2023/10/02 by Krishna, K. Mahesh
#42C15 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2310.01450
Let (Ω, μ), (Δ, ν) be measure spaces and \τα\α∈ Ω, \ωβ\β∈ Δ be 1-bounded continuous Parseval frames for a Hilbert space H. Then we show that (1) log (μ(Ω)ν(Δ))≥ Sτ(h)+Sω(h)≥ -2 log (\frac1+ supα∈ Ω, β∈ Δ|⟨τα, ωβ⟩|2) , ∀ h ∈ Hτ∩ Hω, where amp;Hτ:= \h1 ∈ H: ⟨ h1 , τα⟩ ≠ 0, α∈ Ω\, Hω:= \h2 ∈ H: ⟨ h2, ωβ⟩ ≠ 0, β∈ Δ\,
amp;Sτ(h):= -∫Ω| ⟨ (h)/(‖h‖), τα⟩ |2log | ⟨ (h)/(‖h‖), τα⟩ |2 dμ(α), ∀ h ∈ Hτ,
amp; Sω(h):= -∫Δ| ⟨ (h)/(‖h‖), ωβ⟩ |2log | ⟨ (h)/(‖h‖), ωβ⟩ |2 dν(β), ∀ h ∈ Hω. We call Inequality (1) as Continuous Deutsch Uncertainty Principle. Inequality (1) improves the uncertainty principle obtained by Deutsch [Phys. Rev. Lett., 1983]. We formulate Kraus conjecture for 1-bounded continuous Parseval frames. We also derive continuous Deutsch uncertainty principles for Banach spaces.