Quantum Atomic Synchronization Based on Shared Prior Entanglement

Richard Jozsa2,DanielS.Abrams1, Jonathan P. Dowling1, and Colin P. Williams1 1 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California 91109-8099. 2 Dept. of Computer Science, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol BS8 1UB, UK. (Received: May 3, 2000)

System of Units (SI), the unit of is defined as the We demonstrate that two spatially separated parties (Al- second, which is the duration of exactly 9,192,631,770 ice and Bob) can utilize shared prior , periods of oscillation corresponding to the hyperfine (ra- and classical communications, to establish a synchronized 133 pair of atomic . In contrast to classical synchroniza- dio) transition frequency for the ground-state of the Cs atom [5]. The fact that this frequency is identical for all tion schemes, the accuracy of our protocol is independent of 133 Alice or Bob’s knowledge of their relative locations or of the Cs atoms, which are sufficiently isolated from the en- 133 properties of the intervening medium. vironment, allows anyone to establish a Cs time stan- PACS: 03.67.-a, 03.67.Hk, 06.30.Ft, 95.55.Sh dard of comparable accuracy. In general, any set of identical qubits may be used as In the Special , there are two meth- the time standard in a temporal interferometer, which ods for synchronizing a pair of spatially separated clocks, employs the Ramsey method of separated oscillatory A and B, which are at rest in a common inertial frame. fields [6]. In the language of the- The usual procedure is Einstein Synchronization (ES), ory this Ramsey interferometer corresponds to a sim- which involves an operational line-of-sight exchange of ple with just two gates, acting on one light pulses between two observers, say Alice and Bob, qubit. Specifically, let us suppose that the qubit has who are co-located with their clocks A and B, respec- energy eigenstates |0i and |1i with energy eigenvalues tively [1]. A less commonly used Clock Synchronization E0

1 By monitoring the oscillations of either P0 or P1 as a her measurement, Alice knows the labels belonging to the function of time we get an estimate of the clock phase subensembles I and II but Bob is unable to distinguish Ωt mod 2π and hence of t. them. We now describe our proposed QCS scheme. A fun- As the next step in our QCS protocol, Bob performs damental ingredient will be the entangled singlet state a Hadamard transform on each of his qubits, at some | i √1 | i | i −| i | i ψ0,0 = 2 ( 0 A 1 B 1 A 0 B) where the subscripts time t = tB. Thus he will get an equal mixture of the refer to particles held by Alice and Bob. This singlet states |ψneg(t)i and ψpos(t)i in eqs. (2) and (3). The 1 state is a “dark state” that does not evolve in time pro- corresponding is 2 I, independent of t,so vided A and B undergo identical unitary evolutions. In- no measurement statistic can provide Bob with any tim- deed for any 1-qubit unitary U we have (U ⊗ U)|ψ0,0i = ing information. For Bob to extract a clock, a classical (det U)|ψ0,0i so that |ψ0,0i changes only by an overall un- message from Alice is required. So let us now suppose observable phase. Our protocol below (slightly modified) that Alice post-selects from her entire ensemble the sub- would work equally well using the state ensemble of, say, Type-I qubits. Since the qubits are labelled, she can then tell Bob which subset of his qubits  1 iδ are also Type-I by broadcasting their ordinal labels (say, |ψ0,0(δ)i = √ |0iA|1iB + e |1iA|0iB (5) 2 n =3, 5, 13,...) via any form of classical communiqu´e. Bob is then able to extract his own Type-I and Type-II for any fixed δ. This state still has the essential property subensembles. Choosing the Type-II subensemble, Bob of being constant in time i.e. invariant under U ⊗U where will have a clock exactly in phase with a Type-I clock U is time evolution, diagonal in the {|0i,|1i} basis (but that Alice started at t = 0, the time of her initial mea- unlike the singlet i.e. δ = π, it is not invariant under surement. Bob may either wait for Alice’s message be- more general U’s). fore performing his final Hadamard transform and mea- We imagine that Alice and Bob share an ensemble of a surements, or alternatively, he may do these operations large number of such pairs, labelled n =1, 2, 3,...where earlier and record the outcomes for each label value n. the labels are known to both Alice and Bob. We will refer In other words, Alice and Bob now have clocks that are to a pair of clocks in state |ψ0 0i as an entangled pair of , “ticking” in unison. pre-clocks. Since |ψ0 0i is constant in time the pre-clock , For some applications, such as satellite-based Very pairs could be said to be “idling” – they can provide no Long Baseline Interferometry (VLBI) [7], the fact that direct timing information. We may also write |ψ0 0i as , Alice and Bob’s clocks are phase locked up to only mod- ulo 2π is sufficient. However, there are other appli- | i √1 | i | i −| i | i ψ0,0 = ( pos A neg B neg A pos B)(6)cations, such as the synchronization of satellite-borne 2 atomic clocks in the Global Positioning System (GPS) To start the clocks at some time tA,whichwetaketo [8], where it is important to have a shared origin of time. be t = tA = 0 in Alice’s and Bob’s shared inertial rest For such applications, it is a simple matter to adapt our frame, Alice simultaneously measures all of her pre-clock QCS protocol to construct a common temporal point of pairs in the dual basis {|posi, |negi}. Thus each pair reference. Let us suppose that, in addition to the stan- collapses randomly and simultaneously at A and B into dard clock qubits that run at the frequency Ω, Alice and one of the following states: Bob have an additional set of identical qubits all with a slightly shifted frequency Ω0 = Ω + ∆Ω. For example, if I |ψ i = |posiA|negiB Ω corresponds to the two-level hyperfine transition of the II (7) 133 0 |ψ i = |negiA|posiB standard Cs clock atom, then Ω could correspond to the same transition in the long-lived radioactive isotope 1 135 with equal probability 2 . Alice’s measurement has re- Cs , which is slightly different from Ω due to the very moved the entangled state’s invariance under temporal small isotope shift [9]. evolution, and thus the A and B clocks begin to evolve So now Alice and Bob have two ensembles of entangled in time, in accordance with Eq. (1) – all starting syn- pre-clocks – one with frequency Ω and the other with Ω0 chronously at a time of t = 0 in Alice and Bob’s shared – and the entire protocol is carried out on both ensembles I inertial frame. Comparing the clock states in |ψ i and simultaneously. Bob is then able to extract both a Type- II |ψ i with the evolution in Eq. (1), we see that Alice’s IandaType-I0 clock of his own and observe the two measurement effectively reproduces the result of the first 1 0 evolution probabilities, PB = 2 (1 + cos Ωt)andPB = one-clock Hadamard transform in the Ramsey scheme. 1 0 2 (1 + cos Ω t). He may subtract these two signals to get However the result here is a mixture of two equally − 0 a difference function f(t)=PB (t) PB(t), namely, weighted ensembles I and II. For Alice and Bob these     subensembles are internally synchronised. In addition, 1 1 f(t)=sin ∆Ω(t − t ) sin (Ω + Ω0)(t − t ) (8) each of Bob’s sub-ensembles is running in antisynchrony 2 B 2 B with corresponding subensemble of Alice. As a result of

2 having a slowly varying beat envelope of the form e(t)= as other forms of interferometry. sin ∆Ω(t−tB)/2. If ∆Ω is sufficiently small, Bob can now Finally we wish to point out some intriguing short- determine an origin of time in coincidence with Alice’s. comings and avenues for further development of our QCS Indeed if they have arranged in advance that the whole protocol. We have assumed throughout that Alice and 1 π protocol is completed in time T satisfying 2 ∆Ω T<2 Bob are relatively at rest in a common inertial frame. then the slowly varying beat envelope will be in its first But in most applications we would expect some relative quarter-cycle of oscillation and Bob can uniquely deter- motion to have taken place. If Alice and Bob initially mine Alice’s origin t = tA = 0 by a measurement of the share some |ψ0,0i states then after some relative motions beat intensity. we would expect this state to develop a phase (via dif- There are several immediate applications and advan- fering effects on the evolutions of the en- tages of our QCS protocol. For example, in the GPS ergy eigenstates at A and B), becoming |ψ0,0(δ)i with δ satellite constellation, the ability of the space-borne depending on the entire history of the relative motion. atomic clocks to synchronize with a master atomic clock Our protocol would need to keep track of this phase too, on the ground is limited by the fluctuating refractive in- which emerges as an extra time offset between Alice and dex of the atmosphere. The GPS essentially uses the Bob’s determined synchronisation. More fundamentally, classical ES protocol for establishing synchrony, that is, we know of no general treatment of the concept of en- by exchanging light signals (radio waves) with the ground tanglement for particles in (relativistic) motion (e.g. for station. However, from Earth to Space the fluctuating in- solutions of the ) and an associated theory dex of refraction of the atmosphere causes the speed of of local measurements. A treatment that is sufficiently light to vary randomly, limiting one’s ability to estab- encompassing for our purposes, including both internal lish absolute distance and the resultant timing informa- and degrees of freedom, may need the full ma- tion with high accuracy. This index fluctuation error is chinery of a relativistic quantum field theory. It would the current limiting factor of GPS precision [8]. With also be of great interest to consider other basic protocols our QCS prescription, the properties of the atmosphere of quantum information theory (such as teleportation, do not matter – synchrony is established instantaneously dense coding and entanglement purification) in the con- via the quantum-entangled channel. The uncertainties text of a fully relativistic theory allowing general relative induced by sending timing information through the at- motions. But our QCS protocol may have an especially mosphere simply do not arise in the QCS procedure. In exciting status here – the atomic clocks used in the GPS fact, Alice and Bob need not have exact knowledge of are already sufficiently accurate so that both special and their relative locations. So long as they share prior en- general relativistic corrections must be made, and our tanglement, any broadcast classical message is sufficient protocol may provide an especially good starting point to establish synchrony. for the experimental investigation of the “tension” be- There is another fundamental advantage of using QCS tween the fundamentals of quantum measurement theory over ES. Classical ES requires the exchange and timing of and relativity [12–14]. light pulses, but light is actually a quantum field. Hence The current QCS protocol is also limited by the fact the arrival time of a light pulse is itself subject to quan- that it does not specify the method by which the shared tum fluctuations, limiting the accuracy of the ES proto- prior entanglement between Alice and Bob is estab- col. In contrast, our QCS scheme is unaffected by this lished. One possibility is for Alice and Bob to meet kind of noise. at a common location, create an ensemble of N identi- √1 | i | i −| i | i We also mention that the Ramsey two-pulse temporal cal EPR pairs each in the state 2 ( 0 A 1 B 1 A 0 B) interferometer is isomorphic, via the SU(2) algebra, to and then go their separate ways. This method might an optical or matter-wave Mach-Zehnder interferometer appear to be equivalent to that of classical slow clock (MZI). In this case, the qubit states |0i and |1i now rep- transport (CSCT) (in which a pair of (non-entangled) resent the presence or absence of a single-particle Fock clocks are synchronized at a common location at time state incident on either the lower or upper input port t = 0 and transported slowly to remote locations) but of the MZI, respectively [11]. Hence, the QCS proto- there are essential differences. In CSCT, the clocks be- col may be immediately adapted to the task of phase gin ticking in synchrony but subsequently, because they locking a pair of spatially separated optical or atom in- are non-entangled, their drift with respect to one terferometers, if we identify Ωt ↔ kx,wherex is the another necessitating periodic re-synchronization, after interferometric path-length difference, and k =2π/λ is some characteristic time t = tdrift, via the Einstein syn- the particle wavenumber. We may then use essentially chronization (ES) protocol. Unfortunately, ES is an im- the same procedure given above to develop a nonlocal, perfect procedure because of the inability to measure Quantum Interferometer Phase-Locking protocol, allow- the time of arrival of a photon precisely and the uncer- ing us to phase lock two spatially separated interferom- tainty in the refractive index of the medium separating eters, which may reside at unknown relative locations. the clocks (inducing uncertainty in the speed of light be- This result too has obvious applications to VLBI, as well

3 tween the clocks). Thus CSCT followed by periodic re- tanglement. synchronization leads to imperfect synchrony at all times later than some characteristic time t = tdrift. In QCS, however, Alice and Bob’s pre-clock ensembles ACKNOWLEDGMENTS are entangled, allowing them to retain “idling” non-local clocks (assuming they are able to preserve pure entan- We would like to acknowledge interesting and useful glement as they become physically separated) up until discussions with S. L. Braunstein, N. Cerf, H. J. Kimble, the point at which Alice measures. Hence, in QCS, Al- H. Mabuchi, L. Maleki, J. P. Preskill, and N. Yu. The ice can select the moment at which she wants to achieve research described in this paper was carried out at the synchrony with Bob at some time later than t = tdrift. Jet Propulsion Laboratory, California Institute of Tech- Thus QCS, even with slow clock transport, still permits nology. D.S.A., J.P.D. and C.P.W. are supported by a a functionality that is not possible with CSCT. In many contract with the National Aeronautics and Space Ad- respects this is reminiscent of dense coding [15]. Dense ministration. R. J. is supported by the U.K. Engineering coding is a technique for achieving a 2-to-1 compression and Physical Sciences Research Council. of classical bits to qubits at communication time.Like QCS, dense coding requires shared prior entanglement. If one counts the cost of the communications needed to es- tablish that prior entanglement then a critic could claim that dense coding is no better than classical coding. How- ever, functionally, dense coding is useful because it allows [1] A. Einstein, Ann. D. Physik 17, 891 (1905). Einstein. us to defer the moment at which we send a compressed The Swiss Years: Writings, 1900-1909, Vol. 2 (Princeton message. Likewise, QCS allows us to defer the moment University Press, Princeton, NJ, 1989), pp. 140-171. we achieve synchrony. [2] A.S. Eddington, The Mathematical Theory of Relativity, An alternative scheme for establishing the shared prior 2nd ed., Cambridge University Press, , 1924. entanglement would not require Alice and Bob to meet [3] R. Anderson, I. Vetharaniam, and G.E. Stedman, Phys. at all. Instead, it would involve Alice and Bob each re- Rep. 295, 93 (1998). ceiving corresponding members of EPR pairs from some [4] A.K. Ekert, Phys. Rev. Lett. 67, 661 (1991). common source and then using entanglement purification [5] C. Audoin, Metrologia 29, 113 (1992). to distill them into an ensemble of singlet states as re- [6] N.F. Ramsey, Molecular Beams, Oxford University Press, quired by QCS. Unfortunately, there is a hidden assump- Oxford, U.K., (1969), Sec. V.4. 2nd ed. John Wiley and tion of simultaneity in the actions to be performed by Sons, NY, (1970), Sec. 7.6. [7] G.S. Levy, et al., Acta Astronaut. 15, 481 (1987). Alice and Bob in the current entanglement purification | i | i [8] National Research Council Staff, The Global Position- protocols when the states 0 and 1 evolve differently in ing System: A Shared National Asset, National Academy time [16]. This means ultimately that the existing entan- Press, Washington, DC, (1995). glement purification schemes can only create states of the [9] I.I. Sobelman, Atomic Spectra and Radiative Transitions, form |ψ0,0(δ)i,whereδ is unknown, rather than the true 2nd ed. Springer-Verlag, Berlin, (1992), Sec. 6.2.5. singlets (or states with known δ) needed for QCS. We are [10] M.T. Jaekel and S. Reynaud, Phys. Rev. Lett. 76, 2407 currently investigating whether we can use such states (1996). in a modified version of QCS or indeed whether there [11] J.P. Dowling, Phys. Rev. A 57, 4736 (1998). are alternative (asynchronous) entanglement purification [12] A. Suarez and V. Scarani, Phys. Lett. A 232, 9 (1997). protocols that can produce pure singlets. [13] P. Caban and J. Rembielinski, Phys. Rev. A 59, 4187 In conclusion, we have presented a quantum proto- (1999). [14] H. Zbinden, J. Brendel, W. Tittle, and N. Gisin, quant- col for synchronizing spatially separated atomic-clocks, ph/0002031. which uses only shared prior entanglement and a classical [15] C.H. Bennett and S.J. Wiesner, Phys. Rev. Lett., 69, channel. The two synchronizing parties may be at far- (1992) pp.2881-2884 distant and unknown relative locations and the accuracy [16] J.P. Preskill private communication. of the time transfer is not effected by the distance of sep- aration or by noise on the classical channel. Our protocol has direct applications for use in very long baseline inter- ferometry and remote satellite synchronization, and our procedure can also be mapped directly into a new tech- nique for phase locking remote optical or matter-wave interferometers. In future work we intend to generalize these results to incorporate both special and general rel- ativistic effects and consider further the problem of how to establish and maintain the necessary shared prior en-

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