Manuscript Em06-B@1 · Em06-B-v1.tex · proposition · 61e38859

Proposition: Uncertainty retained at fluctuation scale

Normalized fluctuations: what remains quantum · lines 1113–1136 · proof included or linked

Em06-B@1#prop:uncertainty, Em06-B@1#eq:uncertainty

For a residual state with finite number expectation,

C_{ii}^{ρ} C_{j j}^{ρ} \geq (C_{ij}^{ρ})^{2} + A_{ij}^{2} .

Thus nonzero $A_{ij}$ prevents both normalized variances from vanishing in any such state.

Proof. The Hilbert–Schmidt vectors

(Φ (v_{i}) - b_{i}^{ρ}) ρ^{1/2}, (Φ (v_{j}) - b_{j}^{ρ}) ρ^{1/2}

have squared norms $C_{ii}^{ρ}, C_{j j}^{ρ}$ , and inner product $C_{ij}^{ρ} + i A_{ij}$ , by Proposition prop:covariance. Hilbert-space Cauchy–Schwarz proves the inequality.

Displayed formulas in this passage

Em06-B@1#eq:uncertainty · line 1116
$C_{ii}^{ρ} C_{j j}^{ρ} \geq (C_{ij}^{ρ})^{2} + A_{ij}^{2} .$
unlabeled · line 1127
$(Φ (v_{i}) - b_{i}^{ρ}) ρ^{1/2}, (Φ (v_{j}) - b_{j}^{ρ}) ρ^{1/2}$

Proposition: Uncertainty retained at fluctuation scale

Displayed formulas in this passage

Nearby in this section