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How to project qubits faster using quantum feedback Kurt Jacobs 65 Abraham heights, Nelson 7001, Auckland, NZ When one performs a continuous measurement, whether on a classical or quantum system, the measurement provides a certain average rate at which one becomes certain about the state of the system. Foraquantumsystem thisisan average rateat which thesystem isprojected ontoapure state. We show that for a standard kind of continuous measurement, for a qubit this rate may be increased by applying unitary operations during the measurement (that is, by using Hamiltonian feedback),in contrast totheequivalentmeasurement on a classical bit,where reversibleoperations cannot be used to enhance the rate of entropy reduction. We determine the optimal feedback algorithm and discuss the Hamiltonian resources required. 3 PACSnumbers: 03.67.-a,03.65.Ta,02.50.Tt,02.50.Ey 0 0 2 It was discovered recently ([1, 2]) that the average an example of something which the quantum nature of n amount by which the quantum state of a system is pu- an object makes possible. a rified during a measurement depends, in general, on the Since this feedback algorithm is designed to enhance J basisin whichthe measurementis made. Since changing the properties of the measurement itself, the resulting 3 the basis of a measurement is equivalent to performing measurement process is an example of an adaptive mea- 1 a unitary transformation on the system, we can re-state surement, a term first introduced, we believe, by Wise- this property by saying that a unitary transformation man[3]. Wiseman’sadaptivemeasurementscheme,real- 1 maybeappliedtoasystem,soastoincreasetheaverage izedrecentlybyArmenet.al.[4],wasdesignedtomakea v amount of information that the measurement provides canonicalphase measurementoptimally well,giventhat, 6 5 about the final (post-measurement) state. This restate- in practice, one only has arbitrary quadrature measure- 0 ment makes it particularly clear what is being changed mentsatonesdisposal. Thetopicdiscussedhereisthere- 1 about the measurement in order to enhance the infor- fore a different kindof applicationfor adaptive measure- 0 mation: merely the addition of a separate Hamiltonian ments. 3 evolution. Taking this point of view, the ability to per- Beforewebeginthe developmentofthe feedbackalgo- 0 form unitary transformations, or equivalently, Hamilto- rithm,letussayafewwordsaboutquantumandclassical / h nianevolution,canbeconsideredasaresourcewhichcan measurements,andtherelationshipbetweenthem. Clas- p beusedtoenhancethepropertiesofafixedmeasurement sicalmeasurementsonclassicalsystemsare describedby - t process. Bayesianinference,andmaybewritteninthesameform n Consider nowa continuous measurementof, for exam- as quantum measurements. If one writes the classical a ple, a qubit. This may be represented by a sequence of probability distribution of the quantity one is measuring u q identical ‘finite strength’ measurements, each of which asadiagonaldensitymatrix,thenclassicalmeasurements : partially projects the qubit onto the basis 0 , 1 [1] are in fact a subset of quantum measurements: While v {| i | i} (Wewillrefertothisasthecomputationalbasis,bywhich quantum measurements are described by a set of opera- i X is simply meant the basis in which information is en- torsΩ ,wheretheonlyrestrictionisthat Ω†Ω =1, n n n n r coded). As more measurements are made, the purity classical measurements have the further restriction that a of the state of the qubit increases, until eventually the all the Ω commute with the density maPtrix describing n qubit ends up in one of the basis states. Examining this the classical system (for a fuller discussion see [7]). process, we find that the state that results from a mea- The behavior of a quantum measurement with com- surement in the sequence is not ideal for the purposes of muting operators therefore reduces to that of a classi- purification for the next measurement in the sequence. calmeasurementwheneitherunitarytransformationsare It is therefore possible to perform a unitary transforma- notavailabletorotatethequbitoutofthecomputational tion at the end of each measurement (and depending on basis, or there is a rapid decoherence process which im- the measurement result) to increase the average rate at mediatelydecoheresthequbitinthecomputationalbasis. which the state is purified. In the continuum limit this In that case the unitary transformation becomes merely becomes a continuous feedback process. a classical diffusion process acting on the bit. This fact, while interesting as it provides a technique Here we will be interested in a widely applicable for enhancing a continuousmeasurement,is, if anything, model of continuous measurement, where the measure- more interesting from a fundamental point of view, due ment record is a Wiener process. We will measure in tothefactthatthesamemeasurementprocessperformed the z-basis of a spin-half system (that is, measure the on a classical bit cannot be enhanced by a reversible observable σ ), denoting the basis states as (thus z {|±i} transformation,since the necessarysequence oftransfor- the final result of the measurement will either be |−i mationsrequiresthatsuperpositionstatesofthe compu- or + ). (Any further reference to ‘the computational | i tational basis must be available. Thus, this constitutes basis’ will refer to these states.) To describe the con- 2 tinuous measurementone applies a POVMin eachsmall purifying the state. This somewhatcuriousresult means time interval ∆t, where the POVM is chosen to scale that, if the observer’s state-of knowledge is not maxi- with time in such a way that a sensible continuum limit mallycomplementaryto themeasurementbasis,andthe exists [8]. The resulting continuous measurement is not observer’sobjective is purification, then a unitary trans- merelyamathematicalcuriosity,asitcorrespondstoreal formationshouldbeappliedtothestatepriortothemea- measurements on physical systems [9, 10, 11, 12, 13] (in surement. Itis alsoworthnoting that, when the average particular, Korotkov ([10, 11, 12, 13]) gives explicit ex- reductioninthe entropyis greatest,this reductionis the amples for solid-state qubits). While the approachin [8] sameforbothmeasurementoutcomes. Thus,inthiscase, uses a POVM in each time step which has an infinite the entropy reduction is deterministic, not random. number of outcomes, for measurements on a two-state Now let us consider the consequences of this for a system one can alternatively employ a POVM with two- continuousmeasurement, performedonaninitially com- outcomes. A two-outcome measurement which provides pletelymixedstate. Approximatingthe continuousmea- information about the computational basis is surementbyasequenceofmeasurements ,weseethat M when we make the first measurement, ∆ is maximal. P Ω± = √κ +√1 κ However, the result of the first measurement does not |±ih±| − |∓ih∓| 1 produce a state diagonal in a basis complementary to = (√κ+√1 κ)I (√κ √1 κ)σ , (1) 2 − ± − − z the z basis, and this is generally true of the state result- (cid:2) (cid:3) ing from a measurement in the z-basis. As a result, the We will refer to this measurement in what follows as average purification will not be optimal for at least the . When κ = 1/2 provides no information about majority of the measurements , and thus for essen- tMhe quantum system,Mleaving the state of knowledge un- tially all of the duration of the cMontinuous measurement changed,andwhenκ=0or1theoperatorsΩ± arerank process. one projectors,so that providesthe maximalamount This suggests, therefore, the following procedure: Af- M of information about the final state; this case is an ‘infi- tereachmeasurementweapplyaunitarytransformation nite strength’ measurement by the terminology of [1, 2]. to rotate the state appropriately, so as to achieve max- Setting κ = 1/2 √2γ∆t, and taking the limit of re- imal ∆ for each measurement in the sequence (each − P peated measurements as ∆t 0, the evolution of the measurementstep). Takingthe continuumlimit, this re- → density matrix describing the observers state of knowl- sults in a continuous feedback algorithmwhich increases edge, ρ, is given by the stochastic Schrödinger equation the rate of projection during the continuous measurement. Calculating the behavior of the linear entropy, P, dρ= γ[σz,[σz,ρ]]dt+ 2γ( σz,ρ + 2 σz )dW, (2) asafunctionoftimeisstraightforward,because,asmen- − { } − h i tioned above, when ∆ is maximal, the change in P is p P where dW is the Gaussian stochastic Wiener increment the same for both outcomes of the measurement. Hence, satisfying dW2 =dt. for a finite sequence of steps we have, for the nth step, We are interested here in how the observer’s uncertainty of the quantum state reduces over time. In or- P = 1 b2/n nP , (4) n 0 − der to keep the calculations tractable, we will use as our measure of uncertainty the so called “linear entropy”, whereb2 =(2κ 1). Int(cid:0)hecontinu(cid:1)umlimitthisbecomes − P(ρ)=1 Tr[ρ2]. (That this is a useful measure of un- − P(t) = e−8γtP(0). (5) certainty is due to its concavity [2].) For a completely mixed state, P(ρ)=1/2,and for a pure state, P(ρ)=0. Without the feedback process, the evolution of P(t) de- Ingeneral,the reductioninthe linearEntropywillde- pends on the measurement outcomes, and as a result is pend upon the outcome of the measurement. As such, stochastic. Thus, not only does the Hamiltonian feed- a sensible measure of the purifying power of the mea- back algorithm increase the rate at which the state is surement is therefore the average reduction in the linear purified, but also changes the entropy from being a ran- entropy, over the two outcomes. From [2] we know that dom function of time to a deterministic one. (In fact, the amountby which,onaverage,the two-outcomemea- it is clear that this is true for any entropy, linear, von surement purifies the state (that is, reduces P(ρ)) is Neumann, or otherwise.) Intheabsenceoffeedback,duetoitsstochasticnature, 1 (1 2P)cos2θ the behavior of P(t) is much harder to obtain, even for ∆ = (2κ 1)P − − , (3) P − 1 (1 2P)(2κ 1)2cos2θ thesimplecanonicalcontinuous(classical!) measurement − − − we have here. Using the technique in [14], one obtains, where we have written the initial density matrix as for an initially completely mixed state, ρ=(I+a σ)/2,withθbeingtheanglebetweenaandthe z-axis,and· a2 =(1 2P). Thereductioninuncertainty e−4γt +∞ e−x2/(2t) | | − P (t) = dx, (6) is maximized when θ = π/2. That is, when the basis in c which the density matrix is diagonalis maximally differ- √8πtZ−∞ cosh(√8kx) ent from the z-basis, being the basis in which the mea- where we use the subscriptc to denote the fact that this surement is made, the measurement is most effective in is also the result for a classical continuous measurement 3 1.8 Alternatively, if we were to apply a non-optimal procedure to both cases (both sides of Eq.(7)), then we would 1.7 have, for each of the outcomes i=1,2, 1.6 mal speedup using feedback 111...345 Tophtuims,ablyptrhoeceedXjn=u2d1reposijfgwtjiho(eguils(dePcgo)i)nvde≥stefp(,git(wPo))s.equentialno(9n)- Opti 1.2 2 2 2 p pigi(g (P) p f(g (P)) 1.1 i j j i  ≥ i i i=1 j=1 i=1 X X X 11e-06 1e-05 0.0001 0.001 0.01 0.1 1   f(f(P)) (10) Linear entropy ≥ where the first line uses Eq.(9), and the second uses the FIG. 1: Thespeed-up factor provided by theoptimal Hamil- tonian feedback algorithm, as a function of final purity,for a linearity of f(P). Thus, after two steps the result of continuousmeasurementinagivenbasisofaqubit. Thecase non-optimalmeasurementsinbothstepsgivesanaverage displayed is for an initially completely mixed state. entropy which is higher than the result of two optimal steps. Clearly this procedure canbe repeated n times to obtainthe resultfor n measurementsteps. Thus,we can onaclassicalbit. Ananalyticsolutionforthe integralin write the final result of n non-optimal steps as Eq.(6)doesnotappeartoexist,andwethereforeevaluate it numerically. In figure 1 we plot the speed-up factor 2n p P fo(n)(P) (11) provided by the feedback algorithm in obtaining a given i i ≥ final level of purity, when the initial state is completely Xi=1 mixed. This factor is independent of γ, and increases for some p and P . In order for the procedure which i i with the final purity, tending to 2 as the final entropy uses all optimal steps to render a greater entropy than tends to zero. a procedure that uses some non-optimal steps, then it The Hamiltonian feedback algorithm above has been would have to be possible to apply an optimal step to obtained simply by optimizing the increase in purity of the left hand side of Eq.(11), such that 2 p f(P ) < i=1 i i the final state at each measurement step. We now show fo(n+1)(P). However,since f(P) is linear, this is impos- that this is, in fact, the optimal feedback algorithm for P sible. We can therefore conclude that the Hamiltonian obtaining the maximum purity of the state at any par- feedback algorithm presented above gives the maximum ticular future time. First, let us denote the map which possible entropy reduction for any number of steps, or, takesus fromaninitial linearentropyP, to the finallin- in the continuum limit, the maximum possible entropy ear entropy P′, for a single time step, when we use the reduction for a measurement of any given duration. optimal unitary transformation, as f(P). Thus we have So far we have discussed the effects of the feedback P′ =f(P)=(1 b2)P, so f is linear in P. algorithm, but not given explicitly the algorithm itself. − Now, let us consider a single measurement step , Theunitarytransformationrequiredaftereachmeasure- M where the initial entropy is P. If we use a sub-optimal ment step must be such so as to rotate the qubit so that procedure, then one of two states results, and we can the Bloch vector lies in the xy-plane. The first thing label the linearentropy ofthese as P =g (P) andP = 1 1 2 to note is that the minimum angle of rotation required g (P), respectively. Onthe otherhand,the entropythat 2 to do this is achieved by rotating such that the x and resultsfromtheoptimalprocedureisf(P),andthethree y elements of the Bloch vector remain in the same pro- entropies satisfy portions (i.e. by keeping the angle that the Blochvector makes with the x and y axes the same). The second is n that an application of the measurement also keeps piPi f(P), (7) M ≥ these angles the same. Assuming that the initial state i X is either completely mixed, or has been rotated prior to where p and p are the respective probabilities that the the measurementso that the Blochvector lies in the xy- 1 2 twosub-optimalentropieswereobtained. Ifweapplythe plane (which is the optimal thing to do), then the initial optimalproceduretobothresults(bothsides)thensince az =0, and the state after a measurement is M f(P) is linear, we have ρ = 1(I + 1 b2(a σ +a σ ) bσ ). (12) f 2 − x x y y ± z 2 p p f(g (P)) f(f(P)). (8) The unitary transformation required to rotate such a i i ≥ state so that once again a = 0, using the minimum i=1 z X 4 angle of rotation, is U =exp[ i(α/2)n σ], with since α(0)=π/2. Thus, a real continuous feedback pro- − · cedure will provide a lower rate of projection,depending α = tan−1 b/ √1 2P 1 b2 , (13) ontheavailableHamiltonianresources. Wenotethatthe ± − − divergence of the feedback Hamiltonian for P =1/2 has n = ( cos(hφ),si(cid:16)n(φ),0), p (cid:17)i (14) an analogue in Wiseman’s adaptive phase measurement; − in that scheme, for this state, it is the rate of change of and where we have used the fact that a2 = a2 +a2 = 1 2P,anddefinedφbya = a cos(φ).| S|inceφxremayins the phase estimate which diverges. It is interesting that th−e same throughout thexsequ|e|nce of measurements, n for both schemes this divergence is associated with the fact that the measurement must break the symmetry of remainsunchangedthroughoutthefeedbackprocess,and the state. it is merely α that changes at each feedback step. After the nth measurement P is given by Eq.(4), and hence n While the feedback algorithm projects a qubit onto a final pure state with maximal speed, it is fairly clear α = tan−1 b/( 1 2P (1 b2/n)n 1 b2) . n ± − 0 − − from the construction of the process that the operations h p p i corresponding to the many possible final outcomes will Notethat,whentheinitialstateiscompletelymixed,the be mutually non-orthogonal. As a result, this adaptive rotation angle α, after the first measurement is always measurement will almost certainly not provide full in- π/2, regardless of the strength of (i.e. regardless of M formation regarding the initial preparation of the qubit. the value of b). Thisiswhywerefertothealgorithmasprojecting,rather In the continuum limit, the feedback angle α(t) be- thanmeasuringthequbit,forthesenseoftheterm“mea- comes surement” contains a certain ambiguity: it could mean either obtaining information about the initial prepara- √8γdW α(t) = tan−1 tion, useful in classical and quantum communication, or 1 2P(0)e−8γt! the final state, useful in quantum feedback control and − √8γpdW quantumstate preparation. While in this casethe quan- = , P(t)<1/2. (15) tum state is projected maximally fast, this is at the ex- 1 2P(0)e−8γt − penseoflosinginformationabouttheinitialpreparation. Since the Hamiltponian required to rotate through α is This raises the question of whether there is a trade-off between speed of projection, and loss of initial informa- proportional to α(t)/dt, the Hamiltonian required for tion, in the kind of measurements considered here. In the feedback is proportional to the measurement noise addition,optimalalgorithmsforprojectinghigherdimen- dW/dt. Thus the feedback required to obtain an opti- sional systems, and optimal rates obtainable with fixed malprojectionrateisWiseman-MilburntypeMarkovian Hamiltonian resources, are also open questions for fur- feedback, with the addition of a time dependent factor. ther work. However, this kind of feedback is strictly speaking an idealization valid in the limit of a large Hamiltonian, since dW/dt is infinite. In addition, the feedback Hamil- The author would like to thank Howard Wiseman for tonian diverges for an initially completely mixed state, helpful comments on the manuscript. [1] A.C.Doherty,K.JacobsandG.Jungman,Phys.Rev.A (1993); G.J. Milburn, K. Jacobs, and D.F. Walls, Phys. 63, 062306 (2001). Rev. A 50 5256 (1994); [2] C.A. Fuchs and K. Jacobs, Phys. Rev. A 63, 062305 [10] A.N. Korotkov, Phys. Rev. B 60, 5737 (1999), Eprint: (2001). cond-mat/9909039; [3] H.M. Wiseman, Phys.Rev.Lett. 75, 4587 (1995). [11] A.N. Korotkov, Physica B 280, 412 (2000), Eprint: [4] M.A.Armen,J.K.Au,J.K.Stockton,A.C.Dohertyand cond-mat/9906439; H. Mabuchi, ‘Adaptive homodyne measurement of opti- [12] A.N. Korotkov,Phys.Rev.B 63, 115403 (2001); cal phase’, quant-ph/0204005. [13] A.N. Korotkov, ‘Quantum feedback control of a [5] E.T. Jaynes, in E.T. Jaynes : Papers on Probabil- solid-state qubit’, Phys. Rev. B (in press), Eprint: ity, Statistics, and Statistical Physics, edited by R.D. cond-mat/0204516. Rosenkrantz,(Dordrecht,Holland, 1983). [14] K. Jacobs and P.L. Knight, Phys. Rev. A. 57, 2301 [6] T. Bayes, Phil. Trans. Roy. Soc., 330 (1763); or see, e.g. (1998). S.J. Press, Bayesian statistics : principles, models, and applications, (Wiley,New York,1989). [7] K.Jacobs, Quant.Information Processing 1, 73 (2002). [8] C.M. Caves and G.J. Milburn, Phys. Rev. A 36, 5543 (1987). [9] H.M. Wiseman and G.J. Milburn, Phys.Rev. A 47, 642