1.9 Normalized states and literal unitary transport
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In a twelve-dimensional complex Hilbert space, let \(|0\rangle,\ldots,|11\rangle\) be an orthonormal basis. Define
Its squared norm is \(4(1/2)^2=1\). Define the amplitude states
Both have norm one because \(\sum_nc(n)^2=48\). The second state is a separately specified state, constructed from the absolute-value vector. No universal quantum operation implementing the nonlinear rule \(z\mapsto|z|\) is being asserted.
Theorem QA12: Exact preparation and recovery
With the positive-sign Fourier matrix defined in Section 1.2,
Proof
At coordinate \(n\), the first left-hand side is
This is the defining coordinate of \(\psi\). Applying \(F^\dagger\) and using \(F^\dagger F=I\) proves the second equality. \(\square\)
The normalization relation is \(\sqrt{48}=2\sqrt{12}\). In the exact coordinate implementation the Fourier scale is \(\sqrt3/6=1/\sqrt{12}\), and the state scale is \(\sqrt3/12=1/\sqrt{48}\). The later formal source proves those scale identities and the exponential entry formula, rather than leaving them as informal interpretations of a rational tuple.
This is a mathematical quantum-state construction. An actual realization would additionally require state preparation, a twelve-dimensional unitary implementation, coherence control, and sampling. A twelve-dimensional Fourier matrix is not the ordinary sixteen-dimensional Fourier transformation on four qubits. The unitary direct sum \(F_{12}\oplus I_4\) supplies a mathematical sixteen-dimensional embedding, but it supplies no gate synthesis or resource bound by itself.