Lorentz-invariant look at quantum clock-synchronization protocols based on distributed entanglement

Abstract

Recent work has raised the possibility that quantum information theory techniques can be used to synchronize atomic clocks nonlocally. One of the proposed algorithms for quantum clock synchronization (QCS) requires distribution of entangled pure singlets to the synchronizing parties. Such remote entanglement distribution normally creates a relative phase error in the distributed singlet state which then needs to be purified asynchronously. We present a fully relativistic analysis of the QCS protocol which shows that asynchronous entanglement purification is not possible, and, therefore, that the proposed QCS scheme remains incomplete. We discuss possible directions of research in quantum information theory which may lead to a complete, working QCS protocol. Clock synchronization with shared singletsSuppose a supply of identical but distinguishable twostate systems (e.g. atoms) are available whose betweenstate transitions can be manipulated (e.g. by laser pulses). Let |1 and |0 denote, respectively, the excited and ground states (which span the internal Hilbert space H) of the prototype two-state system, and let the energy difference between the two states be Ω (we will use units in which = c = 1 throughout this letter). Without loss of generality, we can assumewhere H 0 denotes the internal Hamiltonian. Suppose pairs of these two-state systems are distributed to two spatially-separated observers Alice and Bob. The Hilbert space of each pair can be written as H A ⊗ H B , where ⊗ denotes tensor product between vector spaces. A ("pure") singlet is the specific entangled quantum state in this product Hilbert space given by[in what follows, we will omit tensor-product signs in expressions of the kind Eq. (2) unless required for clarity]. Two important properties of the singlet state Ψ are: (i) it is a "dark" state (invariant up to a multiplicative phase factor) under the time evolution U t ≡ exp(itH 0 ), i.e. (U t ⊗ U t )Ψ = e iφ Ψ where e iφ is an overall phase, and (ii) it is similarly invariant under all unitary transformations of the form U ⊗ U , where U is any arbitrary unitary map on H (not necessarily equal to U t ). Both properties are needed for the Quantum Clock Synchronization (QCS) protocol of Jozsa et. al.[1], which assumes a supply of such pure singlet states shared as a resource between the synchronizing parties Alice and Bob. Specifically, consider the unitary (Hadamard) transformation (π/2 -pulse followed by the spin operator σ z ) on H given byUnlike the states |0 and |1 , which are dark under time evolution (they only pick up an overall phase under U t ), the states |+ and |− are "clock states" (in other words, they accumulate an observable relative phase under U t ) because of the energy difference Ω as specified in Eq. (1). Such states can be used to "drive" precision clocks in the following way: Start, for example, with an ensemble of atoms in the state |+ produced by an initial Hadamard pulse at time t 0 , and apply a second Hadamard pulse at a later time t 0 + T . This leads to a final state ...

Review on quantum clock synchronization schemes

“…In the QCS, the two synchronizing parties may be at far-distant and unknown relative locations and the accuracy of the time synchronization is not affected by the distance of separation or by noise on the classical channel. A proof-of-principle experiment on distant clock synchronization has been reported [11] . This experiment not only achieved of picosecond resolution at 3 km distance, but also demonstrated a concept for high-accuracy nonlocal timing and positioning based on the quantum feature of entangled states.…”

Entanglement from longitudinal and scalar photons

“…The role of special relativity and covariance in entanglement and quantum information has been discussed in a number of earlier papers [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25], none of which are based on a covariant description of the photons in the Lorentz gauge. A covariant polarization for the photons has often been used in the Coulomb gauge, which provides a correct description of entangled states and quantum information under Lorentz transformations.…”

Clock synchronization using maximal multipartite entanglement

“…Here, we consider a type of clock synchronization protocol where entangled qubits are distributed to various parties holding the individual clocks [5][6][7][8][9][10][11][12]. Each quantum system is described as a two-level spin precessing around the z-axis at a fixed frequency ω.…”

“…The first is based on the correlations between photon arrival times, or the related arrival times of optical signals detected by homodyne detection [1][2][3][4]. The second approach is based on the internal time evolution of quantum systems [5][6][7][8][9][10][11][12]. Although the latter approach requires an effective suppression of decoherence and is therefore much more challenging to implement, it might be of greater fundamental interest, since it allows a very general treatment of time in quantum mechanics.…”