Nilpotent quantum mechanics, qubits, and flavors of entanglement

TL;DR

This paper introduces a formalism based on nilpotent commuting variables to describe qubit systems, revealing new properties of entanglement, supersymmetry, and composite behaviors, and providing simplified criteria for state separability.

Contribution

It extends the nilpotent variable formalism to analyze qubit entanglement and supersymmetry, offering new tools and criteria for understanding quantum correlations.

Findings

Nilpotent formalism models qubits as composite, boson-like objects.

Supersymmetric fermion-qubit systems can have identical spectra.

Wronskians of eta-functions serve as criteria for entanglement and separability.

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 $\eta$-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.