Apples and Oranges: Matrix Representations

Abstract

I discuss and elaborate on the isomorphism between kets that span the Hilbert space of n-qubits with column matrices of dimension 2ⁿ. 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 × 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 spin and the Bloch sphere are introduced. A qubit interpretation of spin, and the polarization properties of light is discussed.