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 η 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 Schrödinger 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 ηfunctions. 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.

Qubits, superqubits and squbits

“…Despite research based on conventional for quantum mechanics formalism of the Hilbert space, there are atempts to describe entanglement in different ways, using experiences from supermathematics and other algebraical structures. The η-Hilbert space formulation based on commuting nilpotent variables gives new functional tools to study many-qubit entanglement and to realize that many of interesting entangled states can be identified as elementary η-functions [16,18]. On the other hand after four decades of supersymmetric theories there was proposed super-extension of the notion of qubit and super-entanglement of superstates with theoretically more nonlocality then in conventional many-qubit states.…”

Geometric measure of entanglement for pure states and mean value of spin

“…Let the qubit is realized with the help of spin. Without loss of generality we choose basis vectors of the qubit in (21) as follows…”

“…The states |D n,k and |D n,n−k are dual in the sense of the general notion of pure state duality introduced in Refs. [21,22,23,24].…”

“…Recently there were discussed states with interesting properties which in the formalism of nilpotent quantum mechanics [21,24] are naturally defined as trigonometric functions of nilpotent commuting variables [25,26]. The formalism using η-variables (commuting nilpotent variables) is very efficient in the algebraical description of entanglement and reveals many properties of multi-qubit states from the functional point of view.…”

“…Here we shall consider the sin-states and cos-states using only their binary basis representation (the η-function representation is given in Refs. [21,24,25,26] where relation to trigonometric functions is shown).…”