Department of Mathematics and Statistics, College of Science and Technology, Hassan Usman Katsina Polytechnic, Katsina P.M.B 2052, Nigeria
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand
Center of Excellence in Theoretical and Computational Science (TaCS-CoE), SCL 802 Fixed Point Laboratory, Science Laboratory Building, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
The Institute for Fundamental Study(IF), Naresuan University, Phitsanulok 65000, Thailand
Author to whom correspondence should be addressed.
In this paper, we consider an order-preserving mapping T on a complete partial b-metric space satisfying some contractive condition. We were able to show the existence and uniqueness of the fixed point of T. In the application aspect, the fidelity of quantum states was used to establish the existence of a fixed quantum state associated to an order-preserving quantum operation. The method we presented is an alternative in showing the existence of a fixed quantum state associated to quantum operations. Our method does not capitalise on the commutativity of the quantum effects with the fixed quantum state(s) (Luders’s compatibility criteria). The Luders’s compatibility criteria in higher finite dimensional spaces is rather difficult to check for any prospective fixed quantum state. Some part of our results cover the famous contractive fixed point results of Banach, Kannan and Chatterjea.
The early research motivations in the area of fixed point theory were for solving problems in differential equations [1,2,3]. In 1883, Poincaré  established a theorem that was later proved as an equivalence to the Brouwer’s  fixed point theorem. It was in 1912 that Brouwer  published his fixed point theorem of self-continuous mappings on a closed ball, while in the same year (1912), Poincaré  published his fixed point theorem for area-preserving mappings of an annulus, see [7,8]. No doubt, Poincaré understood the early fixed point theorems and was using them as a tool in finding solutions of some differential equations see [3,4,6,9]).
On the other hand, another research motivation can be linked to the work of Picard ; he was utilising systematic application of successive approximations method for finding solutions of different differential equation problems, see . As a consequence, the famous Banach contraction principle  emerged in 1922, see . Moreover, it was the same year that boundary value problems of nonlinear ordinary differential equations prompt Birkhoff–Kellogg  to lead the struggle for extending Brouwer’s fixed point theorem to function space, see .
Another angle of fixed point research emerged with the advent of the Knaster–Tarski Fixed point theorem [12,13]. The idea was first initiated from both authors (Knaster and Tarski) in 1927 , and later Tarski found some improvement of the work in 1939, which he discussed in some public lectures between 1939 and 1942 [13,14]. Finally, in 1955, Tarski  published the comprehensive results together with some applications. A distinctive property of this theorem is that it involves an order relation defined on the space of consideration. Indeed, the order relation serve as an alternative to the continuity and contraction of the mappings as found in Brouwer  and Banach  fixed point theorems, respectively, see .
After the advent of the Brouwer , Banach  and Knaster–Tarski  fixed point theorems, many researchers engage in extension [15,16,17], generalisation [15,18,19,20,21] and improvements [22,23,24,25,26] of the theorems using different spaces and functions. Along the direction of generalising spaces was Bourbaki–Bakhtin–Cezerwik’s b-metric space [27,28,29], Matthews’s partial metric space  and Shukla’s Partial b-metric space .
Looking into the direction of quantum operations, many researchers are interested in finding the condition(s) that guarantees the existence of fixed points/states of quantum operations and the properties attached to the fixed point sets of the quantum operations, see [32,33,34,35,36].
In the area of quantum information theory, qubit is seen as a quantum system, whereas quantum operation can be viewed as measurement of quantum system; it describes the evolution of the system through the quantum states. Measurements use to 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 vital in the field of 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 .
In 1951, Lüders  discussed the compatibility of quantum states in measurements (quantum operations). He also showed that the compatibility of quantum states in measurements is equivalent to commutativity of the states with each quantum effects in the measurement.
In 1998, Busch et al.  proved a proposition that generalises the Lüder’s theorem and shows that a state is invariant under a quantum operation if the state commutes with every quantum effect that described the quantum operation.
In 2002, Arias et al.  studied the fixed point sets of a quantum operation and gave some conditions to which the set is equal to a commutant set of the quantum effects that described the quantum operation.
In 2011, Long and Zhang  studied the fixed point set of quantum operations, they gave some necessary and sufficient conditions for the existence of a non-trivial fixed point set. Similarly, in 2012, Zhang and Ji  studied the existence of a non-trivial fixed point set of a generalised quantum operation.
In 2016, Zhang and Si  investigated the conditions for which the fixed point set of a quantum operation with respect to a row contraction equals to the fixed point set of the power of the quantum operation for some
It is worth noting that the existence of fixed point(s) of a quantum operation in a finite dimensional Hilbert space depends on the compatibility criteria as provided by Lüders ; fixed quantum states must commute with all quantum effects. Therefore, it is difficult to test the compatibility criteria in higher dimensional spaces; testing commutativity of the state with many quantum effects. Thus, the need for other alternatives arises.
In this paper, motivated by Batsari et al. , Du et al.  and Dung et al. , we establish some fixed point results in partial b-metric spaces with a contraction condition that is different from that of Banach , Kannan  and Chatterjea . As an application of our result(s), we consider using some contractive conditions in establishing the existence of fixed point of a depolarising and generalised amplitude damping quantum operations. For, the depolarising quantum operation is an important source of noise/error in quantum communication that can be found in finite dimensional cases when the quantum system interact with the environment, whereas the generalised amplitude damping is used in the description of energy dissipation effects due to loss of energy from a quantum system.
Moreover, the technique we adopted in establishing the existence of fixed point of quantum operation is entirely different to that of Arias et al. , Busch et al.  and Lüders . We do not utilise the properties of quantum effects, rather we utilise the properties of the Bloch vectors associated to the quantum states in consideration. Thus, it is an alternative to the existing methods in the literature. Our results generalise and improve some existing results in the literature.
Let X be a nonempty set, denotes the set of non negative real numbers, denotes the set of real numbers, denotes the partially ordered set on X and is a metric space.
A b-metric on X is a function such that,
There exists a real number , for which
denotes the metric space. It is clear to see that, every metric is a b-metric with (see [27,28,29]).
The converse is not true in general. For example, taking , if , then is a metric with . However, it is not a metric for and , condition fails .
 Let , . Define by . Then, is a b-metric with and is not a metric.
A partial metric or on X is a function such that,
denotes the partial metric space. Observe that every metric is a partial metric with , (see ). However, the converse is not necessary true.
A mapping T is said to be order-preserving on X, whenever implies .
Let be a complete partial b-metric space with , and associated with a partial order ⪯. Suppose an order preserving mapping satisfies
for all comparable , where and . If there exist such that , then T has a unique fixed point such that
Proofof Theorem 1.
First, we will prove the uniqueness of the fixed point assuming it exists. Let be two distinct comparable fixed points of T. Then,
Thus, is a contradiction. Therefore, the fixed point is unique if it exist, for .
Next we prove that if is a fixed point of T, then . Suppose Then,
Thus contradicting the fact that . Therefore,
Now, we proceed to prove the existence of the fixed point of T satisfying (3). Let be such that . If then, is a fixed point of T. Recall that, T is order-preserving and then, we have , , , ⋯, . By transitivity of ⪯, we have .
Suppose , define a sequence by and let . It is clear that if for some natural number n, then is a fixed point of T, i.e., Let Then, we proceed as follows,
Note that, for any value of and , . Thus, the right hand side of (19) is non-negative.
Taking the limit as of both sides in the respective inequalities (9), (14) and (19), we generally conclude that
Thus, . □
Let be a complete b-metric space with , and associated with a partial order ⪯. Suppose an order-preserving mapping satisfies
for all comparable , where and . If there exist such that then, T has a unique fixed point .
Let (X,p) be a complete partial metric space associated with a partial order ⪯. Suppose an order-preserving mapping satisfies
for all comparable , where . If there exist such that , then T has a unique fixed point and
Let be a complete partial b-metric space associated with a partial order ⪯, and . Let an order-preserving mapping comply with
for comparable elements , where and . If there exist such that then, T has a unique fixed point such that,
Proofof Theorem 2.
The proof is similar to that of Theorem 1. □
We can view the difference between Theorems 1 and 2 in the positions that the terms and took in conditions (3) and (22).
Let be a complete b-metric space associated with a partial order ⪯ and . Let the order-preserving mapping comply with
for comparable elements , where and . If there exist such that then, T has a unique fixed point .
Let (X,p) be a complete partial metric space associated with a partial order ⪯. Let an order-preserving mapping comply with
for comparable elements , where . If there exist such that then, T has a unique fixed point such that,
4. Application to Quantum Operations
In quantum systems, measurements can be seen as quantum operations . Quantum operations are very important in describing quantum systems that interact with the environment.
Let be the set of bounded linear operators on the separable complex Hilbert space H; is the state space of consideration. Suppose is a collection of operators ’s satisfying . A map of the form is called a quantum operation , quantum operations can be used in quantum measurements of states. If the ’s are self adjoint then, is self-adjoint.
General quantum measurements that have more than two values are described by effect-valued measures . Denote the set of quantum effects by Consider the discrete effect-valued measures described by a sequence of satisfying where the sum converges in the strong operator topology. Therefore, the probability that outcome i occurs in the state is and the post-measurement state given that i occurs is . Furthermore, the resulting state after the execution of measurement without making any observation is given by
If the measurement does not disturb the state , then we have (fixed point equation).
Furthermore, the probability that an effect A occurs in the state given that, the measurement was performed is
If A is not disturbed by the measurement in any state we have
and by defining , we end up with .
More measurements are frequently used in quantum dynamics, quantum computation and quantum information theory [37,45,46].
Henceforth we will be dealing with a two-level single qubit quantum system. Where a quantum state can be described as
(see ). Considering the representation of a two-level quantum system by the Bloch sphere (Figure 1) above, a quantum state () can be represented with the below density matrix ,
Furthermore, the density matrix can also take below representation ,
where is the Bloch vector with , and for being the Pauli matrices.
Note that the Bloch vectors with norm less than one are associated to the mixed quantum states, whereas Bloch vectors with norm equals one are associated to the pure quantum states.
Let be two quantum states in a two level quantum system. Then, the Bures fidelity  between the quantum states and is defined as
(see ). The Bures fidelity satisfies , it is 1 if and 0 if and have an orthogonal support (perfectly distinguishable) .
Now consider a two-level quantum system X represented with the collection of density matrices . Define the function by
It is easy to show that is a b-metric on X (partial b-metric) with . Define an order relation ⪯ on X by
It is obvious that, the order relation defined above (29) is a partial order.
Let be a complete partial b-metric space associated with the above order ⪯ (29). Suppose an order-preserving quantum operation that satisfies either conditions in Theorems 1 or 2. Then, T has a fixed point.
Below example covers both Theorems 1 and 2. However, we precisely execute the solution procedure in favour of Theorem 1.
Consider the depolarising quantum operation T on the Bloch sphere X; with the depolarising parameter . Let the comparable quantum states satisfy (29).
We will check that, satisfy all the conditions of our theorem(s), as such, it has a unique fixed point.
Now, let . If the order ⪯ is as defined in (29), we will start by showing T is order-preserving. Note that, T is order-preserving if the angle of rotation describing any two comparable quantum states is invariant under T, and the distance from origin to is less than or equal to the distance from origin to , i.e., if then .
Therefore, using the Bloch sphere representation of states in a two-level quantum system below
we proceed as follows,
Clearly, the angles and are not affected by the depolarising quantum operation T. Furthermore, we can deduce that the distance of the quantum state from origin given by is greater than or equal to the distance of the new quantum state from origin given by , . Therefore, for any two comparable quantum states , with respective distances from origin and such that, , the depolarising quantum operation T produces two quantum states , with respective distances from origin and for . As , then . Thus, ; T is order-preserving.
The fidelity of any two quantum states and can take the form
(see, ), where is the inner/dot product between the vectors and . So, for any comparable quantum states and , for being the angle between and . Using Equation (30), one can show that,
; for a pure state and o the completely mixed state(origin/center).
; for a pure state that is separated from .
Thus, for , .
Furthermore, using the condition is imposed on both Theorems 1 and 2. From the known facts and definitions, we proceed as
Taking , condition (3) in Theorem 1 is satisfied. A similar procedure can be used to prove the compliance of condition (22) in Theorem 2. Finally, in reference to Theorem 1, we conclude that T has a unique fixed point (centre). A similar conclusion can be attained using Theorem 2.
Consider the quantum operation known as the generalised amplitude damping on the Bloch sphere X defined as
with damping parameter and . Let the comparable quantum states satisfies (29). Then, T has a fixed point.
In a similar way as we demonstrated in Example 4, one can show the existence of the invariant state for the generalised amplitude damping T as presented in Equation (31). The effect of the generalised amplitude damping is like a flow of states on the Bloch sphere (Unit ball) towards the fixed state . The generalised amplitude damping can be used in description of energy dissipation effects due to loss of energy from a quantum system. Note that, the invariant state is unique for every
The results in this paper cover some part of the famous contractive fixed point results of Banach , Kannan  and Chatterjea . The contractive conditions (3) and (22) presented can be seen as an improvement to the work of Batsari et al. , Du et al.  and Dung et al. ; as the conditions contain both maximum and minimum functions. Moreover, our results are generalisations of many other existing results in terms of the space in consideration (partial-b metric space).
On the other hand, although the fidelity function is not a metric, we have shown how it can be utilised in studying fixed points of some quantum operations. Moreover, the existence of fixed points of some quantum operations can be studied without given much attention to the quantum effects as seen from Examples 4 and 5. Thus, the criteria and procedure we presented can serve as an alternative in guaranteeing the existence and finding the fixed points of some quantum operations respectively if compared with the existing ones provided by Lüders  and Busch et al. . Our choice for using depolarising and generalised amplitude damping quantum operations was related to their importance as source of quantum error and in description of energy dissipation effect respectively.
Conceptualization, U.B.Y., P.K. and S.Y.-K.; methodology, U.B.Y., P.K. and S.Y.-K.; formal analysis, U.B.Y., P.K. and S.Y.-K.; investigation, U.B.Y., P.K. and S.Y.-K.; writing—original draft preparation, U.B.Y.; writing—review and editing, U.B.Y. and S.Y.-K.; supervision, P.K.; funding acquisition, P.K. All authors have read and agreed to the published version of the manuscript.
Umar Batsari Yusuf was supported by the Petchra Pra Jom Klao Ph. D. Research Scholarship from King Mongkut’s University of Technology Thonburi with grant number (Grant No. 55/2559).
The authors acknowledge the financial support provided by King Mongkut’s University of Technology Thonburi, through the “KMUTT 55th Anniversary Commemorative Fund”. The authors also acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT.
Conflicts of Interest
The authors declare no conflicts of interest.
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