Manuscript VII-E-ST@1 · VII-E-ST.tex · theorem · d28d9c5d

Theorem: Finite hybrid structure

Contract G: finite completely positive hybrid evolution · lines 1178–1225 · proof included or linked

VII-E-ST@1#thm:structure, VII-E-ST@1#eq:hybrid-map, VII-E-ST@1#eq:branch-complete

A linear map between finite direct sums is completely positive and trace preserving exactly when it has branch form

(F X)_{h} = g \sum Ψ_{h ∣ g} (X_{g}), Ψ_{h ∣ g} (X) = ν \sum K_{h ∣ g, ν} X K_{h ∣ g, ν}^{†},

with

h, ν \sum K_{h ∣ g, ν}^{†} K_{h ∣ g, ν} = I_{H} for every g .

Its adjoint is

(F^{*} B)_{g} = h \sum Ψ_{h ∣ g}^{*} (B_{h}) .

A canonical channel on the full matrix space has Kraus operators

∣ h ⟩ ⟨ g ∣ \otimes K_{h ∣ g, ν} .

It discards off-diagonal input label blocks and produces block-diagonal output. It is a specified lift, not a reconstruction of quantum geometric coherences.

Proof. After adjoining any matrix ancilla, a positive direct-sum input has positive blocks. Complete positivity of the branches and summation prove complete positivity of $F$ . Equation (eq:branch-complete) gives trace preservation.

Conversely, compose a given completely positive map with injection into input block $g$ and projection onto output block $h$ . These maps give completely positive branches. For a map between matrix algebras, its Choi matrix is positive; diagonalizing that matrix and unvectorizing its rank-one terms gives the displayed Kraus representation Choi. Trace preservation on each input block gives (eq:branch-complete).

The adjoint follows by pairing. Multiplication of the lifted Kraus operators gives the full identity as their sum of squared moduli, and their action uses only diagonal input blocks.

Displayed formulas in this passage

VII-E-ST@1#eq:hybrid-map · line 1182
$(F X)_{h} = g \sum Ψ_{h ∣ g} (X_{g}), Ψ_{h ∣ g} (X) = ν \sum K_{h ∣ g, ν} X K_{h ∣ g, ν}^{†},$
VII-E-ST@1#eq:branch-complete · line 1188
$h, ν \sum K_{h ∣ g, ν}^{†} K_{h ∣ g, ν} = I_{H} for every g .$
unlabeled · line 1194
$(F^{*} B)_{g} = h \sum Ψ_{h ∣ g}^{*} (B_{h}) .$
unlabeled · line 1199
$∣ h ⟩ ⟨ g ∣ \otimes K_{h ∣ g, ν} .$

Theorem: Finite hybrid structure

Displayed formulas in this passage

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