Modular Deutsch Entropic Uncertainty Principle

1. Introduction

Let $\mathcal{H}$ be a finite dimensional Hilbert space. Given an orthonormal basis ${\left\{{\tau }_j\right\}}_{j=1}^n$ for $\mathcal{H}$, the Shannon entropy at a point $h\in {\mathcal{H}}_{\tau }$ is defined as

$$\begin{array}{c}\hfill S_{\tau }:=\left(h\right)-\sum _{j=1}^n{\left|〈h,{\tau }_j〉\right|}^{2}log{\left|〈h,{\tau }_j〉\right|}^2,\end{array}$$

(1)

where ${\mathcal{H}}_{\tau }:=\{h\in \mathcal{H}:\parallel h\parallel =1,\langle h,{\tau }_j\rangle \ne 0,1\le j\le n\}$ [1]. In 1983, Deutsch derived following breakthrough entropic uncertainty principle for Shannon entropy [1].

Theorem 1

 ([1] (Deutsch Entropic Uncertainty Principle)). Let ${\left\{{\tau }_j\right\}}_{j=1}^n$, ${\left\{{\omega }_k\right\}}_{k=1}^n$ be two orthonormal bases for a finite dimensional Hilbert space $\mathcal{H}$. Then

$$\begin{array}{c}\hfill S_{\tau }\left(h\right)+S_{\omega }\left(h\right)\ge -2log\left(\frac{{\displaystyle 1+\underset{1\le j,k\le n}{max}\left|\langle {\tau }_j,{\omega }_k\rangle \right|}}{2}\right),\forall h\in {\mathcal{H}}_{\tau }\cap {\mathcal{H}}_{\omega }.\end{array}$$

(2)

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 $\mathcal{A}$ be a unital C*-algebra. A left module $\mathcal{E}$ over $\mathcal{A}$ is said to be a (left) Hilbert C*-module if there exists a map $\langle ·,·\rangle :\mathcal{E}\times \mathcal{E}\to \mathcal{A}$ such that the following hold.

(i)

$\langle x,x\rangle \ge 0$, $\forall x\in \mathcal{E}$. If $x\in \mathcal{E}$ satisfies $\langle x,x\rangle =0$, then $x=0$.

(ii)

$\langle x+y,z\rangle =\langle x,z\rangle +\langle y,z\rangle$, $\forall x,y,z\in \mathcal{E}$.

(iii)

$\langle ax,y\rangle =a\langle x,y\rangle$, $\forall x,y\in \mathcal{E}$, $\forall a\in \mathcal{A}$.

(iv)

$\langle x,y\rangle ={\langle y,x\rangle }^{*}$, $\forall x,y\in \mathcal{E}$.

(v)

$\mathcal{E}$ is complete w.r.t. the norm $\parallel x\parallel :=\sqrt{\parallel \langle x,x\rangle \parallel }$, $\forall x\in \mathcal{E}$.

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 $\mathcal{E}$ is a Hilbert C*-module over a unital C*-algebra $\mathcal{A}$, then

$$\begin{array}{c}\hfill \parallel \langle x,z\rangle \langle z,y\rangle \parallel \le \frac{1}{2}\left(\parallel x\parallel \parallel y\parallel +\parallel \langle x,y\rangle \parallel \right),\forall x,y,z\in \mathcal{E},\langle z,z\rangle =1.\end{array}$$

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 $\mathcal{E}$ be a Hilbert C*-module over a C*-algebra $\mathcal{A}$. A collection ${\left\{{\tau }_j\right\}}_{j=1}^{\infty }$ in $\mathcal{E}$ is said to be a (modular) Parseval frame for $\mathcal{E}$ if

$$\begin{array}{c}\hfill x=\sum _{j=1}^{\infty }\langle x,{\tau }_j\rangle {\tau }_j,\forall x\in \mathcal{E}.\end{array}$$

A collection ${\left\{{\tau }_j\right\}}_{j=1}^{\infty }$ in a Hilbert C*-module $\mathcal{E}$ over unital C*-algebra $\mathcal{A}$ with identity 1 is said to have unit inner product if

$$\begin{array}{c}\hfill \langle {\tau }_j,{\tau }_j\rangle =1,\forall j\in \mathbb{N}.\end{array}$$

In analogy with Equation (1), given a unit inner product Parseval frame ${\left\{{\tau }_j\right\}}_{j=1}^{\infty }$ for $\mathcal{E}$, we define modular Shannon entropy at a point $x\in {\mathcal{E}}_{\tau }$ is defined as

$$\begin{array}{c}\hfill S_{\tau }\left(x\right)-\sum _{j=1}^{\infty }\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle log\left(\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle \right)\end{array}$$

(3)

where ${\mathcal{E}}_{\tau }:=\{x\in \mathcal{E}:\langle x,x\rangle =1,\langle x,{\tau }_j\rangle \ne 0,\forall j\in \mathbb{N}\}$.

Theorem 3

 (Modular Deutsch Entropic Uncertainty Principle). Let $\mathcal{E}$ be a Hilbert C*-module over a commutative unital C*-algebra $\mathcal{A}$. Let ${\left\{{\tau }_j\right\}}_{j=1}^{\infty }$, ${\left\{{\omega }_k\right\}}_{k=1}^{\infty }$ be two Parseval frames for $\mathcal{E}$. Then

$$\begin{array}{c}\hfill S_{\tau }\left(x\right)+S_{\omega }\left(x\right)\ge -2log\left(\frac{{\displaystyle 1+\underset{j,k\in \mathbb{N}}{sup}\parallel \langle {\tau }_j,{\omega }_k\rangle \parallel }}{2}\right),\forall x\in {\mathcal{E}}_{\tau }\cap {\mathcal{E}}_{\omega }.\end{array}$$

Proof. 

Let $x\in {\mathcal{E}}_{\tau }\cap {\mathcal{E}}_{\omega }$. Using the Parseval frame property, the commutativity of C*-algebra, Theorem 2 and the result that ‘function logarithm is operator monotone’ [11], we get

$$\begin{array}{cc}\hfill S_{\tau }\left(x\right)+S_{\omega }\left(x\right)& =-\sum _{j=1}^{\infty }\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle log\left(\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle \right)-\sum _{k=1}^{\infty }\langle x,{\omega }_k\rangle \langle {\omega }_k,x\rangle log\left(\langle x,{\omega }_k\rangle \langle {\omega }_k,x\rangle \right)\hfill \\ \hfill & =-\sum _{j=1}^{\infty }\sum _{k=1}^{\infty }\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle \langle x,{\omega }_k\rangle \langle {\omega }_k,x\rangle \left[log\left(\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle \right)+log\left(\langle x,{\omega }_k\rangle \langle {\omega }_k,x\rangle \right)\right]\hfill \\ \hfill & =-\sum _{j=1}^{\infty }\sum _{k=1}^{\infty }\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle \langle x,{\omega }_k\rangle \langle {\omega }_k,x\rangle log\left(\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle \langle x,{\omega }_k\rangle \langle {\omega }_k,x\rangle \right)\hfill \\ \hfill & =-\sum _{j=1}^{\infty }\sum _{k=1}^{\infty }\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle \langle x,{\omega }_k\rangle \langle {\omega }_k,x\rangle log\left(\langle {\tau }_j,x\rangle \langle x,{\omega }_k\rangle \langle {\omega }_k,x\rangle \langle x,{\tau }_j\rangle \right)\hfill \\ \hfill & \ge -\sum _{j=1}^{\infty }\sum _{k=1}^{\infty }\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle \langle x,{\omega }_k\rangle \langle {\omega }_k,x\rangle log\left(\frac{[\parallel {\tau }_j\parallel \parallel {\omega }_k\parallel +\parallel \langle {\tau }_j,{\omega }_k\rangle \parallel \left]\right[\parallel {\omega }_k\parallel \parallel {\tau }_j\parallel +\parallel \langle {\omega }_k,{\tau }_j\rangle \parallel ]}{4}\right)\hfill \\ \hfill & =-\sum _{j=1}^{\infty }\sum _{k=1}^{\infty }\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle \langle x,{\omega }_k\rangle \langle {\omega }_k,x\rangle log\left(\frac{(\parallel {\tau }_j\parallel \parallel {\omega }_k\parallel +\parallel \langle {\tau }_j,{\omega }_k\rangle {\parallel )}^2}{4}\right)\hfill \\ \hfill & =-2\sum _{j=1}^{\infty }\sum _{k=1}^{\infty }\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle \langle x,{\omega }_k\rangle \langle {\omega }_k,x\rangle log\left(\frac{\parallel {\tau }_j\parallel \parallel {\omega }_k\parallel +\parallel \langle {\tau }_j,{\omega }_k\rangle \parallel }{2}\right)\hfill \\ \hfill & \ge -2\sum _{j=1}^{\infty }\sum _{k=1}^{\infty }\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle \langle x,{\omega }_k\rangle \langle {\omega }_k,x\rangle log\left(\frac{1+\parallel \langle {\tau }_j,{\omega }_k\rangle \parallel }{2}\right)\hfill \\ \hfill & \ge -2\sum _{j=1}^{\infty }\sum _{k=1}^{\infty }\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle \langle x,{\omega }_k\rangle \langle {\omega }_k,x\rangle log\left(\frac{1+{sup}_{j,k\in \mathbb{N}}\parallel \langle {\tau }_j,{\omega }_k\rangle \parallel }{2}\right)\hfill \\ \hfill & =-2log\left(\frac{1+{sup}_{j,k\in \mathbb{N}}\parallel \langle {\tau }_j,{\omega }_k\rangle \parallel }{2}\right)\sum _{j=1}^{\infty }\sum _{k=1}^{\infty }\langle x,{\tau }_j\rangle \langle {\tau }_j,x\rangle \langle x,{\omega }_k\rangle \langle {\omega }_k,x\rangle \hfill \\ \hfill & =-2log\left(\frac{1+{sup}_{j,k\in \mathbb{N}}\parallel \langle {\tau }_j,{\omega }_k\rangle \parallel }{2}\right)\langle x,x\rangle \langle x,x\rangle \hfill \\ \hfill & =-2log\left(\frac{1+{sup}_{j,k\in \mathbb{N}}\parallel \langle {\tau }_j,{\omega }_k\rangle \parallel }{2}\right).\hfill \end{array}$$

□

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 ${\left\{{\tau }_j\right\}}_{j=1}^n$, ${\left\{{\omega }_k\right\}}_{k=1}^n$ be two orthonormal bases for a finite dimensional Hilbert space $\mathcal{H}$. Then

$$\begin{array}{c}\hfill S_{\tau }\left(h\right)+S_{\omega }\left(h\right)\ge -2log\left({\displaystyle \underset{1\le j,k\le n}{max}\left|\langle {\tau }_j,{\omega }_k\rangle \right|}\right),\forall h\in {\mathcal{H}}_{\tau }\cap {\mathcal{H}}_{\omega }.\end{array}$$

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 ${\left\{{\tau }_j\right\}}_{j=1}^n$, ${\left\{{\omega }_k\right\}}_{k=1}^m$ be two Parseval frames for a finite dimensional Hilbert space $\mathcal{H}$. Then

$$\begin{array}{c}\hfill S_{\tau }\left(h\right)+S_{\omega }\left(h\right)\ge -2log\left({\displaystyle \underset{1\le j\le n,1\le k\le m}{max}\left|\langle {\tau }_j,{\omega }_k\rangle \right|}\right),\forall h\in {\mathcal{H}}_{\tau }\cap {\mathcal{H}}_{\omega }.\end{array}$$

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 $\mathcal{E}$ be a Hilbert C*-module over a commutative unital C*-algebra $\mathcal{A}$. Let ${\left\{{\tau }_j\right\}}_{j=1}^{\infty }$, ${\left\{{\omega }_k\right\}}_{k=1}^{\infty }$ be two Parseval frames for $\mathcal{E}$. Then

$$\begin{array}{c}\hfill S_{\tau }\left(x\right)+S_{\omega }\left(x\right)\ge -2log\left({\displaystyle \underset{j,k\in \mathbb{N}}{sup}\parallel \langle {\tau }_j,{\omega }_k\rangle \parallel }\right),\forall x\in {\mathcal{E}}_{\tau }\cap {\mathcal{E}}_{\omega }.\end{array}$$