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I discuss and elaborate on the Isomorphismisomorphism between kets that span the Hilbert space of n-qubits with column matrices of dimension \(2^{n}\) . The collection of all row matrices is shown to constitute the corresponding dual space. We illustrate how outer products, or operators, are represented by \(n \times n \) square matrices. The various matrix operations that provide the inner, outer, direct or Kronecker, products for the corresponding Hilbert space are introduced and discussed. The concepts of Spinspin and the Bloch sphereBloch sphere are introduced. A qubit interpretation of spin, and the polarization properties of light is discussed.

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Apples and Oranges: Matrix Representations

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  • 发表日期：

    2025年

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  • Bernard Zygelman

摘要

I discuss and elaborate on the Isomorphismisomorphism between kets that span the Hilbert space of n-qubits with column matrices of dimension \(2^{n}\) . The collection of all row matrices is shown to constitute the corresponding dual space. We illustrate how outer products, or operators, are represented by \(n \times n \) square matrices. The various matrix operations that provide the inner, outer, direct or Kronecker, products for the corresponding Hilbert space are introduced and discussed. The concepts of Spinspin and the Bloch sphereBloch sphere are introduced. A qubit interpretation of spin, and the polarization properties of light is discussed.