Manuscript Hawk03-A@1 · Hawk03-A.tex · theorem · 9a109c12

Theorem: Two independent Bell purifiers

Independent-factor AMPS benchmarks › A sharp two-purifier test · lines 438–486 · proof included or linked

Hawk03-A@1#thm:twopurifiers, Hawk03-A@1#eq:projectionmonogamy

Let $A, B, F$ be independent $d$ -dimensional factors, $d \geq 2$ . Define

∣ Φ_{d} ⟩ = \frac{1}{d} j = 0 \sum^{d - 1} ∣ j ⟩ ∣ j ⟩,

and, on $A B F$ ,

P = ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣_{A B} \otimes I_{F}, Q = I_{A} \otimes ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣_{B F} .

For every density operator $ρ_{A B F}$ ,

Tr (P ρ) + Tr (Qρ) \leq 1 + \frac{1}{d} .

The bound is sharp. Consequently, certificates $Tr (P ρ) \geq 1 - η_{1}$ and $Tr (Qρ) \geq 1 - η_{2}$ require

η_{1} + η_{2} \geq 1 - \frac{1}{d} .

Proof. Let

T_{1} x = ∣ Φ_{d} ⟩_{A B} \otimes x_{F}, T_{2} y = y_{A} \otimes ∣ Φ_{d} ⟩_{B F} .

Both are isometries, $P = T_{1} T_{1}^{†}$ and $Q = T_{2} T_{2}^{†}$ , and direct expansion gives $T_{1}^{†} T_{2} = d^{- 1} I$ after identifying $A$ with $F$ . The Gram operator of $T = (T_{1} T_{2})$ is

T^{†} T = (I d^{- 1} I d^{- 1} I I) .

Its largest eigenvalue is $1 + d^{- 1}$ . The nonzero eigenvalues of $T T^{†} = P + Q$ are the same, proving $∥ P + Q ∥_{\infty} = 1 + d^{- 1}$ and hence (eq:projectionmonogamy). A normalized top eigenvector attains the bound. The last statement follows by substituting the two lower bounds.

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Hawk03-A@1#eq:projectionmonogamy · line 451
$Tr (P ρ) + Tr (Qρ) \leq 1 + \frac{1}{d} .$
unlabeled · line 442
$∣ Φ_{d} ⟩ = \frac{1}{d} j = 0 \sum^{d - 1} ∣ j ⟩ ∣ j ⟩,$
unlabeled · line 446
$P = ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣_{A B} \otimes I_{F}, Q = I_{A} \otimes ∣ Φ_{d} ⟩ ⟨ Φ_{d} ∣_{B F} .$
unlabeled · line 458
$η_{1} + η_{2} \geq 1 - \frac{1}{d} .$
unlabeled · line 465
$T_{1} x = ∣ Φ_{d} ⟩_{A B} \otimes x_{F}, T_{2} y = y_{A} \otimes ∣ Φ_{d} ⟩_{B F} .$
unlabeled · line 473
$T^{†} T = (I d^{- 1} I d^{- 1} I I) .$

Theorem: Two independent Bell purifiers

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