A density tensor hierarchy for open system dynamics: retrieving the
  noise
arXiv:0704.0796v4  [quant-ph]  16 Jul 2007
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
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Phone 609-734-8051; FAX 609-924-8399; email adler@ias.edu
http://arxiv.org/abs/0704.0796v4
ABSTRACT
We develop a density tensor hierarchy for open system dynamics, that
recovers information about fluctuations (or “noise”) lost in passing to the re-
duced density matrix. For the case of fluctuations arising from a classical
probability distribution, the hierarchy is formed from expectations of prod-
ucts 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 n tensor, under contraction of appropriate pairs of tensor indices,
to the order n− 1 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 lat-
ter 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 cou-
pled 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 environ-
ment 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 |ψα〉, each drawn from the same system Hilbert space HS , with each
vector assumed to occur in the ensemble with probability wα,
wα = 1. Measurement
of a general self-adjoint operator R for a system prepared in |ψα〉 typically gives a range of
values, the mean of which given by 〈ψα|R|ψα〉. The mean or expectation over the ensemble
of pure state vectors is then given by
wα〈ψα|R|ψα〉 = TrρR , (1a)
with ρ the mixed state or reduced density matrix defined by
wα|ψα〉〈ψα| . (1b)
Breuer and Petruccione point out that there are three variances that are relevant. The
variance of measurements of R over all pure states in the ensemble is given by
Var(R) = Trρ(R − TrρR)2 = TrρR2 − (TrρR)2 . (2a)
This can be written as the sum of two non-negative terms,
Var(R) = Var1(R) + Var2(R) , (2b)
with Var1(R) the ensemble average of the variances of R within each pure state of the
ensemble,
Var1(R) =
wα[〈ψα|R
2|ψα〉 − 〈ψα|R|ψα〉
2] , (2c)
and with Var2(R) the variance of the pure state means of R over the ensemble,
Var2(R) =
wα〈ψα|R|ψα〉
2 − [
wα〈ψα|R|ψα〉]
2 . (2d)
Thus, Var1(R) is an ensemble average of the quantum variances of R, while Var2(R) is a
measure of the spread of the average values of R resulting from the statistical properties of
the ensemble. As Breuer and Petruccione note, neither of the subsidiary variances Var1,2
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 wα. In particular, the
variances Var1,2 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 {wα} 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 ψα are independent of
α, this tensor reduces to an n-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 ψα on α, there are some general
statements that can be made. First of all, the order n density tensor is a symmetric tensor
in its pair indices, and it can be considered as a matrix operator acting on the n-fold tensor
product of the system Hilbert space HS 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 n tensor to the order n−1 tensor when any row index is
contracted with any column index. We show that the variances Var1,2 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 formal-
ism 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, ver-
ify 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 interaction1 H has vanishing expectation in the environment, but its square H2
does not have a vanishing expectation, because the environment is not in an eigenstate of
H . 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
1 What we call H is usually denoted by HI in the open systems literature. To avoid
confusion, all other Hamiltonians will carry subscripts, e.g., HS and HE for the system and
environment Hamiltonians, HTOT for the total Hamiltonian, etc.
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 (n) tensor to the order (n− 1) 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 |ψα〉 by ρα, with
ρα = |ψα〉〈ψα| , (3a)
and its general matrix element between states |i〉 and |j〉 of HS by
ρα;ij ≡ 〈i|ρα|j〉 . (3b)
The unit trace condition on ρα states that
Trρα = 〈ψα|ψα〉 = 1 , (3c)
and the idempotence condition on ρα states that
ρ2α = |ψα〉〈ψα||ψα〉〈ψα| = |ψα〉〈ψα| = ρα . (3d)
We now define the order n density tensor by
i1j1,i2j2,...,injn
wαρα;i1j1ρα;i2j2 ...ρα;injn = E[ρα;i1j1ρα;i2j2 ...ρα;injn ] , (4a)
with E[Fα] a shorthand for
E[Fα] =
wαFα . (4b)
Since
wαρα;ij =
wα〈i|ρα|j〉 , (5a)
we see that this is just the |i〉 to |j〉 matrix element of the reduced density matrix ρ defined
in Eq. (1b),
ij = 〈i|ρ|j〉 , (5b)
and so the density tensor of Eq. (4a) is a natural generalization of the usual reduced density
matrix. When the states |ψα〉 are independent of the label α, the definition of Eq. (4a)
simplifies to
i1j1,i2j2,...,injn
= ρi1j1ρi2j2 ...ρinjn , (5c)
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 E[...] on the right of Eq. (4a) are just ordinary complex numbers, the density
tensor is symmetric under interchange of any index pair iljl with any other index pair imjm.
Consequently, we can define a generating function for the density tensor by
G[aij ] = E[e
ρα;ijaij ] =
ai1j1 ...ainjn
i1j1,...,injn
, (5d)
where repeated indices i, j are summed. It will often be convenient to abbreviate ρα;ijaij by
ρα · a, so that the generating function becomes in this notation G[a] = E[e
ρα ·a].
Although the density tensor for n > 1 is not an operator on HS , it clearly has the
structure of an operator on the n-fold tensor product HS ⊗ HS ⊗ ... ⊗ HS . Motivated by
this, we will often find it convenient to write the definition of Eq. (4a) as
ρ(n) = E
, (5e)
with each factor ρα;ℓ acting on a distinct factor Hilbert space HS;ℓ in the tensor product
ℓ=1 HS;ℓ. 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 il with any column index
jk. There are two basic cases: (i) one can contract a row index il with its corresponding
column index jl, and (ii) one can contract a row index il with a column index jk with k 6= l.
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 i1 with j1 we find
δi1j1ρ
i1j1,i2j2,...,injn
= E[(Trρ)ρα;i2j2 ...ρα;injn ] = E[ρα;i2j2 ...ρα;injn ] = ρ
(n−1)
i2j2,...,injn
, (6a)
where we have used the unit trace condition of Eq. (3c). For the contraction of j1 with i2,
we find
δj1i2ρ
i1j1,i2j2,...,injn
= E[(ρ2)α;i1j2 ...ρα;injn ] = E[ρα;i1j2ρα;i3j3 ...ρα;injn ] = ρ
(n−1)
i1j2,i3j3,...,injn
, (6b)
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,
δi1j1ρ
i1j1,i2j2
δi2j2ρ
i1j1,i2j2
δj1i2ρ
i1j1,i2j2
δj2i1ρ
i1j1,i2j2
Referring to the generating function of Eq. (5d), the general descent equations can be sum-
marized compactly by the two identities,
∂G[aij ]
=E[(Trρα)e
ρα;ijaij ] = G[aij ] ,
∂2G[aij ]
∂amr∂apq
=E[ρmrρrqe
ρα;ijaij ] = E[ρmqe
ρα;ijaij ] =
∂G[aij ]
When the density matrix ρ 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 (n) tensor to an order (n− 1) tensor in which one
factor ρ is replaced by ρ2; 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
Var1(R) =ρ
(R2)j1i1 − ρ
i1j1,i2j2
Rj1i1Rj2i2 ,
Var2(R) =ρ
i1j1,i2j2
Rj1i1Rj2i2 − (ρ
Rj1i1)
Var(R) =ρ
(R2)j1i1 − (ρ
Rj1i1)
with Rji = 〈j|R|i〉. 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 R and S is given by
wα〈ψα|R|ψα〉〈ψα|S|ψα〉 = ρ
i1j1,i2j2
Rj1i1Sj2i2 , (8b)
which can be used, together with information obtained from ρ(1), to calculate the covariance
and correlation of R and S.
3. Isotropic spin-1/2 ensemble
As a simple example of the density tensor formalism, let us follow Breuer and Petruc-
cione [1] and consider the case of an isotropic spin-1/2 ensemble. Let ~v be a vector in three
dimensions, and consider the ensemble of spin-1/2 pure state density matrices
ρ(~v) =
(1 + ~v · ~σ) , (9a)
with ~σ = (σ1, σ2, σ3) the standard Pauli matrices, and with a uniform probability distribution
of ~v over the unit sphere |~v | = 1 specified by
w(~v ) =
δ(|~v | − 1) . (9b)
(Clearly, ~v has the same significance as the label α used in the preceding section.) Defining
E[P (~v )] =
d3vw(~v )P (~v ) , (10a)
a standard calculation gives
E[1] = 1 , E[vsvt] =
δst , ... , (10b)
with all averages of odd powers of ~v vanishing. From Eq. (9a), we have
ρ(~v )ij =
(δij + vrσ
ij) , (11a)
and the general density tensor over this ensemble is defined by
i1j1,...,injn
= E[ρ(~v )i1j1 ...ρ(~v )injn ] . (11b)
From Eq. (10b), the first three tensors in this hierarchy are now easily found to be
δi1j1 ,
i1j1,i2j2
δi1j1δi2j2 +
~σi1j1 · ~σi2j2
i1j1,i2j2,i3j3
δi1j1δi2j2δi3j3 +
(δi1j1~σi2j2 · ~σi3j3 + δi2j2~σi1j1 · ~σi3j3 + δi3j3~σi1j1 · ~σi2j2)
Using the relations Tr~σ = 0 and (~σ 2)ij = 3δij , 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
G[aij ] = E[e
ρ(~v)ijaij ] , (13)
with ρ(~v )ij given by Eq. (11a). Defining the vector ~A by
~σijaij , (14a)
a simple calculation gives
G[aij ] = exp(
sinh | ~A |
| ~A |
= exp(
Tra)[1 +
( ~A 2)2
+ ...] , (14b)
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 |ψ〉 with a time evolution described by a stochas-
tic 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
ρ = |ψ〉〈ψ| 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 α, 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
α, treating the history dependence of ρ as understood. So in this context, the definition of
Eq. (4a) becomes
i1j1,...,injn
= E[ρi1j1 ...ρinjn ] , (15)
with E[...] the stochastic expectation, and the generating function G[aij ] takes the same form
as given in Eq. (5d) but with the subscript α 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 ρ(n), 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 ρ(1) = E[ρ]. We begin by recapitulating the results of ref [7]. The most general
evolution of a density matrix ρ that preserves Trρ = 1 and obeys the complete positivity
condition is the Lindblad form
dρ = dtLρ , (16a)
Lρ ≡ −i[HTOT, ρ] + ckρc
ck, ρ} , (16b)
with {, } denoting the anticommutator, and with the repeated index k summed. The set of
Lindblad operators ck 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 ρ for which E[dρ] reduces to Eqs. (16a) and (16b)
takes the form
dρ = dtLρ+ |dφ〉〈ψ|+ |ψ〉〈dφ| . (17a)
Here |dφ〉 is a state vector that is a pure noise term, so that
E[|dφ〉] = 0 , (17b)
that is orthogonal to |ψ〉, so that
〈ψ|dφ〉 = 0 , (17c)
and that obeys
|dφ〉〈dφ| = dtW . (17d)
The operator W is the Diósi transition rate operator [5] given by
W =Lρ− {ρ,Lρ}+ ρTr(ρLρ)
=(ck − 〈ck〉)ρ(ck − 〈ck〉)
, (18)
where 〈ck〉 is a shorthand for the quantum state expectation 〈ψ|ck|ψ〉 = Trρck. Although
|dφ〉〈dφ| is completely fixed, Wiseman and Diósi show that |dφ〉|dφ〉 is free, with different
choices for this and different phase choices for the ck corresponding to different pure state
evolutions (or “unravelings”) that yield the same evolution of Eqs. (16a) and (16b) for the
reduced density matrix ρ.
Wiseman and Diósi further show that |dφ〉 can be parameterized by complex Wiener
processes by writing
|dφ〉 = (ck − 〈ck〉)|ψ〉dξ
k , (19a)
E[dξk] = E[dξ
k] = 0 (19b)
and with
dξj(t)dξ
k(t) =dtδjk
dξj(t)dξk(t) =dtujk ,
(19c)
where ukj = ujk is a set of arbitrary complex numbers subject to the condition that the
norm of the complex matrix u ≡ [ujk] be less than or equal to 1. (See Eqs. (4.10) and (4.11)
of ref. [7].) In terms of this parameterization of |dφ〉, the pure state evolution of Eq. (17a)
takes the form
dρ = dtLρ+ (ck − 〈ck〉)ρdξ
k + ρ(ck − 〈ck〉)
†dξk , (19d)
and the corresponding stochastic Schrödinger equation for the wave function is [7]
d|ψ〉 =− iHψdt|ψ〉+ (ck − 〈ck〉)dξ
k|ψ〉 ,
−iHψ =− iHTOT −
kck − 2〈ck〉
∗ck + 〈ck〉
∗〈ck〉
(19e)
We proceed now to use pure state evolution of Eq. (19d) to calculate the evolution
equation for the generating function
G[aij ] = E[exp(ρijaij)] . (20a)
To calculate the differential of Eq. (20a), we use the Itô stochastic calculus rule for the
differential of a function f(w) of a stochastic variable w,
df(w) = dwf ′(w) +
(dw)2f ′′(w) . (20b)
Applying this to Eq. (20a), we get
dG[aij ] = E[(dρmramr +
dρmramrdρpqapq) exp(ρijaij)] . (20c)
Substituting Eq. (19d) for dρ, and using Eqs. (19a-c), together with the Itô calculus rule
E[dwf(w)] = 0, we get
dG[aij ] = dtE
amr(Lρ)mr +
amrapqCmr,pq
exp(ρijaij)
, (21a)
with the coefficient of the quadratic term in aij given by
Cmr,pq =Cpq,mr = dρmrdρpq
=〈m|(ck − 〈ck〉)ρ|r〉〈p|ρ(ck − 〈ck〉)
+〈m|ρ(ck − 〈ck〉)
†|r〉〈p|(ck − 〈ck〉)ρ|q〉
+〈m|(ck − 〈ck〉)ρ|r〉〈p|(cℓ − 〈cℓ〉)ρ|q〉u
+〈m|ρ(ck − 〈ck〉)
†|r〉〈p|ρ(cℓ − 〈cℓ〉)
†|q〉ukℓ
(21b)
This expression can be rearranged by using the identity, valid for general operators A,B,
general states |r〉, |m〉, and general pure state (idempotent) density matrix ρ,
ρA|r〉〈m|Bρ = ρ〈m|BρA|r〉 , (22a)
giving an alternative result for Cmr,pq
Cmr,pq =Wmqρpr +Wprρmq
+[(ck − 〈ck〉)ρ]mqu
kℓ[(cℓ − 〈cℓ〉)ρ]pr
+[ρ(ck − 〈ck〉)
†]prukℓ[ρ(cℓ − 〈cℓ〉)
†]mq ,
(22b)
where we have used Eq. (18) defining the operator W , and where we use the subscript
notation of Eq. (3b) for matrix elements, so that in general Amr = 〈m|A|r〉.
From the evolution equation of Eqs. (21a,b) and (22b) for the generating function,
by expansion in powers of a we can read off the evolution equation for the general density
tensor of order n. Employing now the condensed notation of Eq. (5e), in which matrix
indices are not indicated explicitly, we have
dρ(n) =dtE[
(ρ1...ρn)ℓ(Lρ)ℓ
ℓ<m=1
(ρ1...ρn)ℓmCℓm] .
(23a)
Here (ρ1...ρn)ℓ denotes the product
j=1 ρj with the factor ρℓ omitted, and similarly, (ρ1...ρn)ℓm
denotes the product
ρj with the factors ρℓ and ρm omitted.
2 The coefficient Cℓm is
given by
Cℓm = Cmℓ =[(ck − 〈ck〉)ρ]ℓ[ρ(ck − 〈ck〉)
+[ρ(ck − 〈ck〉)
†]ℓ[(ck − 〈ck〉)ρ]m
+[(ck − 〈ck〉)ρ]ℓ[(ck̄ − 〈ck̄〉)ρ]mu
+[ρ(ck − 〈ck〉)
†]ℓ[ρ(ck̄ − 〈ck̄〉)
†]mukk̄ ,
(23b)
which corresponds in an obvious way to Eq. (21b) when matrix elements are written explicitly
between states 〈m| and |r〉 in the Hilbert space labeled by ℓ, and between states 〈p| and |q〉
in the Hilbert space labeled by m. (No relation is implied between the m used as a state
label, and the m used as a Hilbert space label.) Since Cℓm in Eq. (23a), which depends
through the terms involving ukk̄ on the choice of unraveling, is multiplied by two powers of
a, it does not contribute to the evolution equation for the reduced density matrix ρ(1). 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 ρ(n), with n ≥ 2, have
evolution equations that receive contributions from Cℓm, and so contain information that
distinguishes between different unravelings of the Lindblad evolution.
As a simple illustration of how the tensors ρ(n) for n ≥ 2 distinguish between different
unravelings, let us consider the case of real noise, dξk = dξ
k, for which ujk = δjk, and with
2 For n = 1, (ρ1)1 = 1 and (ρ1)ℓm = 0, while for n = 2, (ρ1ρ2)12 = 1.
a single Lindblad c1, which we choose as either c1 = A or c1 = iA, with A a self-adjoint
operator. Both choices of c1 lead to the same Lindblad, since L is invariant under rephasing
of ck, but through the ukk̄ terms they lead to different expressions for Cmr,pq. When ck = A,
we find from Eq. (21b)
Cmr,pq =〈m|{A− 〈A〉, ρ}|r〉〈p|{A− 〈A〉, ρ}|q〉
=〈m|[ρ, [ρ, A]]|r〉〈p|[ρ, [ρ, A]]|q〉 ,
(24a)
while when ck = iA, we have instead
Cmr,pq = −〈m|[A, ρ]|r〉〈p|[A, ρ]|q〉 . (24b)
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
5. Jump process Schrödinger equation
As our next density tensor application we consider the jump process (piecewise de-
terministic process, or PDP) Schrödinger equation, given by
d|ψ〉 = Adt|ψ〉+BkdNk|ψ〉 , (25a)
where a sum over k is understood, with A and the Bk general (non-self-adjoint) operators,
and with the dNk independent discrete random variables obeying
dNjdNk = δjkdNk , dNjdt = 0 . (25b)
Straightforward calculation shows that this process preserves the norm of |ψ〉 and the pure
state condition ρ2 = ρ = |ψ〉〈ψ|, provided that A and B obey the restrictions
〈A+ A†〉 =0,
〈Bk +B
kBk〉 =0 ,
(25c)
with no summation over k on the second line, which must hold individually for each value
of k. Corresponding to Eq. (25a), the density matrix obeys the evolution equation
dρ =(Aρ+ ρA†)dt+QkdNk ,
Qk =Bkρ+ ρB
+BkρB
(25d)
with a sum over k understood in the dNk term on the first line, but no sum over k understood
in the second line.
Let now E|ψ〉[...] denote an expectation conditioned on the current value of the wave
function being |ψ〉, and E[...] 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 A, Bk, and on
E|ψ〉[dNk] ≡ vkdt , (26a)
such that the expectation of dρ takes the Lindblad form of Eq. (16b), that is,
E[dρ] =dtLρ
Lρ =− i[HTOT, ρ] + ckρc
kck, ρ} .
(26b)
Making the Ansatz
ck −Kk
− 1 , (27a)
withKk constants (this Ansatz includes both the standard quantum jump equation (Kk = 0),
and the orthogonal jump equation (Kk = 〈ck〉), 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
vk =〈(ck −Kk)
†(ck −Kk)〉 ,
A =− iHTOT −
kck +
kck〉+ ckK
(〈ck〉K
k + 〈ck〉
∗Kk) .
(27b)
Let us now define the order n density tensor for the jump models by
ρ(n) = E[
ρℓ] , (28a)
where we use the condensed notation of Eq. (5e). For the differential of this, we find
dρ(n) =E[
(ρ1...ρn)ℓdρℓ +
ℓ<m=1
(ρ1...ρn)ℓmdρℓdρm
ℓ<m<p=1
(ρ1...ρn)ℓmpdρℓdρmdρp + ... + dρ1dρ2dρ3...dρn−1dρn] ,
(28b)
where all powers of dρ must be retained because dN2k = dNk. Using the conditional proba-
bility formula p(|ψ〉∩dNk) = p(dNk| |ψ〉)p(|ψ〉), we get the conditional expectation formula,
valid for an arbitrary function F of the state |ψ〉,
E[F (|ψ〉)dNk] = E[F (|ψ〉)E|ψ〉[dNk]] = E[F (|ψ〉)vk] . (29a)
Using this equation to evaluate the higher order terms in Eq. (28b), together with Eq. (26b)
for the leading term, we get
dρ(n) =dtE[
(ρ1...ρn)ℓ(Lρ)ℓ
ℓ<m=1
(ρ1...ρn)ℓmvk(Qk)ℓ(Qk)m
ℓ<m<p=1
(ρ1...ρn)ℓmpvk(Qk)ℓ(Qk)m(Qk)p + ...+ vk(Qk)1(Qk)2(Qk)3...(Qk)n−1(Qk)n] ,
(29b)
with a sum over k in each term containing vk.
Writing the corresponding generating function in compact notation as
G[a] = E[ea·ρ] , (30a)
the evolution equation for G is given , with the k sum now indicated explicitly, by
dG[a] =E[ea·dρea·ρ]− E[ea·ρ]
(a · dρ)p
)ea·ρ]
=dtE[(a · Lρ+
(a ·Qk)
)ea·ρ] .
(30b)
From Eq. (30b), and the identities
which follow, after some algebra, from Eqs. (16b), (25d),
(27a), and (27b)
{ρ,Lρ} =Lρ−
{ρ,Qk} =Qk −Q
(30c)
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 S interacting
with an environment E . In such a situation, one does not have a classical probability distri-
bution wα 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 ρ. We denote the system basis states by |i〉 , as well as |j〉, and
denote the environment basis states by |ea〉, a = 1, 2, ..... A general density matrix element
has the form 〈e1i|ρ|e2j〉, and the standard reduced density matrix, with the environment
traced out, is defined by
ij = (TrEρ)ij =
〈ei|ρ|ej〉 . (31)
In order to recapture fluctuations that are averaged over in the trace in Eq. (31), we define
the density tensor ρ(n) by
i1j1,i2j2,...,injn
e1,e2,...,en
〈e1i1|ρ|e2j1〉〈e2i2|ρ|e3j2〉...〈en−1in−1|ρ|enjn−1〉〈enin|ρ|e1jn〉
=TrEρi1j1ρi2j2 ...ρinjn .
(32a)
Here we have defined ρiℓjℓ as the matrix, labeled by the system state labels iℓ, jℓ, acting on
the environment Hilbert space HE according to
(ρiℓjℓ)e1e2 = 〈e1|ρiℓjℓ |e2〉 = 〈e1iℓ|ρ|e2jℓ〉 . (32b)
The density tensor ρ(n) is again an operator on a tensor product of system Hilbert spaces
ℓ=1 HS;ℓ. Thus, in a condensed notation analogous to that of Eq. (5e), we can also write
Eq. (32a) as
ρ(n) = TrEρ1ρ2....ρn , (32c)
where ρℓ is an operator acting on HE ⊗HS;ℓ.
We have avoided using a product notation
ℓ=1 in Eq. (32c) because the factors ρiℓjℓ
in Eq. (32a) and ρℓ in Eq. (32c) are different operators on the environment for each ℓ and
thus do not commute. Hence the density tensor is not symmetric under permutation of its
pair indices iℓjℓ, but it is symmetric under cyclical permutation of the indices, as a result
of the cyclic symmetry of the trace. For n = 2, cyclic symmetry is equivalent to symmetry
under pair index interchange, and for n = 3, using the identity
TrABC = Tr
([A,B]C + {A,B}C) , (33)
cyclic symmetry is equivalent to the statement that the density tensor ρ(3) can be written
as the sum of two tensors ρ(3) = ρ(3S) + ρ(3A), with ρ(3S) completely symmetric, and ρ(3A)
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 δiℓjℓ does not
lead to a descent condition, since δiℓjℓρiℓjℓ is not unity, but rather TrSρ, the reduced density
matrix that acts on the environment when the system is traced out. Contraction of a general
jk with a general iℓ for k 6= ℓ gives nothing useful, since in general non-commuting factors
stand between ρk and ρℓ. However, when a column index jk is contracted with the adjacent
row index ik+1, the two density matrices to which they are attached are linked to form the
product ρ2 = ρ, and so we get the descent relation of Eq. (6b), and others related to it by
cyclic permutation symmetry,
δj1i2ρ
i1j1,i2j2,...,injn
(n−1)
i1j2,i3j3,...,injn
. (34)
As noted before, even when ρ2 6= ρ, 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
ρα with weights wα. Typically, α 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, α then
can refer to the energies and momenta of each environmental particle. In this case we define
the density tensor by
i1j1,i2j2,...,injn
α;i1j1,i2j2,...,injn
, (35a)
α;i1j1,i2j2,...,injn
= TrEρα;i1j1ρα;i2j2 ...ρα;injn . (35b)
This definition gives information about both the quantum noise or fluctuations contained
within each ρα, and the classical noise or fluctuations associated with the probability distri-
bution wα. 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
(n);CL
i1j1,i2j2,...,injn
TrEρα;iℓjℓ (36)
which would give information only about the classical noise fluctuations associated with the
probability distribution wα. 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 n reduced density matrix factors, each
of the form
wαTrEρα;iℓjℓ .
As already noted, the density tensor ρ(n) is not measurable by any operation on
the system Hilbert space. Its construction requires knowledge of the full system plus envi-
ronment density matrix, which is not experimentally accessible for complex environments.
Nonetheless ρ(n) 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 Var1,2 to the density tensor ρ
in the classical noise case. The closest analog we find to the fluctuation formulas of Eq. (8a)
involves the n = 3 density tensor. The reason for this is that whereas E[1] = 1, 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 ρE ≡ TrSρ, the reduced density
matrix for the environment. This pushes up the order of the density tensor involved from 2
to 3. Specifically, let AS be an operator acting on HS , but which acts as the unit operator on
HE . 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
AE ≡ TrSρAS , which is an operator on the environmental Hilbert space. The trace of this
operator over the environment is TrEAE = TrETrSρAS = TrS(TrEρ)AS = TrSρ
(1)AS , giving
the expectation of the operator AS when the environment is not observed. On the other
hand, the expectation of this operator formed from the environmental reduced density matrix
is TrEρEAE . The mean squared fluctuation of this operator over the environment is positive
semidefinite, and is given by
TrEρE(AE − TrEρEAE)
2 = TrEρEA
E − (TrEρEAE)
2 , (37a)
where we have used the fact that TrEρE = Trρ = 1. Reexpressing Eq. (37a) entirely in terms
of the pure state density matrix ρ, we have
TrETrSρ(TrSρAS)
2 − (TrETrSρTrSρAS)
=δj1i1ASj2i2ASj3i3ρ
i1j1,i2j2,i3j3
− (δj1i1ASj2i2ρ
i1j1,i2j2
(37b)
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 n = 2 density tensor appears
in Eq. (8a), a n = 3 density tensor appears in Eq. (37b), and where a n = 1 density tensor
appears in Eq. (8a), a n = 2 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 massm. We consider the pure state density matrix
corresponding to definite momenta {~ki} of the bath particles, calculate the corresponding
order n 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 ρℓ in Eq. (33d), is
ρTOTℓ = ρℓρE , (38a)
with ρℓ the initial density matrix of the Brownian particle, characterized by its coordinate
matrix elements 〈~Rℓ|ρℓ|~R
ℓ〉, and with ρE the product density matrix for the bath particles,
|~ki〉〈~ki| . (38b)
Since the bath particle scatterings are all independent, we focus on the effect of
the scattering of a single bath particle, of initial momentum ~k, 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
|I〉 =
~R 〉|~k 〉 , (39a)
corresponding to an initial state density matrix
ρI =|I〉〈I|
|~R 〉|~k〉〈~k|〈~R′| .
(39b)
The corresponding Brownian particle matrix element of ρI , which is still an operator on the
bath particle state, takes the form
〈~R|ρI |~R
′〉 = ρ(~R, ~R′)|~k〉〈~k| , (39c)
ρ(~R, ~R′) = c~Rc
. (39d)
Asymptotically, the effect of the scattering is to replace the initial state |I〉 by |F 〉 =
S|I〉, with S the scattering matrix. Substituting Eq. (39a), and using translation invariance
to relate the scattering matrix S with the Brownian particle at a general coordinate, to the
scattering matrix S0 with the Brownian particle at the origin, we get [11]
|F 〉 =S|I〉 =
c~RS|
~R 〉|~k 〉
~R 〉e−i
~kOP·~RS0e
i~kOP·~R|~k 〉 ,
(40a)
with ~kOP the momentum operator for the bath particle. The corresponding final density
matrix is then
ρF =|F 〉〈F |
|~R 〉e−i
~kOP·~RS0e
i~kOP·~R|~k 〉〈~k|e−i
~kOP·~R
i~kOP·~R
〈~R′| ,
(40b)
and the Brownian particle matrix element of ρF , which is again an operator acting on the
bath particle, is
〈~R|ρF |~R
′〉 = ρ(~R, ~R′)e−i
~kOP·~RS0e
i~kOP·~R|~k〉〈~k|e−i
~kOP·~R
i~kOP·~R
. (40c)
Substituting this expression into Eq. (33d), we get
~R1 ~R
,..., ~Rn ~R′n;F
ρ(~Rℓ, ~R
~k|e−i
~kOP·~R
i~kOP·~R
~kOP·~Rℓ+1S0e
i~kOP·~Rℓ+1 |~k〉
ρ(~Rℓ, ~R
i~kOP·(~R
−~Rℓ+1)S0|~k〉e
i~k·(~Rℓ+1−~R
(40d)
with ~Rn+1 = ~R1. 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, ρ
~R, ~R′
) for
collisional decoherence [12]. Writing
〈~k|S
i~kOP·(~R
−~Rℓ+1)S0|~k〉e
i~k·(~Rℓ+1−~R
) = 1 + f(~Rℓ+1 − ~R
ℓ) , (41a)
with f 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
[1 + f(~Rℓ+1 − ~R
ℓ)] ≃ 1 +
f(~Rℓ+1 − ~R
ℓ) . (41b)
We also note that Eq. (39c), when substituted into Eq. (32c), implies that the value of ρ(n)
before the scattering is
~R1 ~R
,..., ~Rn ~R′n;I
ρ(~Rℓ, ~R
ℓ) . (41c)
Thus when the approximation of Eq. (41b) is substituted into Eq. (40d), we get
~R1 ~R
,..., ~Rn ~R′n;F
~R1 ~R
,..., ~Rn ~R′n;I
f(~Rℓ+1 − ~R
~R1 ~R
,..., ~Rn ~R′n;I
. (41d)
At this point our work is essentially finished, since the remaining steps are identical
to the standard calculation [11,12,13] proceeding from the n = 1 case of Eq. (41d), and the
structure of Eq. (41d) makes it clear how to generalize the standard result for ρ(1) to the
case of general ρ(n). 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 N , the scattering particle density. The effect
of the thermal distribution µ(~k) of momenta ~k is taken into account by including an integral
d~kµ(~k), in accordance with the mixed state procedure of Eq. (35a). Finally, expressing
the S matrix in terms of the scattering amplitude f(~k′, ~k), 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 ρ(n). For the
n = 1 case, the standard answer obtained this way is
∂ρ(1)(t)~R~R′
= −F (~R − ~R′)ρ(1)(t)~R~R′ , (42a)
F (~R) = N
d~kµ(~k)
1− ei(
~k−n̂|~k|)·~R)
|f(n̂|~k|, ~k)|2 , (42b)
where n̂ is a unit vector which gives the direction of the scattered particle momentum
~k′ = n̂|~k|. To compare Eq. (42b) with the n = 1 case of Eq. (41d), we replace ~R by
~R2 = ~R1 and ~R
′ by ~R′1. Then we see that the generalization to n ≥ 1 is given by
∂ρ(n)(t)~R1 ~R′1,..., ~Rn ~R′n
F (~Rℓ+1 − ~R
(n)(t)~R1 ~R′1,..., ~Rn ~R′n
. (42c)
This is our final result for collisional Brownian motion, giving the evolution equation
obeyed by the order n density tensor . We see that it has the generic symmetries expected in
the quantum noise case: although not totally symmetric in its pair indices, ρ(n) is symmetric
under cyclic permutation of these indices. As additional checks, we see that for n = 2 the
factor involving F is
F (~R2 − ~R
1) + F (
~R1 − ~R
2) , (43a)
which is symmetric under the interchange 1 ↔ 2, while for n = 3 we have
F (~R2 − ~R
1) + F (
~R3 − ~R
2) + F (
~R1 − ~R
3) = F
S + FA ,
F S =
[F (~R2 − ~R
1) + F (
~R3 − ~R
2) + F (
~R1 − ~R
+F (~R1 − ~R
2) + F (
~R3 − ~R
1) + F (
~R2 − ~R
3)] ,
[F (~R2 − ~R
1) + F (
~R3 − ~R
2) + F (
~R1 − ~R
−F (~R1 − ~R
2)− F (
~R3 − ~R
1)− F (
~R2 − ~R
3)] ,
(43b)
with F S symmetric, and FA antisymmetric, under any of the pair interchanges 1 ↔ 2, or
1 ↔ 3, or 2 ↔ 3. Checking the descent equations is easy. Setting ~R′1 =
~R2, the term
F (~R2− ~R
1) in Eq. (42c) vanishes, so that on integrating over
~R′1 one is left on the right hand
side with a sum F (~R1 − ~R
n) + F (
~R3 − ~R
2) + ... that does not involve
~R′1, times
d~R′1ρ
(n)(t)~R1 ~R′1, ~R
,..., ~Rn ~R′n
, (43c)
and so the descent equation for ρ(n)(t) then implies the descent equation for its time deriva-
tive.
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
ρ(1) for a system S interacting with an environment E . We assume a total system plus envi-
ronment Hamiltonian HTOT = HE +HS +H , with HE and HS respectively the environment
and system Hamiltonians, and with H the system-environment interaction Hamiltonian.
(We omit the customary subscript I 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 HE and HS . Thus the interaction Hamiltonian
carries a time dependence H(t), and the density matrix obeys the equation of motion
dρ(t)
= −i[H(t), ρ(t)] (44a)
which can be integrated to give
ρ(t) = ρ(0)− i
ds[H(s), ρ(s)] . (44b)
Substituting Eq. (44b) back into Eq. (44a) gives the additional evolution equation
dρ(t)
= −i[H(t), ρ(0)]−
ds[H(t), [H(s), ρ(s)]] . (44c)
One then notes that up to an error of order H3, the time argument of the factor ρ(s) in the
double commutator term is irrelevant, so this factor can be approximated as ρ(t), giving
dρ(t)
= −i[H(t), ρ(0)]−
ds[H(t), [H(s), ρ(t)]] , (44d)
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 ρ1ρ2...ρn
that appears in Eq. (32c). By the chain rule, we have
d(ρ1ρ2...ρn)
ρ2...ρn + ρ1...
...ρn + ρ1....
. (45a)
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 H that
appear. Since Eq. (44c) contains at least one factor of H , there are no terms in Eq. (45a)
with no factors of H . The general term in Eq. (45a) with one factor of H comes from the
term in Eq. (44c) with one factor of H , multiplied by the product of the terms from Eq. (44b)
with no factors of H , giving
[H1(t), ρ1(0)]ρ2(0)...ρn(0)+ρ1(0)[H2(t), ρ2(0)]...ρn(0)+ ...+ρ1(0)ρ2(0)...[Hn(t), ρn(0)]
(45b)
The terms in Eq. (45a) with two factors of H are of two types: (1) the quadratic term in
H on the right of Eq. (44c) times factors of ρ(0), and (2) the linear term in H on the right
of Eq. (44c), multiplied by one factor of the linear term on the right of Eq. (44b), times
factors of ρ(0). We now note that up to an error of order H3, in terms that already contain
two factors of H we can replace all factors ρ(0) or ρ(s) by the corresponding ρ(t), since the
differences ρ(t) − ρ(s) and ρ(t) − ρ(0) are all of order H . Collecting everything, we get the
following formula, which gives the needed extension of Eq. (44d),
d(ρ1ρ2...ρn)
ρ1(0)...ρℓ−1(0)[Hℓ(t), ρℓ(0)]ρℓ+1(0)...ρn(0)
ρ1(t)...ρℓ−1(t)
ds[Hℓ(t), [Hℓ(s), ρℓ(t)]]ρℓ+1(t)...ρn(t)
{ρ1(t)...ρℓ−1(t)[Hℓ(t), ρℓ(t)]ρℓ+1(t)...ρm−1(t)
ds[Hm(s), ρm(t)]ρm+1(t)...ρn(t)
+ ρ1(t)...ρℓ−1(t)
ds[Hℓ(s), ρℓ(t)]ρℓ+1(t)...ρm−1(t)[Hm(t), ρm(t)]ρm+1(t)...ρn(t)}+O(H
(45c)
Taking the overall TrE of this expression then gives a formula for the time evolution of ρ
(n)(t)
as defined by Eq.(32c).
We now make two standard assumptions. First of all, we assume at that at the
initial time t = 0, the density matrix factorizes so that ρ(0) = ρEρS , with ρE and ρS
respectively density matrices for the environment and the system which commute with one
another, and with ρE a pure state density matrix obeying ρ
E = ρE . Secondly, we assume
that 〈H〉E = TrEρEH = 0, 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
TrEρ1(0)...ρℓ−1(0)[Hℓ(t), ρℓ(0)]ρℓ+1(0)...ρn(0)
=ρS1...ρSℓ−1[(TrEρEHℓ(t)), ρSℓ]ρSℓ+1...ρSn = 0 .
(46a)
The remaining terms in Eq. (45c) all have two factors of H . Since ρ(t) and ρ(0) differ by
one power of H , in these terms, up to an error of order H3, we can replace all factors ρ(t)
by the factorized approximation
ρ(t) ≃ ρ(0) = ρEρS = ρETrEρ(0) ≃ ρETrEρ(t) = ρEρ
(1)(t) . (46b)
With these simplifications, and remembering that system operator factors ρ
with different
index values ℓ act on different Hilbert spaces HS;ℓ and so commute, Eq. (45c) becomes an
extended version of the Redfield equation,
dρ(n)(t)/dt = −
1 (t)...ρ
n (t))ℓTrEρ
ds[Hℓ(t), [Hℓ(s), ρ
(t)ρE ]]
1 (t)...ρ
n (t))ℓmTrE
n−(m−ℓ)−1
× {[Hℓ(t), ρ
(t)ρE ]ρ
m−ℓ−1
E [Hm(s), ρ
m (t)ρE ] + [Hℓ(s), ρ
(t)ρE ]ρ
m−ℓ−1
E [Hm(t), ρ
m (t)ρE ]} .
(46c)
This is converted to the Born-Markov equation by setting s→ t− s, and then extending the
upper limit of the s integration from t to ∞, giving
dρ(n)(t)/dt = −
1 (t)...ρ
n (t))ℓTrEρ
ds[Hℓ(t), [Hℓ(t− s), ρ
(t)ρE ]]
1 (t)...ρ
n (t))ℓmTrE
n−(m−ℓ)−1
× {[Hℓ(t), ρ
(t)ρE ]ρ
m−ℓ−1
E [Hm(t− s), ρ
m (t)ρE ] + [Hℓ(t− s), ρ
(t)ρE ]ρ
m−ℓ−1
E [Hm(t), ρ
m (t)ρE ]} .
(46d)
We now note that Eq. (46d) can be further simplified, by taking account of the fact
that whenever an H factor is sandwiched between factors of ρE it vanishes, since ρEHρE =
ρE〈H〉E = 0. This eliminates all terms in the sum over ℓ,m that are not adjacent in a cyclic
sense, i.e., that do not either have m = ℓ + 1, ℓ = 1, ..., n − 1, or ℓ = 1, m = n. The latter,
by use of the cyclic properties of the trace, can be rearranged to give the ℓ = n term of the
former set. We thus get a simplified set of Born-Markov equations. For n = 1, we get the
usual starting point for the Born-Markov master equation derivation,
dρ(1)(t)/dt = −TrE
ds[H(t)H(t− s)ρ(1)(t)ρE + ρ
(1)(t)ρEH(t− s)H(t)
−H(t)ρ(1)(t)ρEH(t− s)−H(t− s)ρ
(1)(t)ρEH(t)] ,
(47a)
and for n ≥ 2, with the subscript n+ 1 identified with 1,
dρ(n)(t)/dt = −TrEρE
× {(ρ
1 (t)...ρ
n (t))ℓ[Hℓ(t)Hℓ(t− s)ρ
(t) + ρ
(t)Hℓ(t− s)Hℓ(t)]
1 (t)...ρ
n (t))ℓℓ+1[ρ
(t)Hℓ(t)Hℓ+1(t− s)ρ
ℓ+1(t) + ρ
(t)Hℓ(t− s)Hℓ+1(t)ρ
ℓ+1(t)]} .
(47b)
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 deriva-
tion, 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 Hℓ(t)
has the form
Hℓ(t) =
eiωtA
ℓα(ω)Bα(t) , (48a)
with A
ℓα acting only in the system Hilbert space HS;ℓ and with Bα acting only in the
environment Hilbert space HE , and with the Hermiticity properties A
(ω) = Aℓα(−ω) and
B†α(t) = Bα(t). Since Eqs. (47a,b) are quadratic in H , one uses Eq. (48a) twice; for each
Hk(t− s) (regardless of the value of the index k) one writes
Hk(t− s) =
e−iω(t−s)Akβ(ω)Bβ(t− s) , (48b)
and for each Hk(t) (again regardless of the value of k) one writes
Hk(t) =
(ω′)B†α(t) . (48c)
The rotating wave approximation then consists of neglecting terms in the double sum with
ω′ 6= ω, so that only the diagonal terms ω′ = ω are left. From the trace over the environment,
and the integral over s, one gets correlators of the form
dseiωs〈B†α(t)Bβ(t− s)〉E ≡ Γαβ(ω) ,
dseiωs〈Bβ(t− s)B
α(t)〉E = Γαβ(−ω)
(49a)
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 Γαβ into self-adjoint and anti-self-adjoint parts, according to
Γαβ(ω) =
γαβ(ω) + iSαβ(ω) . (49b)
Proceeding in this fashion, after some algebra one gets the final result, which can be
written as an equation for all n ≥ 1 by including a δn1 to take account of the special nature
of the n = 1 equation,
dρ(n)(t)/dt =
1 (t)...ρ
n (t))ℓi[ρ
Sαβ(ω)A
(ω)Aℓβ(ω)]
γαβ(ω)
1 (t)...ρ
n (t))ℓ
δn1Aℓβ(ω)ρ
(ω)Aℓβ(ω), ρ
1 (t)...ρ
n (t))ℓℓ+1ρ
(ω)Aℓ+1β(ω)ρ
ℓ+1(t)
(50a)
Despite the fact the the n = 1 and n ≥ 2 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 α to be a three-vector index,
so that Aα becomes ~A, which is related to the dipole operator by Eq. (3.182) of ref [1].
Also, one takes Sαβ(ω) = δαβS(ω), with S(ω) given by Eq. (3.205) of ref [1], and γαβ(ω) =
(4ω3/3)[1 + N(ω)]δαβ , with N(ω) = 1/(e
βω − 1) the photon number operator. One gets in
this way the density tensor generalization of the quantum optical master equation,
dρ(n)(t)/dt =
1 (t)...ρ
n (t))ℓi[ρ
S(ω) ~A
(ω) · ~Aℓ(ω)]
(4ω3/3)[1 +N(ω)]
1 (t)...ρ
n (t))ℓ
δn1 ~Aℓ(ω) · ρ
ℓ (t)
ℓ (ω)−
ℓ (ω) ·
~Aℓ(ω), ρ
ℓ (t)}
1 (t)...ρ
n (t))ℓℓ+1ρ
(t) ~A
(ω) · ~Aℓ+1(ω)ρ
ℓ+1(t)
(50b)
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 mo-
tion of a Brownian particle of mass m, moving in a potential V (x), and interacting with
an environment consisting of harmonic oscillators with masses mo and frequencies ωo, and
annihilation operator bo. The interaction Hamiltonian is assumed to be a linear coupling
H = −xB, with
κoxo =
κo(bo + b
o)/(2moωo)
2 (51a)
a weighted sum of the harmonic oscillator coordinates. A counter-term formally of order H2,
Hc = x
2moωo
≡ x2C , (51b)
is included in the calculation, so that the total Hamiltonian is
HTOT = HE +HS +H +Hc , (52a)
with HE and HS respectively the oscillator and particle Hamiltonians,
obo +
+ V (x) .
(52b)
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 H 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 ρ(t) back to Schrödinger picture; it is easy to
see that the effect of this is to replace H(t) by H = H(0), to replace H(t − s) by H(−s)
(with H(−s) still in the interaction picture), and to change d/dt to D/dt, defined by
Dρ(n)(t)/dt = dρ(n)(t)/dt+ i
TrEρ1(t)...ρℓ−1(t)[p
ℓ/(2m) + V (xℓ), ρℓ(t)]ρℓ+1(t)...ρn(t) .
(53a)
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 H . For the analog of Eq. (47a)
for the special case n = 1, we find
Dρ(1)(t)/dt = −i[Hc, ρ
(1)(t)]− TrE
ds[HH(−s)ρ(1)(t)ρE + ρ
(1)(t)ρEH(−s)H
−Hρ(1)(t)ρEH(−s)−H(−s)ρ
(1)(t)ρEH ] ,
(53b)
and for the analog of Eq. (47b) for n ≥ 2, we have
dρ(n)(t)/dt =− i
1 (t)...ρ
n (t))ℓ[Hcℓ, ρ
ℓ (t)]
−TrEρE
1 (t)...ρ
n (t))ℓ[HℓHℓ(−s)ρ
(t) + ρ
(t)Hℓ(−s)Hℓ]
1 (t)...ρ
n (t))ℓℓ+1[ρ
(t)HℓHℓ+1(−s)ρ
ℓ+1(t) + ρ
(t)Hℓ(−s)Hℓ+1ρ
ℓ+1(t)]} .
(53c)
We next note that
Hℓ = −xℓ(0)B(0) , Hℓ(−s) = −xℓ(−s)B(−s) , (54a)
where, using the assumption that the system evolution is slow compared to the oscillator
time scale, we approximate xℓ(−s) by its free particle dynamics,
xℓ(−s) ≃ xℓ −
s . (54b)
Since the right hand sides of Eqs. (53b,c) are quadratic in H , the operator B giving the
coupling to the oscillators appears, after the environmental trace is taken, only through the
correlators
D(s) ≡i〈[B(0), B(−s)]〉E ,
D1(s) ≡〈{B(0), B(−s)}〉E ,
(55a)
so that we have
〈B(0)B(−s)〉E =
[D1(s)− iD(s)] ,
〈B(−s)B(0)〉E =
[D1(s) + iD(s)] .
(55b)
These correlators appear in the following integrals, which are evaluated or approximated in
Sec. 3.6.2 of ref [1],
dsD(s) =2C ,
dsD1(s) =4mγkBT ,
dssD(s) =2mγ ,
dssD1(s) =4mγkBT/Ω ≃ 0 ,
(55c)
with C the constant defined by the counter term of Eq. (51b), with γ a constant determined
by the harmonic oscillator spectral density, with kB and T respectively the Boltzmann con-
stant and environment temperature, and with Ω a frequency cutoff. For a spectral density
J(ω) with a Lorentz-Drude cutoff function, one has
J(ω) =
2moωo
δ(ω − ωo) =
Ω2 + ω2
. (55d)
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 n = 1, we get the
Caldeira–Leggett master equation,
Dρ(1)(t)/dt = −iγ[x, {p, ρ(1)(t)}]− 2mγkBT [x, [x, ρ
(1)(t)]] . (56a)
For the density tensors with n ≥ 2, we correspondingly get
Dρ(n)/dt =
1 (t)...ρ
n (t))ℓ
−2mγkBT{x
ℓ , ρ
ℓ (t)}+ iγ
ℓ (t)pℓxℓ − xℓpℓρ
ℓ (t)
1 (t)...ρ
n (t))ℓℓ+1[4mγkBTρ
(t)xℓxℓ+1ρ
ℓ+1(t) + iγρ
(t)(xℓpℓ+1 − pℓxℓ+1)ρ
ℓ+1(t)] .
(56b)
We also note that the term proportional to γ on the first line of Eq. (56b) can be written in
the alternative form,
(t)pℓxℓ − xℓpℓρ
(t) +
(t), {xℓ, pℓ}] . (56c)
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 n = 1 and the n ≥ 2 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,
d|ψ〉 = −
(A− 〈A〉)2dt|ψ〉+ (A− 〈A〉)dWt|ψ〉 , (57a)
d|ψ〉 = −
A2dt|ψ〉+ iAdWt|ψ〉 , (57b)
with dWt a real Brownian noise obeying dW
t = dt, and where we have dropped the Hamil-
tonian term. These lead to the respective density matrix evolution equations
dρ = −
[A, [A, ρ]]dt+ [ρ, [ρ, A]]dWt , (58a)
dρ = −
[A, [A, ρ]]dt+ i[A, ρ]dWt , (58b)
which correspond to the same Lindblad type evolution equation for the expectation E[ρ],
dE[ρ] = LE[ρ]dt , Lρ = −
[A, [A, ρ]] . (58c)
Let us now consider the effect of the stochastic evolutions of Eqs. (58a,b,c) on the
expectation of the variance V = Var(A) of the operator A,
V =TrρA2 − (TrρA)2 ,
E[V ] =TrE[ρ]A2 −E[ρi1j1ρi2j2 ]Aj1i1Aj2i2
=Trρ(1)A2 − ρ
i1j1,i2j2
Aj1i1Aj2i2 ,
(59a)
where in the final line we have used the density tensor definition of Eq. (15). For the time
evolution of E[V ] we have
dE[V ]/dt =Tr(LE[ρ])A2 − dρ
i1j1,i2j2
Aj1i1Aj2i2
=Tr(LE[ρ])A2 − 2E[ρi1j1(Lρ)i2j2 ]Aj1i1Aj2i2 − E[Ci1j1,i2j2 ]Aj1i1Aj2i2 ,
(59b)
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 Tr[A, [A, ρ]]A = 0 , the terms in Eq. (59b) involving
the Lindblad L all vanish, and so the time derivative of E[V ] comes entirely form the final
term,
dE[V ]/dt = −E[Ci1j1,i2j2 ]Aj1i1Aj2i2 , (59c)
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 A. The tensor Ci1j1,i2j2 corresponding to Eqs. (57b) and (58b) is
given in Eq. (24b), and since the cyclic property of the trace implies that Tr[A, ρ]A = 0, one
has dE[V ]/dt = 0 for this evolution. On the other hand, the tensor Ci1j1,i2j2 corresponding
to Eqs. (57a) and (58a) is given in Eq. (24a), and through Eq. (59c) implies that
dE[V ]/dt = −E[(Tr[ρ, [ρ, A]]A)2] = −E[(Tr([ρ, A])2)2] , (59d)
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 A to zero as t→ ∞, and
hence reduces the state vector to an eigenstate of A, as discussed in detail in refs [15].
Let us now consider a quantum system S, consisting of a microscopic system coupled
to a macroscopic measuring apparatus, interacting with a quantum environment E , 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 A which is a good “pointer observable”. By
definition, a system operator commutes with the environment Hamiltonian HE , and since
the system in this case includes the apparatus and so is macroscopic, the pointer observable
also obeys [17] [A,H ] = 0, with H the system-environment interaction Hamiltonian. Let us
now write the density matrix evolution in Schrödinger picture,
dρ/dt = −i[HTOT, ρ] = −i[HS +HE +H, ρ] , (60a)
with HS 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 A, we have
dV/dt = Tr(dρ/dt)A2 − 2(TrρA)(Tr(dρ/dt)A) , (60b)
which substituting Eq. (60a), and using the cyclic property of the trace and the fact that A
commutes with both HE and H , simplifies to
dV/dt = iTrρ[HS , A
2]− 2i(TrρA)(Trρ[HS , A]) . (60c)
This can be further simplified by using the definition of the reduced density matrix ρ(1) =
TrEρ, together with the fact that the commutators in Eq. (60c) involve only system operators,
giving
dV/dt = iTrSρ
(1)[HS , A
2]− 2i(TrSρ
(1)A)(TrSρ
(1)[HS , A]) . (60d)
We see that, unlike the Itô equation case discussed above, the time derivative of V here is
determined by ρ(1), rather than by ρ(2).
Let us now take the pointer observable to be a pointer center of mass coordinate
A = X , in which case, once the entanglement of the pointer with the microsystem being
measured has been established, the relevant part of the system Hamiltonian HS is P
2/(2M),
with P the total momentum operator for the pointer of macroscopic mass M . Evaluating
the commutators, and writing TrSρ
(1)O = 〈O〉, we see that
dV/dt =(1/M)〈{P,X}〉 − (2/M)〈X〉〈P 〉
=(1/M)〈{X − 〈X〉, P − 〈P 〉}〉 .
(61a)
By the Schwartz inequality, the right hand side of Eq. (61a) is bounded by
(2/M)〈(X − 〈X〉)2〉
2 〈(P − 〈P 〉)2〉
2 = (2/M)∆X∆P . (61b)
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
∆X∆P ≥ 1/2, we get a least upper bound on Eq. (61b) by substituting ∆X∆P ∼ 1/2.
This shows that |dV/dt| can be made as small as ∼ 1/M , which since M is macroscopic, can
be made essentially arbitrarily small.3 Hence the variance of the pointer variable A 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
3 Restoring factors of Planck’s constant, |dV/dt| can be as small as h̄/M , for which
the reduction time dt is at least of order MdV/h̄. For M ∼ 1024mproton and dV ∼ (1cm)
this gives dt ∼ M(1cm)2/h̄ ∼ 1027s ∼ 1010 times the age of the universe. Note that our
argument places no restriction on the mean pointer momentum 〈P 〉 that establishes the time
needed to attain one or the other of the measurement outcomes X starting from the initial
pointer position.
quantum noise cases shows that structures with different kinematical symmetries,4 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.
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
δi1j1ρ
i1j1,i2j2
= δi1j1E[ρi1j1ρi2j2 ] = E[ρi2j2 ] = ρ
whereas in the quantum noise case one instead has
δi1j1ρ
i1j1,i2j2
= δi1j1TrEρi1j1ρi2j2 = TrEρEρi2j2 6= TrEρi2j2 = ρ
with ρE = TrSρ the reduced density matrix of the environment with the system traced out.
Appendix A: Descent equations for the
isotropic spin-1/2 ensemble
Let us write the generating function of Eq. (14b) as
G[aij ] =fg ,
f(x) = sinh x
2 , x = ~A 2 ,
Tra .
Then, we find
δmrG+ gf
′ ~A · ~σmr ,
∂amr∂apq
δpqgf
′ ~A · ~σmr
+gf ′′ ~A · ~σmr ~A · ~σpq +
gf ′~σmr · ~σpq .
Here the primes denote derivatives of f with respect to x, and in this notation f obeys the
second order differential equation
xf ′′ +
f ′ =
f . (A3)
Contracting the first expression in Eq. (A2) with δmr , and using the tracelessness of the Pauli
matrices, gives the first equation in Eq. (7b). Contracting the second expression in Eq. (A2)
with δrp, and using the differential equation of Eq. (A3) together with the Pauli matrix
identities (~σ 2)mq = 3δmq and σ
iσj = δij+ iǫijkσk, which implies ~A ·~σmp ~A ·~σpq = ~A
2δmq , gives
the second equation in Eq. (7b).
Appendix B: Descent equations for the
Itô stochastic Schrödinger equation
We wish here to verify that
dG[a] = dtE
amr(Lρ)mr +
amrapqCmr,pq
obeys the descent equations of Eq. (7b). Since
δmr(Lρ)mr = δmrCmr,pq = δpqCmr,pq = 0 , (B2a)
we have
∂dG[a]
= dtE
amr(Lρ)mr +
amrapqCmr,pq
(Trρ)eρ·a
= dG[a] , (B2b)
giving the first identity in Eq. (7b). Next we calculate
∂dG[a]
(Lρ)mq +
auv(Cmq,uv + Cuv,mq)
auv(Lρ)uv +
auvarsCuv,rs
(B3a)
while for the contraction of the second variation we have (with indices m, q implicit on the
right hand side)
∂2dG[a]
∂amr∂arq
= dtE
S1 + S2 + auv(T1uv + T2uv)
, (B3b)
(Cmr,rq + Crq,mr) ,
S2 ={Lρ, ρ}mq ,
T1uv =
[(Cmr,uv + Cuv,mr)ρrq + ρmr(Cuv,rq + Crq,uv)] ,
T2uv =
(Lρ)uv +
arsCuv,rs
ρmq ,
(B3c)
We see immediately that auvT2uv gives all of the second line of Eq. (B3a). From Eqs. (16b)
and (18) we find
{Lρ, ρ}mq = (Lρ)mq − [(ck − 〈ck〉)ρ(ck − 〈ck〉)
†]mq − ρmq〈(ck − 〈ck〉)
†(ck − 〈ck〉)〉 , (B4a)
while from Eq. (21b) we have
(Cmr,rq + Crq,mr) = [(ck − 〈ck〉)ρ(ck − 〈ck〉)
†]mq + ρmq〈(ck − 〈ck〉)
†(ck − 〈ck〉)〉 . (B4b)
Hence S1 + S2 = (Lρ)mq, giving the Lρ part of the first line of Eq. (B3a). Finally, again
using Eq. (21b) we find that
[(Cmr,uv + Cuv,mr)ρrq + ρmr(Cuv,rq + Crq,uv)] =
(Cmq,uv + Cuv,mq) , (B4c)
and so auvT1uv 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
dG[a] = dtE
(a · Lρ+
(a ·Qk)
)ea·ρ
(C1a)
obeys the descent equations of Eq. (7b). Since Tr(Lρ) = 0 and TrQk = 〈Bk+B
Bk〉 = 0,
we have
∂dG[a]
= dtE
(a · Lρ+
(a ·Qk)
)(Trρ)ea·ρ
, (C1b)
checking the first line of Eq. (7b). Next we calculate the first variation of G,
∂dG[a]
(Lρ)mq +
(a ·Qk)
(p− 1)!
(Qk)mq
+(a · Lρ+
(a ·Qk)
)ρmqe
(C2a)
and the contracted second variation,
∂2dG[a]
∂amr∂arq
= dtE
(S1 + S2 + S3 + S4)e
, (C2b)
(a ·Qk)
(p− 2)!
(Q2k)mq ,
S2 ={Lρ, ρ}mq ,
(a ·Qk)
(p− 1)!
{Qk, ρ}mq ,
a · Lρ+
(a ·Qk)
ρmq .
(C2c)
We see immediately that S4 gives all of the second line of Eq. (C2a). From Eq. (30c), which
we rewrite here,
{Lρ, ρ} =Lρ−
{Qk, ρ} =Qk −Q
(C3a)
we see that the Lρ part of S2 and the Qk part of {Qk, ρ} in S3 give the first line of Eq. (C2a).
To complete the verification, we must show that S1 cancels against the remainder of S2+S3,
which is
(a ·Qk)
(p− 1)!
(Q2k)mq . (C3b)
But separating off the p = 2 term of S1, and making the change of variable p → p + 1
in the remaining sum, we see that S1 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 n = 2 to n = 1,
and then checking the descent from general n > 2 to n − 1. For the n = 2 density tensor
time derivative, writing out all terms in Eq. (50a) explicitly, and using the fact that since
operators labeled with subscripts 2 and 1 act on different Hilbert spaces, the order in which
they are written is irrelevant, we have
dρ(2)(t)/dt =i[ρ
1 (t),
Sαβ(ω)A
1α(ω)A1β(ω)]ρ
2 (t)
1 (t)i[ρ
2 (t),
Sαβ(ω)A
2α(ω)A2β(ω)]
γαβ(ω)
1α(ω)A1β(ω), ρ
1 (t)}ρ
2 (t)
γαβ(ω)ρ
1 (t)
2α(ω)A2β(ω), ρ
2 (t)}
γαβ(ω)[ρ
1 (t)A
1α(ω)A2β(ω)ρ
2 (t) + A1β(ω)ρ
1 (t)ρ
2 (t)A
2α(ω)] .
(D1a)
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
dρ(2)(t)/dt→i[ρ(1)(t),
Sαβ(ω)A
α(ω)Aβ(ω)]ρ
(1)(t)
+ρ(1)(t)i[ρ(1)(t),
Sαβ(ω)A
α(ω)Aβ(ω)]
γαβ(ω)
{A†α(ω)Aβ(ω), ρ
(1)(t)}ρ(1)(t)
γαβ(ω)ρ
(1)(t)
{A†α(ω)Aβ(ω), ρ
(1)(t)}
γαβ(ω)[ρ
(1)(t)A†α(ω)Aβ(ω)ρ
(1)(t) + Aβ(ω)(ρ
(1)(t))2A†α(ω)]
=i[(ρ(1)(t))2,
Sαβ(ω)A
α(ω)Aβ(ω)]
γαβ(ω)
Aβ(ω)(ρ
(1)(t))2A†α(ω)−
{(ρ(1)(t))2, A†α(ω)Aβ(ω)}
(D1b)
which has the structure of dρ(1)(t)/dt and so verifies the 2 → 1 descent.
To verify the n→ n− 1 descent we make some simplifications in notation. We omit
all superscripts (1), since this leads to no ambiguities, as well as all time arguments (t) and
all frequency arguments (ω). We also abbreviate
Sαβ(ω)A
(ω)Aℓβ(ω) ,
γαβ(ω)A
(ω)Aℓβ(ω) .
(D2a)
Our general strategy is to split the sum
ℓ=1 containing (ρ1...ρn)ℓ into
ℓ=2 plus the ℓ = 1
and the ℓ = n terms, and to split the sum
ℓ=1 containing (ρ1...ρn)ℓℓ+1 into
ℓ=2 plus the
ℓ = 1, ℓ = n− 1, and ℓ = n terms. For the part of dρ(n)/dt involving Lℓ, we have
(ρ1...ρn−1)ℓρni[ρℓ, Lℓ] + (ρ2...ρn)i[ρ1, L1] + (ρ1...ρn−1)i[ρn, Ln] , (D2b)
which on contracting the column index associated with the subscript n with the row index
associated with the subscript 1, and relabeling all quantities that had subscript n with
subscript 1, since they act now in the same Hilbert space, gives
(ρ21ρ2...ρn−1)ℓi[ρℓ, Lℓ] + ρ2...ρn−1i(ρ1[ρ1, L1] + [ρ1, L1]ρ1)
(ρ21ρ2...ρn−1)ℓi[ρℓ, Lℓ] + ρ2...ρn−1i[ρ
1, L1] ,
(D2c)
which has the correct structure for the corresponding part of dρ(n−1)/dt, with ρ1 replaced by
ρ21. The remainder of dρ
(n)/dt is
(ρ1...ρn)ℓ
{Mℓ, ρℓ} − (ρ2...ρn)
{M1, ρ1} − (ρ1...ρn−1)
{Mn, ρn}
(ρ1...ρn)ℓℓ+1ρℓA
Aℓ+1βρℓ+1 + ρ3...ρnρ1A
1αA2βρ2
+ ρ1...ρn−2ρn−1A
n−1αAnβρn + ρ2...ρn−1ρnA
nαA1βρ1
(D3a)
Again, contracting the column index associated with the subscript n with the row index
associated with the subscript 1, and relabeling all quantities that had subscript n with
subscript 1, since they act now in the same Hilbert space, gives
(ρ21...ρn−1)ℓ
{Mℓ, ρℓ} − (ρ2...ρn−1)
ρ1M1ρ1(∗) +
{M1, ρ
(ρ21...ρn−1)ℓℓ+1ρℓA
Aℓ+1βρℓ+1 + (ρ3...ρn−1)ρ
1αA2βρ2
+ ρ2...ρn−2ρn−1A
n−1αA1βρ
1 + ρ2...ρn−1ρ1A
1αA1βρ1(∗)
(D3b)
which on canceling the terms marked with (∗) gives the corresponding part of dρ(n−1)/dt,
with ρ1 replaced by ρ
1. This completes the verification of the n→ n− 1 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 (1) and
all time arguments (t). We first verify the n = 2 to n = 1 descent. For the n = 2 case of
Eq. (56b), we have
Dρ(2)/dt =ρ2[−2mγkBT (x
1ρ1 + ρ1x
1) + iγ(ρ1p1x1 − x1p1ρ1)]
+ρ1[−2mγkBT (x
2ρ2 + ρ2x
2) + iγ(ρ2p2x2 − x2p2ρ2)]
+4mγkBT (ρ1x1x2ρ2 + ρ2x2x1ρ1) + iγ[ρ1(x1p2 − p1x2)ρ2 + ρ2(x2p1 − p2x1)ρ1] .
(E1A)
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
Dρ(2)/dt→− 2mγkBT (x
2ρ2 + ρx2ρ) + iγ(ρpxρ− xpρ2)
− 2mγkBT (ρx
2ρ+ ρ2x2) + iγ(ρ2px− ρxpρ)
+4mγkBT (ρx
2ρ+ xρ2x) + iγ[ρ(xp− px)ρ+ pρ2x− xρ2p] .
(E1B)
We see that the terms that have an operator sandwiched between two factors of ρ cancel,
leaving only terms involving ρ2, which have the form of Eq. (56a) with ρ replaced by ρ2.
To check the n > 2 to n−1 descent, we split the sums that occur in the same manner
as in Appendix D. We thus write Eq. (56b) in the form
Dρ(n)/dt =
(ρ1...ρn)ℓ[−2mγkBT{x
ℓ , ρℓ}+ iγ(ρℓpℓxℓ − xℓpℓρℓ)]
+ρ2...ρn[−2mγkBT{x
1, ρ1}+ iγ(ρ1p1x1 − x1p1ρ1)]
+ρ1...ρn−1[−2mγkBT{x
n, ρn}+ iγ(ρnpnxn − xnpnρn)]
(ρ1...ρn)ℓℓ+1[4mγkBTρℓxℓxℓ+1ρℓ+1 + iγρℓ(xℓpℓ+1 − pℓxℓ+1)ρℓ+1]
+ρ3...ρn[4mγkBTρ1x1x2ρ2 + iγρ1(x1p2 − p1x2)ρ2]
+ρ1...ρn−2[4mγkBTρn−1xn−1xnρn + iγρn−1(xn−1pn − pn−1xn)ρn]
+ρ2...ρn−1[4mγkBTρnxnx1ρ1 + iγρn(xnp1 − pnx1)ρ1] .
We now contract the column index associated with the subscript n with the row index
associated with the subscript 1, and relabel all quantities that had subscript n with subscript
1, since they act now in the same Hilbert space. As is readily seen by inspection of Eq. (E2),
this gives Eq. (56b) with n replaced by n−1 and with ρ1 replaced by ρ
1, together with terms
of the wrong structure, that grouped together give (4−2−2)ρ2...ρn−1mγkBTρ1x
1ρ1 = 0 and
(1 − 1)ρ2...ρn−1iγρ1(x1p1 − p1x1)ρ1 = 0, which thus vanish. This completes the verification
of the descent equation for Eq. (56b).
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