Structure of backward quantum Markov chains

1.   Introduction

The development of the concept of quantum Markovianity significantly advances our comprehension of quantum dynamics and serves as a critical link in connecting various related disciplines ^[3,4,5,19]. This concept not only broadens the theoretical landscape of quantum physics but also catalyzes advancements in practical applications. By integrating rigorous mathematical frameworks with empirical investigations, quantum Markovianity facilitates novel insights into the behavior and control of quantum systems ^[7,15,16].

In the seminal works ^[1,2], a comprehensive framework for quantum Markovianity was introduced, extending the classical notion of Markov processes into the quantum domain. This advanced framework provides a nuanced understanding of the temporal evolution of quantum systems under Markovian conditions. Specifically, the authors constructed two pivotal classes of states to exemplify this notion: QMCs and inverse QMCs. These states are defined on infinite tensor products of finite-dimensional type Ⅰ factors with a totally ordered index set, capturing different aspects of quantum temporal dynamics. QMCs represent systems where the evolution from one state to another follows a Markovian process, while inverse QMCs consider the reverse dynamics within the same framework. Building on these foundational contributions, subsequent research has explored various definitions and properties of QMCs, particularly in connection with quantum information theory.

Notable studies have explored the complex relationships between quantum Markovianity and informational concepts, providing insights into how these chains model information flow within quantum systems ^[8,9,13,19]. Specifically, research in ^[9] has developed a potential theory for QMCs, examining their recurrence, transience, and irreducibility with applications to quantum random walks and entangled states. Additionally, the study of Markov chains has been approached from various perspectives, including quantum operations ^[13,14] and quantum conditional mutual information ^[12].

An analysis of the proofs reveals that the core constructions are applicable to infinite tensor products of arbitrary $C^*$-algebras. Notably, in ^[2,6], the distinction between forward and backward QMCs (BQMCs) was deliberately circumvented. This decision was due to an overemphasis on interpreting the index set as time, which could confine the framework. Instead, the term inverse QMC was adopted, providing a more neutral terminology that is applicable to both probabilistic (i.e., dynamical) and statistical mechanics interpretations. Despite this neutral approach, a more nuanced analysis shows that within each category, QMC and inverse QMC, a natural distinction emerges between forward (or inside) and backward (or outside) QMC. This differentiation highlights deeper structural aspects within each class, suggesting that the evolution and properties of quantum systems can exhibit different characteristics depending on the direction of time or perspective of the analysis.

In this paper, the framework of BQMCs and inverse BQMCs is thoroughly examined. It is shown that for any sequence of backward transition expectations $\mathcal{E}^{(n)} : \mathcal{B}_{n}\otimes \mathcal{B}_{n+1}\to \mathcal{B}_{n}$, there exists at least one BQMCs on $\mathcal{A}$ that aligns with this sequence. This result substantiates the feasibility of the backward QMC model, demonstrating that quantum systems adhering to the specified transition expectations can indeed be constructed.

Additionally, the study establishes that for any sequence of backward transition expectations $\mathcal{E}^{(n)} : \mathcal{B}_{n}\otimes \mathcal{B}_{-\infty}\to \mathcal{B}_{-\infty}$, there is at least one inverse BQMCs on $\mathcal{A}$ that accommodates these expectations.Moreover, concrete examples are provided to distinguish between the two Markovian structures under consideration. Additionally, the paper demonstrates the connection between the introduced QMCs and finitely correlated states, as discussed in ^[10,11]. These examples and connections offer valuable insights into the distinctions and relationships among various quantum Markovian models.

This finding broadens the scope of quantum Markovianity by confirming the practicality of inverse BQMCs models, thus expanding the theoretical understanding of quantum systems within this framework. The obtained results extend QMCs into two-sided one-dimensional (1D) lattice.

Our work advances the study of quantum Markov processes by establishing the existence of these models and offering new perspectives on their dynamics. These advancements push the boundaries of both theoretical and applied quantum physics, offering new insights and methodologies that significantly impact our comprehension of quantum processes and their practical applications. Namely, the present work is promising in connection with the recent development on hidden quantum Markov processes ^[3,4,20].

Let us outline the organization of the paper. Following an introduction to the preliminary concepts in Section 2, Section 3 is dedicated to the study of BQMCs. Section 4 focuses on inverse BQMCs. In Section 5, the paper elucidates the distinct structural properties inherent to both inverse and backward Markov chains. Section 6 establishes a connection between the obtained QMCs and finitely correlated states, further elaborating on the relationships and implications of these models. Finally, Section 7 provide concluding remarks.

2.   Tensor BQMCs on 1D lattice

In the following, all $C^{*}$–algebras are unital and separable unless otherwise stated.

Definition 1. Given two $*$–algebras $\mathcal{A}$, $\mathcal{B}$ and an integer $n \in \mathbb{N}$, a $*$-map

$P$: $\mathcal{A}\rightarrow \mathcal{B}$ is called $n$–positive ($n\in\mathbb N^*$) if $\forall\, b_1, \dots, b_n\in\mathcal{B}$, $\forall\, a_1, \dots, a_n\in\mathcal{A}$

$1$–positive maps are called positive; $n$–positive maps for each natural integer $n$ are called completely positive.

Definition 2. A quasi–conditional expectation with respect to the triplet of $\ast$-algebras $\mathcal{C} \subseteq \mathcal{B} \subseteq \mathcal{A}$ is a completely positive linear map $E :\mathcal{A} \to \mathcal B$, such that

If $E(1) = 1$, $E$ is called a normalized-quasi–conditional expectation.

Definition 3. Let $\mathcal{B}$, $\mathcal{C}$ be $C^{*}$–algebras. A completely positive, identity–preserving linear operator $\mathcal E:\mathcal{B}\to\mathcal{C}$ is called a Markov operator. A Markov operator $\mathcal E:\mathcal{B}\otimes\mathcal{C}\to\mathcal{B}$ is called a backward transition expectation from $\mathcal{B}\otimes\mathcal{C}$ to $\mathcal{B}$.

It is notationally convenient to introduce the notations

Every backward transition expectation $\mathcal E:\mathcal{B}\otimes\mathcal{C}\to \mathcal{B}$ defines two Markov operators: the $\mathcal E$–forward Markov operator (acting on the $1$–st factor of the product):

and the $\mathcal E$–backward Markov operator (acting on the $2$-d factor of the product):

In the following $(\mathcal{B}_{n})_{n\in\mathbb Z}$ will denote a sequence of $C^{*}$–algebras and

with their infinite tensor product with respect to a fixed family of cross norms and the associated embeddings (see ^[18] Definition 1.23.11). For example, we can assume that each algebra $\mathcal{B}_{n}$ is realized on a Hilbert space $\mathcal{H}_{n}$ and that the tensor products are those induced by the tensor products of the corresponding spaces.

In the sequel, by $\mathcal{S}(\mathcal{A})$ we denote the set of states on $\mathcal{A}$, $1_{\mathcal{B}_{n}}$ the identity of $\mathcal{B}_{n},$ and $1_{n}: = j_{n}(1_{\mathcal{B}_{n}}) = 1_{\mathcal{A}}$ the identity of $\mathcal{A}_{n}$.

Definition 4. A state $\varphi_{backw}$ on $\mathcal{A}$ is called a BQMC if there exist:

(i) a sequence of backward transition expectations $\mathcal E^{(n)}:\mathcal{B}_{n}\otimes \mathcal{B}_{n+1}\to\mathcal{B}_{n}$ ($n\in\mathbb Z$),

(ii) a sequence $\psi_{n}\in \mathcal S(\mathcal B_{n})$ ($n\in\mathbb Z$), called a sequence of boundary conditions for $\varphi_{backw}$, such that for all $m\le n\in\mathbb{Z}$ and $b_{h}\in \mathcal{B}_{h}$, $h\in\{m, \dots, n\}$:

3.   Backward Markov Chains

Given a sequence of backward transition expectations $\mathcal E^{(n)} : \mathcal{B}_{n}\otimes \mathcal{B}_{n+1}\to\mathcal{B}_{n}$, define iteratively the vector spaces:

and, for $k\in\{1, \dots, n-m\},$

where the righthand side denotes the linear sub–space of $\mathcal{B}_{m}$ generated by $S^{backw}_{[m, n]}$ for $n\ge m$. The above defined spaces are operator spaces in the sense of ^[17] with identity, so it makes sense to speak of states on them. The operator space $S^{backw}_{[m}$ is the domain of the boundary condition $\psi_{m}$ and since

is a measure of the non–surjectivity of the sequence of transition expectations $\{\mathcal E^{(n)}\}_{n\ge m}$. In particular, if all the $\mathcal E^{(n)}_{1}$ and all the $\{\mathcal E^{(n)}\}_{n\ge m}$ are surjective, then $S^{backw}_{[m} = \mathcal{B}_{m}$.

Definition 5.

A sequence $(\varphi_{[0, n]})$ of states on $\mathcal A$ is called convergent in the strongly finite sense if, for any $a\in\mathcal A$, there exists $n_a\in\mathbb N$ such that for any $n\ge n_a$,

Lemma 1. For a sequence of backward transition expectations

$\mathcal E^{(n)} : \mathcal{B}_{n}\otimes \mathcal{B}_{n+1}\to \mathcal{B}_{n}$, the following statements are true.

(B1) For each $n < N\in\mathbb Z$, the map

is a Markov quasi–conditional expectation with respect to the Markov localization

(B2) For each $n < N\in\mathbb Z$, the limit

exists point-wise in the strongly finite sense on $\mathcal{A}_{loc}$ and defines a Markov quasi–conditional expectation with respect to the Markov localization

Moreover,

Proof. (B1) and (B2) follow from standard arguments on QMC. □

Lemma 2. Let $\varphi_{backw}$ be backward QMC on $\mathcal{A}$ with sequence of transition expectations $\{\mathcal E^{(n)}\}_{n\in\mathbb Z}$ and boundary conditions $(\psi_{n})$. Then:

(B3) The sequence $(\psi_{n})$ satisfies the compatibility condition

In particular, if all the $\{\mathcal E^{(n)}_{1}\}_{n\ge m}$ and all the transition expectations $\{\mathcal E^{(n)}\}_{n\ge m}$ are surjective, then $\psi_{m}\circ \mathcal E^{(m)}_{2} = \psi_{m+1}$.

(B4) The marginal distributions of $\varphi_{backw}$ are

(B5) For any $m_{0}\in\mathbb{Z}$ choosing arbitrarily $\psi_{m_{0}}$, the state on $\mathcal{A}_{[m_{0}}$ defined by

where $E_{m_{0}]}$ is defined by Lemma 1, is a one–sided backward QMC with the sequence of boundary conditions $\psi_{m}$ ($m > m_{0}$) defined by (5).

Proof. (B3) follows from (4) and:

(B4) follows from

(B5) follows from standard arguments on QMC. □

Theorem 1. Given a sequence of backward transition expectations

$\mathcal E^{(n)} : \mathcal{B}_{n}\otimes \mathcal{B}_{n+1}\to \mathcal{B}_{n}$, the set of backward QMC on $\mathcal{A}$ admitting $(\mathcal E^{(n)})$ as sequence transition expectations is not empty.

Proof. Let $\psi_{n}\in \mathcal S(\mathcal{B}_{n})$ and $\chi_{n)}\in \mathcal S(\mathcal{A}_{n)})$ ($n\in\mathbb Z$) be two arbitrary sequences of states. Then, because of the Markov property, for any $n\in\mathbb Z$:

Since $\mathcal{A}$ is a $C^*$–algebra with unit, its state space is compact. Therefore the set of limit points of the sequence (7) is nonempty.

Let $(\varphi^{(n_k)})$ be a sub–sequence of (7) and $\varphi\in\mathcal S(\mathcal{A})$ be such that

point-wise on $\mathcal{A}$. Let $a\in\mathcal{A}_{loc}$, then there exists $n_{0} < N_{0}\in\mathbb{N}$ such that $a\in\mathcal{A}_{[n_{0}, N_{0}]}$. Therefore, for $n_{k} < n_{0},$

and since $N_{0}$ is arbitrary, this identity holds for all $a\in\mathcal{A}_{[n_{0}}$. This means that the sequence of states on $S^{backw}_{[n_{0}}\subseteq\mathcal{B}_{n_{0}}$ given by

converges to a linear functional $\phi_{n_{0}}$. Moreover, the identity

shows that $\phi_{n_{0}}$ is a state on $S^{backw}_{[n_{0}}$. By definition of $E_{n_{0}]}$, (8) implies that the identity (3) holds with the replacements

Since $n_{0}$ is arbitrary, it follows that $\varphi$ is a BQMC with transitions expectations $(\mathcal{E}^{(n)})$ and sequence of boundary conditions $(\phi_{n})$. □

4.   Inverse BQMCs

In this section, we are going to define and investigate inverse BQMCs and prove an existence result.

Definition 6.

A state $\varphi$ on $\mathcal{A}$ is called an inverse BQMC if there exist:

(j1) an algebra $\mathcal{B}_{-\infty}$;

(j2) a sequence of backward transition expectations $\mathcal E^{(n)}:\mathcal{B}_{-\infty}\otimes\mathcal{B}_{n} \to\mathcal{B}_{-\infty}$, $n\in\mathbb Z$;

(j3) a state $\psi_{-\infty}\in \mathcal S(\mathcal B_{-\infty})$, such that for all $m\le n\in\mathbb{N}$ and $b_{h}\in M_{d_{h}}(\mathbb C)$, $h\in\{m, \dots, n\}$.

Given a sequence of backward transition expectations $\mathcal E^{(n)}:\mathcal{B}_{-\infty}\otimes\mathcal{B}_{n} \to\mathcal{B}_{-\infty}$, define iteratively the operator spaces:

and, for $k\in\{1, \dots, n-m\},$

where the right hand side denotes the linear sub–space of $\mathcal{B}_{-\infty}$ generated by $S^{backw}_{(m, n)}$ for $n\ge m$. $S^{backw}_{(m}$ is the domain of the boundary condition $\psi_{-\infty}$ and since

is a measure of the non–surjectivity of the sequence of transition expectations $\{\mathcal E^{(n)}\}_{n\ge m}$. In particular, if all the $\mathcal E^{(n)}_{1}$ and all the $\{\mathcal E^{(n)}\}_{n\ge m}$ are surjective, then $S^{backw}_{(m} = \mathcal{B}_{-\infty}$.

Lemma 3. For a sequence of backward transition expectations $\mathcal E^{(n)} : \mathcal{B}_{-\infty}\otimes \mathcal{B}_{n}\to \mathcal{B}_{-\infty},$ the following statements are true.

(IB1) For each $m\in\mathbb Z$, the map

is a Markov quasi–conditional expectation with respect to the Markov localization

(IB2) For each $n\in\mathbb Z,$ the limit

exists point-wise in the strongly finite sense on $\mathcal{A}_{loc}$ and defines a Markov quasi–conditional expectation with respect to the Markov localization

Moreover,

Proof. (IB1) holds because by definition the map $E_{[m, [(m+1)}$ satisfies

(IB2) follows from standard arguments on QMC together with the fact that, as $m\to -\infty$, $\mathcal{A}_{[m}\uparrow \mathcal{A}_{(-\infty} = \mathcal{A}$. □

Lemma 4. Let $\varphi_{inv}$ be an inverse BQMC on $\mathcal{A}$ with sequence of transition expectations $\{\mathcal E^{(n)}\}_{n\in\mathbb Z}$ and boundary condition $\psi_{-\infty}$. Then:

(IB3) $\psi_{-\infty}$ satisfies the compatibility condition

In particular, if all the $\{\mathcal E^{(n)}_{1}\}_{n\ge m}$ and all the transition expectations $\{\mathcal E^{(n)}\}_{n\ge m}$ are surjective, then $\psi_{-\infty}\circ \mathcal E^{(m)}_{2} = \psi_{-\infty}$.

(IB4) The marginal distributions of $\varphi_{inv}$ are

(IB5) For any $m_{0}\in\mathbb{Z}$, choosing arbitrarily $\psi_{-\infty}$, the state on $\mathcal{A}_{[m_{0}}$ defined by

where $E_{m_{0}]}$ is defined by Lemma 3. is a one–sided BQMC with boundary condition $\psi_{-\infty}$ defined by (11).

Proof. (IB3) follows from (10) and the fact that for every $n$ and $b_{j}\in\mathcal B_{j}$, $j\in\{m, \dots, n\}$:

(IB4) follows from

(IB5) follows from standard arguments on QMC. □

Theorem 2. Given a sequence of backward transition expectations

$\mathcal E^{(n)} : \mathcal{B}_{n}\otimes \mathcal{B}_{-\infty}\to \mathcal{B}_{-\infty}$, the set of BQMC on $\mathcal{A}$ admitting $(\mathcal E^{(n)})$ as sequence transition expectations is not empty.

Proof. Let $\psi_{-\infty}\in \mathcal S(\mathcal{B}_{-\infty})$ and $\chi_{(n}\in \mathcal S(\mathcal{A}_{(n})$ ($n\in\mathbb Z$) be two arbitrary sequences of states. Then, because of the Markov property, for any $n\in\mathbb Z$:

Since $\mathcal{A}$ is a $C^*$–algebra with unit, its state space is compact. Therefore, the set of limit points of the sequence (13) is nonempty.

Let $(\varphi^{(n_k)})$ be a sub–sequence of (7) and $\varphi\in\mathcal S(\mathcal{A})$ be such that

point-wise on $\mathcal{A}$. Let $a\in\mathcal{A}_{loc}$, then there exists $n_{0} < N_{0}\in\mathbb{N}$ such that $a\in\mathcal{A}_{[n_{0}, N_{0}]}$. Therefore, for $n_{k} > n_{0},$

and since $N_{0}$ is arbitrary, this identity holds for all $a\in\mathcal{A}_{[n_{0}}$. This means that the sequence of states on $S^{backw}_{[n_{0}}\subseteq\mathcal{B}_{-\infty}$ given by

converges to a linear functional $\phi_{-\infty}$. Moreover, the identity

shows that $\phi_{-\infty}$ is a state on $S^{backw}_{[n_{0}}$. By definition of $E_{[n_{0}}$, (8) implies that the identity (3) holds with the replacements

Since $n_{0}$ is arbitrary, it follows that $\varphi$ is an inverse BQMC with transitions expectations $(\mathcal{E}^{(n)})$ and boundary conditions $\phi_{-\infty}$. □

5.   Example of inverse and BQMCs

Within this section, we aim to elucidate the distinct structural properties inherent in inverse and backward Markov chains, employing a concrete example as an illustrative tool.

Let $\mathcal{M}_2: = M_{2}(\mathbb{C})$ be our starting $C^{*}$-algebra. By $\sigma_{x}$, $\sigma_{y}$, $\sigma_{z}$ we denote the Pauli spin operators, i.e.,

Let $E = \frac{1}{\sqrt{2}}(1, 1)$ and define $\mathcal{E}: \mathcal{M}_2 \otimes \mathcal{M}_2\rightarrow \mathcal{M}_2$, for all $a = (a_{ij})$, $b = (b_{ij})$ by

where

Lemma 5. $\mathcal{E}$ defined by (15) is a backward Markov transition expectation, i.e., a completely positive identity preserving linear map.

Proof. $\mathcal{E}$ is identity preserving, in fact, from (15),

Additionally, $\mathcal{E}$ is expressed in the Kraus representation, indicating its property of being completely positive. □

Lemma 6. For all $n\in \mathbb{Z}$, let $\mathcal{E}_n: = \mathcal{E}$ be defined by (15) and $\psi_{n}$ be a state on $\mathcal{B}_n\equiv\mathcal{M}_2$. Then for all $m\le n\in\mathbb{Z}$ and $b_h = \begin{pmatrix} b_{h, 11} & b_{h, 12} \\ b_{h, 21} & b_{h, 22} \end{pmatrix}\in \mathcal{B}_{h}$, let $h\in\{m, \dots, n\}$

Proof By observing (15), it becomes apparent that

Hence, (16) can be derived through iterative processes. □

Lemma 7. Let $(\mathcal{E}^{(n)} = \mathcal{E})_{n\in \mathbb{Z}}$ denote a sequence of backward transition expectations, defined by (15). Consider $\psi_{-\infty}$ as a state on $\mathcal{B}_{-\infty}\equiv\mathcal{M}_2$. Then, for all $m\leq n\in\mathbb{Z}$ and $b_h = \begin{pmatrix} b_{h, 11} & b_{h, 12} \\ b_{h, 21} & b_{h, 22} \end{pmatrix}\in \mathcal{B}_{h}$, where $h\in \{m, \dots, n\}$, we have the following expression:

Proof By observing (15), it becomes apparent that

then

One can see that $\mathcal{E}(\sigma_z b_m\sigma_z\otimes b_{m+1}) = \mathcal{E}(\sigma_y b_m\sigma_y\otimes b_{m+1})$. Consequently, we obtain:

One more step,

Through the process of iteration, we find that

□

Remark 1. The above example illustrates the difference between the structure of the backward Markov chain and the inverse Markov chains.

6.   Recovering finitely correlated states

In this section, we will check whether, BQMCs and the inverse QMCs have finitely correlated states structure. Recall that, a finitely correlated state (FCS) ^[10,11] is a translation invariant state $\varphi$ on the quasi-local algebra $\mathcal{A} = \bigotimes_{n\in \mathbb{Z}}\mathcal{B}_n$, where each $\mathcal{B}_n$ is a copy of a C$^*$-algebra $\mathcal{B}$ with unit $1$, characterized by the existence of a finite dimensional linear space $\mathcal{V}$, a linear map $\mathbb{E}: A\in \mathcal{B} \mapsto \mathcal{L}(\mathcal{V})$ ($\mathcal{L}(\mathcal{V})$ being the set of linear maps from $\mathcal{V}$ into itself), an element $e\in\mathcal{V},$ and a linear functional $\rho\in\mathcal{V}^*$ such that

and for $n\in\mathbb{Z}, m\in \mathbb{N},$ and $a_j\in \mathcal{B}_j$:

Let $\varphi_{backw} \equiv ((\mathcal{B}_n)_n, (\mathcal{E}^{(n)})_n, (\psi_n)_n)$ be a BQMC whose correlations are given by (3). In the homogenous, for which all the $\mathcal{B}_n$ are copies of finite dimensional C$^*$-algebra, all the transition expectations $\mathcal{E}_n$ are copies of a transition expectation $\mathcal{E}: \mathcal{B}\otimes \mathcal{B}\to\mathcal{B}$, and the functional $\psi_n$ is a copy of a state $\psi\in \mathcal{S}(\mathcal{B})$. By taking $\mathcal{V} = \mathcal{B}, \mathbb{E} = \mathcal{E}, \rho = \psi$ and $e = {{\mathbf 1}\!\!{\rm I}}$, one can see that $\varphi_{backw}$ defines an FCS on $\mathcal{A}$.

Similarly, let $\varphi \equiv ((\mathcal{B}_n)_n, \mathcal{B}_{\infty}, (\mathcal{E}_{n}), \psi_{\infty})$ be an inverse backward Markov chain. Assume that $\mathcal{B}_n$ are copies of a C$^*$-algebra $\mathcal{B}$ and $\mathcal{E}^{(n)}$ are copies of a transition expectation $\mathcal{E}: \mathcal{B}_{-\infty}\otimes\mathcal{B}\to \mathcal{B}_{-\infty}$. By taking $\mathcal{V} = \mathcal{B}_{-\infty}, \rho = \psi_{-\infty}$ and $e = 1$, then the state $\varphi$ generates an FCS. Contrary, to the FCS generated by BQMCs, the linear space $\mathcal{V} = \mathcal{B}_{-\infty}$ is not assumed to coincide with the algebra $\mathcal{B}$.

7.   Conclusions

In conclusion, this paper significantly advances the field of quantum Markovianity by thoroughly examining BQMCs and inverse BQMCs. The results demonstrate that for any given sequence of backward transition expectations, both backward QMCs and inverse BQMCs can be constructed, thus affirming the feasibility and applicability of these models. This work not only clarifies the distinctions between different Markovian structures but also establishes important connections between QMCs and finitely correlated states. By providing concrete examples and elucidating the structural differences, this study enhances our understanding of quantum systems dynamics. The findings contribute to the broader theoretical framework and offer new perspectives that could impact practical applications in quantum information science. The insights gained from this research are expected to influence ongoing studies, including those related to hidden quantum Markov processes, and pave the way for further exploration in quantum dynamics and information theory.

Author contributions

Luigi Accardi: Conceptualization, Methodology, Writing – original draft; El Gheteb Soueidi: Methodology, Writing – original draft; Abdessatar Souissi: Writing – original draft, Writing – review & editing; Mohamed Rhaima: Validation, Funding acquisition; Farrukh Mukhamedov: Conceptualization, Methodology; Farzona Mukhamedova: Visualization, Validation.

Acknowledgments

The authors extend their appreciation to King Saud University in Riyadh, Saudi Arabia for funding this research work through Researchers Supporting Project Number (RSPD2024R683)

Conflict of interest

The authors declare that they have no conflicts of interest.