Quantum Advantage in Identifying the Parity of Permutations with Certainty

• URL: http://arxiv.org/abs/2508.04310v1
• Date: Wed, 06 Aug 2025 10:55:32 GMT
• Title: Quantum Advantage in Identifying the Parity of Permutations with Certainty
• Authors: Arnau Diebra, Santiago Llorens, David González-Lociga, Albert Rico, John Calsamiglia, Mark Hillery, Emili Bagan,
• Abstract summary: We establish a sharp quantum advantage in determining parity of an unknown permutation applied to any number $n ge 3$ of particles.<n>We also assess the minimum entanglement these states need to carry, finding it to be close to maximal, and even maximal in some cases.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We establish a sharp quantum advantage in determining the parity (even/odd) of an unknown permutation applied to any number $n \ge 3$ of particles. Classically, this is impossible with fewer than~$n$ labels, being the success limited to random guessing. Quantum mechanics does it with certainty with as few as $\lceil \sqrt{n}\, \rceil$ distinguishable states per particle, thanks to entanglement. Below this threshold, not even quantum mechanics helps: both classical and quantum success are limited to random guessing. For small $n$, we provide explicit expressions for states that ensure perfect parity identification. We also assess the minimum entanglement these states need to carry, finding it to be close to maximal, and even maximal in some cases. The task requires no oracles or contrived setups and provides a simple, rigorous example of genuine quantum advantage.

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