Factores psicológicos

Here, we present a typical application of the previous results to bilinear control

systems arising in open quantum dynamics. Let her₀(n) be the set of all hermitian

n×n-matrices with trace zero and consider the following bilinear control system on

GL(her₀(n)): ˙ X= (A0−i m ∑ k=1 uk(t)adHk)⋅X , X(0) ∶= Iher0(n), (21)

where all Hkare traceless hermitian n×n-matrices and A0can be for now an arbitrary

linear operator acting on her₀(n). Moreover, let ad_su(n)denote the adjoint action of su(n) of her₀(n), i.e.

ad_su(n)∶= {iadH∶= [iH,⋅] ∶ her0(n) → her0(n) ∣ iH ∈ su(n)}. (22)

Then, Corollary 12 and Proposition 13 imply the following preliminary result.

Theorem 14. Let D∶= gl(her₀(n)) and C ∶= ad_su(n)⊂ gl(her₀(n)). Then (21) is

generically accessible.

The controlled unital Lindblad-Kossakowski master equation

The state of a finite dimensional n-level quantum system is completely described by its density operator ρ. Thus the entire state space is given by the compact convex set

D ∶= {ρ ∈ Cn×n_{∣ ρ = ρ}†_{≥ 0 , Tr(ρ) = 1} (23)}

of all positive semidefinite operators with trace one acting on the Hilbert space

H ∶= Cn_{. In what follows, we consider only open quantum control systems described}

by the Lindblad-Kossakowski master equation [10, 17] with coherent control inputs, i.e. the control inputs enter only the Hamiltonian part of the systems. Precisely, we have ˙ ρ= L(ρ) = −i[H0+ m ∑ k=1uk(t)Hk,ρ]+LD(ρ), ρ(0) = ρ0∈ D (24)

where H0∈ her0(n) and H1,...,Hm∈ her0(n) denote the internal drift Hamiltonian

and external control Hamiltonians, respectively. As before, u(t) ∶= (u1(t),...,um(t))

are admissible time-dependent control signals with values in U∶= Rm. The dissipative

drift term LD, which models various interactions with the environment, can be

expressed as a linear operator of the following form [10, 17]

LD(ρ) = 1 2 n2−1 ∑ j,k=1 ajk([Bj,ρBk]+[Bjρ,Bk]). (25)

Here, without loss of generality, we take(B1,...,Bn2₋₁) to be any orthonormal

basis of her₀(n). Moreover, A ∶= (a_jk)_j,k=1,...,n2₋₁has to be positive semidefinite to

guarantee complete positivity of the semi-flow(etL)

For the definition of complete positivity and issues related to its physical interpreta- tions in open quantum systems, we recommend to consult e.g. [1]. For further issues on completely positive maps and their relations to Lie semigroups, Lie wedges and reachable sets of open quantum systems, see also [7, 15, 16] and the references therein. The Lindblad-Kossakowski master equation (24) is called unital if its flow leaves the density matrix ρ= In/n invariant, i.e. if L(In) = 0. Otherwise, when L(In) ≠ 0, it

is called non-unital. Now, we are ready to state and prove the announced genericity result for the unital Lindblad-Kossakowski master equation.

Theorem 15. The unital n-level Lindblad-Kossakowski master equation with a single coherent control is generically accessible. More precisely, let C∶= ad_su(n)and let D denote the set of all operators acting on her(n) of the form −adiH0+LD, whereiH0

is in su(n) and LDis unital and given by(25). Then, the family of bilinear control

systems described by(24) is generically accessible with respect to D×C.

Proof. Instead of ρ∈ D consider the reduced density matrix ˆρ ∶= ρ −In/n ∈ her0(n).

Since (24) is assumed to be unital, the time evolution of ˆρ follows again (24).

Moreover, if we can show that the group lift of (24) to GL(her₀(n)) given by ˙

X= (LD−iadH0−i

m

∑

k=1uk(t)adHk)⋅X , X(0) ∶= Iher0(n), (26)

is generically accessible then the same holds for (24) as GL(her₀(n)) acts clearly

transitively on her₀(n). Now, by Theorem 14 we know that generic accessibility

holds with respect to gl(her₀(n))×ad_su(n). Since it is known [15] that the set D is a closed convex cone of gl(her₀(n)) with non-empty interior the result follows. A similar result holds for the non-unital case, the interested reader is referred to [15].

Acknowledgments

The work by I. Kurniawan was fully supported by the Elite Network of Bavaria (ENB) from the Bavarian State Ministry of Science, Research and Arts, under the framework of the International Doctorate Program in Engineering and Computer Science: Identification, Optimization and Control with Applications in Modern Technologies.

Bibliography

[1] R. Alicki and K. Lendi. Quantum dynamical semigroups and applications, volume 717 of Lecture Notes in Physics. Springer, second edition, 2007. Cited p. 113.

[2] C. Altafini. Controllability properties for finite dimensional quantum Markovian master equations. J. Math. Phys., 44(6):2357–2372, 2003. Cited p. 104. [3] C. Altafini. Coherent control of open quantum dynamical systems. Phys. Rev.

A, 70:062321, 2004. Cited p. 104.

[4] N. Bourbaki. Lie Groups and Lie Algebras, Chapters 1-3. Springer, second edition, 1989. Cited p. 107.

Festschrift in Honor of Uwe Helmke G. Dirr et al. [5] D. D’Alessandro. Introduction to Quantum Control and Dynamics. Chapman

& Hall/CRC, 2008. Cited p. 104.

[6] G. Dirr and U. Helmke. Lie theory for quantum control. GAMM-Mitteilungen, 31(1):59–93, 2008. Cited p. 104.

[7] G. Dirr, U. Helmke, I. Kurniawan, and T. Schulte-Herbrüggen. Lie-semigroup structures for reachability and control of open quantum systems: Kossakowski- Lindblad generators form Lie wedge to Markovian channels. Rep. Math. Phys., 64(1-2):93–121, 2009. Cited p. 113.

[8] D. Elliott. Bilinear Control Systems: Matrices in Action. Springer, 2009. Cited pp. 103 and 106.

[9] J. Gauthier and G. Bornard. Contrôlabilité des systèmes bilinéaires. SIAM J. Control and Optimization, 20(3):377–384, 1982. Cited p. 110.

[10] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan. Completely positive dynamical semigroups of N-level systems. J. Mathematical Phys., 17(5):821– 825, 1976. Cited pp. 104 and 112.

[11] V. Jurdjevic. Geometric Control Theory. Cambridge University Press, 1997. Cited pp. 103, 104, 105, 106, and 108.

[12] V. Jurdjevic and I. Kupka. Control systems on semi-simple Lie groups and their homogeneous spaces. Ann. Inst. Fourier, Grenoble, 31(4):151–179, 1981. Cited pp. 103, 104, 107, 108, 109, and 110.

[13] V. Jurdjevic and H. J. Sussmann. Control systems on Lie groups. J. Differential Equations, 12:313–329, 1972. Cited p. 106.

[14] A. W. Knapp. Lie Groups Beyond an Introduction. Birkhäuser, second edition, 2002. Cited pp. 108, 109, and 111.

[15] I. Kurniawan. Controllability aspects of the Lindblad-Kossakowski master equations, a Lie-theoretical approach. PhD thesis, University of Würzburg, 2010. Cited pp. 110 and 113.

[16] I. Kurniawan, G. Dirr, and U. Helmke. Controllability aspects of quantum dynamics: A unified approach for closed and open systems. To appear in IEEE Transaction on Automatic Control, special issue on Quantum Control. Cited p. 113.

[17] G. Lindblad. On the generators of quantum dynamical semigroups. Comm. Math. Phys., 48(2):119–130, 1976. Cited pp. 104 and 112.

[18] R. R. Mohler. Bilinear control processes. With applications to engineering, ecology, and medicine. Academic Press, 1973. Cited p. 103.

[19] F. Silva Leite. Pairs of generators for compact real forms of the classical Lie algebras. Lin. Algebra and its Applications, 121:123–133, 1989. Cited p. 110. [20] F. Silva Leite and P. Crouch. Controllability on classical Lie groups. Math.

Control Signals Systems, 1:31–42, 1988. Cited p. 110.

[21] E. Sontag. Mathematical Control Theory. Springer, 1990. Cited pp. 103 and 107.

[22] H. J. Sussmann. Some properties of vector field systems that are not altered by small perturbations. J. Differential Equations, 20:292–315, 1976. Cited p. 103.

Detection of motion direction of targets using a turtle