Manuscript Oai280-C@1 · Oai280-C.tex · proposition · 7d44b8c0

Proposition: Normalization and physical implementation

One connected two-cell encoding · lines 717–740 · proof included or linked

Oai280-C@1#prop:W

Both $F$ and $W$ are isometries. If

C = (⟨ Φ_{2} ∣_{e e^{'}}) (V_{L} \otimes V_{R}),

then $W = 2 C$ and $C^{†} C = I_{L S} /4$ . Equation (eq:W) is a deterministic isometric implementation: prepare one Bell pair and apply the two isometries $F$ . It does not postselect a Bell outcome.

Proof. Single-leg privacy gives

a \sum V_{a, e; l} V_{a, f; m} = δ_{e f} δ_{l m} /2,

so $F^{†} F = I_{4}$ . Equation (eq:W) now immediately gives $W^{†} W = I_{4}$ . Direct contraction with the normalized Bell bra gives $C_{a, c; l, s} = 2^{- 1/2} \sum_{e} V_{a, e; l} V_{c, e; s}$ , proving $W = 2 C$ and the claimed success operator.

Displayed formulas in this passage

unlabeled · line 721
$C = (⟨ Φ_{2} ∣_{e e^{'}}) (V_{L} \otimes V_{R}),$
unlabeled · line 732
$a \sum V_{a, e; l} V_{a, f; m} = δ_{e f} δ_{l m} /2,$

Proposition: Normalization and physical implementation

Displayed formulas in this passage

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