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.
Highlights
Being extremely important resources in quantum information and computation, it is vital to efficiently detect and properly characterize entangled states
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
We study the interesting relations between the entanglement witness eigenstates, tensor products of SO(3) representations and discrete lattice walks with special constraints
Summary
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. 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 known to be somewhat protected against decoherence. A family of generalized Catalan numbers, as well as the corresponding lattice walks are thereby suggested
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