Manuscript Hawk04-A@1 · Hawk04-A.tex · proposition · 526092ba

Proposition: Why a classical key is not the complete complement

Reset, dephasing, and omitted purifiers › Mixed quantum encryption requires a complete purifier · lines 677–713 · proof included or linked

Hawk04-A@1#prop:key

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Let $U_{k}$ , $k = 1, \dots, d^{2}$ , be a Weyl unitary basis. The channel

ρ ⟼ \frac{1}{d ^{2}} k \sum U_{k} ρ U_{k}^{†} \otimes ∣ k ⟩ ⟨ k ∣_{K}

has constant marginals on the ciphertext $R$ and key $K$ , but exact recovery from $R K$ . This does not contradict no-hiding.

Proof. The Weyl twirl gives the ciphertext marginal $I_{R} / d$ ; the key marginal is uniformly distributed. Controlled application of $U_{k}^{†}$ recovers $ρ$ .

A pure complete dilation is

V ∣ ψ ⟩ = \frac{1}{d} k \sum U_{k} ∣ ψ ⟩_{R} ∣ k ⟩_{K} ∣ k ⟩_{F} .

The complete complement of $R$ is $K F$ , not $K$ . Its off-diagonal key-purifier matrix entries contain

\frac{1}{d ^{2}} Tr (U_{k} ρ U_{ℓ}^{†}) ∣ k, k ⟩ ⟨ ℓ, ℓ ∣ .

They are generally input dependent. Since the ciphertext channel is constant, Corollary cor:nohiding proves exact recoverability from the complete $K F$ output.

Displayed formulas in this passage

unlabeled · line 681
$ρ ⟼ \frac{1}{d ^{2}} k \sum U_{k} ρ U_{k}^{†} \otimes ∣ k ⟩ ⟨ k ∣_{K}$
unlabeled · line 697
$V ∣ ψ ⟩ = \frac{1}{d} k \sum U_{k} ∣ ψ ⟩_{R} ∣ k ⟩_{K} ∣ k ⟩_{F} .$
unlabeled · line 704
$\frac{1}{d ^{2}} Tr (U_{k} ρ U_{ℓ}^{†}) ∣ k, k ⟩ ⟨ ℓ, ℓ ∣ .$

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section: Mixed quantum encryption requires a complete purifier