Manuscript En04-A@1 · En04-A.tex · section · bab77211

The declared finite fragment

Typed preparation, coherent programs, and Enter › The declared finite fragment · lines 611–668

En04-A@1#eq:enter

Types are

T ::= I ∣ Q ∣ Bit ∣ T \otimes T,

with observable algebras

A_{I} = C, A_{Q} = M_{2} (C), A_{Bit} = C \oplus C .

Classical registers are block diagonal. No conversion from a quantum register to a classical register is implicit.

A judgement $P : T \to T^{'}$ consumes the entire input interface and returns its declared output interface. Sequential and tensor composition require matching types. A live quantum name is not duplicated or silently omitted. This follows the minimal ownership reading of CheungEn03A, rather than an unprovided larger calculus.

The primitives are pure allocation, specified finite unitaries, destructive computational readout, classical blockwise control, and classical discard:

new : I \to Q, Enter : Q^{\otimes s} \to Bit^{\otimes s}, dropBit : Bit \to I .

A specified unitary constant is a semantic object, not an assertion of exact finite-gate compilation, efficiency, or hardware realization.

For a measured subsystem $S$ and surviving quantum subsystem $T$ , write

M_{x} = ⟨ x ∣_{S} \otimes I_{T} .

The destructive readout instrument is

E_{S} (ρ) = x \sum ∣ x ⟩ ⟨ x ∣_{K} \otimes M_{x} ρ M_{x}^{†} .

The measured names are consumed. For a pure input,

p_{x} = ∥ M_{x} ψ ∥^{2}, ψ_{x} = M_{x} ψ / p_{x} (p_{x} > 0) .

No normalized residual is assigned to a zero-probability branch.

These are the destructive Born–Lüders residual formulas supplied by Landmark I within its stated strategy class CheungI (thm:main). We define the instrument directly; we do not reconstruct a measurement-uniqueness theorem from incomplete hypotheses.

Displayed formulas in this passage

En04-A@1#eq:enter · line 650
$E_{S} (ρ) = x \sum ∣ x ⟩ ⟨ x ∣_{K} \otimes M_{x} ρ M_{x}^{†} .$
unlabeled · line 614
$T ::= I ∣ Q ∣ Bit ∣ T \otimes T,$
unlabeled · line 618
$A_{I} = C, A_{Q} = M_{2} (C), A_{Bit} = C \oplus C .$
unlabeled · line 636
$new : I \to Q, Enter : Q^{\otimes s} \to Bit^{\otimes s}, dropBit : Bit \to I .$
unlabeled · line 646
$M_{x} = ⟨ x ∣_{S} \otimes I_{T} .$
unlabeled · line 657
$p_{x} = ∥ M_{x} ψ ∥^{2}, ψ_{x} = M_{x} ψ / p_{x} (p_{x} > 0) .$

The declared finite fragment

Displayed formulas in this passage

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