3.2 Projectors and quantum packets
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3.2.1 Orthogonal packet decompositions
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Let \({\mathcal H}\) be a finite-dimensional Hilbert space and let \(\{P_\alpha\}_{\alpha\in A}\) be orthogonal projectors satisfying
Every state and operator decompose as
The diagonal blocks act within packets. The off-diagonal blocks transfer amplitude or coherence between packets.
Define the packet-transfer graph of an operator \(X\) by placing an edge \(\beta\to\alpha\) whenever
An exactly symmetry-preserving operator has only diagonal edges. A sparse transfer graph may still permit efficient simulation if the active packets and internal states remain compact.
3.2.2 Conserved observables
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Let
be a self-adjoint observable with spectral projectors \(P_k\).
Theorem 3.4: Invariant quantum packets
If a unitary \(U\) satisfies \([U,Q]=0\), then
and
If \(P_k{\lvert \psi\rangle}={\lvert \psi\rangle}\), then \(P_kU{\lvert \psi\rangle}=U{\lvert \psi\rangle}\).
Proof
Since \(U\) commutes with \(Q\), it commutes with every spectral projector obtained from the functional calculus of \(Q\). Thus
for \(i\neq j\). Summing the diagonal blocks gives the direct-sum form. The final statement follows from
3.2.3 Residue and character packets
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Suppose \(Q\) has integer spectrum. For a modulus \(M\), define
Let
Then
On a \(Q\)-eigenvector, the right side is the roots-of-unity filter. The same finite-character formula selects residue classes in signal polyphase analysis and charge sectors in quantum theory.
3.2.4 Finite-group projectors
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Let a finite group \(G\) act unitarily by \(U_g\). For an irreducible character \(\chi\) of degree \(d_\chi\), the central isotypic projector is
Character orthogonality gives
on the represented space. If an operator \(A\) commutes with every \(U_g\), then
This framework includes momentum, parity, total spin, point-group symmetry, permutation symmetry, and stabilizer syndrome labels. The isotypic projector identifies a symmetry type. It does not necessarily resolve the multiplicity space within that type.
3.2.5 Nested refinements
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If one conserved label is too coarse, intersect commuting projector families. For mutually commuting observables \(Q_1,\ldots,Q_s\), a joint packet has the form
The dimension can decrease dramatically, but each extra label must be efficiently computable and preserved. A refinement that requires exponential preprocessing merely relocates the complexity.
Remark 3.5
A packet decomposition is basis independent as an orthogonal direct sum, although a practical simulator must choose a concrete basis or compressed coordinate system within each active packet.