Prerpints.org logo
Preprint
Article

Modular Deutsch Entropic Uncertainty Principle

Altmetrics

Downloads

123

Views

101

Comments

0

This version is not peer-reviewed

Submitted:

14 June 2024

Posted:

14 June 2024

You are already at the latest version

Alerts
Abstract
Khosravi, Drnov\v{s}ek and Moslehian [\textit{Filomat, 2012}] derived Buzano inequality for Hilbert C*-modules. Using this inequality we derive Deutsch entropic uncertainty principle for Hilbert C*-modules over commutative unital C*-algebras.
Keywords: 
Subject: Computer Science and Mathematics  -   Analysis

MSC:  46L08; 42C15; 46L05

1. Introduction

Let H be a finite dimensional Hilbert space. Given an orthonormal basis { τ j } j = 1 n for H , the Shannon entropy at a point h ∈ H τ is defined as
S τ : = ( h ) − ∑ j = 1 n h , τ j 2 log h , τ j 2 ,
where H τ : = { h ∈ H : ∥ h ∥ = 1 , 〈 h , τ j 〉 ≠ 0 , 1 ≤ j ≤ n } [1]. In 1983, Deutsch derived following breakthrough entropic uncertainty principle for Shannon entropy [1].
Theorem 1
 ([1] (Deutsch Entropic Uncertainty Principle)). Let { τ j } j = 1 n , { ω k } k = 1 n be two orthonormal bases for a finite dimensional Hilbert space H . Then
S τ ( h ) + S ω ( h ) ≥ − 2 log 1 + max 1 ≤ j , k ≤ n | 〈 τ j , ω k 〉 | 2 , ∀ h ∈ H τ ∩ H ω .
The inequality (2) is recently derived for Banach spaces [2]. It is observed very recently that using Buzano inequality (see [3,4,5]) one can provide a simple proof of Theorem 1 (see Corollary 1 in [2]). As Hilbert C*-modules became more important in noncommutative geometry, we are mainly motivated from the following problem. What is the modular version of Theorem 1? Hilbert C*-modules are first introduced by Kaplansky [6] for modules over commutative C*-algebras and later developed for modules over arbitrary C*-algebras by Paschke [7] and Rieffel [8].
Definition 1
 ([6,7,8]). Let  A  be a unital C*-algebra. A left module  E over A  is said to be a (left) Hilbert C*-module if there exists a map  〈 · , · 〉 : E × E → A  such that the following hold.
(i)
〈 x , x 〉 ≥ 0 , ∀ x ∈ E . If x ∈ E satisfies 〈 x , x 〉 = 0 , then x = 0 .
(ii)
〈 x + y , z 〉 = 〈 x , z 〉 + 〈 y , z 〉 , ∀ x , y , z ∈ E .
(iii)
〈 a x , y 〉 = a 〈 x , y 〉 , ∀ x , y ∈ E , ∀ a ∈ A .
(iv)
〈 x , y 〉 = 〈 y , x 〉 * , ∀ x , y ∈ E .
(v)
E is complete w.r.t. the norm ∥ x ∥ : = ∥ 〈 x , x 〉 ∥ , ∀ x ∈ E .
Our prime tool to derive modular Deutsch uncertainty is the following modular Buzano inequality by Khosravi, Drnovšek, and Moslehian [9].
Theorem 2
([9] (Modular Buzano Inequality)). If E is a Hilbert C*-module over a unital C*-algebra A , then
∥ 〈 x , z 〉 〈 z , y 〉 ∥ ≤ 1 2 ∥ x ∥ ∥ y ∥ + ∥ 〈 x , y 〉 ∥ , ∀ x , y , z ∈ E , 〈 z , z 〉 = 1 .
In this paper we derive Theorem 1 for Hilbert C*-modules over commutative unital C*-algebras.

2. Modular Deutsch Entropic Uncertainty Principle

We begin by recalling the definition of frames for Hilbert C*-modules.
Definition 2
([10]). Let E be a Hilbert C*-module over a C*-algebra A . A collection { τ j } j = 1 ∞ in E is said to be a (modular) Parseval frame for E if
x = ∑ j = 1 ∞ 〈 x , τ j 〉 τ j , ∀ x ∈ E .
A collection { τ j } j = 1 ∞ in a Hilbert C*-module E over unital C*-algebra A with identity 1 is said to have unit inner product if
〈 τ j , τ j 〉 = 1 , ∀ j ∈ N .
In analogy with Equation (1), given a unit inner product Parseval frame { τ j } j = 1 ∞ for E , we define modular Shannon entropy at a point x ∈ E τ is defined as
S τ ( x ) − ∑ j = 1 ∞ 〈 x , τ j 〉 〈 τ j , x 〉 log ( 〈 x , τ j 〉 〈 τ j , x 〉 )
where E τ : = { x ∈ E : 〈 x , x 〉 = 1 , 〈 x , τ j 〉 ≠ 0 , ∀ j ∈ N } .
Theorem 3
 (Modular Deutsch Entropic Uncertainty Principle). Let E be a Hilbert C*-module over a commutative unital C*-algebra A . Let { τ j } j = 1 ∞ , { ω k } k = 1 ∞ be two Parseval frames for E . Then
S τ ( x ) + S ω ( x ) ≥ − 2 log 1 + sup j , k ∈ N ∥ 〈 τ j , ω k 〉 ∥ 2 , ∀ x ∈ E τ ∩ E ω .
Proof. 
Let x ∈ E τ ∩ E ω . Using the Parseval frame property, the commutativity of C*-algebra, Theorem 2 and the result that ‘function logarithm is operator monotone’ [11], we get
S τ ( x ) + S ω ( x ) = − ∑ j = 1 ∞ 〈 x , τ j 〉 〈 τ j , x 〉 log ( 〈 x , τ j 〉 〈 τ j , x 〉 ) − ∑ k = 1 ∞ 〈 x , ω k 〉 〈 ω k , x 〉 log ( 〈 x , ω k 〉 〈 ω k , x 〉 ) = − ∑ j = 1 ∞ ∑ k = 1 ∞ 〈 x , τ j 〉 〈 τ j , x 〉 〈 x , ω k 〉 〈 ω k , x 〉 log ( 〈 x , τ j 〉 〈 τ j , x 〉 ) + log ( 〈 x , ω k 〉 〈 ω k , x 〉 ) = − ∑ j = 1 ∞ ∑ k = 1 ∞ 〈 x , τ j 〉 〈 τ j , x 〉 〈 x , ω k 〉 〈 ω k , x 〉 log ( 〈 x , τ j 〉 〈 τ j , x 〉 〈 x , ω k 〉 〈 ω k , x 〉 ) = − ∑ j = 1 ∞ ∑ k = 1 ∞ 〈 x , τ j 〉 〈 τ j , x 〉 〈 x , ω k 〉 〈 ω k , x 〉 log ( 〈 τ j , x 〉 〈 x , ω k 〉 〈 ω k , x 〉 〈 x , τ j 〉 ) ≥ − ∑ j = 1 ∞ ∑ k = 1 ∞ 〈 x , τ j 〉 〈 τ j , x 〉 〈 x , ω k 〉 〈 ω k , x 〉 log [ ∥ τ j ∥ ∥ ω k ∥ + ∥ 〈 τ j , ω k 〉 ∥ ] [ ∥ ω k ∥ ∥ τ j ∥ + ∥ 〈 ω k , τ j 〉 ∥ ] 4 = − ∑ j = 1 ∞ ∑ k = 1 ∞ 〈 x , τ j 〉 〈 τ j , x 〉 〈 x , ω k 〉 〈 ω k , x 〉 log ( ∥ τ j ∥ ∥ ω k ∥ + ∥ 〈 τ j , ω k 〉 ∥ ) 2 4 = − 2 ∑ j = 1 ∞ ∑ k = 1 ∞ 〈 x , τ j 〉 〈 τ j , x 〉 〈 x , ω k 〉 〈 ω k , x 〉 log ∥ τ j ∥ ∥ ω k ∥ + ∥ 〈 τ j , ω k 〉 ∥ 2 ≥ − 2 ∑ j = 1 ∞ ∑ k = 1 ∞ 〈 x , τ j 〉 〈 τ j , x 〉 〈 x , ω k 〉 〈 ω k , x 〉 log 1 + ∥ 〈 τ j , ω k 〉 ∥ 2 ≥ − 2 ∑ j = 1 ∞ ∑ k = 1 ∞ 〈 x , τ j 〉 〈 τ j , x 〉 〈 x , ω k 〉 〈 ω k , x 〉 log 1 + sup j , k ∈ N ∥ 〈 τ j , ω k 〉 ∥ 2 = − 2 log 1 + sup j , k ∈ N ∥ 〈 τ j , ω k 〉 ∥ 2 ∑ j = 1 ∞ ∑ k = 1 ∞ 〈 x , τ j 〉 〈 τ j , x 〉 〈 x , ω k 〉 〈 ω k , x 〉 = − 2 log 1 + sup j , k ∈ N ∥ 〈 τ j , ω k 〉 ∥ 2 〈 x , x 〉 〈 x , x 〉 = − 2 log 1 + sup j , k ∈ N ∥ 〈 τ j , ω k 〉 ∥ 2 .
□
In 1988, Maassen and Uffink (motivated from the conjecture by Kraus made in 1987 [12]) improved Deutsch entropic uncertainty principle.
Theorem 4
 ([13] (Maassen-Uffink Entropic Uncertainty Principle)). Let  { τ j } j = 1 n , { ω k } k = 1 n  be two orthonormal bases for a finite dimensional Hilbert space H . Then
S τ ( h ) + S ω ( h ) ≥ − 2 log max 1 ≤ j , k ≤ n | 〈 τ j , ω k 〉 | , ∀ h ∈ H τ ∩ H ω .
In 2013, Ricaud and Torrésani [14] showed that orthonormal bases in Theorem 4 can be improved to Parseval frames.
Theorem 5
 ([14] (Ricaud-Torrésani Entropic Uncertainty Principle)). Let { τ j } j = 1 n , { ω k } k = 1 m be two Parseval frames for a finite dimensional Hilbert space H . Then
S τ ( h ) + S ω ( h ) ≥ − 2 log max 1 ≤ j ≤ n , 1 ≤ k ≤ m | 〈 τ j , ω k 〉 | , ∀ h ∈ H τ ∩ H ω .
Proofs of Theorems 4 and 5 use Riesz-Thorin interpolation (RTI). To the best of author’s knowledge, RTI does not exists for abstract Hilbert C*-modules. Therefore we end by formulating the following conjecture.
Conjecture 6
 (Modular Kraus Entropic Conjecture). Let E be a Hilbert C*-module over a commutative unital C*-algebra A . Let { τ j } j = 1 ∞ , { ω k } k = 1 ∞ be two Parseval frames for E . Then
S τ ( x ) + S ω ( x ) ≥ − 2 log sup j , k ∈ N ∥ 〈 τ j , ω k 〉 ∥ , ∀ x ∈ E τ ∩ E ω .

References

  1. Deutsch, D. Uncertainty in quantum measurements. Phys. Rev. Lett. 1983, 50, 631–633. [Google Scholar] [CrossRef]
  2. Krishna, K.M. Functional Deutsch Uncertainty Principle. J. Class. Anal. 2024, 23, 11–18. [Google Scholar] [CrossRef]
  3. Buzano, M.L. Generalizzazione della diseguaglianza di Cauchy-Schwarz. Rend. Sem. Mat. Univ. e Politec. Torino 1971/73, 31, 405–409 (1974). [Google Scholar]
  4. Fujii, M.; Kubo, F. Buzano’s inequality and bounds for roots of algebraic equations. Proc. Amer. Math. Soc. 1993, 117, 359–361. [Google Scholar] [CrossRef]
  5. Steele, J.M. The Cauchy-Schwarz master class: An introduction to the art of mathematical inequalities; Cambridge University Press: Cambridge, 2004; pp. x+306. [Google Scholar] [CrossRef]
  6. Kaplansky, I. Modules over operator algebras. Amer. J. Math. 1953, 75, 839–858. [Google Scholar] [CrossRef]
  7. Paschke, W.L. Inner product modules over B*-algebras. Trans. Amer. Math. Soc. 1973, 182, 443–468. [Google Scholar] [CrossRef]
  8. Rieffel, M.A. Induced representations of C*-algebras. Advances in Math. 1974, 13, 176–257. [Google Scholar] [CrossRef]
  9. Khosravi, M.; Drnovšek, R.; Moslehian, M.S. A commutator approach to Buzano’s inequality. Filomat 2012, 26, 827–832. [Google Scholar] [CrossRef]
  10. Frank, M.; Larson, D.R. Frames in Hilbert C*-modules and C*-algebras. J. Operator Theory 2002, 48, 273–314. [Google Scholar]
  11. Chansangiam, P. A survey on operator monotonicity, operator convexity, and operator means. Int. J. Anal. 2015. pp. Art. ID 649839, 8. [Google Scholar] [CrossRef]
  12. Kraus, K. Complementary observables and uncertainty relations. Phys. Rev. D 1987, 35, 3070–3075. [Google Scholar] [CrossRef] [PubMed]
  13. Maassen, H.; Uffink, J.B.M. Generalized entropic uncertainty relations. Phys. Rev. Lett. 1988, 60, 1103–1106. [Google Scholar] [CrossRef] [PubMed]
  14. Ricaud, B.; Torrésani, B. Refined support and entropic uncertainty inequalities. IEEE Trans. Inform. Theory 2013, 59, 4272–4279. [Google Scholar] [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.
Prerpints.org logo

Preprints.org is a free preprint server supported by MDPI in Basel, Switzerland.

facebook logotwitter logolinkedin logo
MDPI Initiatives
Important Links
Subscribe

Disclaimer

Terms of Use

Privacy Policy

© 2024 MDPI (Basel, Switzerland) unless otherwise stated