5.2 Channels, Kraus maps, and Lindblad generators
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5.2.1 Four-index transfer
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A quantum channel with Kraus operators \(K_\mu\) acts by
Relative to operator packets
the channel has blocks
Thus a channel carries four packet labels: an input coherence pair and an output coherence pair.
5.2.2 Symmetry-preserving channels
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If every Kraus operator commutes with the symmetry action, the channel is diagonal in the isotypic labels. More generally, Kraus operators may carry definite charges, producing a sparse selection rule. Amplitude damping transfers population between number sectors. Dephasing damps selected off-diagonal blocks. Particle loss moves weight-\(k\) population to weight \(k-1\).
5.2.3 Lindblad dynamics
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A Markovian open system obeys
Packet selection rules of \(H\) and \(L_a\) determine a sparse directed graph on coherence pairs. If the reachable graph and each block representation remain polynomial, a classical integrator can evolve the reduced system.
5.2.4 Error accounting for truncated channels
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Let \(\Pi_A\) project operator space onto an active set of coherence blocks. A reduced evolution uses
The discarded transfer is
Duhamel's formula yields an error bound analogous to the Hamiltonian leakage theorem. The resulting estimate depends on the induced norm of the discarded superoperator transfer and the integration time.
5.2.5 Syndrome-conditioned recovery
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Error correction naturally fits the four-index model. Syndrome measurement selects a population packet, classical processing chooses a recovery, and the recovery transfers the state toward the code packet. A full simulation must still retain the logical state inside each syndrome sector.
Open program or boundary
A useful open-system collapse theorem should state the norm, time horizon, packet-discovery algorithm, integration precision, and growth of reachable coherence pairs. Sparsity of a displayed block matrix is not enough by itself.