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Quantum Advantage in Identifying the Parity of Permutations with Certainty

Arnau Diebra, Santiago Llorens, David González-Lociga, Albert Rico, John Calsamiglia, Mark Hillery, Emili Bagan

TL;DR

The paper proves a sharp quantum advantage in identifying the parity of an unknown permutation on $n$ particles: perfect parity discrimination is possible with as few as $d \ge \lceil\sqrt{n}\rceil$ distinguishable states per particle, while below this threshold random guessing remains optimal. It achieves this by leveraging Schur–Weyl duality and permutation-group representation theory to construct a parity-detecting entangled state $|\psi_e\rangle$ whose even/odd images are orthogonal under the permutation action, and by providing explicit constructions for small $n$ (3–5). It also analyzes the entanglement cost via the geometric measure of entanglement, giving tight lower bounds $E_{\lambda a}$ that are near maximal for the minimal-dimension cases, and it uses semidefinite programming to bound these quantities rigorously. The work exemplifies a nonasymptotic, ancilla-free quantum advantage based purely on permutation symmetry and multipartite entanglement, with potential paths toward experimental realization and extensions to other symmetry groups.

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.

Quantum Advantage in Identifying the Parity of Permutations with Certainty

TL;DR

The paper proves a sharp quantum advantage in identifying the parity of an unknown permutation on particles: perfect parity discrimination is possible with as few as distinguishable states per particle, while below this threshold random guessing remains optimal. It achieves this by leveraging Schur–Weyl duality and permutation-group representation theory to construct a parity-detecting entangled state whose even/odd images are orthogonal under the permutation action, and by providing explicit constructions for small (3–5). It also analyzes the entanglement cost via the geometric measure of entanglement, giving tight lower bounds that are near maximal for the minimal-dimension cases, and it uses semidefinite programming to bound these quantities rigorously. The work exemplifies a nonasymptotic, ancilla-free quantum advantage based purely on permutation symmetry and multipartite entanglement, with potential paths toward experimental realization and extensions to other symmetry groups.

Abstract

We establish a sharp quantum advantage in determining the parity (even/odd) of an unknown permutation applied to any number of particles. Classically, this is impossible with fewer than labels, being that the success is limited to random guessing. Quantum mechanics does it with certainty with as few as 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 , 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.

Paper Structure

This paper contains 21 sections, 1 theorem, 113 equations, 1 figure, 6 tables.

Table of Contents

  1. The groups $S_n$ and $A_n$ in a nutshell
  2. The symmetric group $S_n$
  3. Conjugacy classes
  4. Representations
  5. Characters and character table
  6. Schur's lemma
  7. The action of $S_n$ and $\operatorname{SU}(d$); Schur–Weyl duality
  8. Irreps of $\operatorname{SU}(d)$
  9. Schur-Weyl duality
  10. Group algebra and projectors
  11. Standard and generalized Young symmetrizers
  12. The alternating group $A_n$
  13. The parity-detecting state $|\psi_e\rangle$ in the computational basis
  14. Three qubits
  15. Four qubits
  16. ...and 6 more sections

Key Result

Theorem I

Perfect parity identification ($P_{\rm s} = 1$) is achievable if Below this threshold, assuming all permutations are equally likely, parity remains indistinguishable and one cannot do better than random guessing ($P_{\rm s} = 1/2$).

Figures (1)

  • Figure 1: The YDs corresponding to the five partitions of 4.

Theorems & Definitions (1)

  • Theorem I