Manuscript Psy03-A@1 · Psy03-A.tex · section · b30eb6e9

Complete record semantics

Complete instruments and operational intervention budgets › Complete record semantics · lines 1447–1479

Psy03-A@1#eq:history

A finite instrument is a family of CP trace-nonincreasing maps ${I_{r}}$ whose sum is trace preserving DaviesLewis. Its complete output is

I (ρ) = r ⨁ I_{r} (ρ) .

It retains both the record and the unnormalized residual state. Failure, preparation loss, and erasure are explicit branches.

For a Markov sequence,

P (r_{1 : m}) = Tr [I_{r_{m}}^{(m)} Λ_{m : m - 1} \dots I_{r_{1}}^{(1)} Λ_{1 : 0} (ρ)] .

Adaptive choices are functions of recorded history. For retained environmental memory, the joint state rather than a reset channel must be propagated.

Equal effects need not mean equal instruments. For example,

I_{r} (ρ) = Tr (E_{r} ρ) ∣ 0 ⟩ ⟨ 0 ∣, J_{r} (ρ) = Tr (E_{r} ρ) ∣ 1 ⟩ ⟨ 1 ∣

have equal record probabilities but different residual states.

Displayed formulas in this passage

unlabeled · line 1453
$I (ρ) = r ⨁ I_{r} (ρ) .$
Psy03-A@1#eq:history · line 1461
$P (r_{1 : m}) = Tr [I_{r_{m}}^{(m)} Λ_{m : m - 1} \dots I_{r_{1}}^{(1)} Λ_{1 : 0} (ρ)] .$
unlabeled · line 1474
$I_{r} (ρ) = Tr (E_{r} ρ) ∣ 0 ⟩ ⟨ 0 ∣, J_{r} (ρ) = Tr (E_{r} ρ) ∣ 1 ⟩ ⟨ 1 ∣$

Complete record semantics

Displayed formulas in this passage

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