Quantum Advantage in Identifying the Parity of Permutations with Certainty

TL;DR

This paper demonstrates a quantum advantage in identifying the parity of permutations with certainty using fewer states than classical methods, highlighting the power of entanglement and quantum mechanics.

Contribution

It establishes the minimal quantum resources needed for perfect parity identification and provides explicit states and entanglement measures for small cases.

Findings

Quantum mechanics achieves certainty with as few as rac{\u221a n}{} states per particle.

Classical success limited to random guessing below this threshold.

Explicit states and entanglement measures for small n cases.

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 that the success is 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.