Nilpotent quantum mechanics, qubits, and flavors of entanglement
Abstract
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 η-wavefunctions, generalized Schr\"odinger 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 η-eigenfunctions for the qubit-qubit system give the set of Bloch vectors as a natural basis. Then the η-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 74, 022331 (2006)], we generalize and extend approach presented there. Our main tool for study the entanglement or separability of states are Wronskians of η-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.