From entanglement witness to generalized Catalan numbers

Abstract

Being extremely important resources in quantum information and computation, it is vital to efficiently detect and properly characterize entangled states. We analyze in this work the problem of entanglement detection for arbitrary spin systems. It is demonstrated how a single measurement of the squared total spin can probabilistically discern separable from entangled many-particle states. For achieving this goal, we construct a tripartite analogy between the degeneracy of entanglement witness eigenstates, tensor products of SO(3) representations and classical lattice walks with special constraints. Within this framework, degeneracies are naturally given by generalized Catalan numbers and determine the fraction of states that are decidedly entangled and also known to be somewhat protected against decoherence. In addition, we introduce the concept of a “sterile entanglement witness”, which for large enough systems detects entanglement without affecting much the system’s state. We discuss when our proposed entanglement witness can be regarded as a sterile one.

Spin multiplicities

“…The simplest illustration is M (0; n; 1/2) for even n. For this particular case, (22) and Stirling's approximation, n! ∼ n→∞ √ 2πn n e n , give directly the main term in the asymptotic behavior,…”

“…The t integral is just a beta function, B m + 1 2 , 3 2 = Γ m + 1 2 Γ 3 2 /Γ (m + 2), which leads back to exactly (22) for s = 0. But rather than using Stirling's approximation, it is more instructive to determine the asymptotic behavior directly from the integral (29) using Watson's lemma.…”

“…Multiplicities such as those in the Table have also attracted some recent attention in the field of quantum computing, ultimately with implications for cryptography. In particular, there are so-called "entanglement witness" (EW) operators that allow the detection of entangled states [20,21,22]. By knowing the degeneracy of the EW eigenstates for an n-particle state, one can determine the fraction of all states for which entanglement is "decidable" -a fraction that is especially of interest in the limit of large n. For systems of n spin j particles, with the EW operator taken to be the Casimir of the total spin, this fraction of decidable states [22] is denoted f j (n).…”

Generation of Cryptographic Keys With Algorithm of Polygon Triangulation and Catalan Numbers

“…Generally, the methods of counting and Catalan numbers in asymmetric cryptographic systems hold an important place in generating keys, the design of a cryptologic algorithm, and the process of cryptanalysis [1][2][3]. In references [4,6,7], the concrete applications of combinatorial problems in cryptography are listed.…”

$s$-Catalan numbers and Littlewood-Richardson polynomials

“…The purpose of this note is to study two generalizations of the well-known Catalan numbers and prove some basic combinatorial properties of these numbers by characterizing them as Littlewood-Richardson coefficients of certain triples of partitions. These numbers first appeared in quantum physics problems about spin multiplicities [5,6], and have also been studied in special cases by Belbachir and Iguerofa [4].…”