The third order correction on Hawking radiation and entropy conservation during black hole evaporation process

Physics Letters B 759 (2016) 293–297

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Physics Letters B www.elsevier.com/locate/physletb

The third order correction on Hawking radiation and entropy conservation during black hole evaporation process Hao-Peng Yan, Wen-Biao Liu ∗ Department of Physics, Institute of Theoretical Physics, Beijing Normal University, Beijing 100875, China

a r t i c l e

i n f o

Article history: Received 27 April 2016 Accepted 24 May 2016 Available online 27 May 2016 Editor: N. Lambert Keywords: Black hole tunneling Entropy correction Entropy conservation Information loss paradox

a b s t r a c t Using Parikh–Wilczek tunneling framework, we calculate the tunneling rate from a Schwarzschild black hole under the third order WKB approximation, and then obtain the expressions for emission spectrum and black hole entropy to the third order correction. The entropy contains four terms including the Bekenstein–Hawking entropy, the logarithmic term, the inverse area term, and the square of inverse area term. In addition, we analyse the correlation between sequential emissions under this approximation. It is shown that the entropy is conserved during the process of black hole evaporation, which consists with the request of quantum mechanics and implies the information is conserved during this process. We also compare the above result with that of pure thermal spectrum case, and find that the non-thermal correction played an important role. © 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

1. Introduction The discovery of Hawking radiation [1] greatly promoted the establishment and development of black hole thermodynamics, and it also triggered a discussion about information loss paradox for more than forty years [2]. This paradox is so important because it would provide a key ingredient in the search for a theory of quantum gravity which is expected to reconcile the disharmony among thermodynamics, relativity and quantum mechanics. Historically, L. Susskind et al. once proposed three postulates called as black hole complementarity (BHC) [3] upon a plain belief that the principle of quantum theory and a phenomenological description of a black hole should be based on, which seems to make this paradox be a little relaxation. However, from detailed searching on the underlying microphysical basis, AMPS(S) recently found that the three statements of BHC is inconsistent and proposed a firewall argument [4,5] as a conservative resolution, which arose intense attention immediately [6]. Since AdS/CFT correspondence [7] has shown that a theory for gravitation must be unitary, some works [8–15] studied this issue from quantum theory (microscopic level). Considering the consistency of physics, it should be benefit to think what clues semiclassical description of a black hole can provide to us. At a macroscopic level, Parikh once pointed out that the claim of information loss paradox rests on two pil-

*

Corresponding author. E-mail address: [email protected] (W.-B. Liu).

lars [16]: an exactly thermal spectrum and the validity of the no-hair theorem. Considering energy conservation and a dynamical geometry of space–time background, Parikh and Wilczek firstly proposed a tunneling model [17] for discussing Hawking radiation, and found the exact spectrum is not thermal and satisfies  ∼ exp( S ) [17,18], in which they have used WKB approximation. What does this non-thermal correction implies for the black hole information puzzle is an interesting problem. Parikh firstly calculated the correlation between sequential emissions and found there is no such correlation [16]. Soon after that, Arzano got the same result after taking quantum correction of black holes entropy into consideration [19]. However, using standard statistical method and distinguishing between statistical dependence or independence of two sequential emissions, B.C. Zhang firstly claimed that there exists correlation between Hawking radiations which is responsible for a process of entropy conservation, so they proposed an argument that the information is conserved [20,21]. After taking into account of the log-correction to Bekenstein–Hawking entropy, Y.X. Chen and K.N. Shao obtained a similar result [22]. Now, we will discuss Hawking radiations from a Schwarzschild black hole under the third order quantum correction and calculate the correlation between them again. It is shown that the information conservation conclusion is still true after the correction. In Sec. 2, we applied WKB approximation at the third order correction to calculate the emission rate of a tunneling particle (S-shell) from a Schwarzschild black hole, and then the expressions for the emission spectrum and black hole entropy to the third order correction are given. In Sec. 3, we first review B.C. Zhang’s

http://dx.doi.org/10.1016/j.physletb.2016.05.079 0370-2693/© 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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H.-P. Yan, W.-B. Liu / Physics Letters B 759 (2016) 293–297

method for calculating the correlation and the information loss paradox under lower order approximation, then apply the method to consider the case at the third order correction. In Sec. 4, we give the conclusion and compare the result with that of pure thermal spectrum. Moreover, we conclude a more general and deep result for the correlation and information conservation. For convenience, we use geometry units (c = G = 1) before Eq. (22) and Planck units (c = G = h¯ = 1) after Eq. (23).

Then we can write the WKB wave function to the third order approximation as

i X (r ) = exp[ ( I 0 − h¯ 2 I 2 ) + I 1 − h¯ 2 I 3 ], h¯ where

r I2 =

2. Tunneling rate at the third order correction

r

J.Y. Zhang first calculated the tunneling rate of Hawking radiation at the second order correction using Parikh–Wilczek tunneling framework [23,24]. Based on their works, we will make a further calculation to get a more precise tunneling rate to the third order correction. For a Schwarzschild black hole, considering the symmetry of a particle (S-wave) created from the neighbor of horizon, the radial equation of motion is

1 d r 2 dr

(r 2

dψ dr

)+

2m h¯ 2

( E − U (r ))ψ = 0.

(1)

By the substitution

X (r )

ψ(r ) =

r

(2)

,

Eq. (1) can be rewritten as

d2 X dr 2

+[

2m h¯ 2

( E − U (r ))] X = 0.

(3)

X (r )

ψ(r ) =

r

=

1 r

exp[

i I (r ) h¯

I3 =

p r dr ,

2I 0 I 1 + I 0 = 0,

(7)

 2



(8)

 

 



(9)

2I 0 I 2 + ( I 1 ) + I 1 = 0, 2I 0 I 3 + 2I 1 I 2 + I 2 = 0, √ where p r = 2m( E − U (r )).

Using Hamilton’s equation r˙ = we can obtain the canonical momentum p r in classically inaccessible region [17]. So,

dp r = 0

dH r˙

dH + dp |r , r

= −i π r ,

(10)

I 1 = − I 2 = −

1 I 0

I 0 = −i π , 1

=− , 2 I 0 2r 1 2I 

I 1 =

(( I 1 )2 + I 1 ) = −(

0

I 3 = −

1 2I 0

i

2

ri i iπ − exp[− ( pr dr − h¯ 2 I 2 ) − ]). h¯ 4

(18)

The connection at r = r i is

r

1

r

1

2

 √ exp[−h¯ I 3 ] exp[− (| h¯ v

p r dr | − h¯ 2 I 2 )], (r > r i )

(19)

ri

and the connection at r = r f is

1

1

2

√ exp[−h¯ I 3 ] exp[− (| h¯ v

r p r dr | − h¯ 2 I 2 )], (r < r f ) rf

r iπ  − √ exp[−h¯ I 3 ] exp[ ( pr dr − h¯ 2 I 2 ) + ], (r > r f ) (20) h¯ 4 v 1

i

2

and the wave function in region III is



I 0 = p r = −i π r ,

(17)

rf

we can obtain I 0 , I 1 , I 2 and I 3 as 

+ C2,

ri 2 1 π √ exp[−h¯ 2 I 3 ] sin[ ( pr dr − h¯ 2 I 2 (r )) + ], (r < ri ) h¯ 4 v

 

pr =

1

(6)





3

16π 2 r 4

(16)

r

(5)

r



I 3 dr = −

+ C1,

ri iπ X I (r ) = √ exp[−h¯ I 3 ](exp[ ( p r dr − h¯ 2 I 2 ) + ] h¯ 4 i v

(4)

Putting Eqs. (4), (5) into Eq. (3), we have

pr

1

r

],

h¯ h¯ h¯ I (r ) = I 0 (r ) + ( ) I 1 (r ) + ( )2 I 2 (r ) + ( )3 I 3 (r ) + · · · i i i

 

3i

16π r 2

where C 1 , C 2 are integration constants. In order to get the tunneling rate of an emitted particle, we need to obtain the ingoing wave function and outgoing wave function. We divide the whole region by two tunneling points r i and r f (r i is the initial horizon radius and r f is the final horizon radius before and after it radiates a particle) into three regions: ingoing and reflection region I, barrier region II, and the outgoing region III. Regions I and III are classically accessible and the wave function is oscillating in these regions, but region II is classically inaccessible and the wave function is exponentially damped in this region. Considering the connections between the oscillating and the exponential solutions at r = r i and r = r f [23], the wave functions at the above three regions and the connections at those two points can be given as follows. In region I, we have [25]

where

I0 = ±

I 2 dr =

1

The WKB wave function of a particle is

(15)

(2I 1 I 2 + I 2 ) =

(11)

1 2r 2

3i

)

1

8π r

3

1

4π 2 r 5

(12)

,

.

, 3

I 2 =

9i 1 8π r

, 4

(13) (14)

1 1 X I I I (r ) = − √ exp[−h¯ 2 I 3 ] exp[− (ImI 0 − h¯ 2 ImI 2 )] h¯ v

r i iπ ], × exp[ ( pr dr − h¯ 2 I 2 ) + h¯ 4 b

pr m

where v = represents velocity of the tunneling particle. The flux density of a wave function is

(21)

H.-P. Yan, W.-B. Liu / Physics Letters B 759 (2016) 293–297

j=−

ih¯ 2m

(ψ ∗

∂ ∂ ψ − ψ ψ ∗ ), ∂r ∂r

(22)

so we can obtain the ingoing flux density j in and the outgoing flux density j out as follows 2 j in = v (r i )|ψin (ri )| =

j out =

1 r i2

2 v (r f )|ψout (r f )|

exp[−2I 3 (r i )],

=

1

 (0) ( E ) = exp[−8π E ( M −

(24)

Therefore, by substituting Eq. (17) into Eqs. (23), (24), the probability of barrier penetration is

j out j in

=

r i2 r 2f

exp[−2(ImI 0 − ImI 2 )]

× exp[

3

(

1

−

8π 2 r 4f

1 r i4

(25)

)],

ImI 2 =

3

1

(

4 Af

−

1 Ai

(26)

),

where A is the area of a black hole’s horizon and subscripts i and f are for initial horizon and final horizon respectively. Substituting Eq. (26) into Eq. (25) and considering (i → f ) = | M f i |2 · (phase space factor) [17], we have

phase space factor = exp[(

−(

Af 4

Ai 4

− ln

− ln

Af

Ai 4

4

+

+

3 4 8 Af

3 4 8 Ai

3

+ (

4

8 Af

3 4

+ (

8 Ai

phase space factor =

Ni

=

eS f e Si

)2 )].

(27)

A 4

− ln

A 4

+

34 8A

A 4

+ α ln

A 4

= e S f −Si .

3 4

+ ( )2 + const , 8 A

4

+ o( ) + const .

(28)

(29)

3. Correlation between Hawking radiations Based on the expression of tunneling probability ( E ) ∼ in Sec. 4 we will use S i ,

middle and final black hole respectively), the correlation between two sequential emissions E 1 and E 2 can be obtained as follows [19]

C ( E 1 + E 2 ; E 1 , E 2 ) = ln ( E 1 + E 2 ) − ln[( E 1 )( E 2 )].

(33) (34)

In order to make a reasonable understanding of this correlation, B.C. Zhang adopts quantum information theory to calculate the mutual information for sequential emission of two particles with energies E 1 and E 2 and finds S (0) ( E 1 : E 2 ) = 8π E 1 E 2 , which is exactly equal to the correlation of Eq. (34). The mutual information [28] in a composite quantum system composed of sub-systems A and B is defined as

(35)

where S ( X ) = − ln ( X ). Considering the contribution of correlations (mutual information entropy), we count the entropy of the total system composed of a black hole and its radiations. After first emitting a particle with an energy E 1 from a black hole of mass M, it is easy to find

S (0) ( E 1 ) + S B H ( M − E 1 ) = S B H ( M ).

(36)

S ( 0) ( E 1 , E 2 ) + S B H ( M − E 1 − E 2 )

= S ( 0) ( E 1 ) + S ( 0) ( E 2 | E 1 ) + S B H ( M − E 1 − E 2 )

(30)

A

e S bh ( f inal) = exp( S bh ) (For simplicity, e S bh (initial) S m and S f as the entropy of an initial,

,

C (0) ( E 1 + E 2 ; E 1 , E 2 ) = 8π E 1 E 2 = 0.

= S B H ( M ).

(37)

Repeating the process until the black hole is completely exhausted, we find

S ( 0) ( E 1 , E 2 , · · · , E n ) =

(31)

Since it is important to distinguish between statistical dependence or independence, we should be careful about distinguishing the notations ( E ), ( E | E  ) and ( E 1 , E 2 ), which represents marginal probability, conditional probability and joint probability respectively [27].

n 

S ( 0) ( E i | M , E 1 , E 2 , · · · , E i −1 )

i =1

which is in agreement with the general formulation of the black hole entropy [19,26]

SQ G =

2

= S ( 0) ( E 1 ) + S ( 0) ( E 2 ) − S ( 0) ( E 1 : E 2 ) + S B H ( M − E 1 − E 2 )

Comparing Eq. (27) with Eq. (28), we get the expression of the black hole entropy to the third order correction

SQ G =

E

If we bear in mind the correlation among Hawking radiations, the total entropy for the first two successive emissions E 1 and E 2 and their accompanying black hole is

)2 )

On the other hand, we have

Nf

(32)

)],

S ( A : B ) ≡ S ( A ) + S ( B ) − S ( A , B ) = S ( A ) − S ( A | B ),

where ImI 0 and ImI 2 can be easily obtained [23] as

1 ImI 0 = − ( A f − A i ), 2

E 2

 (0) ( E | E  ) = exp[−8π E ( M − E  )] −

× exp[−2I 3 (r f )].

p =

Before calculating the correlation at the third order correction, we review the lower order approximation cases in which (0) (1) the black hole entropy takes the form as S bh = S B H and S bh = S B H + α ln S B H , where S B H is Bekenstein–Hawking entropy. (0) For the case of S bh = 4A [20], we have

(23)

exp[−2(ImI 0 − ImI 2 )]

r 2f

295

( 0)

= S B H ( M ).

(38)

Therefore, after a detailed calculation, it is shown that the total entropy carried away by the outing particles plus that of the accompanying black hole remains conserved, which is exactly equal to the Bekenstein–Hawking entropy of the initial black hole. This provides a self-consistent interpretation for information conservation. (1) For the case of S bh = 4A + α ln 4A , we can also find a result similar to Eqs. (34), (38) [21,22]. Now let’s consider the case of the third order approximation. In Sec. 2 we get the tunneling probability Eq. (25) and the black hole entropy Eq. (29) to the third order correction. We have

 (3) ( E ) = exp[−8π E ( M − + +

3

E 2

1

(

32π ( M − E )2 3 1

(

) − ln −

27 π 2 ( M − E )4

1 M2 1

−

( M − E )2 M2

)

M4

)],

(39)

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H.-P. Yan, W.-B. Liu / Physics Letters B 759 (2016) 293–297

(3 )

SQ G =

A 4

− ln

A 4

+

34 8A

3 4

+ ( )2 + const .

(40)

8 A

Putting Eq. (39) into Eq. (31), we can get the correlation

C (3 ) ( E 1 + E 2 ; E 1 , E 2 )

= 8π E 1 E 2 + ln − − − −

3

( M − E 1 )2 ( M − E 2 )2 M 2 ( M − E 1 − E 2 )2 1

(

+

32π ( M − E 1 − E 2 1 1

)2

( M − E 1 )2 3

−

( M − E 2 )2 1

(

27 π 2 ( M − E 1 − E 2 )4 1 1

( M − E 1 )4

−

1 M2

)

+

( M − E 2 )4

1 M4 (41)

).

Then we calculate the mutual information S (3) ( E 1 : E 2 ), it is easy to check S (3) ( E 1 : E 2 ) = C (3) ( E 1 + E 2 ; E 1 , E 2 ). Now, we calculate the total entropy of Hawking radiations and their accompanying black holes during the evaporative process of an initial black hole. After emitting a particle we have (3 )

S (3 ) ( E 1 ) + S Q G ( M − E 1 )

= 8π E 1 ( M − −

3

(

E1 2 1

) + ln

32π ( M − E 1 )2

( M − E 1 )2 M2

−

1 M

)− 2

3 27 π

( 2

1

( M − E 1 )4

−

S ( E ) + S bh ( M − E ) = −( S f − S i ) + S f

1 M

) 4

= S i = S bh ( M ),

3

1

+

32π ( M − E 1 )2 2

2

3

1

27 π 2 ( M − E 1 )4 3 1 3

= 4π M − ln[4π M ] +

32π M 2

+

= −( S m − S i ) − ( S f − S m ) + S f = S i = S bh ( M ),

+ const . 1

27 π 2 M 4

S (E 1, E 2, · · · , En ) = (42)

and after emitting two particles, we have (3 )

(3 )

S (3) ( E 1 , E 2 ) + S Q G ( M − E 1 − E 2 ) = S Q G ( M ).

(43)

Repeating the process until the black hole is completely exhausted, once again we can obtain

S (3 ) ( E 1 , E 2 , · · · , E n ) =

n 

S ( 3 ) ( E i | M , E 1 , E 2 , · · · , E i −1 )

i =1

(3 )

= S Q G ( M ).

(46)

··· ,

+ const .

= S (Q3)G ( M ),

(45)

S ( E 1 ) + S ( E 2 | E 1 ) + S bh ( M − E 1 − E 2 )

+ 4π ( M − E 1 )2 − ln[4π ( M − E 1 )2 ] +

into consideration [20,21]. For high order WKB approximation, the spectrum and the correlation will add high order corrections and the total entropy is still conserved if we take the proper formula of black hole entropy with a corresponding order. This is important because it satisfies the demand of quantum mechanics and implies the information is conserved at least in phenomenological level. Even though we get the above conclusion from specific and tedious calculation in previous sections just like B.C. Zhang does in Refs. [20,21], we ultimately realize the same conclusion can be obtained by several general expressions. If the tunneling probability of a particle with energy E 1 + E 2 is equal to the joint probability of emitting two particles E 1 and E 2 , ( E 1 + E 2 ) = ( E 1 , E 2 ), according to Eqs. (31), (35), the correlation is equal to mutual information, C ( E 1 + E 2 ; E 1 , E 2 ) = S ( E 1 : E 2 ). Since the energy conservation plays a fundamental role in physics, the space–time geometry should be dynamical when black hole emits particles, so the emission spectrum is dynamical as well. Therefore, the tunneling probabilities of different radiations are not statistically independence, ( E | E  ) = ( E ), which leads to a general result S ( E : E  ) = 0. Although the mutual information is nonzero in general, we will later show that the correlation exits only when the non-thermal corrections were taken into consideration and it is equal to the mutual information exactly. Moreover, the entropy conservation can also be obtained from a general derivation. If the tunneling rate satisfies ( E ) = exp( S f − S i ), using S ( E ) = − ln ( E ), we have

(44)

Using the method outlined before, analogously, we find that under the third order WKB approximation the correlations and black hole entropy were both extended in higher order corrections, and the total entropy is still conserved. No information is lost in the Hawking radiation as tunneling when black hole entropy takes a more general formula. 4. Discussions and conclusions Phenomenologically, using a semi-classical treatment of Hawking radiation as tunneling, Parikh and Wilczek first find the emission spectrum or the tunneling probability are non-thermal [17]. B.C. Zhang adopts this spectrum to compute the correlations among Hawking radiations and finds the total entropy of a whole system composed with emissions and their accompany black hole is conserved when taking the contribution of these correlations

n 

S ( E i | M , E 1 , E 2 , · · · , E i −1 )

i =1

= S i = S bh ( M ).

(47)

However, since WKB approximation is a semi-classical method, we are not interested in the final stage of the evaporation process when the mass of a remaining black hole is comparative to that of an emitted particle and the quantum effects are considerable. What we emphasize is that when taking non-thermal correction into consideration, information can leak out through the radiation to avoid the information loss paradox. In order to see the important role that the non-thermal correction plays more clearly, we would like to compare them. For the case of pure thermal spectrum, ( E ) ∼ exp(−8π E M ) = exp( S bh ), thus S ( E 1 , E 2 , · · · , E n ) = S bh ( M ). Also we can find ( E 1 + E 2 ) = ( E 1 , E 2 ), which leads to C ( E 1 + E 2 ; E 1 , E 2 ) = S ( E 1 , E 2 ). This is the reason of the information loss problem, and C ( E 1 + E 2 ; E 1 , E 2 ) = 0 seems that there is totally no correlation among emissions at least at late-times [16]. However, for the cases of non-thermal spectrum as Eqs. (32), (39), ( E 1 + E 2 ) = ( E 1 , E 2 ) is satisfied, and the correlations among Hawking radiations are nonzero means that they are not statistically independent. In addition, the tunneling probability satisfies ( E ) = exp( S f − S i ), and then we have S ( E 1 , E 2 , · · · , E n ) = S bh ( M ) during the process of black hole evaporation, which provides a powerful evidence for information conservation and can be regarded as a resolution for the information loss paradox [20,21] at least at a macroscopic level and from a semi-classical viewpoint. However, the microscopic mechanism how the information transfers from the black hole interior is still unclear [4,14].

H.-P. Yan, W.-B. Liu / Physics Letters B 759 (2016) 293–297

Considering about Hawking radiation and black hole thermodynamics, we have both the macroscopic and microscopic explanation of the related quantities. To be specific, there are two kinds of entropy definition [29]. The macroscopic and thermodynamic entropy is S T = − ln  , while the microscopic definition is entanglement entropy or Von Neumann entropy as S E = −Tr[ρ ln ρ ] [3]. In the information theory and quantum physics, the conservation of Von Neumann entropy means information conservation and unitary. However, since the Parikh–Wilczek tunneling model is just a macroscopic and semi-classical description, it seems insufficient to say that their thermal entropy conservation result supports that Hawking radiation can maintain unitary. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant Nos. 11235003, 11175019, and 11178007). References [1] [2] [3] [4] [5]

S.W. Hawking, Commun. Math. Phys. 43 (3) (1975) 199. S.W. Hawking, Phys. Rev. D 14 (1976) 2460. L. Susskind, L. Thorlacius, J. Uglum, Phys. Rev. D 48 (8) (1993) 3743. A. Almheiri, D. Marolf, J. Polchinski, J. Sully, J. High Energy Phys. 02 (2013) 062. A. Almheiri, D. Marolf, J. Polchinski, D. Stanford, J. Sully, J. High Energy Phys. 09 (2013) 018.

297

[6] R.B. Mann, Black Holes: Thermodynamics, Information, and Firewalls, Springer, 2015. [7] J.M. Maldacena, Int. J. Theor. Phys. 38 (1999) 1113. [8] S.W. Hawking, Phys. Rev. D 72 (2005) 084013. [9] S.D. Mathur, Class. Quantum Gravity 26 (22) (2009) 224001. [10] S.L. Braunstein, S. Pirandola, K. Zyczkowski, Phys. Rev. Lett. 110 (10) (2013) 101301. [11] S.B. Giddings, Phys. Rev. D 85 (2012) 124063. [12] Y. Nomura, J. Varela, S.J. Weinberg, Phys. Rev. D 87 (2013) 084050. [13] D.N. Page, J. Cosmol. Astropart. Phys. 1309 (2013) 028. [14] J. Maldacena, L. Susskind, Fortschr. Phys. 61 (9) (2013) 781. [15] S.W. Hawking, M.J. Perry, A. Strominger, arXiv:1601.00921, 2016. [16] M.K. Parikh, arXiv:hep-th/0402166, 2004. [17] M.K. Parikh, F. Wilczek, Phys. Rev. Lett. 85 (2000) 5042. [18] M.K. Parikh, Int. J. Mod. Phys. D 13 (2004) 2351. [19] M. Arzano, A.J.M. Medved, E.C. Vagenas, J. High Energy Phys. 09 (2005) 037. [20] B. Zhang, Q.y. Cai, L. You, M.s. Zhan, Phys. Lett. B 675 (2009) 98. [21] B. Zhang, Q.y. Cai, M.s. Zhan, L. You, Ann. Phys. 326 (2011) 350. [22] Y.X. Chen, K.N. Shao, Phys. Lett. B 678 (1) (2009) 131. [23] J. Zhang, Phys. Lett. B 668 (2008) 353. [24] J. Zhang, Phys. Lett. B 675 (2009) 14. [25] J. Zeng, Quantum Mechanics, vol. II, Science Press, Beijing, 1997. [26] A. Medved, E.C. Vagenas, Mod. Phys. Lett. A 20 (23) (2005) 1723. [27] G. Grimmett, D. Stirzaker, Probability and Random Processes, Oxford Science Publications, Oxford, UK, 1992. [28] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010. [29] A. Roy, M.H. Rahat, arXiv:1406.3635, 2014.