Manuscript Hawk07-A@1 · Hawk07-A.tex · proposition · f9c0f5da

Proposition: Append-only entropy

Coherent entropy, classical records, and the dimension neck › Enter records cannot turn over · lines 1251–1275 · proof included or linked

Hawk07-A@1#prop:record

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For every fixed prescribed process whose complete record is append only,

H (Y_{\leq k + 1}) = H (Y_{\leq k}) + H (Y_{k + 1} ∣ Y_{\leq k}) \geq H (Y_{\leq k}) .

The statement permits adaptive readout and does not require Haar scrambling.

For fixed-basis Enter on $P_{k}$ ,

H (Y_{\leq k}) = S (Δ_{P_{k}} ρ_{P_{k}}) \geq S (ρ_{P_{k}}),

where $Δ$ denotes basis dephasing.

Proof. The first identity is the classical chain rule. Dephasing is an average of diagonal unitary conjugations, so entropy concavity gives the second inequality.

Displayed formulas in this passage

unlabeled · line 1255
$H (Y_{\leq k + 1}) = H (Y_{\leq k}) + H (Y_{k + 1} ∣ Y_{\leq k}) \geq H (Y_{\leq k}) .$
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$H (Y_{\leq k}) = S (Δ_{P_{k}} ρ_{P_{k}}) \geq S (ρ_{P_{k}}),$

Proposition: Append-only entropy

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