Fixed point theorem in a partial b-metric space applied to quantum operations

Теорема о фиксированной точке в частичном b-метрическом пространстве с применением в квантовых операциях

Теорема фиксне тачке у делимичном b-метричком простору примењена на квантне операције

Rakesh Tiwaria rtiwari@govtsciencecollegedurg.ac.in

Government V. Y. T. Post-Graduate Autonomous College, India

Mohammad Saeed Khanb drsaeed9@gmail.com

Sefako Makgatho Health Sciences University, South Africa

Shoba Ranic shobharaniy89@gmail.com

Government V. Y. T. Post-Graduate Autonomous College, India

Nicola Fabianod nicola.fabiano@gmail.com

University of Belgrade, “Vinča” Institute of Nuclear Sciences - National Institute of the Republic of Serbia, Serbia

This work is licensed under Creative Commons Attribution 4.0 International.

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 and Tarski 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, 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).

After the approach of the Brouwer (Brouwer, 1911), Banach (Banach, 1922) and Knaster-Tarski (Tarski, 1955) fixed point theorems, many researchers become involved in extension (Browder, 1959; Leray & Schauder, 1934; Schauder, 1930), generalization (Batsari et al, 2018; Browder, 1959; Du et al, 2018) and improvements (Batsari et al, 2018; Batsari & Kumam, 2018; Kannan, 1972; Khan et al, 1984) of the theorems using different spaces and functions. In the way of generalizing spaces was Bourbaki-Bakhtin-Cezerwik’s b-metric space (Bakhtin, 1989; Bourbaki, 1974; Czerwik, 1993), Matthews’s partial metric space (Matthews, 1994) and Shukla’s Partial b-metric space (Shukla, 2014).

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 with respect to a row contraction A equals to the fixed point set of the power of the quantum operation 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 such that,

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

(2)

A partial b-metric space (X, p_s) is complete, if every Cauchy sequence {x_n} in (X, p_s) converges to a point x ∈ X such that,

(3)

DEFINITION 2. A mapping T is said to be order-preserving on X, whenever x ⪯ y implies T(x) ⪯ T(y) for all x, y ∈ X.

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

(4)

for all comparable x, y ∈ X, where α, β ∈ [0, θ] and If there exists x₀ ∈ X such that x₀ ⪯ T(x₀), then T has a unique fixed point ∈ X such that p_s(, ) = 0.

Proof. Suppose x₀T(x₀), define a sequence {x_n} ⊆ X b_y x_n = Tⁿ(x₀) and let q_n = p_s(x_n, x_n+1). It is clear that if x_n = x_n+1 for some natural number n, then x_n is a fixed point of T, i.e., x_n+1 = T(x_n) = x_n.

Let x_n+1x_n for all n ∈ N. Then, we proceed as follows:

Thus, we have

which implies

(5)

By simplifying (5), we have

(6)

For , we deduce that

Therefore, from (6), we conclude that p_s(x_n, x_n+1) = q_n ≤ q_n−1 = p_s(x_n−1, x_n). Thus, 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).

Next, we show is Cauchy. Let x_n, x_m ∈ X, for all n, m∈.

(7)

implies that

(8)

Now, taking the limit as n, m → +∞ in (7), we have

Therefore, {x_n} is a Cauchy sequence in X. For X being complete, there exists ∈ X such that

Now, we proceed to prove the existence of the fixed point of T satisfying (1). Let x₀ ∈ X be such that x₀ ⪯ T(x₀). If T(x₀) = x₀ then, x₀ is a fixed point of T. Recall that, T is order-preserving and x₀ ⪯ T(x₀) then, we have x₀ ⪯ T(x₀) = x₁, x₁ ⪯ T(x₁) = x₂, x₂ ⪯ T(x₂) = x₃, · · · , x_n ⪯ T(x_n) = x_n+1. By transitivity of ⪯, we have x₀ ⪯ x₁ ⪯ x₂ ⪯ x₃ ⪯ · · · ⪯ x_n ⪯ x_n+1 ⪯ · · · .

For showing ∈ X is a fixed point of T, we proceed as follows:

(9)

Case I: Suppose max{p_s(x_n, ), p_s(x_n, T()), p_s(, T(x_n))} = p_s(x_n, ). Then, from inequality (9), we have

From the above inequality, we have

which implies

(10)

We can observe that for ,

(11)

If , then, from equality (11) we have

(12)

Similarly, if , then, from equality (11),

(13)

From equalities (12) and (13), we conclude that the right-hand side of (10) is non-negative.

Case II: Suppose max{p_s(x_n, ), p_s(x_n, T()), p_s(, T(x_n))} = p_s(x_n, T()). Then, from inequality (9), we have

from the above inequality, we have

so that

(14)

from the fact that , we have if α > β then by (14), we have

(15)

If by (15), we have

(16)

If by (15), we have

(17)

From inequalities (16) and (17), we conclude that the right-hand side of (10) is non-negative.

If α < β, then by (14), we have

(18)

Similarly for (18), we conclude that the right-hand side of (10) is non-negative.

Case III: Suppose max{ps(x_n, ), p_s(x_n, T()), p_s(, T(x_n))} = p_s(, T(x_n))). Then, from inequality (9), we have

By the simplification of the above equality, we have

(19)

Note that, for any value of α, β ∈ [0, θ) and 4 − s²β − 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() = .

Next, we prove that if ∈ X is a fixed point of T, then p_s(, ) = 0.

Suppose p_s(, ) 0. Then

This is contradicting the fact that p_s(, ) 0. Therefore, p_s(, ) = 0.

Last, we will prove the uniqueness of the fixed point. Let x₁, x₂ ∈ X be two distinct fixed points of T. Then

This is a contradiction. Therefore, the fixed point is unique.

REMARK 2. If we take and p_s(x, T(y)) + p_s(y, T(x)) ≥ p_s(x, T(x)) + p_s(y, T(y)) then we find Theorem 1 of Batsari et al. (Batsari & Kumam, 2020).

COROLLARY 3. Let (X, p) be a complete partial metric space associated with a partial order ⪯. Suppose an order-preserving mapping T : X → X satisfies

(20)

for all comparable x, y ∈ X, where θ ∈ [0, 1]. If there exists x₀ ∈ X such that x₀ ⪯ T(x₀), then T has a unique fixed point ∈ X and p(, ) = 0.

Now we apply our main result similar to (Batsari & Kumam, 2020) as follows:

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 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 satisfying . A map of the form 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, 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 . 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 (Arias et al, 2002). Furthermore, the resulting state after the implementation of measurement without making any consideration is given by

(21)

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

(22)

If A is not interrupted by the measurement in any state we have

and by defining , we end up with ϕ(A) = A.

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

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),

(23)

Also, the density (ρ) matrix is,

(24)

where is the Bloch vector with , and σ = [σ_x, σ_y, σ_z] where

Figure 1
Bloch sphere

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 → as follows:

It is easy to show that ps is a b-metric on X with s taking the value 1 approximately. They also define an order relation ⪯ on X by

(25)

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.

EXAMPLE 0.1. Consider the depolarizing quantum operation T on the Bloch sphere X; with the depolarizing parameter p ∈ [0, 1]. Let the comparable quantum states satisfy (25).

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).

Therefore, as (Batsari & Kumam, 2020) using the Bloch sphere representation of states in a bi-level quantum system below

So

Clearly, the angles θ and ϕ do not change by the depolarizing quantum operation T. Also, we can deduce that the distance of the quantum state ρ from the origin given by µ is greater than or equal to the distance of the new quantum state T(ρ) from the origin given by (1−p)µ, p ∈ [0, 1]. Consequently, for any two quantum states which are comparable ρ, δ ∈ X(ρ ⪯ δ), with respective distances from the origin µ_ρ and µ_δ such that, µ_ρ ≤ µ_δ, the depolarizing quantum operation T constructs two quantum states T(ρ), T(δ) ∈ X, have distances (1 − p)µ_ρ and (1 − p)µ_δ from the origin for p ∈ [0, 1] respectively. Since µ_ρ ≤ µ_δ, then (1 − p)µ_ρ ≤ (1 − p)µ_δ, for all p ∈ [0, 1]. Thus, T(ρ) ⪯ T(δ), which proves T is order-preserving.

The fidelity of any two quantum states and is,

(26)

see (Batsari & Kumam, 2020; Chen et al, 2002), where is the inner dot product between the vectors and . So, for any comparable quantum states and for ϑ being the angle between and. Using Equation (26), we have,

see (Davies, 1976; Göhde, 1965). Thus, for ρ, δ ∈ X. Now, using s = 1 and θ ∈ [0, 1] on Theorems 1. We have

Taking and β = 1, condition (1) in Theorem 1 is satisfied. So T has a unique fixed point in X.

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