A density tensor hierarchy for open system dynamics: retrieving the noise
Stephen L. Adler
TL;DR
This paper introduces a density tensor hierarchy for open quantum systems that captures quantum and classical fluctuations, providing a framework to analyze noise and dynamics beyond the reduced density matrix.
Contribution
It develops a novel hierarchy formalism for open system fluctuations, linking classical and quantum noise descriptions, with applications to stochastic equations and master equations.
Findings
Hierarchy recovers classical fluctuation information lost in reduced density matrices.
Trace formalism applies to quantum noise in system-environment interactions.
Examples include stochastic Schrödinger equations and quantum Brownian motion.
Abstract
We introduce a density tensor hierarchy for open system dynamics, that recovers information about fluctuations lost in passing to the reduced density matrix. For the case of fluctuations arising from a classical probability distribution, the hierarchy is formed from expectations of products of pure state density matrix elements, and can be compactly summarized by a simple generating function. For the case of quantum fluctuations arising when a quantum system interacts with a quantum environment in an overall pure state, the corresponding hierarchy is defined as the environmental trace of products of system matrix elements of the full density matrix. Only the lowest member of the quantum noise hierarchy is directly experimentally measurable. The unit trace and idempotence properties of the pure state density matrix imply descent relations for the tensor hierarchies, that relate the order…
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April, 2007
A density tensor hierarchy for open
system dynamics: retrieving the noise
Stephen L. Adler
Institute for Advanced Study
Princeton, NJ 08540
Send correspondence to:
Stephen L. Adler
Institute for Advanced Study
Einstein Drive, Princeton, NJ 08540
Phone 609-734-8051; FAX 609-924-8399; email [email protected]
ABSTRACT
We develop a density tensor hierarchy for open system dynamics, that recovers information about fluctuations (or “noise”) lost in passing to the reduced density matrix. For the case of fluctuations arising from a classical probability distribution, the hierarchy is formed from expectations of products of pure state density matrix elements, and can be compactly summarized by a simple generating function. For the case of quantum fluctuations arising when a quantum system interacts with a quantum environment in an overall pure state, the corresponding hierarchy is defined as the environmental trace of products of system matrix elements of the full density matrix. Whereas all members of the classical noise hierarchy are system observables, only the lowest member of the quantum noise hierarchy is directly experimentally measurable. The unit trace and idempotence properties of the pure state density matrix imply descent relations for the tensor hierarchies, that relate the order tensor, under contraction of appropriate pairs of tensor indices, to the order tensor. As examples to illustrate the classical probability distribution formalism, we consider a spatially isotropic ensemble of spin-1/2 pure states, a quantum system evolving by an Itô stochastic Schrödinger equation, and a quantum system evolving by a jump process Schrödinger equation. As examples to illustrate the corresponding trace formalism in the quantum fluctuation case, we consider the tensor hierarchies for collisional Brownian motion of an infinite mass Brownian particle, and for the the weak coupling Born-Markov master equation. In different specializations, the latter gives the hierarchies generalizing the quantum optical master equation and the Caldeira–Leggett master equation. As a further application of the density tensor, we contrast stochastic Schrödinger equations that reduce and that do not reduce the state vector, and discuss why a quantum system coupled to a quantum environment behaves like the latter. The descent relations for our various examples are checked in a series of Appendices.
1. Introduction
Increasing attention is being paid to the dynamics of open quantum systems, that is, to quantum systems acted on by an environment. Such systems are of interest for studies of dissipative phenomena, decoherence, backgrounds to quantum computers and to precision measurements, and theories of quantum measurement. A principal tool in studying open quantum systems is the reduced density matrix, obtained from the pure state density matrix by tracing over environment degrees of freedom, or in stochastic models where the environment is represented by a noise term in the Schrödinger equation, by averaging over the noise. As is well-known, this transition from the pure state density matrix to the reduced density matrix is not one-to-one, since information about the total system is lost. For example, in stochastic models, there is known to be a continuum of different unravelings, or pure state density matrix stochastic evolutions, that yield the same master equation for the reduced density matrix. The question that we investigate here is the extent to which one can form objects that refer only to the basis vectors of the system Hilbert space, but that nonetheless recapture information that is lost in passing to the reduced density matrix. In the first part of this paper (Sections 2 through 5), we discuss classical noise arising from fluctuations defined by classical probability distributions. In the second part (Sections 6 through 9), we give an analogous discussion of quantum noise, which appears in the physically important case of a quantum system coupled to a quantum environment in an overall pure state. We also give an extension, making contact with the discussion of the first part, to the case in which the overall system is in a mixed state superposition of pure states. The final section contains a discussion of quantum measurements that relates the material in the first and second parts.
For the case of classical probability distributions, a relevant discussion appears in Chapter 5 of the book The Theory of Open Quantum Systems by Breuer and Petruccione [1], following up on earlier papers by those authors [2], by Wiseman [3] and by Mølmer, Castin, and Dalibard [4]. In simplified form, Breuer and Petruccione introduce an ensemble of pure state vectors , each drawn from the same system Hilbert space , with each vector assumed to occur in the ensemble with probability , . Measurement of a general self-adjoint operator for a system prepared in typically gives a range of values, the mean of which given by . The mean or expectation over the ensemble of pure state vectors is then given by
[TABLE]
with the mixed state or reduced density matrix defined by
[TABLE]
Breuer and Petruccione point out that there are three variances that are relevant. The variance of measurements of over all pure states in the ensemble is given by
[TABLE]
This can be written as the sum of two non-negative terms,
[TABLE]
with the ensemble average of the variances of within each pure state of the ensemble,
[TABLE]
and with the variance of the pure state means of over the ensemble,
[TABLE]
Thus, is an ensemble average of the quantum variances of , while is a measure of the spread of the average values of resulting from the statistical properties of the ensemble. As Breuer and Petruccione note, neither of the subsidiary variances can be expressed as the density matrix expectation of some self-adjoint operator.
Our aim in the first part of this paper is to extend the formalism of ref [1] by utilizing a density tensor hierarchy, which captures the statistical information that is lost in forming the reduced density matrix of Eq. (1b). A density tensor, defined as an ensemble average of density matrices, was first introduced by Mielnik [5], and was applied to discussions of density functions on the space of quantum states and their application to thermalization of quantum systems by Brody and Hughston [6]. These papers, in addition to introducing the concept of a density tensor which is developed further here, also contain the important result that in the case of a continuum probability distribution, the density tensor hierarchy gives all of the information needed to reconstruct the probability function . In particular, the variances for any observable, and more general statistical properties of the ensemble as well, can be expressed as contractions of density tensor matrix elements with appropriate matrix elements of the observable(s) of interest.
The basic construction of the density tensor hierarchy corresponding to a classical probability distribution is given in Sec. 2. Here we generalize the reduced density matrix of Eq. (1b) to a density tensor, formed by taking a product of pure state density matrix elements, and averaging over the ensemble of pure states. When the are independent of , this tensor reduces to an -fold product of reduced density matrices, and so the difference between the density tensor and this product is a measure of the statistical fluctuations in the ensemble. In the generic case of non-trivial dependence of on , there are some general statements that can be made. First of all, the order density tensor is a symmetric tensor in its pair indices, and it can be considered as a matrix operator acting on the -fold tensor product of the system Hilbert space with itself. The symmetry of the density tensor allows construction of a generating function that on expansion gives the density tensors of all orders. Additionally, as a consequence of the unit trace and idempotence conditions obeyed by the pure state density matrix, the density tensor hierarchy satisfies a system of descent equations, relating the order tensor to the order tensor when any row index is contracted with any column index. We show that the variances defined by Breuer and Petruccione can be expressed in terms of appropriate contractions of density tensor elements with operator matrix elements.
In subsequent sections we develop some concrete applications of the general formalism for classical probability distributions. In Sec. 3, we consider an isotropic ensemble of spin-1/2 pure state density matrices, construct the density tensors through order 3, verify the descent equations, and calculate the generating function. In Sec. 4 we apply the formalism to a quantum system evolving under the influence of noise as described by a stochastic Schrödinger equation, with the ensemble defined as the set of all histories of an initial quantum state under the influence of the noise. Assuming white noise described by the Itô calculus, we give the dynamics of the general density tensor in terms of the general unraveling of the Lindblad equation constructed by Wiseman and Diósi [7], and show that the order two and higher density tensors distinguish between inequivalent unravelings that give the same reduced density matrix (i.e., the same order one density tensor). In Sec. 5 we develop an analogous formalism for the case of jump (piecewise deterministic process) unravelings of the Lindblad equation.
We turn next to an analysis of a quantum system coupled to a quantum environment, rather than to an external classical noise source. Here, one is confronted with the problem of discussing the system dissipation associated with the system-environment interaction within a single overall pure state of system plus environment (or in a thermal state that is a weighted average of such pure states). Typically, in master equation derivations, the system-environment interaction††1 What we call is usually denoted by in the open systems literature. To avoid confusion, all other Hamiltonians will carry subscripts, e.g., and for the system and environment Hamiltonians, for the total Hamiltonian, etc. has vanishing expectation in the environment, but its square does not have a vanishing expectation, because the environment is not in an eigenstate of . The associated variance is then a measure of quantum fluctuations associated with the environment state, and is the source of quantum “noise” driving the system dissipation. Our aim in the second part of this paper is to generalize the formalism of the first part to recapture information about this noise that is lost in the passage to the system reduced density matrix. We do this in Sec. 6 by defining a density tensor hierarchy as the trace over the environment of a product of environment operators constructed as the system matrix elements of the total density matrix. Unlike the classical noise construction, which uses only the system density matrix, the construction in the quantum noise case requires knowledge of the full system plus environment density matrix, and so (except for the order one case) does not give a system observable. It is nonetheless computable in any theory of the system plus environment, and is of theoretical, rather than empirical, interest. Because the environment operators entering the construction are non-commutative, this hierarchy is no longer totally symmetric in its system index pairs, but by the cyclic permutation property of the trace, it is symmetric under cyclic permutation of the system index pairs. Also, because the system trace of these environment operators gives only the reduced environment density matrix, rather than unity, there is in general no descent equation associated with taking this trace. However, when indices of adjacent system operators are contracted, one gets the square of the overall density matrix, and so there remains a set of descent relations connecting the order tensor to the order tensor. Finally, in the case of thermal (or other mixed) overall states, we define the appropriate tensor as a weighted sum of pure state tensors, in analogy with the definition of Sec. 2.
In subsequent sections, we give applications of the trace hierarchy formalism to several classic problems discussed in the theory of quantum master equations. In Sec. 7 we consider the quantum Brownian motion (and resulting decoherence) of a massive Brownian particle in interaction with an independent particle bath of scatterers. In Sec. 8 we discuss the tensor hierarchy corresponding to the weak coupling Born–Markov master equation, and it specialization to the quantum optical master equation. Finally in Sec. 9, we give an analogous discussion for the Caldeira–Leggett model of a particle in interaction with a system of environmental oscillators.
We conclude with a discussion that bridges the considerations of the classical noise and the quantum noise cases. In Sec. 10, we contrast two different Itô stochastic Schrödinger equations, both of which have the same Lindblad, but only one of which leads to state vector reduction. We relate this to the fact that the equation giving the time derivative of the stochastic expectation of operator variances involves the order two density tensor, which differs for the two cases. We discuss the analogous equation for the time dependence of the variance of a “pointer operator” in the case of a quantum system coupled to a quantum environment, and show why this does not lead to state vector reduction. Thus we see no mechanism for quantum “noise” in a closed quantum system plus environment to provide a resolution of the quantum measurement problem.
2. The density tensor for classical noise and its kinematical properties
We proceed to establish our notation and to define the density tensor hierarchy in the classical noise case. We denote the pure state density matrix formed from the unit normalized state by , with
[TABLE]
and its general matrix element between states and of by
[TABLE]
The unit trace condition on states that
[TABLE]
and the idempotence condition on states that
[TABLE]
We now define the order density tensor by
[TABLE]
with a shorthand for
[TABLE]
Since
[TABLE]
we see that this is just the to matrix element of the reduced density matrix defined in Eq. (1b),
[TABLE]
and so the density tensor of Eq. (4a) is a natural generalization of the usual reduced density matrix. When the states are independent of the label , the definition of Eq. (4a) simplifies to
[TABLE]
and so the difference between Eq. (4a) and a product of reduced density matrix elements is a reflection of the statistical structure of the ensemble. Since the factors within the expectation on the right of Eq. (4a) are just ordinary complex numbers, the density tensor is symmetric under interchange of any index pair with any other index pair . Consequently, we can define a generating function for the density tensor by
[TABLE]
where repeated indices are summed. It will often be convenient to abbreviate by , so that the generating function becomes in this notation .
Although the density tensor for is not an operator on , it clearly has the structure of an operator on the -fold tensor product . Motivated by this, we will often find it convenient to write the definition of Eq. (4a) as
[TABLE]
with each factor acting on a distinct factor Hilbert space in the tensor product . One can pass easily back and forth from this notation to one in which the system matrix indices are displayed explicitly.
Let us consider next the result of contracting any row index with any column index . There are two basic cases: (i) one can contract a row index with its corresponding column index , and (ii) one can contract a row index with a column index with . Since the density tensor is symmetric in its index pairs, it suffices to consider only one example of each case, since all others can be obtained by permutation. For the contraction of with we find
[TABLE]
where we have used the unit trace condition of Eq. (3c). For the contraction of with , we find
[TABLE]
where now we have used the idempotence condition of Eq. (3d). As an illustration of how this works when all possible index pair contractions are considered, we give the complete set of contractions reducing the second order density tensor to a first order density tensor,
[TABLE]
Referring to the generating function of Eq. (5d), the general descent equations can be summarized compactly by the two identities,
[TABLE]
When the density matrix used to define the density tensor is a mixed state density matrix, the trace descent relation of Eq. (6a) is unchanged, while the indempotency relation of Eq. (6b) relates the contraction an order tensor to an order tensor in which one factor is replaced by ; this is not a member of the original hierarchy, but still gives a useful relation for checking calculations.
To conclude this section, let us return to the variances introduced by Breuer and Petruccione. In terms of the order one and order two density tensors, we evidently have
[TABLE]
with . Clearly, other statistical properties of the ensemble are readily expressed in terms of the density tensor hierarchy. For example, the ensemble average of the product of the expectations of two different operators and is given by
[TABLE]
which can be used, together with information obtained from , to calculate the covariance and correlation of and .
3. Isotropic spin-1/2 ensemble
As a simple example of the density tensor formalism, let us follow Breuer and Petruccione [1] and consider the case of an isotropic spin-1/2 ensemble. Let be a vector in three dimensions, and consider the ensemble of spin-1/2 pure state density matrices
[TABLE]
with the standard Pauli matrices, and with a uniform probability distribution of over the unit sphere specified by
[TABLE]
(Clearly, has the same significance as the label used in the preceding section.) Defining
[TABLE]
a standard calculation gives
[TABLE]
with all averages of odd powers of vanishing. From Eq. (9a), we have
[TABLE]
and the general density tensor over this ensemble is defined by
[TABLE]
From Eq. (10b), the first three tensors in this hierarchy are now easily found to be
[TABLE]
Using the relations and , it is now easy to verify that the descent relations of Eqs. (6a) and (6b) are satisfied by Eq. (12).
For the isotropic spin-1/2 ensemble, the generating function of Eq. (5d) becomes
[TABLE]
with given by Eq. (11a). Defining the vector by
[TABLE]
a simple calculation gives
[TABLE]
from which one can read off the values of the low order density tensors given in Eq. (12). The verification of the descent relations of Eq. (7b) for the generating function of Eq. (14b) is given in Appendix A.
4. Itô stochastic Schrödinger equation
We consider next a state vector with a time evolution described by a stochastic Schrödinger equation, which is a frequently used model approximation to open system dynamics. In this case the state vector and the corresponding pure state density matrix are implicit functions of the noise, which takes a different sequence of values for each history of the system. In the notation of Sec. 2, the different histories are labeled by the subscript , and the expectation of Eq. (4b) is an average over all possible histories. It is customary, however, in discussing stochastic Schrödinger equations to omit the subscript , treating the history dependence of as understood. So in this context, the definition of Eq. (4a) becomes
[TABLE]
with the stochastic expectation, and the generating function takes the same form as given in Eq. (5d) but with the subscript omitted.
Our aim in this section is to derive an equation of motion for the generating function, which on expansion yields equations of motion for all density tensors , taking as input the general pure state density matrix evolution constructed by Wiseman and Diósi [7], that corresponds to a given Lindblad form [8,9] for the time evolution of the reduced density matrix . We begin by recapitulating the results of ref [7]. The most general evolution of a density matrix that preserves and obeys the complete positivity condition is the Lindblad form
[TABLE]
with
[TABLE]
with denoting the anticommutator, and with the repeated index summed. The set of Lindblad operators describes the effects on the system of the reservoir or environment that is modeled by an external classical noise. Wiseman and Diósi show that the most general evolution of the pure state density matrix for which reduces to Eqs. (16a) and (16b) takes the form
[TABLE]
Here is a state vector that is a pure noise term, so that
[TABLE]
that is orthogonal to , so that
[TABLE]
and that obeys
[TABLE]
The operator is the Diósi transition rate operator [5] given by
[TABLE]
where is a shorthand for the quantum state expectation . Although is completely fixed, Wiseman and Diósi show that is free, with different choices for this and different phase choices for the corresponding to different pure state evolutions (or “unravelings”) that yield the same evolution of Eqs. (16a) and (16b) for the reduced density matrix .
Wiseman and Diósi further show that can be parameterized by complex Wiener processes by writing
[TABLE]
with
[TABLE]
and with
[TABLE]
where is a set of arbitrary complex numbers subject to the condition that the norm of the complex matrix be less than or equal to 1. (See Eqs. (4.10) and (4.11) of ref. [7].) In terms of this parameterization of , the pure state evolution of Eq. (17a) takes the form
[TABLE]
and the corresponding stochastic Schrödinger equation for the wave function is [7]
[TABLE]
We proceed now to use pure state evolution of Eq. (19d) to calculate the evolution equation for the generating function
[TABLE]
To calculate the differential of Eq. (20a), we use the Itô stochastic calculus rule for the differential of a function of a stochastic variable ,
[TABLE]
Applying this to Eq. (20a), we get
[TABLE]
Substituting Eq. (19d) for , and using Eqs. (19a-c), together with the Itô calculus rule , we get
[TABLE]
with the coefficient of the quadratic term in given by
[TABLE]
This expression can be rearranged by using the identity, valid for general operators , general states , and general pure state (idempotent) density matrix ,
[TABLE]
giving an alternative result for
[TABLE]
where we have used Eq. (18) defining the operator , and where we use the subscript notation of Eq. (3b) for matrix elements, so that in general .
From the evolution equation of Eqs. (21a,b) and (22b) for the generating function, by expansion in powers of we can read off the evolution equation for the general density tensor of order . Employing now the condensed notation of Eq. (5e), in which matrix indices are not indicated explicitly, we have
[TABLE]
Here denotes the product with the factor omitted, and similarly, denotes the product with the factors and omitted.††2 For , and , while for , . The coefficient is given by
[TABLE]
which corresponds in an obvious way to Eq. (21b) when matrix elements are written explicitly between states and in the Hilbert space labeled by , and between states and in the Hilbert space labeled by . (No relation is implied between the used as a state label, and the used as a Hilbert space label.) Since in Eq. (23a), which depends through the terms involving on the choice of unraveling, is multiplied by two powers of , it does not contribute to the evolution equation for the reduced density matrix . So as expected, the reduced density matrix evolution is given solely by the Lindblad term and is independent of the choice of unraveling. Higher density tensors , with , have evolution equations that receive contributions from , and so contain information that distinguishes between different unravelings of the Lindblad evolution.
As a simple illustration of how the tensors for distinguish between different unravelings, let us consider the case of real noise, , for which , and with a single Lindblad , which we choose as either or , with a self-adjoint operator. Both choices of lead to the same Lindblad, since is invariant under rephasing of , but through the terms they lead to different expressions for . When , we find from Eq. (21b)
[TABLE]
while when , we have instead
[TABLE]
We will return to this example in Sec. 10.
Using the expression of Eq. (21a) for the time evolution of the generating function, the descent equations of Eq. (7b) can be verified; this calculation is carried out in Appendix B.
5. Jump process Schrödinger equation
As our next density tensor application we consider the jump process (piecewise deterministic process, or PDP) Schrödinger equation, given by
[TABLE]
where a sum over is understood, with and the general (non-self-adjoint) operators, and with the independent discrete random variables obeying
[TABLE]
Straightforward calculation shows that this process preserves the norm of and the pure state condition , provided that and obey the restrictions
[TABLE]
with no summation over on the second line, which must hold individually for each value of . Corresponding to Eq. (25a), the density matrix obeys the evolution equation
[TABLE]
with a sum over understood in the term on the first line, but no sum over understood in the second line.
Let now denote an expectation conditioned on the current value of the wave function being , and be the expectation value over the entire history of the jump process (which leads to an ensemble of different current values of the wave function). We wish to find restrictions on , , and on
[TABLE]
such that the expectation of takes the Lindblad form of Eq. (16b), that is,
[TABLE]
Making the Ansatz
[TABLE]
with constants (this Ansatz includes both the standard quantum jump equation (), and the orthogonal jump equation (), as special cases; see Schack and Brun [10] for a concise review), some calculation shows that the conditions of Eqs. (25c) and (26b) are satisfied if we choose
[TABLE]
Let us now define the order density tensor for the jump models by
[TABLE]
where we use the condensed notation of Eq. (5e). For the differential of this, we find
[TABLE]
where all powers of must be retained because . Using the conditional probability formula , we get the conditional expectation formula, valid for an arbitrary function of the state ,
[TABLE]
Using this equation to evaluate the higher order terms in Eq. (28b), together with Eq. (26b) for the leading term, we get
[TABLE]
with a sum over in each term containing .
Writing the corresponding generating function in compact notation as
[TABLE]
the evolution equation for is given , with the sum now indicated explicitly, by
[TABLE]
From Eq. (30b), and the identities which follow, after some algebra, from Eqs. (16b), (25d), (27a), and (27b)
[TABLE]
one can prove that Eq. (30b) obeys the descent equations, as shown in Appendix C.
6. The density tensor for quantum noise and its kinematical properties
Let us now consider a closed quantum system, consisting of a system interacting with an environment . In such a situation, one does not have a classical probability distribution and fluctuations associated with this probability distribution. Instead, one deals with the system plus environment as the only pure state that is given, with the fluctuations that are averaged over in deriving the master equation coming from quantum fluctuations associated with the system-environment interaction. Weighted averages of the sort that we have used in our definition of Eq. (4a) appear only when the total state is a mixture of pure states, such as a thermal state, but in this case, important system quantum fluctuations still occur in each pure state component of this mixture. In order to describe this more general situation, we shall have to generalize our definition of a density tensor hierarchy.
To achieve this, we initially suppose the overall system plus environment to have the pure state density matrix . We denote the system basis states by , as well as , and denote the environment basis states by . A general density matrix element has the form , and the standard reduced density matrix, with the environment traced out, is defined by
[TABLE]
In order to recapture fluctuations that are averaged over in the trace in Eq. (31), we define the density tensor by
[TABLE]
Here we have defined as the matrix, labeled by the system state labels , acting on the environment Hilbert space according to
[TABLE]
The density tensor is again an operator on a tensor product of system Hilbert spaces . Thus, in a condensed notation analogous to that of Eq. (5e), we can also write Eq. (32a) as
[TABLE]
where is an operator acting on .
We have avoided using a product notation in Eq. (32c) because the factors in Eq. (32a) and in Eq. (32c) are different operators on the environment for each and thus do not commute. Hence the density tensor is not symmetric under permutation of its pair indices , but it is symmetric under cyclical permutation of the indices, as a result of the cyclic symmetry of the trace. For , cyclic symmetry is equivalent to symmetry under pair index interchange, and for , using the identity
[TABLE]
cyclic symmetry is equivalent to the statement that the density tensor can be written as the sum of two tensors , with completely symmetric, and completely antisymmetric, under pair index interchange. Also because the density tensor is not totally symmetric in its pair indices, we cannot introduce a generating function by imitating Eq. (5d)
Similarly, because of factor non-commutativity, the density tensor satisfies only a subset of the descent equations of Eqs. (6a), (6b), and (7b). Contraction with does not lead to a descent condition, since is not unity, but rather , the reduced density matrix that acts on the environment when the system is traced out. Contraction of a general with a general for gives nothing useful, since in general non-commuting factors stand between and . However, when a column index is contracted with the adjacent row index , the two density matrices to which they are attached are linked to form the product , and so we get the descent relation of Eq. (6b), and others related to it by cyclic permutation symmetry,
[TABLE]
As noted before, even when , the descent relation corresponding to Eq. (34) is still useful for checking calculations. Since we cannot define a generating function as in Eq. (5d), in the quantum noise case we do not have analogs of the descent equations in the form of Eq. (7b); when verifying the descent equations in the various cases considered below, we will work directly from Eq. (34).
We will also consider a more general definition of the density tensor, corresponding to the case in which the system plus environment is in a mixed state composed of pure states with weights . Typically, refers to an eigenvalue of a conserved quantum number of the total system, such as the energy; when the environment is considered in the independent particle approximation, with the system back reaction on the environment neglected, then can refer to the energies and momenta of each environmental particle. In this case we define the density tensor by
[TABLE]
with
[TABLE]
This definition gives information about both the quantum noise or fluctuations contained within each , and the classical noise or fluctuations associated with the probability distribution . Note that in the mixed state case one could also define a density tensor that is a direct analog of the classical noise definition of Sec. 2, by
[TABLE]
which would give information only about the classical noise fluctuations associated with the probability distribution . In the examples computed in the following sections, where a weak coupling approximation is made, the definition of Eq. (36) typically contains no more information than could be gotten from a product of reduced density matrix factors, each of the form .
As already noted, the density tensor is not measurable by any operation on the system Hilbert space. Its construction requires knowledge of the full system plus environment density matrix, which is not experimentally accessible for complex environments. Nonetheless is computable in any theory of the system-environment interaction, and we believe it to be of conceptual and theoretical interest, even if not of direct empirical relevance.
We close out this section by noting that in the quantum noise case, there is no analog of Eq.(8a), which relates the positive semidefinite variations to the density tensor in the classical noise case. The closest analog we find to the fluctuation formulas of Eq. (8a) involves the density tensor. The reason for this is that whereas , the trace over the environment of unity is the dimension of the environmental Hilbert space; to get a unit trace over the environment we must include a factor of , the reduced density matrix for the environment. This pushes up the order of the density tensor involved from 2 to 3. Specifically, let be an operator acting on , but which acts as the unit operator on . In place of the expectations used in the classical noise discussion of Eqs. (1a) through (2d), in the quantum noise case of system plus environment we consider the expression , which is an operator on the environmental Hilbert space. The trace of this operator over the environment is , giving the expectation of the operator when the environment is not observed. On the other hand, the expectation of this operator formed from the environmental reduced density matrix is . The mean squared fluctuation of this operator over the environment is positive semidefinite, and is given by
[TABLE]
where we have used the fact that . Reexpressing Eq. (37a) entirely in terms of the pure state density matrix , we have
[TABLE]
where we have used the fact that the right hand side of Eq. (37b) involves only the symmetric part of the order 3 density tensor. Thus, as noted above, where a density tensor appears in Eq. (8a), a density tensor appears in Eq. (37b), and where a density tensor appears in Eq. (8a), a density tensor appears in Eq. (37b).
7. Collisional Brownian Motion
As our first application of Eqs. (32a-c) and Eqs. (35a,b), we consider the collisional Brownian motion of a massive Brownian particle immersed in a bath of scattering particles. We work in the approximation of neglecting recoil of the Brownian particle, and of treating the bath as a collection of free particles of mass . We consider the pure state density matrix corresponding to definite momenta of the bath particles, calculate the corresponding order density tensor defined by Eqs. (32a-c), and then average over the thermal distribution of the bath particles as in Eqs. (35a,b). Thus the initial density matrix for the total system, corresponding to the factor in Eq. (33d), is
[TABLE]
with the initial density matrix of the Brownian particle, characterized by its coordinate matrix elements , and with the product density matrix for the bath particles,
[TABLE]
.
Since the bath particle scatterings are all independent, we focus on the effect of the scattering of a single bath particle, of initial momentum , on the Brownian particle, which we take to be in a superposition of position eigenstates. Thus the initial state of the Brownian particle and the bath particle that we are considering is
[TABLE]
corresponding to an initial state density matrix
[TABLE]
The corresponding Brownian particle matrix element of , which is still an operator on the bath particle state, takes the form
[TABLE]
with
[TABLE]
Asymptotically, the effect of the scattering is to replace the initial state by , with the scattering matrix. Substituting Eq. (39a), and using translation invariance to relate the scattering matrix with the Brownian particle at a general coordinate, to the scattering matrix with the Brownian particle at the origin, we get [11]
[TABLE]
with the momentum operator for the bath particle. The corresponding final density matrix is then
[TABLE]
and the Brownian particle matrix element of , which is again an operator acting on the bath particle, is
[TABLE]
Substituting this expression into Eq. (33d), we get
[TABLE]
with . The matrix element appearing in the final line of Eq. (40d) is one that is familiar from the standard calculation of the reduced density matrix (that is, ) for collisional decoherence [12]. Writing
[TABLE]
with proportional to the square of the scattering amplitude, the product of matrix elements in Eq. (40d) can be written, to second order accuracy in the scattering amplitude, as
[TABLE]
We also note that Eq. (39c), when substituted into Eq. (32c), implies that the value of before the scattering is
[TABLE]
Thus when the approximation of Eq. (41b) is substituted into Eq. (40d), we get
[TABLE]
At this point our work is essentially finished, since the remaining steps are identical to the standard calculation [11,12,13] proceeding from the case of Eq. (41d), and the structure of Eq. (41d) makes it clear how to generalize the standard result for to the case of general . In brief, the standard procedure is to multiply the right hand side of Eq. (41d) by the number of scattering particles, which combines with a normalizing factor of the inverse volume to give an overall factor of , the scattering particle density. The effect of the thermal distribution of momenta is taken into account by including an integral , in accordance with the mixed state procedure of Eq. (35a). Finally, expressing the matrix in terms of the scattering amplitude , and noting that the squared delta function for energy conservation gives an overall factor of the elapsed time, Eq. (41d) becomes, in the limit of small elapsed time, a formula for the time derivative of . For the case, the standard answer obtained this way is
[TABLE]
with
[TABLE]
where is a unit vector which gives the direction of the scattered particle momentum . To compare Eq. (42b) with the case of Eq. (41d), we replace by and by . Then we see that the generalization to is given by
[TABLE]
This is our final result for collisional Brownian motion, giving the evolution equation obeyed by the order density tensor . We see that it has the generic symmetries expected in the quantum noise case: although not totally symmetric in its pair indices, is symmetric under cyclic permutation of these indices. As additional checks, we see that for the factor involving is
[TABLE]
which is symmetric under the interchange , while for we have
[TABLE]
with symmetric, and antisymmetric, under any of the pair interchanges , or , or . Checking the descent equations is easy. Setting , the term in Eq. (42c) vanishes, so that on integrating over one is left on the right hand side with a sum that does not involve , times
[TABLE]
and so the descent equation for then implies the descent equation for its time derivative.
8. The weak coupling Born-Markov approximation and the quantum optical
master equation for the density tensor
We turn next to the density tensor extension of the standard weak coupling Born-Markov approximation, that is used to give a master equation for the reduced density matrix for a system interacting with an environment . We assume a total system plus environment Hamiltonian , with and respectively the environment and system Hamiltonians, and with the system-environment interaction Hamiltonian. (We omit the customary subscript on the interaction Hamiltonian to avoid a proliferation of subscripts.) We shall work in this section in interaction picture, in which the operators carry the time dependence associated with and . Thus the interaction Hamiltonian carries a time dependence , and the density matrix obeys the equation of motion
[TABLE]
which can be integrated to give
[TABLE]
Substituting Eq. (44b) back into Eq. (44a) gives the additional evolution equation
[TABLE]
One then notes that up to an error of order , the time argument of the factor in the double commutator term is irrelevant, so this factor can be approximated as , giving
[TABLE]
which is used as the starting point for the standard master equation derivation.
Our first step is to derive a suitable extension of Eq. (44d) for the product that appears in Eq. (32c). By the chain rule, we have
[TABLE]
For each undifferentiated factor on the right of Eq. (45a) we substitute Eq. (44b), and for each time derivative factor we substitute Eq. (44c), with appropriate subscripts added. Let us now organize the terms obtained this way according to the number of factors of that appear. Since Eq. (44c) contains at least one factor of , there are no terms in Eq. (45a) with no factors of . The general term in Eq. (45a) with one factor of comes from the term in Eq. (44c) with one factor of , multiplied by the product of the terms from Eq. (44b) with no factors of , giving
[TABLE]
The terms in Eq. (45a) with two factors of are of two types: (1) the quadratic term in on the right of Eq. (44c) times factors of , and (2) the linear term in on the right of Eq. (44c), multiplied by one factor of the linear term on the right of Eq. (44b), times factors of . We now note that up to an error of order , in terms that already contain two factors of we can replace all factors or by the corresponding , since the differences and are all of order . Collecting everything, we get the following formula, which gives the needed extension of Eq. (44d),
[TABLE]
Taking the overall of this expression then gives a formula for the time evolution of as defined by Eq.(32c).
We now make two standard assumptions. First of all, we assume at that at the initial time , the density matrix factorizes so that , with and respectively density matrices for the environment and the system which commute with one another, and with a pure state density matrix obeying . Secondly, we assume that , that is, we take the interaction Hamiltonian to have a vanishing expectation in the initial environmental state. As a result of these two assumptions, the environmental trace of the first term on the right hand side of Eq. (45c) vanishes, since
[TABLE]
The remaining terms in Eq. (45c) all have two factors of . Since and differ by one power of , in these terms, up to an error of order , we can replace all factors by the factorized approximation
[TABLE]
With these simplifications, and remembering that system operator factors with different index values act on different Hilbert spaces and so commute, Eq. (45c) becomes an extended version of the Redfield equation,
[TABLE]
This is converted to the Born-Markov equation by setting , and then extending the upper limit of the integration from to , giving
[TABLE]
We now note that Eq. (46d) can be further simplified, by taking account of the fact that whenever an factor is sandwiched between factors of it vanishes, since . This eliminates all terms in the sum over that are not adjacent in a cyclic sense, i.e., that do not either have , , or . The latter, by use of the cyclic properties of the trace, can be rearranged to give the term of the former set. We thus get a simplified set of Born-Markov equations. For , we get the usual starting point for the Born-Markov master equation derivation,
[TABLE]
and for , with the subscript identified with 1,
[TABLE]
At this point it is useful to check (and we have done so) that the descent equations are satisfied by Eqs. (47a) and (47b).
The remainder of the derivation follows closely the standard master equation derivation, in the rotating wave approximation, that proceeds from Eq. (47a), so we will only give a sketch. For further details, and in particular a discussion of the physical justification for the approximations involved, see Sec. 3.3 of ref [1] and also ref [13]. One assumes that has the form
[TABLE]
with acting only in the system Hilbert space and with acting only in the environment Hilbert space , and with the Hermiticity properties and . Since Eqs. (47a,b) are quadratic in , one uses Eq. (48a) twice; for each (regardless of the value of the index ) one writes
[TABLE]
and for each (again regardless of the value of ) one writes
[TABLE]
The rotating wave approximation then consists of neglecting terms in the double sum with , so that only the diagonal terms are left. From the trace over the environment, and the integral over , one gets correlators of the form
[TABLE]
where in the second line we have used the definition of the first line and the adjointness properties of the integrand. It is also customary to decompose the reservoir correlation function into self-adjoint and anti-self-adjoint parts, according to
[TABLE]
Proceeding in this fashion, after some algebra one gets the final result, which can be written as an equation for all by including a to take account of the special nature of the equation,
[TABLE]
Despite the fact the the and density tensors have a different structure, the descent equations are satisfied by Eq. (50a), as verified in Appendix D.
Finally, we note that Eq. (50a) is readily converted to the quantum optical master equation and its density tensor generalizations, by taking to be a three-vector index, so that becomes , which is related to the dipole operator by Eq. (3.182) of ref [1]. Also, one takes , with given by Eq. (3.205) of ref [1], and , with the photon number operator. One gets in this way the density tensor generalization of the quantum optical master equation,
[TABLE]
which is our final result of this section.
9. The Caldeira–Leggett model master equation for the density tensor
The Caldeira–Leggett model [14] describes the damping of the one-dimensional motion of a Brownian particle of mass m, moving in a potential , and interacting with an environment consisting of harmonic oscillators with masses and frequencies , and annihilation operator . The interaction Hamiltonian is assumed to be a linear coupling , with
[TABLE]
a weighted sum of the harmonic oscillator coordinates. A counter-term formally of order ,
[TABLE]
is included in the calculation, so that the total Hamiltonian is
[TABLE]
with and respectively the oscillator and particle Hamiltonians,
[TABLE]
Our aim will be to get a description of the effect on the particle motion of the couplings to the oscillator environment, in the high temperature limit. Our derivation of the density tensor generalization of the high temperature master equation closely follows that of Sec. 3.6 of ref [1], to which the reader is referred for a discussion of the physical motivation of the approximations involved.
Since the environmental expectation of the interaction Hamiltonian vanishes, we can proceed directly from the simplified Born-Markov equation of Eqs. (47a) and (47b). The first step is to transform the density matrix back to Schrödinger picture; it is easy to see that the effect of this is to replace by , to replace by (with still in the interaction picture), and to change to , defined by
[TABLE]
It is also necessary to explicitly include commutators arising from the counter term, which is easy since this term is treated as being already quadratic in . For the analog of Eq. (47a) for the special case , we find
[TABLE]
and for the analog of Eq. (47b) for , we have
[TABLE]
We next note that
[TABLE]
where, using the assumption that the system evolution is slow compared to the oscillator time scale, we approximate by its free particle dynamics,
[TABLE]
Since the right hand sides of Eqs. (53b,c) are quadratic in , the operator giving the coupling to the oscillators appears, after the environmental trace is taken, only through the correlators
[TABLE]
so that we have
[TABLE]
These correlators appear in the following integrals, which are evaluated or approximated in Sec. 3.6.2 of ref [1],
[TABLE]
with the constant defined by the counter term of Eq. (51b), with a constant determined by the harmonic oscillator spectral density, with and respectively the Boltzmann constant and environment temperature, and with a frequency cutoff. For a spectral density with a Lorentz-Drude cutoff function, one has
[TABLE]
This completes the specification of the calculation; the rest is just the algebra of assembling all the pieces, and so we pass directly to the result. For , we get the Caldeira–Leggett master equation,
[TABLE]
For the density tensors with , we correspondingly get
[TABLE]
We also note that the term proportional to on the first line of Eq. (56b) can be written in the alternative form,
[TABLE]
Equations (56a) and (56b) are our final results for the Caldeira–Leggett model. As was the case for the master equations derived in the preceding section, despite the differences between the structure of the and the equations, the descent equations are satisfied, as verified in Appendix E.
10. An application to state vector reduction
We turn now to considerations that bridge the discussions given above in the classical and quantum noise cases. We begin with an analysis of two Itô stochastic Schrödinger equations,
[TABLE]
and
[TABLE]
with a real Brownian noise obeying , and where we have dropped the Hamiltonian term. These lead to the respective density matrix evolution equations
[TABLE]
and
[TABLE]
which correspond to the same Lindblad type evolution equation for the expectation ,
[TABLE]
Let us now consider the effect of the stochastic evolutions of Eqs. (58a,b,c) on the expectation of the variance of the operator ,
[TABLE]
where in the final line we have used the density tensor definition of Eq. (15). For the time evolution of we have
[TABLE]
where we have used Eq. (58c) in the first line and Eq. (23a) in the second line. Since the cyclic property of the trace implies that , the terms in Eq. (59b) involving the Lindblad all vanish, and so the time derivative of comes entirely form the final term,
[TABLE]
and thus is determined by the evolution equation for the second order density tensor. This is why the state vector evolutions of Eqs. (57a) and (57b), or equivalently the density matrix evolutions of Eqs. (58a) and (58b), lead to very different results for the evolution of the variance of the operator . The tensor corresponding to Eqs. (57b) and (58b) is given in Eq. (24b), and since the cyclic property of the trace implies that , one has for this evolution. On the other hand, the tensor corresponding to Eqs. (57a) and (58a) is given in Eq. (24a), and through Eq. (59c) implies that
[TABLE]
which is negative definite. Starting from Eq. (59d), some simple inequalities imply that the stochastic evolution of Eqs. (57a) and (58a) drives the variance of to zero as , and hence reduces the state vector to an eigenstate of , as discussed in detail in refs [15].
Let us now consider a quantum system , consisting of a microscopic system coupled to a macroscopic measuring apparatus, interacting with a quantum environment , with the totality forming a closed system. A general result [16], using just the linearity of quantum mechanics, shows that state vector reduction cannot occur in this case. To understand this result through an analysis similar to that just given for Eqs. (57a,b), let us consider the behavior of the variance of a system operator which is a good “pointer observable”. By definition, a system operator commutes with the environment Hamiltonian , and since the system in this case includes the apparatus and so is macroscopic, the pointer observable also obeys [17] , with the system-environment interaction Hamiltonian. Let us now write the density matrix evolution in Schrödinger picture,
[TABLE]
with the system Hamiltonian. We consider the system evolution after a brief interaction has entangled the apparatus states with the microscopic subsystem quantum states that are to be distinguished by the pointer reading. For the time evolution of the variance of the pointer observable , we have
[TABLE]
which substituting Eq. (60a), and using the cyclic property of the trace and the fact that commutes with both and , simplifies to
[TABLE]
This can be further simplified by using the definition of the reduced density matrix , together with the fact that the commutators in Eq. (60c) involve only system operators, giving
[TABLE]
We see that, unlike the Itô equation case discussed above, the time derivative of here is determined by , rather than by .
Let us now take the pointer observable to be a pointer center of mass coordinate , in which case, once the entanglement of the pointer with the microsystem being measured has been established, the relevant part of the system Hamiltonian is , with the total momentum operator for the pointer of macroscopic mass . Evaluating the commutators, and writing , we see that
[TABLE]
By the Schwartz inequality, the right hand side of Eq. (61a) is bounded by
[TABLE]
Let us now determine the minimum value of the bound of Eq. (61b) that is compatible with the parameters of a feasible measurement. Since the uncertainty principle implies that , we get a least upper bound on Eq. (61b) by substituting . This shows that can be made as small as , which since is macroscopic, can be made essentially arbitrarily small.††3 Restoring factors of Planck’s constant, can be as small as , for which the reduction time is at least of order . For and , this gives times the age of the universe. Note that our argument places no restriction on the mean pointer momentum that establishes the time needed to attain one or the other of the measurement outcomes starting from the initial pointer position. Hence the variance of the pointer variable stays essentially constant, and is not forced to reduce to zero in the course of the measurement.
We conclude, in agreement with the arguments of [16], that a quantum apparatus interacting with a quantum environment does not act like the stochastic equation of Eq. (57a) in terms of reducing the state vector. Although a quantum environment acts on a quantum system with a form of “noise”, our analysis of the density tensor hierarchy in the classical and quantum noise cases shows that structures with different kinematical symmetries,††4 The dissimilarities between the symmetries of the classical noise and quantum noise hierarchies are least for the order two density tensor. In the order two case, cyclic symmetry is equivalent to full permutation symmetry, and so the index symmetry properties are the same in the classical and quantum noise cases, and as a consequence the descent equations in the quantum noise case correspond to the idempotence descent equations in the classical noise case. Only the classical noise descent equation implied by the unit trace condition has no precise quantum noise counterpart: in the classical case, one has
whereas in the quantum noise case one instead has
with the reduced density matrix of the environment with the system traced out. different dynamical evolutions, and different implications for the measurement process are involved. As a result, the quantum noise in a closed quantum system does not mimic the action of the classical noise in objective reduction models, and cannot be invoked to give a resolution of the quantum measurement problem within the framework of unmodified quantum mechanics.
Acknowledgments
I wish to thank Angelo Bassi, Todd Brun, Lajos Diósi, Larry Horwitz, and Lane Hughston for instructive conversations over a number of years that helped motivate this study, and Francesco Petruccione for the gift some years ago of a copy of ref [1]. This work was supported in part by the Department of Energy under Grant #DE–FG02–90ER40542.
Appendix A: Descent equations for the
isotropic spin-1/2 ensemble
Let us write the generating function of Eq. (14b) as
[TABLE]
Then, we find
[TABLE]
Here the primes denote derivatives of with respect to , and in this notation obeys the second order differential equation
[TABLE]
Contracting the first expression in Eq. (A2) with , and using the tracelessness of the Pauli matrices, gives the first equation in Eq. (7b). Contracting the second expression in Eq. (A2) with , and using the differential equation of Eq. (A3) together with the Pauli matrix identities and , which implies , gives the second equation in Eq. (7b).
Appendix B: Descent equations for the
Itô stochastic Schrödinger equation
We wish here to verify that
[TABLE]
obeys the descent equations of Eq. (7b). Since
[TABLE]
we have
[TABLE]
giving the first identity in Eq. (7b). Next we calculate
[TABLE]
while for the contraction of the second variation we have (with indices implicit on the right hand side)
[TABLE]
with
[TABLE]
We see immediately that gives all of the second line of Eq. (B3a). From Eqs. (16b) and (18) we find
[TABLE]
while from Eq. (21b) we have
[TABLE]
Hence , giving the part of the first line of Eq. (B3a). Finally, again using Eq. (21b) we find that
[TABLE]
and so gives the remainder of the first line of Eq. (B3a), completing the check of the descent equations.
Appendix C: Descent equations for the
jump Schrödinger equation
We verify here that
[TABLE]
obeys the descent equations of Eq. (7b). Since and , we have
[TABLE]
checking the first line of Eq. (7b). Next we calculate the first variation of ,
[TABLE]
and the contracted second variation,
[TABLE]
with
[TABLE]
We see immediately that gives all of the second line of Eq. (C2a). From Eq. (30c), which we rewrite here,
[TABLE]
we see that the part of and the part of in give the first line of Eq. (C2a). To complete the verification, we must show that cancels against the remainder of , which is
[TABLE]
But separating off the term of , and making the change of variable in the remaining sum, we see that is exactly the negative of Eq. (C3b), completing the argument.
Appendix D: Descent equations for the Born-Markov
master equation
We wish here to verify that Eq. (50a) obeys the descent equations of Eq. (34). We separate the verification into two parts, first checking the descent from to , and then checking the descent from general to . For the density tensor time derivative, writing out all terms in Eq. (50a) explicitly, and using the fact that since operators labeled with subscripts and act on different Hilbert spaces, the order in which they are written is irrelevant, we have
[TABLE]
Contracting the column index associated with the subscript 1 with the row index associated with the subscript 2, and dropping the subscripts since all operators now act in the same Hilbert space, we get
[TABLE]
which has the structure of and so verifies the descent.
To verify the descent we make some simplifications in notation. We omit all superscripts , since this leads to no ambiguities, as well as all time arguments and all frequency arguments . We also abbreviate
[TABLE]
Our general strategy is to split the sum containing into plus the and the terms, and to split the sum containing into plus the , , and terms. For the part of involving , we have
[TABLE]
which on contracting the column index associated with the subscript with the row index associated with the subscript , and relabeling all quantities that had subscript with subscript , since they act now in the same Hilbert space, gives
[TABLE]
which has the correct structure for the corresponding part of , with replaced by . The remainder of is
[TABLE]
Again, contracting the column index associated with the subscript with the row index associated with the subscript , and relabeling all quantities that had subscript with subscript , since they act now in the same Hilbert space, gives
[TABLE]
which on canceling the terms marked with gives the corresponding part of , with replaced by . This completes the verification of the descent.
Appendix E: Descent equations for the
Caldeira–Leggett model
We verify here that Eqs. (56a) and (56b) obey the descent equations of Eq. (34). As in the preceding appendix, we simplify the notation by omitting all superscripts and all time arguments . We first verify the to descent. For the case of Eq. (56b), we have
[TABLE]
Contracting the column index associated with the subscript 1 with the row index associated with the subscript 2, and dropping subscripts since all operators now act in the same Hilbert space, we get
[TABLE]
We see that the terms that have an operator sandwiched between two factors of cancel, leaving only terms involving , which have the form of Eq. (56a) with replaced by .
To check the to descent, we split the sums that occur in the same manner as in Appendix D. We thus write Eq. (56b) in the form
[TABLE]
We now contract the column index associated with the subscript with the row index associated with the subscript , and relabel all quantities that had subscript with subscript , since they act now in the same Hilbert space. As is readily seen by inspection of Eq. (E2), this gives Eq. (56b) with replaced by and with replaced by , together with terms of the wrong structure, that grouped together give and , which thus vanish. This completes the verification of the descent equation for Eq. (56b).
References
[1] Breuer H-P and Petruccione F (2002) The Theory of Open Quantum Systems (Oxford: Oxford University Press) [2] Breuer H-P and Petruccione F (1996) Phys. Rev. A 54 1146 [3] Wiseman H M (1993) Phys. Rev. A 47 5180 [4] Mølmer K, Castin Y and Dalibard J (1993) J. Opt. Soc. Am. B 10 524 [5] Mielnik, B (1974) Commun. Math. Phys. 37 221; see especially p 240. I wish to thank Lane Hughston for bringing this reference, and ref [6] as well, to my attention. [6] Brody, D C and Hughston, L P (1999) J. Math. Phys. 40 12, Eqs. (31) and (32); Brody, D C and Hughston, L P (1999) Proc. Roy. Soc. A 455 1683, Sec. 2(e); Brody, D C and Hughston, L P (2000) J. Math. Phys. 41, 2586, Eq. (9) and subsequent discussion. [7] Wiseman H M and Diósi L (2001) Chem. Phys. 268 91. See also Diósi L (1986) Phys. Lett. A 114 451 for the transition rate operator. [8] Lindblad G (1976) Commun. Math. Phys. 48 119 [9] Gorini V, Kossakowski A and Sudarshan E C G (1976) J. Math. Phys. 17 821 [10] Schack R and Brun T A (1997) Comp. Phys. Commun. 102 210 [11] Gallis M R and Fleming G N (1990) Phys. Rev. A 42 38 [12] Diósi L (1995) Europhys. Lett. 30 63; Dodd P J and Halliwell J J (2003) Phys. Rev. D 67 105018; Hornberger K and Sipe J E (2003) Phys. Rev. A 68 012105; Adler S L (2006) J. Phys. A: Math. Gen. 39 14067 [13] Hornberger K (2006) Introduction to decoherence theory, arXiv: quant-ph/0612118 [14] Caldeira A O and Leggett A J (1983) Physica A 121 587 [15] Ghirardi G C, Pearle P and Rimini A (1990) Phys. Rev. A 42 78; Hughston L P (1996) Proc. Roy. Soc. A 452 953; Adler S L and Horwitz L P (2000) J. Math. Phys. 41 2485; Adler S L, Brody D C, Brun T A and Hughston L P (2001) J Phys. A: Math. Gen. 34 8795; Adler S L (2004) Quantum Theory as an Emergent Phenomenon (Cambridge UK: Cambridge University Press) Sec. 6.2 [16] Bassi A and Ghirardi G C Phys. Lett. A 275 373 [17] Zurek W H (1981) Phys. Rev. D 24 1516; Schlosshauer M (2004) Rev. Mod. Phys. 75 1267, p. 1280 [18] For reviews of stochastic reduction models, see Bassi A and Ghirardi G C (2003) Phys. Reports 379 257; Pearle P (1999) Collapse models, in Open Systems and Measurements in Relativistic Quantum Field Theory, Lecture Notes in Physics 526, Breuer H-P and Petruccione F eds. (Berlin: Springer-Verlag)