Cycle Index Polynomials and Generalized Quantum Separability Tests

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.

arXiv comment26 pages, 7 figures

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Cited by in corpus (6)

• Testing symmetry on quantum computers
• Introduction to Quantum Error Correction with Stabilizer Codes
• Quantum Computational Complexity and Symmetry
• Resource-efficient algorithm for estimating the trace of quantum state powers
• Quantum Entanglement & Purity Testing: A Graph Zeta Function Perspective
• Dubious Identities: A Visit to the Borwein Zoo