Manuscript Oai272-A@1 · Oai272-A.tex · theorem · a40828a0

Theorem: A PPT pair gives a 2d+1-dimensional channel

The channel-completion interface · lines 1227–1287 · proof included or linked

Oai272-A@1#thm:completion, Oai272-A@1#eq:completion, Oai272-A@1#eq:corner

Let $F, G : M_{d} (C) \to M_{d} (C)$ be PPT. There are explicit $s, t > 0$ and a trace-preserving PPT channel $Θ : M_{2 d + 1} (C) \to M_{2 d + 1} (C)$ whose squared Choi matrix has a coordinate-local corner equal to $s t J (GF)$ . If $J (GF)$ is entangled, $Θ^{2}$ is not entanglement breaking.

Proof. Set

A = F^{†} (I), B = G^{†} (I), s = \frac{1}{1 + Tr A}, t = \frac{1}{1 + Tr B},

D = I - s A, E = I - tB .

Since $A, B ⪰ 0$ ,

D ⪰ s I ≻ 0, E ⪰ t I ≻ 0.

Use the real coordinate inclusions $V_{0}, V_{1}$ in

C^{2 d + 1} = C^{d}_{0} \oplus C^{d}_{1} \oplus C ∣ ⋆ ⟩, P_{⋆} = ∣ ⋆ ⟩ ⟨ ⋆ ∣.

Write $X_{j j} = V_{j}^{†} X V_{j}$ . Define

Θ (X) = s V_{1} F (X_{00}) V_{1}^{†} + t V_{0} G (X_{11}) V_{0}^{†} + [Tr (D X_{00}) + Tr (E X_{11}) + ⟨ ⋆ ∣ X ∣ ⋆ ⟩] P_{⋆} .

The moving terms are CP. Each flag term is a positive-effect measurement followed by a fixed preparation, hence CP. The identities $s A + D = tB + E = I$ prove trace preservation.

Output transpose replaces $F, G$ by $T_{d} F, T_{d} G$ in the moving branches and fixes $P_{⋆}$ . All branches remain CP, so $Θ$ is PPT. This uses ordinary complex-linear transpose; it does not conjugate the scalar flag coefficient.

The flag is absorbing. For every $Y \in M_{d} (C)$ ,

V_{0}^{†} Θ^{2} (V_{0} Y V_{0}^{†}) V_{0} = s t GF (Y) .

Expanding in input matrix units gives

(V_{0}^{†} \otimes V_{0}^{†}) J (Θ^{2}) (V_{0} \otimes V_{0}) = s t J (GF) .

Local compression preserves separability. Therefore a separable $J (Θ^{2})$ would force separability of its entangled corner, a contradiction.

Displayed formulas in this passage

Oai272-A@1#eq:completion · line 1255
$Θ (X) = s V_{1} F (X_{00}) V_{1}^{†} + t V_{0} G (X_{11}) V_{0}^{†} + [Tr (D X_{00}) + Tr (E X_{11}) + ⟨ ⋆ ∣ X ∣ ⋆ ⟩] P_{⋆} .$
Oai272-A@1#eq:corner · line 1279
$(V_{0}^{†} \otimes V_{0}^{†}) J (Θ^{2}) (V_{0} \otimes V_{0}) = s t J (GF) .$
unlabeled · line 1238
$A = F^{†} (I), B = G^{†} (I), s = \frac{1}{1 + Tr A}, t = \frac{1}{1 + Tr B},$
unlabeled · line 1242
$D = I - s A, E = I - tB .$
unlabeled · line 1246
$D ⪰ s I ≻ 0, E ⪰ t I ≻ 0.$
unlabeled · line 1250
$C^{2 d + 1} = C^{d}_{0} \oplus C^{d}_{1} \oplus C ∣ ⋆ ⟩, P_{⋆} = ∣ ⋆ ⟩ ⟨ ⋆ ∣.$
unlabeled · line 1275
$V_{0}^{†} Θ^{2} (V_{0} Y V_{0}^{†}) V_{0} = s t GF (Y) .$

Theorem: A PPT pair gives a 2d+1-dimensional channel

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