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Article

A Note on Injective g-Frames in Quaternionic Hilbert Spaces

1
School of Intelligent Engineering, Henan Institute of Technology, Xinxiang 453003, China
2
School of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
3
School of Science, Henan Institute of Technology, Xinxiang 453003, China
*
Author to whom correspondence should be addressed.
Axioms 2024, 13(12), 851; https://doi.org/10.3390/axioms13120851
Submission received: 6 November 2024 / Revised: 1 December 2024 / Accepted: 2 December 2024 / Published: 3 December 2024
(This article belongs to the Section Geometry and Topology)

Abstract

:
Motivated by recent advancements in the quantum detection problem employing both discrete and continuous frames, this paper delves into a quantum detection problem utilizing g-frames within the context of quaternionic Hilbert spaces. We offer several equivalent representations of injective g-frames in separable quaternionic Hilbert spaces. By normalizing the trace, we establish a classification for the g-frame injectivity problem. Additionally, we propose a method to derive an injective g-frame by leveraging an injective frame within quaternionic Hilbert spaces. Furthermore, we demonstrate that the injectivity of a g-frame remains intact under a linear isomorphism, while injective g-frames exhibit instability in infinite-dimensional scenarios.

1. Introduction

Recall that set { f k } i = 1 is a frame for a quaternionic Hilbert space K , if there exist constants α , β > 0 such that, for all f K ,
α f 2 k = 1 | f k f | 2 β f 2 .
The numbers α and β are referred to as the lower and upper frame bounds, respectively. In the particular scenario where α equals β , the frame is known as a tight frame. Additionally, when both α and β are equal to 1, this is termed a Parseval frame.
It is worth remembering that Khokulan, Thirulogasanthar, and Srisatkunarajah, in their work [1], were the first to introduce the notion of frames within finite-dimensional quaternion Hilbert spaces. Subsequently, Sharma and Goel, in [2], extended this concept to separable quaternionic Hilbert spaces. For a deeper understanding of frames in quaternion Hilbert spaces, readers may consult [3,4,5] and the references cited therein. Over the years, numerous extensions of the frame theory have been explored. Several are presented as particular instances of the refined g-frame theory first introduced by Sun in [6].
We recall the definition of the g-frame in quaternionic Hilbert spaces [7].
Definition 1. 
Let L ( K , K k ) denote the set of all bounded linear operators from the right quaternionic Hilbert space K to the right quaternionic Hilbert space K k . A family { T k L ( K , K k ) } k = 1 is termed a g-frame for K relative to { K k } k = 1 if there exist constants A , B > 0 such that, for every f K , the inequality
A f 2 k = 1 T k f 2 B f 2 ,
holds. The constants A and B are referred to as the g-frame bounds. A g-frame { T k } k = 1 is classified as a tight g-frame when A = B and as a Parseval g-frame when A = B = 1 .
G-frames integrate operator theory with frame theory, allowing for considerable flexibility in selecting the spaces K k and the corresponding operators { T k L ( K , K k ) } k = 1 . For further information on g-frames, refer to [6,8].
Frames have attracted significant attention in recent years within the realm of signal processing, particularly in phase-retrieval [9,10]. The phase-retrieval problem involves distinguishing pure states based on their measurements utilizing a positive operator-valued measure (POVM). Conversely, the quantum detection problem aims to distinguish all states from their measurements. In simpler terms, this problem entails uniquely identifying a density operator (or state) from quantum measurements represented by POVMs acting on a density operator. More specifically, the quantum detection problem seeks to delineate the properties of a POVM ϕ such that if, for every measurable set E,
tr ( ρ 1 ϕ ( E ) ) = tr ( ρ 2 ϕ ( E ) )
implies ρ 1 = ρ 2 , where ρ 1 , ρ 2 L ( K ) are states.
The issue of quantum detection was addressed by Botelho-Andrade and colleagues [11], primarily focusing on frames that are either finite or infinite but discrete, while Han et al. [12] tackled the continuous frame scenario by constructing specific types of frame positive operator-valued measures. Hong and Li provided multiple characterizations for S p -injective continuous frames using discrete representations of these frames [13]. Furthermore, Han and co-authors examined the quantum detection problem through the utilization of multi-window Gabor frames [14]. The aim of this article is to explore the quantum detection problem within infinite-dimensional separable quaternionic Hilbert spaces using g-frames. For historical context and recent advancements in the quantum detection problem, we recommend consulting [15,16,17,18] and their referenced literature.
The structure of the remainder of this article is outlined as follows: In Section 2, we present foundational knowledge and context regarding the quantum detection problem. We will introduce the notion of quantum injectivity for g-frames within quaternionic Hilbert spaces and demonstrate an equivalent representation of the g-frame. Moving on to Section 3, we offer equivalent formulations for injective g-frames and categorize the problem of g-frame injectivity, revealing that injective g-frames are unstable in the context of infinite-dimensional quaternionic Hilbert spaces.

2. Preliminaries

The quaternion algebra W represents an extension of the complex numbers into a four dimensional space, defined as
W = { | w = w 0 + w 1 x + w 2 y + w 3 z , where w 0 , w 1 , w 2 , w 3 R }
with the elements x , y , z fulfilling the following relations:
x y = y x = z , x 2 = y 2 = z 2 = 1 and x y z = 1 .
Define the space 2 ( W ) as:
2 ( W ) = { w k } k I W : k I | w k | 2 < .
This space carries out the operation of multiplying on the right using quaternion scalars and is furnished with a quaternionic inner product on 2 ( W ) , which is specified by
v w = k I v k ¯ w k , v = { v k } k I and w = { w k } k I .
Quaternion algebra and the right quaternionic Hilbert space are explored extensively in papers dedicated to operator theory. For further reading on these topics, the curious reader may also consult References [7,19,20].
We now introduce the notion of operator-valued measures with respect to quaternionic Hilbert spaces. Throughout the paper, K will denote the right quaternionic Hilbert space.
Definition 2. 
Let Ξ denote a locally compact Hausdorff space, and denote, using Υ, the σ-algebra comprising the Borel sets of Ξ. A map ϕ : Υ L ( K ) is defined as an operator-valued measure if, for any countable collection { U k } k I Υ , such that
U k U j = , k j ,
it fulfills the condition
ϕ k I U k = k I ϕ ( U k ) ,
where the convergence on the right-hand side is interpreted in the weak operator topology. Additionally, ϕ is referred to as a positive operator-valued measure if it is positive and ϕ ( Ξ ) = I K .
As previously stated, the quantum detection problem involves investigating the presence of a quantum measurement executed by a POVM, denoted as ϕ , which is capable of distinguishing states within a quaternionic Hilbert space K . Define L ( Υ , R ) as the collection of bounded functions on Υ , and S ( K ) as the set of density operators on the quaternionic Hilbert space K , satisfying
{ M : M = M * 0 , tr ( M ) = 1 } .
For a given system, the quantum detection problem poses the following question: Is there a specific type of POVM ϕ that ensures the following map is injective?
Q : S ( K ) L ( Υ , R ) , Q ( ρ ) ( X ) = tr ( ρ ϕ ( X ) ) , X Υ ,
The application of the frame to the above problem entails recognizing a Parseval frame that results in a frame positive operator-valued measure, thereby guaranteeing that the associated map Q is injective. We further explore the generalization of this frame-centric problem, wherein we directly interact with general operators. Specifically, the g-frame quantum detection problem is articulated as follows: under the conditions of a g-frame { T k L ( K , K k ) } k = 1 , does the mapping
M : S ( K ) , ( M ( M ) ) k = tr ( T k * M T k ) , k = 1 , 2 ,
become injective?
Definition 3. 
We say that a g-frame { T k L ( K , K k ) } k = 1 provides quantum injectivity when the map M that is related to { T k } k = 1 is injective.
For k = 1 , 2 , , express tr ( T k * M T k ) as M T k , T k . Let S 1 ( K ) represent the collection of trace class operators on K , and use M 1 to denote the following linear operator:
M 1 : S 1 ( K ) , ( M 1 ( M ) ) k = M T k , T k , k = 1 , 2 , .
Using · 1 to signify the 1 -norm, we can derive an equivalent formulation of the g-frame, as described below.
Proposition 1. 
The g-frame condition can be expressed equivalently as
A f 2 M 1 ( f f ) 1 B f 2 .
Proof. 
It is evident that f f belongs to S 1 ( K ) . For any element f in K , it is easy to observe that the i-th entry of M 1 ( f f ) is provided by T k ( f ) 2 , and thus
M 1 ( f f ) 1 = k = 1 T k ( f ) 2 .
Consequently, a sequence constitutes a g-frame if and only if the condition
A f 2 M 1 ( f f ) 1 B f 2
holds true. □

3. Injective g-Frames

In the following, we will provide some equivalent formulations for injective g-frames and present a categorization of the g-frame injectivity problem.
It is evident that the quantum injectivity of a g-frame is equivalent to the statement that, given a self-adjoint trace class operator M with zero trace, if
M T k , T k = 0 , k = 1 , 2 , ,
then M = 0 . Analogously, { T k } k = 1 is deemed injective if, for any self-adjoint trace-class operator M satisfying
M T k , T k = 0 , k = 1 , 2 , ,
it follows that M = 0 . Notably, injectivity entails quantum injectivity. Our discussion begins with the fundamental observation that working with positive operators is unnecessary. Let S 1 + * ( K ) denote the positive trace class and self-adjoint operators on K and S 1 * ( K ) the trace class and self-adjoint operators on K .
Theorem 1. 
Given a g-frame { T k } k = 1 for K , the following are equivalent:
1. 
If M , N S 1 + * ( K ) and
M T k , T k = N T k , T k , k = 1 , 2 , ,
then M = N .
2. 
If M , N S 1 * ( K ) and
M T k , T k = N T k , T k , k = 1 , 2 , ,
then M = N .
3. 
{ T k } k = 1 is an injective g-frame.
Proof. 
1 ⇒ 2 Let M , N S 1 * ( K ) , such that
M T k , T k = N T k , T k , k = 1 , 2 , .
Set L = M N . Then, R S 1 * ( K ) . Let { e k } k = 1 be a Hilbert basis for K and let { u k } k = 1 be an eigenbasis for R with respect to the eigenvalues { λ k } k = 1 . Define operators P and Q on K using
P e k = u k , Q e k = λ k e k , k = 1 , 2 , .
Therefore, P is unitary and Q S 1 * ( K ) . Since
P Q P * u k = P Q e k = λ k P e k = λ k u k = L u k , k = 1 , 2 , ,
we have
L = U V U * .
Let
p k = | λ k | and q k = | λ k | λ k , k = 1 , 2 , .
Clearly, λ k = p k q k . Let Q 1 and Q 2 be operators defined by
Q 1 e k = p k e k , Q 2 e k = q k e k , k = 1 , 2 , .
Note that L S 1 ( K ) , therefore,
k = 1 | λ k | < .
Hence Q 1 , Q 2 S 1 + * ( K ) and we have
L = P Q 1 P * P Q 2 P * .
Moreover, P Q 1 P * , P Q 2 P * S 1 + * ( K ) . Since
L T k , T k = tr ( T k * L T k ) = tr ( T k * ( P Q 1 P * P Q 2 P * ) T k ) = P Q 1 P * T k , T k P Q 2 P * T k , T k = 0 ,
we can obtain that
P Q 1 P * = P Q 2 P * .
Therefore, L = 0 , and thus P = Q .
2 ⇒ 3 Assume that M S 1 * ( K ) , such that
M T k , T k = 0 , k = 1 , 2 , .
It follows that
M T k , T k = 0 T k , T k , k = 1 , 2 , .
Hence, M = 0 . Thus, { T k } k = 1 is an injective g-frame.
3 ⇒ 1 Suppose that M , N S 1 + * ( K ) , such that
M T k , T k = N T k , T k , k = 1 , 2 , .
It follows that
( M N ) T k , T k = 0 , k = 1 , 2 , .
As M N is a self-adjoint operator and the sequence { T k } k = 1 is an injective g-frame, we conclude that M = N . □
If we normalize the trace and additionally stipulate that our operators possess a trace of one, we can categorize the g-frame injectivity problem.
Theorem 2. 
Let { T k } k = 1 be a g-frame for K ; the following are equivalent:
1. 
If M , N S 1 + * ( K ) , tr ( M ) = tr ( N ) = 1 and
M T k , T k = N T k , T k , k = 1 , 2 , ,
then M = N .
2. 
If M , N S 1 * ( K ) , tr ( M ) = tr ( N ) = 1 and
M T k , T k = N T k , T k , k = 1 , 2 , ,
then M = N .
3. 
If M S 1 * ( K ) , tr ( M ) = 0 and
M T k , T k = 0 , k = 1 , 2 , ,
then M = 0 .
Proof. 
1 ⇒ 2 Let
M , N S 1 * ( K ) and tr ( M ) = tr ( N ) = 1
such that
M T k , T k = N T k , T k , k = 1 , 2 , .
Set L = M N . Then,
L S 1 * ( K ) and tr ( L ) = 1
Let { e k } k = 1 be a Hilbert basis for K and let { u k } k = 1 be an eigenbasis for L with respect to the eigenvalues { λ k } k = 1 . Then,
k = 1 λ k = 0 .
Define operators P and Q on K by
P e k = u k , Q e k = λ k e k , k = 1 , 2 , .
Then, P is unitary and Q S 1 * ( K ) , tr ( Q ) = 0 and L = P Q P * . Let
λ = 1 + k = 1 | λ k | , t 1 = 1 + | λ 1 | λ , s 1 = 1 + | λ 1 | λ 1 λ ,
and
t k = | λ k | λ , s k = | λ 1 | λ 1 λ , i = 2 , 3 .
Hence, λ , t k , s k are non-negative numbers and λ k = t k s k , k = 1 , 2 , . Let V 1 , V 2 be operators on K defined by
V 1 e k = t k e k , V 2 e k = s k e k , k = 1 , 2 , .
Then,
V 1 , V 2 S 1 * ( K , tr ( V 1 ) = tr ( V 2 ) = 1
and we have
L = P V 1 P * P V 2 P * .
Moreover, P V 1 Q * , P V 2 Q * S 1 + * ( K ) and tr ( P V 1 P * ) = tr ( P V 2 P * ) = 1 . Since
L T k , T k = P V 1 P * T k , T k P V 2 P * T k , T k = 0 ,
we obtain
P V 1 P * = P V 2 P * .
It follows that L = 0 , and hence, M = N .
2 ⇒ 3 Assume that
M S 1 * ( K ) and tr ( M ) = 0
such that
M T k , T k = 0 , k = 1 , 2 , .
Let N be a operator on K defined by
N e 1 = e 1 , N e k = 0 , i = 2 , 3 , .
Then
M , M + N S 1 * ( K ) and tr ( M ) = tr ( M + N ) = 1 .
Since
( M + N ) T k , T k = N T k , T k , k = 1 , 2 , ,
this implies that M + N = N and therefore M = 0 .
3 ⇒ 1 Suppose that
M , N S 1 + * ( K ) and tr ( M ) = tr ( N ) = 1
such that
M T k , T k = N T k , T k , k = 1 , 2 , .
Thus,
( M N ) T k , T k = 0 , k = 1 , 2 , .
It follows that M = N . □
The theorem below demonstrates a method to derive an injective g-frame utilizing an injective frame. It is worth noting that a frame { f k } k = 1 is considered injective if, for any self-adjoint trace class operator M that fulfills the condition
M f k | f k = 0 , k = 1 , 2 , ,
it implies that M = 0 .
Theorem 3. 
Let { f k } k = 1 be a frame for K . Define
T k = f k f k , k = 1 , 2 , .
Then, { T k } k = 1 constitutes an injective g-frame for K if, and only if, { f k } k = 1 is an injective frame for K .
Proof. 
Let M S 1 * ( K ) . Then,
M T k , T k = M ( f k f k ) , f k f k = f k 2 M f k | f k .
These identities, along with standard arguments, suffice to prove the theorem. □
Theorem 4. 
Let { T k } k = 1 be an injective g-frame for a K . Then, for each invertible operator Θ on K , the sequence { S k } k = 1 , defined by S k = T k Θ , k = 1 , 2 , , is likewise an injective g-frame for K .
Proof. 
Firstly, let A and B be the g-frame bounds for the sequence { T k } k = 1 . Consequently, for any f belonging to K , we have the inequality:
A Θ f 2 k = 1 S k f 2 B Θ f 2 .
Furthermore, given that Θ is invertible, we can derive:
A Θ 1 2 f 2 k = 1 S k f 2 B Θ 2 f 2 .
Hence, { S k } k = 1 constitutes a g-frame for K .
Secondly, consider M S 1 * ( K ) such that
M T k Θ , T k Θ = tr ( Θ * T k * M T k Θ ) = 0 , k = 1 , 2 , .
The injectivity of { T k } k = 1 leads to the conclusion that Θ * M Θ = 0 , and consequently, M = 0 . □
Corollary 1. 
Let { T k } k = 1 be an injective g-frame for K . Then, the canonical dual g-frame { T k S 1 } k = 1 of { T k } k = 1 is likewise an injective g-frame, where S is the g-frame operator of { T k } k = 1 .
Example 1. 
Let { e k } k = 1 be the Hilbert basis for 2 ( W ) and let
a k 0 , b k 0 for k = 1 , 2 ,
be such that
k = 1 | a k | 2 < and k = 1 | b k | 2 < .
Define
x k = a k ( e 1 + e k + 1 ) , y k = b k ( e 1 + i e k + 1 ) for k = 1 , 2 , .
Let S r be the right shift operator on 2 ( W ) and let T k = e k e k for k = 1 , 2 , and
{ Γ i k } i = 0 , k = 1 , = { 1 2 i S r i x k 1 2 i S r i x k } i = 0 , k = 1 , ,
{ Ω i k } i = 0 , k = 1 , = { 1 2 i S r i y k 1 2 i S r i y k } i = 0 , k = 1 , .
Then, Y = { T k } k = 1 { Γ i k } i = 0 , k = 1 , { Ω i k } i = 0 , k = 1 , is an injective g-frame for 2 ( W ) .
The stability of injective frames has attracted significant attention in the research. We proceed to investigate the stability characteristics of injective g-frames. Initially, a definition is required.
Definition 4. 
Given g-frames Λ = { T k } k = 1 and Γ = { T k } k = 1 for 2 ( W ) , we define the distance between them using
d 2 ( Λ , Γ ) = k = 1 T k Γ k 2 .
The theorem below demonstrates the instability of injective g-frames in the context of infinite dimensions.
Theorem 5. 
As in Example 1, let Y = { T k } k = 1 { Γ i k } i = 0 , k = 1 , { Ω i k } i = 0 , k = 1 , be an injective g-frame for 2 ( W ) . Then, for every ε > 0 , there is a g-frame Z , such that d ( Y , Z ) < ϵ , and Z is not injective.
Proof. 
For any f belonging to 2 ( W ) , the following inequality holds:
i = 0 k = 1 | f , 1 2 i S r i x k | 2 i = 0 k = 1 1 4 i 4 a k 2 f 2 { a k } k = 1 2 f 2 .
In other words, the sequence { 1 2 i S r i x k } i = 0 , k = 1 , constitutes a Bessel sequence. Consequently,
i = 0 k = 1 1 2 i S r i x k 1 2 i S r i x k 2
converges. Therefore, there is an N such that, for every given ε ,
i = N + 1 k = 1 1 2 i S r i x k 1 2 i S r i x k 2 < ε 2 .
Define Φ i k = 1 2 i S r i x k 1 2 i S r i x k for i = 0 , 1 , , N and k = 1 , 2 , and set Φ i k = 0 for other values of i and k. According to Theorem 3.13 in [11], the set Z = { T k } k = 1 { Φ i k } i = 0 , k = 1 , { Ω i k } i = 0 , k = 1 , does not ensure injectivity. This concludes the proof. □
Example 2. 
Let { e k } k = 1 be the Hilbert basis for 2 ( W ) and let a k 0 for k = 1 , 2 , be such that
k = 1 a k 2 < .
Define x k = a k ( e 1 + e k + 1 ) for k = 1 , 2 , . Let S r be the right shift operator on 2 ( W ) and let
{ T k } k = 1 = { e k e k } k = 1 , { Γ i k } i = 0 , k = 1 , = { 1 2 i S r i x k 1 2 i S r i x k } i = 0 , k = 1 , .
Then, X = { T k } k = 1 { Γ i k } i = 0 , k = 1 , is an injective g-frame for 2 ( W ) .

4. Conclusions

In this paper, we focus on the quantum detection problem in the framework of right quaternionic Hilbert spaces by employing g-frames. Several equivalent representations are provided for injective g-frames in right quaternionic Hilbert spaces. By normalizing the trace, the article classifies the problem of g-frame injectivity. Moreover, a method is proposed to derive an injective g-frame by leveraging an existing injective frame within quaternionic Hilbert spaces. The study further reveals that the injectivity of a g-frame is preserved under linear isomorphism, whereas injective g-frames exhibit instability in infinite-dimensional contexts. These findings not only extend the current knowledge base but also pave the way for future research by suggesting directions for exploring the applications of g-frames in more complex and diverse quaternionic Hilbert spaces.

Author Contributions

Formal analysis, J.Z., F.G. and G.H.; funding acquisition, J.Z. and G.H.; investigation, J.Z. and G.H.; methodology, J.Z., F.G. and G.H.; software, J.Z., F.G. and G.H.; validation, J.Z., F.G. and G.H.; writing—original draft, J.Z. and G.H. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by National Natural Science Foundation of China (12301149; 62272079), Henan Provincial Department of Science and Technology Research Project (232102210187) and Key Scientific Research Projects of Colleges and Universities in Henan Province (23B210005).

Data Availability Statement

Data are contained within the article.

Acknowledgments

The authors would like to extend their sincere gratitude to the referees for their careful reading of this article and valuable comments, which helped to greatly improve the readability of this article.

Conflicts of Interest

The authors declare no conflicts of interest.

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Zhang, J.; Gao, F.; Hong, G. A Note on Injective g-Frames in Quaternionic Hilbert Spaces. Axioms 2024, 13, 851. https://doi.org/10.3390/axioms13120851

AMA Style

Zhang J, Gao F, Hong G. A Note on Injective g-Frames in Quaternionic Hilbert Spaces. Axioms. 2024; 13(12):851. https://doi.org/10.3390/axioms13120851

Chicago/Turabian Style

Zhang, Jianxia, Fugen Gao, and Guoqing Hong. 2024. "A Note on Injective g-Frames in Quaternionic Hilbert Spaces" Axioms 13, no. 12: 851. https://doi.org/10.3390/axioms13120851

APA Style

Zhang, J., Gao, F., & Hong, G. (2024). A Note on Injective g-Frames in Quaternionic Hilbert Spaces. Axioms, 13(12), 851. https://doi.org/10.3390/axioms13120851

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