Nilpotent quantum mechanics, qubits, and flavors of entanglement

• Published in: arXiv.org 2008-10
• Main Author: Frydryszak, Andrzej M
• Format: Article
• Language: English
• Subjects: Eigenvectors; Elementary particles; Fermions; Flavor (particle physics); Formalism; Pauli exclusion principle; Properties (attributes); Quantum entanglement; Quantum mechanics; Quantum theory; Qubits (quantum computing); Supersymmetry; Wave functions
• ISSN: 2331-8422

We address the question of description of qubit system in a formalism based on the nilpotent commuting variables. In this formalism qubits exhibit properties of composite objects being subject of the Pauli exclusion principle, but otherwise behaving boson-like. They are not fundamental particles. In such an approach the classical limit yields the nilpotent mechanics. Using the space of \(\eta\)-wavefunctions, generalized Schr\"{o}dinger equation etc. we study properties of pure qubit systems and also properties of some composed, hybrid models: fermion-qubit, boson-qubit. The fermion-qubit system can be truly supersymmetric, with both SUSY partners having identical spectra. It is new and very interesting that SUSY transformations relate here only nilpotent object. The \(\eta\)-eigenfunctions for the qubit-qubit system give the set of Bloch vectors as a natural basis. Then the \(\eta\)-formalism is applied to the description of the pure state entanglement. Nilpotent commuting variables were firstly used in this context in [A. Mandilara, et. al., Phys. Rev. A \textbf{74}, 022331 (2006)], we generalize and extend approach presented there. Our main tool for study the entanglement or separability of states are Wronskians of \(\eta\)-functions. The known invariants and entanglement monotones for systems of \(n=2,3,4\) qubits are expressed in terms of the Wronskians. This approach gives criteria for separability of states and insight into the flavor of entanglement of the system and simplifies description.