )u (q ) < ur  us |H1 |un  um > =∈ { r 1 s 2 + un (q1 |λ1 , α1 )um (q2 |λ2 , α2 )} Enm (λ, α ) − Ers (λ, α )

(19)

(r,s ) =(n,m )

All these are standard results from time independent perturbation theory [6]. The approximate quantum evolution operator U0 (t ) = exp(−itH0 ) for the unperturbed Hamiltonian H0 is now given by application of the spectral theorem:

U0 (t ) =



|un  um + δ unm > exp(−it (Enm + Enm )) < un  um +  unm |

(20)

n,m

This formula should be understood in the following way: The operator U0 (t) acting on any square integrable function f(q1 , q2 ) of the angles is given by

(U0 (t ) f )(q1 , q2 ) =



exp(−it (Enm + Enm ))(un (q1 )um (q2 ) + δ unm (q1 , q2 ))



f (q 1 , q 2 )(u¯ n (q 1 )u¯ m (q 2 )

n,m

+ δ u¯ nm (q 1 , q 2 ))dq 1 dq 2

(21)

3.2. Dyson series expansion for the evolution operator of a quantum robot with external torque The Hamiltonian of a 2-link quantum robot in the presence of external torque processes τ 1 (t), τ 2 (t) applied at the joints is given by

H (t ) = H0 − τ1 (t )q1 − τ2 (t )(q2 − q1 )

(22)

where

H0 =

1 T p M (q )−1 p + V (q ) 2

(23)

as earlier. Writing

V1 (t ) = −τ1 (t )q1 − τ2 (t )(q2 − q1 )

(24)

R. Singla and H. Parthasarathy / Commun Nonlinear Sci Numer Simulat 83 (2020) 105142

5

we express the solution to the Schrodinger equation

idU (t )/dt = H (t )U (t ) = (H0 + V1 (t ))U (t )

(25)

U (t ) = U0 (t )W (t )

(26)

as

so that

iW (t ) = V˜1 (t )W (t ), V˜1 (t ) = U0 (t )V1 (t )U0 (−t )

(27)

The solution is the Dyson series

W (t ) = I +

∞ 

(−i )n

 0
n=1

V˜1 (t1 ) . . . V˜ (tn )dt1 . . . dtn

(28)

which converges provided

max0≤s≤t

 V1 (s ) < ∞

(29)

In this case, we have the upper bound

 W (t ) ≤ exp(



0

t

 V1 (s )  ds )

(30)

the operator norm used here is the spectral norm. 3.3. Quantum robot in the presence of compound poisson noise: computation of moment generating functional of the wave function and transition probabilities Let Yk , k = 1, 2, . . . be a sequence of i.i.d. two dimensional random vectors and we replace the torque process potential V1 (t) above by the process w (t)V0 where V0 is an operator and w (t ) = YN (t )+1 dN (t )/dt where N(t) is a Poisson process with intensity λ independent of the Yk s. Equivalently, we have

w(t ) =



t 0

w (s )ds =

N (t )



Yk

(31)

k=1

It is clear that w(t) is a Levy process, i.e a process with independent increments and w (t) is therefore a non-Gaussian white noise process [7,8]. The Schrodinger equation with this random perturbation can be expressed as

dU (t ) = −i(H0 dt + YN (t )+1 dN (t )V0 )U (t )

(32)

where Yn , n = 1, 2, . . . are i.i.d. r.v’s, N(.) is Poisson process independent of {Yn } and V0 is a matrix. Note that

w(t ) =



T 0

YN (s )+1 dN (s ) =

N (t )



Yk

(33)

k=1

is a Levy process and dw(t ) = YN (t )+1 dN (t ). Then the above Schrodinger Eq. (32) is the same as

dU (t ) = −i(H0 dt + V0 dw(t ))U (t )

(34)

and for U(t) to be unitary, we require that

d (U ∗ (t )U (t )) = 0

(35)

dU ∗ .U + U ∗ .dU + dU ∗ .dU = 0

(36)

2 iYN+1 (V0∗ − V0 ) + YN+1 V0∗V0 = 0, ∀N ≥ 1

(37)

or

or

This can be satisfied for example by taking {YN } to be Bernoulli random variables assuming values {0, 1} only and further

i(V0∗ − V0 ) + V0∗V0 = 0

(38)

Let iV0 = I − S. Then Eq. (38) reduces to

−(I − S )∗ − (I − S ) + (I − S∗ )(I − S ) = 0

(39)

−I + S + S∗ − S − S∗ + S∗ S = 0

(40)

or

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R. Singla and H. Parthasarathy / Commun Nonlinear Sci Numer Simulat 83 (2020) 105142

or S∗ S = I, i.e S is an isometry, and

V0 = −i(I − S )

(41)

dU (t ) = (−iH0 dt + (S − I )YN (t )+1 dN (t ))U (t )

(42)

Let S be close to I. Then the Levy process term is a small perturbation to the Hamiltonian H0 . Let U (t ) = U0 (t )W (t ), U0 (t ) = exp(−itH0 ).Then,

dW (t ) = U0 (−t )(S − I )U0 (t )W (t )YN (t )+1 dN (t ) = (U0 (−t )SU0 (t ) − I )W (t )YN (t )+1 dN (t )

(43)

The approximate solution is

W (t ) ≈ I +

 0

t

YN (t )+1 (U0 (−t )SU0 (t ) − I )dN (t ) ≈ I +

N (t )



Yk Ek

(44)

k=1

where Ek = U0 (−tk − )SU0 (tk − ) − I ≡ E (tk − ) and E (t ) = U0 (−t )SU0 (t ) − I, and {tk } are the jump times of N(t). We now compute the approximate transition probabilities for our quantum robot perturbed by Levy noise. Before doing so, we derive for the sake of completeness, a method by which the MGF (Moment generating functional) of the wave function can be evaluated [9,10]. We have up to first order in the noise amplitude

|ψ (t ) > = U0 (t )|ψ (0 ) > +U0 (t )

N (t )



Yk E (tk − )|ψ (0 ) >= |ψ0 (t ) > +

k=1

N (t )



YkU0 (t )E (tk − )|ψ (0 ) >

(45)

k=1

The moment generating functional of |ψ (t) > , 0 ≤ t ≤ T is given by

E exp(



T 0

< ϕ (t )|d (U0∗ (t )ψ (t )) > ) = E{exp(

N (t )



Yk < ϕ (tk − )|E (tk − )|ψ (0 ) > )}

k=1

 ∞ n  exp(−λT )(λT )n × [ p. exp(< ϕ (tk )|E (tk )|ψ (0 ) > ) + 1 − p]dt1 . . . dtn n!T n [0,T ]n n=0 k=1  T  λn = exp(−λT )[ ( p exp(< ϕ (t )|E (t )|ψ (0 ) > ) + 1 − p)dt]n n! 0 n≥0  T = exp{λ p (exp(< ϕ (t )|E (t )|ψ (0 ) > ) − 1 )dt } =

(46)

0

Note that

< ϕ (t )|d (U0∗ (t )ψ (t )) >=< ϕ (t )|U0∗ (t )|dψ (t ) > + < ϕ (t )|U0∗ (t )|ψ (t ) > dt =< U0 (t )ϕ (t )|dψ (t ) > +i < ϕ (t )|H0U0∗ (t )|ψ (t ) > dt

(47)

So that, an alternate way to express the MGF is to compute the MGF of the random perturbation to the non-random wave function:

|ψ (t ) > −|ψ0 (t ) > |ψ1 (t ) >= U0 (t )

N (t )



Yk E (tk − )|ψ (0 ) >

(48)

k=1

d|ψ1 (t ) > = dt (t )

N (t )



YkU0 (t )E (tk )|ψ (0 ) > + U0 (t )E (t )|ψ (0 ) > dN (t )YN (t )+1

(49)

k=1

 0

T

< ϕ (t )d|ψ1 (t ) > = =



N (t )

T

dt 0



Yk < ϕ (t )|U0 (t )E (tk )|ψ (0 ) > +

k=1

N (T ) n 

 Yk

n=1 k=1



tn−1

 N (t )

N (T )

=

Yk

k=1

tk−1

tn



T 0

YN (t )+1 < ϕ (t )|U0 (t )E (t )ψ (0 ) > dN (t )

< ϕ (t )|U0 (t )E (tk )|ψ (0 ) > dt +

N (T )



Yn+1 < ϕ (tn )|U0 (tn )E (tn )|ψ (0 ) >

n=0

< ϕ (t )|U0 (t )E (tk )|ψ (0 ) > dt +

N (T )



Yn+1 < ϕ (tn )|U0 (tn )E (tn )|ψ (0 ) >

(50)

n=0

This formula can be used to calculate the moment generating functional of |ψ 1 (t) > , 0 ≤ t ≤ T:

E[exp(



T 0

< ϕ (t )|dψ1 (t ) > )]

(51)

R. Singla and H. Parthasarathy / Commun Nonlinear Sci Numer Simulat 83 (2020) 105142

7

Now we address the transition probability computation. Let |n >, n = 0, 1, 2 . . . be the energy eigenstates of H0 with energy eigenvalues En , n = 0, 1, 2 . . .. Then, for n = m, the transition probability from |n > → |m > in time T is approximately

PT (|n >→ |m > ) = E|

N (t )



Yk < m|E (tk − )|n > |2

k=1

 ∞ r   exp(−λT )(λT )r = E| Yk < m|E (τk )|n > |2 dτ1 . . . dτr r r!T [0,T ]r r=1

= exp(−λT )



E(YkYq ) =

p p2

k=1



r!

r≥1

Now

λ

r

r  [0,T ]r

{E(YkYq ) < m|E (τk )|n > × < n|E (τq )|m >}dτ1 . . . dτr

(52)

k,q=1

k=q k = q

if if

where P {Yk = 1} = p, P {Yk = 0} = 1 − p. Thus

PT (|n >→ |m > ) = exp(−λT )



r!

r≥1

= exp(−λT )

 r≥1

λr λr r!

 [0,T ]r

{p

r  k=1

{r pT r−1



T 0



| < m|E (τk )|n > |2 + 2 p2

< m|E (τk )|n >< n|E (τ j )|m >}dτ1 . . . dτr

1≤k< j≤r

| < m|E (τ )|n > |2 dτ + p2 r (r − 1 )T r−2 |



T 0

< m|E ( τ )|n > d τ |2 }

(53)

Since

 r≥1



λr r!

rT r−1 = λ

 (λT )r−1 = λ exp(λT ), ( r − 1 )!

(54a)

r≥1

λr r (r − 1 )T r−2 r!

r≥1

= λ2

it follows that

PT (|n >→ |m > ) = λ p

 (λT )r−2 = λ2 exp(λT ) ( r − 2 )!

(54b)

r≥2



T 0

| < m|E (τ )|n > |2 dτ + λ2 p2 |



T 0

< m|E ( τ )|n > d τ |2

(55)

This formula can be used to estimate parameters of the Hamiltonian from transition probability measurements. 3.4. A special case of poisson noise driven Schrodinger equation In our simulation studies which we display later, we have made software experiments using a special case of the model discussed in section 2.3. The model studied is noisy Schrodinger equation

dU (t ) = ((−iH0 dt − iV0 dN (t )))U (t )

(56)

where H0 is a Hermitian operator,

H0 =

1 2

pT M (q )−1 p + V (q )

(57)

p = −i h ¯ ∇q being the Hamiltonian of the quantum robot in a gravitational field and V0 being another operator that dictates the intensity and shape of the quantum jerks. is V0 must be selected by the Ito differential rule for Poisson processes (dN (t ))2 = dN (t ) in such a way that U(t) is a unitary operator for all time:

0 = d (U ∗U ) = dU ∗ .U + U ∗ .dU + dU ∗ .dU = U ∗ (t )(i(V0∗ − V0 )dN (t ) + V0∗V0 dN (t ))U (t )

(58)

This means that V0 must necessarily satisfy

i(V0∗ − V0 ) + V0∗V0 = 0

(59)

We require to select the N2 × N2 complex matrix V0 to meet this criterion. Suppose for example V0 is Hermitian. Then, we require that V02 = 0. Since V0 is Hermitian, this is possible only if V0 = 0 so there is no noise. At the other extreme, suppose V0 is skew Hermitian, i.e, V0∗ = −V0 . Then the above requirement becomes

−2iV0 − V02 = 0

(60)

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R. Singla and H. Parthasarathy / Commun Nonlinear Sci Numer Simulat 83 (2020) 105142

or writing V0 = iQ where Q is Hermitian, this requirement becomes

2Q + Q 2 = 0

(61)

If Q is non-singular, then we must have Q = −2I. Now more generally, assume that V0 = Q + iP where both Q and P are Hermitian. Then the requirement becomes

2P + (Q − iP )(Q + iP ) = 0

(62)

2P + Q 2 + P 2 + i(QP − P Q ) = 0

(63)

Thus,

If in particular, Q and P commute, we get

2P + P 2 + Q 2 = 0

(64)

In this case, we may thus choose an orthonormal basis in which both P and Q are diagonal with P = diag[ p1 , . . . , pK ], Q = diag[q1 , . . . , qK ] and then get

2 pk + p2k + q2k = 0, k = 1, 2, . . . , K

(65)

Since the p k s and q k s are real numbers (eigenvalues of Hermitian matrices) we must take pk so that 2 pk + p2k ≤ 0. If pk > 0,

then this becomes

pk ≤ −2 which is impossible while if pk < 0, this becomes 2 + pk > 0, i.e, −2 < pk < 0. Assuming this, we get qk = ± −2 pk − p2k .

Our simulation example is based on taking V0 = −2iI and hence the jerky Schrodinger equation is

dU (t ) = (−iH0 dt − 2I.dN (t ))U (t )

(66)

Equivalently, the wave function satisfies

dψ (t ) = (−iH0 dt − 2IdN (t ))ψ (t )

(67)

If the noise is to be much smaller than the signal amplitude, then we may scale H0 appropriately. It should be noted that when the intensity of the Poisson noise is high, the wave function evolution will show sharp spikes which means that sudden transitions from one state to the other are possible. These sudden transitions should ideally be determined by the stochastically averaged value of the quantum transition probabilities [11,12]. These transition probabilities are computed using the Levy quantum model for the evolution operator U(t) as

dU (t ) = (−iH0 dt − 2IdN (t ))U (t )

(68)

the solution to which is given by the Poisson chaos Dyson series

U (t ) = U0 (t ) +

 n≥1

(−2 )n



0
U0 (t − t1 )U0 (t1 − t2 ) . . . × U0 (tn−1 − tn )U (tn )dN (t1 ) . . . dN (tn )

(69)

Instead of evaluating the multiple Poisson integral, we assume an ansatz

U (t ) = exp(−iH0 t + iaN (t )) = exp(−iH0 t ) exp(iaN (t ))

(70)

where a is a constant. Then the Ito rule for Poisson processes gives

dU (t ) = −iH0U (t )dt + (exp(ia ) − 1 )U (t )dN (t )

(71)

so to get agreement with the Poisson Schrodinger Eq. (68), we require that

exp(ia ) − 1 = −2

(72)

exp(ia ) = −1

(73)

or

or a = π . Thus

U (t ) = exp(−iH0 t ) exp(iπ N (t ))

(74)

Let |n > , |m > be two distinct eigenstates of H0 with eigenvalues En and Em respectively. Then the exact transition probability amplitude from state |n > to state |m > in time t is given by

< m| exp(−iH0 t ) exp(iπ N (t ))|n >= exp(−iEm t ) exp(iπ N (t ))δ [m − n] = 0

(75)

so it becomes clear that no transitions can take place. However suppose we modify the Hamiltonian dynamics to the general case

dU (t ) = (−iH0 dt − iV0 dN (t ))U (t )

(76)

R. Singla and H. Parthasarathy / Commun Nonlinear Sci Numer Simulat 83 (2020) 105142

9

where V0 is a non-Hermitian operator. Then, for unitary dynamics, we require that

i(V0∗ − V0 ) + V0∗V0 = 0

(77)

Then we require the full Dyson series

U (t ) = U0 (t ) +



(−i )n

n≥1



0
U0 (t − t1 )V0U0 (t1 − t2 ) × V0U0 (t2 − t3 ) . . . V0U0 (tn )dN (t1 ) . . . dN (tn )

(78)

where U0 (t ) = exp(−itH0 ). If we truncate to one term in the Dyson series then

U (t ) ≈ U0 (t ) − i



t 0

U0 (t − t1 )V0U (t1 )dN (t1 )

(79)

so the stochastically averaged transition probability from state |n > to state |m > is approximately given by

E| < m|



t 0

F (t, t1 )dN (t1 )|n > |2 = E| =λ





t 0 t

0

< m|F (t, t1 )|n > dN (t1 )|2

| < m|F (t, t1 )|n > |2 dt1 + λ2 |



t 0

< m|F (t, t1 )|n > dt1 |2

(80)

where

F (t, t1 ) = U0 (t − t1 )V0U (t1 )

(81)

3.5. Quantum evolution of the robot in the presence of both white gaussian and white compound poisson noise The following remark may be made regarding the extension of the theory developed in the paper for the analysis of quantum systems with Levy noise. Suppose the noise consists of both a Gaussian and a compound Poisson component [13]. We propose the following model for the evolution operator in the Schrodinger equation subject to such noise:

U (t ) = [−(iH0 +

q q p   1 2 Lk )dt − i Lk dBk (t ) − i Vk XNk (t )+1 (k )dNk (t )]U (t ) 2 k=1

k=1

(82)

k=1

where the Vk s are operators acting in the system Hilbert space, L k s are Hermitian operators in the system Hilbert space, Nk (t) s are independent Poisson processes with λ k s as the respective intensities. The B k s are standard independent Brownian motion processes independent of the Poisson processes. Xn (k) s are independent Bernoulli random variables assuming values one and zero with probabilities P (Xn (k ) = 1 ) = pk , P (Xn (k ) = 0 ) = 1 − pk . We then find that

d (U (t )∗U (t )) = dU (t )∗U (t ) + U (t )∗ dU (t ) + dU (t )∗ dU (t ) = 0

(83)

provided that

i(Vk − Vk∗ ) + Vk∗Vk = 0

(84)

or equivalently that the operator

Sk = I + iVk

(85)

is unitary for every k. To prove these, we use the Ito formulae

dBk .dB j = δk, j dt, dNk dN j = δk, j dNk

(86)

( k )2

We also make use of the identity Xn = Xn (k ) for a Bernoulli random variable that assumes values only zero and one. Let ρ 0 be the initial mixed state of the system. After time t, it evolves to the random mixed state

ρ (t ) = U (t )ρ0U (t )∗

(87)

and the density matrix/state of the system component alone obtained after averaging over the bath variables is given by

ρs (t ) = E(ρ (t ))

(88)

A simple computation based on It’s formulae for the Brownian motion and Poisson processes then gives

ρs (t ) = −i[H, ρs (t )] −

 1 2 (Lk ρs (t ) + ρs (t )L2k − 2Lk ρs (t )Lk ) + λk pkVk ρs (t )Vk∗ 2 k

where

H = H0 +



λk pkVk

(89)

k

(90)

k

This equation can be solved perturbatively by replacing Lk with  o Lk and Vk with  o Vk . It is instructive to carry out this exercise and calculate ρ s (t) up to O(o2 ). Remark 1. The perturbation parameter  o used in this section is specific to time dependent perturbation theory and is used to evaluate the effects of noise on the Schrodinger Hamiltonian dynamics.

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4. Simulation studies 4.1. Quantum robot simulation with non-random torque and with gravity present: In the forthcoming sections, we describe a discrete algorithm for the quantized robot dynamics using the HamiltonianSchrodinger approach in the presence of a classical torque which may have a Levy noise component. The Hamiltonian of the system is expressed as

H (t ) =

1 T p M (q )−1 p + V (q ) − τ1 (t )q1 − τ2 (t )(q2 − q1 ) 2

(91)

where p = −i h ¯ ∇ q is the two dimensional momentum operator in quantum mechanics. To simulate the evolution of a wave function ψ (t, q) where q = (q1 , q2 ) we consider a discretization of the position space:

q1 = 2π k1 /N, q2 = 2π k2 /N, 0 ≤ k1 , k2 ≤ N − 1 Writing δ q = 2π /N we have q1 = k1 δ q, q2 = k2 δ q where δ q = 2π /N. The mass moment of inertia matrix of the system is



M (q ) =

a b.cos(q2 − q1 )



b.cos(q2 − q1 ) c

(92)

where a, b, c are of order ml2 where m is the typical mass of a link and l is its typical length. We find that

pT M (q )−1 p = (q )−1 (cp21 + ap22 − 2bp1 p2 cos(q1 − q2 ))

(93)

(q ) = det (M (q )) = ac − b2 cos2 (q2 − q1 )

(94)

During our simulation process, we replace the ordered pair of angular position variables (q1 , q2 ) by the single integer variable Nk1 + k2 + 1 where q1 = k1 δ q, q2 = k2 δ q. Thus, our wave function ψ (q1 , q2 ) is replaced by an N2 × 1 complex vector whose Nk1 + k2 + 1th entry is ψ (k1 δ q, k2 δ q) with 0 ≤ k1 , k2 ≤ N − 1. We note in passing that as k1 , k2 take values 0, 1, . . . , N − 1, Nk1 + k2 + 1 varies over 1, 2, . . . , N 2 in a one-to-one way. If ψ [k] represents the elements of such an N2 × 1, vector, then the action of p1 on such a vector should be represented by −i times a finite difference w.r.t the first variable k1 . This means that a column vector ψ whose entries are transforms to a vector p1 ψ

( p1 ψ )(Nk1 + k2 + 1 ) = −i(ψ (N (k1 + 1 ) + k2 + 1 ) − ψ (Nk1 + k2 + 1 ))/δ q

(95)

Likewise,

p2 ψ (Nk1 + k2 + 1 ) = −i(ψ (Nk1 + k2 + 2 ) − ψ (Nk1 + k2 + 1 ))/δ q

(96)

These operations can be described by replacing p1 by the matrix −i(ZN − IN )  IN /δ q and p2 by the matrix −iIN  (ZN − IN )/δ q where

ZN = ((δ [n − m + 1] ))1≤n,m≤N

(97)

is the spatial advancement operator for one variable [14,15]. We denote these matrices by P1 , P2 respectively. The multiplication operators (q2 − q1 ) and cos(q2 − q1 ) are respectively replaced by N2 × N2 diagonal matrices −1 and C whose (Nk1 + k2 + 1, Nk1 + k2 + 1 )th entries are (k2 − k1 )δ q ) and cos(k2 − k1 )δ q ). The time dependent Hamiltonian at time t is then an N2 × N2 matrix

H (t ) = (c/2 )P1 −1 P1 + (a/2 )P2 −1 P2 ) − b(P1 −1CP2 + P2 −1CP1 ) − τ1 (t )Q1 − τ2 (t )Q2

(98)

where

Q1 = Q  IN , Q2 = IN  Q with

Q = diag[kδ q :, k = 0, 1, . . . , N − 1] being the one dimensional discretized position matrix. 4.2. Scaling considerations Our simulation results show plots of the wave function with time for an initially chosen wave function. These simulations are based on the space-time discretized Schrodinger equation

ψ [t + 1] = ψ [t] − iδt.H (t )ψ [t]

(99)

Some plots of the wave function evolution have been shown in Figs. 1–8. The interpretation of these plots is that at each time t, |ψ [Nk1 + k2 + 1]|2 is proportional to the probability of the robot link angles (q1 (t), q2 (t)) to be concentrated around the point (k1 δ q, k2 δ q). It should be noted that the actual quantum robot dynamics must be obtained by scaling down the

R. Singla and H. Parthasarathy / Commun Nonlinear Sci Numer Simulat 83 (2020) 105142

11

Quantum Probability Distribution

1

Probability

0.8 0.6 0.4 0.2 0 0

10

20

30

40

50

60

70

80

90

100

Time ( in steps of delta = 0.001 seconds) Fig. 1. Link angles quantum probability density |ψ t (q1 , q2 )|2 (modulus square of wave function) at different times with low Levy noise.

Quantum Average Angular Position

Fig. 2. Link angles quantum probability density |ψ t (q1 , q2 )|2 (modulus square of wave function) with time and angular position on different axes, with low Levy noise.

Quantum Position Average

3

link 1 link 2

2.5 2 1.5 1 0.5 0 0

10

20

30

40

50

60

70

80

90

100

Time (in steps of delta = 0.001 seconds) Fig. 3. Angular position average (∫q|ψ t (q)|2 d2 q) with respect to quantum pdf |ψ t (q)|2 (modulus square of wave function) with low Levy noise.

masses and lengths of the links and correspondingly space and time [16,17]. For example, if we consider the Schrodinger equation

ih ¯

∂ψ (t, q ) = (− h ¯ 2 /2 )∇qT M (q )−1 ∇q ψ (t, q ) + V (q )ψ (t, q ) − (τ (t ), q )ψ (t, q ) ∂t

(100)

where M(q) is of the order ml2 with m the order of link masses and l the order of link length and V(q) is of order ml We replace m by α m and l by β l where α , β are scale factors very small compared to unity so that α m is the characteristic

R. Singla and H. Parthasarathy / Commun Nonlinear Sci Numer Simulat 83 (2020) 105142

Mean Square Angular Position Fluctuations

12

Angular Position Quantum Dispersion

3.5

link 1 link 2

3 2.5 2 1.5 1 0.5 0 0

10

20

30

40

50

60

70

80

90

100

Time (in steps of delta = 0.001 seconds)   Fig. 4. Variance ( q2k |ψt (q )|2 d2 q − ( qk |ψt (q )|2 d2 q )2 ) of quantum pdf |ψ t (q)|2 (modulus square of wave function) with low Levy noise.

Quantum Probability Distribution

Probability

0.8 0.6 0.4 0.2 0 0

10

20

30

40

50

60

70

80

90

100

Time ( in steps of delta = 0.001 seconds) Fig. 5. Link angles quantum probability density |ψ t (q1 , q2 )|2 (modulus square of wave function) at different times with high Levy noise.

Fig. 6. Link angles quantum probability density |ψ t (q1 , q2 )|2 (modulus square of wave function) with time and angular position on different axes, with high Levy noise.

atomic/molecular mass and α l is the characteristic atomic length, then the above Schrodinger equation becomes

ih ¯

∂ψ (t, q ) = (− h ¯ 2 /2αβ 2 )∇qT M (q )−1 ∇q ψ (t, q ) + αβ V (q )ψ (t, q ) − (τ (t ), q )ψ (t, q ) ∂t

(101)

we next define the space-time scaled wave function

φ (t, q ) = ψ (δ1 .t, δ2 .q )

(102)

R. Singla and H. Parthasarathy / Commun Nonlinear Sci Numer Simulat 83 (2020) 105142

Quantum Position Average

3.5

Quantum Average Angular Position

13

link 1 link 2

3 2.5 2 1.5 1 0.5 0 0

10

20

30

40

50

60

70

80

90

100

Time (in steps of delta = 0.001 seconds)

Mean Square Angular Position Fluctuations

Fig. 7. Angular position average (∫q|ψ t (q)|2 d2 q) with respect to quantum pdf |ψ t (q)|2 (modulus square of wave function) with high Levy noise.

Angular Position Quantum Dispersion

4

link 1 link 2

3.5 3 2.5 2 1.5 1 0.5 0 0

10

20

30

40

50

60

70

80

90

100

Time (in steps of delta = 0.001 seconds)   Fig. 8. Variance ( q2k |ψt (q )|2 d2 q − ( qk |ψt (q )|2 d2 q )2 ) of quantum pdf |ψ t (q)|2 (modulus square of wave function) with high Levy noise.

Then the scaled Schrodinger equation becomes

(i h¯ /δ1 )

∂φ (t, q ) ˜ (q )−1 ∇q φ (t, q ) + αβ V˜ (q )φ (t, q ) − δ2 (τ (t ), q )φ (t, q ) = (− h ¯ 2 /(2αβ 2 δ22 ))∇qT M ∂t

(103)

This equation can be rearranged as

(iαβ 2 δ22 / h¯ δ1 )

∂φ (t, q ) ˜ (q )−1 ∇q φ (t, q ) + (α 2 β 3 δ22 / h = (−1/2 )∇qT M ¯ 2 )V˜ (q )φ (t, q ) − (αβ 2 δ23 / h ¯ 2 )(τ (t ), q )φ (t , q ) ∂t

(104)

Thus if our scaling parameters are chosen in a suitable system of units so that

αβ 2 δ22 / h¯ δ1 = 1, α 2 β 3 δ22 / h¯ 2 = 1 and the torque process is replaced by the scaled torque

τ˜ (t ) = (αβ 2 δ23 / h¯ 2 )τ (t )

(105)

then we obtain the scaled Schrodinger equation which we have actually simulated:

i

∂φ (t, q ) ˜ (q )−1 ∇q φ (t, q ) + V˜ (q )φ (t, q ) − (τ˜ (t ), q )φ (t, q ) = (−1/2 )∇qT M ∂t

(106)

In these expressions, we have defined

˜ (q ) = M (δ2 q ), V˜ (q ) = V (δ2 q ) M These are the spatially (angular) scaled mass moment of inertia and gravitational or any other potential [18,19].

(107)

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R. Singla and H. Parthasarathy / Commun Nonlinear Sci Numer Simulat 83 (2020) 105142

4.3. Simulation with poisson noise The next equation simulated is the noisy Schrodinger Eq. (67). Fig. 1 shows a plot of the quantum probability density |ψ (t, q)|2 with position for low Levy noise. The position pair q = (q1 , q2 ) has been encoded as explained above into a single value. Fig. 1 shows that the probability attains a peak at a given position and decays with change in this position. Further, the peak is attained at a position which varies slowly with time showing approximate classical behavior. Some of the plots are nearly flat showing nearly equidistribution of the position. This means that after a sufficiently long time, the system displays quantum behavior. The low Levy noise amplitude is confirmed by the non-spiky behavior of the probability density. Fig. 2 displays the same data as Fig. 1 but with both time and space, ie, it is a plot of the probability density as a function of both time and position. Fig. 3 is a plot of the quantum average position for both the links for the same data as for Fig. 1 It shows that with progress in time, the quantum average stabilizes at a constant value. This justifies the fact that external torque has been applied to counter the torque produced by the gravitational field and further the system is not very small, so that it may be regarded as being approximately classical. When gravitational torque and external torque nearly cancel each other and in addition when the Levy noise is low, then the angular position is approximately constant in the classical case. In the quantum case, cancellation of the torques implies that only small quantum fluctuations will remain with the average remaining nearly constant. Fig. 4 displays the quantum mean square value of the link angular positions. It shows that this mean square value converges nearly to zero confirming approximate classical behavior. This happens because of the larger size and masses of the links (see the detailed explanation in the Remark 1 regarding Feynman path integrals). Fig. 5 shows the quantum probability density as a function of position at different times, corresponding to high Levy noise. The rapid fluctuations in this density are due to the high intensity Levy noise and most of the plots are increasing showing that the PDF’s begin to peak at larger values of the angular positions. This means that classical probabilistic behavior occurs in the vicinity of larger angles. Fig. 6 is a 3D plot of the PDF as a function of time and position. It shows that as time increases, the PDF peaks at smaller values of the angular positions and that this peak is highly dominant. The interpretation is that as time progresses the behavior becomes more classical. Fig. 7 displays a plot of the quantum position average versus time for the same data as for Fig. 5. It shows initial increase and then stabilization about a constant value. This confirms the fact that as time progresses, the cancellation of the torque causes stabilization about a fixed position. The plots also show small classical fluctuations arising from the high intensity Levy noise. This is confirmed by Fig. 8 which shows the quantum mean square value of the position with time. Initially this dispersion is large owing to both classical Levy and quantum random effects. As time progresses only the classical Levy random effect remains and the quantum fluctuations become small owing to the system size being larger compared to the Planck scale. The final residual dispersion is due to classical Levy noise. The figures demonstrate the following well known facts: (1) The wave function magnitude square at each time shows a peak at some position and decays on both sides showing that the angular position coordinates are more likely to take these fixed values. This confirms the fact that when the size of the body is much larger than the Planck’s scale, then the object behaves classically ie it is likely to hover around a fixed trajectory. On the other hand, when the size of the body becomes much smaller comparable to the Planck’s scale, then at each time, there is a well pronounced probability distribution over the different angles, ie, the trajectory of the object becomes random. (2) When the size of the object is large, the mean square quantum fluctuations in the angular variables is small owing to the body following classical mechanics while when the size is smaller, the mean square fluctuation becomes larger owing to dominance of quantum effects. The mean and mean square fluctuation in the position at any time t are computed by numerically discretizing the integrals

< q1 > (t ) = < q2 > (t ) =

 

< q21 > (t ) = < q22 > (t ) =

q1 |ψ (t, q1 , q2 )|2 dq1 dq2 , q2 |ψ (t, q1 , q2 )|2 dq1 dq2  

(108)

(q1 − < q1 > (t ))2 |ψ (t , q1 , q2 )|2 dq1 dq2 , (q2 − < q2 > (t ))2 |ψ (t , q1 , q2 )|2 dq1 dq2

(109)

The strength of the quantum fluctuations as a function of time is measured by the ”noise to signal ratio”

< q2k > (t ) < qk > (t )2

K

, k = 1, 2

(110)

In our simulation studies, we have computed these values at different times. Table 1 shows the nsr’s defined by

2 t=1 < (qk ) > (t )  K 2 t=1 (t )

for k = 1, 2 and for low and high intensity Levy noise [20]. When the intensity of the Levy noise is low in

the system, the wave function magnitude square does not show jerky discontinuous fluctuations, rather the wave function is smooth and peaks around a given angular coordinate pair. When the intensity of Levy noise is high, the wave function magnitude square shows sharp discontinuous fluctuations which means that sudden transitions from one state to the other are possible.

R. Singla and H. Parthasarathy / Commun Nonlinear Sci Numer Simulat 83 (2020) 105142

15

Table 1 NSR for quantum fluctuations. Noise Intensity

NSR q1

NSR q2

Low Levy Noise High Levy Noise

0.3297 0.3284

0.0668 0.3224

5. Conclusions We have in this paper presented a theoretical analysis of a quantum 2-link robot in a gravitational field in the presence of Levy noise perturbation. The Levy noise chosen is a compound Poisson process. Using It’s formula for the Poisson process, the general form of Schrodinger equation that guarantees unitary evolution has been derived. Simulation of the wave function which is a random field have been performed. The primary conclusion of these plots is that larger the robot, the more classical stochastic behavior id displayed by the wave function and smaller the robot, the more quantum behavior is displayed by the robot. Classical behavior corresponds to spikes in the wave function around large magnitudes peaked at an isolated point and quantum behavior corresponds to nearly equally large wave function magnitudes at different point with classical Levy spikes everywhere. Further, we have made computations on the statistics of the wave function as well as transition probabilities between two stationary states. The quantum analysis has been based on using time independent perturbation theory to evaluate the approximate evolution operator of the unperturbed robot. Finally, we remark that the importance of this work stems from the fact that a quantum robot having dimensions on the Planck’s scale appears in the form of bonded molecules and the length and angles of these bonds can be controlled by external torque induced using electromagnetic fields. Thus the chemical properties of such molecules can be altered leading to manufacture of new kinds of molecular medicine. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] Ticozzi F, Viola L. Analysis and synthesis of attractive quantum Markovian dynamics. Automatica 20 09;45:20 02–9. doi:10.1016/j.automatica.2009.05. 005. [2] Parthasarathy KR. An introduction to quantum stochastic calculus. Birkhuser; 1992. ISBN 978-3-0348-0566-7 [3] Zhang W, Chen B-S. Stochastic affine quadratic regulator with applications to tracking control of quantum systems. Automatica 2008;44:2869–75. doi:10.1016/j.automatica.2008.03.024. [4] van Kampen NG. A new approach to noise in quantum mechanics. J Stat Phys 2004;115:1057–72. doi:10.1023/B:JOSS.0000022383.06086.6c. [5] Schiff LI. Quantum mechanics. 3rd ed. McGraw Hill Higher Education; 1968. ISBN: 978-0070552876 [6] Thiemann T. Modern canonical quantum general relativity. 1st ed. Cambridge University Press; 2008. ISBN: 978-0521741873 [7] Hao-Han W, Fu-Jiang J, Lian-You L, Liang W. A stochastic filtering algorithm using Schrodinger equation. Acta Autom Sin 2014;40:2370–6. doi:10.1016/ S1874- 1029(14)60366- 9. [8] Gaoa Z, Shib X. Observer-based controller design for stochastic descriptor systems with Brownian motions. Automatica 2013;49:2229–35. doi:10.1016/ j.automatica.2013.04.001. [9] Qi B. A two-step strategy for stabilizing control of quantum systems with uncertainties. Automatica 2013;49:834–9. doi:10.1016/j.automatica.2013.01. 011. [10] Bonnabel S, Mirrahimi M, Rouchon P. Observer-based hamiltonian identification for quantum systems. Automatica 2009;45:1144–55. doi:10.1016/j. automatica.20 08.12.0 07. [11] Itami T. Nonlinear optimal control as quantum mechanical eigenvalue problems. Automatica 2005;41:1617–22. doi:10.1016/j.automatica.20 05.04.0 07. [12] Kuang S, Cong S. Lyapunov control methods of closed quantum systems. Automatica 2008;44:98–108. doi:10.1016/j.automatica.2007.05.013. [13] Singla R, Agarwal V, Parthasarathy H. Statistical analysis of tracking and parametric estimation errors in a 2-link robot based on Lyapunov function. Nonlinear Dyn 2015:1–22. doi:10.1007/s11071- 015- 2151- 9. [14] Khandekar DC, deFalco D. Applications of white noise calculus to the computation of Feynman integrals. Stoch Process Appl 1988;29:257–66. doi:10. 1016/0304-4149(88)90041-5. [15] Mirrahimi M, Rouchon P, Turinici G. Lyapunov control of bilinear Schrodinger equations. Automatica 2005;41:1987–94. doi:10.1016/j.automatica.2005. 05.018. [16] Beauchard K, Coron JM, Mirrahimi M, Rouchon P. Implicit Lyapunov control of finite dimensional Schrodinger equations. Syst Control Lett 2007;56:388– 95. doi:10.1016/j.sysconle.2006.10.024. [17] Eldar YC, Forney GD. Optimal tight frames and quantum measurement, information theory. IEEE Trans 2002;48:599–610. doi:10.1109/18.985949. [18] Yanagisawa M, Kimura H. Transfer function approach to quantum control-part II: control concepts and applications. Autom Control IEEE Trans 2003;48:2121–32. doi:10.1109/TAC.2003.820065. [19] Davies PCW, Betts DS. Quantum Mechanics. 2nd ed. CRC Press; 1994. ISBN: 978-0748744466 [20] Singla R, Parthasarathy H, Agarwal V. Classical robots perturbed by Levy processes: analysis and Levy disturbance rejection methods, 89; 2017. p. 553–75.