Naimark-Type Results Using Frames

Sharma, Raksha; Khanna, Nikhil

doi:https://doi.org/10.1155/2024/8588361

Journal of Function Spaces

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Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 8588361 | https://doi.org/10.1155/2024/8588361

Naimark-Type Results Using Frames

Raksha Sharma¹and Nikhil Khanna²

Academic Editor: Calogero Vetro

Received29 Oct 2023

Revised06 Feb 2024

Accepted09 Feb 2024

Published22 Feb 2024

Abstract

In this article, a modified version of frame called frame associated with a sequence of scalars (FASS) is defined. This modified version of frame is used to study quantum measurements. Also, using FASS, some Naimark-type results are obtained. Finally, a formula to give the average probability of an incorrect measurement using FASS is obtained.

1. Introduction

Duffin and Schaeffer [1] formalised the definition of frames for Hilbert space in 1952 to examine some challenging issues involving nonharmonic Fourier series. In order to explore signal processing, Duffin and Schaeffer essentially abstracted the basic Gabor concept. However, until the seminal study by Daubechies et al. [2] in 1986, it did not seem that the concepts of Duffin and Schaeffer attracted much attention outside of the nonharmonic Fourier series. Although not to the level of the extremely quick growth of wavelets, the idea of frames started to be researched more extensively after this groundbreaking breakthrough. Frames have historically been utilised in sampling theory, data compression, image processing, and signal processing. The theory is now being used in a growing number of fields, including filterbanks, optics, signal detection, and the study of Besov spaces and Banach space theory.

Let denote a separable Hilbert space equipped with inner product . A sequence of elements in is called a frame for , if there exist positive constants and such that

The scalars and are called frame bounds and they are not unique. If , the frame is called a tight frame, whereas if , the frame is called a Parseval frame. For the frame , the inequality in (1) is known as the frame inequality. The operator defined by is called the preframe operator or the synthesis operator, and its adjoint operator is called the analysis operator which is given by

Composing the operators and , we obtain another operator called the frame operator which is given by

The frame operator is a positive, self-adjoint, and invertible operator on . Thus, the reconstruction formula for all is given by

For more details related to frames and some of their generalizations, one may refer to [3–8].

Eldar and Forney [9] investigated the connection between tight frames and rank-one quantum measurement. Additionally, they described frame matrices by comparing them to the measurement matrices of quantum mechanics. They extended tight frames to orthonormal bases utilising Neumark’s theorem [10, 11]. They constructed the optimal tight frames by drawing inspiration from least squares measurement of quantum mechanics. In this study, we demonstrate various Naimark-type results and derive a formula to calculate the average probability of an incorrect measurement when utilising FASS.

2. Frames Associated with a Sequence of Scalars

We begin this section by defining a modified version of frame called frame associated with a sequence of scalars.

Definition 1. Let , where are finite subsets of with for all , and let be any sequence of scalars. A sequence in Hilbert space is called a frame associated with a sequence of scalars with respect to , if there exist constants and such that

If , then is called a tight frame associated with a sequence of scalars with respect to . If , then is called a Parseval frame associated with a sequence of scalars with respect to .

It should be noted that is a FASS for associated with a sequence of scalars with respect to if and only if is a frame for .

Definition 2. Let be a sequence of subsets of as defined in Definition 1. For an orthonormal basis of , let us consider

One may observe that are subspaces of . An inner product defined on is given by

One can handily corroborate that is a Hilbert space.

It can easily be substantiated that the operator given by is bounded and is called the synthesis operator of the frame associated with a sequence of scalars . Also, the bounded operator given by is called the analysis operator of the frame associated with a sequence of scalars . By composing operators and , we obtain frame operator given by

Let be a frame associated with a sequence of scalars . Let and be synthesis and analysis operators, respectively, and let be the frame operator of the frame . One may promptly observe that

Therefore, we have . So, is the pseudoinverse of , and is a projection from onto .

If is a Parseval frame for with respect to , then ; that is, is isometry.

Given a Parseval frame for with respect to , the following result establishes that there exist a Hilbert space containing and an orthonormal frame for with respect to such that the orthogonal projection of onto is for each . One may observe that (Proposition 1.1 in [12]) a classic result in frame theory is closely related to Theorem 3.

Theorem 3. Let be a Parseval frame associated with a sequence of scalars for with respect to , where scalars are nonzero. Then, there exist a Hilbert space with as a subspace of and an orthonormal frame associated with a sequence of scalars for with respect to such that , for all , where is a projection from onto .

Proof. Let and be orthogonal projection from onto . Define , , and . Further, define as This gives Thus, is an isometry that is . Also, we know that is a projection from onto . And we have

3. Quantum Measurement

According to the well-known spectral theorem, the projection-valued measures (PVMs) or spectral measures correlate one to one with the self-adjoint operators. In conventional quantum mechanics, quantum observables are represented by PVMs. PVMs are defined in [9, 13–16] as follows.

3.1. Projection-Valued Measure (PVM)

A PVM on Hilbert space is any set of operators on which satisfies the following: (i)Each operator is a self-adjoint projection for all (ii), (iii)

Let be a -algebra of the subsets of a locally compact space and be the set of bounded operators on Hilbert space . A positive operator-valued measure (POVM) is a function such that (i)For all , is a positive self-adjoint operator(ii)(iii)For all disjoint subsets , we have(iv)

The representation of quantum observables by POVMs is found to be more appropriate than by spectral measures. In 1940s, POVMs were defined to study some extensions of operators (symmetric). Later, around 1970s, POVMs were used as a tool to describe the quantum measurements. It was observed that POVMs are an extension of quantum observable that are embodied by a spectral measure (PVMs). Presently, POVMs are used as a basic tool in the study of quantum information theory [17] and quantum optics. In [18], Ali studied certain geometrical properties of POVM, defined on the Borel sets of locally compact space , taking values in the set of all bounded operators on a separable Hilbert space. In terms of POVM for observables, a thorough examination of the basic ideas of quantum theory as well as current experiments connected to it is presented in [19]. The broad statistical (convex) approach framework in [20] presents a purely statistical characterization of measurements of observables (characterized by spectral measures in standard quantum mechanics formalism). In [21], Prugovečki studied the stochastic quantum mechanics. The necessity of using non-normalized POVM is also described in order to understand the idea of quantum localization in spacetime. POVM is precisely defined in [13, 14, 22] as follows.

3.2. Positive Operator-Valued Measure (POVM)

A POVM on Hilbert space is any set of operators on which satisfies the following conditions: (i)Each operator is positive, for all (ii)

The quantum observables delineated by POVMs are like a generalization of the basic or standard quantum observables. So POVMs are generally called unsharp observables or generalized observables. A very important result which discusses the interconnection between POVMs and PVMs is the Naimark dilation theorem. However, its interpretation from physical perspective is not very clear, and so there is some awkwardness in interpreting the Hilbert space .

It is well known that a Parseval frame defines a POVM (see, for example, [23]). For the convenience of the reader, we state and prove this result in the particular case of Parseval frames associated with a sequence of scalars.

Theorem 4. Let be Parseval frame associated with a sequence of scalars for with respect to , and let . Then, is a POVM on .

Proof. First, we show that each is a positive operator. For , we obtain To establish the completeness relation, let . Then, In the following result, we show that POVM in a Hilbert space can give rise to a Parseval frame associated with a sequence of scalars for .

Theorem 5. Let be a POVM on a Hilbert space . Then, there exist a disjoint partition of with finite for all and a sequence of scalars and a sequence in such that is a Parseval frame associated with a sequence of scalars for with respect to .

Proof. Let be disjoint partitions of with finite for all . Note that is positive and self-adjoint operator. So, in view of the spectral theorem of positive operator, there exists an orthonormal set in and positive numbers such that for all , we have But , and so, we obtain Taking and , for , we get We shall now prove Naimark-type results using frames associated with a sequence of scalars. More precisely, we prove that an orthonormal frame associated with a sequence of scalars represents projection-valued measure in Hilbert spaces.

Theorem 6. Let be an orthonormal frame associated with a sequence of scalars for with respect to . Also, let , for . Then, is a projection-valued measure on .

Proof. It is clear that is self-adjoint for and We are now left to show that is a projection for . Let . Then, we have Finally, we show that a Parseval frame associated with a sequence of scalars can also give a PVM through dilation theorem.

Theorem 7. Let be a frame associated with a sequence of scalars for with respect to . Then, there exist a Hilbert space with as a subspace of , with , for all , and a sequence of operators on such that is a projection-valued measure on , where is a projection from onto and , for .

Proof. Proof follows from Theorem 3 and Theorem 6.

A quantum system in a pure state is characterized by a normalized vector in a Hilbert space . Information about a quantum system is extracted by subjecting the system to a measurement. In quantum theory, the outcome of a measurement is inherently probabilistic, with the probabilities of the outcomes of any conceivable measurement determined by the state vector . Now, we will show how Parseval frame associated with a sequence of scalars can be used in quantum measurement. Let be Parseval frame associated with a sequence of scalars for with respect to . Take , for . Suppose the measurement is performed upon a quantum system in a pure state . Then, the probability of the outcome is given by

Moreover, one can notice that

Let be unit normed states in Hilbert space with corresponding probabilities that sum to 1. If the state of the system is , for , then the measurement provides us the information that the system is in the state with high probability of , given by

The probability that the output of our measurement device will be is if the state of the system state is . Consequently, is the probability of correct measurement. Since each occurs with probability , the average probability of a successful measurement is

Thus, the average probability of incorrect measurement is given by

Note that the probability that we measure the system erroneously to be is if the state of the system is for and . Hence, the following relation yields the average probability of an inaccurate measurement:

Next, we use a frame associated with a sequence of scalars for with respect to to provide the average probability of an inaccurate measurement.

Theorem 8. Let be unit normed states in Hilbert space with corresponding probabilities that sum to 1 and be Parseval frame associated with a sequence of scalars for with respect to . Then, the average probability of an incorrect measurement is given by

Proof. We know that Thus, we obtain

4. Conclusion

Positive operator-valued measures (POVMs) have long been the subject of the study. Later, POVMs were used as a tool to delineate the quantum measurements. According to the Naimark dilation theorem, POVMs are seen as an extension of a quantum observable that is represented by spectral measures (or PVMs). POVMs are currently employed in the research of quantum information theory and quantum optics as a basic tool. In this article, we proved various Naimark-type results using frames associated with a sequence of scalars. The average probability of an incorrect measurement is then obtained using a frame associated with a sequence of scalars.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Both authors contributed equally to this paper and read and approved the final manuscript.

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Copyright

Copyright © 2024 Raksha Sharma and Nikhil Khanna. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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