FIXED POINT THEOREM IN A PARTIAL b-METRIC SPACE APPLIED TO QUANTUM OPERATIONS

Introduction/purpose: A fixed point theorem of an order-preserving mapping on a complete partial b-metric space satisfying a contractive condition is constructed. Methods: Extension of the results of Batsari et al. Results: The fidelity of quantum states is used to construct the existence of a fixed quantum state. Conclusions: The fixed quantum state is associated to an order-preserving quantum operation.


Introduction and preliminaries
A partial metric space is a generalized metric space in which each object does not necessarily have a zero distance from itself (Aamri & El Moutawakil, 2002). Another angle of fixed point research emerged with the approach of the Knaster-Tarski fixed point theorem (Knaster, 1928; Tarski, 1955. The idea was first initiated from Knaster andTarski in 1927 (Knaster, 1928), and later Tarski found some improvement of the work in 1939, which he discussed in some public lectures between 1939 and 1942 (Tarski, 1955(Tarski, , 1949. Finally, in 1955, Tarski (Tarski, 1955 published the comprehensive results together with some applications. A different property of this theorem is that it involves an order relation defined on the space of consideration. Indeed, the order relation serves as an alternative to the continuity and contraction of the mappings as found in the Brouwer (Brouwer, 1911) and Banach (Banach, 1922) fixed point theorems, respectively, see (Tarski, 1955).
In the area of the quantum information theory, a qubit is seen as a quantum system, whereas a quantum operation can be inspected as the measurement of a quantum system; it describes the development of the system through the quantum states. Measurements have some errors which can be corrected through quantum error correction codes. The quantum error correction codes are easily developed through the information-preserving structures with the help of the fixed points set of the associated quantum operation. Therefore, the study of quantum operations is necessary in the field of the quantum information theory, at least in developing the error correction codes, knowing the state of the system (qubit) and the description of energy dissipation effects due to loss of energy from a quantum system (Nielsen & Chuang, 2000).
In 1951, Luders (Lüders, 1950) discussed the compatibility of quantum states in measurements (quantum operations). He also proved that the compatibility of quantum states in measurements is equivalent to the commutativity of the states with each quantum effects in the measurement. In 1998, Busch et al. (Busch & Singh, 1998 generalized the Luders theorem. He also showed that a state is unchanged under a quantum operation if the state commutes with every quantum effect that relates the quantum operation. In 2002, Arias et al. (Arias et al, 2002) studied the fixed point sets of a quantum operation and gave some conditions for which the set is equal to a commutate set of the quantum effects that described the quantum operation. In 2011, Long and Zhang (Zhang & Ji, 2012) deliberated the fixed point set for quantum operations, they presented some necessary and sufficient conditions for the existence of a non-trivial fixed point set. Similarly, in 2012, Zhang and Ji (Long & Zhang, 2011) deliberated the existence of a non-trivial fixed point set of a generalized quantum operation. In 2016, Zhang and Si (Zhang & Si, 2016) explored the conditions for which the fixed point set of a quantum operation (ϕ A ) with respect to a row contraction A equals to the fixed point set of the power of the quantum operation (ϕ j A ) for some 1 ≤ j < +∞. Other useful references are (Agarwal et al, 2015; Debnath et al, 2021; Kirk & Shahzad, 2014. DEFINITION 1. (Shukla, 2014) A partial b-metric on the set X is a function For all x, y ∈ X, p s (x, y) = p s (y, x) (4) There exists a real number s ≥ 1 such that, for all x, y, z ∈ X, p s (x, z) ≤ s[p s (x, y) + p s (y, z)] − p s (y, y). (X, p s ) denotes the partial b-metric space. Note that every partial metric is a partial b-metric with s = 1. Also, every b-metric is a partial b-metric with p s (x, x) = 0, for all x, y ∈ X.
A sequence {x n } in the space (X, p s ) converges with respect to the topology τ b to a point x ∈ X, if and only if (1) The sequence {x n } is Cauchy in (X, p s ) if the below limit exists and is finite lim DEFINITION 2. A mapping T is said to be order-preserving on X, whenever

Main result
The objective of this work is to establish a fixed point theorem in a complete partial b-metric space. THEOREM 1. Let (X, p s ) be a complete partial b-metric space with s ≥ 1 and associated with a partial order ⪯. Suppose an order preserving mapping T : X → X satisfies p s (T (x), T (y)) ≤ α max{p s (x, y), p s (x, T (y)), p s (y, T (x))} Then, we proceed as follows: Thus, we have By simplifying (5), we have Therefore, from (6), we conclude that n=1 is a monotone non-increasing sequence of real numbers and bounded below by 0. Therefore, lim n→+∞ q n = 0, see Chidume et al. (Chidume & Chidume, 2014).
implies that Now, taking the limit as n, m → +∞ in (7), we have For showingx ∈ X is a fixed point of T , we proceed as follows: Case I: Suppose max{p s (x n ,x), p s (x n , T (x)), p s (x, T (x n ))} = p s (x n ,x). Then, from inequality (9), we have From the above inequality, we have We can observe that for β ∈ min{ 1 s 3 , 2 s+1 }, If β = 1 s 3 , then, from equality (11) we have Similarly, if β = 2 s+1 , then, from equality (11), From equalities (12) and (13), we conclude that the right-hand side of (10) is non-negative.
If α < β, then by (14), we have Similarly for (18), we conclude that the right-hand side of (10) is non-negative.
Case III: Suppose max{p s (x n ,x), p s (x n , T (x)), p s (x, T (x n ))} = p s (x, T (x n ))). Then, from inequality (9), we have By the simplification of the above equality, we have Note that, for any value of α, β ∈ [0, θ) and 4 − s 2 β − sβ ≥ 0. Thus, the right-hand side of (10) is non-negative. Taking the limit as n → +∞ of both sides in the respective inequalities (10), (14) and (19), we conclude that Thus, T (x) =x. Next, we prove that ifx ∈ X is a fixed point of T , then p s (x,x) = 0.
This is contradicting the fact that p s (x,x) ̸ = 0. Therefore, p s (x,x) = 0.
Last, we will prove the uniqueness of the fixed point. Let x 1 , x 2 ∈ X be two distinct fixed points of T . Then This is a contradiction. Therefore, the fixed point is unique.
Now we apply our main result similar to (Batsari & Kumam, 2020) as follows: Application to quantum operations In quantum systems, measurements can be seen as quantum operations (Seevinck, 2003). Quantum operations are very important in narrating quantum systems that collaborate with the environment.

Let B(H) be the set of bounded linear operators on the separable complex Hilbert space H; B(H) is the state space of consideration. Suppose
is called a quantum operation (Arias et al, 2002), quantum operations can be used in quantum measurements of states. If the A i 's are self adjoint then, ϕ A is self-adjoint.
General quantum measurements that have more than two values are narrated by effect-valued measures (Arias et al, 2002). Denote the set of quantum effects by ε(H) = {A ∈ B(H) : 0 ≤ A ≤ I}. Consider the discrete effect-valued measures narrated by a sequence of E i ∈ ε(H), i = 1, 2, . . . satisfying E i = I where the sum converges in the strong operator topology. Therefore, the probability that outcome i eventuates in the state ρ is ρ (E i ) and the post-measurement state given that i eventuates is E (Arias et al, 2002). Furthermore, the resulting state after the implementation of measurement without making any consideration is given by If the measurement does not disturb the state ρ, then we have ϕ(ρ) = ρ. Furthermore, the probability that an effect A eventuates in the state r given that the measurement was conducted is From now, we will be dealing with a bi-level (|0⟩, |1⟩) single qubit quantum system where a quantum state |Ψ⟩ can be narrated as |Ψ⟩ = a|0⟩ + b|1⟩, with a, b ∈ C and |a| 2 + |b| 2 = 1 see (Batsari & Kumam, 2020; Nielsen & Chuang, 2000. Considering the characterization of a bi-level quantum system by the Bloch sphere ( Figure  1) above, a quantum state (|Ψ⟩) can be represented with the density matrix below (ρ), Also, the density (ρ) matrix is, where r ρ = [r x , r y , r z ] is the Bloch vector with ∥r ρ ∥ ≤ 1, and σ = [σ x , σ y , σ z ] where Let ρ, σ be two quantum states in a bi-level quantum system. Then, the Bures fidelity (Bures, 1969) between the quantum states ρ and σ is defined as The Bures fidelity satisfies 0 ≤ F (ρ, σ) ≤ 1, if ρ = σ it takes the value 1 and 0 if ρ and σ have an orthogonal support (Nielsen & Chuang, 2000). Now consider a two-level quantum system X represented with the collection of density matrices {ρ : ρ is as defined in Equation (24)}. Define the function p s : X × X → R + as follows: It is easy to show that p s is a b-metric on X with s taking the value 1 approximately. They also define an order relation ⪯ on X by ρ ⪯ δ iff the line from the origin joining the point r δ passes through r ρ .
It is easy to show that the order relation defined above is a partial order (Batsari & Kumam, 2020). As in (Batsari & Kumam, 2020), we find the following corollary.
COROLLARY 4. Let (p s , X) be a complete partial b-metric space associated with the above order ⪯. Suppose an order-preserving quantum operation T : X → X that satisfies conditions in Theorems 1. Then, T has a fixed point.
The following example validates our main result. We examine that T : X → X satisfies all the conditions of our theorem. Now, let ρ, δ ∈ X. We show that T is order preserving with definition (25). For this, we will prove that if ρ ⪯ δ then T (ρ) ⪯ T (δ).
Taking α = 1 2 and β = 1, condition (1) in Theorem 1 is satisfied. So T has a unique fixed point in X.