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Seth Lloyd† and Lorenza Viola d’Arbeloff Laboratory for Information Systems and Technology, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 We investigate the control resources needed to effect arbitrary quantum dynamics. We show that the ability to perform measurements on a quantum system, combined with the ability to feed back the measurement results via coherent control, allows one to control the system to follow any desired open-system evolution. Such universal control can be achieved, in principle, through the repeated application of only two coherent control operations and a simple “Yes-No” measurement.

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Ever since the discovery of quantum mechanics more stroboscopic, matching up to the desired one at discrete than a century ago, the problem of controlling quantum intervals, and approximate. But the error of approxi- systems has been an important experimental issue [1]. A mation can be made as small as desired by making ∆t variety of techniques are available for controlling quan- sufficiently small. Since typically only a small number tum systems, and a detailed theory of quantum control of Hamiltonians (two, e.g.) are required to generate all has been developed [2,3]. Methods of geometric and co- possible Hamiltonians via commutation, arbitrary coher- herent control are particularly powerful for controlling ent transformations can be built up from a small set of the coherent dynamics of quantum systems [3,4]. In simple building blocks. This coherent control technique coherent control, a series of semiclassical potentials de- is open-loop because no information about the state of scribed by Hamiltonians Hi are applied to the system. the system is required to enforce the desired dynamics. A basic result in coherent{ control} is that, for a finite Now consider open-system dynamics. The simplest dimensional quantum system, only a small set of Hamil- way to describe an open quantum system is to embed it in tonians is required to enact any desired unitary transfor- a closed quantum system. Let the quantum system and mation on the system [2,3]. In general, only two different environment be described by a joint density matrix ρSE. Hamiltonians will suffice to achieve such universal control The time evolution of ρSE is given by a unitary trans- [5]. Thus, arbitrarily complicated time evolutions can be formation ρSE UρSEU †. Before the transformation built up out of simple building blocks. the system on its7→ own is described by a reduced density In this Letter we investigate the corresponding ques- matrix ρS =trEρSE. Afterward, its state is trEUρSEU †. tion for open-system, incoherent dynamics. Can an arbi- It is then immediately clear that if one has the ability to trarily complicated open-system dynamics be built up by enact any coherent transformation on the system and en- applying coherent control together with only a few basic vironment taken together, then one can enact any desired open-system operations? Unlike the case of Hamiltonian open-system transformation of the system on its own [6]. dynamics, where complete control can be attained purely In general, however, one does not have control over in terms of open-loop manipulations as implied above, the system’s environment, but only over the system it- this turns out to be possible only if closed-loop control self. If the system and the environment are initially un- (i.e., feedback) is allowed. We show that only one sim- correlated, so that ρSE(0) = ρS ρE, the time evolution ple coupling to a measurement apparatus, combined with of S can be described by a completely-positive,⊗ trace- the ability to coherently manipulate the system and feed preserving map ρS AkρSA† ,where A† Ak = 11 back the measurement results by the repeated applica- 7→ Pk k Pk k and the so-called Kraus operators Ak can be derived tion of a few basic control operations, allows one to enact from the embedding of the previous paragraph [7]. The arbitrary open-system dynamics. most general infinitesimal limit of such a time evolution Let us first review the basic result of coherent is given by the Lindblad equation [8]: control theory mentioned above. Consider an n- dimensional quantum system, S, and suppose that one dρ S = i [H, ρ ] can apply potentials corresponding to a set of Hamil- dt − S tonians Hi , so that one can apply unitary op- 1 {± } iHin tn Hi1 t1 a F †F ρ + ρ F †F 2F ρ F † , (1) erators of the form e± ...e± .Notethat X ij i j S S i j j S i iH ∆t iH ∆t iH ∆t iH ∆t [H ,H ]∆t2 3 − 2 −
e 2 e 1 e− 2 e− 1 = e 1 2 + O(∆t ), i,j where [H1,H2] is the commutator of H1 and H2.Re- peating this procedure shows that applying Hamiltonians where H = H† is the effective system Hamiltonian (the from the set allows one to enact any coherent evolution natural Hamiltonian possibly renormalized by a Lamb- iAt of the form e− ,whereA belongs to the algebra gener- shift term), F is a basis for the space of bounded { i} ated from the set by commutation. The time evolution is traceless operators on S,andA = A† = a is a positive { ij }

1 semi-definite matrix. After diagonalizing A,Eq.(1)can by letting the system to be measured interact with an be cast in the equivalent canonical form auxiliary environment (or ancilla) and then performing a conventional von Neumann measurement on the latter dρS [12]. In terms of Kraus operators, a generalized mea- = i [H, ρS] dt − surement corresponds to a set of operators Akm such 1 L† L ρ + ρ L† L 2L ρ L† , (2) that km Akm† Akm = 11. Suppose that the state of S − 2 X k k S S k k − k S k
P k is ρS as above. The measurement then gives the result k with probability pk =tr m AkmρSAkm† , leaving S in for a set of bounded, traceless Lindblad operators Lk P { } the state ρk = m AkmρSAkm† /pk. The information that [8]. The Lindblad equation is a Markovian master equa- can be gatheredP about a quantum system is maximized tion that defines a quantum dynamical semigroup: in- for so-called “pure” measurements [12,13], in which case tegrating the Lindblad equation over time t defines an a single value value of m occurs. We will restrict to pure open-system transformation Λt such that ΛtΛs =Λt+s measurements henceforth. for 0. t, s A particularly interesting class of pure measurements Suppose≥ that, in analogy with the infinitesimal open- occurs when the Kraus operators are positive (hence Her- loop prescription of the closed-system case, one attempts mitian). Such measurements, which can be realized by to build up arbitrary open-system evolutions by applying coupling the system to a suitable “pointer variable” of coherent operations associated with Hamiltonians Hi the measuring apparatus, are the “least disturbing” mea- together with some set of open-system infinitesimal{ op-} surements that can be effected on S [13]. As such, they erations described by Lindblad operators . Clearly, Lk supply a natural analogue of von Neumann’s model of the implementation of the coherent part of{ the} dynam- projective measurements. Suppose that one can perform ics, corresponding to , poses no problem. However, one H such a generalized measurement corresponding to a set of quickly arrives at an impasse while trying to enact the positive operators B , and feed back the results by apply- open-system operation corresponding to an arbitrary A, k ing a coherent transformation U when the k-th outcome as the infinitesimal transformations Λ are not invert- k ∆t is found. The net effect on the system is to apply the ible in general. This has a drastic effect in determining transformation the set of matrices A that can be reached by composi- tion of infinitesimal open-loop transformations. If we can ρ U B ρ B†U † . (3) apply infinitesimal operations corresponding to two dif- S 7→ X k k S k k k ferent Lindblad equations specified by A and A0,thenwe can generate an infinitesimal operation corresponding to That is, feedback gives rise to an open-system opera- A + A0. But it is not in general possible to build up an ar- tion corresponding to Kraus operators Ak = UkBk.But, bitrary positive semi-definite matrix by adding together from the polar decomposition theorem, any operator can a finite set of positive semi-definite matrices. Indeed, it be written A = U A ,where A = A† A is pos- has been shown recently that even in the case of a sim- k k| k| | k| q k k ple two-state quantum system a continuous set of dis- itive and Uk is defined to be 11onthekernel of Ak 1 K tinct non-unitary open-loop transformations is required and Ak Ak − on the orthogonal subspace ⊥ [14]. As | | K to generate an arbitrary Markovian dynamics [9]. a consequence, the ability to make a generalized mea- Turn now to closed-loop control. In closed-loop con- surement corresponding to an arbitrary positive opera- trol one performs measurements on the system and feeds tor ( Ak ) together with the ability to apply an arbitrary | | back the results of these measurements by applying op- coherent unitary transformation conditioned on the re- erations that are a function of the measurement results. sult of that measurement (Uk), allows one to enact an Although both the system and the measurement appa- arbitrary open-system operation (Ak = Uk Ak ). | | ratus are quantum-mechanical in principle, the feedback The polar decomposition provides a natural separation operations we consider here involve feeding back classi- of a general quantum operation in terms of a measure- cal, not quantum information. The case of feeding back ment step, followed by a feedback step. This decom- quantum information via fully coherent quantum feed- position, which has been implicitly used throughout the back was addressed in [10]. Note that closed-loop control history of quantum measurement [1,7,13], has recently is intrinsically non-Markovian in that the feedback loop proven useful to investigate the trade-off between in- retains memory of the system’s state at previous times formation and disturbance in quantum feedback control [11]. As will now be shown, such non-Markovian feature [15]. In our context, since we have assumed the ability is essential to allow for the generation of arbitrary open- to build up arbitrary coherent transformations, we need system dynamics, including arbitrary Markovian Lind- only to find a way to perform measurements correspond- blad evolutions, based on a few simple operations. ing to arbitrary positive operators in order to enact an To analyze closed-loop control, we need to describe arbitrary open-system time evolution. quantum measurements. Quantum measurement is a Consider the simplest possible example of a quantum special case of an open-system time evolution. A gen- measurement, a so-called “Yes-No” measurement [7] in eralized quantum measurement can always be described which the measurement apparatus M consists of a single

2 two-level quantum system with states 0 , 1 . Assume Generalized measurements with more than two out- | i | i that ρM = 0 0 initially, and couple the system to the comes can be constructed in a straightforward way [7]. apparatus via| ih an| interaction Hamiltonian H = γX Y , For example, if one can make measurements correspond- ⊗ where γ>0 is a coupling constant, and X, Y are Her- ing to arbitrary A0,A1, a measurement corresponding to 2 2 2 mitian operators acting on S, M respectively. We choose arbitrary Hermitian B0,B1,B2,withB0 + B1 + B2 = 11, X to be a positive operator projecting onto a pure state can be enacted as follows. Perform a measurement cor- of S, and let Y = 0 1 + 1 0 = σ . The state of the responding to B ,B = B2 + B2. If the result of x 0 10 p 1 2 system and apparatus| ih after| | theyih | interact for a time t is this measurement is 0, do nothing. If the result of this measurement is 1, perform a second measurement cor- ρSM(t)= cos(γtX) i sin(γtX) σx ρS ρM 1 1 − ⊗
⊗ responding to A0 = B1B10 − , A1 = B2B10 − (note that 1 cos(γtX)+i sin(γtX) σx . (4) ⊗
B10 − is well-defined on the set of states that result when Now make a von Neumann measurement of 0, 1onthe the first measurement has outcome 1). This measure- ment is generally not Hermitian, but the previous para- ancilla. The result 0 occurs with probability p0 = tr cos2(γtX)ρ , in which case S is in the state ρ = graph shows that we can do arbitrary generalized mea- S 0 surements with two outcomes. The outcome 0 then cor- cos(γtX)ρS cos(γtX)/p0. The result 1 occurs with prob- 2 responds to B0, the outcome 10 corresponds to B1,and ability p1 =trsin(γtX)ρS, leaving S in the state ρ1 = the outcome 11 corresponds to B2. Arbitrary measure- sin(γtX)ρS sin(γtX)/p1. In other words, this Yes-No measurement corresponds to Hermitian Kraus operators ments with multiple (possibly non-Hermitian) outcomes cos(γtX), sin(γtX). can be built up by allowing feedback and by following an The ability to perform arbitrary coherent transforma- analogous procedure. tions on the system can now be used to transform this To summarize, the ability to perform a single simple simple Yes-No measurement into an arbitrary Yes-No measurement on the system, together with the ability measurement. This can be accomplished by applying an to apply coherent control to feed back the measurement average Hamiltonian technique [16] during the system’s results, allows one to enact an arbitrary finite-time open- coupling to the auxiliary quantum system in the course system evolution ρS AkρS A† . 7→ k k of the measurement. Before the von Neumann measure- Now turn to the case ofP infinitesimal open-system time ment is made on the ancilla, perform the following se- evolutions governed by the Lindblad equation. To ad- quence of rapid coherent transformations on the system: dress infinitesimal time evolutions, we can imagine that Apply a unitary transformation V1, wait for a time ∆1, the interaction between the system and the ancilla repre- then perform the transformation V1†; now apply a second senting the measurement apparatus only takes place for a transformation V2, wait for a time ∆2, and perform the short period of time. Accordingly, the unitary propagator transformation V †; etc.,forN periods. Assume that the exp( iHt) evolving the system and apparatus together 2 − Vi’s are effected on a time scale short compared with the can be expanded to second order in time. By writing ∆ , and let ∆ =∆t. Repeat N = t/∆t times. If N H = γX Y as above, one gets i i i ⊗ is large, thenP to lowest order in ∆t the net effect of these repeated transformations is to replace X in Eq. (4) by ρSM(t)=ρS ρM iγt(XρS YρM ρSX ρM Y ) 2 ⊗2 − ⊗ − ⊗ X = (∆i/∆t)V †XVi. γ t Pi i X2ρ Y 2ρ 2Xρ X Yρ Y Starting from a given projector X onto a pure state, − 2 S ⊗ M − S ⊗ M any positive operator with unit trace can be written in 2 2 + ρSX ρM Y . (5) the form X for some Vi,∆i 0. But with an appropriate ⊗
choice of the operator X, and≥ the parameters γ, t,any pair of positive Kraus operators can be represented in the The time evolution for the system on its own is obtained by tracing over the measurement apparatus and taking form B0 =cos(γtX), B1 = sin(γtX). So our simple mea- surement procedure together with the ability to perform the small-time limit (i.e., t small but finite [17]). This coherent control translates into the ability to make arbi- results in the Lindblad equation for the system, trary two-outcome minimally disturbing measurements. dρS 2 2 Now add feedback. Performing the von Neumann mea- = ia [X, ρS] b (X ρS 2XρSX + ρSX ), (6) surement on the ancilla and feeding back the result of of dt − − − the measurement by applying U0 if the result is 0 and 2 2 U1 if the result is 1 gives an open-system time evolution where a = γ trM YρM =0andb =(γ t/2) trM Y ρM = 2 2for = 0 0 , = . That is, comparing with ρS A0ρSA† + A1ρSA†,whereA0 = U0 cos(γtX)and γ t/ ρM Y σx 7→ 0 1 (2), the simple coupling| ih | above results in a single Lindblad A1 = U1 sin(γtX). But, by polar decomposition, any set of two Kraus operators can be written in this form. As a operator L = √2bX. result, the ability to perform coherent control combined To complete the infinitesimal measurement and feed with the ability to perform a single, simple measurement back the classical information, continue just as in the on the system translates into the ability to enact an ar- non-infinitesimal case, by making a von Neumann mea- bitrary two-operator open-system time evolution. surement on the auxiliary system. If the result is 0, do

3 nothing; if the result is 1, apply the coherent transforma- the results of the measurements allows any desired open- tion U to S. By using Eq. (5) and tracing over M,the system transformation to be enacted using only the co- resulting state of the system is herent tools required for Hamiltonian control together with a single, simple quantum measurement. The ability 2 2 γ t 2 2 to implement a similar control strategy in the appropri- ρ (t)=ρ X ρ 2UXρ XU† + ρ X . (7) S S − 2 S − S S
ate small-time limit allows the construction of any de- sired continuous-time Markovian evolution described by In the small-time limit, this gives rise to a Lindblad equa- a Lindblad master equation as well. tion (2) with L = √2bUX. That is, feeding back the result of the infinitesimal measurement yields a Lindblad This work was supported by DARPA/ARO under the equation with a single Lindblad operator proportional to QUIC initiative. UX,whereX is a unit-trace positive operator and U is † Corresponding author: [email protected] of our choosing. The average Hamiltonian techniques described above can now be used to construct an infinitesimal measure- ment corresponding to an arbitrary positive operator X. Here, care must be taken to insure that the sequence of average Hamiltonian operations Vi can be performed within the small “coarse-graining” time t above. Note [1] Information Complexity and Control in Quantum that what is important in the derivation of the Lindblad Physics, edited by A. Blaquiere, S. Diner, G. Lochak equation (6) is not that t be small but rather that γt be (Springer, New York, 1987). small: accordingly, even though the relevant control time [2] G. Huang, T. Tarn, and J. Clark, J. Math. Phys. 24, scale ∆t is not vanishingly small, γ can always be made 2608 (1983). sufficiently small that the averaging operations can be fit [3] A. Peirce, M. Dahleh, and H. Rabitz, Phys. Rev. A 37, within a time t for which the derivation holds. Feeding 4950 (1988); M. Dahleh, A. Peirce, and H. Rabitz, ibid. back the results of the measurement allows one to en- 42, 1065 (1990); W. Warren, H. Rabitz, and M. Dahleh, act a Lindblad equation governed by an arbitrary single Science 259, 1581 (1993); V. Ramakrishna et al., ibid. Lindblad operator L = √2bUX. 51, 960 (1995). The same conclusion could have been also estab- [4] R. W. Brockett, SIAM J. Control 10, 265 (1972); Differ- lished directly from the ability to implement arbitrary ential Geometric Control Theory, edited by Brockett, R. two-operator open-system evolutions. In fact, in the Millman, and H. Sussman (Birkhauser, Boston, 1983). small-time limit discussed above, the quantum opera- [5] S. Lloyd, Phys. Rev. Lett. 75, 346 (1995). [6] S. Lloyd, Science 273, 1073 (1996). 2 2 2 tion specified by A0 = U0 cos(γtX) 11 (γ t /2) X , [7] K. Kraus, States, Effects, and Operations (Springer, A = U sin(γtX) γtU X,withU' =−11,U = U,is 1 1 ' 1 0 1 Berlin, 1983). formally equivalent to the Lindblad equation (2) with [8] R. Alicki and K. Lendi, Quantum Dynamical Semigroups L = √2bUX. In terms of the matrix A appearing in (1), and Applications (Springer-Verlag, Berlin, 1987). the ability to generate an arbitrary Lindblad operator [9] D. Bacon et al., lanl e-print quant-ph/0008070. L = ciFi, ci C, translates into the ability to gener- [10] S. Lloyd, Phys. Rev. A 62, 022108 (2000). Pi ∈ ate any rank-one matrix A = c c∗ [9]. As noted above, [11] However, closed-loop control can give rise to an effec- { i j } Lindblad equations with multiple Lk can be enacted by tively Markovian dynamics for the system considered on performing the operations corresponding to each of the its own in the limit where the controller only retains memory of a single time step. See H. Wiseman and G. J. Lk in succession. So by making a simple infinitesimal measurement, manipulating it by average Hamiltonian Milburn, Phys. Rev. Lett. 70, 548 (1993); H. Wiseman, techniques, and coherently feeding back the results of Phys. Rev. A 49, 2133 (1994). the measurement, one can enact in principle an arbitrary [12] A. Peres, Quantum Theory: Concepts and Methods Lindblad equation. The enactment is scaled in time, stro- (Kluwer Academic, Dordrecht, 1993). [13] H. N. Barnum, Ph.D. Thesis, University of New Mex- boscopic, and approximate, but the scaled time interval ico (1998), available at http://wwwcas.phys.unm.edu/ of the stroboscopic evolution and the error in the approx- ˜hbarnum/homepage.html. imation can be made as small as possible by increasing [14] H. L¨utkenpohl, Handbook of Matrices (John Wiley & the number of control operations per step. Sons, New York, 1996). In conclusion, coherent control combined with feed- [15] A. C. Doherty, K. Jacobs, and G. Jungman, lanl e-print back allows one to enact an arbitrary open quantum sys- quant-ph/0006013. tem dynamics. In contrast to the case of closed quan- [16] L. Viola, E. Knill, and S. Lloyd, Phys. Rev. Lett. 82, tum systems, in which an arbitrary Hamiltonian time 2417 (1999); L. Viola, S. Lloyd, and E. Knill, Phys. Rev. evolution can be enacted using only a few open-loop op- Lett. 83, 4888 (1999). erations, the open-loop control of open-system dynam- [17] Physically, the small time t can be regarded as a coarse- ics requires in general an infinite repertoire of indepen- graining time resulting from the memory time scale over dent operations. But closing the loop and feeding back which the auxiliary system resets its state.

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