Foundations of Quantum Programming

7.5 Recovering symmetry and antisymmetry

The last section developed the techniques for solving quantum recursive equations in the free Fock space. However, the solutions found in the free Fock space are still not what we really need because they may not preserve symmetry or antisymmetry and thus cannot directly apply to the symmetric Fock space for bosons or the antisymmetric Fock space for fermions. In this section, we introduce the technique of symmetrization that allows us to transform every solution in the free Fock space to a solution in the bosonic or fermionic Fock spaces.

7.5.1 Symmetrization Functional

Let us first isolate a special subdomain of $\mathcal{O}(\mathcal{G}({\mathcal{H}}_C)\otimes \mathcal{H})$, namely the domain of symmetric operators. As in Subsection 7.4.1, let $\mathcal{H}$ be the ...