Optimal four-copy detection of the three-tangle and its fault-tolerant cost
Abstract
The three-tangle, τ3, 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 τ32/48, while previous four-copy constructions achieve τ32/256 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 τ32/96. 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.
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BibTeX
@misc{perez2026optimal,
title = {{Optimal four-copy detection of the three-tangle and its fault-tolerant cost}},
author = {Federico Tomás B. Pérez and Berta Casas and I-Chi Chen and Su Yeon Chang and Philipp Schleich and M. Cerezo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27260},
eprint = {2610.27260},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27260},
}
RIS
TY - UNPB
TI - Optimal four-copy detection of the three-tangle and its fault-tolerant cost
AU - Federico Tomás B. Pérez
AU - Berta Casas
AU - I-Chi Chen
AU - Su Yeon Chang
AU - Philipp Schleich
AU - M. Cerezo
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27260
UR - https://arxiv.org/abs/2610.27260
AB - The three-tangle, τ_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 τ_3^2/48, while previous four-copy constructions achieve τ_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 τ_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.
ER -