Hilbert-Schmidt operators and the conjugate of a complex Hilbert space: Dirac's bra-ket formalism revisited

Frank Oertel

Published 2023-08-08Version 1

We reveal in detail how the definition of the inner product on a given complex Hilbert space - usually used in mathematics (where linearity is assumed in the first component and semilinearity in the second) - directly links to Dirac's powerful bra-ket formalism in quantum physics. To this end, we just have to make use of the conjugate of a complex Hilbert space (by which an analysis of semilinear operators can be handled by means of linear operator theory) and re-apply the theorem of Fr\'{e}chet-Riesz accordingly. Applications are specified, including a self-contained and simple description of the tensor product of two complex Hilbert spaces $H \otimes K$ (answering a related question of B. K. Driver) and a purely linear algebraic description of the quantum teleportation process (Example 3.8). In doing so, we provide an explicit construction of a canonical isometric isomorphism between the Hilbert spaces $H \otimes (K \otimes L)$ and $(H \otimes K) \otimes L$ (Theorem 3.7).