Quantum smooth uncertainty principles for von Neumann bi-algebras

Abstract

In this article, we prove various smooth uncertainty principles on von Neumann bi-algebras, which unify numbers of uncertainty principles on quantum symmetries, such as subfactors, and fusion bi-algebras etc, studied in quantum Fourier analysis. We also obtain Widgerson-Wigderson type uncertainty principles for von Neumann bi-algebras. Moreover, we give a complete answer to a conjecture proposed by A. Wigderson and Y. Wigderson.

Quantum convolution inequalities on Frobenius von Neumann algebras

“…In 2021, A. Wigderson and Y. Wigderson [27] unified various classical (smooth) uncertainty principles related to discrete Fourier transforms. In [9], the authors unified various quantum (smooth) uncertainty principles on quantum symmetries related to quantum Fourier transforms.…”

“…For any positive operator x ∈ M, p ∈ [1, ∞], ǫ ∈ [0, 1], the (p, ǫ) smooth entropy of x is defined byH p ǫ (x) := sup{H(y) : y ∈ M, y ≥ 0, y − x p ≤ ǫ}. Remark 4.1.We refer to another smooth entropy studied in quantum smooth uncertainty principles in[9] and smooth Renyi entropy studied by R. Renner and S. Wolf in quantum information in[23].4.2.2. Continuity of convolution entropy.…”

“…|λ j (x i ) − λ j (y i )| ≤ r/2.By Lemma 3.26 in[9], we haveλ(x) − λ(y) 1 = x i ) − λ j (y i )| ≤ x − y 1 . (4.13)Let f (t) = −t log t. We have|H(x) − H(y)| = m i=1 δ i Tr i (x i log x i − y i log y i ) (λ j (x i )) − f (λ j (y i )) |λ j (x i ) − λ j (y i )| + 2| log r||λ j (x i ) − λ j (y i )| Lemma 4.f ( λ(x) − λ(y) 1 ) + (log d + 2| log r|) λ(x) − λ(y) 1 Jensen's inequality.By Hölder's inequality, we haveλ(x) − λ(y) 1 ≤ x − y 1 ≤ d 1−1/p x − y p ≤ d 1−1/p ǫ.…”