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Quantum Atomic Clock Synchronization Based on Shared Prior Entanglement

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Quantum Atomic Clock Synchronization Based on Shared Prior Entanglement Quantum Atomic Clock 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 time 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 quantum entanglement, periods of oscillation corresponding to the hyperfine (ra- and classical communications, to establish a synchronized 133 pair of atomic clocks. 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 Theory of Relativity, 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 quantum information 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 quantum circuit 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 <E1 respectively. We introduce the dual basis | i √1 | i | i | i √1 | i−| i Protocol (CSP) is that of Eddington, namely, Slow Clock pos = 2 ( 0 + 1 )and neg = 2 ( 0 1 ) and write 1 − Transport (SCT). In the SCT scheme, the two clocks A Ω= h¯ (E1 E0) which will define our unit of time. The and B are first synchronized locally, and then they are Hadamard transform (or π/2 pulse) is defined by the op- transported adiabatically (infinitesimally slowly) to their eration |0i→|posi and |1i→|negi. The dual basis final separate locations in the common inertial frame evolves in time (up to an overall unobservable phase) as [2,3]. In this paper we propose a third CSP that uti- 1 − Ω 2 Ω 2 lizes the resource of shared prior entanglement between |pos(t)i = √ e i t= |0i + ei t= |1i 2 (1) | i √1 −iΩt=2| i− iΩt=2| i the two synchronizing parties. neg(t) = 2 e 0 e 1 Our proposed method of Quantum Clock Synchro- nization (QCS) has features in common with Ekert’s To construct a qubit atomic clock via the Ramsey method entanglement-based quantum key-distribution protocol [6], we apply a Hadamard transform to an ensemble of [4] in which Alice and Bob initially share only prior- N identical qubits all in state |0i at some time t =0. entangled qubit pairs. The key does not exist initially but This generates an ensemble of |posi states which then is created from the ensemble of entangled pairs through a evolve as in eq. (1). After a time t, we apply a second series of measurements and classical messages. Similarly Hadamard transform to the evolving ensemble of qubits. for our QCS protocol below, no actual clocks exist ini- This gives a final state for each qubit: tially but rather only “entangled clocks” in a global state which does not evolve in time. The synchronized clocks | pos(t)i =cos(Ωt=2) |0i−i sin (Ωt=2) |1i (2) are then extracted via the measurements and classical communications performed by Alice and Bob. In this way (For later purposes we also note that if we had started | i our QCS scheme establishes synchrony without having to with an ensemble of neg states we would finally get the transport timing information between Alice and Bob. In states contrast, for the classical ES and SCT synchronization | (t)i = −i sin (Ωt=2) |0i +cos(Ωt=2) |1i (3) schemes, synchrony information must transmitted from neg Alice to Bob over some classical channel, which can limit which is just π/2 out of phase with | pos(t)i). At the accuracy of the synchronization. this point we simultaneously measure each qubit in the We first review how an atomic clock operates, in the {|0i; |1i} basis. The probabilities of obtaining |0i or |1i language of quantum information theory. An atomic are given by clock consists of an ensemble of identical two-level sys- tems (qubits) whose temporal evolution rate is taken as 1 1 P0 = (1 + cos (Ωt)) and P1 = (1 − cos (Ωt)) (4) the time standard. For example, in the International 2 2 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 0;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 | neg(t)i and pos(t)i in eqs. (2) and (3). The 1 state is a “dark state” that does not evolve in time pro- corresponding density matrix 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)| 0;0i = ing information. For Bob to extract a clock, a classical (det U)| 0;0i so that | 0;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, | 0;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 | 0 0i as an entangled pair of ; “ticking” in unison. pre-clocks. Since | 0 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 | 0 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 0;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.
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