Phase assumption-free multiparty quantum clock synchronization

TL;DR

This work tackles the challenge of establishing a universal clock across many parties without a shared phase reference, by developing a phase-assumption-free multiparty quantum clock synchronization protocol based on supersinglet states purified via LOCC. The central idea is to distribute a central time signal using a supersinglet that remains invariant under SU(2) and to remove phase offsets through entanglement purification, enabling deterministic timing information across all nodes with a signal that scales favorably with the number of parties. The authors derive explicit amplitudes for each party, demonstrate the optimality of the supersinglet over other spin-zero states, and quantify the timing error as a function of fidelity and shot noise, including realistic dephasing and potential residual phase effects. The proposed approach promises practical advantages over classical synchronization (e.g., TWSTFT) and is well-suited for long-distance operation and integration into a future quantum internet, provided reliable long-distance entanglement distribution and offline entanglement purification can be achieved. $\delta t \approx \frac{1}{\omega} \sqrt{ \frac{1}{2M} + 4(1 - F_{\text{super}}) }$ captures the core error scaling, with additional systematic and SQL contributions discussed.

Abstract

We investigate methods to broadcast timing information from a central clock to all other clocks by the use of multipartite entanglement. This task is a necessary step in establishing a coordinated universal time, currently performed using classical synchronization methods. Using an entanglement-based method has the advantage that the timing results are independent of the intervening medium. We generalize existing bipartite quantum clock synchronization methods and take special care to address issues of different phase conventions being adopted at each node (the ``Preskill phase problem''). Using supersinglet purification, we show that this allows for a scalable method with a time signal that is a constant with respect to the number of nodes.

Figures (5)

• Figure 1: Quantum time broadcasting using supersinglet states. Alice wishes to share her timing information with the remaining parties, each holding a qubit depicted by a Bloch sphere (we show the $N = 4$ case). The full protocol proceeds as follows. Step 1: Distribute singlet pairs $|\Psi^- \rangle$ between parties following symmetry of supersinglet state (dashed lines). Interchange of qubits within Group I ($n \in [1,N/2]$) and Group II ($n \in [N/2+1,N]$) leaves the distribution symmetric. Step 2: Perform entanglement purification following Ref. bennett1996 or similar. Step 3: Check correlation of $\langle Z_i Z_j \rangle$ of singlet pairs, if not consistent with singlet state, then apply a $\pi$ rotation around $Z$ axis and repeat Step 2. At this point the Preskill phase is removed. Step 4: Perform purification protocol in Ref. ahmad2025distillation to obtain a supersinglet state $| {\cal S} \rangle$ (wavy lines). Step 6: Alice measures her state in the $X$ basis and classically broadcasts the result. Step 7: Each party measures their local qubit in the $X$ basis, and adjusts the amplitude according to Alice's outcome. Step 8: Parties infer the time of Alice's measurement using (\ref{['qcsamp']}) and (\ref{['supersingletamplitudes']}).
• Figure 2: Performance of the multiparty quantum clock synchronization. (a) Optimization with respect to singlet states for the $N = 4$ case using parametrization $\ket{\psi} = \cos(\theta)\ket{\Psi^-}_{12} \ket{\Psi^-}_{34} + e^{i\phi}\sin(\theta)\ket{\mathcal{S}_4}$. As the function to be optimized, we take $\left( \prod_{n=2}^{N} |A_n | \right)^{\frac{1}{N-1}}$, the geometric mean of the QCS amplitudes. (b) Error estimate (\ref{['totalerror']}) of multiparty QCS for various supersinglet fidelities $F_{\text{super}}$.
• Figure 3: Entanglement purification of supersinglet states with the Preskill phase. Fidelities are defined as $F_{\text{singlet}} = \langle \Psi^- | \rho | \Psi^- \rangle$ and $F_{\text{super}} = \langle {\cal S} | \rho | {\cal S} \rangle$. We target the $N = 4$ supersinglet. Purification iterations are shown for (a) singlet Bell pairs using BBPSSW purification bennett1996; (b) supersinglets using methods in Ref. ahmad2025distillation. The initial state for the singlet Bell pair purification of (a) is $\exp (i \varphi Z_1/2) | \Psi^- \rangle$, where $\varphi$ is the Preskill phase. The output of BBPSSW purification after 10 rounds is used as the input for the supersinglet purification in (b).
• Figure 4: Angular momentum coupling chart for $N$ spin-$1/2$ (qubits). Each path corresponds to a irreducible representation of the total angular momentum.
• Figure 5: Signal amplitudes $A_n = \text{Tr}( \rho X_1 X_n )$ after the dephasing channel (\ref{['dephasingchannel']}) as a function of $p$. The solid curve show the signal of parties in Group I ($n \in [2,N]$), which remains invariant with respect to the total number of parties $N$. The dashed curves represent the signal of Group II ($n \in [N/2+1,N]$), for different system sizes $N$.