2026-10-21 · arXiv Quantum Physics
arXiv:2610.27260v1 Announce Type: new
Abstract: The three-tangle, $\tau_3$, is a measure of genuine tripartite entanglement that distinguishes GHZ-type from W-type entanglement in pure three-qubit states. Here, we identify the statistically optimal one-sided four-copy detector of the three-tangle and study its implementation cost. The optimal detector has click probability $\tau_3^2/48$, while previous four-copy constructions achieve $\tau_3^2/256$ and therefore require asymptotically $16/3$ times as many experimental repetitions. Using the representation-theoretic structure of the problem, we construct an ancilla-free implementation of the optimal detector using 15 CNOT gates, together with an 11-CNOT detector with click probability $\tau_3^2/96$. These results reveal a tradeoff between statistical efficiency and implementation cost, since reducing the number of repetitions requires increasingly expensive circuits. We then show that this tradeoff persists once the circuits are encoded. In a fully fault-tolerant setting, the increased click probability comes with a substantial non-Clifford synthesis cost, while an early fault-tolerant implementation based on $T$ gates and STAR rotations reduces this overhead at the expense of additional encoded resources. We quantify these costs using both the Steane and $[[8,3,2]]$ codes, providing explicit resource estimates for implementing the optimal detector across different computational regimes.
原文链接：https://arxiv.org/abs/2610.27260
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