Manuscript Oai280-B@1 · Oai280-B.tex · proposition · 6d92ab75

Proposition: Six-leg perfectness

Privacy, perfectness, and exact access-dependent recovery › The six-leg perfect state · lines 456–475 · proof included or linked

Oai280-B@1#prop:perfect

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Every reduction of $∣ T ⟩$ to three of its six qubit legs is $I_{8} /8$ . Consequently every reduction to at most three legs is maximally mixed.

Proof. A triple containing $Q$ consists of $Q$ and two boundary legs $B$ . Lemma lem:privacy, applied to the Bell matrix-unit expansion, gives

ρ_{QB} = \frac{I _{Q}}{2} \otimes \frac{I _{B}}{4} = \frac{I _{8}}{8} .

A triple not containing $Q$ has a complementary triple containing $Q$ . Since $∣ T ⟩$ is pure and both triple spaces have dimension eight, their reduced states have the same eigenvalues. The first calculation therefore makes the second triple maximally mixed too. Smaller reductions follow by partial trace.

Displayed formulas in this passage

unlabeled · line 467
$ρ_{QB} = \frac{I _{Q}}{2} \otimes \frac{I _{B}}{4} = \frac{I _{8}}{8} .$

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section: The six-leg perfect state