Manuscript Oai069-A@1 · Oai069-A.tex · section · a4913aab

The encoded local-system multiplicity

Phase 2: positive realization and access-restricted recovery › The encoded local-system multiplicity · lines 650–675

Oai069-A@1#eq:encoding

Let $R_{1}, R_{2}$ be qubits and $B ≅ K$ an inaccessible gauge register. Let $H$ , $X$ , and $Z$ denote the usual qubit Hadamard and Pauli matrices. Define

ϵ_{k} = k mod 2, U_{k} = CNOT_{1 \to 2} (H Z^{ϵ_{k}} \otimes I) .

The encoder is the isometry

V : ∣ k ⟩ ∣ ψ ⟩_{C} ∣ ξ ⟩_{K} ⟼ v_{k}_{S} \otimes U_{k} (∣ ψ ⟩_{R_{1}} ∣ 0 ⟩_{R_{2}}) \otimes ∣ ξ ⟩_{B} .

The receiver gets $S, R_{1}, R_{2}$ , not $B$ .

The source logical label has monodromy $ω^{k}$ . The encoded monodromy acts as $S_{1}$ on $S$ , with identity on the multiplicity and gauge registers. Thus $V$ intertwines the declared local systems.

Displayed formulas in this passage

Oai069-A@1#eq:encoding · line 661
$V : ∣ k ⟩ ∣ ψ ⟩_{C} ∣ ξ ⟩_{K} ⟼ v_{k}_{S} \otimes U_{k} (∣ ψ ⟩_{R_{1}} ∣ 0 ⟩_{R_{2}}) \otimes ∣ ξ ⟩_{B} .$
unlabeled · line 655
$ϵ_{k} = k mod 2, U_{k} = CNOT_{1 \to 2} (H Z^{ϵ_{k}} \otimes I) .$

The encoded local-system multiplicity

Displayed formulas in this passage

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