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Uncertainty principles on $C^{*}$-algebras

Saptak Bhattacharya

TL;DR

The paper addresses uncertainty relations for observables in a unital C*-algebra, focusing on obtaining non-vanishing, n-independent lower bounds even for odd numbers of observables. It provides an elementary short proof of Robertson's Standard Uncertainty Principle and derives new lower bounds using the sum/difference of positive matrices and the matrix geometric mean, including the bound det|M - M^T| ≤ det Cov_phi(tilde{x}). It then yields norm-based and trace-based inequalities, notably a k-observable bound in trace norm ||[x_i,x_j]||_1 ≤ (k-1) tr({x_i,x_j}) and an extension to von Neumann algebras with a faithful trace tau. These results deliver robust, non-vanishing uncertainty relations with potential impact in quantum information and operator-algebra contexts.

Abstract

In this paper we prove some uncertainty bounds for commutators and anti-commutators of observables in a $C^*$-algebra. We give a short, elementary proof of Robertson's Standard Uncertaity Principle in this setting. We also prove some other uncertainty relations for which the lower bound doesn't vanish for any number of observables.

Uncertainty principles on $C^{*}$-algebras

TL;DR

The paper addresses uncertainty relations for observables in a unital C*-algebra, focusing on obtaining non-vanishing, n-independent lower bounds even for odd numbers of observables. It provides an elementary short proof of Robertson's Standard Uncertainty Principle and derives new lower bounds using the sum/difference of positive matrices and the matrix geometric mean, including the bound det|M - M^T| ≤ det Cov_phi(tilde{x}). It then yields norm-based and trace-based inequalities, notably a k-observable bound in trace norm ||[x_i,x_j]||_1 ≤ (k-1) tr({x_i,x_j}) and an extension to von Neumann algebras with a faithful trace tau. These results deliver robust, non-vanishing uncertainty relations with potential impact in quantum information and operator-algebra contexts.

Abstract

In this paper we prove some uncertainty bounds for commutators and anti-commutators of observables in a -algebra. We give a short, elementary proof of Robertson's Standard Uncertaity Principle in this setting. We also prove some other uncertainty relations for which the lower bound doesn't vanish for any number of observables.
Paper Structure (2 sections, 9 theorems, 63 equations)

This paper contains 2 sections, 9 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Main results

Key Result

Theorem 1

Let $A$ be a positive definite density matrix. Let $\{H_j\}_{j=1}^k$ be Hermitian and let $f:(0,\infty)\to\mathbb{R}$ be a positive operator monotone function satisfying $f(1)=1$ and $xf(\frac{1}{x})=f(x)$ for all $x\in (0,\infty)$. Then

Theorems & Definitions (19)

  • Theorem
  • Lemma 1
  • proof
  • Remark
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • proof
  • ...and 9 more