Quantum Clock Synchronization Under Decoherence Effect

In this paper, we study a quantum clock synchronization for a three observer locating system, using entangled qubits and discuss the eﬀect of the environmental noise on the phase of the system using decoherence theory. We will show that the error on the ﬁnal result of the clock synchronization can be less than the current classical synchronizations, under suitable circumstances. Finally, with an appropriate modelling of the environment we can correct the error of the synchronization for a more accurate locating.


I. INTRODUCTION
The theory of quantum mechanics, that has revolutionized our understanding of the nature, is now fundamentally changes the development of new technologies inspired by its own basic principles.In this regard, so many interests have risen to develop quantum technologies in communication.The goal in quantum communication is to outperform its classical counterpart, and also brings new functions that previously were totally undefined in classical regime.
Many technologies in quantum communication are being developed and also, some are in test.For example, quantum key distribution [1] allows two parties to share a common key with provable unconditional security.This level of secrecy is completely unachievable in classical communication.Also, quantum communication will be the backbone of the quantum network and quantum internet [2].A network of devices that will revolutionize how we perform our communications and computational tasks [3].
Recently several experiments have been done demonstrating the feasibility and implementation of quantum communication technologies even with satellite-toground links [4].For example, the Chinese Academy of Science satellite Micius, has applied fundamental tests in space, quantum communication protocols and also satellite-to-ground quantum key distribution.The development of such quantum satellite technologies prepare a situation that quantum communication can be developed for a broader use in our life [5,14].
One of the main issues in developing quantum communication technologies, is the clock synchronization between bases [7].In classical mechanics, when special relativity is taken into account, two basic methods are available to synchronize clocks.Einstein's synchronization [8] and Eddington's slow clock transport [9].However, in spite of the superb stabilities of the next generation of atomic clocks [10], the question of how best to synchronize clocks with high precision is one that frequently addressed, such as: time transfer laser links for the Einstein protocol [11], and quantum adaptations of Eddington's protocol [12].Accordingly, a third method of clock synchronization proposed by Josza and co-workers [13],based on quantum entanglement.In this method an entangled quantum state shares between the spatial locations and uses the non-local features of quantum mechanics for clock synchronization.In all classical methods, light or matter is exchanged between locations.Also, using entanglement the two party protocol can be extended to multi-parties and several experiment verification have been reported [14,15].Despite the reasonable results of clock synchronization wtih quantum entanglement, in real cases the effect of the environmental noise and the dissipation of the quantum states is causing problems.The environment effect on quantum states due to decoherence theory.With more distance between the locations, decoherence acts more effectively and quantum properties decays faster and clock synchronization shows more error.In long distances and high environmental noise, quantum clock synchronization can not be reliable at all.
In this paper, we review the modern quantum clock synchronization model and describe the decoherence process in spin-1/2 model.Furthermore, we calculate the dissipation effect of the decoherence on our clock synchronization model through the master equation.At last we numerically solve the master equation and study the effect of the environmental noise on the system in different environmental conditions.This paper organized as follows: in section II we define the model of clock synchronization with quantum entanglement.In section III, we describe the decoherence theory and calculate the master equation.In section IV, we numerically solve the master equation and discuss the results on a practical case.Finally, in last section we conclude the paper.

II. CLOCK SYNCHRONIZATION MODEL
In this section we simply describe the quantum entanglement clock synchronization protocol between two locations [16], where Alice and Bob want to synchronize their clocks.Alice and Bob share the well-known entangled Bell-state with each other: We assume that the state prepared at Alice location and one of the particles is sent to Bob.Though, the definitions of the states are respect to Alice's basis Where, Indexes under the states shows which particle is sent to Bob and which one remain in Alice's location.
The appropriate basis to view the state in Alice and Bob's respective local bases are where for simplicity we chose the irrelevant global phase θ Here, by performing an entanglement purification circuit [16], on both sides we can reach the desired entangled state shared between Alice and Bob: As our time evolution operator (U) is diagonal in ♣0⟩, ♣1⟩ basis, our state is stationary.By applying a Hadamard gate on (4), we have in which we have For a system in ♣pos⟩ state, by measuring the system in σ 1 basis we will have probabilities below to find the system in ♣0⟩ and ♣1⟩ states respectively, where, After sharing the state (5), at t = 0 Alice simultaneously measures all particles in σ 1 basis (♣pos⟩ and ♣neg⟩).In this situation, both A and B particles collapse to one of the states below: The probability of collapsing to each one of these states is 1/2.Alice and Bob's clocks start to evolve according to equations (8).Both set of particles started evolving at t = 0.
Here, we have to subsets of states I, II that Alice has full information on them regarding her measurements, but Bob has no knowledge about the subsets.In such situation, Bob cannot gain any knowledge about time by random measurements.Therefore, Alice needs to send Bob a classical message.For example, Alice chooses I subset of qubits.Now Alice can inform Bob with the subsets label.Now, Bob can choose his particles among the I and II subsets.With choosing II subset, Bob will have particles in a same phase as Alice's.Then, Bob will measures his particles in σ 1 basis and and observes P 0 (t) oscillations.In this case, Alice and Bob are synchronized.
However, the presence of the environmental noise change the state's phase and prevents Bob from synchronizing his clock with Alice's.In the next section we describe the decoherence model for calculating the environmental noise via the quantum Brownian motion master equation.

III. LINDBLAD MASTER EQUATION OF QUANTUM BROWNIAN MOTION
In this section, we are employing the model of the quantum Brownian Motion provided by Maniscalco et.al. [17], Who applied Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation [18].In this systemnoise (environment) model the total Hamiltonian is Ĥ = Ĥs + Ĥε + Ĥint , (13) where in this equation Ĥs , Ĥε and Ĥint are system, environment and system-environment interaction, respectively.We consider the central system as a two level harmonic oscillator (equivalent to spin-1/2 system), and a thermal bath of electromagnetic harmonic oscillators as the environment.Considering the system and environment as explained above, we can write the total Hamiltonian as: where Ω (ω) is the system (the environment) frequency, and P and X (p and q) are momentum and position operators of the system (the environment), respectively.For simplicity we removed mass from the equations and considered ℏ = 1.The interaction Hamiltonian defines as where the position coordinate of the central system X linearly couples to the position qi of the i-th thermal bath oscillator with the coupling strength c i .Here, Ê is the environment operator.
By defining ρ as the total density matrix, the following assumptions are in order: 1.The system and the environment are supposed to be uncorrelated at t=0; i.e., ρ(0) = ρs (0) ⊗ ρε (0) where ρs and ρε are the system and the environment density matrices, respectively.2. The environment is stationary; i.e., [ Ĥε , ρε (0)] = 0, and also the expectation value of Ê is zero; i.e., Tr E [ρ E (0)] = 0. 3. The system-environment coupling is weak and we neglect the effect of the oscillator frequency renormalization since it is negligible under the weak coupling.Therefor, Hence, by averaging over rapidly oscillating terms, one gets the following secular approximated master equation [17] where â = ( X + i P )/ √ 2 and â † = ( X − i P )/ √ 2 are the annihilation and creation operators, respectively.he time-dependent coefficients γ(t) and ∆(t) represent the classical damping and diffusive terms, define as where and are noise and dissipation kernels, respectively.If the quantity ∆ ± γ remains positive in all times, The master equation ( 16) will be in the Lindblad form [19].For environment frequency distribution we consider the case of an Ohmic spectral density for the bath with Lorentz-Drude cutoff [20] J(ω) = 2γ 0 ω π where Λ is the cut-off frequency and the dimensionless factor γ 0 describes the system-environment effective coupling strength.Thus, for the expressions of noise and dissipation kernels, we have )cos(ωτ ) ( 22) where k is the Boltzmann constant and T denotes the temperature.In long-time limit assumption the coefficients ∆(t) and γ(t) approach their stationary values.So, their expressions up to the second order of coupling constant are with r = Λ/Ω, where the master equation ( 16) becomes similar to the well-known Markovian master equation of damped harmonic oscillator in which Γ = γ 2 0 Ωr 2 /(1 + r 2 ) and n = (e Ω/kT − 1) −1 .The positiveness of the coefficients ∆ ± γ assures us that the master equation ( 16) is in the Lindblad form.
Regarding GKSL master equation ( 26), for a dissipative spin-1/2 master equation we can write where independent of time Hamiltonian reads as in which c is the oscillation (tunneling) rate.Also, L is Lindblad super operator and reads as where, S + , S − are ladder operators and are similar and act similarly to creation and annihilation operators: Here we write the equations in interaction picture.For density matrix and tunneling Hamiltonian (H 1 ) in the interaction picture we have ρ D (t) = e itH0 ρ s (t)e −itH0 (33) For master equation (31) in the interaction picture we have Applying a desired density matrix and writing equations in matrix form, we reach the set of equations below where, in the equations above we have In the following section, we solve the equation numerically and discuss the effect of the noise on the clock synchronization error and will compare it to a practical classical clock synchronization method.

IV. RESULTS AND DISCUSSION
After the entangled states are sent to Bob's location, evolving with time, environmental noise acts on the state through decoherence theory and effect the states in two way.First, noise make the system to dephase through time and after a specific time known as decoherence time, that depends on the environment properties and the strength of the environment and system interaction, the system lose its quantum properties.Second the dissipation effect of decoherence make the system lose energy over time.Both of the effects result in change in state phase and an error in clock synchronization.Let us have a look on the order of the magnitude of the decoherence time.By using the thermal de Broglie wavelength λ dB = 1/ √ 2mk B T , we can define a corresponding decoherence time as where ∆X is the dispersion in position space and we have In the case of ours the decoherence time can be small,but in suitable conditions, like a desired temperature and isolating system to have a weak interaction with environment can rise and be acceptable in our case.As it is relevant in equation ( 42), increasing the frequency of the photons of the chosen electromagnetic wave, can significantly increase the decoherence time.So waves with high frequencies have higher decoherence time respectively.To show how the decoherence affect the clock synchronization, after solving the equations (37), we sketch the changes on the probability of Bob, finding the system in states ♣0⟩ and ♣1⟩ versus the time that system interacts with environment in variety of oscillation (tunneling) rate.Oscillation rates in the system heavily depends on Temperature and central system frequency After Bob receiving the particles and chose his subset to measure, After measuring an acceptable number of particles he can define the probabilities P 0 (t) and P 1 (t) and synchronize his clocks.However, as we showed in Fig. 1, system interacting with environment, decohere the states and results is probabilities to decay and applies error to Bob's clock.In this model the oscillating rates in the system Hamiltonian have main role on the environmental noise.As we can see in Fig 1, with decreasing the oscillating rates, the error resulting from environmental noise can be negligible.
Considering a practical case, in which, Alice and Bob with the distance equal to 100km between them, want to synchronize their clocks.In a classical manner, according to Sathyamoorty and co-workers [21], showed that in perfect scenario, the classical clock synchronization would have about 0.3% error in clock synchronization.However, in the case of ours, in room temperature with central system frequency equal to Ω = 1 × 10 12 s −1 , the clock synchronization error is about 0.1% that can easily be improved by controlling the temperature and isolate the system, weaken the system-environment interaction and lower the oscillation rates.

V. CONCLUSION
In this paper we addressed the problem of environmental noise on quantum clock synchronization.First we discussed a quantum model for clock synchronization using quantum entangled states.Then, we describe a model for decoherence effect on the system.
In this regard, we used a Lindblad quantum Brownian motion master equation and discussed both dephasing and dissipation effect of the decoherence on the system.Also we considered the tunneling effect in this model, and studied that the error due to the environmental noise can depends on many parameters such as temperature, the system electromagnetic wave frequency, the strength of the system and environment coupling and the oscillation rates.Finally, we showed that by preparing a desired condition, the error in quantum model of clock synchronization can be far negligible in compare to classical methods.
FIG. 1.(a) The changes in P1(t) and (b) in P2(t) with T = 300K and Ω = 1 × 10 12 s −1 , in variety of oscillation rates.The probabilities deviation from value 0.5 represents the error resulted from environmental noise.In lower oscillation rates, the effect of decoherence and environmental noise become negligible