Optimal Measurement and Control in Quantum Dynamical Systems.

Transcription

1 Optimal Measurement an Control in Quantum Dynamical Systems. V P Belavin Institute of Physics, Copernicus University, Polan. (On leave of absence from MIEM, Moscow, USSR) Preprint No 411, Torun, February 1979 Abstract A Marovian moel for an open quantum ynamical system with input an output channels an a feebac is escribe. A multi-stage version of the theory of quantum measurement an statistical ecisions applie to the optimal control problem for quantum ynamical iscrete-time objects is evelope. Quantum analogies of Stratonovich non-stationary ltering an Bellman quantum ynamical programming for the time being iscrete are obtaine. The Gaussian case of quantum one-imensional linear Marovian ynamical system with a quantum linear transmission line is stuie. The optimal quantum multi-stage ecision rule consisting of the classical linear optimal control strategy an quantum optimal ltering proceure is foun. The latter contains the optimal quantum coherent measurement on the output of the line an the recursive processing by Kalman{Bucy lter. All the results are illustrate by an example of the optimal control problem for a quantum open oscillator at the input of a quantum wave transmission line. 1 Introuction High perspectives of applying quantum coherent electromagnetic generators of optical an infra-re frequency ban for communication an control of quantum ynamical objects stimulates an increase of the interest in theoretical investigations of potential possibilities of information systems containing quantum channels. Due to funamental limitations of quantum-mechanical irect measurement a specic quantum problem of optimal inirect observation on the output of a quantum channel arises in such investigations. Here we consier such a problem for a quantum open systems with a xe output quantum noisy channel an input feebac channel for optimal control of quantum ynamics of the system base on the results of the optimal quantum observations. As for the criteria of optimality for inirect quantum measurement we tae the quality criteria of feebac control corresponing to the classical optimal control theory. It is essential in quantum feebac control theory that the systems uner the continuous

2 temporal observation is ept continuously open, i.e. it shoul match with the channel of observation, in orer not to be emolishe by them, by letting out an information. Such matching property, which we call nonemolition conition, is formulate in this paper. First we iscover this conition for the simplest case of quantum open controlle system: quantum open oscillator matche to a pair of transmission lines, an then we emonstrate its use for an explicit solution of the linear-quaratic problem of quantum optimal feebac control by eriving quantum optimal Kalman-type lter. Although is shown in this example the nonemolition conition allows a natural extension of the orthoox quantum theory of static or instantaneous measurements into the continuous temporalspacial omain, our main concern here will be the formulation of the time-iscrete theory of quantum temporal measurements an the corresponing multi-stage quantum ecision an control theory. The problem of quantum measurement optimization, as it formulate by K. Helstrom [1] primarily for etection an estimation in the static quantum communication systems, has been stuie so far only in non-temporal setting [2{7]. The solutions of the problem of optimal etection an estimation in quantum channels, which have been foun within this framewor of the single-stage (static) quantum-statistical ecision theory, are typical noncausal, an therefore they cannot be realize in real time of the temporal omain for optimal of quantum feebac control an ltering. The ynamical problem of quantum temporal measurement { quantum optimal ltering for quantum communication has consiere recently by the author [8] in the iscrete time Marov chain setting. As it was shown in this paper, the quantum ynamical programming for multi-stage optimal measurement problem can be consierably simplie ue to assumption that not only the processing of the measurement results but also the quantum measurement itself may epen on all previous measurement results. It correspons to the assumption that we can choose a quantum measurement apparatus on the basis of the previous measurement ata separately at every instant in time. Though in reality it is possible to imagine such a situation only for a nite number of stages an a nite set of measurement results (time an measuring scale being iscrete), this extension of amissible measurement an ecision proceures is mathematically very convenient an from the physical point of view is not contraictory. The choice of the measuring apparatus an of the observe ata processing accoring to all previous measurement results on the whole enes the strategy in multi-stage quantum ecision theory escribe here. Within the framewor of such an approach the problem of quantum ltering of ranom signal was reuce in [8] to the well-stuie problem of the static optimal quantum measurement on every xe stage with conitional a priori istribution epening on the previous observe ata. Here we escribe the multi-stage quantum statistical ecision theory applie to the problem of optimal control of a quantum Marovian iscrete time system with a matche quantum channel. This theory may be consiere as a quantum (operational) analogue of the stochastic control theory, base on Stratonovich theory of conitional Marovian processes [9], an Bellman ynamic programming [10]. The optimal ltering an the control strategy are foun here in case of one-imensional quantum linear Marovian system with quantum Gaussian noises an the mean-square loss function both in the iscrete an continuous time. In orer to pose the problem of measurement an control correctly from the physical

3 point of view, let us consier the following motivating example. 2 Controlle quantum open oscillator with quantum transmission line We are going to give a Marovian moel of the simplest quantum system with a communication channel: the quantum open oscillator matche with a transmission line. It is an excellent mathematical moel of a single-moe antenna for quantum raiophysics an optical control an communication. Let x be an operator of complex amplitue of a quantum oscillator with Hamiltonian x x, which satises the canonical commutation relations with x being an ajoint operator [x; x ] = xx x x = h1 (2.1) where 1 is the unit operator, an h > 0 is the Planc constant. Assume that in general case this oscillator is controlle by the complex amplitue u by means of a quantum-mechanical transmission line with wave resistance =2, where the operator of the wave y s t c travelling from the oscillator into the line is measure. In the simplest case of ieal conjugation between the line an the measuring apparatus, when there is no reection of the wave travelling from the oscillator, i.e. in case of the matche line, x (t) an y (t) are escribe by the pair of linear equations [16] x (t) =t + x(t) = u(t) + v (t) ; x (0) = x; (2.2) y (t) = x (t) x (t) =t = (x (t) u (t)) v (t) ; (2.3) where, generally speaing, is a complex number with xe real part, + =, an with arbitrary imaginary part epening on the choice of the representation, v t + s c the amplitue operator of the wave travelling out of line towars the oscillator, this operator is responsible for the commutator preservation. Uner natural for super-high an optical frequencies assumption of narrowness of the frequency ban which we eal with the commutators for v (t) in the representation of \rotating waves" have elta-function form [17]: [v (t) ; v (t 0 )] = 0; h v (t) ; v (t 0 ) i = h1 (t t 0 ) : (2.4) Integrating equation (2.2) an taing into account that v (t) oes not epen on x(t 0 ) when t > t 0, it is easy to verify that the commutator [x (t) ; x (t) ] is constant, moreover, x (t) commutes both with y (t 0 ) an y (t 0 ) when t > t 0, an the commutators for y (t) ; y (t 0 ) ; y (t 0 ) coincie with (2.4). The latter means that consiering von Neumann reuction which appears as a result of some quantum measurement of y (t) at previous instants of time t 0 < t oes not aect the future behaviour of x (t 1 ) ; y (t 1 ) ; t 1 > t, so that equations (2.2), (2.3) remain unchange. This fact together with the Marovianity hypothesis of the quantum process x (t) which hol for quantum thermal equilibrium states of the wave v(t) in case of narrow ban approximation [17] simplies to the large

4 extent optimal measurement an control problems for the simplest quantum ynamical system mentione above. Let us assume, that the initial state x is Gaussian with the mathematical expectation hxi = z an h(x z) (x z)i = 0; h(x z) (x z)i = h; v (t) is the quantum Gaussian white noise, which is escribe by the following correlations hv (t) v (t 0 )i = 0; hv (t) v (t 0 )i = h (t t 0 ) ; with = (exp (h=t ) 1) 1 for the equilibrium state with the temperature T, where > 0 is the Boltzmann constant. As an example, let us try to choose the optimal measurement of the controlle quantum oscillator (2.2) with transmission line (2.3), so that to minimize its energy hx (t) x (t)i at the nal instant of time t = by means of the control strategy the norm R 0 j u (t) j2 t of which shoul not be too great. If we want also to force the quantum amplitue x(t) to follow the classical process u(t), this problem can be characterize by the quality criterion hx () x ()i + 0 hu (t) u (t) +! (x (t) u (t)) (x (t) u (t))i t: (2.5) Here ;! 0 are parameters responsible for the measurement quality: when = = 0 (2.5) correspons to the problem of pure ltration, when! = 0; 6= 0, it correspons to the pure control problem. It will be shown below (see x5) that the optimal measurement minimizing criterion (2.5) is statistically equivalent to the measurement of the stochastic process z (t) = bx (t)+ x (t) escribe by Kalman{Bucy lter: ^x (t) =t + ^x (t) = u (t) + (t) (y (t) (^x (t) u (t))) : (2.6) Here ^x (0) = z; (t) = ( (t) ) = ( + ) ; (t) is the solution of the equation (t) =t = ( (t)) ( + (t)) = ( + ) ; (0) = ; x (t) =t + x (t) = (t) (v (t) x (t)) ; x (0) = 0; (2.7) where v (t) is the amplitue operator with commutators [v (t) ; v (t 0 )] = 0; h v (t) ; v (t 0 ) i = h(t t 0 ) which change the quantum process ^x (t) into the classical (commutative) iusion complex process, an with correlations of vacuum noise of the intensity = 0 if 0 an = if > 0: hv (t) v (t 0 )i = 0; hv (t) v (t 0 )i = h (t t 0 ) : (2.8) For instance, such measurement taes place by the heteroyning [7] where v (t) stans for a stanar wave. In this case the optimal control strategy u o (t) coincies with the classical one: u o (t) = (t) z (t), where (t) = ( (t)!) = ( +!), an (t) is a solution of the equation:

5 (t) =t = (! (t)) ( + (t)) = ( +!) ; () = ; (2.9) which together with (2.7) enes the minimum quantity of losses (2.5): h (0) + (t) + ( (t)!) 2 (t) = ( +!) t + (0) j z j 2 : 0 By setting = 0;! = 0, we obtain in particular the solution of the terminal control problem for an oscillator with thermal noise equal to zero. But in this case unlie the classical one the optimal measurement remains inirect an the equation (2.7) remains regular corresponing to the white noise in the channel of intensity j j h. Thus to consier the quantum measurement postulates is statistically equivalent to the aing of white noise into the channel of intensity j j h what exclues the singular case of pure measurement of the amplitue ^x. It is interesting to note that, in the stationary case = =, e.g. in the case of thermal equilibrium when > 0; T > 0 an = (exp fh=t g 1) 1, the optimal amplication coecient (t) equals to zero which means the possibility of optimal control of the quantum oscillator without measurement. It also hols when! =, the solution of equation (2.9) is stationary an optimal fee-bac coecient (t) equals to zero. But in the contrary case < 0; T < 0 which correspons to the active meium of the oscillator (laser) the optimal coecients (t) ; (t) are strictly negative an non-zero even for the stationary solution (t) = 0; (t) = = j j of equations (2.7), (2.9). 3 Quantum ynamical ltering Now let us give a rigorous setting of the quantum ynamical observation problem for the optimal control of a quantum-mechanical object when time is iscrete t 2 ft g =0; 1;:::. Let A be von Neumann algebras on a Hilbert space H, each is generate by one or a few ynamical variables (operators) x = (x i ) i2i in H. One can consier a quantummechanical object in the Heisenberg picture at the instant of time t +1 > t > 0 with x = x (t ), such that all algebras A are equivalent to the initial algebra A 0 = A, generate by the positions an momentums x = (q; p) at t = 0. Let B ; = 1; 2; ::: be von Neumann algebras of observables generate in H by output ynamical variables y = j2j, y j by means of which this object can be observe in a nonemolition way say, on the time intervals (t 1 ; t ]. As it has been shown above on the example of the matche transmission line, the output observables b 2 B in the matche channels shoul commute with all present an future operators a 1 2 A 1; 1 of the ynamical system, but not necessarily with the past ones a 0 2 A 0, 0 <. This commutativity conition together with the commutativity b 0 b 0 = b 0b 0 for all b 0 2 B, b 0 2 B = will be referre as the nonemolition conition. Let us enote P ; R the ual spaces to A ; B with respect to some stanar pairings < :; : >, say the subspaces of trace class operators 2 A ; 2 B which are ual to the simple algebras of all boune operators a 2 A ; b 2 B on the corresponing Hilbert spaces with respect to the bilinear trace-forms < ; a >= tr [ a ] ; < ; b >= tr [ b ] ;

6 an enote S the corresponing subspace ual to the von Neumann algebra B _ A generate by the commutating B an A. We shall use the operational terminology, briey summarize in the Appenix. Thus we shall call the positive normalize elements 2 P ; 2 R an 2 S, which are usually escribe by the statistical ensity operators, the statistical states of the quantum object at the instants of time t, the states of the channel on the interval (t 1 ; t ], an the joint state of the object an channel at the moment t respectively, or simply the states on A ; B an B _ A B A. Now we aopt the hypothesis of Marovianity of the Heisenberg ynamics, restricte to the escribe quantum object an output channel in H, with respect to a given state of the whole system!. Let all the inuce states =!j (B _ A ) ; = 1; 2 : : : an their restrictions ; on B ; A be ene by the initial state 0 = on A 0 = A an by a family fm g 1;2::: of statistical morphisms 1 7! = 1 M. These transition maps P 1! S can be escribe as the pre-ual to positive normalize superoperators M : B A! A 1 having for the simple algebras the form M c = tr B [ c ] ; 8c 2 B A : Here ; = 1; 2; : : : are states on some algebras B, for which the simple algebras B A are isomorphic to the von Neumann tensor proucts A 1 B, an tr B is the partial trace on B such that M [a 1 b ] =< ; b > a 1 for all a 1 2 A 1 ; b 2 B. This assumption correspons to the requirement that the channel shoul be matche with the object an implies the semigroup ynamics [20] 1 7! = tr B f M g of the quantum-mechanical object with iscrete time. Furthermore, we shall suppose that every morphism M may epen on the results = f 0g 0 < of previous measurement ata 0 2 ; 0 <, say via epenence of some controlle parameters u 2 U of the sequence f 0g 0 < ue to a feebac 7! u. The nonemolition measurements uring the time intervals (t 1 ; t ] are escribe by positive operator-value measures b () 2 B ; = 1; 2; : : : on the ata space 3 with a given Borel structure of the measurable subsets z such that b () = 1 is the ientity operator of B. We shall assume that every -measurement b () also may epen on all preceing measurement results 1 ; : : : ; 1, an not only ue to a epenence on u 2 U an the feebac, but irectly, being aaptive in time. The functions 7! (M ; b ; ) are suppose to be wealy measurable in the sense that for all 1 2 P 1 an a 2 A an all Borel subsets the complex functions 7!< 1 M ; b ; a > are Borel functions on = Q 0 < 0; where = ; 0 = U. We shall call every sequence n b ; o of such \conitional", or aaptive measurements the measurement strategy. =1;2;::: Let us enote B ; the conitional transition measures P 1! P, that is the operational-value conitional measures on, ene as the preual to superoperator values B ; : A! A 1 by the formula a 7! B ; a = M h b ; a i ; (3.1)

7 an enote +1 the P - value measures on +1 obtaine for = 1; 2; : : : by the recurrence = 1 B ; (3.2) with the initial conition 0 ( 1 ) = (u 0 ; 1 ) if 1 = U. Lemma 1 All the measures +1 are positive in the sense that < +1 ; a +1 > 0 for all A - value positive measurable functions a +1 0 an are normalize, < +1 ; 1 >= 1, where 1 is the ientity operator of A. Proof As the superoperator-value measures B ; are positive an normalize in the sense that R B ; h a ; i 0 for all a +1 0 an B ; 1 = 1, the lemma can be easily prove by inuction, using the positivity an normalization of 0. Thus the measure +1, obtaine by the recurrence (3.2), escribes the total statistical state on the algebra A an on the expaning space +1 = Let us ene a posteriori state of the object at time t as P {value Raon-Nioym erivative 1 = 1 = < 1 ; 1 > (3.3) which exists in the wea sense ue to absolute continuity of 1 with respect to < 1 ; 1 >. Theorem 2 The a posteriori states +1 ; = 1; 2; : : : can be obtaine by the nonlinear recurrency ; = 1 T ; ; 1 ; 0 1 = ; where T +1 ; 1 is the (P 1! P )-value Raon-Nioym erivative T ; ; 1 = B ; = < 1 B ; ; 1 > : Proof The nonlinear transition operations T are ene in the wea sense almost everywhere by the Raon-Nioym erivatives < 1 T +1 ; 1 ; a >=< 1 B ; ; a > = < 1 B ; ; 1 > : The proof of the theorem follows immeiately by inuction ue to the Bayes formula < ; 1 > = < 1 ; 1 >=< 1 B ; ; 1 >; from the enitions (3.2), (3.3) Note that the equation (3.4), escribing the conitional Marovian evolution of a posteriori state of a quantum-mechanical object, can be regare as a quantum generalization of Stratonovich nonlinear lter equation with iscrete time. A semi-quantum case when a partially observe object is escribe by a classical Marovian process fx g =0;1;::: an the channel is non-classical, was consiere in [12].

8 4 Quantum ynamical programming Let us consier the problem of optimization of the observation strategy n b ; o on the xe iscrete time interval [0; K]. The optimal strategy fb o g 2[0;K) is ene as a strategy, which minimizes the average cost =< K ; a K > + KX =1 < 1 ; c 1 >; (4.1) given by a self-ajoint semi-boune operator a K 2 A K of nal losses, an by similar operator-value functions +1 7! c +1 2 A, = 0; : : : ; K 1. (In the case of unboune a K an c +1 only their spectral measures shoul belong to A K an A.) Let us remar that the cost (4.1) oes not epen on the last measurement b K K ; which can be chosen arbitrarily, an K = K K+1. As it follows from enitions (3.1), (3.2) the P 0 =1 in (4.1) for any = 1; : : : ; K is inepenent of the measures b 1 1 ; for 1. Hence in orer to n the optimal {measurement b ; from some < K it is enough to vary the future average observation cost functional =< K ; a K > + KX 0 =+1 < ; c > : (4.2) Lemma 3 The explicit epenence of on b ; is ane = < ; ; b ; >; (4.3) where ; = 1 A ;. Here A +1 is a (P 1! R ) {value function on +1 which is ene as preual to the superoperators b 7! A +1 b = M h b a +1 i ; 8b 2 B ; (4.4) where a +1 is an operator-value function on satisfying the linear inverse-time recurrency a 1 = B ; a ; + c 1 ; (4.5) = 1; :::; K with the bounary conition K K+1 = a K. Proof First let us prove that the future losses (4.2) can be represente as = < ; a ; >; where a +1 2 A is the solution to the equation (4.5). It is obviously vali for = K, an if it is true for a < K, then substituting (3.2) into this representation of, we obtain < +1 ; a +1 > + < 1 ; c 1 > = < 1 ; B ; a ; > +c 1 :

9 So this is also vali for 1 with a 1 given in (4.5), an by using the inverse-time inuction, it is vali for any 2 [0; K). Now we can obtain (4.3) by < ; a ; >=< 1 B ; ; a ; > =< 1 M ; b ; a ; > =< 1 A ; b ; >=< ; ; b ; >; where we use the enitions (3.1) an (4.4) for the operations B an A Theorem 4 If the strategy n b o ; o is optimal for the cost functional (4.1), it 2[1;K) satises the following system of equations ; b o ; = 0; (4.6) 2 [1; K), where = ; b o ; : These equations together with the system of inequalities ; ; 2 [1; K) (4.7) give the necessary an sucient conitions of the optimality for quantum measurement strategy b o, = 1; :::K 1 corresponing to the minimal values of the future average costs (4.2). Proof o = < ; 1 > (4.8) As the variables b ; ; = 1; 2; : : : ; K of the functional (4.2) are inepen- ent, the optimal measure b o ; minimizes the ane functional separately for every xe family n b 1 1 ; o. The necessary an sucient conitions (4.6), (4.7) of 1> optimality for b o ;, minimizing the ane functional (4.3) with a xe, follow immeiately by the linear programming metho, as it was note in the single-stage theory of optimal quantum measurements [2, 4{7] Note that the minimal value o of the total average cost (4.1) is given by the solution a o = a o 0 of the recurrency (4.5) with B = B o at = 0 as o =< ; a o >. Let us note that with the help of the a posteriori states, one can write conitions (4.6), (4.7) in the following form ; b o ; = 0; (4.9) +1 ; 2 [1; K); (4.10)

10 We examine a Marovian one-imensional quantum ynamical system, escribe at iscrete instants t = by the algebras A an B, which are generate by the non- where +1 = 1 A +1. Accoring to the Bellman ynamical programming metho [15] the verication of the optimality conition formulate above can be carrie out sequentially in inverse time = K 1; : : : ; 1 applying the recurrence (4.5) for the superoperator A +1 after solving the ltering recurrent equation (3.4). The optimal control of Marovian partially observe quantum-mechanical object can be reuce to the optimal measurement problem investigate above as follows. Let M (u 1 ) : P 1! R P be the quantum statistical morphisms (transitions) controlle by some parameters u 2 U; = 0; : : : ; K 1. A control strategy f g <K is given by a choice of the feebac, ene by a measurable epenence of each u on all measurement ata 0 2 Y; 0, an also on the preceing controls u 0; 0 <. The optimal control for a xe measurement strategy is suppose to minimize the average cost ene by a nal operator a K an operator-value cost functions c (u ) ; = 0; : : : ; K 1. Denoting 1 = u 0, = (u 0 ; 1 ; u 1 ; : : : ; 1 ; u 1 ), = (; u) ; the average cost functional even with ranom control strategies can be represente in the form (4.1), given by the quantum measurement strategy n b ; o on = Y U of the form b o ; u = b o ; o ; ; u (4.11) an c 0 ( 1 ) = c 0 (u 0 ), c +1 = c (u ). The quantum optimal control problem can be formulate then as one of searching for the optimal Y U {measurements b o ; ; 2 (1; ) ; an an optimal initial control u o corresponing to the minimal value o = inf u < 1 (u) ; 1 > + < 0 ; c 0 (u) > of average cost (4.1). In general, the optimal measurement strategy may not be in the prouct form (4.11), but if there exists a non-ranomize strategy u o = o ; ; 2 [1; K) for some Y -measurements b o ; for which the Y U - measurements are optimal, where (; ) is the Dirac - measure, then the ata spaces Y may be calle the sucient spaces. The optimal measurements b o ; on sucient ata spaces Y satisfy obviously the equations ; ; o ; b o ; = 0; 2 [1; K); where = ; ; o ; b o ; ; which together with the inequalities (4.10) are necessary an sucient for the nonranomize control strategy f o g. 5 Quantum ltering in Boson linear Marovian system in a Gaussian state

11 selfajoint operators x 6= x an y 6= y respectively, satisfying the canonical commutation relations. Let us suppose that they act in the same Hilbert space H, where they satisfy the linear quantum stochastic equations x = x 1 + u 1 + v (5.1) y = x 1 + u 1 + w : (5.2) Here ; ; ; are some, in general complex parameters, the controls u can also accept complex values, x 0 = x is the initial operator in H, generating the algebra A, an v ; w are some operators in H; generating the algebras B. To obtain the Marov ynamics, we nee to assume the inepenence of x an all the pairs (v ; w ) such that the algebras A an B, corresponing to ierent instants of time t, commutate, an the joint state! is the prouct of the states on A an all B ; = 1; 2 : : :. We shall ene the canonical commutation relations for the generating operators x; v ; w with their ajoints as following: [x; x ] = h1 [v ; v ] = 1 j j 2 h1; [w ; w ] = " j j 2 h1; [w ; v ] = h1; (5.3) where h > 0 an 1 is the ientity in H ( other, unwritten commutators, incluing all those corresponing to ierent instants of time to be equal to zero.) Here the choice of the commutator [w ; v ], responsible for the commutativity [y ; x ] = 0 is essential, the other nonzero commutators are chosen so that the commutators [x ; x ] = h1; [y ; y ] = "h1 shoul be constant. The escribe system we shall call the iscrete linear Marovian quantum open oscillator. Let us escribe the states 2 P by the Glauber [21] istributions p (), 2 C, normalize on the complex plane C with respect to the Lebesgue measure = ReIm=h. In the representation escribe in the Appenix, the Marovian morphisms P 1! P, corresponing to the linear equations (5.1), (5.2), transform the istributions p 1 () into the two-imensional istributions g (; ) = q 1 u; 1 u p ; (5.4) where q (; ) are some other (not necessarily Glauber) istributions on C 2, which escribe the inepenent states on algebras B. When " = 0, the operators y ; y are simultaneously measurable, an the a posteriori states on A uner the xe spectral values y 0 = 0 an u 0, 0 < are ene recurrently by the a posteriori Glauber istribution p j 1 accoring the Bayes formula p j 1 = g ; j =r j : Here g ; j are the istributions obtaine by substitution of p 1 j into (5.4) instea of p 1 (), an r j = g ; j

12 are the probability istributions, escribing the complex values, which arise as the results of the irect measurements of y uner the xe = (u 0 ; 1 ; u 1 ; : : : ; 1 ; u 1 ). When " 6= 0, only inirect measurement of y are possible which are escribe, for instance, by the B {value measures b () = #m ( y ) #; (5.5) represente by some istributions m () on C as it is escribe in the Appenix (A.3). In this case in orer to calculate a posteriori Glauber istribution one shoul change q (; ) in formula (5.4) for the istribution q 1 (; ) = m 1 q ; 1 1 : (5.6) Theorem 5 Let the initial state of the quantum oscillator be escribe by the Glauber istribution of Gaussian type p () = 1 exp n j z j 2 =h o ; (5.7) the istributions q (; ), escribing the transitions (5.4), be also Gaussian: q (; ) = exp f ( j j2 +2Re + j j 2 ) =h ( j j 2 )g j j 2 ; (5.8) an the measures b are escribe as in (5.5), by the Gaussian istributions m () = 1 exp n j j 2 =h o : (5.9) Then a posteriori states (3.3) at each instant = 1; 2; : : : ; are given by the conitional Glauber istributions of Gaussian form p j +1 = 1 exp n j z j 2 =h o ; (5.10) where z ; are ene by the recurrent equations of the complex Kalman lter: z = z 1 + u 1 + ( z 1 u 1 ) ; z 0 = z; (5.11) where =j j j j 2 ; 0 = ; (5.12) = ( 1 ) = =j j ; 1 = + : Proof Due to the chosen representation, the proof is similar to the erivation of the classical one-imensional Kalman lter for the complex Gaussian process x given by (5.1) an y 1 = y + w, where w are inepenent Gaussian variables with zero mean values an the covariances ". (For this proof see, for instance, [22].) One shoul only tae into account that istributions (5.6) are also Gaussian of the type (5.8) with the

13 [x; x ] = h1; [v (t) ; v (t) ] = ( + ) ht1; parameter 1 = + instea of. Substituting q (; ) in (5.4) by q 1 (; ) an p by the conitional istribution p 1 j of type (5.10), we obtain 1 () g 1 ; j = p j 1 r 1 j ; where p j 1 is the Gaussian istribution (5.10) with the parameters (5.11), (5.12), an r 1 j = 1 exp n j z 1 j 2 =h o : (5.13) Thus the quantum Gaussian ltering is controlle by the classical Kalman lter for the complex amplitue in the Glauber representation Note, that in istinction from the classical case, the covariance matrix of istributions (5.8), (5.9) shoul not only be non{negative enite but shoul also satisfy the Heisenberg uncertainty principle! j! j2 1 j j 2 ; "; (5.14) " as it follows from inequality (A.5). In particular it exclues the case = 0 of the irect observation of y when " > 0. As shown in the next paragraph, a posteriori mathematical expectations z with = max (0; ") appear to be the optimal estimates u o = z of the operators x with respect to the square quality criterion c (u ) =:j x u j 2 : with the minimal mean square error h. In the commutative case [x ; x ] = 0 this optimality was prove in [11]. Note, that instea of calculating z by means of the recurrent formula (5.11) using the results ( 1 ; :::; ) of the inirect measurement (5.5) one may regar z itself as a results of the measurement escribe by the B {value measure: where an b ; z = #n (z ^x ) #z; (5.15) n (z) = 1 j j 2 m (z= j j) ; ^x = z 1 + u 1 + (y z 1 ) (5.16) is an operator, epening on the values z 1 ; u ; an inepenent of the preceing measurement an control results. It is interesting to consier the time continuous limit, when the quantum oscillator (5.1), (5.2) is escribe by the quantum stochastic ierential equations x (t) + x (t) t = u (t) t + v (t) ; (5.17) y (t) = x (t) t + u (t) t + w (t) ; (5.18) i.e. by equations (5.1), (5.2) with x (t ) = x ; y (t ) = y ; ' 1 ; ' ; ' ; " ' "; where (t ) = t t 1 = tens to zero. In aition to that the commutation relations (5.3) change in the following way

14 [w (t) ; w (t) ] = "ht1; [w (t) ; (t) ] = ht1; an the other commutators incluing those corresponing to the ierent instants of time are equal to zero. By passing to the limit as! 0 when ' ; ' ; ', it is easy to obtain uner the assumptions of the Theorem 3 that a posteriori state (t; t ) is escribe by the Glauber istribution p (t; j t ) of Gaussian type (5.10) with the parameters z (t) ; (t) which correspon to the Kalman{Busy lter z (t) + z (t) t = u (t) t + (t) ( (t) (z (t) u (t)) t) : (5.19) Here (t) = ( (t) ) = 1 ; 1 = v + ; z (0) = z; (0) = ; (t) =t + ( + ) (t) = j (t) j 2 1 ; an (t) are the results of the corresponing inirect measurement of y (t) which are realize by the measurement of the sum y (t) + w (t), where w (t) is an inepenent quantum white noise, ene by the coecients "; : [w (t) ; w (t) ] = "ht1; hw (t) w (t)i = ht: As shown at the en of the next paragraph, such \continuous" measurement appears to be also optimal in the Gaussian case when = max (0; "). 6 Optimal measurement an control in quantum open linear system In the following theorem it is not require that the istributions p 0 ; q an m shoul be Gaussian an it is assume only that they shoul have the zero mathematical expectations, an the covariances shoul coincie with the covariances ; ; ; ; of the istributions (5.7) { (5.9) respectively, not necessary being of the form (5.11). Theorem 6 Let the operator of nal losses be quaratic: a K = x Kx K, where 0, an c (u ) =!x x #u x #u x + # 1 j u j 2 ;! 0; # 1 > 0 (6.1) be quaratic loss operators for all 2 [0; K). Suppose u = z ; 2 [0; K) is a linear control strategy, where z are the linear estimates (5.11) base on the results of the inirect measurement (5.5), an = +1 # = ; (6.2) with =j j # 1 an satisfying the following equation =j j ! j j 2 ; K = : (6.3) Then the operators of future losses (4.5) are also quaratic: a +1 = 1 + j u + z j 2 + x x

15 the operator ^x, ene in (5.16), is linear with respect to y, an r o j is the istribution (5.14) with the parameters o = + o, where o = max (0; "). where an + (z x ) (z x ) 2Re (u + z ) (z x ) ; (6.4) KX = h i + i + 2Rexi + 1 j i j 2 ; i=+1 =j j 2 + j +1 j 2 +1; = 0; (6.5) = +1 #: Proof In the representation a K =: K (x K ) :; c (u ) =: (x ; u ) :; a +1 =: +1 x ; +1 : the recurrent equation (4.6) has the form ; +1 = +1 ; +1 ; ; u q 1 ( u ; u ) + ( ; u ) where u = +1 z; z = z + u + +1 ( z ) : (6.6) Let us assume that the function () has the quaratic form (6.4); in particular, it has this form at = K, namely K () = j j 2. Inserting the latter into (6.6) an integrating, we obtain, that the function 1 is of the same form with 1 = # 1 + j j 2 an 1 ; 1 given by (6.3) an (6.5), an 1 = + h + + 2Re + 1 j j 2 : Summing K i= ( i 1 i ) an taing into account that K = 0 an K = 0, we obtain (6.4) also for 1 Lemma 7 Let us assume that starting from the instant + 1, the controls u are chosen to be linear u 1 = 1z 1 with the coecients (6.2), where z 1; 1 > epen linearly on the results of the subsequent inirect measurement +1 ; : : : ; K 1 by virtue of the formula (5.11) with the initial conition z = z. Let also the inirect measurements be escribe by the Gaussian istributions (5.9) up to the. Then the operator ;, ene in (4.9), has the following normal form ; = + where + : j u + ^x j 2 + j +1 j 2 +1 j z ^x j 2 r o y j :; (6.7) =: j ^x j 2 + (h ( + ) + ) 1 r o y j :;

16 Proof Inee, the operator +1 similar to the ensity operator 2 R is ene by the istribution r ; +1 = ; +1 q ( 1 u 1 ; 1 u 1 ) p 1 1 j 1 : (6.8) It is a symbol of the contrary orer (see the Appenix), which is normal when " < 0 an antinormal when " > 0. In the former case, inserting the operator symbol (6.4) into (6.9) an integrating with respect to the Gaussian type of the istribution p 1 1 j, we obtain (6.7), where r o j coincies with the istribution r j of the Gaussian type (5.13) with the parameter v 1 =. In the latter case " > 0, the normal symbol of the operator +1 is obtaine from (6.9) by means of the convolution of type (A.2) with the istribution (5.9) with = ", an in the result of the parameter increases for ". In this case r o j is also the normal symbol of the conitional ensity operator on B Theorem 8 Let the quantum oscillator (5.1), (5.2) be escribe by the Gaussian initial an transitional istributions of the Gaussian form (5.7), (5.8), an the quality criterion (4.2) be ene by the quaratic nal an transitional operators K = x Kx K an c (u ) of form (6.1) respectively. Then the optimal strategy is linear: u = z, where is ene by (6.2), an z are optimal linear estimates (5.11) base on the results f i g i of the coherent measurements (5.5) which are escribe by the istributions (5.9) with the minimal value of the parameter = o. Proof We shoul verify the necessary an sucient optimality conitions (4.10), (4.11) for the operator (6.7) an the mentione above measurement at each instant. As ; 0, an the ensity operator : r y j : is non{negative enite, the ierences +1 are non-negative enite operators as well. It remains to verify the equations (4.13) for the optimal strategy o ; = z of the coherent measurements (5.5) or, what is the same, of the measurements (5.15) with the Gaussian istributions n o (z), corresponing to the case = o. Inserting u = z into (6.7) an taing into account (6.5), we obtain ; ; o ; = :j z ^x j 2 r y j : : Thus, equations (4.13) with " > 0 can be written in the form an the ajoint ones can be written for " < 0 also as (z ^x ) #n o (z ^x ) # = 0; (6.9) #n o (z ^x ) # (z ^x ) = 0: (6.10) The operators #n o (z ^x ) # escribe by the Gaussian istributions n o (z), which realize the lower boun of the Heisenberg inequality, are well nown as proportional to

17 coherent projectors [8]. The operators ^x when " > 0, are proportional to the annihilation operators, an when " < 0, they are proportional to the creation operators, for which the coherent projectors are the right an the left eigen-projectors respectively. Hence, the equations (6.9) is satise in the case " > 0, an the equation (6.10) is satise if " < 0. Note that, in the antinormal case when the coherent projectors are escribe by the Dirac istributions (z) on C, these equations are written as the ientities (z ^x ) # (z ^x ) # = # (z ^x ) (z ^x ) # = 0; " > 0; # (z ^x ) # (z ^x ) = # (z ^x ) (z ^x ) # = 0; " < 0: The minimal losses, corresponing to the optimal quantum strategy are ene by the following formula o = 0 j z j 2 +h 0 + KX + 1 #! 1 ; (6.11) =1 where ; are ene by (6.2), (6.3), an ; by (5.11), (5.12) with = max (0; ") Let us also obtain the solution to the corresponing time-continuous optimal control problem for the quantum open system, escribe by the linear stochastic ierential equations (5.17), (5.18) an the quaratic integral criterion :j x () j 2 : +! :j x (t) j 2 : 2Re#u (t) x (t) + # 1 j u (t) j 2 t: 0 This criterion is obtaine by setting! '!; # ' #; # 1 ' # 1 in the conitions of the Theorem 6.1; an passing to the limit as! 0. So, the solution to the quantum optimal control problem for the time continuous quantum open system (5.17), (5.18) with quantum white noises v; w is ene as the limit of the solution to the iscrete problem at! 0. The optimal strategy, obtaine in this limit, is obviously linear with respect to the optimal estimate z (t) of x (t) as in the classical case [20]: u (t) = (t) z (t), where (t) = (t) # =# 1 ; () = ; an (t) satises the equation: (t) =t + ( + ) (t) =! j (t) j 2 # 1 : The optimal estimate z (t) is obtaine by coherent measurements, corresponing to the case = max (0; ") in the time-continuous Kalman lter, an the minimal mean square losses are ene by the integral o = 0 j z j 2 +h 0 + (t) + (t) (t) # (t) t : 0 In particular, when = = " = + > 0; = = ; # 1 = # + ; # =!, we obtain the solution to the optimal control problem for the quantum open oscillator matche with the transmission line (2.3) of the wave resistance =2 which was consiere as the motivating example in x2. In this case the equations (5.17), (5.18) are reuce to (2.2), (2.3), where the generalize erivatives v (t) = v (t) =t; y (t) = y (t) =t represent the irect an reverse waves on the input of the open oscillator. Acnowlegements. It is a great pleasure to than Professor R. S. Ingaren for his interest in this wor. I wish also to than Mr P. Staszewsi for the help uring the preparation of the English version of this paper.

18 A APPENDIX Let A, B be von{neumann algebras, i.e. selfajoint wealy close subalgebras of operators in a complex Hilbert space H incluing the ientity operator 1, an P, R be preual spaces of ultra wealy continuous functionals on A an B, respectively. The elements 2 P an 2 R are calle states on A an B respectively if < ; a > 0, < ; b > 0 8a 0; b 0 (a; b 0 means the non-negative eniteness of the operators a 2 A an b 2 B), an if < ; 1 > = 1, < ; 1 > = 1. Linear operators transforming operators b 2 B into operators a 2 A are calle superoperators, an the preual linear maps P! R are calle operations. The typical example of a superoperator gives a representation b 7! u bu, where u is a unitary operator. An operation M : 7! M 2 R is calle the (statistical) morphism if the ual superoperator b 7! Mb 2 A is positive 1 Mb 0; 8b 0 an M1 = 1 (it is convenient to enote the morphisms an ual superoperators by the same symbol with the right an the left action respectively: < M; b >=< ; Mb >.) A B{value measure b () on some Borel space 3 is calle {measurement, if b () 0 for any Borel an R b () = 1 in the same sense. If M : P! R is a morphism escribing a quantum channel, {the state on its input an b () {the measurement on its output, then the probability istribution on is calculate by any of the formulas P () =< M; b () >=< ; Mb () > : (A.1) Let, for instance, the subalgebras A an B be generate by the operators x an y respectively with the canonical commutation relations [x; y] = 0; [x; x ] = h1; [y; x ] = h1; [y; y ] = "h1; where 2 C; " 2 R an h > 0 is a constant. It may be assume that y = x + v hols, where v is an operator in H commuting with x an x, but not commuting with the ajoint one: [v; v ] = (" j j 2 ) h1, an the algebra generate by the pair x; y can be represente in the form of the tensor prouct A B, where B is the von{neumann algebra generate by the operator v. We shall write the operators, generate by the operators x an v in the form #' (x; v) #, where ' (; ) are complex{value functions of ; 2 C, calle symbols, an the notation # # inicates such orer of action for the operators between them, that rst act the operators x; v, an then their conjugate. For instance, # j x j 2 # = x x. In a suciently wie class of symbols any operator from A B can be represente in such a form, an this representation is single-value an injective. In the case y = x + v the operators a 2 A are escribe by the symbols ' (; ) = () an the operators b 2 B by the symbols ' (; ) = ( )as in the classical commutative case h = 0. The states in this quasi-classical representation are escribe by istributions q (; ), generalizing the probability ensities an representing the ensity operators as the symbols of the contrary orer, which are ual to the orer for the symbols ' (; ). Due to h > 0, x = p h is the stanar creation operator, an x= p h is the stanar annihilation operator, so 1 For a physical realization of the statistical morphisms by conitional expectations of the representations a stronger conition of complete positivity [Mb i ] i;=1:::n 0; 8n, where [b i ] i;=1:::n 0 is any non-negative enite operator-matrix with the elements b i 2 B, shoul be impose on the morphisms.

19 In accorance with formula (A.1) such a measurement is escribe by the B {value measure b () = #n ( y) #; (A.4) that the representation a = # (x) # of operators a 2 A is normal [19], escribe by the holomorphic symbols () with respect to both ;. The corresponing symbols p () of the states on A are escribe by the Glauber istributions p (), which are ene as the linear functionals < ; a >= p () () ( = ReIm=h) ; escribing the symbols of the ensity operator, appropriate to the antinormal orer. The normal orer is enote by the parentheses : :, so we have # (x) # =: (x) : when [x; x ] 0. Note, that the antinormal symbol p () of the ensity operator an the normal symbol p o () are connecte by the convolution [21] p o () = exp n j 1 j 2 =h o p 1 1 : (A.2) The appropriate representation of the algebra B ; an hence A B, is normal only if " >j j 2, when [v; v ] > 0. If m () is a istribution which enes a state on B an there is no statistical epenence, a state on A B is escribe by the prouct p () m () an a state on the sub-algebra B by the convolution r () = m ( ) p () : (A.3) A superoperator B! A, which is ual to a morphism (A.3), is escribe by the symbol transformation () = () m ( ) : For the normality of the appropriate representation b = # (y) # of the operators b 2 B with the istribution (A.3) being Glauber, it is sucient, that " > 0. When " < 0, the istribution r () is the normal symbol of the appropriate ensity operator = #r (y) #. Let us consier the complex measurements, escribe by the measurements of the sum y + w = z, where w is an operator in H, which commutes with y an y, but oes not commute with the ajoint one w : [w; w ] = " j j 2 h1; so that [z; z ] = 0 (it is assume that the space H is chosen suciently wie, otherwise such an operator in H may not exist.) If () is a istribution escribing a state on the algebra B 1 generate by the operator w, then the probability istribution of the results of such a measurement on the output of the channel is escribe by the norme with respect to the Lebesgue measure ensity s () = n ( ) m ( ) p () :

20 an the istribution n () satises the conition j j 2 n () max n " j j 2 h; 0 o (A.5) in accorance with the inequality w w f[w ; w] ; 0g. When " > 0 an representation (A.4) is normal, inequality (A.5) prohibits, in particular, istributions of Dirac {form.

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