Entangled Quantum Clocks for Measuring Proper-Time Difference

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provided by CERN Document Server ? WonYoung Hwang∗, Doyeol (David) Ahn†, and Sung Woo Hwang Institute of Quantum Information Processing and Systems, University of Seoul, 90 Jeonnong, Tongdaemoon, Seoul 130-743, Korea

Yeong Deok Han Division of Semiconductor, Electricity, and Automobile Engineering, Woosuk University, 490 Hujeong, Samrye, Wanju, Cheonbuk 565-701, Korea

how the proposed scheme is more precise than a scheme We report that entangled pair of non-degenerate quan- where two separate quantum clocks are employed. Then tum bits (clocks) can be used as a specialized detector for we consider using the proposed scheme in precision test precisely measuring difference of proper-times that each con- of relativistic time-dilation effects. stituent quantum bit experiences. We describe how the pro- Let us assume that we have entangled pair of qubits in posed scheme is more precise than a scheme of two-separate- the state quantum-clocks. We consider possibilities that the proposed scheme can be used in precision test of the relativity theory. Ψ− = 0 1 1 0 , (0.1) | i | iA| iB −| iA| iB 03.67.-a, 06.30.Ft, 04.80.Cc where A and B respectively corresponds to each quantum clock whose proper-time difference will be compared and normalization factor is omitted as would be through- Quantum clock synchronization is a new possible ap- out this paper. We also assume that the two mutually plication of quantum entanglement: two remotely sepa- orthogonal state of qubit 0 and 1 (α = A, B) are rated participants Alice and Bob can synchronize their | iα | iα respectively eigenstates of its Hamiltonian Hα with eigen- clocks, provided that they share pairs of quantum bits 0 1 values Eα and Eα. Since the overall phase factor can be (qubits) in a maximally entangled state with a known neglected, time evolution during time t of each qubit is relative phase [1]. However, as noted in the original [1] in general given by the unitary operation and other [2]- [5] works, the problem of the scheme is that it is difficult to prepare the needed entangled states Uα(t) 0 α = 0 α without synchronized clocks between Alice and Bob. The | i | iiE t U (t) 1 = e− α 1 , (0.2) task of transporting and preserving one part of entangled α | iα | iα 1 0 qubits is physically equivalent to that of doing so for a where Eα = Eα Eα andh ¯ is set to be one. When two non-entangled qubit (separate quantum clock) [2], due clocks make different− space-time trajectories, the time for to linearity of quantum mechanics. Thus in this case each clock is given by it’s own proper-time. First let us there is no improvement over classical technique where consider a simple case EA = EB = E. (We will later non-entangled quantum clocks should be very carefully consider a general case where Eα’s are time-dependent transported and preserved. If Alice and Bob can pre- and thus are not the same.) After proper-times tA and pare the required entangled qubits by the purification tB has respectively elapsed for A and B qubits, the initial protocol [7], it would be an improvement since require- qubits in Eq. (0.1) becomes ment of the careful manipulations of qubits is relaxed while qubits are being delivered and stored. However, UA(tA)UB(tB) Ψ− iEt | i it does not seem to be possible for Alice and Bob to = U (t )(e− B 0 1 1 0 ), A A | iA| iB −| iA| iB do so without synchronized clocks between them [1,4]. iEtB iEtA = e− 0 A 1 B e− 1 A 0 B. (0.3) Some authors have considered methods for overcoming | i | i − | i | i this problem [3,5]. Others gave Lorentz-invariant look In order that the state remains in the initial one, as re- at the scheme [4] and discussed the scheme in connec- quired in the quantum clock synchronization [1], a relation with quantum error correction [5]. We also note tion that tA = tB should be satisfied. (We do not con- that an efficient quantum clock-transport scheme that sider the case where the terms differ by 2nπ, n is integer.) enable clock synchronization in a different way has been This condition can be fulfilled by slow transportation of proposed [6]. In this paper, we propose another applica- qubits. In this case the relativistic effect is something to tion of the entangled pairs of non-degenerate qubits. The be suppressed by careful manipulation (slow transporta- resulting scheme is a specialized detector that precisely tion) of qubits. Oppositely, we can utilize the property measures difference of proper-times that each qubit expe- in Eq. (0.3) of the state in measuring the relativistic riences. We consider the case where acceleration’s effects time-dilation effect. In the proposed scheme we initially on qubit’s time evolution is non-negligible. We suggest a prepare qubits pairs in the state ψ− at one site. Then solution and rather a utilization of this case. We describe we make each party (labeled by A| oriB) depart and fol- low its own space-time trajectory and then gather them

1 again. Then we perform some (collective) measurement [8]. Measure the phase difference ∆φ and calculate each on the qubits and get information about the proper-time party’s proper-time using formula of special relativity. difference ∆t = tA tB, as shown in the following. Ne- The difference between them is the acceleration effect. glecting overall phase− factor, we can rewrite Eq. (0.3) Now let us describe how the proposed scheme is more as, precise than a method where proper-times of two separate quantum clocks that have traveled through different iE∆t UA(tA)UB(tB ) Ψ− = 0 A 1 B e− 1 A 0 B. (0.4) space-time routes are simply compared to estimate the | i | i | i − | i | i difference between them. It is clear that accuracy of the As we see, the proper-time difference ∆t contribute to two-separate-clocks scheme is limited by certain uncer- relative phase in the qubit pair. Thus we can determine tainty of each constituent separate clock. That is, the ∆t by measuring the relative phase of the qubit pair. In proper-time difference cannot be measured more accu- other words, the difference ∆t of the proper-times that rately than the uncertainty of each clock. One origin of each qubit experiences since they departed, is accumulat- the uncertainty can be explained as follows. Accuracy edly recorded in the relative phase of the non-degenerate of a quantum clock is roughly a product of the period qubit pair in Eq. (0.4), which can be read out by collec- T of a phase rotation and uncertainty in phase measure- tively measuring the qubits at one site. ment δφ,thatis, Tδφ. One may to try to decrease the Let us now consider the general case. Eα may be term Tδφ by decreasing∼ T or by increasing the speed of time-dependent due to small perturbation by interaction phase rotation (or Eα). However, in this case δφ become with environments or acceleration that each qubit suffers larger instead, because the faster it rotates the more dif- during the round trip in space-time. As we did above, ficult it becomes to accurately measure a phase φ:in here we assume that the environmental effects can be re- real measurement we would inevitably measure an av- moved by shielding the qubits from environment, which erage of the phase during a finite time interval δt,not is possible at least in principle. It is not well known that of an instance. Thus we have δφ δt/T and then yet how quantum clocks are perturbed by acceleration the uncertainty Tδφ is given by δt. In∼ contrast, in the except for the fact that the effect would be very small proposed scheme the relative phase in Eq. (0.4) to be [10]. The condition above that EA = EB = E assumes measured does not rotate when the qubits are being mea- that acceleration effect can be removed by some meth- sured. Thus the above reciprocal relation between T and ods, e.g., careful choice of the system to be used as quan- δφ is no longer valid in the proposed scheme. Namely, we tum clocks or simply making the acceleration very small. can increase speed of phase rotation of each qubit thus Here we consider the case where the acceleration effects decreasing T in the proposed scheme, without increasing on the qubit’s time evolution is non-negligible (while en- the uncertainty of phase δφ to be measured. Therefore we vironment effect is negligible). Here the phase difference can determine the proper-time difference ∆t more accu- tA tB ∆φ = R E (t0)dt0 R E (t)dt that we would measure rately in the proposed scheme than in the two-separate- 0 A − 0 B in the proposed scheme is combined results of relativistic clocks method. Let us consider quantum clocks whose and acceleration effects, which cannot be discriminated phase rotation can be turned on or off, as we like by from each other in the measurement result. Nevertheless, some operation. For example, spin precession of particles we can still utilize the proposed scheme for measuring by applied magnetic field can used as quantum clocks. proper-time difference by making each quantum clock to Here we can make the clocks turned on (off) by applying suffer the same acceleration effect while following differ- nonzero (zero) magnetic field. In this case accuracy of ent path in space-time. For example, we can consider the the two-separate-clocks scheme would not be limited by following case almost similar to the twin-paradox exper- the reciprocal relation since we may turn off both clocks iment [8]. One (the other) party makes a short (long) when we are measuring them. However, in this case the trip. However their accelerating sections in the space- magnetic field instead must be precisely controlled to the time map are the same with each other. In this case level of required accuracy of the scheme, which would be the acceleration effect cancels with each other and thus a much more difficult task than attaining the required we can estimate pure relativistic time-dilation effect from accuracy with two naturally given energy eigenstates. the measurement result. Let us consider another exam- The precision that the entangled quantum clocks ple, gravitational time-dilation [9,10] where we can also scheme affords is estimated to be order of period T ,as- make the acceleration effects to cancel with each other: suming δφ 1. The period T of hyperfine transition ∼ 10 first prepare the two parties of entangled qubits. Then is of order of 10− second. However, in the proposed lift both of them to a higher place and then bring down scheme a system with more rapidly rotating phase can one party to the original place. After waiting for a long be employed since the phase become stationary when it time, bring down the other party with the same magni- is being measured, as noted above. By choosing some tude of acceleration and velocity as the first one. On the qubits whose energy difference Eα is of an order of one 14 other hand, we can make use of the proposed scheme for electron volt, we can obtain T 10− second. How- ∼ measuring the acceleration effect. Let one party to be at ever, the greater energy difference Eα becomes, the more rest and another party to makes a trip with some acceler- probable the higher energy state spontaneously makes a ation, as we do in the original twin-paradox experiment transition to the lower one in general. This problem may

2 be avoided by adopting some metastable states as qubits, [4] U. Yurtsever and J.P. Dowling, quant-ph/0010097. although this problem would limit the accuracy of the [5] J. Preskill, quant-ph/0010098. proposed scheme. It is interesting to note that at least in [6] I.L. Chuang, Phys. Rev. Lett. 85, 2006 (2000). principle the proposed scheme may used to detect time- [7] C.H. Bennett, D.P. Divincenzo, J.A. Smolin, and W.K. dilation effect that gravitational wave cause, in a setting Wootters, Phys. Rev. A 54, 3824 (1996). similar to a non-mechanical gravitational wave detector [8] E.F. Taylor and J.A. Wheeler, Spacetime Physics (W.H. proposed by Braginsky and Menskii [11,10]: fix two com- Freeman and Company, San Francisco and London, ponent quantum clocks on edges of a disk, making an 1966). angle π/2 with the origin of the disk. The disk is free- [9] R.V. Pound and G.A. Rebka, Phys. Rev. Lett. 4, 337 falling and constantly rotating in phase with a frequency (1960). component of gravitational wave and the axis of rotation [10] C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravita- tion, (W.H. Freeman and Company, New York, 1973). is pointing to the source of the wave. Then one clock’s [11] V.B. Braginsky and M.B. Menskii, Sov. Phys.-JETP time is constantly dilated than the other one’s, due to Lett. 13, 417 (1971). gravitational field of the wave. However, it still seems to be a formidable task to detect gravitational wave using the proposed scheme. In conclusion, we reported that entangled pair of non- degenerate qubits (clocks) can be used as a specialized detector for precisely measuring difference of proper-times that each constituent qubit experiences. Acceleration’s effects on qubit’s time evolution may be non-negligible. In this case we considered some experiments where the acceleration effect cancels with each other. Oppositely, we also considered making use of the proposed scheme in detecting the acceleration effect. Then we described how the proposed scheme is more precise than a scheme where readings of two separate quantum clocks are compared. The proposed scheme can be used in precision test of relativistic time dilation effects-the twin paradox effect [8], gravitation time-dilation [9,10], and possibly time-dilation due to gravitational wave.

ACKNOWLEDGMENTS

This work was supported by the Korean Ministry of Science and Technology through the Creative Research Initiatives Program under Contract No. 98-CR-01-01- A-20. We are very grateful to Dr. LEE Jinhyoung for valuable comments.

∗ [email protected] † Also with Department of Electrical Engineering, University of Seoul, Seoul 130-743, Korea; [email protected] ? Permanent address: Department of Electronics Engineer- ing, Korea University, 5-1 Anam, Sungbook-ku, Seoul 136-701, Korea. [1] R. Jozsa, D.S. Abrams, J.P. Dowling, and C.P. Williams, Phys.Rev.Lett.85, 2010 (2000). [2] E.A. Burt, C.R. Ekstrom, and T.B. Swanson, quant- ph/0007030 (available at http://xxx.lanl.gov). [3] M.S. Shahriar, quant-ph/0010007.