A Note on Injective g-Frames in Quaternionic Hilbert Spaces

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 ${\left\{f_k\right\}}_{i=1}^{\infty }$ is a frame for a quaternionic Hilbert space $\mathbb{K}$, if there exist constants $\alpha ,\beta >0$ such that, for all $f\in \mathbb{K}$,

$${\alpha \parallel f\parallel }^2\le \sum _{k=1}^{\infty }|⟨f_k\mid f⟩{|}^2\le \beta {\parallel f\parallel }^2.$$

The numbers $\alpha$ and $\beta$ are referred to as the lower and upper frame bounds, respectively. In the particular scenario where $\alpha$ equals $\beta$, the frame is known as a tight frame. Additionally, when both $\alpha$ and $\beta$ 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 $\mathcal{L}(\mathbb{K},{\mathbb{K}}_k)$ denote the set of all bounded linear operators from the right quaternionic Hilbert space $\mathbb{K}$ to the right quaternionic Hilbert space ${\mathbb{K}}_k$. A family ${\{T_k\in \mathcal{L}(\mathbb{K},{\mathbb{K}}_k)\}}_{k=1}^{\infty }$ is termed a g-frame for $\mathbb{K}$ relative to ${\left\{{\mathbb{K}}_k\right\}}_{k=1}^{\infty }$ if there exist constants $A,B>0$ such that, for every $f\in \mathbb{K}$, the inequality

$${A\parallel f\parallel }^2\le \sum _{k=1}^{\infty }\parallel T_k{f\parallel }^2\le B{\parallel f\parallel }^2,$$

holds. The constants A and B are referred to as the g-frame bounds. A g-frame ${\left\{T_k\right\}}_{k=1}^{\infty }$ 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 ${\mathbb{K}}_k$ and the corresponding operators ${\{T_k\in \mathcal{L}(\mathbb{K},{\mathbb{K}}_k)\}}_{k=1}^{\infty }$. 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 $\varphi$ such that if, for every measurable set E,

$$\mathrm{tr}\left({\rho }_1\varphi \left(E\right)\right)=\mathrm{tr}\left({\rho }_2\varphi \left(E\right)\right)$$

implies ${\rho }_1={\rho }_2$, where ${\rho }_1,{\rho }_2\in \mathcal{L}\left(\mathbb{K}\right)$ 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 ${\mathcal{S}}_p$-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 $\mathbb{W}$ represents an extension of the complex numbers into a four dimensional space, defined as

$$\mathbb{W}=\left\{\right|w=w_0+w_{1}x+w_{2}y+w_{3}z,\mathrm{where}{w}_0,w_1,w_2,w_3\in \mathbb{R}\}$$

with the elements $x,y,z$ fulfilling the following relations:

$$xy=-yx=z,x^2=y^2=z^2=-1\mathrm{and}xyz=-1.$$

Define the space ${\ell }_2\left(\mathbb{W}\right)$ as:

$${\ell }_2\left(\mathbb{W}\right)=\left\{{\left\{w_k\right\}}_{k\in \mathbb{I}}\subset \mathbb{W}:\sum _{k\in \mathbb{I}}{\left|w_k\right|}^2<\infty \right\}.$$

This space carries out the operation of multiplying on the right using quaternion scalars and is furnished with a quaternionic inner product on ${\ell }_2\left(\mathbb{W}\right)$, which is specified by

$$⟨v\mid w⟩=\sum _{k\in \mathbb{I}}\overline{v_k}{w}_k,v={\left\{v_k\right\}}_{k\in \mathbb{I}}\mathrm{and}w={\left\{w_k\right\}}_{k\in \mathbb{I}}.$$

(1)

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, $\mathbb{K}$ 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 $\varphi :\mathsf{{\rm Y}}\to \mathcal{L}\left(\mathbb{K}\right)$ is defined as an operator-valued measure if, for any countable collection ${\left\{U_k\right\}}_{k\in \mathbb{I}}\subseteq \mathsf{{\rm Y}}$, such that

$$U_k\cap U_j=\varnothing ,k\ne j,$$

it fulfills the condition

$$\varphi \left(\bigcup _{k\in \mathbb{I}}{U}_k\right)=\sum _{k\in \mathbb{I}}\varphi \left(U_k\right),$$

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 $\varphi \left(\Xi \right)=I_{\mathbb{K}}$.

As previously stated, the quantum detection problem involves investigating the presence of a quantum measurement executed by a POVM, denoted as $\varphi$, which is capable of distinguishing states within a quaternionic Hilbert space $\mathbb{K}$. Define $\mathcal{L}(\mathsf{{\rm Y}},\mathbb{R})$ as the collection of bounded functions on $\mathsf{{\rm Y}}$, and $\mathcal{S}\left(\mathbb{K}\right)$ as the set of density operators on the quaternionic Hilbert space $\mathbb{K}$, satisfying

$$\{M:M=M^{*}\ge 0,\mathrm{tr}\left(M\right)=1\}.$$

For a given system, the quantum detection problem poses the following question: Is there a specific type of POVM $\varphi$ that ensures the following map is injective?

$$\mathbb{Q}:\mathcal{S}\left(\mathbb{K}\right)\to \mathcal{L}(\mathsf{{\rm Y}},\mathbb{R}),\mathbb{Q}\left(\rho \right)\left(X\right)=\mathrm{tr}\left(\rho \varphi \right(X\left)\right),\forall X\in \mathsf{{\rm Y}},$$

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 $\mathbb{Q}$ 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 ${\{T_k\in \mathcal{L}(\mathbb{K},{\mathbb{K}}_k)\}}_{k=1}^{\infty }$, does the mapping

$$\mathbb{M}:\mathcal{S}\left(\mathbb{K}\right)\mapsto {\ell }^{\infty },{\left(\mathbb{M}\left(M\right)\right)}_k=\mathrm{tr}\left(T_k^{*}M{T}_k\right),\forall k=1,2,\cdots$$

become injective?

Definition 3. 

We say that a g-frame ${\{T_k\in \mathcal{L}(\mathbb{K},{\mathbb{K}}_k)\}}_{k=1}^{\infty }$ provides quantum injectivity when the map $\mathbb{M}$ that is related to ${\left\{T_k\right\}}_{k=1}^{\infty }$ is injective.

For $k=1,2,\cdots ,$ express $\mathrm{tr}\left(T_k^{*}M{T}_k\right)$ as $⟨M{T}_k,T_k⟩$. Let ${\mathcal{S}}_1\left(\mathbb{K}\right)$ represent the collection of trace class operators on $\mathbb{K}$, and use ${\mathbb{M}}_1$ to denote the following linear operator:

$${\mathbb{M}}_1:{\mathcal{S}}_1\left(\mathbb{K}\right)\mapsto {\ell }^{\infty },{\left({\mathbb{M}}_1\left(M\right)\right)}_k=⟨M{T}_k,T_k⟩,k=1,2,\cdots .$$

Using ${\parallel ·\parallel }_1$ to signify the ${\ell }^1$-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

$${A\parallel f\parallel }^2\le \parallel {\mathbb{M}}_1{(f\otimes f)\parallel }_1\le B{\parallel f\parallel }^2.$$

Proof. 

It is evident that $f\otimes f$ belongs to ${\mathcal{S}}_1\left(\mathbb{K}\right)$. For any element f in $\mathbb{K}$, it is easy to observe that the i-th entry of ${\mathbb{M}}_1(f\otimes f)$ is provided by $\parallel T_k{\left(f\right)\parallel }^2$, and thus

$$\parallel {\mathbb{M}}_1{(f\otimes f)\parallel }_1=\sum _{k=1}^{\infty }{\parallel T_k\left(f\right)\parallel }^2.$$

Consequently, a sequence constitutes a g-frame if and only if the condition

$${A\parallel f\parallel }^2\le \parallel {\mathbb{M}}_1{(f\otimes f)\parallel }_1\le B{\parallel f\parallel }^2$$

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

$$⟨M{T}_k,T_k⟩=0,\forall k=1,2,\cdots ,$$

then $M=0$. Analogously, ${\left\{T_k\right\}}_{k=1}^{\infty }$ is deemed injective if, for any self-adjoint trace-class operator M satisfying

$$⟨M{T}_k,T_k⟩=0,\forall k=1,2,\cdots ,$$

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 ${\mathcal{S}}_{1+}^{*}\left(\mathbb{K}\right)$ denote the positive trace class and self-adjoint operators on $\mathbb{K}$ and ${\mathcal{S}}_1^{*}\left(\mathbb{K}\right)$ the trace class and self-adjoint operators on $\mathbb{K}$.

Theorem 1. 

Given a g-frame ${\left\{T_k\right\}}_{k=1}^{\infty }$ for $\mathbb{K}$, the following are equivalent:

1. 

If $M,N\in {\mathcal{S}}_{1+}^{*}\left(\mathbb{K}\right)$ and

$$⟨M{T}_k,T_k⟩=⟨N{T}_k,T_k⟩,\forall k=1,2,\cdots ,$$

then $M=N$.

2. 

If $M,N\in {\mathcal{S}}_1^{*}\left(\mathbb{K}\right)$ and

$$⟨M{T}_k,T_k⟩=⟨N{T}_k,T_k⟩,\forall k=1,2,\cdots ,$$

then $M=N$.

3. 

${\left\{T_k\right\}}_{k=1}^{\infty }$ is an injective g-frame.

Proof. 

1 ⇒ 2 Let $M,N\in {\mathcal{S}}_1^{*}\left(\mathbb{K}\right)$, such that

$$⟨M{T}_k,T_k⟩=⟨N{T}_k,T_k⟩,\forall k=1,2,\cdots .$$

Set $L=M-N$. Then, $R\in {\mathcal{S}}_1^{*}\left(\mathbb{K}\right)$. Let ${\left\{e_k\right\}}_{k=1}^{\infty }$ be a Hilbert basis for $\mathbb{K}$ and let ${\left\{u_k\right\}}_{k=1}^{\infty }$ be an eigenbasis for R with respect to the eigenvalues ${\left\{{\lambda }_k\right\}}_{k=1}^{\infty }$. Define operators P and Q on $\mathbb{K}$ using

$$P{e}_k=u_k,Q{e}_k={\lambda }_k{e}_k,\forall k=1,2,\cdots .$$

Therefore, P is unitary and $Q\in {\mathcal{S}}_1^{*}\left(\mathbb{K}\right)$. Since

$$PQ{P}^{*}{u}_k=PQ{e}_k={\lambda }_{k}P{e}_k={\lambda }_k{u}_k=L{u}_k,\forall k=1,2,\cdots ,$$

we have

$$L=UV{U}^{*}.$$

Let

$$p_k=|{\lambda }_k|\mathrm{and}{q}_k=\left|{\lambda }_k\right|-{\lambda }_k,\forall k=1,2,\cdots .$$

Clearly, ${\lambda }_k=p_k-q_k$. Let $Q_1$ and $Q_2$ be operators defined by

$$Q_1{e}_k=p_k{e}_k,Q_2{e}_k=q_k{e}_k,\forall k=1,2,\cdots .$$

Note that $L\in {\mathcal{S}}_1\left(\mathbb{K}\right)$, therefore,

$$\sum _{k=1}^{\infty }\left|{\lambda }_k\right|<\infty .$$

Hence $Q_1,Q_2\in {\mathcal{S}}_{1+}^{*}\left(\mathbb{K}\right)$ and we have

$$L=P{Q}_1{P}^{*}-P{Q}_2{P}^{*}.$$

Moreover, $P{Q}_1{P}^{*},P{Q}_2{P}^{*}\in {\mathcal{S}}_{1+}^{*}\left(\mathbb{K}\right)$. Since

$$\begin{array}{cc}⟨L{T}_k,T_k⟩\hfill & =\mathrm{tr}\left(T_k^{*}L{T}_k\right)\hfill \\ & =\mathrm{tr}\left(T_k^{*}(P{Q}_1{P}^{*}-P{Q}_2{P}^{*})T_k\right)\hfill \\ & =⟨P{Q}_1{P}^{*}{T}_k,T_k⟩-⟨P{Q}_2{P}^{*}{T}_k,T_k⟩=0,\hfill \end{array}$$

we can obtain that

$$P{Q}_1{P}^{*}=P{Q}_2{P}^{*}.$$

Therefore, $L=0$, and thus $P=Q$.

2 ⇒ 3 Assume that $M\in {\mathcal{S}}_1^{*}\left(\mathbb{K}\right)$, such that

$$⟨M{T}_k,T_k⟩=0,\forall k=1,2,\cdots .$$

It follows that

$$⟨M{T}_k,T_k⟩=⟨0T_k,T_k⟩,\forall k=1,2,\cdots .$$

Hence, $M=0$. Thus, ${\left\{T_k\right\}}_{k=1}^{\infty }$ is an injective g-frame.

3 ⇒ 1 Suppose that $M,N\in {\mathcal{S}}_{1+}^{*}\left(\mathbb{K}\right)$, such that

$$⟨M{T}_k,T_k⟩=⟨N{T}_k,T_k⟩,\forall k=1,2,\cdots .$$

It follows that

$$⟨(M-N)T_k,T_k⟩=0,\forall k=1,2,\cdots .$$

As $M-N$ is a self-adjoint operator and the sequence ${\left\{T_k\right\}}_{k=1}^{\infty }$ 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 ${\left\{T_k\right\}}_{k=1}^{\infty }$ be a g-frame for $\mathbb{K}$; the following are equivalent:

1. 

If $M,N\in {\mathcal{S}}_{1+}^{*}\left(\mathbb{K}\right)$, $\mathrm{tr}\left(M\right)=\mathrm{tr}\left(N\right)=1$ and

$$⟨M{T}_k,T_k⟩=⟨N{T}_k,T_k⟩,\forall k=1,2,\cdots ,$$

then $M=N$.

2. 

If $M,N\in {\mathcal{S}}_1^{*}\left(\mathbb{K}\right)$, $\mathrm{tr}\left(M\right)=\mathrm{tr}\left(N\right)=1$ and

$$⟨M{T}_k,T_k⟩=⟨N{T}_k,T_k⟩,\forall k=1,2,\cdots ,$$

then $M=N$.

3. 

If $M\in {\mathcal{S}}_1^{*}\left(\mathbb{K}\right)$, $\mathrm{tr}\left(M\right)=0$ and

$$⟨M{T}_k,T_k⟩=0,\forall k=1,2,\cdots ,$$

then $M=0$.

Proof. 

1 ⇒ 2 Let

$$M,N\in {\mathcal{S}}_1^{*}\left(\mathbb{K}\right)\mathrm{and}\mathrm{tr}\left(M\right)=\mathrm{tr}\left(N\right)=1$$

such that

$$⟨M{T}_k,T_k⟩=⟨N{T}_k,T_k⟩,\forall k=1,2,\cdots .$$

Set $L=M-N$. Then,

$$L\in {\mathcal{S}}_1^{*}\left(\mathbb{K}\right)\mathrm{and}\mathrm{tr}\left(L\right)=1$$

Let ${\left\{e_k\right\}}_{k=1}^{\infty }$ be a Hilbert basis for $\mathbb{K}$ and let ${\left\{u_k\right\}}_{k=1}^{\infty }$ be an eigenbasis for L with respect to the eigenvalues ${\left\{{\lambda }_k\right\}}_{k=1}^{\infty }$. Then,

$$\sum _{k=1}^{\infty }{\lambda }_k=0.$$

Define operators P and Q on $\mathbb{K}$ by

$$P{e}_k=u_k,Q{e}_k={\lambda }_k{e}_k,\forall k=1,2,\cdots .$$

Then, P is unitary and $Q\in {\mathcal{S}}_1^{*}\left(\mathbb{K}\right)$, $\mathrm{tr}\left(Q\right)=0$ and $L=PQ{P}^{*}.$ Let

$$\lambda =1+\sum _{k=1}^{\infty }\left|{\lambda }_k\right|,t_1={\displaystyle \frac{1+|{\lambda }_1|}{\lambda }},s_1={\displaystyle \frac{1+|{\lambda }_1|-{\lambda }_1}{\lambda }},$$

and

$$t_k={\displaystyle \frac{|{\lambda }_k|}{\lambda }},s_k={\displaystyle \frac{|{\lambda }_1|-{\lambda }_1}{\lambda }},\forall i=2,3\cdots .$$

Hence, $\lambda ,t_k,s_k$ are non-negative numbers and ${\lambda }_k=t_k-s_k,\forall k=1,2,\cdots .$ Let $V_1,V_2$ be operators on $\mathbb{K}$ defined by

$$V_1{e}_k=t_k{e}_k,V_2{e}_k=s_k{e}_k,\forall k=1,2,\cdots .$$

Then,

$$V_1,V_2\in {\mathcal{S}}_1^{*}(\mathbb{K},\mathrm{tr}\left(V_1\right)=\mathrm{tr}\left(V_2\right)=1$$

and we have

$$L=P{V}_1{P}^{*}-P{V}_2{P}^{*}.$$

Moreover, $P{V}_1{Q}^{*},P{V}_2{Q}^{*}\in {\mathcal{S}}_{1+}^{*}\left(\mathbb{K}\right)$ and $\mathrm{tr}\left(P{V}_1{P}^{*}\right)=\mathrm{tr}\left(P{V}_2{P}^{*}\right)=1$. 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

$$M\in {\mathcal{S}}_1^{*}\left(\mathbb{K}\right)\mathrm{and}\mathrm{tr}\left(M\right)=0$$

such that

$$⟨M{T}_k,T_k⟩=0,\forall k=1,2,\cdots .$$

Let N be a operator on $\mathbb{K}$ defined by

$$N{e}_1=e_1,N{e}_k=0,\forall i=2,3,\cdots .$$

Then

$$M,M+N\in {\mathcal{S}}_1^{*}\left(\mathbb{K}\right)\mathrm{and}\mathrm{tr}\left(M\right)=\mathrm{tr}(M+N)=1.$$

Since

$$⟨(M+N)T_k,T_k⟩=⟨N{T}_k,T_k⟩,\forall k=1,2,\cdots ,$$

this implies that $M+N=N$ and therefore $M=0$.

3 ⇒ 1 Suppose that

$$M,N\in {\mathcal{S}}_{1+}^{*}\left(\mathbb{K}\right)\mathrm{and}\mathrm{tr}\left(M\right)=\mathrm{tr}\left(N\right)=1$$

such that

$$⟨M{T}_k,T_k⟩=⟨N{T}_k,T_k⟩,\forall k=1,2,\cdots .$$

Thus,

$$⟨(M-N)T_k,T_k⟩=0,\forall k=1,2,\cdots .$$

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 ${\left\{f_k\right\}}_{k=1}^{\infty }$ is considered injective if, for any self-adjoint trace class operator M that fulfills the condition

$$⟨M{f}_k|f_k⟩=0,\forall k=1,2,\cdots ,$$

it implies that $M=0$.

Theorem 3. 

Let ${\left\{f_k\right\}}_{k=1}^{\infty }$ be a frame for $\mathbb{K}$. Define

$$T_k=f_k\otimes f_k,\forall k=1,2,\cdots .$$

Then, ${\left\{T_k\right\}}_{k=1}^{\infty }$ constitutes an injective g-frame for $\mathbb{K}$ if, and only if, ${\left\{f_k\right\}}_{k=1}^{\infty }$ is an injective frame for $\mathbb{K}$.

Proof. 

Let $M\in {\mathcal{S}}_1^{*}\left(\mathbb{K}\right)$. Then,

$$⟨M{T}_k,T_k⟩=⟨M(f_k\otimes f_k),f_k\otimes f_k⟩={\parallel f_k\parallel }^2⟨M{f}_k|f_k⟩.$$

These identities, along with standard arguments, suffice to prove the theorem. □

Theorem 4. 

Let ${\left\{T_k\right\}}_{k=1}^{\infty }$ be an injective g-frame for a $\mathbb{K}$. Then, for each invertible operator Θ on $\mathbb{K}$, the sequence ${\left\{S_k\right\}}_{k=1}^{\infty }$, defined by $S_k=T_k\mathsf{\Theta },k=1,2,\cdots ,$ is likewise an injective g-frame for $\mathbb{K}$.

Proof. 

Firstly, let A and B be the g-frame bounds for the sequence ${\left\{T_k\right\}}_{k=1}^{\infty }$. Consequently, for any f belonging to $\mathbb{K}$, we have the inequality:

$${A\parallel \mathsf{\Theta }f\parallel }^2\le \sum _{k=1}^{\infty }\parallel S_k{f\parallel }^2\le B{\parallel \mathsf{\Theta }f\parallel }^2.$$

Furthermore, given that $\mathsf{\Theta }$ is invertible, we can derive:

$${\displaystyle \frac{A}{\parallel {\mathsf{\Theta }}^{-1}{\parallel }^2}}{\parallel f\parallel }^2\le \sum _{k=1}^{\infty }\parallel S_k{f\parallel }^2\le {B\parallel \mathsf{\Theta }\parallel }^2{\parallel f\parallel }^2.$$

Hence, ${\left\{S_k\right\}}_{k=1}^{\infty }$ constitutes a g-frame for $\mathbb{K}$.

Secondly, consider $M\in {\mathcal{S}}_1^{*}\left(\mathbb{K}\right)$ such that

$$⟨M{T}_k\mathsf{\Theta },T_k\mathsf{\Theta }⟩=\mathrm{tr}\left({\mathsf{\Theta }}^{*}{T}_k^{*}M{T}_k\mathsf{\Theta }\right)=0,\forall k=1,2,\cdots .$$

The injectivity of ${\left\{T_k\right\}}_{k=1}^{\infty }$ leads to the conclusion that ${\mathsf{\Theta }}^{*}M\mathsf{\Theta }=0$, and consequently, $M=0$. □

Corollary 1. 

Let ${\left\{T_k\right\}}_{k=1}^{\infty }$ be an injective g-frame for $\mathbb{K}$. Then, the canonical dual g-frame ${\left\{T_k{S}^{-1}\right\}}_{k=1}^{\infty }$ of ${\left\{T_k\right\}}_{k=1}^{\infty }$ is likewise an injective g-frame, where S is the g-frame operator of ${\left\{T_k\right\}}_{k=1}^{\infty }$.

Example 1. 

Let ${\left\{e_k\right\}}_{k=1}^{\infty }$ be the Hilbert basis for ${\ell }_2\left(\mathbb{W}\right)$ and let

$$a_k\ne 0,b_k\ne 0\mathit{for}k=1,2,\cdots$$

be such that

$$\sum _{k=1}^{\infty }|a_k{|}^2<\infty \mathit{and}\sum _{k=1}^{\infty }{\left|b_k\right|}^2<\infty .$$

Define

$$x_k=a_k(e_1+e_{k+1}),y_k=b_k(e_1+\mathrm{i}{e}_{k+1})\mathit{for}k=1,2,\cdots .$$

Let $S_r$ be the right shift operator on ${\ell }_2\left(\mathbb{W}\right)$ and let $T_k=e_k\otimes e_k$ for $k=1,2,\cdots$ and

$${\left\{{\mathsf{\Gamma }}_{ik}\right\}}_{i=0,k=1}^{\infty ,\infty }={\{{\displaystyle \frac{1}{2^i}}{S}_r^i{x}_k\otimes {\displaystyle \frac{1}{2^i}}{S}_r^i{x}_k\}}_{i=0,k=1}^{\infty ,\infty },$$

$${\left\{{\mathsf{\Omega }}_{ik}\right\}}_{i=0,k=1}^{\infty ,\infty }={\{{\displaystyle \frac{1}{2^i}}{S}_r^i{y}_k\otimes {\displaystyle \frac{1}{2^i}}{S}_r^i{y}_k\}}_{i=0,k=1}^{\infty ,\infty }.$$

Then, $\mathcal{Y}={\left\{T_k\right\}}_{k=1}^{\infty }\cup {\left\{{\mathsf{\Gamma }}_{ik}\right\}}_{i=0,k=1}^{\infty ,\infty }\cup {\left\{{\mathsf{\Omega }}_{ik}\right\}}_{i=0,k=1}^{\infty ,\infty }$ is an injective g-frame for ${\ell }_2\left(\mathbb{W}\right)$.

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 $\mathsf{\Lambda }={\left\{T_k\right\}}_{k=1}^{\infty }$ and $\mathsf{\Gamma }={\left\{T_k\right\}}_{k=1}^{\infty }$ for ${\ell }_2\left(\mathbb{W}\right)$, we define the distance between them using

$$d^2(\mathsf{\Lambda },\mathsf{\Gamma })=\sum _{k=1}^{\infty }{\parallel T_k-{\mathsf{\Gamma }}_k\parallel }^2.$$

The theorem below demonstrates the instability of injective g-frames in the context of infinite dimensions.

Theorem 5. 

As in Example 1, let $\mathcal{Y}={\left\{T_k\right\}}_{k=1}^{\infty }\cup {\left\{{\mathsf{\Gamma }}_{ik}\right\}}_{i=0,k=1}^{\infty ,\infty }\cup {\left\{{\mathsf{\Omega }}_{ik}\right\}}_{i=0,k=1}^{\infty ,\infty }$ be an injective g-frame for ${\ell }_2\left(\mathbb{W}\right)$. Then, for every $\epsilon >0$, there is a g-frame $\mathcal{Z}$, such that $d(\mathcal{Y},\mathcal{Z})<ϵ,$ and $\mathcal{Z}$ is not injective.

Proof. 

For any f belonging to ${\ell }_2\left(\mathbb{W}\right)$, the following inequality holds:

$$\sum _{i=0}^{\infty }\sum _{k=1}^{\infty }|⟨f,{\displaystyle \frac{1}{2^i}}{S}_r^i{x}_k⟩{|}^2\le \sum _{i=0}^{\infty }\sum _{k=1}^{\infty }{\displaystyle \frac{1}{4^i}}4a_k^2{\parallel f\parallel }^2\le \parallel {\left\{a_k\right\}}_{k=1}^{\infty }{\parallel }^2{\parallel f\parallel }^2.$$

In other words, the sequence ${\left\{{\displaystyle \frac{1}{2^i}}{S}_r^i{x}_k\right\}}_{i=0,k=1}^{\infty ,\infty }$ constitutes a Bessel sequence. Consequently,

$$\sum _{i=0}^{\infty }\sum _{k=1}^{\infty }\parallel {\displaystyle \frac{1}{2^i}}{S}_r^i{x}_k\otimes {\displaystyle \frac{1}{2^i}}{S}_r^i{x}_k{\parallel }^2$$

converges. Therefore, there is an N such that, for every given $\epsilon$,

$$\sum _{i=N+1}^{\infty }\sum _{k=1}^{\infty }\parallel {\displaystyle \frac{1}{2^i}}{S}_r^i{x}_k\otimes {\displaystyle \frac{1}{2^i}}{S}_r^i{x}_k{\parallel }^2<{\epsilon }^2.$$

Define ${\mathsf{\Phi }}_{ik}={\displaystyle \frac{1}{2^i}}{S}_r^i{x}_k\otimes {\displaystyle \frac{1}{2^i}}{S}_r^i{x}_k$ for $i=0,1,\cdots ,N$ and $k=1,2,\cdots$ and set ${\mathsf{\Phi }}_{ik}=0$ for other values of i and k. According to Theorem 3.13 in [11], the set $\mathcal{Z}={\left\{T_k\right\}}_{k=1}^{\infty }\cup {\left\{{\mathsf{\Phi }}_{ik}\right\}}_{i=0,k=1}^{\infty ,\infty }\cup {\left\{{\mathsf{\Omega }}_{ik}\right\}}_{i=0,k=1}^{\infty ,\infty }$ does not ensure injectivity. This concludes the proof. □

Example 2. 

Let ${\left\{e_k\right\}}_{k=1}^{\infty }$ be the Hilbert basis for ${\ell }_2\left(\mathbb{W}\right)$ and let $a_k\ne 0$ for $k=1,2,\cdots$ be such that

$$\sum _{k=1}^{\infty }{a}_k^2<\infty .$$

Define $x_k=a_k(e_1+e_{k+1})$ for $k=1,2,\cdots .$ Let $S_r$ be the right shift operator on ${\ell }_2\left(\mathbb{W}\right)$ and let

$${\left\{T_k\right\}}_{k=1}^{\infty }={\{e_k\otimes e_k\}}_{k=1}^{\infty },{\left\{{\mathsf{\Gamma }}_{ik}\right\}}_{i=0,k=1}^{\infty ,\infty }={\{{\displaystyle \frac{1}{2^i}}{S}_r^i{x}_k\otimes {\displaystyle \frac{1}{2^i}}{S}_r^i{x}_k\}}_{i=0,k=1}^{\infty ,\infty }.$$

Then, $\mathcal{X}={\left\{T_k\right\}}_{k=1}^{\infty }\cup {\left\{{\mathsf{\Gamma }}_{ik}\right\}}_{i=0,k=1}^{\infty ,\infty }$ is an injective g-frame for ${\ell }_2\left(\mathbb{W}\right)$.

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.