# Apollonius Representation of Qubits

## TL;DR

This paper introduces a novel geometric representation of qubits using Apollonius circles, revealing constant entropy and concurrence along these circles, and connects these geometric features with quantum entanglement measures and fidelity.

## Contribution

The paper develops a new Apollonius circle-based geometric framework for representing qubits, linking entropy, concurrence, and fidelity in a unified visual model.

## Key findings

- Shannon entropy is constant along Apollonius circles for single qubits.
- Concurrence remains constant on the same Apollonius circle for two qubit states.
- Fidelity between states is constant along Apollonius circles, generalizing to n-qubit states.

## Abstract

We introduce the qubit representation by complex numbers on the set of Apollonius circles with common symmetric points at $0$ and $1$, related with $|0\rangle$ and $|1\rangle$ states. For one qubit states we find that the Shannon entropy as a measure of randomness is a constant along Apollonius circles. For two qubit states, the concurence as a characteristic of entanglement is taking constant value for the states on the same Apollonius circle. Geometrical meaning of concurence as an area and as a distance in the Apollonius representation are found. Then we generalize our results to arbitrary $n$-qubit Apollonius states and show that the fidelity between given state and the symmetric one, as reflected in an axes, is a constant along Apollonius circles. For two qubits it coinsides with the concurence. For generic two qubit states we derived Apollonius representation by three complex parameters and show that the determinant formula for concurence is related with fidelity for symmetric states by two reflections in a vertical axis and inversion in a circle. We introduce the complex concurence and an addition formula for Apollonius states and show that for generic two qubit states its modulus satisfies the law of cosine. Finally, we show that for two qubit Apollonius state in bipolar coordinates, the complex concurence is decribed by static one soliton solution of the nonlinear Schr\"odinger equation.