Quantum Injectivity of Frames in Quaternionic Hilbert Spaces

Abstract

A quantum injective frame is a frame capable of differentiating states based on their respective frame measurements, whereas the quantum-detection problem associated with frames endeavors to delineate all such frames. In the present paper, the concept of injective frames in infinite dimensional quaternionic Hilbert spaces is introduced. Further, some properties of injective frames such as the invariance of injective frames under invertible operators are discussed and several solutions to the frame quantum-detection problem are given. Finally, by employing operator theory and frames theory in quaternionic Hilbert spaces, some characterizations and classifications of frames for solving the injectivity problem are given.

1. Introduction

Quantum theory emerged as a means to address the outcomes of physical measurements that remained unexplained within the conventional framework [1]. The retrieval of data from quantum systems is systematically executed in accordance with the principles of quantum-measurement theory [2]. This measurement process is facilitated by a quantum instrument, aimed at accurately determining a quantum state—a prerequisite for the operation of quantum information-processing devices, including quantum teleporters and quantum computers [3].

The quaternion system, initially formulated by Hamilton in 1843, was subsequently applied within the realm of mathematics. Initially, quaternions encountered skepticism due to their non-commutative multiplication property. However, their distinctive structures were harnessed to represent rotational and translational movements in geometry. Quaternions have found widespread applications in classical Newtonian physics and quantum physics. Notably, in recent times, they have been instrumental in deriving kinematic and dynamic formulations employed in robotics and animation.

Duffin and Schaeffer were the pioneers in introducing frames in the context of studying nonharmonic Fourier series, as cited in [4]. Subsequently, Young [5] and Daubechies, Grossmann and Meyer [6] revitalized the concept by proposing them as viable substitutes for bases in $L^2\left(\mathbb{R}\right)$ and various other Hilbert spaces. Frames have garnered extensive application across multiple domains, notably in signal and data analysis. Distinguishing themselves from orthogonal bases, a pivotal characteristic of frames lies in their ability to furnish redundant representations of signals, thereby enabling the resolution of numerous practical challenges.

Khokulan, Thirulogasanthar and Srisatkunarajah initially presented the frames of finite dimensional quaternionic Hilbert spaces in [7]. Sharma and Virender [8] subsequently delved into various types of dual frames corresponding to a given frame within a finite dimensional quaternionic Hilbert space. These investigations sparked the generalization of frame concepts to separable quaternionic Hilbert spaces, as discussed in [9]. Recently, Chen, Dang and Qian [10] examined the frames of Hardy spaces, contextualizing them within quaternionic spaces and Euclidean spaces of Clifford algebras. Ellouz, in [11], focused on studying properties of K-frames in quaternionic Hilbert spaces and further provided an exact characterization of the dual K-Bessel sequence for a specified K-frame in such spaces, as reported in [12]. Motivated by the requirement for constructing continuous frames of rank n on right quaternionic Hilbert spaces, Khokulana, Thirulogasanthar and Muraleetharan [13] analyzed the S-spectrum of right quaternion operators. Additionally, Zhang and Li [14] introduced the notion of approximate dual frames in quaternionic Hilbert spaces and elucidated their nature. In this paper, our focus is on exploring the frame quantum-detection problem within the realm of quaternionic Hilbert spaces.

The subsequent sections of this paper are structured as follows: In Section 2, we introduce some necessary notations and the definition of quaternionic Hilbert spaces, and give the background of the frame quantum-detection problem. In Section 3, the injective frame in quaternionic Hilbert spaces is defined and several properties of injective frames are derived. Notably, if a Parseval frame possesses quantum injectivity, it automatically qualifies as an injective frame. In Section 4, we present multiple solutions to the frame quantum-detection problem by leveraging operator theory and frame theory in quaternionic Hilbert spaces. The results indicate that our focus can be narrowed down to self-adjoint right $\mathbb{Q}$-linear operators. By normalizing the trace, we provide a classification for the injectivity problem. Moreover, we offer several frame characterizations aimed at addressing the injectivity problem. Section 5 summarizes the work of this article.

2. Preliminaries

To articulate precisely what quantum detection entails, it is imperative to delve into the fundamentals of quantum detection within the realm of quaternionic Hilbert spaces. Denote the algebra of quaternions by $\mathbb{Q}$ and its natural basis by $\{u_1,u_2,u_3,u_4\}$. The multiplication of $\mathbb{Q}$ is defined as follows:

$$u_i\times u_i=-u_4,u_4=1(1\le i\le 3),u_i\times u_j=-u_j\times u_i=u_k(1\le i,j\le 3),$$

(1)

where $\left(ijk\right)$ is an even permutation of (123). An element of $\mathbb{Q}$ is called a quaternion. Each element of the set is expressed as $q=a{u}_1+b{u}_2+c{u}_3+d{u}_4$, where $a,b,c,d$ are ordinary numbers and $u_i(1\le i\le 4)$ is the orthonormal basis for $\mathbb{Q}$ and they satisfy the relations given in (1).

Let $\mathbb{H}$ be a vector space defined over the skew field of quaternions $\mathbb{Q}$, meaning $\mathbb{H}$ forms an additive group and the scalar multiplication of vectors $x\in \mathbb{H}$ by scalars $\alpha ,\beta \in \mathbb{Q}$ from the right adheres to the axioms of associativity and distributivity. Furthermore, $\mathbb{H}$ is designated as a quaternionic inner product space if there exists a Hermitian quaternionic scalar product $⟨·\mid ·⟩:\mathbb{H}\times \mathbb{H}\mapsto \mathbb{Q}$ that fulfills the subsequent properties:

• $⟨x\mid y⟩=\overline{⟨y\mid x⟩}$, ∀$x,y\in \mathbb{H}$;
• $⟨x\mid x⟩>0$, ∀$x\in \mathbb{H}$ unless $x=0$;
• $⟨x\mid yp+zq⟩>=⟨x\mid y⟩p+⟨x\mid z⟩q$, ∀$x,y,z\in \mathbb{H}$.

If the vector space $\mathbb{H}$ is complete with respect to the norm defined as

$${\parallel x\parallel }_{\mathbb{H}}=\sqrt{⟨x\mid x⟩},x\in \mathbb{H},$$

then it is designated as a quaternionic Hilbert space. It is noted in references [15,16] that quaternionic Hilbert spaces exhibit numerous standard properties akin to those of complex Hilbert spaces, including the existence of a Hilbert basis, adherence to the Cauchy–Schwarz inequality, and satisfaction of the parallelogram identity.

Let $\mathbb{I}$ be a countable set serving as an index, and let $\mathbb{H}$ and $\mathbb{K}$ denote two quaternionic Hilbert spaces. A right $\mathbb{Q}$-linear operator is defined as a mapping $T:\mathbb{H}\mapsto \mathbb{K}$ which satisfies the condition

$$T(xp+yq)=\left(Tx\right)p+\left(Ty\right)q\mathrm{if}x,y\in \mathbb{H}\mathrm{and}p,q\in \mathbb{Q}.$$

The operator T is termed bounded if there exists a positive constant M such that, for every $x\in \mathbb{H}$, ${\parallel Tx\parallel }_{\mathbb{K}}\le M{\parallel x\parallel }_{\mathbb{H}}$. The adjoint operator $T^{*}$ of T is defined through the relationship $⟨x\mid Ty⟩=⟨T^{*}x\mid y⟩,\forall x,y\in \mathbb{H},$ and T is said to be self-adjoint if it holds that $T=T^{*}$.

For the quaternion field $\mathbb{Q}$, which is non-commutative, we define the space ${𝓁}_2\left(\mathbb{Q}\right)$ as follows:

$${𝓁}_2\left(\mathbb{Q}\right)=\left\{{\left\{q_i\right\}}_{i\in \mathbb{I}}\subset \mathbb{Q}:\sum _{i\in \mathbb{I}}{\left|q_i\right|}^2<\infty \right\}$$

endowed with right multiplication by quaternion scalars and equipped with a quaternionic inner product on ${𝓁}_2\left(\mathbb{Q}\right)$ given by

$$⟨p\mid q⟩=\sum _{i\in \mathbb{I}}\overline{p_i}{q}_i,p={\left\{p_i\right\}}_{i\in \mathbb{I}}\mathrm{and}q={\left\{q_i\right\}}_{i\in \mathbb{I}}.$$

(2)

It is straightforward to note that ${𝓁}_2\left(\mathbb{Q}\right)$ forms a right quaternionic Hilbert space with respect to the quaternionic inner product defined in (2). For more details about Quaternion analysis, we refer to [17,18,19,20,21].

Let $\mathcal{B}\left(\mathbb{H}\right)$ denote the space of bounded right $\mathbb{Q}$-linear operators acting on an infinite dimensional quaternionic Hilbert space $\mathbb{H}$. Consider ${\left\{e_i\right\}}_{i\in \mathbb{I}}$ to be an orthonormal basis spanning $\mathbb{H}$. For any finite rank right $\mathbb{Q}$-linear operator T defined on $\mathbb{H}$, the trace of T is formulated as:

$$\mathrm{tr}\left(T\right)=\sum _{i\in \mathbb{I}}⟨e_i\mid T{e}_i⟩,$$

where the sum is finite and remains unchanged regardless of the choice of the orthonormal basis.

The trace induces a scalar product defined as ${⟨T\mid S⟩}_{HS}=\mathrm{tr}\left(S{T}^{*}\right),$ where $\mathrm{tr}$ denotes the trace operation. The closure of the set of finite rank right $\mathbb{Q}$-linear operators with respect to this scalar product, denoted by ${\mathcal{S}}_2\left(\mathbb{H}\right)$, represents the space of Hilbert–Schmidt right $\mathbb{Q}$-linear operators on $\mathbb{H}$. For any $T\in \mathcal{B}\left(\mathbb{H}\right)$, where $\mathcal{B}\left(\mathbb{H}\right)$ is the space of bounded right $\mathbb{Q}$-linear operators on $\mathbb{H}$, we define $\left|T\right|=\sqrt{T{T}^{*}}$ as the positive square root of $T{T}^{*}$. An operator T is said to be a trace class right $\mathbb{Q}$-linear operator if $\mathrm{tr}\left(\right|T\left|\right)<\infty$. The collection of all such trace class right $\mathbb{Q}$-linear operators is denoted by ${\mathcal{S}}_1\left(\mathbb{H}\right)$ and constitutes a Banach space under the trace norm ${\parallel ·\parallel }_1$ defined as ${\parallel T\parallel }_1=\mathrm{tr}\left(\right|T\left|\right)$.

In quantum mechanics, the statistical results of quantum state tomography are characterized by the positive operator value measure [22]. The formal definition of the positive operator value measure is as follows.

Definition 1 

([22]). Let Ω be a locally compact Hausdorff space, and Σ the σ-algebra of the Borel set of Ω. The mapping $\nu :\Sigma \mapsto \mathcal{B}\left(\mathbb{H}\right)$ is an operator valued measure, if there is a countable set ${\left\{E_k\right\}}_{k\in \mathbb{I}}\subseteq \Sigma$ that satisfies $E_i\cap E_j=Ø$, $i\ne j$,

$$\nu \left(\bigcup _{k\in \mathbb{I}}{E}_k\right)=\sum _{k\in \mathbb{I}}\nu \left(E_k\right).$$

(3)

The series on the right side of Equation (3) converges according to weak operator topology. If ν is positive and $\nu \left(\mathrm{\Omega }\right)=I_{\mathcal{H}}$, ν is called a positive operator valued measure.

The collection of states residing in the quaternionic Hilbert space is formulated as

$$\mathcal{S}\left(\mathbb{H}\right)=\{T:T\in {\mathcal{S}}_1\left(\mathbb{H}\right),T=T^{*}\ge 0,\mathrm{tr}\left(T\right)=1\},$$

serving as the repository of quantum states for any quantum system. For a given quantum state $\rho$, the quantum measurement executed by a positive operator-valued measure $\nu$ is encapsulated in the mapping $\pi :\Sigma \mapsto \mathbb{R}$, which is defined by

$$\pi \left(E\right)=\rho \left(\nu \right(E\left)\right)=\mathrm{tr}\left(\nu \right(E\left)T\right),$$

where $T\in \mathcal{S}\left(\mathbb{H}\right)$ belongs to the equivalence class associated with $\rho$.

Let $\mathcal{L}(\Sigma ,\mathbb{Q})$ be the bounded set of functions on $\Sigma$. Given the positive operator valued measure $\nu$ related to the quaternionic Hilbert space $\mathbb{H}$, define the following mapping

$$\Pi :\mathcal{S}\left(\mathbb{H}\right)\mapsto \mathcal{L}(\Sigma ,\mathbb{Q}),\Pi \left(\rho \right)\left(E\right)=\rho \left(\nu \right(E\left)\right),\forall E\in \Sigma .$$

The quantum-detection problem asks when is the mapping $\Pi$ injective?

Recall that the collection ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ is designated as a frame [9] for a quaternionic Hilbert space $\mathbb{H}$, provided that there exist positive constants A and B, such that for all $x\in \mathbb{H}$, the inequality

$${A\parallel x\parallel }^2\le \sum _{i\in \mathbb{I}}\mid ⟨f_i\mid x⟩{\mid }^2\le B{\parallel x\parallel }^2$$

holds. Here, A and B are termed the lower and upper frame bounds, respectively. In the special case where $A=B$, the frame is referred to as a tight frame. Furthermore, when $A=B=1$, it is specifically called a Parseval frame.

Example 1. 

A frame is a set of generated vectors that allows for stable expansion and reconstruction of vectors. However, unlike orthogonal bases, frame vectors do not need to be linearly independent. This provides design flexibility that orthogonal bases do not possess. For instance, let ${\left\{e_i\right\}}_{i\in \mathbb{I}}$ be a Hilbert basis for a right quarenionic Hilbert space $\mathbb{H}$. Let ${\left\{u_i\right\}}_{i\in \mathbb{I}}$ be a sequence in $\mathbb{H}$ defined as $u_1=e_1,u_i=e_{i-1},i\ge 2$. Then, ${\left\{u_i\right\}}_{i\in \mathbb{I}}$ is a non-Parseval frame for $\mathbb{H}$. According to the definition of frame, we can expand any element in $\mathbb{H}$ using ${\left\{u_i\right\}}_{i\in \mathbb{I}}$.

We define the analysis operator associated with the frame ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ as $T:\mathbb{H}\mapsto {𝓁}_2\left(\mathbb{Q}\right)$ by

$$Tx={\left\{⟨f_i\mid x⟩\right\}}_{i\in \mathbb{I}},\forall x\in \mathbb{H}.$$

The synthesis operator $T^{*}$ is given by:

$$T^{*}\left({\left\{q_i\right\}}_{i\in \mathbb{I}}\right)=\sum _{i\in \mathbb{I}}{f}_i{q}_i,\forall {\left\{q_i\right\}}_{i\in \mathbb{I}}\in {𝓁}_2\left(\mathbb{Q}\right).$$

The frame operator is defined as $S=T^{*}T$, which is a positive, self-adjoint invertible right $\mathbb{Q}$-linear operator on $\mathbb{H}$ satisfying:

$$Sx=\sum _{i\in \mathbb{I}}{f}_i⟨f_i\mid x⟩,\forall x\in \mathbb{H}.$$

Example 2. 

It is acknowledged that given any frame ${\left\{f_i\right\}}_{i\in \mathbb{I}}$, the transformed frame ${\left\{S^{-1}{f}_i\right\}}_{i\in \mathbb{I}}$ constitutes a Parseval frame. Furthermore, it is a well-established fact that a frame is a Parseval frame precisely when its associated frame operator coincides with the identity operator. Let ${\left\{e_i\right\}}_{i\in \mathbb{I}}$ be a Hilbert basis for a right quarenionic Hilbert space $\mathbb{H}$ as in Example 1. Let ${\left\{u_i\right\}}_{i\in \mathbb{I}}$ be a sequence in $\mathbb{H}$ defined as

$$u_i=u_{2i-1}=e_i,i\in \mathbb{I}.$$

Then, ${\left\{u_i\right\}}_{i\in \mathbb{I}}$ is a tight frame for $\mathbb{H}$ with bound $A=2$ and ${\left\{2^{-1}{u}_i\right\}}_{i\in \mathbb{I}}$ is Parseval frame.

If ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ is a Parseval frame for a quaternionic Hilbert space $\mathbb{H}$, it naturally induces a positive operator valued measure $\nu$ on $\mathrm{\Omega }=\mathbb{I}$ with $\Sigma =2^{\mathbb{I}}$ (the power set of $\mathbb{I}$):

$$\nu \left(E\right)=\sum _{i\in E}{f}_i{f}_i^{*},\mathrm{where}{f}_i^{*}:\mathbb{H}\mapsto \mathbb{Q},f_i^{*}\left(f\right)=⟨f_i\mid f⟩.$$

with strong convergence. Given a state $T\in \mathcal{S}\left(\mathbb{H}\right)$, the frame induced quantum measurement is given by the function

$$\tau :\Sigma \mapsto \mathbb{Q},\tau \left(E\right)=\sum _{i\in E}\mathrm{tr}\left(T{f}_i{f}_i^{*}\right)=\sum _{i\in E}⟨f_i\mid T{f}_i⟩$$

In this case, the quantum-detection problem asks: Is there a Parseval frame ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ on quaternionic Hilbert space $\mathbb{H}$ so that the map

$$\mathbb{M}:\mathcal{S}\left(\mathbb{H}\right)\mapsto \mathcal{L}(\Sigma ,\mathbb{Q}),\mathbb{M}\left(T\right)\left(E\right)=\sum _{i\in E}⟨f_i\mid T{f}_i⟩,\forall E\in \mathbb{I},$$

is injective? We say that a frame ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ gives quantum injectivity (or is quantum injective) if the mapping $\mathbb{M}$ associated with ${\left\{f_i\right\}}_{i=1}^{\infty }$ is injective. For more information about the quantum-detection problem, we refer the readers to [23,24,25,26].

3. Injective Frame in Quaternionic Hilbert Spaces

We will embark on tackling a substantially broader quantum-detection challenge. Specifically, our focus will encompass:

(i)

Right $\mathbb{Q}$-linear self-adjoint operators that potentially lack positivity.

(ii)

Right $\mathbb{Q}$-linear operators which, while not necessarily trace-one, adhere to the Hilbert–Schmidt property.

(iii)

Frames that deviate from the Parseval condition.

It will be demonstrated that addressing this generalized formulation of the problem concurrently resolves the original query. Prior to our endeavor, a precise definition is imperative to establish.

Definition 2. 

A family of vectors ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ in a quaternionic Hilbert space $\mathbb{H}$ is said to be injective if whenever a positive Hilbert–Schmidt self-adjoint right $\mathbb{Q}$-linear operator T satisfies

$$⟨f_i\mid T{f}_i⟩=0,\forall i\in \mathbb{I},$$

then $T=0$.

It is clear that injective frame must have quantum injectivity. The following theorem states that when a Parseval frame has quantum injectivity, then it is also an injective frame.

Theorem 1. 

Suppose that ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ is a quantum injective frame for a quaternionic Hilbert space $\mathbb{H}$, then ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ is injective.

Proof. 

Let ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ be a quantum injective Parseval frame for a quaternionic Hilbert space $\mathbb{H}$. Then

$$I_{\mathbb{H}}=\sum _{i\in \mathbb{I}}{f}_i{f}_i^{*}.$$

Therefore

$$\mathrm{tr}\left(T\right)=\sum _{i\in \mathbb{I}}\mathrm{tr}\left(T{f}_i{f}_i^{*}\right).$$

Assume that $\mathrm{tr}\left(T{f}_i{f}_i^{*}\right)=0,i=1,2,\dots$ for some Hilbert–Schmidt self-adjoint right $\mathbb{Q}$-linear operator T. Then $\mathrm{tr}\left(T\right)=0$. Moreover, frame ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ has quantum injectivity, so $T=0$, and thus ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ is injective. □

Now we will prove that for quantum-detection problem, we do not need to find Parseval frames. If we have a frame that provides injectivity, then its canonical Parseval frame is injective.

Proposition 1. 

Let ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ be a frame for a quaternionic Hilbert space $\mathbb{H}$ which gives injectivity. If S is a bounded invertible right $\mathbb{Q}$-linear operator on $\mathbb{H}$, then ${\left\{S{f}_i\right\}}_{i\in \mathbb{I}}$ also gives injectivity.

Proof. 

Let T be a Hilbert–Schmidt self-adjoint right $\mathbb{Q}$-linear operator such that

$$⟨S{f}_i\mid TS{f}_i⟩=0,\forall i\in \mathbb{I}.$$

Then $⟨f_i\mid S^{*}TS{f}_i⟩=0,\forall i\in \mathbb{I}.$ Note that $S^{*}TS$ is also a Hilbert–Schmidt self-adjoint right $\mathbb{Q}$-linear operator. Therefore, $S^{*}TS=0$ and hence $T=0$. □

Corollary 1. 

Let ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ be a frame with frame operator S. If ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ gives injectivity, then the canonical Parseval frame ${\left\{S^{-\frac{1}{2}}{f}_i\right\}}_{i\in \mathbb{I}}$ also gives injectivity.

For the frames ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ and ${\left\{g_i\right\}}_{i\in \mathbb{I}}$, if there is a bounded invertible right $\mathbb{Q}$-linear operator S such that $f_i=S{g}_i,i\in \mathbb{I},$ then we say that two frames ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ and ${\left\{g_i\right\}}_{i\in \mathbb{I}}$ are similar. The following corollary indicates that similar frames preserve injectivity.

Corollary 2. 

Suppose that ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ and ${\left\{g_i\right\}}_{i\in \mathbb{I}}$ are similar injective frames for a quaternionic Hilbert space $\mathbb{H}$. Then, ${\left\{f_i\right\}}_{i\in \mathbb{I}}$ is injective if and only if ${\left\{g_i\right\}}_{i\in \mathbb{I}}$ is injective.

4. Characterizations of Injective Frame in Quaternionic Hilbert Spaces

In this section, our objective is to resolve the injectivity problem pertaining to infinite dimensional quaternionic Hilbert spaces. We commence by demonstrating that our focus can be narrowed down to solely working with self-adjoint right $\mathbb{Q}$-linear operators.

Theorem 2. 

The following statements are equivalent for a family of vectors ${\left\{f_i\right\}}_{i=1}^{\infty }$ in a quaternionic Hilbert space $\mathbb{H}$

(1) 

For any two Hilbert–Schmidt, positive and self-adjoint right $\mathbb{Q}$-linear operators $T,S$, if

$$⟨f_i\mid T{f}_i⟩=⟨f_i\mid S{f}_i⟩,\forall i=1,2,\dots ,$$

then $T=S$.

(2) 

For any two Hilbert–Schmidt and self-adjoint right $\mathbb{Q}$-linear operators $T,S$, if

$$⟨f_i\mid T{f}_i⟩=⟨f_i\mid S{f}_i⟩,\forall i=1,2,\dots ,$$

then $T=S$.

(3) 

Frame ${\left\{f_i\right\}}_{i=1}^{\infty }$ is injective.

Proof. 

(1) ⇒ (2) Let T and S be Hilbert–Schmidt and self-adjoint right $\mathbb{Q}$-linear operators such that

$$⟨f_i\mid T{f}_i⟩=⟨f_i\mid S{f}_i⟩,\forall i=1,2,\dots ,$$

Let R be defined as $R=T-S$. Consequently, R is a Hilbert–Schmidt and self-adjoint right $\mathbb{Q}$-linear operator. Consider ${\left\{e_i\right\}}_{i=1}^{\infty }$ to be an orthonormal basis for $\mathbb{H}$, and let ${\left\{u_i\right\}}_{i=1}^{\infty }$ be an eigenbasis for R corresponding to the eigenvalues ${\left\{{\lambda }_i\right\}}_{i=1}^{\infty }$. Define two right $\mathbb{Q}$-linear operators U and V on $\mathbb{H}$ as follows:

$$U{e}_i=u_i,V{e}_i=e_i{\lambda }_i,\forall i=1,2,\dots .$$

Then, U is a unitary right $\mathbb{Q}$-linear operator, and V is a trace class and self-adjoint right $\mathbb{Q}$-linear operator. Since

$$UV{U}^{*}{u}_i=UV{e}_i=U{e}_i{\lambda }_i=u_i{\lambda }_i=R{u}_i,\forall i=1,2,\dots ,$$

we have

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

Now let $t_i=\left|{\lambda }_i\right|$ and $s_i=\left|{\lambda }_i\right|-{\lambda }_i,\forall i=1,2,\dots .$ Obvious, ${\lambda }_i=t_i-s_i$. Let $V_1$ and $V_2$ be right $\mathbb{Q}$-linear operators defined by

$$V_1{e}_i=e_i{t}_i,V_2{e}_i=e_i{s}_i,\forall i=1,2,\dots .$$

Note that R is a Hilbert–Schmidt operator, we have ${\sum }_{i=1}^{\infty }{\left|{\lambda }_i\right|}^2<\infty$. Hence, $V_1$ and $V_2$ are positive Hilbert–Schmidt and self-adjoint right $\mathbb{Q}$-linear operators and we obtain

$$R=U{V}_1{U}^{*}-U{V}_2{U}^{*}.$$

Moreover, $U{V}_1{U}^{*}$ and $U{V}_2{U}^{*}$ are Hilbert–Schmidt positive and self-adjoint right $\mathbb{Q}$-linear operators. Since

$$\begin{array}{c}⟨f_i\mid R{f}_i⟩=⟨f_i\mid U{V}_1{U}^{*}{f}_i⟩-⟨f_i\mid U{V}_2{U}^{*}{f}_i⟩=0\hfill \end{array}$$

it follows that $U{V}_1{U}^{*}=U{V}_2{U}^{*}.$ Thus $R=0$ and hence $T=S$.

(2) ⇒ (3) Let T be any Hilbert–Schmidt and self-adjoint right $\mathbb{Q}$-linear operator such that

$$⟨f_i\mid T{f}_i⟩=0.$$

This implies that

$$⟨f_i\mid T{f}_i⟩=⟨f_i\mid 0f_i⟩,\forall i=1,2,\dots .$$

It follows that $T=0$. Thus ${\left\{f_i\right\}}_{i=1}^{\infty }$ is injective.

(3) ⇒ (1) Let T and S be any positive Hilbert–Schmidt and self-adjoint right $\mathbb{Q}$-linear operators such that

$$⟨f_i\mid T{f}_i⟩=⟨f_i\mid S{f}_i⟩,\forall i=1,2,\dots .$$

Then

$$⟨f_i\mid (T-S)f_i⟩=0,\forall i=1,2,\dots .$$

Since $T-S$ is a self-adjoint right $\mathbb{Q}$-linear operator and ${\left\{f_i\right\}}_{i=1}^{\infty }$ is injective, we arrived at $T=S$. □

Remark 1. 

The operators used in the definition of quantum injective frame are positive and self-adjoint operators, and the above theorem proves that the same conclusion can be obtained if the operators are required to be self-adjoint. Therefore, we only need to deal with self-adjoint operators for quantum-detection problems.

If the operator is a trace class, then we will obtain the following theorem. Note that for the trace class right $\mathbb{Q}$-linear operator R, the proof of above theorem is still valid.

Theorem 3. 

Given a family of vectors ${\left\{f_i\right\}}_{i=1}^{\infty }$ in a quaternionic Hilbert space $\mathbb{H}$, the following are equivalent:

(1) 

For any two trace class, positive and self-adjoint right $\mathbb{Q}$-linear operators $T,S$, if

$$⟨f_i\mid T{f}_i⟩=⟨f_i\mid S{f}_i⟩,\forall i=1,2,\dots ,$$

then $T=S$.

(2) 

For any two trace class and self-adjoint right $\mathbb{Q}$-linear operators $T,S$, if

$$⟨f_i\mid T{f}_i⟩=⟨f_i\mid S{f}_i⟩,\forall i=1,2,\dots ,$$

then $T=S$.

(3) 

Frame ${\left\{f_i\right\}}_{i=1}^{\infty }$ is injective.

By standardizing the trace value, we can categorize the injectivity issue provided that our operators are additionally constrained to have a trace of one. Under this setting, it is generally possible to eliminate one measurement.

Theorem 4. 

Given a frame ${\left\{f_i\right\}}_{i=1}^{\infty }$ for a quaternionic Hilbert space $\mathbb{H}$, the following are equivalent:

(1) 

If T and S are positive trace class right $\mathbb{Q}$-linear operators and self-adjoint of trace one and

$$⟨f_i\mid T{f}_i⟩=⟨f_i\mid S{f}_i⟩,\forall i=1,2,\dots ,$$

then $T=S$.

(2) 

If T and S are trace class right $\mathbb{Q}$-linear operators and self-adjoint of trace one and

$$⟨f_i\mid T{f}_i⟩=⟨f_i\mid S{f}_i⟩,\forall i=1,2,\dots ,$$

then $T=S$.

(3) 

If T is trace class self-adjoint right $\mathbb{Q}$-linear operator of trace zero and

$$⟨f_i\mid T{f}_i⟩=0,\forall i=1,2,\dots ,$$

then $T=0$.

Proof. 

(1) ⇒ (2) Let T and S be self-adjoint right $\mathbb{Q}$-linear operators and trace class of trace one such that

$$⟨f_i\mid T{f}_i⟩=⟨f_i\mid S{f}_i⟩,\forall i=1,2,\dots ,$$

Let R be defined as $R=T-S$. Consequently, R emerges as a self-adjoint trace class right $\mathbb{Q}$-linear operator with a trace equal to zero. Suppose ${\left\{e_i\right\}}_{i=1}^{\infty }$ constitutes an orthonormal basis for $\mathbb{H}$, and ${\left\{u_i\right\}}_{i=1}^{\infty }$ represents an eigenbasis for R corresponding to the eigenvalues ${\left\{{\lambda }_i\right\}}_{i=1}^{\infty }$. Under these conditions, it holds that: ${\sum }_{i=1}^{\infty }{\lambda }_i=0.$ We now define two right $\mathbb{Q}$-linear operators, U and V, on $\mathbb{H}$ as follows:

$$U{e}_i=u_i,V{e}_i=e_i{\lambda }_i,\forall i=1,2,\dots .$$

Here, U is a unitary right $\mathbb{Q}$-linear operator, whereas V is a self-adjoint right $\mathbb{Q}$-linear operator with a trace of zero. Furthermore, these operators satisfy the relation $R=UV{U}^{*}.$ Let

$$\xi =1+\sum _{i=1}^{\infty }\left|{\lambda }_i\right|,t_1=\frac{1+|{\lambda }_1|}{\xi },s_1=\frac{1+|{\lambda }_1|-{\lambda }_1}{\xi },$$

and

$$t_i=\frac{|{\lambda }_i|}{\xi },s_i=\frac{|{\lambda }_1|-{\lambda }_1}{\xi },\forall i=2,3\dots .$$

Then, the numbers $\xi ,t_i,s_i$ are all non-negative quaternions and ${\lambda }_i=t_i-s_i,\forall i=1,2,\dots .$ Now define right $\mathbb{Q}$-linear operators $V_1,V_2$ on $\mathbb{H}$ by

$$V_1{e}_i=e_i{t}_i,V_2{e}_i=e_i{s}_i,\forall i=1,2,\dots .$$

Then, $V_1,V_2$ are self-adjoint and trace class right $\mathbb{Q}$-linear operators of trace one and we have

$$R=U{V}_1{U}^{*}-U{V}_2{U}^{*}.$$

Moreover, $U{V}_1{U}^{*}$, $U{V}_2{U}^{*}$ are self-adjoint right $\mathbb{Q}$-linear operators and positive trace class of trace one. Since

$$\begin{array}{c}⟨f_i\mid R{f}_i⟩=⟨f_i\mid U{V}_1{U}^{*}{f}_i⟩-⟨f_i\mid U{V}_2{U}^{*}{f}_i⟩=0,\hfill \end{array}$$

we obtain $U{V}_1{U}^{*}=U{V}_2{U}^{*}$. Therefore $R=0$, and hence $T=S$.

(2) ⇒ (3) Let T be any trace class right $\mathbb{Q}$-linear operator of trace zero such that

$$⟨f_i\mid T{f}_i⟩=0,\forall i=1,2,\dots .$$

Define operator S on $\mathbb{H}$ by

$$S{e}_1=e_1,S{e}_i=0,\forall i=2,3,\dots .$$

Then, S and $T+S$ are self-adjoint and trace class right $\mathbb{Q}$-linear operators of trace one. Since

$$⟨f_i\mid (T+S)f_i⟩=⟨f_i\mid S{f}_i⟩,\forall i=1,2,\dots ,$$

we have $T+S=S$ and thus $T=0$.

(3) ⇒ (1) Let $T,S$ be self-adjoint and positive trace class right $\mathbb{Q}$-linear operators of trace one such that

$$⟨f_i\mid T{f}_i⟩=⟨f_i\mid S{f}_i⟩,\forall i=1,2,\dots .$$

Then $⟨f_i\mid (T-S)f_i⟩=0,\forall i=1,2,\dots .$ This implies that $T=S$. □

We are now poised to undertake a classification of the injectivity problem pertaining to right $\mathbb{Q}$-linear operators with a trace equal to one. Initially, we delineate a subspace within the real quaternionic space ${𝓁}_1\left(\mathbb{Q}\right)$ as per the following specification:

$$\mathbb{W}=\left\{({\lambda }_1,{\lambda }_2,{\lambda }_3,\dots )\in {𝓁}_1\left(\mathbb{Q}\right),\sum _{j=1}^{\infty }{\lambda }_j=0\right\}.$$

Theorem 5. 

Given a frame ${\left\{f_i\right\}}_{i=1}^{\infty }$ for a quaternionic Hilbert space $\mathbb{H}$, the following are equivalent:

(1) 

If T is a self-adjoint, right $\mathbb{Q}$-linear operator belonging to the trace class, with a trace equal to zero, such that

$$⟨f_i\mid T{f}_i⟩=0,\forall i=1,2,\dots .$$

then $T=0$.

(2) 

For any orthonormal basis ${\left\{e_j\right\}}_{j=1}^{\infty }$ of $\mathbb{H}$ and for any $\lambda =({\lambda }_1,{\lambda }_2,{\lambda }_3,\dots )\in \mathbb{W}$, if

$$\sum _{j=1}^{\infty }{\lambda }_j{\left|⟨e_j\mid f_i⟩\right|}^2=0,i=1,2,\dots ,$$

then $\lambda =0$.

Proof. 

(1) ⇒ (2) Suppose that assumption (2) if false. Then, there is an orthonormal basis ${\left\{e_j\right\}}_{j=1}^{\infty }$ for $\mathbb{H}$ and an $\lambda =({\lambda }_1,{\lambda }_2,{\lambda }_3,\dots )\in \mathbb{W}$ such that

$$\sum _{j=1}^{\infty }{\lambda }_j{\left|⟨e_j\mid f_i⟩\right|}^2=0,i=1,2,\dots ,$$

but $\lambda \ne 0$. Define a right $\mathbb{Q}$-linear operator on $\mathbb{H}$ by

$$T{e}_j=e_j{\lambda }_j,j=1,2,\dots .$$

Therefore T is a non-zero self-adjoint right $\mathbb{Q}$-linear operator. Since

$$⟨f_i\mid T{f}_i⟩=\sum _{j=1}^{\infty }{\lambda }_j{\left|⟨e_j\mid f_i⟩\right|}^2=0,i=1,2,\dots ,$$

this contradicts with assumption (1).

(2) ⇒ (1) Assuming that T is a self-adjoint, right $\mathbb{Q}$-linear operator belonging to the trace class and having a trace of zero, satisfying the condition: $⟨f_i\mid T{f}_i⟩=0,i=1,2,\dots .$ Let ${\left\{e_j\right\}}_{j=1}^{\infty }$ denote an eigenbasis of T corresponding to its eigenvalues ${\left\{{\lambda }_j\right\}}_{j=1}^{\infty }$. For any $i=1,2,\dots$, we have

$$\sum _{j=1}^{\infty }{\lambda }_j{\left|⟨e_j\mid f_i⟩\right|}^2=⟨f_i\mid T{f}_i⟩=0.$$

Since T is right $\mathbb{Q}$-linear operator and trace class, we obtain

$$\sum _{j=1}^{\infty }|{\lambda }_j|=\sum _{j=1}^{\infty }\left|⟨e_j\mid T{e}_j⟩\right|<\infty .$$

Moreover,

$$\sum _{j=1}^{\infty }{\lambda }_j=0.$$

Hence $\lambda =({\lambda }_1,{\lambda }_2,{\lambda }_3,\dots )\in \mathbb{W}$. From assumption (2), we get $\lambda =0$, so $T=0$. □

We end this section with the following theorem, which gives another classification of frames for solving the injectivity problem in quaternionic Hilbert spaces.

Theorem 6. 

Given a frame ${\left\{f_i\right\}}_{i=1}^{\infty }$ in a quaternionic Hilbert space $\mathbb{H}$, the following are equivalent:

(1) 

The sole trace class and self-adjoint right $\mathbb{Q}$-linear operator T such that $⟨f_i\mid T{f}_i⟩=0,$ $i=1,2,\dots$ is $T=0$.

(2) 

For any orthonormal basis ${\left\{e_j\right\}}_{j=1}^{\infty }$ for $\mathbb{H}$ and for any sequence $\lambda =({\lambda }_1,{\lambda }_2,{\lambda }_3,\dots )\in {𝓁}_1\left(\mathbb{Q}\right)$ and, if

$$\sum _{j=1}^{\infty }{\lambda }_j{\left|⟨e_j\mid f_i⟩\right|}^2=0,i=1,2,\dots ,$$

then $\lambda =0$.

Proof. 

(1) ⇒ (2) Assume that statement (2) is false. Hence, there is an orthonormal basis ${\left\{e_j\right\}}_{j=1}^{\infty }$ and an $\lambda =({\lambda }_1,{\lambda }_2,{\lambda }_3,\dots )\in {𝓁}_1\left(\mathbb{Q}\right)$ such that

$$\sum _{j=1}^{\infty }{\lambda }_j{\left|⟨e_j\mid f_i⟩\right|}^2=0,i=1,2,\dots ,$$

but $\lambda \ne 0$. Define a right $\mathbb{Q}$-linear operator on $\mathbb{H}$ by

$$T{e}_j=e_j{\lambda }_j,j=1,2,\dots .$$

Obviously, T is a non-zero right $\mathbb{Q}$-linear and self-adjoint operator and

$$\left|T\right|e_j=\sqrt{T{T}^{*}}{e}_j=\left|{\lambda }_j\right|e_j,j=1,2,\dots .$$

Thus,

$$\sum _{j=1}^{\infty }|⟨e_j\mid \left|T\right|e_j⟩|=\sum _{j=1}^{\infty }\left|{\lambda }_j\right|<\infty ,$$

and thus T is a non-zero right $\mathbb{Q}$-linear and self-adjoint operator of trace class. In addition,

$$⟨f_i\mid T{f}_i⟩=\sum _{j=1}^{\infty }{\lambda }_j{\left|⟨e_j\mid f_i⟩\right|}^2=0,i=1,2,\dots ,$$

This contradicts with condition (1). Therefore, statement (2) is established.

(2) ⇒ (1) Assuming that the operator T is a self-adjoint, right $\mathbb{Q}$-linear operator belonging to the trace class, with the property that $⟨f_i\mid T{f}_i⟩=0,i=1,2,\dots$. Then, there exists an eigenbasis ${\left\{e_j\right\}}_{j=1}^{\infty }$ for T, accompanied by corresponding eigenvalues ${\left\{{\lambda }_j\right\}}_{j=1}^{\infty }$. Consequently, for any integer $i=1,2,\dots$, we obtain that

$$\sum _{j=1}^{\infty }{\lambda }_j{\left|⟨e_j\mid f_i⟩\right|}^2=⟨f_i\mid T{f}_i⟩=0.$$

In addition,

$$\sum _{j=1}^{\infty }|{\lambda }_j|=\sum _{j=1}^{\infty }\left|⟨e_j\mid T{e}_j⟩\right|<\infty ,$$

that is, $\lambda =({\lambda }_1,{\lambda }_2,{\lambda }_3,\dots )\in {𝓁}_1\left(\mathbb{Q}\right)$. From condition (2), it can be concluded that $\lambda =0$, therefore $T=0$. □

5. Conclusions

In this paper, we work on a much more general quantum-detection problem. We have defined quantum injective and injective frames on quaternionic Hilbert spaces and obtained some properties of injective frames. In particular, if a Parseval frame has quantum injectivity, then it is also an injective frame. As the main results, we give several solutions to the frame quantum-detection problem by employing operator theory and frames theory in quaternionic Hilbert spaces. The results show that we only need to work with self-adjoint right $\mathbb{Q}$-linear operators. By normalizing the trace, we give a classification for the injectivity problem. Under this setting, it is generally possible to eliminate one measurement. Moreover, some characterizations of frames for solving the injectivity problem are given. The method established in this article can be used to extend the classical results of vector values to the context of operator values in quaternionic Hilbert spaces.