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ОРИГИНАЛНИ НАУЧНИ РАДОВИ ОРИГИНАЛЬНЫЕ НАУЧНЫЕ СТАТЬИ ORIGINAL SCIENTIFIC PAPERS
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.
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FIXED POINT THEOREM IN A PARTIAL £ b-METRIC SPACE APPLIED TO f
QUANTUM OPERATIONS !
Rakesh Tiwaria, Mohammad Saeed Khanb Shoba Ranic, Nicola Fabianod
CT
Government V. Y. T. Post-Graduate Autonomous College, o
Department of Mathematics, Durg, Chhattisgarh, Republic of India e-mail: rtiwari@govtsciencecollegedurg.ac.in, q.
ORCID iD: ©https://orcid.org/0000-0002-6112-0116 ro
b ^
b Sefako Makgatho Health Sciences University, Department of Mathematics and Applied mathematics, %
Ga-Rankuwa, Republic of South Africa,
e-mail: drsaeed9@gmail.com, a
ORCID iD: ©https://orcid.org/0000-0003-0216-241X c Government V. Y. T. Post-Graduate Autonomous College, ro
Department of Mathematics, Durg, Chhattisgarh, Republic of India, e-mail: shobharaniy89@gmail.com, ORCID iD: ©https://orcid.org/0000-0001-9143-5410 d University of Belgrade, "Vinca" Institute of Nuclear Sciences - National Institute of the Republic of Serbia, Belgrade, Republic of Serbia, o
e-mail: nicola.fabiano@gmail.com, corresponding author, ORCID iD: ©https://orcid.org/0000-0003-1645-2071
DOI: 10.5937/vojtehg70-37109;https://doi.org/10.5937/vojtehg70-37109 -o FIELD: Mathematics *
ARTICLE TYPE: Original scientific paper 10
Abstract.
E
° Key words: partial b-metric space, order-preserving mapping, quantum
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operation, fidelity of quantum state, Bloch vector.
o 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 yy Moutawakil, 2002). Another angle of fixed point research emerged with 3 the approach of the Knaster-Tarski fixed point theorem (Knaster, 1928; o 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 o 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 w theorems, respectively, see (Tarski, 1955).
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After the approach of the Brouwer (Brouwer, 1911), Banach (Banach, 1922) and Knaster-Tarski (Tarski, 1955) fixed point theorems, many w researchers become involved in extension (Browder, 1959; Leray & o Schauder, 1934; Schauder, 1930), generalization (Batsari et al, 2018; o 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; Bour-baki, 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 (Luders, 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 (j) 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
ps: X x X ^ R+ such that,
(1) For all x,y e X, x = y iff ps(x,x) = Ps(x,y) = Ps(y,y)
(2) For all x,y e X, ps(x, x) < ps(x, y)
(3) For all x,y e X, ps(x, y) = Ps(y, x)
(4) There exists a real number s > 1 such that, for all x,y,z e X, ps(x, z) < s[Ps(x,y) + ps (y, z)] - ps (y,y).
(X,ps) denotes the partial b-metric space. Note that every partial metric is
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ps(x, x) = 0, for all x,y € X.
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A sequence {xn} in the space (X,ps) converges with respect to the topology Tb to a point x € X, if and only if
lim ps(xn,x)= Ps(x,x). (1)
n—
^ The sequence {xn} is Cauchy in (X,ps) if the below limit exists and is
o finite
lim ps(xn,xm) < (2)
x A partial b-metric space (X,ps) is complete, if every Cauchy sequence
uü {xn} in (X,ps) converges to a point x e X such that,
^
< lim ps(xn ,xm)= ps(x,x). (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 e X.
^ Main result
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The objective of this work is to establish a fixed point theorem in a com-w plete partial b-metric space.
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Theorem 1. Let (X, ps) 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
ps(T(x),T(y)) < amax{ps(x,y),ps(x,T(y)),ps(y,T(x))}
0
+2 min{ps(x, T(y)) + ps(y, T(x)),ps(x, T(x)) + ps(y, T(y))} (4)
for all comparable x,y e X, where a,0 e [0,0] and d = min{, s+i}. If there exists x0 e X such that x0 ^ T(x0), then T has a unique fixed point x e X such that ps(x, x) = 0.
Proof. Suppose x0 = T(x0), define a sequence {xn} c X by xn = Tn(x0) and let qn = ps(xn,xn+i). It is clear that if xn = xn+i for some natural
number n, then xn is a fixed point of T, i.e., xn+1 = T(xn) = xn. Let xn+i = xn for all n e N. Then, we proceed as follows:
Qn = Ps(xn,xn+l) = Ps(T (xn-l),T (xn))
< a max{ps(xn-i,xn),Ps(xn-1, T(xn)),Ps(xn, T(xn-1))}
p
+2 min{ps(xn-1, T(xn)) + Ps (x n, T(x l)),Ps(x n— 1 , T (xn-1)) + Ps (xn,T (xn))} = a max{Ps (xn- 1,xn),Ps (xn-1, xn+1), Ps (xn, xn)}
p
+ 2 min{Ps(xn-1,xn+1) + Ps(xn,xn),Ps(xn-1,xn) + Ps(xn,xn+1)}
= a max{Ps(xn-1,xn),s[Ps(xn-1,xn) + Ps(xn ,xn+1)]} p r s[Ps(xn-1,xn) + Ps(xn,xn+1)] + Ps(xn-1,xn) + Ps (x„,x„+1) ]
+f[
2
= a(s[Ps(xn-1,xn) + Ps(xn,xn+1)]) +
P / \ rPs(xn- 1, xn) + Ps(xn,xn+1 ) ]
2(s +2 J
p(s + 1) , , , , ,,
= as +--;-(Ps(xn-1,xn )+ Ps(xn,xn+1))
4
4as + Ps + p
4
(Ps(xn-1,xn) + Ps(xn,xn+1))
4as + ps + p
4 (
j-1 + Qn).
Thus, we have
4as + ps + p Qn < --(Qn-1 +
which implies
.4 - 4as - ps - p. .4as + ps + p.
(-4-)Qn < (-4-)Qn-1
(5)
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By simplifying (5), we have
( 4as + ps + p .
Qn < (4-4a—s-p)Qn-1
(6)
For d e min{ , s+r}, we deduce that
4as + ps + p
0 <-—< 1.
4 - 4as - ps - p
529
° Therefore, from(6), we conclude that ps(xn,xn+i) = qn < qn—l =
0 ps(xn—l,xn). Thus, {qn}'+=^l is a monotone non-increasing sequence of real numbers and bounded below by 0. Therefore, qn = 0, see
Chidume et al.(Chidume & Chidume, 2014).
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Next, we show {xn}+=°l is Cauchy. Let xn, xm e X, for all n,m e N. Then,
Ps(xn,xm)= Ps(Tn xo,Tm xo) = Ps(T (xn-l),T (xm-i)) ^t — a max{ps(xn—l) xm—l) ,ps(xn—li xm) ,ps (xn,xm—1)}
P
+ 2 min ps (xn— l, xm) + ps (xm-1, xn), ps (xn-1, xn) + ps (xm-1, xm)
>_ = a max{s(ps(xn—l,xn) + ps(x n) xm— l)),ps(x n—l) xm), ps (xn, xm— l)}
a:
pi , P <ps (x n— l xm ) + ps (x m— 1, xn ) + ps(x n— l xn ) + ps(xm— l,xm)-,
2{ 2 } = a max{s(ps(xn—l,xn) + s(ps(xn,xm) + ps(xm,xm—l))),ps(xn—l,xn)
P
< +ps(x m l , xm
)} + 4(s(ps (xn—l,xn) + ps(xn,xm)) + s(ps(xm— l,xm)
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+ps(xm,xn)) + ps(x n— l , xn ) + ps(xm—l,xm) — as(ps (xn—l, xn) + s(ps(xni xm) + ps(xm) xm—l))
m P
q +4 (sps(xn— l,xn) + 2sps(x n, xm) + sps (xm— l, xm)
Z 4
O +ps(xn— l,xn) + p.s(x m— l i xm))
fy < (as + f + 4)p.(xn— l,xn) + (as2 + y )p.(xn,xm)
+ (as2 + ^P + 4)p.(xm—l,xm), (7)
implies that
(1 - (as2 + y ))p.(xn, xm) < (as + + 4)p.(xn— l, xn) +
(as2 + + 4)p.(xm—l,xm)
2 sP P
< 2 - 2as2 - sP((as + T + 4)ps(xn—l'xn)
+ (as2 + + 4)p.(xm— l,xm)). (8)
530
Now, taking the limit as n,m ^ in (7), we have
lim Ps(xn,xm)=0.
Ps(x, T (x)) < s[Ps(x,xn+1) + Ps (xn+1, T (x)] - Ps(xn+1,xn+1)
< s[Ps(x,xn+1) + Ps(T(xn),T(x)]
< s[Ps(x,xn+1) + amax{Ps(xn,x),Ps(xn,T(x)),Ps(x,T(xn))}
p
+ 2 min{Ps(xn,T (x)) + Ps(x,T (xn )),Ps(xn, T (xn)) + Ps(x,T (x))}]
< s[Ps(x,xn+1) + amax{Ps(xn,x),Ps(xn,T(x)),Ps(x,T(xn))}
+4(Ps(xn, T(x)) + Ps(x, T(xn)) + Ps(xn, T(xn)) + Ps(x, T(x)))]. (9)
Case I: Suppose max{Ps(xn,x),Ps(xn,T(x)),Ps(x,T(xn))} = Ps(xn,x). Then, from inequality (9), we have
Ps(x, T(x)) < s [Ps(x, xn+1) + aPs(xn, x) +p(Ps(xn,T (x)) + Ps(x,T (xn)) + Ps(xn, T (xn)) + Ps(x,T (x)))] 2 ^ 2 ^
sp
< sPs(x,xn+1) + sa'Ps(xn, x) + — (s(Ps(xn,x) + Ps(x, T(x)))
+Ps(x, xn+1) + Ps(xn, xn+1) + Ps(x, T(x)))
= (s + "4")Ps(x, xn+1) + (sa + ^)Ps(xn, x)
.s2p sp. _ rT1,„, sp . .
+ (— + ~4~)Ps(x,T (x) + — Ps(xn,xn+1).
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Therefore, {xn} is a Cauchy sequence in X. For X being complete, there exists x e X such that
lim Ps(xn,x)= lim Ps(xn,xm)= Ps(x,x) = 0.
n—^+^0 n,m—+xi
Now, we proceed to prove the existence of the fixed point of T satisfying (1). Let x0 e X be such that x0 ^ T(x0). If T(x0) = x0 then, x0 is a o 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, •••, xn ^ T(xn) = xn+1. By transitivity of we have x0 ^ x1 ^ x2 ^ x3 ^ • • • ^ xn xn+1 <••• .
For showing x e X is a fixed point of T, we proceed as follows:
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(1 - s4~ - sP)ps(x,T(x)) < (s + ^)ps (x,xn+l) +
£ s2P sP
^ (sa + —-)p.(xn ,x) + —ps (xn,xn+l),
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which implies
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O ps(x, T (x) < 4 - s2^ - sP [(s + S~J )ps (x, xn+l) +
< . s2P^ , sP
(sa + —)ps(xn,x) + — ps(xn,xn+l)]. (10)
l2
m We can observe that for P e min{ , s+y},
* 4 - s2P - sP = 4 - s2P - sP
= 4 - sP(s + 1). (11)
If P = sl, then, from equality (11) we have
u 4 - s P - sP = 4 - sP (s + 1)
=4 - s(s + 1)-
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s+1
s2
o >0 for all s > 1. (12)
Similarly, if P = +, then, from equality (11),
4 - s2P - sP = 4 - sP(s + 1)
2
= 4 - (s + 1)s
s+1
<4 - s(s + 1)^ s3
> 0 for all s > 1. (13)
From equalities (12) and (13), we conclude that the right-hand side of (10) is non-negative.
Case II: Suppose max{ps(xn,x),ps(xn ,T (x)),ps(x,T (s„))} ps(xn,T(x)). Then, from inequality (9), we have
Ps(x, T(x)) < s [ps(x, xn+i) + aps(xn, T(x)) + P (ps(xn,T(x)) + ps(x,T(xn)) + ps(xn,T(xn)) + ps(x,T(x)) + 2^ 2 ^
< s[ps(x,xn+i) + as(ps(xn,x) + ps(x,T(x)))
P,
+ ^(s(ps(xn,x) + ps(x,T (x))) + ps(x,xn+i) + ps(xn,xn+i) + ps(x,T (x)))]
P P
< (s + -s)ps(x, xn+i) + (s2a + -s2)ps(xn, x)
4
2P2P
+(s2a + -s2 + -s)ps(x, T(x)) + ysps(xn, xn+i),
44
from the above inequality, we have
4
4
p p t-
(1 - s 2a - -s 2 - -s)ps(x, T(x)) < (s + -s)ps(x, xn+i)
P
4
4
4
+(s2a + 4 s2)ps(xn,x) + 4 sps(xn,xn+i),
4
so that
ps(x,T(x)) <
4 - 4s2a - Ps2 - Ps [(s + 4s)ps(x' xn+i)
p
p
+(s2a + -s2)ps(xn, x) + -sps(xn, xn+i)]
(14)
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from the fact that d e min{ i, 5+ }, we have if a> P then by (14), we have
4 - 4s2a - Ps2 - ps = 4 - 4s2p + Ps2 + ps = 4 - (5s + 1)sp.
(15)
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If P = -3 by (15), we have
4 - 4s2a - Ps2 - Ps = 4 - (5s + 1)sP
= 4 - (5s + 1)s^
> 0 for all s> 1.
(16)
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P If P = 5+r by (15), we have
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4 - 4s2a - Ps2 - Ps = 4 - (5s + 1)sP
2
= 4 - (5s + 1)s-
a < 4 - (5s + 1)s^r
> 0 for all s > 1. (17)
s+1 1
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g From inequalities (16) and (17), we conclude that the right-hand side of
^ (10) is non-negative.
If a < P, then by (14), we have
X
w 4 - 4s2a - Ps2 - Ps = 4 - 4s2a + as2 + Ps
=4 - 5s2a + sa
* =4 - (5s + 1)sa. (18)
Similarly for (18), we conclude that the right-hand side of (10) is non-negative.
Case III: Suppose max{p.(xn,x),p.(xn ,T (x)),p.(x,T (xn))} =
y ps(x,T(xn))). Then, from inequality (9), we have
p.(x,T(x)) < sps(x,xn+l) + ap.(x, T(xn)))
P
o +7(ps(xn, T(x)) + p.(x, T(xn)) + p.(xn, T(xn)) + ps(x, T(x)))]
> 4
< sps(x,xn+l) + asp.(x, xn+l)
P
+4(s(p.(xn,x) + p.(x,T (x))) + p.(x,xn+l) + ps(xn,xn+l) + p.(x,T (x)))]
4'
< (s + as + -s)ps(x,xn+l) ^s p.(xn,x)
+(4 s2 + 4 s)p.(x,T (x)) + 4 sps(xn,xn+l).
By the simplification of the above equality, we have
4 P
p.(x, T(x) < 4 - s2P - sP [(s + as + 4s)p.(x, xn+l)
+4 s2 p.(xn,x) + 4 sps(xn,xn+l)]. (19)
534
Note that, for any value of a, 3 G [0,0) and 4 - s23 - s( > 0. Thus, the ^
2
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right-hand side of (10) is non-negative. Taking the limit as n ^ of both 1 sides in the respective inequalities (10), (14) and (19), we conclude that
=0.
Thus, T(x) = x.
Next, we prove that if x e X is a fixed point of T, then ps(x, x) = 0. Suppose ps(x, x) = 0. Then
ps(x, x) = ps(T(x,x)) < amax{ps(x, x),ps(x, x)),ps(x, x))}
/3
+2 min{ps(x, x)) + ps(x,T(x)),ps(x,x)) + ps(x,T(x))}
= (a + 3)ps(x,x) = 0ps(x,x)
< Ps(X, X).
This is contradicting the fact that ps(X, X) = 0. Therefore, ps(X, X) = 0.
Last, we will prove the uniqueness of the fixed point. Let xi, x2 g X be two distinct fixed points of T. Then
Ps(Xl,X2) = Ps(T (Xi),T (X2 )) < amax{ps(xi,X2),Ps(xi,T(x2)),Ps(x2,T(xi))}
/3
+2 min{Ps(xi,T(X2)) + Ps(X2,T(xi)),Ps(xi,T(xi)) + Ps(X2,T(X2))}
= a max{Ps(xi,X2),Ps(xi, X2),Ps(xi, X2)}
/3
+ 2 min{Ps(xi, X2) + Ps(Xi,X2),Ps(Xi,X2) + Ps(Xi,X2)} = (a + 3)Ps(xi,X2) = 0'Ps (xi, X2) < Ps(Xi,X2).
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° This is a contradiction. Therefore, the fixed point is unique. □
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Remark 2. If we take a = f and ps(x, T(y)) + ps(y, T(x)) > ps(x, T(x)) + ps(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 of with a partial order < Suppose an order-preserving mapping T : X ^ X □E satisfies
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Ps(T(x),T(y)) < amax{ps(x,y),Ps(x,T(y)),ps(y,T(x))}
0
+2 min ps(x, T (y)) + ps (y, T (x)),ps(x, T (x)) + ps(y, T (y)) (20)
for all comparable x,y e X, where 9 e [0,1]. If there exists x0 e X such
£ that x0 ^ T(xo), then T has a unique fixed point X e X and p(X, X) = 0.
<c
Now we apply our main result similar to (Batsari & Kumam, 2020) as follows:
3 Application to quantum operations
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g In quantum systems, measurements can be seen as quantum op-
erations (Seevinck, 2003). Quantum operations are very important in dd narrating quantum systems that collaborate with the environment.
o
o Let B(H) be the set of bounded linear operators on the separable com-
plex Hilbert space H; B(H) is the state space of consideration. Suppose A = {Ai, A* : i = 1,2,3,...} is a collection of operators Ai s e B(H) satisfying ^ AiA* < I .A map 0 : B(H) ^ B(H) of the form 0A(B) = £ AiBA* is called a quantum operation (Arias et al, 2002), quantum operations can be used in quantum measurements of states. If the Ai'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 e(H) = {A e B(H) : 0 < A < I}. Consider the discrete effect-valued measures narrated by a sequence of Ei e e(H),i = 1,2,... satisfying Ei = I where the sum converges in the strong operator topology. Therefore, the probability that outcome i eventuates in the state p is p(Ei)
E 2 pE 2
and the post-measurement state given that i eventuates is ¡rpEi (Arias et al, 2002). Furthermore, the resulting state after the implementation of measurement without making any consideration is given by
$(p) = £ Ei pEi
(21)
If the measurement does not disturb the state p, then we have 0(p) = p. Furthermore, the probability that an effect A eventuates in the state r given that the measurement was conducted is
P>(p)(A) = t^^Ei pEi] = tr(j2Ei pEi p) (22)
If A is not interrupted by the measurement in any state we have
EEi pEi = A
and by defining 0(A) = ^ E2 pE2, we end up with 0(A) = A.
From now, we will be dealing with a bi-level (|0), |1)) single qubit quantum system where a quantum state can be narrated as
= a|0) + bl1), with a, b e 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 (|^)) can be represented with the density matrix below (p),
l*> = P =Ö
1 + n cos 0 n cos $ sin 0 — in sin $ sin 0^
n cos $ sin 0 + in sin $ sin 0 1 — n cos 0
n € [0,1], 0 < 0 < n, and 0 < $ < 2n.
(23)
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Also, the density (p) matrix is,
1rr__, 1
p = 2[I + rPa] = 2
1+ rz
rx + iry
1 rz
(24)
where rp = [rx, ry, rz] is the Bloch vector with \\rp\\ < 1, and a = [ax, ay, az]
where
ax
(0 0) a=(° 0)>az=(0 —!,)
537
_ r> iy>
x ir y
It is easy to show that the order relation defined above is a partial order
(Batsari & Kumam, 2020). 1
As in (Batsari & Kumam, 2020), we find the following corollary.
Corollary 4. Let (ps,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 Si
o
point.
The following example validates our main result.
Example 0.1. Consider the depolarizing quantum operation T on the Bloch sphere X; T(p) = 2p + (1 - p)p with the depolarizing parameter p e [0,1]. ik. Let the comparable quantum states satisfy (25).
We examine that T : X ^ X satisfies all the conditions of our theorem. Now, let p,5 e X. We show that T is order preserving with definition (25). For this, we will prove that if p ^ 5 then T (p) ^ T (5).
Therefore, as (Batsari & Kumam, 2020) using the Bloch sphere representation of states in a bi-level quantum system below
1 / 1+ f cos 6 f cos $ sin 6 — if sin $ sin 6
2 yf cos $ sin 6 + if sin $ sin 6 1 — f cos 6
f £ [0,1], 0 < 6 < n, and 0 < $ < 2n,
)
So,
:)+
1 — pf 1 + f cos 6 f cos $ sin 6 — if sin $ sin 6
2 ¡ cos $ sin 6 + i¡ sin $ sin 6 1 — ¡ cos 6
)
1 (p ^ + 1+ f cos 6 f cos $ sin 6 — if sin $ sin 6\
2 y0 pj 2 \f cos $ sin 6 + if sin $ sin 6 1 — f cos 6 J
1 / p + pf cos 6 pf cos $ sin 6 — ipf sin $ sin 6\
2 \pf cos $ sin 6 + ipf sin $ sin 6 p — pf cos 6 J
_ 1i 1+(1 — p)^ cos 9 (1-p)[/U cos 0 sin 9—i^ sin 0 sin 9] \
2\ (1— p)[p cos 0 sin 9+i^ sin 0 sin 9] 1—(1—p)jU cos 9. J
<M LT) £± Œ
co c o
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p
° Clearly, the angles d and 0 do not change by the depolarizing quantum
0 operation T. Also, we can deduce that the distance of the quantum state p from the origin given by p is greater than or equal to the distance of the new quantum state T(p) from the origin given by (1 -p)p, p e [0,1]. Consequently, for any two quantum states which are comparable p,5 e X(p < 5), with respective distances from the origin pp and fi¿ such that, pp < fi¿, the depolarizing quantum operation T constructs two quantum states ^ T(p),T(5) e X, have distances (1 - p)pp and (1 - p)p¿ from the origin g for p e [0,1] respectively. Since pp < ¡j,s, then (1 - p)pp < (1 - p)p¿,
° for all p e [0,1]. Thus, T(p) ^ T(5), which proves T is order-preserving.
<
o
o >
CN (M o (M
The fidelity of any two quantum states p = 2(I2 + rp ■ a) and 5 = 1 (I2 +
n ■ a) is,
x
o
LU
I— _ _
| F(p,5) = 2[l + rp ■ n + - \\rp\\y 1 -||rp||2] (26)
see (Batsari & Kumam, 2020; Chen et al, 2002), where rp ■rs is the inner dot product between the vectors rp and n■ So, for any comparable quantum ^ states p = 1 (I2 + rp ■a) and 5 = 1 (I2 + n ■ a), rp ■ rs = \\rp\\\rS\\ cos § for § being the angle between rp and rs■ Using Equation (26), we have, (i). F(p,p) = 1.
0 (ii) F(p, o) = 1; for p a pure state and o the completely mixed state.
1 (iii) F(p, p-) = 0; for p a pure state that is 180° separated from p, £ see (Davies, 1976; Gohde, 1965). Thus, 1.000 < e6(1-F(p>s)) < 1.181 for
p,5 e X. Now, using s = 1 and d e [0,1] on Theorems 1. We have
_ ..................11 11^11 ie 5 (1-F (T(p),Tm
Ps(T(p),T(5))=maX{|M, \\}e5 1 4
= _, ^ Me 5 (.i-F(.np),Tm
< 4(\\5\\e 1 (1-F(t(p)'S)) + \\p\\e5(1-F(T(P)'P))}
= 1 ( 1(Ps(T (p),5) + Ps(T (p),p))) = 1 (2 max{ps(p, ô),ps(p, T(S)),Ps(T(p),S)}
+ . ps(p,T(5))+ Ps(T(p),5) Ps(p,T(p)) + ps(5,T(5)) + min{ 2 ' 2 }'
Taking a = 2 and 3 = 1, condition (1) in Theorem 1 is satisfied. So T has a unique fixed point in X.
540
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ТЕОРЕМА О ФИКСИРОВАННОЙ ТОЧКЕ В ЧАСТИЧНОМ Ь-МЕТРИЧЕСКОМ ПРОСТРАНСТВЕ С ПРИМЕНЕНИЕМ В КВАНТОВЫХ ОПЕРАЦИЯХ
Ракеш Тивариа, Мохаммад Саид Кханб, Шоба Раниа, Никола Фабианов
а Государственный автономный колледж последепипломного образования VYT , кафедра математики, г. Дург, Чхаттисгарх, Республика Индия б Университет медицинских наук Сефако Макгато, кафедра математики и прикладной математики, г Га-Ранкува, Южно-Африканская Республика
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в Белградский университет, Институт ядерных наук Винча -Национальный институт Республики Сербия, г. Белград, Республика Сербия, корреспондент
РУБРИКА ГРНТИ: 27.35.33 Математические модели
электродинамики и оптики 27.35.57 Математические модели квантовой физики
27.39.27 Нелинейный функциональный анализ
ВИД СТАТЬИ: оригинальная научная статья Резюме:
Введение/цель: Сконструирована теорема о неподвижной точке с сохранением порядка в полном и частичном Ь-метрическом пространстве при выполнении условий сжатия.
Методы: В данной статье применен метод расширения результатов Батсари и др.
Результаты: Точность квантового состояния используется для построения неподвижного квантового состояния.
Выводы: Неподвижное квантовое состояние связано с квантовой операцией, сохраняющей порядок.
Ключевые слова: частичное Ь-метрическое пространство, отображение с сохранением порядка, квантовая операция, точность квантового состояния, вектор Блоха.
ТЕОРЕМА ФИКСНЕ ТАЧКЕ У ДЕЛИМИЧНОМ Ь-МЕТРИЧКОМ ПРОСТОРУ ПРИМЕНЕНА НА КВАНТНЕ ОПЕРАЦШЕ
Ракеш Тивариа, Мохамед Саид Канб, Шоба Раниа, Никола Фабианог
а Државни колец за последипломске студне В.и.Т, Департман за математику, Дург, Чатисгар, Република Инди]а
б Универзитет здравствених наука Сефако Макгато, Департман за математику и приме^ену математику, Га-Ранкува, иужноафричка Република в Универзитет у Београду, Институт за нуклеарне науке "Винча" -Национални институт Републике Срби]е, Београд, Република Срби]а, ауторза преписку
ВРСТА ЧЛАНКА: оригинални научни рад Сажетак:
ОБЛАСТ: математика ^
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Увод/цил: Конструисана ¡е теорема фиксне тачке мапи-раъа с очуваъем редоследа на комплетном парци]алном Ь-метричком простору уз задовоъаваъе контрактивног услова.
Методе: Применен метод проширен¡е резултатима Бат-сари]а и других.
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Резултати: Верност квантног ста^а користи се за кон-струиса^е фиксног квантног ста^а.
Закъучак: Фиксно квантно ста^е повезано jе са квантном операциям ща чува редослед. ^
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Къучне речи: парц^ални b-метрички простор, мапира^е о с очува^ем редоследа, квантна операц^а, верност квантног ста^а, Блохов вектор.
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Paper received on / Дата получения работы / Датум приема чланка: 22.03.2022. Manuscript corrections submitted on / Дата получения исправленной версии работы / Датум достав^а^а исправки рукописа: 21.06.2022. §
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Paper accepted for publishing on / Дата окончательного согласования работы / Датум о коначног прихвата^а чланка за об]ав^ива^е: 23.06.2022.
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(http://vtg.mod.gov.rs, http://BTr.M0.ynp.cp6). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license oi
(http://creativecommons.Org/licenses/by/3.0/rs/).
© 2022 Авторы. Опубликовано в "Военно-технический вестник / Vojnotehnicki glasnik / Military го
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