Einstein synchronisation by quantum teleportation

We present a protocol of Einstein’s synchronization of two spatially separated clocks on the basis of quantum teleportation in which the Einstein-Podolsky-Rosen (EPR) state plays a central role. Our scheme does not contain the frame defining problem nor the phase contamination problem for the EPR state which appeared in the quantum clock synchronization (Jozsa et al 2000 Phys. Rev. Lett. 85, 2010).


Introduction
In the seminal paper on the special relativity published in 1905 [1], Einstein considered a thought experiment for two spatially separated clocks to synchronize by exchanging light signals back and forth between them. The first light signal starts from one of the observers Alice at the time t A0 as her clock indicates, reaches the other observer Bob at the time t B1 measured by his clock and then the information of the numerical value t B1 is sent back to Alice by the light signal, who receives the information of t B1 at the time t A2 by her clock. By this communication Alice obtains all of the necessary data t t ,  Einstein then demanded that the clock synchronization above should hold also in an arbitrary inertial frame with a relative velocity from the rest frame assuming the frame independence of the light velocity and arrived at the Lorentz transformation between the coordinates of the inertial and the rest frames. He chose the criterion (1) on the basis of the equality of the light velocity in both ways.
Nowadays, the Einstein clock synchronization is also practically important to implement the GPS system. We notice that, although Einstein did not explicitly mention, Bob has to send back the information of t B1 in some way to convince Alice that her clock is synchronized to Bob's by checking the criterion (1). Some people (e.g., [2,3])use the terminology 'Einstein synchronization' to broadly mean the date adjustment of the two spatially separated clocks by exchanging signals even without the criterion (1). In this paper the meaning of Einstein synchronization is closer to the original one in the sense that the criterion (1) is required as the validation. We take the criterion (1) as an operational one leaving aside its physical meaning. We stress that checking that the two classical clocks are synchronized is conceptually different from creating the two synchronized clocks, or quantum synchronization if they were quantum clocks. For the latter we will quickly review the recent development.
Initially Alice and Bob share an EPR state: which remains the same up to an overal phase factor in the time evolution. This choice is important because the protocol is time-blind i.e., insensitive to the time when it starts. The singlet state | ñ EPR can also be written as is the eigenstate of the observable σ x with the eigenvalue ±1. Suppose Alice measures her σ x to obtain |ñ A . Then the state of Bob jumps to |ñ B , correspondingly 1 evolves in time duration t as, Alice sends a classical message which asks Bob to do the Hadamard transformation H B . After that Bob will have a clock state: when Bob's clock indicates that the time is t=t 1 . Alice correspondingly has her clock state: Alice's state is identical to Bob's state if she flips her bit.
To implement the creation of the clock states of Alice and Bob, Alice sends the classical information±of the outcome when she has performed the first measurement of σ x to Bob, then Bob can see the time indicated by his own clock. Indeed the QCS proposal is not the check of the synchronization of the two clocks but rather an automatic synchronization of the two quantum clocks.
However, unfortunately there are at least two defects in the original QCS protocol.
(i) As remarked in their own paper [2] and briefly pointed out in the above, the spatial axis x at Bob cannot be definitely arranged relative to the x-axis of Alice so that the Hadamard transformation of Bob in the protocol is ambiguous and so is the resultant state by Alice's observation 2 . Figure 1. The Einstein synchronization An observer A (Alice) needs to be told the arriving time t B1 measured by an observer B (Bob) to confirm that the spatially separated clocks are synchronized by checking the criterion (1). 1 At this stage we have to be careful about the coordinate frame of Bob. The outcome state |ñ B may not be the x-component of B's state, because the 'x-direction' can be any direction perpendicular to the z-axis. As we shall see later, this frame arbitrariness becomes the drawback of the original QCS [2]. 2 The authors of [2] proposed a modification of the protocol using two different two-level systems with different excitation energies. This might work in practice but the complication would be a severe shortcoming as a fundamental protocol for an advanced theory, say, relativistic quantum information.
(ii) The QCS has a drawback similar to Eddinton's synchronization [6]. Namely, the shared entangled state is contaminated by an uncontrollable phase δ in general, because the environments of spatially separated Alice and Bob are different in general and therefore have different interactions [3]. This extra phase factor invalidates its isotropy, i.e.,the equation (3) and therefore the break-down of whole scheme. Recently, Ilo-Okeke et al [5] proposed a modification of QCS to resolve the frame ambiguity problem via entanglement purification but not the phase contamination problem. The motivation of the present work is to propose a protocol for the validation of synchronization of classical clocks in the same sense of Einstein's original paper but a transmission of the time information from Bob to Alice's is done by using quantum teleportation, which does not have the two drawbacks of QCS mentioned above in spite of the use of the EPR pair state. The key to the resolution is the Bell measurements.

The protocol
Here we present a precise protocol of clock synchronization by the quantum teleportation.
(I) Prepare the initial state where the | ñ EPR is the state of EPR defined as before in equation (2) 3 and Ψ 4 are the Bell states defined by A 0 2 knowing the dates t A0 and t A2 by her own clock. Then her clock state becomes Here we have tentatively assumed that the Hadamard transformation H A of Alice is the same as the Hadamard transformation H B of Bob for the simplicity of explanation. Later we will generalise it taking the frame ambiguity into account.
(VI) Finally Alice measures her qubit. The probability to obtain | ñ 0 A by the bit measurement under the condition that the Bell measurement resulted in |Y ñ 1 is which is equal to unity if the synchronization condition is met. Namely, if Alice gets the state | ñ 0 A with a high probability she can convince herself that her clock is synchronized with Bob's. If the result of the Bell measurement was different from the |Y ñ 1 , Alice can arrange different unitary transformation to check the synchronization criterion by the bit measurement | ñ 0 A and then check similarly as before.

A solution to the phase contamination and the frame adjustment problems
The different environments of Alice and Bob induces the uncontrollable phase δ to the EPR state to give Note the sign difference in front of δ from the case of P 1 . For |Y ñ 4 , the result P 4 is the same as P 3 . Eliminating the unwanted δ in Eqs.(23, 24), we obtain an expression for q q - The right hand side is experimentally accessible. If its value is close to unity, the synchronization is validated. Therefore, the phase contamination problem has been overcome by the combination of the more than two Bell measurements.
Let us turn to the frame adjustment problem. Note that the uncontrollable phases ζ and δ appear only in the combination of δ−ζ. Therefore, the formula (25) remains to work for the check of the Einstein synchronisation. This is not a coincidence, because the selective spin rotation of Alice only effectively induces the phase contamination δ. We admit that the Bell measurement is idealized in the sense that it can be done instantaneously. More practically, it needs many repeated measurements, during which the uncontrollable phase δ and ζ might change. We have not estimated the inaccuracy of the synchronization due to this effect.

Summary
To summarize, we have presented a quantum protocol to transmit the time information by teleportation for the Einstein synchronization of two distant clocks, which is free from the frame adjustment and the phase contamination problems. However,there remain points to improve the QCS further. We add a comment on the Eddington synchronisation [6],an alternative of the Einstein synchronization. A quantum version of the Eddinton synchronization in which a clock state is physically exchanged has also been proposed [7].