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Hamilton-Jacobi-Bellman equations for
Quantum Optimal Feedback Control
J. Gougha, V.P. Belavkinb, and O.G. Smolyanovc
aSchool of Computing & Informatics,
Nottingham-Trent University, NG1, 4BU, UK
bSchool of Mathematics,
Nottingham University, NG7 2RD, UK
cFaculty of Mathematics and Mechanics,
Moscow State University, Russia
J Opt. B: Quantum Semiclass. Opt. 7 (2005) S237ñS
Abstract We exploit the separation of the Öltering and control aspects of quan- tum feedback control to consider the optimal control as a classical sto- chastic problem on the space of quantum states. We derive the corre- sponding Hamilton-Jacobi-Bellman equations using the elementary argu- ments of classical control theory and show that this is equivalent, in the Stratonovich calculus, to a stochastic Hamilton-Pontryagin setup. We show that, for cost functionals that are linear in the state, the theory yields the traditional Bellman equations treated so far in quantum feed- back. A controlled qubit with a feedback is considered as example.
1 Introduction
When engineers set about to control a classical system with incomplete data, they can evoke the celebrated Separation Theorem which allows them to treat the problem of estimating the state of the system (based on typically partial observations) from the problem of how to optimally control the system (through feedback of these observations into the system dynamics), see for instance [17]. Remarkably, this approach may also be carried over to the quantum world which cannot be in principle completely observed: this was Örst pointed out by Belavkin in [3],[5], see also the later [12],[15]. Quantum measurement, by its very nature, leads always to partial information about a system in the sense that some quantities always remain uncertain, and due to this the measurement typically alters the prior to a posterior state in process. The Belavkin nondemo- lition principle [4, 6] states that this state reduction can be e§ectively treated within a non-demolition scheme [6],[7] when measuring the system over time.
Hence we may apply a quantum Ölter for either discrete [2] or time-continuous [4] non-demolition state estimation, and then consider feedback control based on the results of this Öltering. The general theory of continuous-time nondemoli- tion estimation developed in [7],[9],[10],[11] derives for quantum posterior states a stochastic Öltering evolution equation not only for di§usive but also for count- ing measurements, however we will consider here the special case of Belavkin quantum state Öltering equation based on a di§usion model described by a single white noise innovation, see e.g. [8],[33],[16]. We should also emphasize that the continuous-time Öltering equation can be obtained as the limit of a discrete-time state reduction based on von Neumann measurements [23],[24],[29],[30], however this time-continuous limit goes beyond the standard von Neumann projection postulate, replacing it by quantum Öltering equation as a stochastic Master equation. Once the Öltered dynamics is known, the optimal feedback control of the system may then be formulated as a distinct problem. Modern experimental physics has opened up unprecedented opportunities to manipulate the quantum world, and feedback control has been already successfully implemented for real physical systems [1],[22]. Currently, these activities have attracted interest in the related mathematical issues such as stability, observability, etc., [15], [25], [19], [26]. The separation of the classical world from the quantum world is, of course, the most notoriously troublesome task faced in modern physics. At the very heart of this issue is the very di§erent meanings we attach to the word state. What we want to exploit is the fact that the separation of the control from the Öltering problem gives us just the required separation of classical from quantum features. By the quantum state we mean the von Neumann density matrix which yields all the (stochastic) information available about the system at the current time - this we also take to be the state in the sense used in control engineering. All the quantum features are contained in this state, and the Öltering equation it satisÖes may then to be understood as classical stochastic di§erential equation which just happens to have solutions that are von Neumann density matrix valued stochastic processes. The ensuing problem of determining optimal control may then be viewed as a classical problem, albeit on the unfamiliar state space of von Neumann density matrices rather than the Euclidean spaces to which we are usually accustomed. Once we get accustomed to this setting, the problem of dynamical programming, Bellmanís optimality principle, etc., can be formulated in much the same spirit as before. We shall consider optimization for cost functions that are non-linear func- tionals of the state. Traditionally quantum control has been restricted to linear functions where - given the physical meaning attached to a quantum state - the cost functions are therefore expectations of certain observables. In this situa- tion, which we consider as a special case, we see that the distinction between classical and quantum features may be blurred: that is, the classical information about the measurement observations can be incorporated as additional random- ness into the quantum state. This is the likely reason why the separation does not seem to have been taken up before.
In quantum control theory it is necessary to consider time-dependent genera- tors L (t), through an integrable time dependence of the controlled Hamiltonian H (t), and, more generally, due to a square-integrable time dependence of the coupling operators R (t). We shall always assume that these integrability con- ditions, ensuring existence and uniqueness of the solution % (t) to the quantum state Master equation
d dt
% (t) = L^0 (t; % (t)) # (t; % (t)) ; (5)
for all for t t 0 with given initial condition % (t 0 ) = % 0 2 S, are fulÖlled. Let F = F [] be a (nonlinear) functional % 7! F [%] on A? (or on S A?), then we say it admits a (Frechet) derivative if there exists an A-valued function r%F [] on A? (T 0? -valued functional on T 0 ) such that
lim h! 0
h
fF [ + h ] F []g = h ; r%F []i ; (6)
for each 2 A? (for each 2 T 0 ). In the same spirit, a Hessian r (^) % 2 r% r% can be deÖned as a mapping from the functionals on S to the A (^) sym^2 -valued functionals, via
lim h;h^0! 0
hh^0
fF [ + h + h^0 0 ] F [ + h ] F [ + h^0 0 ] + F []g
= h 0 ; r% r%F []i : (7)
and we say that the functional is twice continuously di§erentiable whenever r (^) % 2 F [] exists and is continuous in the trace norm topology. Likewise, a functional f : X 7! f [X] on A is said to admit an A?-derivative if there exists an A?-valued function rX f [] on A such that
lim h! 0
h ff [ + hA] f []g = hrX f [] ; Ai (8)
for each A 2 B (h). The derivative rX f [] has zero trace, rX f [A] 2 T 0 for each A 2 A, if and only if the functional f [X I] does not depend on , i.e. is essentially a function f (p) of the class p [X] 2 T 0?. With the customary abuses of di§erential notation, we have for instance
r%f (h%; Xi) = f 0 (h%; Xi) X; rX f (h%; Xi) = f 0 (h%; Xi) %;
for any di§erentiable function f of the scalar x = h%; Xi. Typically, we shall use r% more often, and denote it by just r.
3 Quantum Filtering Equation
The state of an individual continuously measured quantum system does not coincide with the solution of the deterministic master equation (5) but instead
depends on the random measurement output! in a causal manner. We take the output to constitute a white noise process f (t) : t 0 g, in which case it is math- ematically more convenient to work with the integrated process fw (t) : t 0 g, given formally by w (t) =
R (^) t 0 ^ (s)^ ds.^ It is then natural to model^ w^ (t).as a Wiener process and here we take ( ; F; P) to be the canonical probability space: that is, is the space of all continuous paths! = f! (t) : t 0 g with! (0) = 0, and w (t) is the co-ordinate process w (t) ! (t), for each outcome !. The process fw (t) : t 0 g is then the innovations process. We then view the state as an S-valued stochastic process % (t) :! 7! %! (t), depending on the particular observations! = f! (t)g 2. (Here we shall use the symbol as subscript to indicate that the kernel symbol describes a random variable when we do not want to display !.) Causality is reáected through the requirement that the state process be adapted: that is % (t) is measurable variable with respect to the sigma-algebra generated by the Wiener output upto and including time t for each t 0. The Belavkin quantum Öltering equation giving the evolution of the Öltered state in this case is [8],[11],[33],[16]
d% (t) = # (t; % (t)) dt + (% (t)) dw (t) (9)
where dw (t) = w (t + dt) w (t), the time coe¢ cient is
(t; %) = i [%; H (t)] + L^0 R (%) + L^0 L (%) ; (10)
with L^0 L (%) of the form given
L^0 L (%) = L%Ly^
%LyL
LyL%;
and the áuctuation coe¢ cient is
(%) = L% + %Ly^ %; L + Ly^ %: (11)
Here L is a bounded operator describing the coupling of the system to the measurement apparatus. The time coe¢ cient # consists of three separate terms: The Örst term is Hamiltonian and depends on time through the dependence of H on a steering parameter u (t) (belonging to some parameter space U) which we must specify at each time; the second term is the adjoint of a general Lindblad generator LR due to a reservoir coupling which describes the uncontrolled, typically dissipative, e§ect of the environment; the Önal term is adjoint to the time independent Lindblad generator L which is related to the coupling operator L with the measurement apparatus. The maps # and are required to be Lipschitz continuous in all their com- ponents: for L constant and bounded, this will be automatic for the %-variable with the notion of trace norm topology. We remark that tr f (%)g = 0 if tr% = 1 and, by conservativity, tr f# (t; %)g = 0 for all % 2 A?. This implies that the normalization tr% = 1 is conserved under the stochastic evolution (9) and so that q! (t) = % 0 %! (t) 2 T 0 for all t t 0 if %! (t 0 ) = % 0.
4 Optimal Control
From now on we will assume that the Hamiltonian H and therefore # (and ) are functions of a controlled parameter u 2 U depending on t such that the time dependence of the generator L is of the form L (u (t)). Moreover, we do not require at this stage the linearity of # (u; %) with respect to %, as well as the quadratic dependence (%), which means that what follows below is also applicable to more general quantum stochastic kinetic equations
d% (t) = # (u (t) ; % (t)) dt + (% (t)) dw (t)
of Vlassov and Boltzmann type, with only the positivity and trace preservation requirements tr f# (u; %)g = 0 = tr f (%)g. A choice of the control function fu (r) : r 2 [t 0 ; t]g is required before we can solve the Öltering equation (9) at the time t for a given initial state % 0 at time t 0. From what we have said above, this is required to be a U-valued function which we take to be continuous for the moment. The cost for a control function fu (r)g over any time-interval [t; T ] is random and taken to have the integral form
J! [fu (r)g ; t; %] =
Z T
t
C (u (r) ; %! (r)) dr + G (%! (T )) (18)
where f% (r) : r 2 [t; T ]g is the solution to the Öltering equation with initial condition % (t) = %. We assume that the cost density C and the terminal cost, or bequest function, G will be continuously di§erentiable in each of its arguments. In fact, due to the statistical interpretation of quantum states, we should consider only the linear dependence
C (u; %) = h%; C (u)i ; G (%) = h%; Gi (19)
of C and G on the state % as it was already suggested in [5],[7],[12]. We will ex- plicitly consider this case later, but for the moment we will not use the linearity of C and G. We refer to C (u) 2 A as cost observable for u 2 U and G 2 A as the bequest observable. The feedback control u (t) is to be considered a random variable u! (t) adapted with respect to the innovation process w (t), in line with our causality requirement, and so we therefore consider the problem of minimizing its average cost value with respect to fu (t)g. To this end, we deÖne the optimal average cost on the interval [t; T ] to be
S (t; %) := inf fu(r)g
E [J [fu (r)g ; t; %]] ; (20)
where the minimum is considered over all measurable adapted control strategies fu (r) : r tg. The aim of feedback control theory is then to Önd an optimal control strategy fu (t)g and evaluate S (t; %) on a Öxed time interval [t 0 ; T ]. Ob- viously that the cost S (t; %) of the optimal feedback control is in general smaller
than the minimum of E [J [fug ; t; %]] over nonstochastic strategies fu (r)g only, which gives the solution of the open loop (without feedback) quantum control problem. In the case of the linear costs (19) this open-loop problem is equivalent to the following quantum deterministic optimization problem which can be tack- led by the classical theory of optimal deterministic control in the corresponding Banach spaces.
4.1 Bellman & Hamilton-Pontryagin Optimality
Let us Örst consider nonstochastic quantum optimal control theory assuming that the state % (t) 2 S obeys the master equation (5) where # (u; %) is the adjoint L^0 (u) of some Lindblad generator for each u with, say, the control being exercised in the Hamiltonian component i [; H (u)] as before. (More generally, we could equally well consider a nonlinear quantum kinetic equation.) The control strategy fu (t)g will be here non-random, as will be any speciÖc cost J [fug ; t 0 ; % 0 ]. As for S (t; %) = inf J [fug ; t; %] at the times t < t + " < T , one has
S (t; %) = inf fug
(Z
t+"
t
C (u (r) ; % (r)) dr +
Z T
t+"
C (u (r) ; % (r)) dr + G (% (T ))
Suppose that fu^ (r) : r 2 [t; T ]g is an optimal control when starting in state % at time t, and denote by f%^ (r) : r 2 [t; T ]g the corresponding state trajectory starting at state % at time t. Bellmanís optimality principle [13],[17] observes that the control fu^ (r) : r 2 [t + "; T ]g will then be optimal when starting from %^ (t + ") at the later time t + ". It therefore follows that
S (t; %) = inf fu(r)g
Z (^) t+"
t
C (u (r) ; % (r)) dr + S (t + "; % (t + "))
For " small we expect that % (t + ") = % + # (u (t) ; %) " + o (") and provided that S is su¢ ciently smooth we may make the Taylor expansion
S (t + "; % (t + ")) =
@t
S (t; %) + o (") : (21)
In addition, we approximate Z (^) t+"
t
C (u (r) ; % (r)) dr = "C (u (t) ; %) + o (")
and conclude that (note the convective derivative!)
S (t; %) = inf u 2 U
C (u; %) +
@t
S (t; %)
where now the inÖmum is taken over the point-value of u (t) = u 2 U. In the limit "! 0 , one obtains the equation
@t
S (t; %) = inf u2U fC (u; %) + h# (u; %) ; rS (t; %)ig ; (22)
Thus the Pontryagin maximum principle for the quantum dynamical system is the observation that the optimal quantum control problem is equivalent to the Hamiltonian problem for state and co-state fqg and fpg respectively, leading to optimality K# (u; q; p) H# (q; p) with equality for u = u^ (q; p) maximizing K# (u; q; p).
4.2 Bellman Equation for Filtered Dynamics
We now consider the stochastic di§erential equation (9) for the Öltered state in place of the master equation (5). This time, the cost is random and we consider the problem of computing the minimum average cost as in (20). The Bellman principle can however be applied once more. As before, we let fu! (t)g be a stochastic adapted control leading to optimality and let %! (r) be the corresponding state trajectory (now a stochastic process) starting from % at time t. Again choosing t < t + " < T , we have by the Bellman principle
E [S (t + "; % (t + "))] + o (") = S (t; %)
@S
@t (t; %) + C (u; %) + D (u; %) S (t; %)
Taking the limit "! 0 yields the di§usive backward Bellman equation
@S
@t
= inf u2U
fC (u; %) + D (u; %) S (t; %)g : (28)
This equation is to be solved backward with the terminal condition S (T; %) = G (%). Using the Hamiltonian function (24) this can be written in the Hamilton- Jacobi form as
@S
@t
(%) + H# (q; rS (%)) =
h (%) (%) ; r rS (%)i ; (29)
where we have replaced the co-state p [rS (%)] by its representative rS (%) 2 A and omitted with the customary abuse of notation the argument q in % (q) but not in H (q; p). Note that since the di§erence
D (u; %) S (t; %) h# (u; %) ; rS (t; %)i =
D
(%) 2 ; r 2 S (t; %)
E
is assumed to be independent of u, the solution u^ to the minimization problem
inf u2U fC (u; %) + D (u; %) S (t; %)g =
D
(%) 2 ; r 2 S (t; %)
E
H# (q; rS (t; %))
in (28) coincides with the solution u^ (q; p) of the corresponding nonstochastic problem (26) for q = % 0 % and p = p [rS (t; %)].
5 Stochastic Hamilton-Jacobi-Bellman Equation
An alternative approach to deriving the equation (29) will now be formulated, in which the control strategy is a prori not assumed to be nonanticipating. First of all we make a Wong-Zakai approximation [34] to the Stratonovich Öltering
equation (13). This is achieved by introducing a di§erentiable process w !( )(t) = R (^) t 0
() (^) (r) dr converging to the Wiener noise w (t) as ! 0 almost surely and
uniformly for t 2 [0; T ]. (For instance, we may take
n ()^ (t)
o to be Gaussian
with a Önite auto-correlation time which vanishes as ! 0 .) We may then
expect the same type of convergence for the solution,
n % () ! (t)
o , to the random
ODE d dt
%( ! )(t) =
u (t) ; %( ! )(t)
%( ! )(t)
()^ (t)
with non-random initial condition % () ! (t 0 ) =^ % 0 , as for the solution^ f%! (t)g with the same initial data %! (t 0 ) = % 0. If we Öx the output! 2 , then we have an equivalent non-random dynami- cal system for which we will have a minimal cost function and we denote this as S( ! )(t 0 ; % 0 ). Note that this depends on the assumed realization of the measure- ment output process and on the approximation parameter . The HJB equation for S( ! )(t; %) will be (25) with (u) now replaced by (t; u) = (u) + ()^ (t):
@t
S( ! )(t) + H
q; rS( ! )(t)
D
; rS( ! )(t)
E
()^ (t) ;
where we omitted the argument % in S, and in q = % 0 % q (%). In the limit ! 0 we obtain the Stratonovich SDE
dS! (%) + H (% 0 %; rS! (%)) dt = h (%) ; rS! (%)i dw: (30)
which may be called a stochastic Hamilton-Jacobi-Bellman equation. Note that since (%) ()^ (t) doesnít depend on u, the corresponding optimal strategy u! (t) as the solution of the optimization problem
inf u2U
n C (u) +
D
(u) + !()^ (t) ; X
Eo (q) =
D
!()^ (t) ; X
E
(q) H (q; p [X])
is the same function u^ (q; p) of q = % 0 % () ! (t)^ and^ p^ =^ p
h rS () ! (t)
i as in (26)
independent of ()^ (t). Moreover, due to independence of the di§erence
L (%) + (%) Ly^ (%) ; L + Ly^ % %; L + Ly^ (%)
on u, the function # (u) in (26) may technically be even replaced by the function (u).
where H~v (p) = Hv ( p). We then have the forward-time equation ~S
S~
+ H~
r~S ( )
D
; rS~ ( )
E
w ~
w~
= o (") ;
and using the ItÙ-Stratonovich transformation
D ; r~S ( )
E
w ~
w~
D
; r~S ( )
E
[ ~w ( + ") w~ ( )]
D
; r
D
; r~S ( )
EE
we get by substitution S^ ~
~S
D
; r
D
; rS~ ( )
EE
+ H~
rS~ ( )
D
; r~S ( )
E
[ ~w ( + ") w~ ( )] = o (") :
or, in the backward form for the original S (t; %) = ~S (T t; %), S
t +
S
t
h; r h; rS (t)ii "
H (rS (t)) " + h; rS (t)i [w (t) w (t ")] = o (") :
In the di§erential form this clearly is the same as (31).
If we denote by E expectation on , then E [h; rS (t)i d w~ (t)] = 0 since the backward solution S (t), and its derivatives, with nonstochastic terminal con- dition S (T ) = S are independent of the mean-zero past-point ItÙ di§erentials d w~ (t). We then have as a corollary that the averaged cost S (t; %) := E [S (t; %)] will satisfy the equivalent di§usive Hamilton-Jacobi equation
@S
@t
h; r h; rSii (32)
which is the Hˆrmander form of the Bellman equation (29) for optimal cost S (t; %). This proves that the optimal control strategy realizing the stochastic Hamilton Jacoby equation 30 is in average not better than the nonaticipat- ing strategy realizing the equation 28. (In fact it coincides with the optimal nonanticipating strategy in this case.)
6 Linearñquadratic State Cost
A tractable special case, applicable to quantum mechanics, occurs when C (u; %) and G (%) are both linear (19) in the state %, with quadratic dependence of C on u.
Let us specify a cost observable with control parameter u =
u^1 ; ; un
2 Rn and having a quadratic dependence of the form (Einstein index convention!)
C (u) =
g u u + u F + C 0
where (g ) are the components of a symmetric positive deÖnite metric with inverse denoted
g
and F 1 ; ; Fn; C 0 are Öxed bounded operators. We take control Hamiltonian operator to be
H (u) = u Q
where Q 1 ; ; Qn are Öxed controlled coordinates (bounded observables). Our aim is to Önd the optimal value u^ for each pair (q; p) giving a minimum to h%; C (u)i + h# (u; %) ; pi for % = % (q): we will have
@u
fh%; C (u)i + h# (t; u; %) ; Xig
= g u + hP; F i + hi [P; Q ] ; Xi :
Thus the optimal control u^ (q; p (X)) is given by the components
u = g h% (q) ; F + i [Q ; p [X]]i ;
where p [X] is any operator X I. This yields a unique point of inÖmum and on substituting we determine that
H# (q; p) =
g h% (q) ; F + i [Q ; p]i h% (q) ; F + i [Q ; p]i
h% (q) ; C 0 + LR (p) + LL (p)i :
As a result, the Hamilton-Jacobi-Bellman equation takes the form
@S
@t
g h%; F + i [Q ; rS]i h%; F + i [Q ; rS]i
= h%; C 0 + LR (rS) + LL (rS)i +
h (%) (%) ; (r r) Si :
The terminal condition being that S (%; T ) = h%; Gi.
6.1 Controlled Qubit
Let us illustrate the above for the case of a qubit (two-state system). The feedback control problem we consider is similar to the one formulated in [15] with qubit Öltering equation derived in [10]. Choosing a basis f g in the tangent space T 0 of zero trace 2 2 matrices, say given by Pauli spin vector ~ = ( (^) x; (^) y ; (^) z ) with
(^) x =
; (^) y =
0 i i 0
; (^) z =
Putting everything together, we Önd that the Hamilton-Jacobi-Bellman equa- tion is
@S
@t
~p r~S
2
x
@S
@x
@S
@y
^2
x^2 z^2
@^2 S
@x^2
@^2 S
@y^2
1 z^2
2 @^2 S
@z^2
@^2 S
@x@y
xz
1 z^2
@^2 S
@x@z
yz
1 z^2
@^2 S
@y@z
7 Discussion
In our analysis we have sought to think of the quantum state of a controlled system (that is, its von Neumann density matrix) in the same spirit as classical control engineers think about the state of the system. This is possible since quantum (mixed) state is normally a su¢ cient coordinate for closed, as well as open quantum systems under the Markov approximation, and this remains true in the appropriate stochastic sense even if the open system is under a continuous nondemolition observation. The advantage of this is that all the quantum fea- tures of the problem are essentially tied up in the state: once the measurements have been performed the information obtained can be treated as essentially clas- sical, as can the problem of using this information to control the system in an optimal manner. The disadvantage is that we have to deal with a stochastic di§erential equation on the inÖnite dimensional space of quantum states. Nev- ertheless, the Bellman principle can then be applied in much the same spirit as for classical states and we are able to derive the corresponding Hamilton- Jacobi-Bellman theory for a wider class of cost functionals than traditionally considered in the literature. When restricted to a Önite-dimensional represen- tation of the state (on the Bloch sphere for the qubit) with the cost being a quantum expectation, we recover the class of Bellman equations encountered as standard in quantum feedback control. Another quantum optimal feedback control problem in the Önite-dimensional space of the su¢ cient coordinates of the Gaussian Bosonic states was considered in [12].
Acknowledgment We would like to thank Luc Bouten, Ramon van Handel, Hideo Mabuchi, Aubrey Truman for useful discussions. J.G. would like to acknowledge the sup- port of EPSRC research grant GR/R78404/01, and V.P.B. acknowledges EEC support through the ATESIT project IST-2000-29681 and the RTN network QP&Applications.
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