PHYSICA ELSEVIER
Physica A 248 (1998) 393 418
Decoherence and localization in quantum twolevel systems Ting Yu * Theoretical Physics Group, Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, UK Received 13 May 1997
Abstract
We study and compare the decoherent histories approach, the environmentinduced decoherence and the localization properties of the solutions to the stochastic Schr6dinger equation in quantumjump simulation and quantumstate diffusion approaches, for a quantum twolevel system model. We show, in particular, that there is a close connection between the decoherent histories and the quantumjump simulation, complementing a connection with the quantumstate diffusion approach noted earlier by Di6si, Gisin, Halliwell and Percival. In the case of the decoherent histories analysis, the degree of approximate decoherence is discussed in detail. In addition, the various time scales regarding the decoherence and localization are discussed. By using the yon Neumann entropy, we also discuss the predictability and its relation to the upper bounds of degree of decoherence.
PACS: 03.65.Bz; 03.65.Ca; 42.50. Lc Keyword~: Decoherence; Localization; Quantum jumps
I. Introduction
Two primary paradigms  the environmentinduced decoherence approach, proposed by Zurek [1 5], and the consistent histories approach by Griffiths [6] and later by Omnbs [ 7  1 2 ] and by GellMann and Hartle [ 1315]  have been recently developed to solve the fundamental issues in quantum theory, especially, quantum measurement problems and the transition from quantum to classical. The environmentinduced decoherence emphasizes the division between the system and its environment. The interaction of the system with its environment is responsible for the decay of the quantum coherence of the system. The decoherent histories approach is designed to provide the most general descriptions for a closed system by using the concept of history  a * Email:
[email protected] 03784371/98/$19.00 Copyright @ 1998 Elsevier Science B,V. All rights reserved
PHS03784371(97)005542
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sequence of events at a succession of times. Both approaches are applicable to open quantum systems. Another set of viable theories within the framework of quantum mechanics are the various unravellings of the master equation as stochastic Schrrdinger equations for the single member of the ensemble. Among others, the quantumstate diffusion and the quantumjump simulation approaches have been extensively studied in recent years (e.g., see Refs. [ 1622]). As phenomenological theories, these stochastic approaches are not only of theoretical interest but also of practical value. The open quantum system provides a unified framework to exhibit the properties of the various approaches we have mentioned above. The master equation which describes the evolution of the open quantum system plays a central role in the investigations into the decay of quantum coherence due to interaction with a much larger environment. However, it does not tell us how an individual member of an ensemble evolves in a dissipative environment. The unravelling of the master equation as stochastic Schr6dinger equation could provide such a description within its domain of applicability. Corresponding to the decoherence process in the density operator formalism, in stochastic Schrrdinger equation approaches, the solution to the stochastic Schr6dinger equation often possesses a very remarkable property  the solution tends to localize at some special states after a localization time scale. For quantum diffusion approach, this localization property has been justified in many different situations [2126]. It is worth emphasizing that a key point in these approaches is the mutual influence between the system of interests and its environment. This mutual influence is the common sources of the many different phenomena such as dissipation, fluctuation, decoherence, localization, etc. Analysis of decoherence and localization properties is usually rather involved. The entanglement of the complicated mathematics and the subtle conceptual issues often tends to make a detailed scrutiny of the basic concepts impossible. The attractiveness of twolevel system model is that, perhaps it is one of the simplest yet physically meaningful models. The purpose of this paper is to employ a widely used twolevel system model as a unified framework to examine the dynamics of the open quantum system by the decoherent histories, the environmentinduced decoherence and the stochastic Schr6dinger equations. The main aim of the present paper is to establish the connections between the decoherence process and the localization process. The various time scales concerning these processes are discussed. In particular, we have shown a close connection between the decoherence histories and quantumjump simulation, complementing a connection with quantumstate diffusion noted earlier by Dirsi et al. [25]. As byproduct, by using this simple model, we have compared the decoherent histories and environmentinduced decoherence approaches in some details. In the case of the approximate decoherence, we provide a detailed analysis of the degree of decoherence of the twolevel models, which is important for a real physical process. The plan of this paper is as follows. In Section 2, we briefly present our twolevel system model and its basic properties. We study the consistent histories approach and
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its relation to the environmentinduced decoherence, the degree of decoherence and predictability by using von Neumann entropy in Sections 3 and 4, respectively. We study the unravelling of master equation and the localization properties of the solutions to the stochastic Schr6dinger equations in both quantumjump simulations and quantumstate diffusions in Section 5. We summarize and conclude in Section 6. In the appendix we present a proof of the Theorem given in Section 4.
2. The model
A fundamental building block in quantum theory is the twolevel system. We consider a twolevel atom system, which is radiatively damped by its interaction with the many modes of a radiation field in thermal equilibrium at temperature T. The upper level and lower level are denoted by 12> and l l>, respectively. This is a typical example of the socalled systemplusreservoir models in which quantum mechanics is fully implemented. In the case of our model, the system of interest is a single atom, and the reservoir is represented by the quantum radiation. Under some conditions, one may derive the master equation for the twolevel atom by tracing out over the radiation variables. This master equation describes a Markovian process and consequently takes the standard Lindblad form (in the Schr6dinger picture) (e.g., see Refs. [19,27,28]): F; =  ~i [ H , p ] +
~(fi + 1) (2apa t

atap
+ 2~(2a~pa  a d p  p a d ) .

pata) (2.1
)
Here, the Hamiltonian of the atom is given by H = or=, 2
(2.2)
where ~2 is the renormalized frequency. The Lindblad operators, which model the effects of the environment in this situation, are L, = ~ l ) a ,
L2 = X f l ~ a ~.
(2.3)
The transition rate from [2) ~ [1) is described by the term proportional to (y/2)(fi+ 1 ) which contains both the simulated transitions and spontaneous transitions rates, and the transition rate from I1) + 12) is described by the term proportional to (7/2)fi which gives the rate of absorptive transition caused by taking thermal photons from the radiation. The damping constant 7 is the Einstein Acoefficient, and ~ = ~(og, T) is chosen as 
1 e h°~/k~T I
(2.4)
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We use ax, ay and az to denote Pauli matrices and a , a t atomic lowering and raising operators, which are defined in the usual way ,
ox=
oy=
0
'
oz=
_
(2.5)
and a = l ( a x  i~Ty),
a t = ½(ax + ia),).
(2.6)
The master equation, Eq. (2.1), has been widely discussed in many places and is of importance in many quantum optical problems [19,27]. A convenient way to solve the master equation (Eq. (2.1)) is to write down it in the basis 12), I1): /J22 =  7 ( ~ + 1 )P22 q 7nPll ,
(2.7)
/)11 = ~ ( n ~ l ) P 2 2  7 n P l l ,
(2.8)
/J21 = 
[~(2~+ l)+ico]p2,,
(2.9)
/912 = 
[ ~ ( 2 "  I  1)  i~o] pl2 •
(2.10)
The first two equations are the wellknown Einstein rate equations. The general solutions to Eqs. (2.7) and (2.10) are as follows: p22(t) =   B l
~+1
+ B2e ;'~2n+l)t
(2.11)
p l l ( t ) = B1  B2e ~'(2~+j)t ,
(2.12)
P21 (t) = B3e [(~'/2)(2~+1)+i,,)lt,
(2.13)
p l z ( t ) = B4e [e'/2)(2~+1 )i,,,k,
(2.14)
w h e r e B i (i = l, 2, 3, 4) are arbitrary constants which can be easily determined once the
initial condition is given. For the initial density matrix with Tr(p0) = 1 we easily get B~
H+I 2~+ 1 '
B2 ~ P22(0)
(2.15) 2~+ 1 '
(2.16)
B3 = P 2 1 ( 0 ) ,
(2.17)
B4 = P 1 2 ( 0 ) •
(2.18)
It immediately follows from the solutions of the master equation that the density operator p tends to the stationary density operator Ps as t + oc: P ~ P" =
0
0
~
"
(2.19)
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It is seen from the above that the offdiagonal elements, which represent the quantum coherence between the excited state and the ground state o f the atom, vanish in the stationary state. This fact implies that, due to the influence o f the random noise, the quantum coherence decays exponentially as time evolves. This is an elementary example o f environmentinduced decoherence (e.g., see Refs. [ 1  3 ] ) . Diagonalization occurs in the basis [2), [1). it has been shown by Joos and Zeh [29] in their seminal paper that the decoherence processes are typically very effective in a wide variety o f situations. In the case o f our model, from Eqs. (2.9) and (2.10), it follows that the decoherence time scale tD is given by 1 tD ~ 7 ( 2 ~ + 1) "
In particular, when h e ) ~
(2.20) k~T,
h (2.21)
to ~ ",'ksT
While for h~o ~> kB T, i.e., at the lowtemperature limit: 1 tL~ ~  •
(2.22)
For the time being, we are mainly concerned with the two types o f time scales: ~ One is the decoherence time  which is the time scale on which the offdiagonal elements o f the density matrix are suppressed; and the other is the relaxation time which is time scale on which the system approaches thermal equilibrium. It might be useful to note that these two types o f time scales coincide in our twolevel model. This is, certainly, a special feature o f this simple model, which is unlikely shared by other models. 2 It has been found that, for example, the decoherence time in quantum Brownian motion model is typically much shorter than the relaxation time [26]. It is easily shown that the master equation (2.1) is invariant under unitary transtbrmations o f the Lindblad operators: a H UaU*,
a t ~+ U a * U t ,
(2.23)
where U is a unitary matrix. Correspondingly, the density operator p transforms in the same way: p ~
UpU t .
(2.24)
Thus, when t * oc p ~
Up, U ~ .
(2.25)
Generally, the density matrix Up.,.U t is no longer diagonal. This indicates that environmentinduced decoherence does not occur in other bases. This property is useful i Another two time scales  decoherent history time and localization time are to be discussed in Sections 3 and 4, respectively. 2 It is not difficult to contemplate a twolevel model in which those two time scales are different.
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on comparison between the environmentinduced decoherenee and decoherent histories approaches.
3. Decoherent histories in the twolevel system model
3.1. Decoherent histories The decoherent histories approach [615] offers a sensible way to assign probabilities to a sequence of properties of a quantum system without referring to the measurements or to a classical domain. A history is defined in general as a sequence of properties of a closed system occurring at different times, which is denoted as
C~
=
P(n)tt ~,, ~ n ) ~ ,...,
p(l)l, :q ~ t l )~,
(3.1)
where P~i)(ti) are the projection operators in the Heisenberg picture at times t~: p ( i ) ( t i ) = ei/h(t,to )H p(i) ei/h(t,to )H
(3.2)
here, H is the Hamiltonian of the closed system. These projection operators satisfy exhaustive and exclusive conditions:
P(~i)(ti)P~'i)(ti) = 6~,13,P~i)(ti) .
P(~i)(ti) : I,
(3.3)
The superscript (i) labels the set of projections used at time ti and ~i denotes the particular alternative. A natural way to assign the probability to a history is
p(C~) = Tr(C~p(to)C~)  Tr(P(~)(t.) 
...
p(l)'t )p(to)P~')(t,) • 'P~,, (") (tn)) ~ l ~ l
(3.4)
However, one finds that Eq. (3.4) generally does not satisfy the usual probability sum rules. The necessary and sufficient condition to guarantee that the probability sum rules hold is that the real part of the decoherence functional D[~, a t] defined by D[~,_~'] = Tr(C~p(to)C~,)
(3.5)
vanishes for any two different histories C= and Ca,, i.e., ReD[c~,_~'] • 0,
V~ ¢; ~ .
(3.6)
The sets of histories satisfying Eq. (3.6) are said to be consistent (or weakly decoherent). Physical mechanisms causing Eq. (3.6) to be satisfied typically lead also to the stronger condition D[~,_~'] = 0,
V~ ¢ _~',
(3.7)
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which is called medium decoherence [14]. (In this paper, we simply refer it as decoherence). Although the decoherent histories approach was primarily designed for a closed system, the approach is of particular importance for the open system which may be regarded as a subsystem of a large closed system. For the open quantum system, a natural coarsegraining is to focus only on the properties of the distinguished system whilst ignoring the environment. In this case, a natural selection of projections at each time is of the form P~ ® I f', where P~ is a projection onto the distinguished subsystem and I ~' denotes the identity projection on the environment. In the Markovian regime, the decoherence functional could be constructed entirely in terms of the reduced density matrix of the system [30]: D[ Z,Z'] = Tr(P~,,(")Kt,,_, ~ "[p(,, L_~,, 1)•
..K[~[P~')K~'[po]P(',)]...P ( ' '  I h p~,(,~' ) ~j~,, ,
,,
(3.8)
where the trace is taken over the distinguished system only. The quantity K[j ,[.] is superpropagator for the reduced density operator: Pt  K~[p0]. In what follows, we shall make a detailed analysis of the decoherent histories in the twolevel model described by the master equation (Eq. (2.1)) which depicts a Markovian process. First, let us consider the projection operators represented by PI = II)
and
P2 =
I2)<21
•
(3.'))
Obviously, {Pi,
i = 1,2} form a set of complete and exclusive projection operators. Physically, P1 may represent that the atom emits a photon whereas P2 may represent that the atom absorbs a photon. Then the decoherence functional at two time points is given by D[Z,Z'] = 0"~ : , Tr(P~2Kt; 2 ~ [P~,I K 0~,[P0]P~,, l ]) •
(3.10)
It is easily shown that, for any 2 × 2 matrix A, the matrix PiApj (i ¢ j) is an upper (or a lower) triangle matrix. From Eqs. (2.9) and (2.10) we know that K/"~[ .] propagates the matrix with zero diagonal elements into the matrix with zero diagonal elements. So, for any initial density matrix P0, the trace in Eq. (3.10) is exactly zero tbr any different pairs of histories (Z ¢ Z') and for any interval t2  tl. This demonstrates that the set of histories consisting of projectors, Eq. (3.9), are exactly decoherent. The generalization to n time points is straightforward. The exact decoherence for any time interval is slightly surprising. (The density matrix, by contrast, only becomes exactly diagonal as t ~ oc). This exactness is due to the simplicity of the model and we do not expect it to be a generic feature. Next, consider more general projection operators which correspond to the projection to any direction. With any direction denoted by a unit vector n = (sin 0 cos qS, sin 0 sin 4~, cos qS), we associate a vector In) which belongs to the Hilbert space of the twolevel system In)=cos OI1)e
i¢~ " 0
s,n~12) .
(3.11)
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Then one can define the following projection operators on the Hilbert space of the system: P+ = In)(nl, where
In')
P _ = In')(n' I ,
(3.12)
is the orthogonal complimentary of In):
In') = e
i~b • 0
0
sm~[1)+cos~12).
(3.13)
We shall show that a set of histories consisting of the projection operators P+ and P _ are approximately decoherent. To this end, first, note that for any 2 x 2 matrix A,
Tr(P+AP_ ) = Tr(P_AP+ )
= 0.
(3.14)
Hence, from Eqs. (2.11 )(2.14), it can be seen that, after the propagation of Ki''+j [], all of the diagonal elements of matrix K;'~[p±Kt~ [P0]P~:] contain an exponential damping factor Damping factor
=
(3.15)
e 7(2~+1)t
Thus, we conclude
V_~¢_~'.
O[~,_~'] ,~ 0,
(3.16)
This proves that the set of histories consisting of P+,P_ are approximately decoherent if time interval between tk and tk+~ is larger than the characteristic time scale: 1
tdecoherence
y(2~ + 1 )
(3.17)
This is an expected result. We will give a more detailed estimate of the degree of decoherence in the next section. Note that /decoherence decreases as the coupling 7 is made stronger. From Eq. (2.4), it is easy to see that the decoherence is more effective if the temperature of the bath, T, increases. Conversely, decreasing temperature will make the system spend more time to decohere. The maximum decoherence time for a set of histories is 1/7 which corresponds to zero temperature of bath. In this case, the damping is caused by only spontaneous emission, and then the decoherence process is not very effective. In summary, we find exact decoherence of histories characterized by the projections onto I1) and 12), and approximate decoherence in any other basis. Finally, we examine the probabilities for two times histories consisting of projections Eq. (3.9). These are given by p ( 1 , 2 ) = Tr(Pi p(2, 1 ) =
K[{ [P2Kid [Po ]P2 ])
Tr(P2Kt~ [P, Ki'~[P0]P, ])
~+1  2 ~+ 1 (1 
6)p22(t~),
(3.18)

6)pll(h ),
(3.19)
2H+ 1
(1 
T Yu/Physica A 248 (1998) 393 418 p(1,
1) =
Tr(P1K[~[P,K~I [ P o ] P 1 ] )

p(2,
2)
Tr(P2K[~[P2K[~[ p o ] P 2 ] )

=
 2if+
1 1
 
2if+ 1
401
[ff + (ff + 1 )6]P22(tl ),
(3.20)
[(if+ 1 ) + ~ 6 ] p l l ( t l ) .
(3.21)
where ff [ P22(tl)  2ff + ~ + P22(0)
ff ] 6 , 2ff + 1
(3.22)
p ll(tl)  2~ ~ ++1~
2ffff+ ~ 1 6 ,
(3.23)
IP22(0)
and 6 = e x p {  7 ( 2 f f + l ) A t } ( A t = t i  t i i , i = 1,2). As for the n times histories, the calculation for the elementary probabilities will be straightforward. For instance, p(1, 1, • "" 1) = Tr(PIK~',"
l
(2if+l)"
, [ P 1 K,,,' ....: [Pl
I
[~+(~+
• • • K[~ [P, K[~ [Po]Pi]
1 ),~]"
P22(tl).
• • • P, ]P, ] )
(3.24)
Similarly, one may calculate the transition probabilities, etc. 3.2. Decoherent history vs environmenthtduced decoherence
Both decoherent histories and environmentinduced decoherence approaches are designed to solve the fundamental problems in quantum theory, in particular, quantum measurement problems and transition from quantum to classical. The comparison on the two approaches is of interest and of importance. Although it is believed that there is a close interrelationship between them, they are by no means equivalent. First of all, they differ in the conceptual aspect [4]. The environmentinduced decoherence emphasizes the division between the system of interest and its environment. The mutual interaction is responsible for the decay of the quantum coherence. While the decoherent histories approach to quantum theory permits prediction to be made in genuinely closed systems, such as the whole universe. Apart from this conceptual difference, with this twolevel system we will be able to see some other interesting differences. One such difference is that, for any direction In), the sets of histories consisting of the projections P~ In) (nl and P_ = [n')(n' I are decoherent (approximately) for any initial state. However, we have seen from Section 3 that the density matrix tends to become diagonal only in a particular basis I1), 12). Another intriguing difference is the time scales in two formulations. As was shown in the previous section, the histories consisting of the projection onto two levels I1) and 12) are exactly decoherent for any time, moreover, it is independent of the initial density matrices. Whereas environmentinduced decoherence in the same basis 11), 12) could only occur after a certain time, which is dependent on the initial state. From the above we see that those two formalisms differ enormously in this twolevel model.
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T. YulPhysica A 248 (1998) 393 418
It should be noted that the differences exhibited here may not be of generality. It is likely that these differences are purely due to the simplicity of the model or due to the approximations employed in the derivation of master equation (such as BornMarkov approximations). However, at any rate, we have shown, within the domain of applicability of the model presented here, the differences between these two formalisms are significant. Moreover, our discussions here could serve as useful hints for further studies on more realistic models.
4. Degree of decoherence and predictability In this section, we will explicitly estimate the degree of decoherence. We show that the degree of decoherence is determined by the largest and the smallest eigenvalues of the projection operators and density matrix at a certain time (see below Section 4.1). By using von Neumann entropy, we also discuss the predictability and its relation to the upper bounds of degree of decoherence.
4.1. Approximate decoherence Physically, one would not expect the decoherence takes place exactly. Therefore, the investigation of the approximate decoherence is of importance. In practical problems, one can, at best, only expect that probability sum rules are satisfied up to order ~, for some constant ~ < 1. Namely, the interference terms do not have to be exactly zero, but small than probabilities by a factor of e. One simple inequality which turns out to be very useful to the study of the degree of decoherence is [31,32]:
ID[ _~.0~t]]2 ~
a2D[_~._~]D[_~'._~'].
(4.1)
We say that a system decoheres to order a if the decoherence functional satisfies Eq. (4.1). As shown in Ref. [31], such a condition implies that the most probability sum rules will then be satisfied to order a. Based on this twolevel model we will study the degree of decoherence in some detail. To begin with, we establish the following trace inequality which is useful to our studies of the approximate decoherence. Theorem. Suppose that M and N are two n x n positive definite matrices. Let P and Q be two n × n Hermitian matrices satisfying QP = PQ = O.
(4.2)
IT r ( M P N Q )12 <~~2T r ( M P N P ) T r ( M Q N Q ) ,
(4.3)
Then
403
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where c = min{~a¢, gX }, here gM = (2ma xM _ Amin~m)/(Ama x~M + /"minZM),'~t."N = (/'max'N _ Xmin'N)/ "N N M M ~N "N (/~ma~ + )'rain)' and are the maximal and the minimal eigenvalues J[max, )~min /'max, Amin of M and N, respectively.
Remark. In fact, the condition that both M and N are the positivedefinite matrices could be generalized to that one is positive definite, say M, while the other N is positive semidefinite. In this case, ~: = e M. It is hoped that the above theorem is also useful in some other cases. The theorem is proved in the appendix. For a general initial state represented by P0 (pure or mixed state), the decoherence functional of two time points may be written as D[ Z,_~']
Tr(P±K[~[P K~'[po]P+]).
(4.4)
We now write A = K0'[p0],
(4.5)
B
(4.6)
/~rl [P±].
Then Eq. (4.4) may be rewritten, in the new notation, as
D[ ~_,or'] = Tr(BP_AP+ ) .
(4.7)
Note that /£ in Eq. (4.6) is the superpropagator for the projection operators
P(t) = Ko[P(O)]. t
(4.8)
The evolution equation for the projection operators is given by
15 ~[H,
+~(fi+l)(2atPa . ataP. Pata)+2fi(2aPa* . .
aatP aatP), (4.9)
where H, a, a f are defined as before. Note that the evolution equation for the projection operator P is different from that for the density operator p, Eq. (2.1). This reflects the difference between the Schr6dinger and Heisenberg pictures in the density operator formalism. The explicit form of Eq. (4.9) may be written as P22 =
'~'(fi
Jr 1 )(Pll

(4.10)
P22),
(4.11)
Pll = "/~(P22  P I I ) , /521 =  [~(2n )'  + 1 ) 
io)]
P21
P,2 =  [ ' 2 (2fi + 1 ) + i(o] P,2.
,
(4.12) (4.13)
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The general solutions to the above equations are P22(t) = C1 + Cze 7(2~+1)t ,
Pll(t)
,.~ _~,(2~+l)t _~ _}_~£t~2~
= CI 
(4.14) (4.15)
P21(t) = C3 e[;'/2(z~+l)i~°]t ,
(4.16)
Pl2(t) = C4e [''/z(2~+l )+io~]t,
(4.17)
where Ci (i = 1,2, 3, 4) are arbitrary constants. For given initial values, these constants can be expressed as ~+1 C1  2ff q _ ~ P 2 2 ( 0 ) Jr 2 ~   ]  P I I ( 0 ) ,
(4.18)
~+1 C2   (P22(0)  P l l ( 0 ) ) , 2~+ 1
(4.19)
C3 = P 2 1 ( O ) ,
(4.20)
C4 = P I 2 ( 0 )
.
(4.21)
From the definitions given by Eqs. (4.5) and (4.6), it is easy to see that in general, both A and B could be positivedefinite matrices, and since P _ and P+ are projection operators, so the condition given by Eq. (4.2) is automatically satisfied. Using the theorem above, we immediately arrive at
[Tr( BP+AP_ )]2 ~
(4.22)
That is, ID[z,Z,][2 ~<82D [ Z , z ] D [ z , ,Z , ].
(4.23)
where e = min{c A, e~}, CA =
[2 A
ca _
I;~f  ,~fl
(4.24)
 )~A[,
2f + ) , f
"
(4.25)
Here 2 A (i = 1,2) and 22 (i = 1,2) are two eigenvalues of A and B, respectively. (Note that 2~ + 22a = 1.) From Eqs. (4.24) and (4.25), it is easily seen that the degree of decoherence may depend on both the projection operators we use and the initial state of the system. This is also an expected result. For the twolevel system, EA and e 8 can be calculated exactly. Consider, first, the eigenvalues of A. Since Eq. (2.1) preserves the trace, 8A can be written as gA
= V/1  4ZIA2A = X/I  4 d e t A .
(4.26)
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The determinant of A can be explicitly evaluated from the general solutions Eqs. (2.11)(2.14): ~(B+I) d e t A   ( ~2 ~ ++l ) +
[ ~ B+I I.~~Pll(0)+f~P22(0)
p22(0)
2B+ 1
p~(0)
2~(~+1) ( 2 ~ + 1) 2
2 ~ + lJ
"
] p21(0)p12(0)A ,~ (4.27)
where (5 = exp{~/(2B + l)tl}. In order that a set of histories are to be decoherent, one expects that ,5 should be small. Similarly, c 8 can be expressed as
cB =
1
4 det B (Tr B)2 .
(4.28)
From Eqs. (4.14)(4.17), T r B and detB can be easily obtained: 1
T r B = 2Ct + ~ i  C 2 6 1 detB = C f +
k~+
GC2
,
(4.29)
 C3C4
]
(51    C 2 6 1
~+
l
,
(4.30)
where 6t  e x p {  7 ( 2 ~ + 1 )(t2  tl )}. The above discussions show explicitly how the degree of decoherence is related to the projection operators, the initial states and the temperature of bath, as well as the timespacing interval, in accordance with our general expectations. It may be helpful to consider some special cases in which the simpler expressions for ~:.4 and c e may be obtained. In the longtime limit, the density matrix will tend to the stationary density matrix. Then we may get a much simpler expression for ~:A: 1
C4 ~  2~+ 1
(4.3 1 )
As mentioned before, for the decoherent histories, /i and ~l should be small. If we only keep the terms up to the first order of ,51, then t:8 becomes 1/2 ~,B ~~
C3C461 1 C 2 + ~~C1C261
(4.32)
Similarly, the expression for e A can be obtained from Eq. (4.27). It is seen from the above expressions that the degree of decoherence improves as the bath temperature increases. We also see that the projections with the smaller offdiagonal elements will give a better degree of decoherence. For a given system with the initial state, then the matter for investigation is to determine which histories, i.e., which string of projections, will lead to the decoherence condition being satisfied. Therefore, we see that eB serves as the main criterion for the degree of decoherence.
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4.2. yon Neumann entropy and predictability It is also of interest to compute the von Neumann entropy of p(t) [4,5,33]. We will discuss how the initial density matrix and the yon Neumann entropy are related to the upper bounds of the degree of decoherence. We will also discuss the preferred states by using the yon Neumann entropy rather than linear entropy, in this twolevel model. In the case of systemplusreservoir model, the pure states of the system, due to the interaction with environment, will typically deteriorate into the mixtures with the different rates. The rate at which pure initial states evolve into the mixtures reflects the stability of those pure states which are continuously monitored by the environment. The von Neumann entropy provides a convenient measure of the loss of predictability: S =  T r ( p In p ) ,
(4.33)
By definition, the more predictable state (pure state) may have less increase of the entropy in a fixed time period. This characterization process of predictability is called the predictability sieve (coined by Zurek [4,5]) which has been studied recently in quantum Brownian motion model by using the linear entropy [4,5,33,34]. We will see that the twolevel system serves as a very nice toy model to employ this "predictability sieve" by directly using the von Neumann entropy. For the purpose of the evaluating the entropy, we choose a special basis in which p is diagonal. Let 21 and 22 be the eigenvalues of p, then Eq. (4.33) reduces to 2
S =  Z
)~i In 2 i
(4.34)
.
i=1
Obviously, 21 and 22 can be expressed as 1 + ~A
21
~ ,
1  ~:A
22 =
2
(4.35)
Hence, Eq. (4.34) can be rewritten as S=

[ ( 1 2 ~ ) l n ( l +) ~ ~ A 

+
~(1eA)ln( 1 ~ A ) ] 2
A t r i v i a l o b s e r v a t i o n shows that the y o n N e u m a n n e n t r o p y
"
S(eA) is
(4.36) a m o n o t o n i c a l l y de
creasing function of e A. Here, we find an interesting relation between the predictability of initial state and ~;A, which is an upper bound of the degree of decoherence. Namely, the von Neumann entropy provides a restriction on the upper bound of the degree of decoherence. Precisely, the initial density matrix which leads to larger entropy production may give smaller cA. This relation between the predictability and the degree of the decoherence is a physically expected result. To obtain a higher degree of the decoherence one would expect that the environment has stronger influence on the system of interest, such as increasing the temperature of the bath. Then the predictability of the state, correspondingly, decreases.
T. Yu/Physica A 248 (1998) 393 418
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I would like to point out here that the actual degree of decoherence could be much smaller than the upper bound eA, since it is often typically undercut by the lesser upper bound e,8. Moreover, the matter for investigation in histories approach is to determine which histories will satisfy the given degree of decoherence. In contrast, our goal here is merely to see how the initial states are related to the upper bounds of the degree of decoherence, hence, we do not take any particular set of histories into account. Next, by using the yon Neumann entropy we shall find the most predictable states, those states will, by definition, generate the least entropy for a given time interval. Since the entropy S (Eq. (4.36)) is the monotonically decreasing function of ~:~1 it is equivalent to find the states which give rise to the largest ~;A. From Eqs. (4.26) and (4.27) it is easily seen that I~) = e V ~
I1) :k
12)
(4.37)
minimize the yon Neumann entropy, and therefore are the preferred states. This is slightly surprising from both decoherent histories and environmentinduced decoherence points of view. At first sight, one might expect that I 1) and 12) would be the preferred states, since this basis plays a very special role in both formalisms. However, from Eqs. (2.9) and (2.10) it is easy to see that the pure states ]1) and 12) will immediately deteriorate into the mixed states. Namely, those two states are most vulnerable to the influence of the environment. This explains why the basis [1) and [2) are not the preferred states. Also, we see from the above discussions that the states, which diagonalize the density matrix, are not necessarily same as the preferred states that are sorted by the Zurek's predictability sieve.
5. Unravelling of the master equation The master equation provides an ensemble description of a quantum system. The unravelling of master equation as the stochastic Schr6dinger equation for the state vector has provided many insights into the foundation of quantum theory, especially in quantum measurement and the useful tools to study various practical problems in the quantum optics (e.g., see Refs. [23,24]). In this section, we will study the localization in the two different unravellings of the master equation quantumjump simulation and quantumstate diffusion approaches. The former uses the discrete random variables whereas the latter uses the continuous random variables. 5,1. Quantumjump simulation
In the measurement schemes, such as direct photodetection, the master equation, which models the measurement process, in some sense describes the lack of information of the systems. Namely, it describes the measurement process in which the results of
T. YulPhysica
408
measurement
are not extracted.
A 248 (1998)
The quantumjump
393418
simulations,
by contrast, mimic that
which may be observed in a single run of the experiment. The state of the system in this situation is represented by a wave function. The whole physical process under consideration
is the combinations
are characterized the individual stochastic
of continuous
by the discrete
system
umavellings
random
is usually
governed
evolutions
variables.
by a stochastic
are said to be equivalent
and abrupt jumps
Therefore,
which
the wave function
differential
equation.
to the master equations
of The
if the former,
after the stochastic average, could reproduce the latter. The alternative description by a single wave function is not confined in the measurement processes. In general, any master equation with Lindblad form could be unravelled into the stochastic Schriidinger equation. For the master following form:
equation,
Id+) = ;H
Eq. (2. l), the stochastic
Schrodinger
equation
takes the
,$) dt +
(5.1) are the Lindblad operators representing the inHere LI = dma, L2 = fiat fluence of the environment and Ni = LtL; (i = 1,2). (N;) = ($lNil$) represents quantum average and M represents the ensemble average. The real random variables dWi(i = 1,2) satisfy dW*dWj
= JijdWi,
M(dWi)
= (Ni) dt
(5.2) (i = 1,2).
(5.3)
Under condition, Eq. (5.2) it is easy to see that dWi only take two values: 0 and 1. The master equation (Eq. (2.1)) can be recovered from the stochastic Schrodinger equation (Eq. (5.1)) in the sense that if I$) is the solution to Eq. (5.1) then p = Ml$)($l satisfies the master equation (Eq. (2.1)). In what follows,
we shall discuss
the “localization”
properties
of the singlejump
trajectories. Here, by “localization” we mean that the quantumstate vector generated by the stochastic Schriidinger equation will converge to some fixed states in the mean square. More precisely, let A be an operator (not necessarily quantum meansquare deviation as a(A,A) If the solution
Hermitian),
then we define the
= (AtA)  (‘4+)(A). of the stochastic
Schrodinger
(5.4) equation
(Eq. (5.1))
satisfies (5.5)
namely, the dispersion of the operator A tends to decrease as time evolves. Then we say that the solution localizes at the eigenstates of the operator A (A is sometimes called the collapse operator).
T. Yu/Physica A 248 (1998) 393 418
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For the stochastic Schr6dinger equation for the quantumjump simulation in a twolevel system, the collapse operator is a:. Then quantum meansquare deviation in this case is (Ao~) 2 = 1
15.6)
(o~) 2 .
In order to prove the localization, we should first derive the evolution equation of the expectation value of o: by using the following tbrmula: d (m) = (~,]AldO) + (dtp]A]~9) + (dtPlAldt)) ,
(5.7)
where A is an operator. From Eq. (5.1), it is straightforward to arrive at the following equation: d (o:) = (1  @ ~ ) ) d W l
 (1 + (Oz))dW2 + [@:) (N, + N2} + (NL  N 2 ) ] d t .
(5.8) Notice that d( A g z ) 2 ~  2 (~Tz) d (ffz)  (d (o'z)) 2 .
(5.9)
Then, inserting Eq. (5.8) into the above equation, taking the ensemble means and remembering Eq. (5.3), we obtain M d ( A a ~ ) 2 =  ~ 1° / ( n + 1)(1

(O':))2(1
÷
(O'z))

7~(1 + (o:))2(1  ( a : ) ) . (5.10)
The righthand side of Eq. (5.10) is nonpositive, and that it vanishes if and only if 10) is 12) or I1). Hence, we conclude that the solution to the stochastic Schr6dingcr equation (Eq. (5.1)) will localize at 12) or I1) after a certain time. That is, any initial state (which will be a superposition of 11) and 12)) will tend to a solution in which the atom undergoes stochastic jumps between I1 ) and 12). Let us now estimate this localization time. From Eq. (5.10), a few manipulations directly give M d ( A a z ) 2 ~  7(2n + 1 ) ( A ~ ) 2 • So the localization rate tlocalization ~
/localization
(5.11)
is
1
7(2~ + 1) '
(
5.12 )
which agrees with the decoherence time scale Eq. (2.20). Note that this is the minimum localization time. The actual time for localization might be larger than this time. In some sense, that the localization in quantumjump simulation chooses the basis tl ), 12) appears to be natural, since it correspond to the trajectories that would actually observed in an individual experiment. As expected, the set of histories consisting of projection onto the basis give the best degree of decoherence. In addition, we have seen
T. Yu/PhysicaA 248 (1998) 393418
410
that density matrix become diagonal in this basis. Here, we have demonstrated a close connection between the different approaches. This is one of the main results in the paper. The connection we have established here bridges the two different approaches  decoherent histories and quantumjump simulations. The former is regarded as a fundamental theory with a wide range of applicability, whilst the latter is mainly seen as a tool with great practical values, in particular, in the computational aspects.
5.2. Quantumstate diffusion In this subsection, we will illustrate the localization process in another unravelling of the master equation  the quantumstate diffusion approach, which was introduced by Gisin and Percival [16] to describe the quantum open system by using a stochastic Schrrdinger equation (which is often called the LangevinIto stochastic differential equation) for the normalized purestate vector of an individual system of the ensemble. Similar to the quantumjump simulation, a solution of the LangevinIto equation for the diffusion of a pure quantum state in statespace represents a single member of an ensemble whose density operator satisfies the corresponding master equation. Generally, if the master equation takes the standard Lindblad form: /~ =  ~ i[ H , p ] + ~
(LipL~  ~Li I t Lip  ~pL~Li )
(5.13)
Then, correspondingly, the LangevinIto stochastic equation can be written as
Id~l) z ~n I ~l) dt ] ~i ( (Z] lZi  ~Z]Z i  ~ IZ] l (Zi) ) I~l)dt + Z ( L i  (Li)) ]~) d~i,
(5.14)
i
where H is a Hamiltonian (of the open system) and Li a r e Lindblad operators, as before, (Li) = (~'lLilq~). The complex Wiener processes d~i satisfy
M(d~i) = 0,
M(d~id~j) = 0,
M(d~TdCj) = 6ijdt,
(5.15)
where M denotes a mean over the ensemble. Quantumstate diffusion reproduces the master equation in the mean: p = M I~') (ffl,
(5.16)
where [ff) satisfy the quantumstate diffusion equation (Eq. (5.14)), then it can be shown that p satisfies the master equation (Eq. (5.13)). In order to show the localization properties of the LangevinIto equation, we now consider the simplest case which is assumed that the bath temperature is zero (~i = 0). In this case the master equation Eq. (2.1) reduces to
i f~ = ~[n,p] + ~(2apa t  atap  pata) .
(5.17)
T. Yu/Physica A 248 (1998) 393 418
411
Then the corresponding Langevinlto equation is given by I~') dt + Z(2 2 (at) a  ata  (a t) (a))Itk)dt
i Id,/,) =  ~ H
+xfT(a (a))[email protected],
(5.18)
where d~ is the complex Wiener process satisfying
M(d~) = O,
M(d~d~) = O, M(d~*d~) = dt,
(5.19)
where M denotes a mean over probability distribution. The evolution of the quantum average of operators can be calculated by using the following formula:
d(G) = ~ i( [ H , GI) d t  ~ Z1
(L~[Li, \ " G]+[G, Lt]Li)dt i
+ Z(O(G t, Li) d~i + O(L~,G) d~,* ),
(5.20)
i
where
a(A,B) = (AtB)  (A t ) (B).
(5.21)
Using Eq. (5.20), it is straightforward to get the following equations: d(a,)=

(a,,)~(ax
dt
+ 2~;[l + (o~,) (at) 2 +i(ox)(ay)]d~ +~[1
[email protected])=
+ (o:) 
~(ax)+
[email protected])
(5.22)
dt
[i(l+(a:))+i(ov)2(av}@v)]d~
+~[i(1
d (a:)
i (O'r)(Gv)]d~* ,
(ox) 2 
+ (o))  i (o),)2
(o,) (a,.)] d~*,
(5.23)
[(a:) 7 + 7]dt °'~/(1 + (o:))((ar)  i(av))d¢
~'~/(1 +
@~) )( (ax)
+
i (av) )d~* .
(5.24)
Moreover, we need to calculate the higherorder moments. For any Hermitian operator A we have from Eq. (5.4),
d(AA) 2 = d((A 2)  (A) x)  d (A 2)  2 (A)d (A)  (d
(A)) 2 .
(5.25)
72 Yu/Physica A 248 (1998) 393 418
412 Then, we easily obtain
Md(A~x)
2 = 2 ~ (O'x)(O'y)+ 7 (O'x) 2  ~(Ao'x) 4  7(Ao'x) 2 (O'z) 3' 2 (O'z)2
27 ((O'x) (O'v))2 ,
(5.26)
~ )4  7(AOy) 2 (O'z) M d (Aoy) 2 =  2 ( o (crx) (Cry) / g (O'y) 2  ~(AOy 2 (O'z)2  i ((O'x) (0"3))2 "
(5.27)
Now, we are in the position to consider the localization of solutions to Eq. (5.18). Using the master equation, it is very easy to see that the atom will soon collapse into the lower state I1) and keeps there forever. Here, we shall demonstrate that any solution to the LangevinIto equation given by Eq. (5.18) will localize at the lower state after a localization time. The collapse operator in this case is (5.28)
A = o~ + i~h,. Then by using Eq. (5.4) we get
a(A,A) = (Ao'x) 2 + (Affr) 2 + 2 (az).
(5.29)
Hence, we have
M d a ( A , A ) = 7 (ax)2 + 7 (ffy)2 ~(A~Yx)4 ~7(A~yy) 4 )'(AO'x) 2 (~Yz)  7(AtYy) 2 (O'z)  7 (O'z) 2  2 7 ( @ z ) + 1)  7((ax)
(O'y))2.
(5.30)
In order to prove that the lefthand side of Eq. (5.30) as nonpositive, let us denote
(AO'x) 2 = 1   Y ,
(5.31)
(Aoy) 2 = I + Y ,
(5.32)
(~) = 1 +Z.
(5.33)
Substituting Eqs. (5.31(5.33) into Eq. (5.30), we have
M
a(At,A)
[
7 R2X
1
Y2Z~(YZ)
,
25(XZ)2
],
(5.34)
where R = (ox)(a),). Note that
X + Y + 2Z = a(A,A)>~O.
(5.35)
Then, we show that
Mda(At,A)<~O
(5.36)
T. Yu/Physica A 248 (1998) 393 418
413
and the equality holds if and only if X Y=Z=0.
(5.37)
That is, the average in the lefthand sides of Eqs. (5.31)(5.33) is taken over the ground state I1). This proves that the solution to Eq. (5.18) will localize at the ground state when the evolution time is larger than the localization time. Finally, let us estimate the localization rate of the quantum state evolution. Using Eqs. (5.34) and (5.35), we immediately obtain d M~i~a(A,A)<~
 7 ( t T ( A , A ) )
So the localization rate 1 tlocalization ~'~ 
7
tlocalization
2 .
(5 38)
is
(5.39)
In summarizing this section, we have shown the localization process in both quantumjump simulation and quantumstate diffusion. Those localizations have been extensively discussed in the quantumstate diffusion approaches. Here, we have seen that a similar localization process could also occur in the quantumjump simulation. It should be noted that, in the case of the zerotemperature of our twolevel model, for any initial state of the system, the atom will eventually localize at the ground state. Therefore, the system will always evolve from a pure state into the pure state. In this sense, we say that the decoherence and localization are basically trivial in this case. However, the above demonstration of localization can still be regarded as a useful example for showing that quantumstate diffusion picture provides a consistent description with the density matrix formalism and decoherence approach. It is interesting to compare the master equation formulation with their stochastic unravellings. Clearly, the master equations provide a fundamental description of the quantum open systems. But numerical simulation of the manyfreedom problems seems rather awkward as it requires a large memory. Moreover, it cannot provide a description for an individual system. The quantum trajectories approach  the unravelling of master equation as the stochastic Schr6dinger equation  could do this job and have advantages over the master equation in computational aspect [16,35]. For this twolevel model, the merit of stochastic unravellings is mainly in the conceptual aspects. Generally, the localization process is very difficult to show analytically, if not impossible. Obviously, the unravelling of the master equation is not unique, Quantumjump simulations and quantumstate diffusions are only two wellknown examples, which lie in our interests in this paper. These stochastic unravellings are often connected with certain measurement schemes. For instance, the quantumjump simulation can be associated with the direct photodetection, and quantumstate diffusion corresponds to the heterodyne detection. In a quantumjump process, quantumjump simulation may be a natural candidate for description of the process. The quantumstate diffusion by nature is a continuous diffusion process. However, if the transition is so fast that the "diffusion"
414
T. Yu/Physica A 248 (1998) 393 418
from one level to the other level of atom can be regarded as an instantaneous process, then quantumstate diffusion could also give rise to the "jump" process [20,23]. It should be noted that the applicability of quantumjump simulation and quantumstate diffusion are different. The preference of these stochastic approaches are largely dependent on the physical models employed and the problems to be solved. In general, the relation between those two approaches is by no means obvious. Undoubtedly, the researching into this relation would be of importance and of interest [36].
6. Summary and conclusions In this paper, based on the twolevel system models, we have studied and compared in detail the decoherent histories approach, environmentinduced decoherence approach, quantumjump simulation and quantumstate diffusion. We have demonstrated the localization in both quantumjump simulations and quantumstate diffusion approaches. Here, we conclude with a summary and a few remarks. We have shown that there are a number of sets of decoherent histories in this twolevel model. Clearly, these decoherent histories are not equally important from the physical point of view. Among those, the most natural one is that which consist of the projections onto 11) and 12). We have proven that this set of histories give the best degree of decoherence. Note that the density matrix in the basis 11) and 12) will become diagonal after a typically short time. Moreover, we have shown that the solutions to the stochastic SchrSdinger equation in the quantumjump simulation will localize at 11) or 12) after certain time which is basically same as the decoherence time. Also, We have shown the localization process in quantumstate diffusion in the case of zero temperature. In addition, we have found that the environmentinduced decoherence, decoherent histories and the localization process are more effective as the bath temperature increases. Physically, this is an expected result as the bath at a higher temperature would have stronger influence on the system. These results are in agreement with former studies on the quantum Brownian models [26] as well as on the quantumoptical models [37]. Despite the similarity and agreement mentioned above, it is important to note that there also exists the significant difference. Among others, as was shown in this paper, time scale for decoherent histories could be quite different from that for environmentinduced decoherence, and also different from the time scale for localization. It seems that the investigation into the relationship between those two rival formalisms is a delicate task, and the similarity and difference between the two approaches should be studied on the casebycase basis. There is no reason to believe that in more realistic physical problems the two approaches would agree each other in every aspect. In connection with the stochastic evolutions, we have shown a close relationship between the decoherent histories and quantumjump simulations. The solution generated by stochastic Schr6dinger equation (Eq. (5.1)) will randomly jump between the two
72 YuIPhysica A 248 (1998) 393 418
415
levels I1) and 12). However, due to the influence of the bath, the localization process occurs in this basis after a certain time. Meanwhile, as was shown, the set of histories consisting of the projection onto these two levels are perfectly decoherent. Here, we have seen that quantumjump simulation is entirely compatible with the history point of view. This is a very nice result. Similar results in the quantumstate diffusion have been discussed before [25,26]. The approximate decoherence is of basic importance in practical physical process. By using this twolevel system model we can clearly see what determines the degree of decoherence. For a given set of histories, the only adjustable parameters are the temperature of bath, the timespacing interval and the initial state of the system, in accordance with our general expectations. We have studied the predictability of the pure states in this twolevel model. The von Neumann entropy in this situation serves as predictability sieve to sort out the preferred states which yield the smallest entropy production. As byproduct, we also see an interesting relation between the upper bound of degree of decoherence and the initial density operators through the yon Neumann entropy. It is important to notice that, as phenomenological theories, both quantumstate diffusion and quantumjump simulation must be used under some conditions (e.g. see Ref. [20]). The comparison between different approaches therefore must be made in caution, since the correspondence between them is by no means mathematically onetoone correspondence. Rather, we emphasize that, underlying the quantum open system, the mutual influence between the system and its environment is the common theoretical base of all of those approaches and both decoherence and localization are nothing more than the different manifestations of a single entity. Finally, there are several special features of our model which are worth pointing out explicitly. First, the Hamiltonian of the system is diagonal in the basis I1), 12',,. We see that the evolution equations for the diagonal elements and the offdiagonal elements of the density matrix are decoupled in this situation. One consequence of this is the exactness of decoherence histories of the projections onto I1), 12). An immediate generalization of our model is to consider the Hamiltonian that does not enjoy this property. One such example is that, in addition to a thermal radiation field, the twolevel atom is applied a coherent driving field. Notice that, in this situation, the evolution equations for the diagonal and offdiagonal elements of density matrix are no longer decoupled. The possible effect of making this change is that, instead of decoherence, the quantum coherence could be generated due to the influence of the coherent driving field [19]. At last, let us note that the coarsegraining in our twolevel system is made by using projection operators on the system whilst ignoring the environment. It would be interesting to consider the general ndimensional model in which the effect of a further coarsegraining on the degree of decoherence can be discussed. Work towards to this aspect is in progress. The environmentinduced decoherence, decoherent histories as well as various stochastic Schr6dinger equations have provided many important insights into the understanding of fundamental problems in quantum theory. The investigation into the similarity and
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T Yu/Physwa A 248 (1998) 393 418
difference between the different approaches is of importance. More thorough studies in this aspect would be useful. Note added. After completion of the paper, I become aware the related work concerning decoherent histories and quantum jumps carried out independently by T. Brun [38]. His results are in tune with ours in Section 5.
Acknowledgements The author would like to express his sincere thanks to Jonathan Halliwell for suggesting this project, for encouragements, and for many suggestions which are critically important for the ideas in this paper. He is grateful to Todd Brun for useful conversations and to Lajos Di6si for many useful conversations and encouragements. He is also grateful to Bernhard Meister and Andreas Zoupas for interesting discussions. This work was supported by the SBFSS scholarship from the British Council.
Appendix A. Proof of theorem In this appendix, we shall give a proof of Theorem in Section 3. Since both M and N are positivedefinite matrices, one of them, say, N can be decomposed as
N =StS,
(A.1)
where S is an n × n matrix. After an arrangement, the righthand side of Eq. (4.3) becomes
ITr(MPSt SQ )I = ITr( SQMPSt )[ .
(A.2)
Suppose Xm are an orthonormal basis in ndimensional space V. Then
ITr(MPNQ)I = ~m Xrm(SQMPSt)xm ,
(m.3)
r is the transpose of Xm. Now, we set where x m Ym = Q g t x m ,
(a.4)
PS?xm •
(A.5)
Zm :
Then the trace in Eq. (A.3) may be rewritten as
Tr(MPNQ) = Z
T ymMzm "
(A.6)
m
Since ym,Zm are orthogonal vectors and M is a positivedefinite matrix, then it is not difficult to arrive at the following inequality (see Ref. [39]):
[(yrmMzm)l ~ 3 M( Y mT M y m ) 1/2 (zmMZm) r 1/2 ,
(A.7)
T. Yu/Physica A 248 (1998) 393418
417
where e M : ()~max m "M ' M x Jr2min),Amax M ",~/ ~M n are the largest and the smallest Amin)/(Ama and Ami eigenvalues of M, respectively. Combining Eq. (A.7) with Cauchy's inequality
ambm \
m
2 ant
~ /
t~
b zm "
(A.8)
m
then Eq. (A.3) becomes
r ITr(MPNQ)[ = ~ Xrm(SQMPSt)xm ~ < ~ lymMz.,I ttl
~.M
Ym
Ym
(A.9)
It is easy to identify that
Tr(MPNP) = Z ymMym, r
(A. lO)
m
Tr(MQNQ) = Z
rMz m . zm
(A.I1)
ttl
This proves that
ITr(MPNQ)I <~eM[Tr(MPNP)]I/2[Tr(MQNQ)]I/2.
(A. 12 )
Since M and N are in the completely symmetric position, so a similar result is true also for ~,N. Then it completes the proof of the theorem. []
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