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JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 43, NUMBER 11
NOVEMBER 2002
Efficient simulation of quantum state reduction
Dorje C. Brodya)
Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom
Lane P. Hughstonb)
Department of Mathematics, King’s College London, The Strand, London WC2R 2LS,
United Kingdom
共Received 12 July 2002; accepted 17 July 2002兲
The energy-based stochastic extension of the Schrödinger equation is a rather special nonlinear stochastic differential equation on Hilbert space, involving a single
free parameter, that has been shown to be very useful for modeling the phenomenon of quantum state reduction. Here we construct a general closed form solution
to this equation, for any given initial condition, in terms of a random variable
representing the terminal value of the energy and an independent Brownian motion.
The solution is essentially algebraic in character, involving no integration, and is
thus suitable as a basis for efficient simulation studies of state reduction in complex
systems. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1512975兴
The standard energy-based stochastic extension of the Schrödinger equation is given by the
following stochastic differential equation:
d兩 ␺ t 典 ⫽⫺iĤ 兩 ␺ t 典 dt⫺ 81 ␴ 2 共 Ĥ⫺H t 兲 2 兩 ␺ t 典 dt⫹ 21 ␴ 共 Ĥ⫺H t 兲 兩 ␺ t 典 dW t ,
共1兲
with initial condition 兩 ␺ 0 典 . Here 兩 ␺ t 典 is the state vector at time t, Ĥ is the Hamiltonian operator,
W t denotes a one-dimensional Brownian motion, and
H t⫽
具 ␺ t 兩 Ĥ 兩 ␺ t 典
具 ␺ t兩 ␺ t典
共2兲
is the expectation of Ĥ in the state 兩 ␺ t 典 . The parameter ␴, which has the units ␴
⬃ 关 energy兴 ⫺1 关 time兴 ⫺1/2, governs the characteristic timescale ␶ R associated with the collapse of
the wave function induced by 共1兲. This is given by ␶ R ⫽1/␴ 2 V 0 , where V 0 is the initial value of
the squared energy uncertainty, which at time t is
V t⫽
具 ␺ t 兩 共 Ĥ⫺H t 兲 2 兩 ␺ t 典
.
具 ␺ t兩 ␺ t典
共3兲
The stochastic equation 共1兲 is perhaps the simplest known dynamic model for state reduction
in quantum mechanics consistent with both the Born probability rules and the principle of energy
conservation. Although the mathematical and phenomenological properties of 共1兲 have been studied extensively,1– 4 it has hitherto been necessary to resort to numerical methods to solve dynamical equations of this type,5 and exact solutions have been unavailable except in very simple cases.
The purpose of this article is to present a general analytic solution for the dynamics of 兩 ␺ t 典 .
We begin with a brief overview of the stochastic framework implicit in the extended Schrödinger dynamics given by Eq. 共1兲, following a line of argument developed in Ref. 4. Specifically,
a兲
Electronic mail: [email protected]
Electronic mail: [email protected]
b兲
0022-2488/2002/43(11)/5254/8/$19.00
5254
© 2002 American Institute of Physics
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J. Math. Phys., Vol. 43, No. 11, November 2002
Efficient simulation of quantum state reduction
5255
we introduce the key notions of filtration, conditional expectation, martingale, and supermartingale, and show how these concepts can be used effectively to characterize the reductive properties
of 共1兲. We then proceed to establish, by novel use of a nonlinear filtering technique, that the
energy expectation process 共2兲 can be expressed as the conditional expectation of a random
variable representing the terminal value of the energy. As a consequence, we are led to simple
analytic expressions for the energy 共20兲 and the state vector 共33兲 in terms of a pair of independent
state variables. These results open up the possibility of efficiently simulating the reduction process
for a variety of models. Finally, we illustrate the practical advantages of our method by analyzing
the timescale associated with the reduction process in the case of a two-state system.
The dynamics of 兩 ␺ t 典 are defined on a probability space 共⍀, F, P兲 with filtration 兵 Ft 其 (0
⭐t⬍⬁). Here ⍀ is the sample space over which F is a ␴-field of open sets upon which the
probability measure P is defined.
The filtration determines for each t⭓0 the information available at that time. More specifically, the filtration consists of a family 兵 Ft 其 of ␴-subfields of F such that s⭐t implies Fs 傺Ft .
Given a random variable X on 共⍀, F, P兲, we write E关 X 兩 Ft 兴 for the conditional expectation of X
with respect to the ␴-subfield Ft 傺F. Intuitively, conditioning with respect to Ft means giving all
the information available up to time t. The nesting Fs 傺Ft for s⭐t thus embodies a notion of
causality. For convenience, we use the abbreviation Et 关 X 兴 ⫽E关 X 兩 Ft 兴 , and we note that the conditional expectation satisfies the tower property Es 关 Et 关 X 兴兴 ⫽Es 关 X 兴 for s⭐t. If Et 关 X 兴 ⫽X, we say
that X is Ft -measurable.
The conditional expectation operation allows us to introduce the concept of a martingale, the
stochastic analog of a conserved quantity. A process X t is said to be an 兵 Ft 其 -martingale if
E关 兩 X t 兩 兴 ⬍⬁ and Es 关 X t 兴 ⫽X s for all 0⭐s⭐t⬍⬁. In other words, X t is an 兵 Ft 其 -martingale if it is
integrable and if its conditional expectation, given information up to time s, is the value X s of the
process at that time.
For a concise mathematical representation of the state reduction process, we also require the
concept of a supermartingale. A process X t is an 兵 Ft 其 -supermartingale if E关 兩 X t 兩 兴 ⬍⬁ and
Es 关 X t 兴 ⭐X s for all 0⭐s⭐t⬍⬁. Intuitively, a supermartingale is on average a nonincreasing process.
The filtration 兵 Ft 其 with respect to which the state vector 兩 ␺ t 典 evolves is generated in a
standard way by the Wiener process W t . We signify this by writing 兵 Ft 其 ⫽ 兵 FW
t 其 . It is straight-martingale,
and
that
the
variance process
forward to verify that the energy process H t is an 兵 FW
其
t
-supermartingale.
That
is
to
say,
V t is an 兵 FW
其
t
Es 关 H t 兴 ⫽H s ,
共4兲
Es 关 V t 兴 ⭐V s .
共5兲
and
These relations can be deduced by applying Ito’s lemma to 共2兲 and 共3兲, from which we infer that
dH t ⫽ ␴ V t dW t ,
共6兲
dV t ⫽⫺ ␴ 2 V 2t dt⫹ ␴ ␤ t dW t ,
共7兲
and that
where
␤ t⫽
具 ␺ t 兩 共 Ĥ⫺H t 兲 3 兩 ␺ t 典
具 ␺ t兩 ␺ t典
共8兲
is the skewness of the energy distribution at time t. The fact that 共6兲 has no drift shows that H t is
a martingale, and the fact that the drift in 共7兲 is negative shows that V t is a supermartingale.
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J. Math. Phys., Vol. 43, No. 11, November 2002
D. C. Brody and L. P. Hughston
In the case of the ordinary Schrödinger equation with a time-independent Hamiltonian, the
energy process 共2兲 is constant. This is usually interpreted as the quantum mechanical expression of
an energy conservation law. However, if a system is in an indefinite state of energy, then it is not
immediately evident what is meant by energy conservation. The martingale condition Es 关 H t 兴
⫽H s can be interpreted as a generalized energy conservation law applicable if a system is in an
indefinite state of energy. In particular, it implies that once state reduction has occurred, the
probabilistic average of the outcome for the energy must equal the initial expectation.
The supermartingale property satisfied by V t is the essence of what is meant by a reduction
process. In fact, it follows from Eq. 共7兲 that the asymptotic behavior of V t is given by
lim E关 V t 兴 ⫽0,
共9兲
t→⬁
which signifies the collapse of the wave function. A positive supermartingale with the property
that its expectation vanishes in the limit as t goes to infinity is called a potential process. Writing
H ⬁ for the random terminal value of the energy, one can then prove as a consequence of 共6兲 and
共7兲 that
H t ⫽Et 关 H ⬁ 兴
共10兲
V t ⫽Et 关共 H ⬁ ⫺H t 兲 2 兴 .
共11兲
and that
That is to say, the processes H t and V t are respectively the FW
t -conditional mean and variance of
H⬁ .
With these facts in hand, we now present a method for solving the stochastic equation 共1兲. Our
approach is based on the theory of nonlinear filtering.6 Filtering techniques have been shown to be
useful in quantum optics in connection with the theory of continuous observations, and in some
situations phenomenological equations similar to 共1兲 for the a posteriori dynamics of a continuously observed system can be derived.7 We shall, however, regard the dynamics of 兩 ␺ t 典 as being
given, and use the filtering methodology with a different end in view: namely, to construct the
solution of 共1兲.
The setup is as follows. We denote by E i (i⫽1,2,...) the eigenvalues of the Hamiltonian of a
given quantum system, and write
␲ i⫽
兩 具 ␺ 0兩 ␺ i典 兩 2
具 ␺ 0 兩 ␺ 0 典具 ␺ i 兩 ␺ i 典
共12兲
for the transition probability from the given initial state 兩 ␺ 0 典 to the eigenstate 兩 ␺ i 典 with energy E i .
If the spectrum of Ĥ is degenerate, then 兩 ␺ i 典 denotes the Lüders state, i.e., the projection of 兩 ␺ 0 典
onto the linear subspace of states corresponding to the eigenvalue E i .
Now let the probability space 共⍀, F, P兲 be given, and on it specify a random variable H that
takes the values E i with probabilities ␲ i . We assume that 共⍀, F, P兲 comes equipped with a
filtration 兵 Gt 其 with respect to which a standard Brownian motion B t is specified, and that H and B t
are independent. We assign no a priori physical significance to H and B t , which are introduced
here simply as an ansatz for obtaining a solution for 共1兲. Next we define the process
␰ t ⫽ ␴ Ht⫹B t .
共13兲
Intuitively, we can think of ␰ t as giving us a ‘‘noisy’’ representation of the information encoded in
the random variable H.
We let 兵 F␰t 其 denote the filtration generated by the process ␰ t , i.e., the information generated
by ␰ t as time progresses, and consider the conditional expectation
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J. Math. Phys., Vol. 43, No. 11, November 2002
Efficient simulation of quantum state reduction
H t ⫽E关 H 兩 F␰t 兴 .
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共14兲
Clearly, F␰t 傺Gt since knowledge of H together with the path 兵 B s 其 0⭐s⭐t implies knowledge of the
path 兵 ␰ s 其 0⭐s⭐t , although the converse is not the case.
It follows from the tower property that H t is an 兵 Ft␰ 其 -martingale. One can think of H t as
representing an estimate for the value of H given the history of the process ␰ s from time 0 up to
time t. More precisely, an 兵 F␰t 其 -measurable random variable H t minimizes the expectation of the
squared deviation of H from H t , given the history of ␰ s from time 0 to time t, if and only if 共14兲
holds. This can be seen by varying the expression E关 (H⫺H t ) 2 兩 F␰t 兴 with respect to H t .
We shall now establish the remarkable fact that the process H t defined by (14) is statistically
indistinguishable from the energy process (2) associated with the stochastic extension of the
Schrödinger equation (1).
The argument goes as follows. First, because ␰ t is a Markov process satisfying
lim t ⫺1 ␰ t ⫽ ␴ H,
共15兲
t→⬁
we can replace 共14兲 with the simpler relation H t ⫽E关 H 兩 ␰ t 兴 . In other words, to determine the
conditional expectation of H given the path 兵 ␰ s 其 0⭐s⭐t , it suffices to condition on the value ␰ t of
the process at the end of the path.
To calculate E关 H 兩 ␰ t 兴 , we require a version of the Bayes formula applicable when we consider
the probability of a discrete random variable conditioned on the value of a continuous random
variable. This is given by
P共 H⫽E i 兩 ␰ t 兲 ⫽
␲ i ␳ 共 ␰ t 兩 H⫽E i 兲
,
兺 i ␲ i ␳ 共 ␰ t 兩 H⫽E i 兲
共16兲
where
␲ i ⫽P共 H⫽E i 兲 .
共17兲
Here ␳ ( ␰ t 兩 H⫽E i ) denotes the conditional probability density for the continuous random variable
␰ t given that H⫽E i . Since B t is a standard Brownian motion, the conditional density for ␰ t is
␳ 共 ␰ t 兩 H⫽E i 兲 ⫽
1
冑2 ␲ t
冉
exp ⫺
冊
1
共 ␰ ⫺ ␴ E it 兲2 .
2t t
共18兲
It follows from the Bayes law 共16兲 that the conditional probability for the random variable H is
P共 H⫽E i 兩 ␰ t 兲 ⫽
␲ i exp共 ␴ E i ␰ t ⫺ 21 ␴ 2 E 2i t 兲
兺 i ␲ i exp共 ␴ E i ␰ t ⫺ 12 ␴ 2 E 2i t 兲
.
共19兲
Therefore, multiplying each side of 共19兲 by E i and summing over i, we deduce that the conditional
expectation of the random variable H given ␰ t is
H t⫽
兺 i ␲ i E i exp共 ␴ E i ␰ t ⫺ 12 ␴ 2 E 2i t 兲
兺 i ␲ i exp共 ␴ E i ␰ t ⫺ 12 ␴ 2 E 2i t 兲
.
共20兲
In order to show that H t is the energy process associated with 共1兲, one further key result is
required: namely, that the process W t defined by
W t⫽ ␰ t⫺ ␴
冕
t
0
H s ds
共21兲
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5258
J. Math. Phys., Vol. 43, No. 11, November 2002
D. C. Brody and L. P. Hughston
is an 兵 Ft␰ 其 -Brownian motion. To verify this, it suffices, by virtue of Lévy’s characterization of
Brownian motion,6 to demonstrate 共a兲 that (dW t ) 2 ⫽dt and 共b兲 that W t is an 兵 F␰t 其 -martingale. To
verify 共a兲 we note that 共13兲 implies
d␰ t ⫽ ␴ Hdt⫹dB t ,
共22兲
and thus (d␰ t ) 2 ⫽dt. On the other hand, 共21兲 implies that dW t ⫽d␰ t ⫺ ␴ H t dt, and hence (dW t ) 2
⫽(d␰ t ) 2 . To establish 共b兲, let 共20兲 define a function H( ␰ ,t) of two variables such that H t
⫽H( ␰ t ,t):
H 共 ␰ ,t 兲 ⫽
兺 i ␲ i E i exp共 ␴ E i ␰ ⫺ 12 ␴ 2 E 2i t 兲
兺 i ␲ i exp共 ␴ E i ␰ ⫺ 12 ␴ 2 E 2i t 兲
共23兲
.
Then applying Ito’s lemma and using the relation (d␰ t ) 2 ⫽dt, we obtain
dH t ⫽ 共 ⳵ t H 共 ␰ t ,t 兲 ⫹ 21 ⳵ 2␰ H 共 ␰ t ,t 兲兲 dt⫹ ⳵ ␰ H 共 ␰ t ,t 兲 d␰ t ,
共24兲
where ⳵ t H( ␰ t ,t) denotes ⳵ H( ␰ ,t)/ ⳵ t valued at ␰ ⫽ ␰ t , and so on. A calculation making use of
共21兲, 共23兲, and 共24兲 then shows that
dH t ⫽ ␴ V 共 ␰ t ,t 兲 dW t ,
共25兲
where the function V( ␰ ,t) is
V 共 ␰ ,t 兲 ⫽
兺 i ␲ i 共 E i ⫺H 共 ␰ ,t 兲兲 2 exp 共 ␴ E i ␰ ⫺ 21 ␴ 2 E 2i t 兲
兺 i ␲ i exp 共 ␴ E i ␰ ⫺ 12 ␴ 2 E 2i t 兲
.
共26兲
Because H t is an 兵 F␰t 其 -martingale, we conclude that W t is also an 兵 F␰t 其 -martingale, and that
establishes 共b兲. We thus deduce that W t is an 兵 F␰t 其 -Brownian motion, with respect to which ␰ t is
a diffusion process satisfying
d␰ t ⫽ ␴ H 共 ␰ t ,t 兲 dt⫹dW t .
共27兲
Now let 兩 ␺ 0 典 be the initial normalized state vector of the quantum system, and let P̂ i denote
for each value of i the projection operator onto the Hilbert subspace corresponding to the energy
eigenvalue E i . We let
兩 ␺ i 典 ⫽ ␲ ⫺1/2
P̂ i 兩 ␺ 0 典
i
共28兲
denote the Lüders state corresponding to E i , and write
⌸ it ⫽P共 H⫽E i 兩 ␰ t 兲
共29兲
for the process defined by 共19兲. Then we can establish our main result, that
兩 ␺ t典 ⫽
兺i e ⫺iE t ⌸ 1/2
it 兩 ␺ i 典
i
共30兲
satisfies the stochastic extension of the Schrödinger equation 共1兲 with the given initial condition.
In particular, by applying Ito’s lemma to 共19兲 and using 共27兲 we obtain
d⌸ it ⫽ ␴ 共 E i ⫺H t 兲 ⌸ it dW t .
共31兲
With another application of Ito’s lemma we deduce that
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J. Math. Phys., Vol. 43, No. 11, November 2002
Efficient simulation of quantum state reduction
1 2
1
1/2
2 1/2
d⌸ 1/2
it ⫽⫺ 8 ␴ 共 E i ⫺H t 兲 ⌸ it dt⫹ 2 ␴ 共 E i ⫺H t 兲 ⌸ it dW t .
5259
共32兲
A short calculation then shows that 共30兲 satisfies the stochastic equation 共1兲, and that the expectation of the operator Ĥ in the state 兩 ␺ t 典 is the process 共20兲.
In summary, the stochastic equation 共1兲 can be solved as follows. We let H be a random
variable taking the values E i with the probabilities ␲ i defined by 共12兲. Letting B t denote an
independent Brownian motion, we define ␰ t as in 共13兲. The solution of 共1兲 is then given by
兩 ␺ t典 ⫽
兺 i 冑␲ i exp共 ⫺iE i t⫹ 21 ␴ E i ␰ t ⫺ 14 ␴ 2 E 2i t 兲 兩 ␺ i 典
共 兺 i ␲ i exp 共 ␴ E i ␰ t ⫺ 12 ␴ 2 E 2i t 兲兲 1/2
,
共33兲
where the 兵 F␰t 其 -Brownian motion W t driving 兩 ␺ t 典 in 共1兲 is given by 共21兲. Expression 共20兲 for H t
follows at once from 共33兲 since 具 ␺ i 兩 Ĥ 兩 ␺ j 典 ⫽E i ␦ i j , and for the variance of the energy we deduce
that V t ⫽V( ␰ t ,t). To obtain a realization of the process 兩 ␺ t 典 , i.e., to carry out a simulation, we
simply choose a value for H in accordance with the given probability law, and then let B t run its
course.
The fact that 共20兲 is indeed a reduction process can be verified directly as follows. Suppose,
in a particular realization of the process H t , the random variable H takes the value E j for some
choice of the index j. Writing ␻ i j ⫽E i ⫺E j and seting ␰ t ⫽ ␴ E j t⫹B t , we obtain
H t⫽
␲ j E j ⫹ 兺 i 共 ⫽ j 兲 ␲ i E i exp 共 ␴␻ i j B t ⫺ 12 ␴ 2 ␻ 2i j t 兲
␲ j ⫹ 兺 i 共 ⫽ j 兲 ␲ i exp 共 ␴␻ i j B t ⫺ 12 ␴ 2 ␻ 2i j t 兲
,
共34兲
for the corresponding realization of H t . However, the exponential martingale M i jt defined for i
⫽ j by
M i jt ⫽exp 共 ␴␻ i j B t ⫺ 12 ␴ 2 ␻ 2i j t 兲
共35兲
that appears in expression 共34兲 has the property
lim P共 M i jt ⬎0 兲 ⫽0.
共36兲
t→⬁
Given that
H t⫽
␲ j E j ⫹ 兺 i 共 ⫽ j 兲 ␲ i E i M i jt
,
␲ j ⫹ 兺 i 共 ⫽ j 兲 ␲ i M i jt
共37兲
we see that H t converges to the value E j with probability one. A similar argument shows that if
H⫽E j , then for each value of i we have limt→⬁ ⌸ it ⫽0 unless i⫽ j, which allows us to verify that
兩 ␺ t 典 converges to the Lüders state corresponding to the energy eigenvalue E j with probability
one.4
Therefore, we see that the random variable H can be identified with the terminal value H ⬁ of
the energy process. The fact that H is not FW
t -measurable for t⬍⬁ indicates that the ‘‘true value’’
of H is ‘‘hidden’’ until the reduction process is complete. On a related point we note that in
stochastic models for state reduction it is sometimes assumed that the driving process W t is in
some way ‘‘external’’ to the quantum system. This assumption, however, is unnecessary: the
filtrations associated with W t , ␰ t , H t , and 兩 ␺ t 典 all coincide, and it is thus consistent to regard the
innovation process W t as being endogenous.
The advantage of expressions 共20兲 and 共33兲 from a computational point of view is that H t and
兩 ␺ t 典 are expressed algebraically in terms of the underlying random variable H and the Brownian
motion B t . These quantities can be thought of as representing independent state variables for the
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5260
J. Math. Phys., Vol. 43, No. 11, November 2002
D. C. Brody and L. P. Hughston
reduction dynamics. As a consequence, we are able to investigate properties of the process 共1兲
directly without having to resort to numerical integration. In particular, by use of 共33兲, a numerical
simulation of the state reduction of complex systems is feasible, including cases for which the
Hamiltonian has a nondiscrete spectrum. It should be emphasized that in our simulation methodology there is no need at any stage for the introduction of a change of probability measure.
In conclusion we present a probabilistic analysis of the timescale associated with the reduction
process, in the case of a two-state system with energies E 1 and E 2 . The initial state is given by
兩 ␺ 0 典 , and the transition probabilities to the energy eigenstates 兩 E 1 典 and 兩 E 2 典 are given by ␲ 1 and
␲2 .
Suppose a measurement of the energy is made, and we condition on the outcome of the
measurement being E 1 . In that case, according to 共37兲, we have
H t⫽
␲ 1 E 1 ⫹ ␲ 2 E 2 M 21t
,
␲ 1 ⫹ ␲ 2 M 21t
共38兲
where
2
t 兲.
M 21t ⫽exp 共 ␴␻ 21B t ⫺ 12 ␴ 2 ␻ 21
共39兲
2
Writing ␤ ⫽ 41 ␴ 2 ␻ 21
for the parameter that determines the characteristic rate of reduction, we can
work out the probability that M 21t ⬍e ⫺n for some value of n. Since B t is normally distributed with
zero mean and variance t, we find
P共 M 21t ⬍e ⫺n 兲 ⫽N 共共 ␤ t 兲 1/2⫺ 21 n 共 ␤ t 兲 ⫺1/2兲 ,
共40兲
where N(x) is the standard normal distribution function. Therefore, for example, we see that
provided t⬎5 ␶ R , we have
P共 M 21t ⬍e ⫺10兲 ⬎ 21 ,
共41兲
where ␶ R ⫽1/␤ . In particular, as H t draws near E 1 we have the relation
H t ⫺E 1 ⬃
␲2
共 E ⫺E 1 兲 M 21t .
␲1 2
共42兲
Thus, after only a relatively few multiples of the characteristic reduction timescale, the amount by
which H t differs from E 1 will with high probability be reduced to a tiny fraction of the energy
difference E 2 ⫺E 1 .
ACKNOWLEDGMENTS
D.C.B. acknowledges support from The Royal Society. L.P.H. acknowledges the hospitality
and support of the Institute for Advanced Study, Princeton, where part of this work was carried
out. We are grateful to S. L. Adler and T. A. Brun for stimulating discussions.
N. Gisin, Helv. Phys. Acta 62, 363 共1989兲; G. C. Ghirardi, P. Pearle, and A. Rimini, Phys. Rev. A 42, 78 共1990兲; I. C.
Percival, Proc. R. Soc. London, Ser. A 447, 189 共1994兲; 451, 503 共1995兲.
2
L. P. Hughston, Proc. R. Soc. London, Ser. A 452, 953 共1996兲.
3
S. L. Adler and L. P. Horwitz, J. Math. Phys. 41, 2485 共2000兲; S. L. Adler and I. Mitra, Phys. Rev. E 62, 4386 共2000兲;
S. L. Adler and T. A. Brun, J. Phys. A 34, 4797 共2001兲; S. L. Adler, in Chance in Physics, Foundations and Perspectives,
edited by J. Bricmont et al. 共Springer, Berlin, 2001兲; J. Phys. A 35, 841 共2002兲; D. C. Brody and L. P. Hughston, Proc.
R. Soc. London, Ser. A 458, 1117 共2002兲.
4
S. L. Adler, D. C. Brody, T. A. Brun, and L. P. Hughston, J. Phys. A 34, 8795 共2001兲.
5
R. Schack, T. A. Brun, and I. C. Percival, J. Phys. A 28, 5401 共1995兲; N. Gisin and M. Rigo, ibid. 28, 7375 共1995兲; R.
Schack and T. A. Brun, Comput. Phys. Commun. 102, 210 共1997兲; I. C. Percival, Quantum State Diffusion 共Cambridge
1
Downloaded 24 Oct 2002 to 155.198.17.120. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jmp/jmpcr.jsp
J. Math. Phys., Vol. 43, No. 11, November 2002
Efficient simulation of quantum state reduction
5261
University Press, Cambridge, 1998兲; M. B. Plenio and P. L. Knight, Rev. Mod. Phys. 70, 101 共1998兲.
R. S. Liptser and A. N. Shiryaev, Statistics of Random Processes, 2nd ed. 共Springer, Berlin, 2000兲, Vols. I and II.
7
L. Diosi, Phys. Lett. A 129, 419 共1988兲; A. Barchielli and V. P. Belavkin, J. Phys. A 24, 1495 共1991兲; A. Barchielli, Rep.
Math. Phys. 33, 21 共1993兲; A. S. Holevo, Statistical Structure of Quantum Theory 共Springer, Heidelberg, 2001兲.
6
Downloaded 24 Oct 2002 to 155.198.17.120. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jmp/jmpcr.jsp