Geometry and Entanglement of Super-Qubit Quantum States

TL;DR

This paper introduces the concept of super-qubits, representing quantum states with extended geometry involving super-Bloch spheres, and explores their entanglement, coherence, and geometric properties, including connections to Fibonacci sequences and the Golden Ratio.

Contribution

It presents a novel geometric framework for super-qubits using super-Bloch spheres, constructs super-coherent states, and links entanglement measures to geometric distances, incorporating Fibonacci and Golden Ratio properties.

Findings

Super-qubits are represented by points on two super-Bloch spheres.

Entanglement (concurrence) is geometrically interpreted as distances on the spheres.

Fibonacci oscillating circles relate to uncertainty ratios approaching the Golden Ratio.

Abstract

We introduce the super-qubit quantum state, determined by superposition of the zero and the one super-particle states, which can be represented by points on the super-Bloch sphere. In contrast to the one qubit case, the one super-particle state is characterized by points in extended complex plain, equivalent to another super-Bloch sphere. Then, geometrically, the super-qubit quantum state is represented by two unit spheres, or the direct product of two Bloch spheres. By using the displacement operator, acting on the super-qubit state as the reference state, we construct the super-coherent states, becoming eigenstates of the super-annihilation operator, and characterized by three complex numbers, the displacement parameter and stereographic projections of two super-Bloch spheres. The states are fermion-boson entangled, and the concurrence of states is the product of two concurrences, corresponding to two Bloch spheres. We show geometrical meaning of concurrence as distance from point-state on the sphere to vertical axes - the radius of circle at horizontal plane through the point-state. Then, probabilities of collapse to the north pole state and to the south pole state are equal to half-distances from vertical coordinate of the state to corresponding points at the poles. For complimentary fermion number operator, we get the flipped super-qubit state and corresponding super-coherent state, as eigenstate of transposed super-annihilation operator. The infinite set of Fibonacci oscillating circles in complex plain, and corresponding set of quantum states with uncertainty relations as the ratio of two Fibonacci numbers, and in the limit at infinity becoming the Golden Ratio uncertainty, is derived.