Symplectic space and orthogonal space of n qubits

2010/11/14 by Jianwei Xu, Jian-Wei Xu, Xu, Jian-Wei
Computer Science · Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #FOS: Physical sciences #Graph theory and applications #Matrix Theory and Algorithms #Quantum Physics (quant-ph) #quant-ph

paper · pdf · doi:10.48550/arxiv.1011.3187

8 pages

arxiv created 2010/11/14 · openalex publication_date 2010/11/14 · arxiv updated 2010/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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