Functional Deutsch Uncertainty Principle

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Functional Deutsch Uncertainty Principle

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K. MAHESH KRISHNA^*

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Submitted:

14 July 2023

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17 July 2023

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Abstract

Let $\{f_j\}_{j=1}^n$ and $\{g_k\}_{k=1}^m$ be Parseval p-frames for a finite dimensional Banach space $\mathcal{X}$. Then we show that \begin{align}\label{UE} \log (nm)\geq S_f (x)+S_g (x)\geq -p \log \left(\displaystyle\sup_{y \in \mathcal{X}_f\cap \mathcal{X}_g, \|y\|=1}\left(\max_{1\leq j\leq n, 1\leq k\leq m}|f_j(y)g_k(y)|\right)\right), \quad \forall x \in \mathcal{X}_f\cap \mathcal{X}_g, \end{align} where \begin{align*} &\mathcal{X}_f\coloneqq \{z\in \mathcal{X}: f_j(z)\neq 0, 1\leq j \leq n\}, \quad \mathcal{X}_g\coloneqq \{w\in \mathcal{X}: g_k(w)\neq 0, 1\leq k \leq m\},\\ &S_f (x)\coloneqq -\sum_{j=1}^{n}\left|f_j\left(\frac{x}{\|x\|}\right)\right|^p\log \left|f_j\left(\frac{x}{\|x\|}\right)\right|^p, \quad S_g (x)\coloneqq -\sum_{k=1}^{m}\left|g_k\left(\frac{x}{\|x\|}\right)\right|^p\log \left|g_k\left(\frac{x}{\|x\|}\right)\right|^p, \quad \forall x \in \mathcal{X}_g. \end{align*} We call Inequality (1) as \textbf{Functional Deutsch Uncertainty Principle}. For Hilbert spaces, we show that Inequality (1) reduces to the uncertainty principle obtained by Deutsch \textit{[Phys. Rev. Lett., 1983]}. We also derive a dual of Inequality (1).

Keywords:

Subject: Computer Science and Mathematics - Analysis

MSC: 42C15

1. Introduction

Let

d \in N

and

\hat{} : L^{2} (R^{d}) \to L^{2} (R^{d})

be the unitary Fourier transform obtained by extending uniquely the bounded linear operator

\begin{matrix} \hat{} : L^{1} (R^{d}) \cap L^{2} (R^{d}) ∋ f \mapsto \hat{f} \in C_{0} (R^{d}); \hat{f} : R^{d} ∋ ξ \mapsto \hat{f} (ξ) \int_{R^{d}} f (x) e^{- 2 π i 〈 x, ξ 〉} d x \in C . \end{matrix}

The Shannon entropy at a function

f \in L^{2} (R^{d}) ∖ {0}

is defined as

\begin{matrix} S (f) - \int_{R^{d}} {|\frac{f (x)}{∥ f ∥}|}^{2} \log {|\frac{f (x)}{∥ f ∥}|}^{2} d x \end{matrix}

(with the convention

0 \log 0 = 0

) [1]. In 1957, Hirschman proved the following result [2].

Theorem 1.1.

[2] (Hirschman Inequality) For all

f \in L^{2} (R^{d}) ∖ {0}

\begin{matrix} S (f) + S (\hat{f}) \geq 0 . \end{matrix}

(2)

In the same paper [2] Hirschman conjectured that Inequality (2) can be improved to

\begin{matrix} S (f) + S (\hat{f}) \geq d (1 - \log 2), f \in L^{2} (R^{d}) ∖ {0} . \end{matrix}

(3)

Inequality (3) was proved independently in 1975 by Beckner [3] and Bialynicki-Birula and Mycielski [4].

Theorem 1.2.

[3,4] (Hirschman-Beckner-Bialynicki-Birula-Mycielski Uncertainty Principle) For all

f \in L^{2} (R^{d}) ∖ {0}

\begin{matrix} S (f) + S (\hat{f}) \geq d (1 - \log 2) . \end{matrix}

Now one naturally asks whether there is a finite dimensional version of Shannon entropy and uncertainty principle. Let

H

be a finite dimensional Hilbert space. Given an orthonormal basis

{τ_{j}}_{j = 1}^{n}

for

H

, the (finite) Shannon entropy at a point

h \in H_{τ}

is defined as

\begin{matrix} S_{τ} (h) - \sum_{j = 1}^{n} {|〈\frac{h}{∥ h ∥}, τ_{j}〉|}^{2} \log {|〈\frac{h}{∥ h ∥}, τ_{j}〉|}^{2} \geq 0, \end{matrix}

where

H_{τ} {h \in H : 〈 h, τ_{j} 〉 \neq 0, 1 \leq j \leq n}

[5]. In 1983, Deutsch derived following uncertainty principle for Shannon entropy which is fundamental to several developments in Mathematics and Physics [5].

Theorem 1.3.

[5] (Deutsch Uncertainty Principle) Let

{τ_{j}}_{j = 1}^{n}

{ω_{j}}_{j = 1}^{n}

be two orthonormal bases for a finite dimensional Hilbert space

H

. Then

\begin{matrix} 2 \log n \geq S_{τ} (h) + S_{ω} (h) \geq - 2 \log (\frac{1 + max_{1 \leq j, k \leq n} | 〈 τ_{j}, ω_{k} 〉 |}{2}) \geq 0, \forall h \in H_{τ} . \end{matrix}

(4)

Recently, author derived Banach space versions of Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani uncertainty principle [6], Donoho-Stark approximate support uncertainty principle [7] and Ghobber-Jaming uncertainty principle [8]. We then naturally ask what is the Banach space version of Inequality (4)? In this paper, we are going to answer this question.

2. Functional Deutsch Uncertainty Principle

In the paper,

K

denotes

C

R

and

X

denotes a finite dimensional Banach space over

K

. Dual of

X

is denoted by

X^{*}

. We need the notion of Parseval p-frames for Banach spaces.

Definition 2.1.

[9,10] Let

X

be a finite dimensional Banach space over

K

. A collection

{f_{j}}_{j = 1}^{n}

X^{*}

is said to be a Parseval p-frame(

1 \leq p < \infty

) for

X

\begin{matrix} {∥ x ∥}^{p} = \sum_{j = 1}^{n} {| f_{j} (x) |}^{p}, \forall x \in X . \end{matrix}

(5)

Note that (5) says that

∥ f_{j} ∥ \leq 1

for all

1 \leq j \leq n

. Given a Parseval p-frame

{f_{j}}_{j = 1}^{n}

for

X

, we define the (finite) p-Shannon entropy at a point

x \in X_{f}

\begin{matrix} S_{f} (x) - \sum_{j = 1}^{n} {|f_{j} (\frac{x}{∥ x ∥})|}^{p} \log {|f_{j} (\frac{x}{∥ x ∥})|}^{p} \geq 0, \end{matrix}

where

X_{f} {x \in X : f_{j} (x) \neq 0, 1 \leq j \leq n}

. Following is the fundamental result of this paper.

Theorem 2.2

(Functional Deutsch Uncertainty Principle) Let

{f_{j}}_{j = 1}^{n}

and

{g_{k}}_{k = 1}^{m}

be Parseval p-frames for a finite dimensional Banach space

X

. Then

\begin{matrix} \frac{1}{{(n m)}^{\frac{1}{p}}} \leq sup_{y \in X, ∥ y ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | f_{j} (y) g_{k} (y) |) \end{matrix}

and

\begin{matrix} \begin{matrix} \log (n m) \geq S_{f} (x) + S_{g} (x) \geq - p \log (sup_{y \in X_{f} \cap X_{g}, ∥ y ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | f_{j} (y) g_{k} (y) |)) > 0, \\ \forall x \in X_{f} \cap X_{g} . \end{matrix} \end{matrix}

(6)

Proof.

Let

z \in X

be such that

∥ z ∥ = 1

. Then

\begin{matrix} 1 & = (\sum_{j = 1}^{n} {| f_{j} (z) |}^{p}) (\sum_{k = 1}^{m} {| g_{k} (z) |}^{p}) = \sum_{j = 1}^{n} \sum_{k = 1}^{m} {| f_{j} (z) g_{k} (z) |}^{p} \\ \leq \sum_{j = 1}^{n} \sum_{k = 1}^{m} {(sup_{y \in X, ∥ y ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | f_{j} (y) g_{k} (y) |))}^{p} \\ = {(sup_{y \in X, ∥ y ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | f_{j} (y) g_{k} (y) |))}^{p} m n \end{matrix}

which gives

\begin{matrix} \frac{1}{m n} \leq {(sup_{y \in X, ∥ y ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | f_{j} (y) g_{k} (y) |))}^{p} . \end{matrix}

Since

1 = \sum_{j = 1}^{n} {|f_{j} (\frac{x}{∥ x ∥})|}^{p}

for all

x \in X ∖ {0}

1 = \sum_{k = 1}^{m} {|g_{k} (\frac{x}{∥ x ∥})|}^{p}

for all

x \in X ∖ {0}

and log function is concave, using Jensen’s inequality (see [11]) we get

\begin{matrix} S_{f} (x) + S_{g} (x) & = \sum_{j = 1}^{n} {|f_{j} (\frac{x}{∥ x ∥})|}^{p} \log (\frac{1}{{|f_{j} (\frac{x}{∥ x ∥})|}^{p}}) + \sum_{k = 1}^{m} {|g_{k} (\frac{x}{∥ x ∥})|}^{p} \log (\frac{1}{{|g_{k} (\frac{x}{∥ x ∥})|}^{p}}) \\ \leq \log (\sum_{j = 1}^{n} {|f_{j} (\frac{x}{∥ x ∥})|}^{p} \frac{1}{{|f_{j} (\frac{x}{∥ x ∥})|}^{p}}) + \log (\sum_{k = 1}^{m} {|g_{k} (\frac{x}{∥ x ∥})|}^{p} \frac{1}{{|g_{k} (\frac{x}{∥ x ∥})|}^{p}}) \\ = \log n + \log m = \log (n m), \forall x \in X_{f} \cap X_{g} . \end{matrix}

Let

x \in X_{f} \cap X_{g}

. Then

\begin{matrix} S_{f} (x) + S_{g} (x) & = - \sum_{j = 1}^{n} \sum_{k = 1}^{m} {|f_{j} (\frac{x}{∥ x ∥})|}^{p} {|g_{k} (\frac{x}{∥ x ∥})|}^{p} [\log {|f_{j} (\frac{x}{∥ x ∥})|}^{p} + \log {|g_{k} (\frac{x}{∥ x ∥})|}^{p}] \\ = - \sum_{j = 1}^{n} \sum_{k = 1}^{m} {|f_{j} (\frac{x}{∥ x ∥})|}^{p} {|g_{k} (\frac{x}{∥ x ∥})|}^{p} \log {|f_{j} (\frac{x}{∥ x ∥}) g_{k} (\frac{x}{∥ x ∥})|}^{p} \\ = - p \sum_{j = 1}^{n} \sum_{k = 1}^{m} {|f_{j} (\frac{x}{∥ x ∥})|}^{p} {|g_{k} (\frac{x}{∥ x ∥})|}^{p} \log |f_{j} (\frac{x}{∥ x ∥}) g_{k} (\frac{x}{∥ x ∥})| \\ \geq - p \sum_{j = 1}^{n} \sum_{k = 1}^{m} {|f_{j} (\frac{x}{∥ x ∥})|}^{p} {|g_{k} (\frac{x}{∥ x ∥})|}^{p} \log (sup_{y \in X_{f} \cap X_{g}, ∥ y ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | f_{j} (y) g_{k} (y) |)) \\ = - p \log (sup_{y \in X_{f} \cap X_{g}, ∥ y ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | f_{j} (y) g_{k} (y) |)) \sum_{j = 1}^{n} \sum_{k = 1}^{m} {|f_{j} (\frac{x}{∥ x ∥})|}^{p} {|g_{k} (\frac{x}{∥ x ∥})|}^{p} \\ = - p \log (sup_{y \in X_{f} \cap X_{g}, ∥ y ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | f_{j} (y) g_{k} (y) |)) . \end{matrix}

□

Corollary 2.3.

Theorem 1.3 follows from Theorem 2.2.

Proof.

Let

{τ_{j}}_{j = 1}^{n}

{ω_{j}}_{j = 1}^{n}

be two orthonormal bases for a finite dimensional Hilbert space

H

. Define

\begin{matrix} f_{j} : H ∋ h \mapsto 〈 h, τ_{j} 〉 \in K; g_{j} : H ∋ h \mapsto 〈 h, ω_{j} 〉 \in K, \forall 1 \leq j \leq n . \end{matrix}

Now by using Buzano inequality (see [12,13]) we get

\begin{matrix} sup_{h \in H, ∥ h ∥ = 1} (max_{1 \leq j, k \leq n} | f_{j} (h) g_{k} (h) |) & = sup_{h \in H, ∥ h ∥ = 1} (max_{1 \leq j, k \leq n} | 〈 h, τ_{j} 〉 | | 〈 h, ω_{k} 〉 |) \\ \leq sup_{h \in H, ∥ h ∥ = 1} (max_{1 \leq j, k \leq n} ({∥ h ∥}^{2} \frac{∥ τ_{j} ∥ ∥ ω_{k} ∥ + | 〈 τ_{j}, ω_{k} 〉 |}{2})) \\ = \frac{1 + max_{1 \leq j, k \leq n} | 〈 τ_{j}, ω_{k} 〉 |}{2} . \end{matrix}

□

Theorem 2.2 brings the following question.

Question 2.4.

Given p, m, n and a Banach space

X

, for which pairs of Parseval p-frames

{f_{j}}_{j = 1}^{n}

and

{g_{k}}_{k = 1}^{m}

for

X

, we have equality in Inequality (6)?

Next we derive a dual inequality of (6). For this we need dual of Definition 2.1.

Definition 2.5.

[14,15,16] Let

X

be a finite dimensional Banach space over

K

. A collection

{τ_{j}}_{j = 1}^{n}

X

is said to be a Parseval p-frame (

1 \leq p < \infty

) for

X^{*}

\begin{matrix} {∥ f ∥}^{p} = \sum_{j = 1}^{n} {| f (τ_{j}) |}^{p}, \forall f \in X^{*} . \end{matrix}

(7)

Note that (7) says that

\begin{matrix} ∥ τ_{j} ∥ = sup_{f \in X^{*}, ∥ f ∥ = 1} | f (τ_{j}) | \leq sup_{f \in X^{*}, ∥ f ∥ = 1} {(\sum_{j = 1}^{n} {| f (τ_{j}) |}^{p})}^{\frac{1}{p}} = sup_{f \in X^{*}, ∥ f ∥ = 1} ∥ f ∥ = 1, \forall 1 \leq j \leq n . \end{matrix}

Given a Parseval p-frame

{τ_{j}}_{j = 1}^{n}

for

X^{*}

, we define the (finite) p-Shannon entropy at a point

f \in X_{τ}^{*}

\begin{matrix} S_{τ} (f) - \sum_{j = 1}^{n} {|\frac{f (τ_{j})}{∥ f ∥}|}^{p} \log {|\frac{f (τ_{j})}{∥ f ∥}|}^{p} \geq 0, \end{matrix}

where

X_{τ}^{*} {f \in X^{*} : f (τ_{j}) \neq 0, 1 \leq j \leq n}

. We now have the following dual to Theorem 2.2.

Theorem 2.6.

(Functional Deutsch Uncertainty Principle) Let

{τ_{j}}_{j = 1}^{n}

and

{ω_{k}}_{k = 1}^{m}

be two Parseval p-frames for the dual

X^{*}

of a finite dimensional Banach space

X

. Then

\begin{matrix} \frac{1}{{(n m)}^{\frac{1}{p}}} \leq sup_{g \in X^{*}, ∥ g ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | g (τ_{j}) g (ω_{k}) |) \end{matrix}

and

\begin{matrix} \begin{matrix} \log (n m) \geq S_{τ} (f) + S_{ω} (f) \geq - p \log (sup_{g \in X_{τ}^{*} \cap X_{ω}^{*}, ∥ g ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | g (τ_{j}) g (ω_{k}) |)) > 0, \\ \forall f \in X_{τ}^{*} \cap X_{ω}^{*} . \end{matrix} \end{matrix}

(8)

Proof.

Let

h \in X^{*}

be such that

∥ h ∥ = 1

. Then

\begin{matrix} 1 & = (\sum_{j = 1}^{n} {| h (τ_{j}) |}^{p}) (\sum_{k = 1}^{m} {| h (ω_{k}) |}^{p}) = \sum_{j = 1}^{n} \sum_{k = 1}^{m} {| h (τ_{j}) h (ω_{k}) |}^{p} \\ \leq \sum_{j = 1}^{n} \sum_{k = 1}^{m} {(sup_{g \in X^{*}, ∥ g ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | g (τ_{j}) g (ω_{k}) |))}^{p} \\ = {(sup_{g \in X^{*}, ∥ g ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | g (τ_{j}) g (ω_{k}) |))}^{p} m n \end{matrix}

which gives

\begin{matrix} \frac{1}{m n} \leq {(sup_{g \in X^{*}, ∥ g ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | g (τ_{j}) g (ω_{k}) |))}^{p} . \end{matrix}

Since

1 = \sum_{j = 1}^{n} {|\frac{f (τ_{j})}{∥ f ∥}|}^{p}

for all

f \in X^{*} ∖ {0}

1 = \sum_{k = 1}^{m} {|\frac{f (ω_{k})}{∥ f ∥}|}^{p}

for all

f \in X^{*} ∖ {0}

and log function is concave, using Jensen’s inequality we get

\begin{matrix} S_{τ} (f) + S_{ω} (f) & = \sum_{j = 1}^{n} {|\frac{f (τ_{j})}{∥ f ∥}|}^{p} \log (\frac{1}{{|\frac{f (τ_{j})}{∥ f ∥}|}^{p}}) + \sum_{k = 1}^{m} {|\frac{f (ω_{k})}{∥ f ∥}|}^{p} \log (\frac{1}{{|\frac{f (ω_{k})}{∥ f ∥}|}^{p}}) \\ \leq \log (\sum_{j = 1}^{n} {|\frac{f (τ_{j})}{∥ f ∥}|}^{p} \frac{1}{{|\frac{f (τ_{j})}{∥ f ∥}|}^{p}}) + \log (\sum_{k = 1}^{m} {|\frac{f (ω_{k})}{∥ f ∥}|}^{p} \frac{1}{{|\frac{f (ω_{k})}{∥ f ∥}|}^{p}}) \\ = \log n + \log m = \log (n m), \forall f \in X_{τ}^{*} \cap X_{ω}^{*} . \end{matrix}

Let

f \in X_{τ}^{*} \cap X_{ω}^{*}

. Then

\begin{matrix} S_{τ} (f) + S_{ω} (f) & = - \sum_{j = 1}^{n} \sum_{k = 1}^{m} {|\frac{f (τ_{j})}{∥ f ∥}|}^{p} {|\frac{f (ω_{k})}{∥ f ∥}|}^{p} [\log {|\frac{f (τ_{j})}{∥ f ∥}|}^{p} + \log {|\frac{f (ω_{k})}{∥ f ∥}|}^{p}] \\ = - \sum_{j = 1}^{n} \sum_{k = 1}^{m} {|\frac{f (τ_{j})}{∥ f ∥}|}^{p} {|\frac{f (ω_{k})}{∥ f ∥}|}^{p} \log {|\frac{f (τ_{j})}{∥ f ∥} \frac{f (ω_{k})}{∥ f ∥}|}^{p} \\ = - p \sum_{j = 1}^{n} \sum_{k = 1}^{m} {|\frac{f (τ_{j})}{∥ f ∥}|}^{p} {|\frac{f (ω_{k})}{∥ f ∥}|}^{p} \log |\frac{f (τ_{j})}{∥ f ∥} \frac{f (ω_{k})}{∥ f ∥}| \\ \geq - p \sum_{j = 1}^{n} \sum_{k = 1}^{m} {|\frac{f (τ_{j})}{∥ f ∥}|}^{p} {|\frac{f (ω_{k})}{∥ f ∥}|}^{p} \log (sup_{g \in X^{*}, ∥ g ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | g (τ_{j}) g (ω_{k}) |)) \\ = - p \log (sup_{g \in X_{τ}^{*} \cap X_{ω}^{*}, ∥ g ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | g (τ_{j}) g (ω_{k}) |)) \sum_{j = 1}^{n} \sum_{k = 1}^{m} {|\frac{f (τ_{j})}{∥ f ∥}|}^{p} {|\frac{f (ω_{k})}{∥ f ∥}|}^{p} \\ = - p \log (sup_{g \in X_{τ}^{*} \cap X_{ω}^{*}, ∥ g ∥ = 1} (max_{1 \leq j \leq n, 1 \leq k \leq m} | g (τ_{j}) g (ω_{k}) |)) . \end{matrix}

□

Theorem 2.6 again gives the following question.

Question 2.7.

Given p, m, n and a Banach space

X

, for which pairs of Parseval p-frames

{τ_{j}}_{j = 1}^{n}

and

{ω_{k}}_{k = 1}^{m}

for

X^{*}

, we have equality in Inequality (8)?

Author is aware of the improvement of Theorem 1.3 by Maassen and Uffink [17] (cf. [18]) (motivated from a conjecture of Kraus [19]) but unable to derive Maassen-Uffink uncertainty principle from Theorem 2.2.

References

Shannon, C.E. A mathematical theory of communication. Bell System Tech. J. 1948, 27, 379–423, 623–656. [Google Scholar] [CrossRef]
Hirschman, I.I., Jr. A note on entropy. Amer. J. Math. 1957, 79, 152–156. [Google Scholar] [CrossRef]
Beckner, W. Inequalities in Fourier analysis. Ann. of Math. (2) 1975, 102, 159–182. [Google Scholar] [CrossRef]
Bialynicki-Birula, I.; Mycielski, J. Uncertainty relations for information entropy in wave mechanics. Comm. Math. Phys. 1975, 44, 129–132. [Google Scholar] [CrossRef]
Deutsch, D. Uncertainty in quantum measurements. Phys. Rev. Lett. 1983, 50, 631–633. [Google Scholar] [CrossRef]
Krishna, K.M. Functional Donoho-Stark-Elad-Bruckstein-Ricaud-Torrésani Uncertainty Principle. arXiv 2023, arXiv:2304.03324v1. [Google Scholar]
Krishna, K.M. Functional Donoho-Stark Approximate-Support Uncertainty Principle. arXiv arXiv:2307.01215v1.
Krishna, K.M. Functional Ghobber-Jaming Uncertainty Principle. arXiv 2023, arXiv:2306.01014v1. [Google Scholar] [CrossRef]
Aldroubi, A.; Sun, Q.; Tang, W.S. p-frames and shift invariant subspaces of L^p. J. Fourier Anal. Appl. 2001, 7, 1–21. [Google Scholar] [CrossRef]
Christensen, O.; Stoeva, D.T. p-frames in separable Banach spaces. Adv. Comput. Math. 2003, 18, 117–126. [Google Scholar] [CrossRef]
Steele, J.M. The Cauchy-Schwarz master class : An introduction to the art of mathematical inequalities; AMS/MAA Problem Books Series; Mathematical Association of America: Washington, DC; Cambridge University Press: Cambridge, 2004; p. x+306. [Google Scholar] [CrossRef]
Buzano, M.L. Generalizzazione della diseguaglianza di Cauchy-Schwarz. Rend. Sem. Mat. Univ. e Politec. Torino 1971/73, 31, 405–409. [Google Scholar]
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]
Terekhin, P.A. Frames in a Banach space. Funktsional. Anal. i Prilozhen. 2010, 44, 50–62. [Google Scholar] [CrossRef]
Casazza, P.; Christensen, O.; Stoeva, D.T. Frame expansions in separable Banach spaces. J. Math. Anal. Appl. 2005, 307, 710–723. [Google Scholar] [CrossRef]
Terekhin, P.A. Representation systems and projections of bases. Mat. Zametki 2004, 75, 944–947. [Google Scholar] [CrossRef]
Maassen, H.; Uffink, J.B.M. Generalized entropic uncertainty relations. Phys. Rev. Lett. 1988, 60, 1103–1106. [Google Scholar] [CrossRef] [PubMed]
Dembo, A.; Cover, T.M.; Thomas, J.A. Information-theoretic inequalities. IEEE Trans. Inform. Theory 1991, 37, 1501–1518. [Google Scholar] [CrossRef]
Kraus, K. Complementary observables and uncertainty relations. Phys. Rev. D (3) 1987, 35, 3070–3075. [Google Scholar] [CrossRef] [PubMed]

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Functional Deutsch Uncertainty Principle

Abstract

1. Introduction

2. Functional Deutsch Uncertainty Principle

References

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