Cycle Index Polynomials and Generalized Quantum Separability Tests

TL;DR

This paper introduces a family of quantum separability tests based on cycle index polynomials of finite groups, providing exact acceptance probabilities and efficient circuit implementations for testing quantum state separability.

Contribution

It derives a novel connection between quantum separability tests and cycle index polynomials, enabling explicit formulas and efficient quantum circuit constructions.

Findings

Acceptance probability expressed via cycle index polynomials

Tests for symmetric and cyclic groups have efficient implementations

Explicit quantum circuits with polynomial gate complexity

Abstract

The mixedness of one share of a pure bipartite state determines whether the overall state is a separable, unentangled state. Here we consider quantum computational tests of mixedness, and we derive an exact expression of the acceptance probability of such tests as the number of copies of the state becomes larger. We prove that the analytical form of this expression is given by the cycle index polynomial of the symmetric group $S_k$, which is itself related to the Bell polynomials. After doing so, we derive a family of quantum separability tests, each of which is generated by a finite group; for all such algorithms, we show that the acceptance probability is determined by the cycle index polynomial of the group. Finally, we produce and analyze explicit circuit constructions for these tests, showing that the tests corresponding to the symmetric and cyclic groups can be executed with $O(k^2)$ and $O(k\log(k))$ controlled-SWAP gates, respectively, where $k$ is the number of copies of the state being tested.