Manuscript Em03-H@8 · Em03-H-v8.tex · theorem · 91e9e3bd

Theorem: Leakage and best compression

Quench leakage, best compression, and departure · lines 643–666 · proof included or linked

Em03-H@8#thm:leakage

Λ_{N} = V^{†} N_{B} (I - P) N_{B} V ⪰ 0, L_{N} = ∥ (I - P) N_{B} V ∥ = λ_{m a x} (Λ_{N}) .

Furthermore,

E_{code} = λ_{m a x} (ν), A \in B (M K) in f ∥ Ω N_{B} V - V A ∥ = Ω L_{N},

attained at $A = Ω ν$ .

Proof. Insert $P + (I - P)$ between number operators. The retained term is $ν^{2}$ , and the remainder is the Gram matrix of $(I - P) N_{B} V$ . Projection onto $I - P$ gives the compression lower bound; choosing $A = Ω ν$ removes the retained component. Rayleigh optimization gives the scalar quantities.

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unlabeled · line 645
$Λ_{N} = V^{†} N_{B} (I - P) N_{B} V ⪰ 0, L_{N} = ∥ (I - P) N_{B} V ∥ = λ_{m a x} (Λ_{N}) .$
unlabeled · line 652
$E_{code} = λ_{m a x} (ν), A \in B (M K) in f ∥ Ω N_{B} V - V A ∥ = Ω L_{N},$

Theorem: Leakage and best compression

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