Manuscript Hawk06-A@1 · Hawk06-A.tex · section · c59fe67c

An exact non-Haar island with an explicit decoder

An exact non-Haar island with an explicit decoder · lines 1292–1322

Hawk06-A@1#eq:padV

The Haar theorem is not the only finite existence result. The following construction supplies an exact channel and decoder.

Let $d = 2^{ℓ}$ , $h = d^{2}$ , $f = 1$ , and label $H, E$ by $d^{2}$ Pauli keys $z$ . Let ${W_{z}}$ be the $ℓ$ -qubit Pauli operators, ignoring their irrelevant overall phases. Prepare

∣ Φ_{d^{2}} ⟩_{H E} = \frac{1}{d} z \sum ∣ z ⟩_{H} ∣ z ⟩_{E} .

Apply the controlled unitary

U = z \sum W_{z}^{L} \otimes ∣ z ⟩ ⟨ z ∣_{H},

and emit the $ℓ$ ciphertext qubits, stopping before emitting any key qubit.

The endpoint complete isometry is

V ∣ ψ ⟩ = \frac{1}{d} z \sum W_{z} ∣ ψ ⟩_{R} ∣ z ⟩_{E} ∣ z ⟩_{B}, b = dim B = d^{2} > 1.

Displayed formulas in this passage

Hawk06-A@1#eq:padV · line 1314
$V ∣ ψ ⟩ = \frac{1}{d} z \sum W_{z} ∣ ψ ⟩_{R} ∣ z ⟩_{E} ∣ z ⟩_{B}, b = dim B = d^{2} > 1.$
unlabeled · line 1302
$∣ Φ_{d^{2}} ⟩_{H E} = \frac{1}{d} z \sum ∣ z ⟩_{H} ∣ z ⟩_{E} .$
unlabeled · line 1307
$U = z \sum W_{z}^{L} \otimes ∣ z ⟩ ⟨ z ∣_{H},$

An exact non-Haar island with an explicit decoder

Displayed formulas in this passage

Nearby in this section