Optimal Quantum Clocks

VOLUME 82, NUMBER 10 PHYSICAL REVIEW LETTERS 8MARCH 1999

V. Buzek,ˇ 1,2 R. Derka,2 and S. Massar 3 1Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, 842 28 Bratislava, Slovakia 2Faculty of Mathematics and Physics, Comenius University, Mlynská dolina F2, Bratislava, Slovakia 3Service de Physique Théorique, Université Libre de Bruxelles, CP 225, Bvd du Triomphe, B1050 Brussels, Belgium (Received 17 August 1998) A quantum clock must satisfy two basic constraints. The first is a bound on the time resolution of the clock given by the difference between its maximum and minimum energy eigenvalues. The second follows from Holevo’s bound on how much classical information can be encoded in a quantum system. We show that asymptotically, as the dimension of the Hilbert space of the clock tends to infinity, both constraints can be satisfied simultaneously. The experimental realization of such an optimal quantum clock using trapped ions is discussed. [S0031-9007(99)08612-3]

PACS numbers: 03.67.–a, 03.65.Bz, 42.50.Ar

Recent technical advances in the laser cooling and trap- jc͘ onto the symmetric subspace without changing its ˆ ping of ions suggest that coherent manipulations of trapped energy: U a cmajm, a͘ ෇ cmjm͘ where jm͘ is as before ions will be performed in the not too far future [1]. Apart the symmetric state with energy m. Since Uˆ commutes P from various important applications such as quantum in- with the Hamiltonian the performance of the clock based formation processing or improving high-precision spec- on jc͘ is identical to the clock whose initial state is the troscopy these techniques also allow us to test fundamental symmetric state jcsym͘ ෇ Uˆ jc͘. concepts of quantum theory. In particular, much deeper After the preparation stage, the ions evolve in insight into the problem of quantum measurement can be time according to the Hamiltonian evolution Vˆ ͑t͒ ෇ obtained. Uˆ ͑t͒Vˆ Uˆ y͑t͒, Uˆ ͑t͒ ෇ exp͕2itHˆ ͖. The task is to deter- In this Letter we study the problem of building an mine the elapsed time t by carrying out a measurement optimal quantum clock from an ensemble of N ions. To on the ions. Note that because of the indeterminism of be specific let us assume an ion trap with N two-level ions quantum mechanics it is impossible, given a single set of all in the ground state jC͘ ෇ j0͘ ≠ ··· ≠j0͘. This state N two-level ions, to determine the elapsed time with cer- is an eigenstate of the free Hamiltonian and thus cannot tainty. The best we can do is to estimate the elapsed time record time. Therefore the first step in building a clock is based on the result of a measurement on the system [2]. to bring the system to an initial state Vˆ which is not an Making a good quantum clock requires a double opti- energy eigenstate. For instance one can apply a Ramsey mization. First of all one can optimize the measurement. pulse whose shape and duration is chosen such that it puts This aspect has been studied in detail in [2] where the all the ions in the product state best measuring strategy was derived. But one can also optimize the initial state Vˆ of the system. It is this sec- Vˆ ෇ rˆ ≠ ··· ≠ rˆ ; prod ond optimization that is studied in this Letter. 1 Before turning to the problem of optimizing the initial rˆ ෇ ͑j0͘ 1 j1͒͘ ͑͗0j 1 ͗1j͒ . (1) ˆ 2 state V, it is instructive to review the fundamental limitations on the performance of quantum clocks. Let We shall also consider more general states, but shall us first consider a simple classical clock that can then always take them to belong to the symmetric subspace be generalized to the quantum case. Our classical clock N of the ions. The basis vectors of this space will be consists of a set of n registers. Each register is either in jm m N denoted ͘, ෇ 0, 1, . . . , . They are the completely the 0 or the 1 state. Thus the classical clock consists of N m symmetrized states of two-level ions with ions in the n bits, and can be in 2n different states. The dynamics of N 2 m excited state and ͑ ͒ ions in the ground state. The the clock is as follows: the first register flips from 0 to 1 or jm E m states ͘ have energy m ෇ (this defines our unit of from 1 to 0 every 2p22n, the second register flips every h energy, setting ¯ ෇ 1 then defines our unit of time). 2p22n11, etc. The last register flips every p. This clock The reason we can restrict ourselves to the symmetric thus measures time modulo 2p. Note that this clock has subspace is that we can map any clock state onto an inherent uncertainty since it cannot measure time with the symmetric subspace without affecting its dynamics. a precision better than 2p22n. Throughout this Letter the V jc cj Indeed consider an initial state ෇ ͗͘ that does time uncertainty is defined as not belong to the symmetric subspace of the atoms. We can decompose jc͘ ෇ m a cmajm, a͘ where jm, a͘, Dt2 t 2 t mod 2p 2 . (2) N ෇ ͓͑ estimate true͒͑ ͔͒ a ෇ 1, . . . , ͑ m ͒ denote a basis of the states with energy p P P 2n m. Consider the unitary operator Uˆ that maps the state For the classical clock Dtclass ෇ p2 ͞ 3.

It is straightforward to replace the classical clock by have better resolution. But is it possible, by making an a quantum version. The quantum clock consists of n optimal measurement and choosing in an astute manner two-level systems (qubits). The first qubit has an energy the initial state of the clock, to make a quantum clock splitting between the levels of 2n21 so that it has the same with similar performances to the classical one? Our main period as the corresponding classical register. The second result is to show that this is indeed the case for clocks qubit has an energy splitting of 2n22 and so on up to the built out of symmetric states of N ions and to provide an last qubit that has an energy splitting of 1. Considered algorithm for constructing such an optimal clock. together these n qubits constitute a quantum system with The problem of constructing quantum clocks has been 2n equally spaced energy levels. considered previously in [7,8]. However, the best clocks The mapping between the Hilbert space of this abstract considered in these papers are based on the phase state quantum clock and the symmetric subspace of N two- jC0͘ described below. As we shall see for thesep clocks level ions is straightforward when N 1 1 ෇ 2n. Indeed the time uncertainty is very large Dt Ӎ͑1͞ n͒and is in this case the dimension and energy spectrum of both very far from reaching equality in Eq. (4). Recently, Hilbert spaces coincide. Note, however, that this is Vaidman and Belkind [9] considered the problem of a not a mapping between the qubits of the clock and clock for which equality holds in Eq. (3). They showed the ions individually, but between energy eigenstates. that in the limit of large N the product states satisfy This comparison between classical and quantum clocks this condition. However, for the productp state the energy suggests that a quantum clock built out of N ions cannot uncertainty is very small: DE ෇ N͞2 hence they also behave better then a classical clock built out of ln͑N 1 1͒ do not saturate Eq. (4). Furthermore, clocks based on registers. That this is indeed the case follows from two product state also do not attain Holevo’s bound. fundamental constraints: A similar approach to the one used here, namely The first constraint is a bound on the time resolution of optimizing both the initial state and the measurement on a the clock that results from its energy spectrum. Indeed system of N ions was considered in [10] with the aim the time-energy uncertainty [3,4] of using the ions as an improved frequency standard. This problem can be rephrased in the following way: one 1 DtDE $ , (3) disposes of a classical but noisy clock which provides 2 some a priori knowledge about the time t and one wants to improve the knowledge of t by using the N ions. 2 ˆ 2 ˆ ˆ ˆ 2 where DE ෇ Tr͑H V͒ 2 ͓Tr͑HV͔͒ relates the uncer- On the other hand, in the present Letter we suppose tainty in the estimated time [defined by Eq. (2)] to the that there is no prior knowledge about t. The other spread in energy of the clock. In the present case there difference with the present work is that our aim is to study is a state withp a maximum energy uncertainty, namely the fundamental structure of quantum mechanics. We j j j c1͘ ෇ ͑1͞ 2͒͑N͘ 1 0͒͘ for which DE ෇ N͞2. In- therefore neglect the effect of noise during the preparation serting in Eq. (3) shows that for any clock built out of and measurement stages and decoherence during the N ions evolution. On the other hand, taking these effects into 1 account was central to [10]. Dt $ . (4) In order to find how to build an optimal clock we N must delve in detail into the functioning of this device. Note that one cannot attain equality in Eq. (3). Indeed the We first recall Holevo’s results concerning the optimal state jc1͘ evolves with a period 2p͞N, hence it necessar- measurement strategy [2]. The measurement is described ily has a large time uncertainty. Thus Eq. (4) is only a by a positive operator measurement (POVM), that is a ˆ R lower bound and maximizing the energy uncertainty as in set ͕Or ͖r෇1 of positive Hermitian operators such that ˆ [5] is not necessarily an optimal procedure. r Or ෇ 1. To each outcome r of the measurement The second fundamental constraint the clock must we associate an estimate t of the time elapsed. The P r obey is a bound on the total information it can carry. difference between the estimated time tr and the true time Indeed Holevo [6] has shown that one cannot encode t is measured by a cost function f͑tr 2 t͒. Here we note and subsequently retrieve reliably more than n bits of that because of the periodicity of the clock, f has to be classical information into n qubits. Letting a clock evolve periodic. We also take f͑t͒ to be an even function. The for a given time interval t can be viewed as trying to task is to minimize the mean value of the cost function encode information about the classical variable t into 2p dt the state of the clock. Hence a measurement on our f¯ ෇ Tr͓Oˆ r Vˆ ͑t͔͒ f͑tr 2 t͒ . (5) r 0 2p model clock [consisting of n ෇ ln͑N 1 1͒ qubits] cannot X Z retrieve more than ln͑N 1 1͒ bits of information about t. To proceed we expand the cost function in Fourier series: Together these two bounds imply that the quantum ` clock cannot perform better than the corresponding classi- f͑t͒ ෇ w0 2 wk cos kt . (6) cal clock: it cannot carry more information and it cannot k෇1 X

2208 VOLUME 82, NUMBER 10 PHYSICAL REVIEW LETTERS 8MARCH 1999

The essential hypothesis made by Holevo is positivity Let us first consider the cost function f ෇ 4 sin2 t͞2. of the Fourier coefficients: wk $ 0, ͑k ෇ 1, 2, . . .͒.He The advantage of this cost function is that the matrix F also supposes that the initial state Vˆ ෇ jv͗͘vj, jv͘ ෇ is particularly simple in this case. Furthermore, for errors 2 m amjm͘ is a pure state (and makes a phase convention much smaller than 2p, f and Dt as defined in Eq. (2) such that am is real and positive). He then shows that coincide. Hence the first constraint on clock resolution P ` ¯ 2 1 Eq. (4) can be approximately replaced by f $ 1͞N . f¯ $ w0 2 wk amam0 , (7) If the initial state is the product state Eq. (1) then 2 0 k෇1 m,m the mean cost is given by the expression f¯ 2 1 2 X jm2Xm0j෇k ෇ ͓ 2N N21 N N and equality is attained only if the measurement is of the 2 i෇0 ͑i ͒͑i11͔͒ which for large N decreases as form f¯ 1 N. If the initial state is the phase state jC , ӍP͞ q 0͘ ˆ 2 itr H then direct calculation shows that f¯ . Thus for Oˆ r ෇ pr jCr ͗͘Crj; pr $ 0; jCr ͘ ෇ e jC0͘ , ෇ N11 21͞2 N both states Dt Ӎ N and one is very far from attaining 1 equality in Eq. (4). jC0͘ ෇ p jm͘ , (8) N 1 1 m෇0 However, neither of these two states is optimal. To X ˆ ˆ ˆ find the optimum we note that the matrix F ෇ 2dmm0 2 with the completeness relation r Or ෇ 1. Several remarks about this result are called for: dmm011 2dm11m0 can be viewed as the discretized second P ˆ 2 2 (1) Holevo supposed that the initial state is a pure derivative operator F Ӎ 2d ͞dx with von Neumann boundary conditions. The lowest eigenvalue of Fˆ is state. If the initial state is mixed, Vˆ ෇ p jc ͗͘c j, i i i i p2 N 1 2 then one finds that the corresponding cost is bounded by therefore approximately lmin Ӎ ͑͞ 1͒ and the P the average of the bounds Eq. (7). This shows that in corresponding eigenvector is building a good quantum clock one should always take p 2 p͑m 1 1͞2͒ the initial state to be pure. jCopt͘Ӎp sin jm͘ . (10) N11 m N 1 1 (2) Holevo considered covariant measurements in X which the times tr takes the continuum of values between Thus in this case the cost decreases for large N as f¯opt Ӎ 0 and 2p. But as shown in [11] the completeness relation p2 p can also be satisfied by taking a discreet set of times ͑N11͒2 corresponding to Dtopt Ӎ ͑N11͒ . Therefore, up to 2pj a factor of p the optimal clock attains the bound Eq. (4). tj ෇ N11 , j ෇ 0,...,N. These “phase” states jCj͘ [12] form an orthonormal basis of the Hilbert space, and this It is also interesting to consider the situation where the measurement is therefore a von Neumann measurement. cost function is the delta function f ෇ 2d͓t͑mod 2p͔͒. 1 0 Thus it is not necessary to use an ancilla to make the In this case Fmm0 ෇ 2 2p for all m, m . One checks that optimal measurement in this case. the phase state jC0͘ is the eigenvector of Fˆ with minimal N (3) In Eq. (7) only the values k ෇ 0,...,N intervene eigenvalue l ෇ 2 2p . Note that one could also have because of the condition jm 2 m0j ෇ k. That is, only the taken a smeared delta function since only the first N 1 1 first N 1 1 Fourier coefficients of the cost function are terms intervene in Eq. (7). The smeared delta function is meaningful. approximately zero everywhere except in an interval of (4) Because of the condition of positivity of wk, not all about 1͞N around zero where it is equal to N. Thus for cost functions are covered by this result, but several impor- this cost function one wants to maximize the frequency 2 t 1 tant examples are 4 sin 2 ෇ 2͑1 2 cos t͒, jt͑mod 2p͒j, with which the estimated value of t is within about N of t jsin 2 j, 2d͓t͑mod 2p͔͒. The most notable absence from the true value. But there is no extra cost if the estimated this list is the quadratic deviation t2 [as defined in Eq. (2)] value is very far from the true one. It is for this reason but it can be well approximated by the first cost function that taking jC0͘ as the initial state when the cost function 2 t 2 2 since 4 sin 2 Ӎ t for jtjøp. is 4 sin t͞2 is bad since making estimates that are wildly We now turn back to the central problem of this Letter, off is strongly penalized in that case. namely how to optimize the initial state of the system The mean value of a cost function gives only par- so that the time estimate is as good as possible. This tial information about the sensitivity of a clock. The corresponds to minimizing the right-hand side of Eq. (7) full information is encoded in the probability P͑tjtr ͒ ෇ ˆ ˆ ˆ N11 with respect to am. To this end let us define the matrix P͑Or jt͒P͑t͒͞P͑Or ͒ ෇ P͑Or jt͒ 2p that the true time is t 1 N given that the readout of the measurement is tr . In the Fmm0 ෇ w0dm,m0 2 wk͑dm,m01k 1dm1k,m0͒. case of the optimal state jCopt͘, one finds that 2 k෇1 X (9) Popt͑tjtr ͒ We must then minimize f¯ ෇ aT Faˆ under the condition T ͓1 1 cos͑N 1 1͒T͔͑1 1 cos T͒ a a ෇ 1. Using a Lagrange multiplier we find the eigen- Ӎ N p p , value equation ͑Fˆ 2l1ˆ͒a෇0, and the task is therefore ͓1 2 cos͑T 1 N11 ͔͒ ͓1 2 cos͑T 2 N11 ͔͒ to find the smallest eigenvalue and eigenvector of Fˆ . (11)

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3.5 In summary, we have seen that for different cost functions there are different optimal clocks. There is, of course, 3.0 no cost function that is in an absolute sense better than an- other, and the choice of a particular cost function depends 2.5 on the physical context. Nevertheless, the experimental ) realization of a quantum clock based on the state jCopt͘ 2.0

= seems particularly desirable because it combines the attrac- r tive features that the a posteriori probability P tjt has a 1.5 ͑ r ͒

P(t|t tight central peak and rapidly decreasing tails. 1.0 Carrying out such an experiment with trapped ions presents two main difﬁculties. The ﬁrst is the preparation ˆ 0.5 of the initial state V. Such coherent manipulation of trapped ions is one of the current experimental challenges. 0.0 A possibly important simplifying feature of the quantum 0123456 clock is that since it is symmetric in the N ions one does t not need to address each ion individually. The second

FIG. 1. A plot of the a posteriori probability P͑tjtr ͒ that problem is to realize the optimal phase measurement as the time was t given that the measurement yielded outcome discussed in the present Letter. Recent experiments [13] tr ෇ p for different initial states (N ෇ 20). The dotted line suggest that this type of coherent preparation and specific corresponds to the product state Eq. (1), the solid line to the projective measurements are possible for systems with a phase state jC0͘ (8), and the dashed line to the optimal state jCopt͘ (10). moderate number of trapped ions. Therefore, one may hope that it will be feasible to make optimal quantum clocks in the not too distant future. p where N Ӎ 4͑N11͒3 and T ෇ t 2 tr . This distribution Our special thanks to Lev Vaidman for his criticism and (see Fig. 1) has a central peak of width 3p͑͞N 1 1͒ careful rereading of the manuscript. We also thank Win 21 and tails which decrease for 2p ¿jt2trj¿N as van Dam, Susana Huelga, Peter Knight, and Christopher 23 24 Popt͑tjtr ͒ӍN jt2trj . Monroe for helpful discussions. This work was in part For the phase state jC0͘ one finds supported by the Royal Society and by the Slovak Acad- emy of Sciences. V. B. and S. M. thank the Benasque 1 ͓1 2 cos͑N 1 1͒T͔ Center for Physics where part of this work was carried PjC ͑tjtr ͒ ෇ . (12) 0͘ 2p͑N 1 1͒ ͑1 2 cos T͒ out. S. M. would like to thank Utrecht University where most of this work was carried out; he is a chercheur quali- This distribution has a slightly tighter central peak of fié du FNRS. width 2p͑͞N 1 1͒ but the tails of the distribution de- 21 22 crease as N jt 2 tr j . It is these slowly decreasing tails that give the main contribution to Dt Ӎ N21͞2. For the product state Eq. (1) the distribution Pprod͑tjtr ͒ p1 [1] D. J. Wineland et al., quant-ph/9710025. has a very wide central peak of width Ӎ N . In this case it is the wide central peak that gives rise to the large time [2] A. S. Holevo, Probabilistic and Statistical Aspects of uncertainty. Quantum Theory (North-Holland, Amsterdam, 1982). Using these distributions it is possible to calculate the [3] L. Mandelstam and I. E. Tamm, Izv. Akad. Nauk. SSSR number of bits of information about time that is encoded Fiz. 9, 122 (1945); J. Phys. (Moscow) 9, 249 (1945). [4] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, in the outcomes of the measurement. This is given by the 3439 (1994); S. L. Braunstein, C. M. Caves, and G. J. mutual information Milburn, Ann. Phys. 247, 135 (1996). [5] J. J. Bollinger et. al., Phys. Rev. A 54, R4649 (1996). I ෇ 2 dtP͑t͒ ln P͑t͒ [6] A. S. Holevo, Probl. Pereda. Inf. 9, 3 (1973) [Probl. Inf. Transm. 9, 177 (1973)]. Z [7] H. Salecker and E. P. Wigner, Phys. Rev. 109, 571 (1958). 1 P͑tr ͒ dtP͑tjtr ͒ ln P͑tjtr ͒ . (13) [8] A. Peres, Am. J. Phys. 48, 552 (1980). r X Z [9] L. Vaidman (private communication); O. Belkind, M.Sc. In all cases the integral in the second term is dominated thesis, Tel Aviv University, 1998. [10] S. F. Huelga et al., Phys. Rev. Lett. 79, 3865 (1997). by the central peak. Thus one finds that for the product 1 [11] R. Derka, V. Buzek,ˇ and A. K. Ekert, Phys. Rev. Lett. 80, state only 2 ln N bits of information about time are 1571 (1998). obtained, whereas for both the states jC0͘ and jCopt͘ [12] D. T. Pegg and S. M. Barnett, Europhys. Lett. 6, 483 one obtains approximately ln N bits, thereby saturating (1988). Holevo’s bound. [13] Q. A. Turchette et al., Phys. Rev. Lett. 81, 3631 (1998).

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