Quantum injectivity of G-frames in Hilbert spaces

Guoqing Hong; Linbin Wan; Jianxia Zhang

doi:10.1515/dema-2025-0175

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Quantum injectivity of G-frames in Hilbert spaces

Guoqing Hong , Linbin Wan and Jianxia Zhang

Published/Copyright: September 24, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

Inspired by some recent work on the quantum detection problem by discrete frames and continuous frames, in this article, we examine the quantum detection problem with G-frames. By employing operator theory and G-frames theory, we give some equivalent descriptions for injective g-frames and classifications of G-frames for solving injectivity problem. Also, we provide an approach to obtain an injective G-frame by using an injective frame. Moreover, we show that the injectivity of a G-frame is preserved under a linear isomorphism. In addition, injective G-frames exhibit instability in infinite-dimensional settings. Some examples are also provided to demonstrate the existence of injective G-frames.

Keywords: quantum detection; quantum injectivity; G-frame; operator-valued measure

MSC 2010: 42C15; 46L10; 47A05; 46C10

1 Introduction

Recall that the notation of frames was first introduced by Duffin and Schaeffer [1], viewed as some kind of “overcomplete basis” as each element can be represented via a frame but the representation might not be unique [2]. Hence, frames have applications in a wide range of areas including sampling theory, operator theory, nonlinear sparse approximation, pseudo-differential operators, wavelet theory, wireless communications, data transmission with erasures, signal processing, and quantum computing.

A sequence { f i } i = 1 ∞ of elements in a Hilbert space ℋ is a frame for ℋ if there exist constants A , B > 0 such that

A ‖ f ‖ 2 ≤ ∑ i = 1 ∞ ∣ ⟨ f , f i ⟩ ∣ 2 ≤ B ‖ f ‖ 2 , ∀ f ∈ ℋ .

The numbers A , B are called frame bounds. A frame is tight if we can choose A = B as frame bounds; a tight frame with bound A = B = 1 is called a Parseval frame.

Over the years, various extensions of the frame theory have been investigated. Several of these are contained as special cases of the elegant theory for G-frames that was introduced by Sun in [3].

Definition 1

Let ℬ ( ℋ , ℋ i ) be the collection of all bounded linear operators from ℋ to ℋ i . A sequence { Λ i ∈ ℬ ( ℋ , ℋ i ) } i = 1 ∞ is called a G-frame for ℋ with respect to { ℋ i } i = 1 ∞ if there exist positive constants A and B such that for any f ∈ ℋ ,

A ‖ f ‖ 2 ≤ ∑ i = 1 ∞ ‖ Λ i f ‖ 2 ≤ B ‖ f ‖ 2 .

The numbers A and B are called G-frame bounds. A G-frame { Λ i } i = 1 ∞ is called tight G-frame if A = B and Parseval G-frame if A = B = 1 .

G-frames share many useful properties with frames, but not all the properties are shared. For example, exact G-frames are not equivalent to G-Riesz bases [3]. G-frames combine operator theory with frame theory and have a large freedom in the choices of the spaces ℋ i and corresponding operators { Λ i ∈ ℬ ( ℋ , ℋ i ) } i = 1 ∞ . Hence, G-frames have attracted more and more attention from many researchers. G-frame theory covers many generalizations of frame theory, such as bounded quasi-projection frame, pseudo-frame, fusion frame, outer frame, oblique frame, and a class of time-frequency localization operators. The G-frame has also been shown to be equivalent to the stable spatial splitting studied in [4]. Therefore, it is more general to consider the quantum detection problem from the perspective of G-frames. For more details about G-frames, see [3,5].

Frames have been extensively studied in the context of signal processing, in particular, phase-retrieval in recent years [6–9]. While the phase-retrieval problem asks to distinguish the pure states from their quantum measurements with a positive operator-valued measure, the quantum detection problem asks to distinguish all the states from their measurements. In other words, this problem is to uniquely determine a density operator (state) from quantum measurements described as positive operator-valued measures acting on a state. More precisely, the quantum detection problem asks to characterize the positive operator-valued measure ν with property if tr ( ρ 1 ν ( E ) ) = tr ( ρ 2 ν ( E ) ) for every measurable set E implies that ρ 1 = ρ 2 , where ρ 1 , ρ 2 ∈ ℬ ( ℋ ) are density operators.

The problem of quantum detection was settled by Botelho-Andrade et al. [10] mainly for finite or infinite but discrete frames and Han et al. [11] for continuous frames by construction some kinds of frame positive operator-valued measures. The authors presented several versions of characterizations for S p -injective continuous frames in terms of discrete representations of continuous frames [12]. Han et al. have begun to examine the quantum detection problem with multiwindow Gabor frames [13]. The purpose of this article is to investigate the quantum detection problem in infinite-dimensional spaces by G-frames. We refer to [14–18] and the reference therein for some historic background of the problem and some recent developments on this topic.

The remainder of this article is organized as follows. Section 2 establishes foundational context by reviewing essential theoretical underpinnings of quantum detection problems, followed by the formal introduction of quantum injectivity within G-frame structures. Building upon this groundwork, Section 3 systematically develops multiple equivalent characterizations for injective G-frames and proposes novel classification schemes specifically designed to resolve injectivity challenges. We show that injective G-frames are unstable in infinite-dimensional cases. The article concludes with illustrative examples showing the existence of injective G-frames across diverse mathematical settings – spanning both real/complex field configurations and finite/infinite-dimensional frameworks – thereby validating theoretical results through concrete constructions.

2 Preliminaries

The density matrix is a mathematical tool used to describe the state of a quantum system, encapsulating the probability distribution information of all possible states of the system. In quantum state tomography, experimenters measure the quantum system by selecting different measurement matrices [19]. These measurement results are utilized to infer the density matrix of the quantum system, thereby revealing the true state of the quantum system. This technique is fundamental for understanding the characteristics of quantum devices and distinguishing between different quantum states [20]. In the aforementioned process, positive operator-valued measures play a pivotal role [21]. The standard definition of a positive operator-valued measure is as follows [22].

Definition 2

Let ℬ ( ℋ ) be the space of all bounded linear operators on a Hilbert space ℋ , and let Σ be a σ -algebra of sets of Ω . A positive operator-valued measure (POVM) is a mapping ν : Σ ↦ ℬ ( ℋ ) such that:

ν ( E ) is a positive self-adjoint operator for all E ∈ Σ ;
ν ( ∅ ) = 0 ;
⟨ ν ( ⋃ k = 1 ∞ E k ) x , y ⟩ = ∑ k = 1 ∞ ⟨ ν ( E k ) x , y ⟩ for all disjoint { E k } k = 1 ∞ ⊆ Σ , x , y ∈ ℋ ;
ν ( Ω ) = I ℋ .

The advantage of using POVMs to generalize a von Neumann measurement is that it allows one to distinguish more accurately among elements of a set of nonorthogonal quantum states. POVMs occupy an important position in quantum measurement theory [23], serving as one of the informational bridges between quantum systems and the classical world. Quantum measurements are represented through POVMs, where these operators act on the state space of the system [24]. The sequence of measurement operators corresponds to the different possible outcomes of the measurement, each associated with a probability such that the sum of the probabilities of all possible outcomes equals 1, aligning with the probabilistic interpretation of quantum mechanics [25]. This probabilistic interpretation grants POVMs glamorous advantages in distinguishing and identifying quantum states.

As mentioned earlier, the quantum detection problem is to explore the existence of the quantum measurement performed by a POVM ν , which can distinguish states on some Hilbert space ℋ . Indeed, let ℒ ( Σ , R ) be the set of bounded function on Σ and let S ( ℋ ) be the set of states or density operators on some Hilbert space ℋ , i.e., S ( ℋ ) = { T ∈ ℬ ( ℋ ) : T = T * ≥ 0 , tr ( T ) = 1 } . Given a quantum system, the quantum detection problem asks the following question:

Is there a “prefect” quantum measurement or a certain type of underlying POVM ν such that the following mapping

Q : S ( ℋ ) → ℒ ( Σ , R ) , Q ( ρ ) ( E ) = tr ( ρ ν ( E ) ) , ∀ E ∈ Σ

is injective?

The frame method to address the quantum detection problem is to find a Parseval frame, which induces a POVM such that the corresponding map Q is injective. We consider the generalization of frame quantum detection problem, that is, we directly work with general operators. Let Σ be the σ -algebra of all subsets of Ω = { 1 , 2 , … , ∞ } . If { Λ i } i = 1 ∞ is a Parseval G-frame for a Hilbert space ℋ , it naturally induces a POVM

ν : Σ ↦ ℬ ( ℋ ) , ν ( E ) = ∑ i ∈ E Λ i * Λ i ,

with strong convergence for any E ∈ Σ .

Given a state T ∈ S ( ℋ ) (i.e., a unit-trace, positive, self-adjoint operator on ℋ ), the quantum measurement induced by a G-frame { Λ i } i = 1 ∞ is given by the mapping M : S ( ℋ ) ↦ ℒ ( Σ , R ) ,

M ( ρ ) ( E ) = tr ( ρ ν ( E ) ) = tr ρ ∑ i ∈ E Λ i * Λ i = ∑ i ∈ E tr ( Λ i * Λ i ρ ) = ∑ i ∈ E ⟨ ρ Λ i , Λ i ⟩ , ∀ E ∈ Σ .

The G-frame quantum detection problem is reformulated as follows: under what properties of a G-frame { Λ i } i = 1 ∞ is the mapping M injective? We say that a G-frame { Λ i } i = 1 ∞ gives quantum injectivity (or is quantum injective) if the mapping M associated with { Λ i } i = 1 ∞ is injective.

3 Injective G-frames

In this section, some equivalent expressions for the injective G-frames will be given and classifications for the G-frame injectivity problem will be shown.

It is not hard to see that quantum injectivity of a G-frame is equivalent to the condition that if ⟨ T Λ i , Λ i ⟩ = 0 , ∀ i = 1 , 2 , … , for a self-adjoint trace class operator T with trace zero, then T = 0 . Similarly, we say that { Λ i } i = 1 ∞ is injective if whenever a self-adjoint trace class operator T satisfies ⟨ T Λ i , Λ i ⟩ = 0 , ∀ i = 1 , 2 , … , then T = 0 . Obviously, injectivity implies quantum injectivity. We start with following elementary fact, which shows that we do not need to work with positive operators.

Theorem 1

Given a G-frame { Λ i } i = 1 ∞ for a Hilbert space ℋ , the following are equivalent:

If T and S are positive trace class and self-adjoint and
⟨ T Λ i , Λ i ⟩ = ⟨ S Λ i , Λ i ⟩ , ∀ i = 1 , 2 , … ,
then T = S .
If T and S are trace class and self-adjoint and
⟨ T Λ i , Λ i ⟩ = ⟨ S Λ i , Λ i ⟩ , ∀ i = 1 , 2 , … ,
then T = S .
G-frame { Λ i } i = 1 ∞ is injective.

Proof

1 ⇒ 2 Let T and S be trace class self-adjoint operators such that

⟨ T Λ i , Λ i ⟩ = ⟨ S Λ i , Λ i ⟩ , ∀ i = 1 , 2 , … .

Set R = T − S . Then R is also a trace class self-adjoint operator. Let { e i } i = 1 ∞ be an orthonormal basis for ℋ and let { u i } i = 1 ∞ be an eigenbasis for R with respect to the eigenvalues { λ i } i = 1 ∞ . Define operators U and V on ℋ by

U e i = u i , V e i = λ i e i , ∀ i = 1 , 2 , … .

Then U is a unitary operator and V is a trace class self-adjoint operator. Since

U V U * u i = U V e i = λ i U e i = λ i u i = R u i , ∀ i = 1 , 2 , … ,

we have

R = U V U * .

Now let t i = ∣ λ i ∣ and s i = ∣ λ i ∣ − λ i , ∀ i = 1 , 2 , … . Obvious, λ i = t i − s i . Let V 1 and V 2 be operators defined by

V 1 e i = t i e i , V 2 e i = s i e i , ∀ i = 1 , 2 , … .

Note that R is trace class, so ∑ i = 1 ∞ ∣ λ i ∣ < ∞ . Hence, V 1 and V 2 are positive trace class self-adjoint operators, and we have

R = U V 1 U * − U V 2 U * .

Moreover, U V 1 U * and U V 2 U * are trace class positive self-adjoint operators. Since

⟨ R Λ i , Λ i ⟩ = tr ( Λ i * R Λ i ) = 0 ,

we obtain by that U V 1 U * = U V 2 U * . Thus R = 0 and hence T = S .

2 ⇒ 3 Let T be any trace class self-adjoint operator such that

⟨ T Λ i , Λ i ⟩ = 0 , ∀ i = 1 , 2 , … .

This implies that

⟨ T Λ i , Λ i ⟩ = ⟨ 0 Λ i , Λ i ⟩ , ∀ i = 1 , 2 , … .

It follows that T = 0 . Thus, { Λ i } i = 1 ∞ is injective.

3 ⇒ 1 Let T and S be any positive trace class self-adjoint operators such that

⟨ T Λ i , Λ i ⟩ = ⟨ S Λ i , Λ i ⟩ , ∀ i = 1 , 2 , … .

Then

⟨ ( T − S ) Λ i , Λ i ⟩ = 0 , ∀ i = 1 , 2 , … .

Since T − S is a self-adjoint operator and { Λ i } i = 1 ∞ is injective, we arrived at T = S .□

By normalizing the trace, we can give a classification for the G-frame injectivity problem if we require further that our operators are trace one.

Theorem 2

Given a G-frame { Λ i } i = 1 ∞ for a Hilbert space ℋ , the followings are equivalent:

If T and S are positive trace class and self-adjoint of trace one and
⟨ T Λ i , Λ i ⟩ = ⟨ S Λ i , Λ i ⟩ , ∀ i = 1 , 2 , … ,
then T = S .
If T and S are trace class and self-adjoint of trace one and
⟨ T Λ i , Λ i ⟩ = ⟨ S Λ i , Λ i ⟩ , ∀ i = 1 , 2 , … ,
then T = S .
If T is trace class self-adjoint of trace zero and
⟨ T Λ i , Λ i ⟩ = 0 , ∀ i = 1 , 2 , … ,
then T = 0 .

Proof

1 ⇒ 2 Let T and S be self-adjoint trace class operators of trace one such that

⟨ T Λ i , Λ i ⟩ = ⟨ S Λ i , Λ i ⟩ , ∀ i = 1 , 2 , … .

Set R = T − S . Then R is a self-adjoint trace class operator of trace zero. Let { e i } i = 1 ∞ be an orthonormal basis for ℋ and let { u i } i = 1 ∞ be an eigenbasis for R with respect to the eigenvalues { λ i } i = 1 ∞ . Then ∑ i = 1 ∞ λ i = 0 . Define operators U and V on ℋ by

U e i = u i , V e i = λ i e i , ∀ i = 1 , 2 , … .

Then U is a unitary operator and V is a self-adjoint operator of trace zero and R = U V U * . Let

ξ = 1 + ∑ i = 1 ∞ ∣ λ i ∣ , t 1 = 1 + ∣ λ 1 ∣ ξ , s 1 = 1 + ∣ λ 1 ∣ − λ 1 ξ

and

t i = ∣ λ i ∣ ξ , s i = ∣ λ i ∣ − λ i ξ , ∀ i = 2 , 3 … .

Then the numbers ξ , t i , s i are all nonnegative numbers and λ i = t i − s i , ∀ i = 1 , 2 , … . Now define operators V 1 , V 2 on ℋ by

V 1 e i = t i e i , V 2 e i = s i e i , ∀ i = 1 , 2 , … .

Then V 1 , V 2 are self-adjoint trace class 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 positive trace class of trace one. Since

⟨ R Λ i , Λ i ⟩ = ⟨ U V 1 U * Λ i , Λ i ⟩ − ⟨ U V 2 U * Λ i , Λ i ⟩ = 0 ,

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 operator of trace zero such that

⟨ T Λ i , Λ i ⟩ = 0 , ∀ i = 1 , 2 , … .

Define operator S on ℋ by

S e 1 = e 1 , S e i = 0 , ∀ i = 2 , 3 , … .

Then S and T + S are self-adjoint trace class operators of trace one. Since

⟨ ( T + S ) Λ i , Λ i ⟩ = ⟨ S Λ i , Λ i ⟩ , ∀ i = 1 , 2 , … ,

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

3 ⇒ 1 Let T , S be self-adjoint positive trace class of trace one such that

⟨ T Λ i , Λ i ⟩ = ⟨ S Λ i , Λ i ⟩ , ∀ i = 1 , 2 , … .

Then ⟨ ( T − S ) Λ i , Λ i ⟩ = 0 , ∀ i = 1 , 2 , … . This implies that T = S .□

The following result is another classification of G-frames for solving the injectivity problem.

Theorem 3

Given a G-frame { Λ i } i = 1 ∞ for a Hilbert space ℋ , the following are equivalent:

For any λ = ( λ 1 , λ 2 , λ 3 , … ) ∈ ℓ 1 and for any orthonormal basis { e j } j = 1 ∞ for ℋ , if
∑ j = 1 ∞ λ j ‖ Λ i e j ‖ 2 = 0 , i = 1 , 2 , … ,
then λ = 0 .
If T is a trace class and self-adjoint operator such that tr ( Λ i * Λ i T ) = 0 , i = 1 , 2 , … , then T = 0 .

Proof

1 ⇒ 2 Suppose that operator T is a trace class and self-adjoint operator such that tr ( Λ i * Λ i T ) = 0 , i = 1 , 2 , … . There is an eigenbasis { e j } j = 1 ∞ for T with respect to eigenvalues { λ j } j = 1 ∞ . Hence, for any i = 1 , 2 , … , we have

∑ j = 1 ∞ λ j ‖ Λ i e j ‖ 2 = tr ( Λ i * Λ i T ) = 0 .

In addition,

∑ j = 1 ∞ ∣ λ j ∣ = ∑ j = 1 ∞ ∣ ⟨ T e j , e j ⟩ ∣ < ∞ ,

that is, λ = ( λ 1 , λ 2 , λ 3 , … ) ∈ ℓ 1 . From condition 2, it can be concluded that λ = 0 , therefore, T = 0 .

2 ⇒ 1 Assume that the statement 2 is false. Then there exist an λ = ( λ 1 , λ 2 , λ 3 , … ) ∈ ℓ 1 and an orthonormal basis { e j } j = 1 ∞ such that

∑ j = 1 ∞ λ j ‖ Λ i e j ‖ 2 = 0 , i = 1 , 2 , … ,

but λ ≠ 0 . Define an operator T on ℋ by

T e j = λ j e j , j = 1 , 2 , … .

Obviously, T is a nonzero self-adjoint operator and

∣ T ∣ e j = T T * e j = ∣ λ j ∣ e j , j = 1 , 2 , … .

Therefore,

∑ j = 1 ∞ ∣ ⟨ ∣ T ∣ e j , e j ⟩ ∣ = ∑ j = 1 ∞ ∣ λ j ∣ < ∞ ,

and thus, T is a nonzero self-adjoint and trace class operator. Moreover,

tr ( Λ i * Λ i T ) = ∑ j = 1 ∞ ⟨ Λ i T e j , Λ i e j ⟩ = ∑ j = 1 ∞ λ j ⟨ Λ i e j , Λ i e j ⟩ = ∑ j = 1 ∞ λ j ‖ Λ i e j ‖ 2 = 0 , i = 1 , 2 , … ,

This contradicts with the condition 1. Therefore, statement 2 is established.□

The following theorem is a classification of injectivity problem concerning operators of trace one. First, we define a subspace of the real space ℓ 1 as follows:

W = ( λ 1 , λ 2 , λ 3 , … ) ∈ ℓ 1 , ∑ j = 1 ∞ λ j = 0 .

Theorem 4

Given a G-frame { Λ i } i = 1 ∞ for a Hilbert space ℋ , the following are equivalent:

For any orthonormal basis { e j } j = 1 ∞ of ℋ and for any λ = ( λ 1 , λ 2 , λ 3 , … ) ∈ W , if
∑ j = 1 ∞ λ j ‖ Λ i e j ‖ 2 = 0 , i = 1 , 2 , … ,
then λ = 0 .
If T is a trace zero and self-adjoint operator such that
tr ( Λ i * Λ i T ) = 0 , i = 1 , 2 , … ,
then T = 0 .

Proof

1 ⇒ 2 Suppose that T is a trace zero and self-adjoint operator such that tr ( Λ i * Λ i T ) = 0 , i = 1 , 2 , … . Let { e j } j = 1 ∞ be an eigenbasis for T with respect to eigenvalues { λ j } j = 1 ∞ . For any i = 1 , 2 , … , we have

∑ j = 1 ∞ λ j ‖ Λ i e j ‖ 2 = tr ( Λ i * Λ i T ) = 0 .

Since T is a trace class operator, we obtain

∑ j = 1 ∞ ∣ λ j ∣ = ∑ j = 1 ∞ ∣ ⟨ T e j , e j ⟩ ∣ < ∞ .

Also,

∑ j = 1 ∞ λ j = 0 .

Hence, λ = ( λ 1 , λ 2 , λ 3 , … ) ∈ W . From assumption 2, we obtain λ = 0 , so T = 0 .

2 ⇒ 1 Suppose that assumption 2 if false. Then there exist an orthonormal basis { e j } j = 1 ∞ for ℋ and an λ = ( λ 1 , λ 2 , λ 3 , … ) ∈ W such that

∑ j = 1 ∞ λ j ‖ Λ i e j ‖ 2 = 0 , i = 1 , 2 , … ,

but λ ≠ 0 . Define an operator T on ℋ by

T e j = λ j e j , j = 1 , 2 , … .

Then T is a nonzero self-adjoint operator. It follows that

tr ( Λ i * Λ i T ) = ∑ j = 1 ∞ λ j ‖ Λ i e j ‖ 2 = 0 , i = 1 , 2 , … .

This contradicts with assumption 1.□

The following theorem shows that if a Parseval G-frame has quantum injectivity, then it is also an injective G-frame.

Theorem 5

Suppose that { Λ i } i = 1 ∞ is a quantum injective Parseval G-frame for ℋ , then { Λ i } i = 1 ∞ is injective.

Proof

Let { Λ i } i = 1 ∞ be a quantum injective Parseval G-frame for ℋ . Then

I ℋ = ∑ i = 1 ∞ Λ i * Λ i .

Therefore,

tr ( T ) = ∑ i = 1 ∞ tr ( Λ i * Λ i T ) .

Assume that tr ( Λ i * Λ i T ) = 0 , i = 1 , 2 , … for some self-adjoint trace class operator T . Then tr ( T ) = 0 . Also, { Λ i } i = 1 ∞ has quantum injectivity, so T = 0 , and thus, { Λ i } i = 1 ∞ is injective.□

The following theorem shows a way to obtain an injective G-frame by using an injective frame.

Theorem 6

Let { f i } i = 1 ∞ be a frame for ℋ . Define

Λ i = f i ⊗ f i , ∀ i = 1 , 2 , … .

Then { Λ i } i = 1 ∞ is an injective G-frame for ℋ if and only if { f i } i = 1 ∞ is a an injective frame for ℋ .

Proof

Let T be any trace class self-adjoint operator on ℋ . Then

⟨ T Λ i , Λ i ⟩ = ⟨ T ( f i ⊗ f i ) , f i ⊗ f i ⟩ = ‖ f i ‖ 2 ⟨ T f i , f i ⟩ .

These identities and routine arguments prove the theorem.□

The injectivity of a G-frame is preserved under a linear isomorphism. Specifically, we show the following.

Theorem 7

Let { Λ i } i = 1 ∞ be a G-frame for a Hilbert space ℋ . Then for any invertible operator Θ on ℋ , the sequence { Γ i } i = 1 ∞ defined by Γ i = Λ i Θ , i = 1 , 2 , … is injective if and only if { Λ i } i = 1 ∞ is injective.

Proof

Let A and B be G-frame bounds for { Λ i } i = 1 ∞ . Then for any f ∈ ℋ , we have

A ‖ Θ f ‖ 2 ≤ ∑ i = 1 ∞ ‖ Γ i f ‖ 2 ≤ B ‖ Θ f ‖ 2 .

Moreover, since Θ is an invertible operator on ℋ , we obtain

A ‖ Θ − 1 ‖ 2 ‖ f ‖ 2 ≤ ∑ i = 1 ∞ ‖ Γ i f ‖ 2 ≤ B ‖ Θ ‖ 2 ‖ f ‖ 2 .

Thus { Γ i } i = 1 ∞ is a G-frame for ℋ . Let T be a self-adjoint trace class operator on ℋ and

⟨ T Λ i Θ , Λ i Θ ⟩ = tr ( Θ * Λ i * T Λ i Θ ) = 0 , ∀ i = 1 , 2 , … .

The injectivity of { Λ i } i = 1 ∞ implies that Θ * T Θ = 0 . Since Θ is invertible, it follows that T = 0 .

Conversely, suppose that { Γ i = Λ i Θ } i = 1 ∞ is injective, and T is an arbitrary self-adjoint trace class operator. If for every i = 1 , 2 , … ,

tr ( Λ i * Λ i T ) = tr ( Θ * Λ i * Λ i Θ Θ − 1 T ( Θ * ) − 1 ) = 0 ,

then Θ − 1 T ( Θ * ) − 1 = 0 , and thus, T = 0 . Hence, { Λ i } i = 1 ∞ is injective.□

For the G-frames { Λ i } i = 1 ∞ and { Γ i } i = 1 ∞ , if there is a bounded invertible operator Θ such that Γ i = Λ i Θ , i = 1 , 2 , … , then we say that two G-frames { Λ i } i = 1 ∞ and { Γ i } i = 1 ∞ are similar. The following corollary indicates that similar G-frames preserve injectivity.

Corollary 1

Suppose that { Λ i } i = 1 ∞ and { Γ i } i = 1 ∞ are similar injective G-frames for ℋ . Then { Λ i } i = 1 ∞ is injective if and only if { Γ i } i = 1 ∞ is injective.

In fact, it is not necessary to find Parseval injective G-frame for G-frame quantum detection problem. If there is an injective G-frame, then its canonical Parseval G-frame must be injective.

Corollary 2

Let { Λ i } i = 1 ∞ be an injective G-frame for ℋ . Then its canonical Parseval G-frame { Λ i S Λ − 1 2 } i = 1 ∞ is also injective, where S Λ is the G-frame operator of { Λ i } i = 1 ∞ .

The stability of injective frames has been a hot topic of research. We now examine the stability property for injective G-frames. Following [3], we will use the following metric to measure the distance between G-frames.

Definition 3

Given G-frames Λ = { Λ i } i = 1 ∞ and Γ = { Γ i } i = 1 ∞ for a Hilbert space ℋ , we define the distance between them by

d 2 ( Λ , Γ ) = ∑ i = 1 ∞ ‖ Λ i − Γ i ‖ 2 .

The following theorem shows that injective G-frames are unstable in infinite-dimensional cases.

Theorem 8

Let Y = { Λ i } i = 1 ∞ ∪ { Γ i k } i = 0 , k = 1 ∞ , ∞ ∪ { Ω i k } i = 0 , k = 1 ∞ , ∞ be the injective G-frame for the complex space ℓ 2 as in Example 4. Then for any ε > 0 , there is a G-frame Z such that d ( Y , Z ) < ε , and Z is not injective.

Proof

Let { 1 2 i S r i x k } i = 0 , k = 1 ∞ , ∞ be the sequence defined in Example 4 of Sect. 4. Then, for any f ∈ ℓ 2 , we have

∑ i = 0 ∞ ∑ k = 1 ∞ ∣ ⟨ f , 1 2 i S r i x k ⟩ ∣ 2 ≤ ∑ i = 0 ∞ ∑ k = 1 ∞ 1 4 i ‖ f ‖ 2 ‖ S r i x k ‖ 2 ≤ ∑ i = 0 ∞ ∑ k = 1 ∞ 1 4 i 4 a k 2 ‖ f ‖ 2 ≤ ∑ i = 0 ∞ 1 4 i − 1 ∑ k = 1 ∞ a k 2 ‖ f ‖ 2 ≤ 16 3 ‖ { a k } k = 1 ∞ ‖ 2 ‖ f ‖ 2 .

That is, { 1 2 i S r i x k } i = 0 , k = 1 ∞ , ∞ is a Bessel sequence, and thus,

∑ i = 0 ∞ ∑ k = 1 ∞ 1 2 i S r i x k ⊗ 1 2 i S r i x k 2

converges. Therefore, for any ε , there exists N such that

∑ i = N + 1 ∞ ∑ k = 1 ∞ 1 2 i S r i x k ⊗ 1 2 i S r i x k 2 < ε 2 .

Let

Φ i k = 1 2 i S r i x k ⊗ 1 2 i S r i x k , i = 0 , 1 , … , N ; k = 1 , 2 , … , 0 , otherwise .

By Theorem 6, Z = { Λ i } i = 1 ∞ ∪ { Φ i k } i = 0 , k = 1 ∞ , ∞ ∪ { Ω i k } i = 0 , k = 1 ∞ , ∞ cannot give injectivity. This completes the proof.□

Remark 1

As indicated earlier, there exist quantum injective G-frames that are unstable in infinite-dimensional spaces. Therefore, we only need to consider the stability of injective G-frames in finite-dimensional spaces. It is useful to point out that injectivity and quantum injectivity coincide if ℋ is a finite-dimensional Hilbert space. We can establish the stability for finite-dimensional injective G-frames. The proof strategy for this observation is similar to that of [11, Theorem 5], so we omit it.

4 Examples

In this section, we intend to discuss some concrete examples. First, we construct two injective G-frames in finite-dimensional cases.

Example 1

Let { Λ i } i = 1 6 be matrixes on R 3 defined by

Λ 1 = 1 0 0 0 0 0 0 0 0 , Λ 2 = 0 0 0 0 1 0 0 0 0 , Λ 3 = 0 0 0 0 0 0 0 0 1 , Λ 4 = 1 1 0 1 1 0 0 0 0 , Λ 5 = 0 0 0 0 1 1 0 1 1 , Λ 6 = 1 1 1 1 1 1 1 1 1 .

Assume that T is a self-adjoint matrix on R 3 . It is straightforward to see that if

tr ( Λ i * Λ i T ) = 0 , ∀ i = 1 , 2 , 3 , 4 , 5 , 6 .

Then T = 0 . That is, { Λ i } i = 1 6 is an injective G-frame for R 3 .

Example 2

Let { Λ i } i = 1 6 be matrixes on C 2 defined by

Λ 1 = e 2 π i 3 e 5 π i 6 e 5 π i 6 e π i , Λ 2 = e 2 π i 3 − e 5 π i 6 − e 5 π i 6 e π i , Λ 3 = e π i e 5 π i 6 e 5 π i 6 e 2 π i 3 , Λ 4 = e π i − e 5 π i 6 − e 5 π i 6 e 2 π i 3 , Λ 5 = 1 0 0 0 , Λ 6 = 0 0 0 1 .

Clearly, { Λ i } i = 1 6 is a G-frame for C 2 . Let T = a b b ¯ c ∈ S ( C 2 ) . If we set ⟨ T Λ i , Λ i ⟩ = 0 , for i = 1 , 2 , … , 6 , then we obtain a = b = c = 0 , which means that the G-frame { Λ i } i = 1 6 gives injectivity.

The following is an example for the real Hilbert space ℓ 2 .

Example 3

Let { e i } i = 1 ∞ be the canonical basis for the real Hilbert space ℓ 2 and let a i ≠ 0 for i = 1 , 2 , … be such that ∑ i = 1 ∞ a i 2 < ∞ . Define x k = a k ( e 1 + e k + 1 ) for k = 1 , 2 , … . Let S r be the right shift operator on ℓ 2 , and let

{ Λ i } i = 1 ∞ = { e i ⊗ e i } i = 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 = { Λ i } i = 1 ∞ ∪ { Γ i k } i = 0 , k = 1 ∞ , ∞ is an injective G-frame for ℓ 2 .

The complex version of Example 3 looks like:

Example 4

Let { e i } i = 1 ∞ be the canonical basis for the complex Hilbert space ℓ 2 , and let a i ≠ 0 , b i ≠ 0 for i = 1 , 2 , … be such that ∑ i = 1 ∞ a i 2 < ∞ and ∑ i = 1 ∞ b i 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 ℓ 2 and let Λ i = e i ⊗ e i for i = 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 = { Λ i } i = 1 ∞ ∪ { Γ i k } i = 0 , k = 1 ∞ , ∞ ∪ { Ω i k } i = 0 , k = 1 ∞ , ∞ is an injective G-frame for ℓ 2 .

Acknowledgments

The authors are very grateful to the referee for carefully reading the manuscript and providing many helpful comments and suggestions that helped improve the representation of this article.

Funding information: This research was supported by National Natural Science Foundation of China (12301149) and Key Research and Development Project of Henan Province (241111210100).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

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Received: 2024-03-14

Revised: 2025-03-24

Accepted: 2025-08-20

Published Online: 2025-09-24

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2025-0175

Keywords for this article

quantum detection; quantum injectivity; G-frame; operator-valued measure

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