States, measurements, and open dynamics
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5.1 Density operators: populations and coherences
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5.1.1 Two packet indices
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A pure state uses one packet label. A density operator uses two:
The diagonal block \(P_\alpha\rho P_\alpha\) is population inside packet \(\alpha\). The off-diagonal block \(P_\alpha\rho P_\beta\) is coherence between packets \(\alpha\) and \(\beta\).
Discarding off-diagonal blocks is not merely a change of coordinates. It is a physical operation such as dephasing. An exact simulator must preserve all coherence blocks relevant to the final observable.
5.1.2 Hilbert–Schmidt decomposition
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Orthogonality of the left-right operator packets gives
This identity supplies an exact coherence-energy ledger. It can be used to decide which blocks are large, to audit dephasing, or to bound truncation errors in approximate packet simulations.
5.1.3 Symmetry-preserving unitary dynamics
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If \(U\) commutes with every packet projector, then
The left and right packet labels are preserved. Populations and coherences evolve independently by their declared labels.
5.1.4 Observable sparsity
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For an observable \(M\),
Only operator packets coupled by \(M\) contribute. A measurement that is diagonal in the symmetry labels depends only on populations. A phase-sensitive measurement can require off-diagonal coherences.
5.1.5 Polynomial operator-space collapse
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If only polynomially many density blocks are reachable and every block has a polynomial representation, then mixed-state evolution can also be simulated in polynomial time. The representation size is not the square of the full Hilbert dimension. It is the total cost of the active left-right blocks.
Derived theorem or corollary
For a polynomial active set \(A_n\) of packet labels, a density-matrix simulator has at most \(|A_n|^2\) label pairs. Polynomial-time simulation follows when each active coherence block has a polynomial classical representation and channel updates remain sparse and efficient.