Symplectic space and orthogonal space of n qubits

TL;DR

This paper introduces symplectic and orthogonal spaces in the Hilbert space of n qubits, exploring their properties and applications in quantum physics, including basis structures and group mappings.

Contribution

It defines symplectic and orthogonal spaces for n qubits and analyzes their properties, linking local operations to these mathematical structures.

Findings

Homomorphic mapping of local operations to symplectic or orthogonal groups

Identification of the generalized 'magic basis' as a bi-orthonormal basis

Application example demonstrating relevance in physics

Abstract

In the Hilbert space of n qubits, we introduce the symplectic space (n odd) and the orthogonal space (n even) via the spin-flip operator. Under this mathematical structure we discuss some properties of n qubits, including homomorphically mapping the local operations of n qubits into the symplectic group or orthogonal group, and prove that the generalized ``magic basis'' is just the bi-orthonormal basis (that is, the orthonormal basis of both Hilbert space and the orthogonal space ). Finally, an example is given to discuss the application in physics of this mathematical structure.