Research proof index

The defining property of purification, that tracing out the purifying system gives the original mixed state.

Spectrum of direct product. There is a permutation σ so that the spectrum of the direct product of ρ₁ and ρ₂, as permuted under σ, is the pairwise products of the spectra of ρ₁ and ρ₂.

The specturm of a pure state is (1,0,0,...), i.e. a constant distribution.

The summatory Möbius function has sublinear growth: as $x\to\infty$, $\sum_{n<\lfloor x\rfloor}\mu(n)=o(x).$ This is the Möbius-function form of the prime number theorem.

Multidistance is unchanged when a finite family of random variables is reindexed along an equality $m'=m$. This says that the quantity depends on the family, not on the particular equal presentation of its finite index type.

Let m ≥ 2, and let X_[m] be a tuple of G-valued random variables. Let W := ∑ X_i. Then d[W;-W] ≤ 2 D[X_i].