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Continuous Deutsch Uncertainty Principle and Continuous Kraus Conjecture

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26 July 2023

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27 July 2023

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Abstract
Let $(\Omega, \mu)$, $(\Delta, \nu)$ be measure spaces and $\{\tau_\alpha\}_{\alpha\in \Omega}$, $\{\omega_\beta\}_{\beta \in \Delta}$ be 1-bounded continuous Parseval frames for a Hilbert space $\mathcal{H}$. Then we show that \begin{align}\label{UE} \log (\mu(\Omega)\nu(\Delta))\geq S_\tau(h)+S_\omega (h)\geq -2 \log \left(\frac{1+\displaystyle \sup_{\alpha \in \Omega, \beta \in \Delta}|\langle\tau_\alpha , \omega_\beta\rangle|}{2}\right) , \quad \forall h \in \mathcal{H}_\tau \cap \mathcal{H}_\omega, \end{align} where \begin{align*} &\mathcal{H}_\tau := \{h_1 \in \mathcal{H}: \langle h_1 , \tau_\alpha \rangle \neq 0, \alpha \in \Omega\}, \quad \mathcal{H}_\omega := \{h_2 \in \mathcal{H}: \langle h_2, \omega_\beta \rangle \neq 0, \beta \in \Delta\},\\ &S_\tau(h):= -\displaystyle\int\limits_{\Omega}\left|\left \langle \frac{h}{\|h\|}, \tau_\alpha\right\rangle \right|^2\log \left|\left \langle \frac{h}{\|h\|}, \tau_\alpha\right\rangle \right|^2\,d\mu(\alpha), \quad \forall h \in \mathcal{H}_\tau, \\ & S_\omega (h):= -\displaystyle\int\limits_{\Delta}\left|\left \langle \frac{h}{\|h\|}, \omega_\beta\right\rangle \right|^2\log \left|\left \langle \frac{h}{\|h\|}, \omega_\beta\right\rangle \right|^2\,d\nu(\beta), \quad \forall h \in \mathcal{H}_\omega. \end{align*} We call Inequality (\ref{UE}) as \textbf{Continuous Deutsch Uncertainty Principle}. Inequality (\ref{UE}) improves the uncertainty principle obtained by Deutsch \textit{[Phys. Rev. Lett., 1983]}. We formulate Kraus conjecture for 1-bounded continuous Parseval frames. We also derive continuous Deutsch uncertainty principles for Banach spaces.
Keywords: 
Subject: Computer Science and Mathematics  -   Analysis

1. Introduction

Let H be a finite dimensional Hilbert space. Given an orthonormal basis { ω j } j = 1 n for H , the (finite) Shannon entropy at a point h ∈ H τ is defined as
S τ ( h ) − ∑ j = 1 n h ∥ h ∥ , τ j 2 log h ∥ h ∥ , τ j 2 ≥ 0 ,
where H τ { h ∈ H : 〈 h , τ j 〉 ≠ 0 , 1 ≤ j ≤ n } . In 1983, Deutsch derived following uncertainty principle for Shannon entropy [3].
Theorem 1.1. 
(Deutsch Uncertainty Principle) [3] Let { τ j } j = 1 n , { ω j } j = 1 n be two orthonormal bases for a finite dimensional Hilbert space H . Then
2 log n ≥ S τ ( h ) + S ω ( h ) ≥ − 2 log 1 + max 1 ≤ j , k ≤ n | 〈 τ j , ω k 〉 | 2 ≥ 0 , ∀ h ∈ H τ ∩ H ω .
In 1988, followed by a conjecture of Kraus [9] made in 1987, Maassen and Uffink improved Inequality (1) [12].
Theorem 1.2. (Kraus Conjecture/Maassen-Uffink Uncertainty Principle) [9,12] Let { τ j } j = 1 n , { ω j } j = 1 n be two orthonormal bases for a finite dimensional Hilbert space H . Then
2 log n ≥ S τ ( h ) + S ω ( h ) ≥ − 2 log max 1 ≤ j , k ≤ n | 〈 τ j , ω k 〉 | ≥ 0 , ∀ h ∈ H τ ∩ H ω .
In 2013, Ricaud and Torrésani [13] showed that Theorem 1.2 holds for Parseval frames.
Theorem 1.3. (Maassen-Uffink-Ricaud-Torrésani Uncertainty Principle) [13] Let { τ j } j = 1 n , { ω j } j = 1 n be two Parseval frames for a finite dimensional Hilbert space H . Then
2 log n ≥ S τ ( h ) + S ω ( h ) ≥ − 2 log max 1 ≤ j , k ≤ n | 〈 τ j , ω k 〉 | ≥ 0 , ∀ h ∈ H τ ∩ H ω .
Recently, Banach space versions of Deutsch uncertainty principle have been derived in [11]. To formulate them, we need some notions. Given a Parseval p-frame { f j } j = 1 n for X , we define the (finite) p-Shannon entropy at a point x ∈ X f as
S f ( x ) − ∑ j = 1 n f j x ∥ x ∥ p log f j x ∥ x ∥ p ≥ 0 ,
where X f { x ∈ X : f j ( x ) ≠ 0 , 1 ≤ j ≤ n } . On the other way, given a Parseval p-frame { τ j } j = 1 n for X * , we define the (finite) p-Shannon entropy at a point f ∈ X τ * as
S τ ( f ) − ∑ j = 1 n f ( τ j ) ∥ f ∥ p log f ( τ j ) ∥ f ∥ p ≥ 0 ,
where X τ * { f ∈ X * : f ( τ j ) ≠ 0 , 1 ≤ j ≤ n } .
Theorem 1.4. 
[11] (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
1 ( n m ) 1 p ≤ sup y ∈ X , ∥ y ∥ = 1 max 1 ≤ j ≤ n , 1 ≤ k ≤ m | f j ( y ) g k ( y ) |
and
log ( n m ) ≥ S f ( x ) + S g ( x ) ≥ − p log sup y ∈ X f ∩ X g , ∥ y ∥ = 1 max 1 ≤ j ≤ n , 1 ≤ k ≤ m | f j ( y ) g k ( y ) | > 0 , ∀ x ∈ X f ∩ X g .
Theorem 1.5. 
(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
1 ( n m ) 1 p ≤ sup g ∈ X * , ∥ g ∥ = 1 max 1 ≤ j ≤ n , 1 ≤ k ≤ m | g ( τ j ) g ( ω k ) |
and
log ( n m ) ≥ S τ ( f ) + S ω ( f ) ≥ − p log sup g ∈ X τ * ∩ X ω * , ∥ g ∥ = 1 max 1 ≤ j ≤ n , 1 ≤ k ≤ m | g ( τ j ) g ( ω k ) | > 0 , ∀ f ∈ X τ * ∩ X ω * .
In this paper, we derive continuous versions of Theorem 1.1, Theorem 1.4 and Theorem 1.5. We also formulate a conjecture based on Theorem 1.2. We wish to say that functional continuous uncertainty principles are derived in [10].

2. Continuous Deutsch Uncertainty Principle and Continuous Kraus Conjecture

In the paper, K denotes C or R and H (resp. X ) denotes a Hilbert space (resp. Banach space) (need not be finite dimensional) over K . We use ( Ω , μ ) to denote a measure space. Continuous frames are introduced independently by Ali, Antoine and Gazeau [1] and Kaiser [8]. In the paper, K denotes C or R and H denotes a finite dimensional Hilbert space.
Definition 2.1. 
[1,8] Let ( Ω , μ ) be a measure space. A collection { τ α } α ∈ Ω in a Hilbert space H is said to be a continuous Parseval frame for H if the following conditions hold.
(i) 
For each h ∈ H , the map Ω ∋ α ↦ 〈 h , τ α 〉 ∈ K is measurable.
(ii) 
∥ h ∥ 2 = ∫ Ω | 〈 h , τ α 〉 | 2 d μ ( α ) , ∀ h ∈ H .
We consider the following subclass of continuous Parseval frames.
Definition 2.2. 
A continuous Parseval frame { τ α } α ∈ Ω for H is said to be 1-bounded if
∥ τ α ∥ ≤ 1 , ∀ α ∈ Ω .
Note that if { τ j } j = 1 n is a Parseval frame for a Hilbert space H , then ∥ τ j ∥ ≤ 1 , ∀ 1 ≤ j ≤ n (see Remark 3.12 in [7]). We are unable to derive this for continuous frames. Given a continuous 1-bounded Parseval frame { τ α } α ∈ Ω for H , we define the continuous Shannon entropy at a point h ∈ H τ as
S τ ( h ) − ∫ Ω h ∥ h ∥ , τ α 2 log h ∥ h ∥ , τ α 2 d μ ( α ) ≥ 0 ,
where H τ { h ∈ H : 〈 h , τ α 〉 ≠ 0 , α ∈ Ω } . Following is the first fundamental result of this paper.
Theorem 2.3. (Continuous Deutsch Uncertainty Principle) Let ( Ω , μ ) , ( Δ , ν ) be measure spaces and { τ α } α ∈ Ω , { ω β } β ∈ Δ be 1-bounded continuous Parseval frames for a Hilbert space H . Then
log ( μ ( Ω ) ν ( Δ ) ) ≥ S τ ( h ) + S ω ( h ) ≥ − 2 log 1 + sup α ∈ Ω , β ∈ Δ | 〈 τ α , ω β 〉 | 2 ≥ 0 , ∀ h ∈ H τ ∩ H ω .
Proof. 
Since 1 = ∫ Ω h ∥ h ∥ , τ α 2 d μ ( α ) for all h ∈ H ∖ { 0 } , 1 = ∫ Δ h ∥ h ∥ , ω β 2 d ν ( β ) for all h ∈ H ∖ { 0 } and log is concave, using Jensen’s inequality (cf. [6]) we get
S τ ( h ) + S ω ( h ) = ∫ Ω h ∥ h ∥ , τ α 2 log 1 h ∥ h ∥ , τ α 2 d μ ( α ) + ∫ Δ h ∥ h ∥ , ω β 2 log 1 h ∥ h ∥ , ω β 2 d ν ( β ) ≤ log ∫ Ω h ∥ h ∥ , τ α 2 1 h ∥ h ∥ , τ α 2 d μ ( α ) + log ∫ Δ h ∥ h ∥ , ω β 2 1 h ∥ h ∥ , ω β 2 d ν ( β ) = log ( μ ( Ω ) ) + log ν ( Δ ) ) = log ( μ ( Ω ) ν ( Δ ) ) , ∀ h ∈ H τ ∩ H ω .
Let h ∈ H τ ∩ H ω . Then using Buzano inequality [2,5] we get
S τ ( h ) + S ω ( h ) = − ∫ Δ ∫ Ω h ∥ h ∥ , τ α 2 h ∥ h ∥ , ω β 2 log h ∥ h ∥ , τ α 2 + log h ∥ h ∥ , ω β 2 d μ ( α ) d ν ( β ) = − ∫ Δ ∫ Ω h ∥ h ∥ , τ α 2 h ∥ h ∥ , ω β 2 log h ∥ h ∥ , τ α h ∥ h ∥ , ω β 2 d μ ( α ) d ν ( β ) = − 2 ∫ Δ ∫ Ω h ∥ h ∥ , τ α 2 h ∥ h ∥ , ω β 2 log h ∥ h ∥ , τ α h ∥ h ∥ , ω β d μ ( α ) d ν ( β ) ≥ − 2 ∫ Δ ∫ Ω h ∥ h ∥ , τ α 2 h ∥ h ∥ , ω β 2 log h ∥ h ∥ 2 ∥ τ α ∥ ∥ ω β ∥ + | 〈 τ α , ω β 〉 | 2 d μ ( α ) d ν ( β ) = − 2 ∫ Δ ∫ Ω h ∥ h ∥ , τ α 2 h ∥ h ∥ , ω β 2 log 1 + | 〈 τ α , ω β 〉 | 2 d μ ( α ) d ν ( β ) ≥ − 2 ∫ Δ ∫ Ω h ∥ h ∥ , τ α 2 h ∥ h ∥ , ω β 2 log 1 + sup α ∈ Ω , β ∈ Δ | 〈 τ α , ω β 〉 | 2 d μ ( α ) d ν ( β ) = − 2 log 1 + sup α ∈ Ω , β ∈ Δ | 〈 τ α , ω β 〉 | 2 ∫ Δ ∫ Ω h ∥ h ∥ , τ α 2 h ∥ h ∥ , ω β 2 d μ ( α ) d ν ( β ) ≥ − 2 log 1 + sup α ∈ Ω , β ∈ Δ | 〈 τ α , ω β 〉 | 2 .
□
Theorem 2.3 promotes following question.
Question 2.4. 
Let ( Ω , μ ) , ( Δ , ν ) be measure spaces, H be a Hilbert space. For which pairs of 1-bounded continuous Parseval frames { τ α } α ∈ Ω and { ω β } β ∈ Δ for H , we have equality in Inequality (2)?
Based on Theorems 1.3 and Theorem 2.3 we formulate following conjecture.
Conjecture 2.5. (Continuous Kraus Conjecture) Let ( Ω , μ ) , ( Δ , ν ) be measure spaces and { τ α } α ∈ Ω , { ω β } β ∈ Δ be 1-bounded continuous Parseval frames for a Hilbert space H . Then
S τ ( h ) + S ω ( h ) ≥ − 2 log sup α ∈ Ω , β ∈ Δ | 〈 τ α , ω β 〉 | ≥ 0 , ∀ h ∈ H τ ∩ H ω .
Next we derive continuous Deutsch uncertainty for Banach spaces. We need a definition.
Definition 2.6. 
[4] Let ( Ω , μ ) be a measure space and X be a Banach space over K . A collection { f α } α ∈ Ω in X * is said to be acontinuous Parseval p-frame( 1 ≤ p < ∞ ) for X if the following conditions hold.
(i) 
For each x ∈ X , the map Ω ∋ α ↦ f α ( x ) ∈ K is measurable.
(ii) 
∥ x ∥ p = ∫ Ω | f α ( x ) | p d μ ( α ) , ∀ x ∈ X .
If ∥ τ α ∥ ≤ 1 , ∀ α ∈ Ω , then we say that the frame { f α } α ∈ Ω is 1-bounded.
Similar to Definition 2.2, we set the following.
Definition 2.7. 
A continuous Parseval p-frame { f α } α ∈ Ω for X is said to be 1-bounded if
∥ f α ∥ ≤ 1 , ∀ α ∈ Ω .
Given a 1-bounded continuous Parseval p-frame { f α } α ∈ Ω for X , we define the continuous p-Shannon entropy at a point x ∈ X f as
S f ( x ) − ∫ Ω f α x ∥ x ∥ p log f α x ∥ x ∥ p d μ ( α ) ≥ 0 ,
where X f { x ∈ X : f α ( x ) ≠ 0 , α ∈ Ω } .
Theorem 2.8. (Functional Continuous Deutsch Uncertainty Principle ) Let ( Ω , μ ) , ( Δ , ν ) be measure spaces and { f α } α ∈ Ω , { g β } β ∈ Δ be 1-bounded continuous Parseval p-frames for a Banach space X . Then
1 ( μ ( Ω ) ν ( Δ ) ) 1 p ≤ sup y ∈ X , ∥ y ∥ = 1 sup α ∈ Ω , β ∈ Δ | f α ( y ) g β ( y ) |
and
S f ( x ) + S g ( x ) ≥ − p log sup y ∈ X , ∥ y ∥ = 1 sup α ∈ Ω , β ∈ Δ | f α ( y ) g β ( y ) | ≥ 0 , ∀ x ∈ X f ∩ X g .
Proof. 
Let z ∈ X be such that ∥ z ∥ = 1 . Then
1 = ∫ Ω | f α ( z ) | p d μ ( α ) ∫ Δ | g β ( z ) | p d ν ( β ) = ∫ Ω ∫ Δ | f α ( z ) g β ( z ) | p d ν ( β ) d μ ( α ) ≤ ∫ Ω ∫ Δ sup y ∈ X , ∥ y ∥ = 1 sup α ∈ Ω , β ∈ Δ | f α ( y ) g β ( y ) | p d ν ( β ) d μ ( α ) = sup y ∈ X , ∥ y ∥ = 1 sup α ∈ Ω , β ∈ Δ | f α ( y ) g β ( y ) | p μ ( Ω ) ν ( Δ )
which gives
1 μ ( Ω ) ν ( Δ ) ≤ sup y ∈ X , ∥ y ∥ = 1 sup α ∈ Ω , β ∈ Δ | f α ( y ) g β ( y ) | p .
Let x ∈ X f ∩ X g . Then
S f ( x ) + S g ( x ) = − ∫ Ω ∫ Δ f α x ∥ x ∥ p g β x ∥ x ∥ p log f α x ∥ x ∥ p + log g β x ∥ x ∥ p d ν ( β ) d μ ( α ) = − ∫ Ω ∫ Δ f α x ∥ x ∥ p g β x ∥ x ∥ p log f α x ∥ x ∥ g β x ∥ x ∥ p d ν ( β ) d μ ( α ) = − p ∫ Ω ∫ Δ f α x ∥ x ∥ p g β x ∥ x ∥ p log f α x ∥ x ∥ g β x ∥ x ∥ d ν ( β ) d μ ( α ) ≥ − p ∫ Ω ∫ Δ f α x ∥ x ∥ p g β x ∥ x ∥ p log sup y ∈ X , ∥ y ∥ = 1 sup α ∈ Ω , β ∈ Δ | f α ( y ) g β ( y ) | d ν ( β ) d μ ( α ) = − p log sup y ∈ X , ∥ y ∥ = 1 sup α ∈ Ω , β ∈ Δ | f α ( y ) g β ( y ) | ∫ Ω ∫ Δ f α x ∥ x ∥ p g β x ∥ x ∥ p d ν ( β ) d μ ( α ) = − p log sup y ∈ X , ∥ y ∥ = 1 sup α ∈ Ω , β ∈ Δ | f α ( y ) g β ( y ) | .
□
Corollary 2.9. 
Theorem 1.1 follows from Theorem 2.8.
Proof. 
Let ( Ω , μ ) , ( Δ , ν ) be measure spaces and { τ α } α ∈ Ω , { ω β } β ∈ Δ be 1-bounded continuous Parseval frames for a Hilbert space H . Define
f α : H ∋ h ↦ 〈 h , τ α 〉 ∈ K ; ∀ α ∈ Ω , g β : H ∋ h ↦ 〈 h , ω β 〉 ∈ K , ∀ β ∈ Δ .
Now by using Buzano inequality [2,5] we get
sup h ∈ H , ∥ h ∥ = 1 sup α ∈ Ω , β ∈ Δ | f α ( h ) g β ( h ) | = sup h ∈ H , ∥ h ∥ = 1 sup α ∈ Ω , β ∈ Δ | 〈 h , τ α 〉 | | 〈 h , ω β 〉 | ≤ sup h ∈ H , ∥ h ∥ = 1 sup α ∈ Ω , β ∈ Δ ∥ h ∥ 2 ∥ τ α ∥ ∥ ω β ∥ + | 〈 τ α , ω β 〉 | 2 = 1 + sup α ∈ Ω , β ∈ Δ | 〈 τ j , ω k 〉 | 2 .
□
Theorem 2.8 brings the following question.
Question 2.10. 
Let ( Ω , μ ) , ( Δ , ν ) be measure spaces, X be a Banach space and p > 1 . For which pairs of continuous Parseval p-frames { f α } α ∈ Ω , and { g β } β ∈ Δ for X , we have equality in Inequality (4)?
Next we derive a dual inequality of (4). For this we need dual of Definition 2.6.
Definition 2.6. 
Let ( Ω , μ ) be a measure space and X be a Banach space over K . A collection { τ α } α ∈ Ω in X is said to be a continuous Parseval p-frame ( 1 ≤ p < ∞ ) for X * if the following conditions hold.
(i) 
For each ϕ ∈ X * , the map Ω ∋ α ↦ ϕ ( τ α ) ∈ K is measurable.
(ii) 
∥ f ∥ p = ∫ Ω | f ( τ α ) | p d μ ( α ) , ∀ f ∈ X * .
f ∥ τ α ∥ ≤ 1 , ∀ α ∈ Ω , then we say that the frame { τ α } α ∈ Ω is 1-bounded.
Given a 1-bounded continuous Parseval p-frame { τ α } α ∈ Ω for X * , we define the continuous p-Shannon entropy at a point f ∈ X τ * as
S f ( x ) − ∫ Ω f ( τ α ) ∥ f ∥ p log f ( τ α ) ∥ f ∥ p d μ ( α ) ≥ 0 ,
where X τ * { f ∈ X : f ( τ α ) ≠ 0 , α ∈ Ω } } . We now have the following dual to Theorem 2.8.
Theorem .(Continuous Deutsch Uncertainty Principle for Banach spaces) Let ( Ω , μ ) , ( Δ , ν ) be measure spaces and { τ α } α ∈ Ω , { ω β } β ∈ Δ be 1-bounded continuous Parseval p-frames for the dual X * of a Banach space X . Then
1 ( μ ( Ω ) ν ( Δ ) ) 1 p ≤ sup g ∈ X * , ∥ g ∥ = 1 sup α ∈ Ω , β ∈ Δ | g ( τ α ) g ( ω β ) |
and
S τ ( f ) + S ω ( f ) ≥ − p log sup g ∈ X * , ∥ g ∥ = 1 sup α ∈ Ω , β ∈ Δ | g ( τ α ) g ( ω β ) | ≥ 0 , ∀ f ∈ X τ * ∩ X ω * .
Proof. 
Let h ∈ X * be such that ∥ h ∥ = 1 . Then
1 = ∫ Ω | h ( τ α ) | p d μ ( α ) ∫ Δ h ( ω β ) | p d ν ( β ) = ∫ Ω ∫ Δ | h ( τ α ) h ( ω β ) | p d ν ( β ) d μ ( α ) ≤ ∫ Ω ∫ Δ sup g ∈ X * , ∥ g ∥ = 1 sup α ∈ Ω , β ∈ Δ | g ( τ α ) g ( ω β ) | p d ν ( β ) d μ ( α ) = sup g ∈ X * , ∥ g ∥ = 1 sup α ∈ Ω , β ∈ Δ | g ( τ α ) g ( ω β ) | p μ ( Ω ) ν ( Δ )
which gives
1 μ ( Ω ) ν ( Δ ) ≤ sup g ∈ X * , ∥ g ∥ = 1 sup α ∈ Ω , β ∈ Δ | g ( τ α ) g ( ω β ) | p .
Let f ∈ X τ * ∩ X ω * . Then
S τ ( f ) + S ω ( f ) = − ∫ Ω ∫ Δ f ( τ α ) ∥ f ∥ p f ( ω β ) ∥ f ∥ p log f ( τ α ) ∥ f ∥ p + log f ( ω β ) ∥ f ∥ p d ν ( β ) d μ ( α ) = − ∫ Ω ∫ Δ f ( τ α ) ∥ f ∥ p f ( ω β ) ∥ f ∥ p log f ( τ α ) ∥ f ∥ f ( ω β ) ∥ f ∥ p d ν ( β ) d μ ( α ) = − p ∫ Ω ∫ Δ f ( τ α ) ∥ f ∥ p f ( ω β ) ∥ f ∥ p log f ( τ α ) ∥ f ∥ f ( ω β ) ∥ f ∥ d ν ( β ) d μ ( α ) ≥ − p ∫ Ω ∫ Δ f ( τ α ) ∥ f ∥ p f ( ω β ) ∥ f ∥ p log sup g ∈ X * , ∥ g ∥ = 1 sup α ∈ Ω , β ∈ Δ | g ( τ α ) g ( ω β ) | d ν ( β ) d μ ( α ) = − p log sup g ∈ X * , ∥ g ∥ = 1 sup α ∈ Ω , β ∈ Δ | g ( τ α ) g ( ω β ) | ∫ Ω ∫ Δ f ( τ α ) ∥ f ∥ p f ( ω β ) ∥ f ∥ p d ν ( β ) d μ ( α ) = − p log sup g ∈ X * , ∥ g ∥ = 1 sup α ∈ Ω , β ∈ Δ | g ( τ α ) g ( ω β ) | .
□
Theorem 2.12 again gives the following question.
Question 2.13. 
Let ( Ω , μ ) , ( Δ , ν ) be measure spaces, X be a Banach space and p > 1 . For which pairs of continuous Parseval p-frames { τ α } α ∈ Ω , and { ω β } β ∈ Δ for X * , we have equality in Inequality (5)

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