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 one. 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 analyse 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.

Quantum entanglement & purity testing: A graph zeta function perspective

“…In the next section, we will show that the separability tests in [3] and [22] corresponding to the symmetric group are equivalent to the conditions…”

“…The separability tests outlined in [3,22] are examples of G-Bose symmetry tests, where G is some finite group. Let U : G → U(H) be a unitary representation of G on the Hilbert space H.…”

“…In [3], this test is generalized to any finite group G. Moreover, the acceptance probability of this generalized test is shown to be given by the cycle index polynomial of G, which is defined by…”

“…In recent decades, a number of criteria for separability have been developed, including the positive partial transpose (PPT) criterion [18,24] and k-extendibility [7,26], and the problem of determining whether a state is separable or entangled has been shown to be NP-hard in many cases [13]. Additionally, quantum algorithms which test for separability, such as the SWAP test [1] and its generalizations [3], are under continuous development.…”

“…In Section 2, we review the separability tests defined in [3,22] related to previous work in [8,9,26]. We then review previous constructions of the density matrix of a graph but ultimately take a different approach by naturally assigning an arbitrary density matrix to a weighted graph in Section 3.…”