Paper VII: Cyclic symmetry packets

I.3 Cyclic symmetry packets

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I.3.1 Character projectors

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Let \(N\ge1\) be an integer, and let \(S\) be a unitary symmetry whose order divides \(N\):

\[ S^N=I. \]

Set \(\omega=e^{2\pi i/N}\).

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Theorem I.3.1: Cyclic quantum projectors

The operators

\[ \boxed{ P_j = \frac1N\sum_{r=0}^{N-1}\omega^{-jr}S^r, \qquad 0\le j<N, } \]

are self-adjoint, pairwise orthogonal projectors satisfying

\[ P_jP_k=\delta_{jk}P_j, \qquad \sum_{j=0}^{N-1}P_j=I, \qquad SP_j=\omega^jP_j. \]

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Proof of Theorem I.3.1

Using \(S^{r+s}=S^{r+s\bmod N}\) and the finite character orthogonality relation,

\[ \frac1N\sum_{r=0}^{N-1}\omega^{(k-j)r}=\delta_{jk}, \]

we obtain \(P_jP_k=\delta_{jk}P_j\). Summing over \(j\) cancels every term indexed by \(r\ne0\) and retains the \(r=0\) term, which proves completeness. Unitarity and the change of residue \(r\mapsto-r\) give

\[ P_j^*=\frac1N\sum_{r=0}^{N-1}\omega^{jr}S^{-r}=P_j. \]

Finally, multiplication by \(S\) shifts the summation index and contributes the factor \(\omega^j\).

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For parity,

\[ S=Z^{\otimes n}, \qquad P_{\mathrm{even}}=\frac{I+S}{2}, \qquad P_{\mathrm{odd}}=\frac{I-S}{2}. \]

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I.3.2 Tensor refinement

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Fix \(N\ge1\). Suppose every factor \(a\) has a complete orthogonal family \(P^{(a)}_0,\ldots,P^{(a)}_{N-1}\). The fine tensor packets are

\[ P_{\mathbf j}^{\mathrm{fine}} = P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}. \]

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Theorem I.3.2: Tensor/polyphase refinement

For each residue \(r\in{\mathbb Z}/N{\mathbb Z}\), define

\[ \boxed{ P_r^{\mathrm{coarse}} = \sum_{j_1+\cdots+j_m\equiv r\pmod N} P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}. } \]

Then the \(P_r^{\mathrm{coarse}}\) are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.

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Proof of Theorem I.3.2

The fine tensor projectors are mutually orthogonal and sum to the identity. Grouping the fine addresses by their total residue partitions the full address set. Sums over disjoint sets of orthogonal projectors remain orthogonal projectors, and the union of the residue classes is the full fine address set.

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Three-qubit computational addresses refine the even and odd parity packets. — The eight computational packets of three qubits refine the two total-parity packets exactly.

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Numerical certificate I.3.3: Exact parity refinement

For three qubits, the local computational projectors were grouped by the parity of their Hamming weight. The resulting coarse projectors agreed with \((I\pm Z^{\otimes3})/2\) with residual \(0\).

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