Geometry and Dynamics of Quantum State
Diffusion
Nikola Burić ∗
Institute of Physics, P.O. Box 57, 11001 Belgrade, Serbia.
November 26, 2018
Abstract
Riemannian metric on real 2n-dimensional space associated with
the equation governing complex diffusion of pure states of an open
quantum system is introduced and studied. Examples of a qubit under
the influence of dephasing and thermal environments are used to show
that the curvature of the diffusion metric is a good indicator of the
properties of the environment dominated evolution and its stability.
PACS: 03.65.Yz
∗e-mail: buric@phy.bg.ac.yu
http://arxiv.org/abs/0704.1365v1
1 Introduction
The states of an open quantum system are commonly described by a density
matrix ρ̂. In many cases, the evolution of ρ̂(t) is governed by a master
equation of the Linblad form [1],[2] for the density matrix ρ̂(t)
dρ̂(t)
= −i[Ĥ, ρ̂] + 1
[L̂lρ̂, L̂
l ] + [L̂l, ρ̂L̂
l ], (1)
where the Linblad operators L̂l describe the influence of the environment.
The equation (1) represents the general form of an evolution equation for a
quantum system which satisfies Markov property.
However, this theoretical approach to the dynamics of open quantum
systems is not unique. In real experiments it is often useful to understand and
model the dynamics of pure quantum states [3],[4],[5]. Indeed, the evolution
of an open system can be described directly in terms of the dynamics of the
system’s pure state. The corresponding evolution equation is a stochastic
modification of the unitary Schroedinger equation. In fact, the density matrix
ρ̂ can be written, in different but equivalent ways, as a convex combination
of pure states. Each of these results in a stochastic differential equation for
|ψ(t) > in the Hilbert spaceH. Such stochastic Schroedinger equations (SSE)
are called stochastic unravelling [6],[7],[2] of the Linblad master equation for
the reduced density matrix ρ̂(t). There are many different forms of nonlinear
and linear SSE that have been used in the context of open systems [8],[3],
[2],[6], [7] or suggested as fundamental modifications of the Schroedinger
equation [9] [10],[11],[6] [12],[13]. They are all consistent with the requirement
that the solutions of (1) and of SSE satisfy
ρ̂(t) = E[|ψ(t) >< ψ(t)|]. (2)
where E[|ψ(t) >< ψ(t)|] is the expectation with respect to the distribution
of the stochastic process |ψ(t) >. The advantages of the description in terms
of the pure states and SSE over the description by ρ̂ are twofold. On the
practical side, the computations are much more practical, as soon as the size
of the Hilbert space is moderate or large [14]. On the theoretical side, the
stochastic evolution of pure states provides valuable insides which can not
be inferred from the density matrix approach [15],[16],[6],[2],[5],[17].
There are two main approaches to the unravelling of the Linblad master
equation: the method of quantum state diffusion [6] and the relative state
method [3], [2], with specific advantages associated with each of the methods.
The relative state method is usually used do describe the situations when the
measurement is the dominant interaction with the environment. The method
offers particular flexibility in that the master equation can be unravelled into
different stochastic equations conditioned on the results of measurement.
On the other hand the correspondence between the QSD equations and the
Linblad master equations is unique, and is not related to a particular mea-
surement scheme, or the form of the Markov environment. The resulting
SSE is always of the form of a diffusion process on the Hilbert space of pure
states, which is its main property to be explored in this paper.
We shall concentrate on the unique unravelling of the master equation
given by the quantum state diffusion equation, and explore the fact that it
represents a diffusion process. QSD equation is the unique unravelling of (1)
which preserves the norm of the state vector and has the same invariance as
(1) under the unitary transformations of the environment operators {L̂l}, [6].
The equation is given by the following formula:
|dψ > = −iĤ|ψ > dt
2 < L̂
l > L̂l − L̂
l L̂l− < L̂
l >< L̂l >
|ψ(t) > dt
(L̂l− < Ll >)|ψ(t) > dWl (3)
where <> denotes the quantum expectation in the state |ψ(t) > and dWl are
independent increments (indexed by l) of complex Wiener c-number processes
Wl(t).
The equation (3) represent a diffusion process on a complex vector space.
We shall utilize the diffusion matrix of this process to define a Riemannian
metric on the corresponding real space. We shall then study the properties of
this diffusion metric as a field fixed by the environment and in relation to the
stochastic evolution of the state vector, for different types of the environment.
It will be shown, using examples of the dephasing and thermal environments
and the measurement of an observable, that the curvature of the diffusion
metric is a good indicator of the properties of the environment dominated
evolution and its stability. We shall see that the curvature maxima of the
diffusion metric coincide with the states that are preferred by the particular
type of the environment. Furthermore, if the maxima are sharp and positive
the stochastic dynamics governed by the environment and a Hamiltonian
perturbation that does not commute with L̂l, is likely to be attracted to the
state with the maximal (positive) curvature. On the other hand the states
that correspond to the negative values of the curvature are unstable. Our
analyzes of the QSD equation, and the results, are strictly related to the
fact that the equation represent a norm-preserving diffusion process, and in
this sense are applicable to the stochastic modifications of the Schroedinger
equation that describe a norm-preserving diffusion on the Hilbert space of
pure states, like the QSD equation and, for example, the equations of the
spontaneous collapse models [13].
The structure of the paper is as follows. We shall first discuss, in the next
section, a way to relate a Riemannian metric on a real space R2n to a complex
diffusion process on Cn. Then, in section 3, we shall apply this procedure to
define the Riemannian metric associated with QSD, and than the properties
of this metric for various types of environments will be studied. Finally, in
section 4, we shall summarize and discuss our results.
2 Riemannian metric of a complex diffusion
Using the following notation
f(|ψ >) = −iĤ |ψ > (4)
2 < ψ|L̂†l |ψ > L̂l − L̂
l L̂l− < ψ|L̂
l |ψ >< ψ|L̂l|ψ >
|ψ >,
B(|ψ >)dW =
(L̂l− < ψ|Ll|ψ >)|ψ > dWl. (5)
the QSD equation (3) assumes the standard form of a stochastic differen-
tial equation (SDE) for an n-dimensional autonomous (stationary) complex
diffusion process:
d|ψ >= f(|ψ >)dt+B(|ψ >)dW. (6)
|ψ(t) > and f(|ψ(t) >) are complex vectors of complex dimension n, and dW
are differential increments of an m-dimensional complex Wiener process:
E[dWl] = E[dWldWl′ ] = 0,
E[dWldW̄ l′ ] = δl,l′dt,
l = 1, 2 . . .m, (7)
where E[·] denotes the expectation with respect to the probability distribu-
tion given by the (m-dimensional) process W , and W̄l is the complex conju-
gate of Wl. B(|ψ >) is n × m matrix, where m is at most n2 − 1, and the
diffusion matrix is
G = BB†. (8)
Thus, G(|ψ >) is Hermitian and nonnegative-definite. Notice that, unlike the
case of a general SDE, the dissipative part of the drift (4) and the diffusion
term (5) are determined by the same operators L̂l, and related in such a way
that the diffusion equation preserves the norm of the state vector.
The complex n-dimensional equation (3) generates 2n-dimensional real
diffusion. Let us introduce the following real n dimensional vectors
(ψ̄ − ψ), q =
(ψ̄ + ψ)
(q + ip), ψ̄ =
(q − ip), (9)
and a 2n dimensional vector X = (q, p). Similarly, we introduce real and
imaginary parts of the vector f and order them as components of a 2n real
vector F = (fR, f I), and introduce real and imaginary parts of the incre-
ments of the complex m-dim Wiener process dW by
dWi = (dW
i + idW
2, i = 1, 2, . . .m (10)
It is easily checked that the real and the imaginary parts are increments
of a real 2m-dimensional process, i.e.
E(dWRi dW
j ) = E(dW
j ) = δi,jdt, E(dW
j ) = 0. (11)
With this notation we have
dψ̄ + dψ
dψ̄ − dψ
. (12)
Substitution of the complex equation (3) and its complex conjugate, leads to
the following 2n dimensional real SDE:
fR(p, g)
f I(p, q)
BR −BI
BI BR
, (13)
The matrix B of dimension 2n× 2m
B = 1√
BR −BI
BI BR
, (14)
where
(B)ij = (B
R)ij + i(B
I)ij (15)
gives the diffusion matrix G for the real 2n dimensional diffusion described
by the process (13)
G = BBT = 1
(BR)(BR)T + (BI)(BI)T (BR)(BI)T − (BI)(BR)T
(BI)(BR)T − (BR)(BI)T (BI)(BI)T + (BR)(BR)T
We can write the matrix G in terms of real and imaginary components of
the n× n complex matrix G = BB† as follows
G = 1
GR GI
−GI GR
where −GI = (GI)T , since the matrix G is Hermitian. Furthermore, one
can see that, besides the equalities between the entries corresponding to the
symmetry of the matrix, there are other equalities
(G)i,j = (G)i+n,j+n, i, j = 1, 2 . . . n (18)
The matrix G is symmetric and nonnegative, but it could be singular.
However, the matrix Diag{1/2, 1/2, . . . , 1/2}+ G gives a Riemannian metric
on the real 2n dimensional vector space. The factor 1/2 of the Euclidian
part is chosen in order that the Euclidian norm of a vector corresponding to
a complex n-vector of unit norm is also unity.
Once the diffusion metric Diag{1/2, 1/2, . . . , 1/2} + G is calculated the
standard formulas [18] give the connection coefficients Γkµν of the Levi-Civita
connection for this metric in terms of the coefficients gµν = δµν/2 + (G)µν
Γkµν =
gkλ(∂µgλν + ∂νgλµ − ∂λgµν) (19)
Curvature tensor, Ricci tensor and the scalar curvature of the diffusion metric
are also given by the standard formulas [18]:
Rkλµν = ∂µΓ
νλ − ∂νΓkµλ + Γ
µη − Γ
νη, (20)
Ricµν = R
µλν , R = gµνRicµν . (21)
Before we present the results of calculations of the diffusion metric and
its curvature for different types of environments, we would like to consider
briefly real representation of the QSD equation in the case when the Linblad
operators are Hermitian. This includes, for example, the dephasing environ-
ment or measurement, or the primary QSD [19], [6] and other fundamental
stochastic modifications of the Schroedinger equation [9],[10],[12]. The goal
of this digression is to point out to the connection between the general QSD
equation (3) and some other stochastic modifications of the Schroedinger
equation that have the form of a norm-preserving diffusion equation, and
that consequently the construction of the diffusion metrics and its proper-
ties are applicable to these equations also. In the case of Hermitian Linblad
operators the real representation of (3) assumes a specially simple and illumi-
nating form. Applying the same derivation as from equation (9) to equation
(13) one obtains the following:
dpi = −Hijqjdt+ (2 < L > Lij − (L2)ij− < L2 > δij)pjdt
(Lij− < L > δij)pjdWR +
(Lij− < L > δij)qjdW I , (22)
where we have, for reasons of simplicity, included only one Linblad operator
and the summation over repeated indexes is assumed. Noticing that for an
arbitrary linear operator B
Bijqj = δij
∂ < B >
, Bijpi = δij
∂ < B >
equation (23) becomes
dpi = −
∂ < H >
∂ < L >
dWR +
∂ < L >
, (24)
where ∆2L =< L2 > − < L >2. There is an analogous equation for dqi. The
two sets of equations represent a diffusion process on R2n, consisting of the
drift given by a Hamiltonian dynamical system on R2n with the Hamilton’s
function < H > and the dissipative part determined by ∆2L =< L2 > − <
L >2 and the diffusion term determined by< L >. The drift and the diffusion
are such that the norm of the vectors in R2n is preserved. Furthermore, the
equations are invariant under a global gauge transformation corresponding
to the multiplication of vectors |ψ > by a phase factor. Takeing into the
account the norm invariance and the global phase symmetry the equations
can be written as a diffusion equation on the phase space S2n−1/S1 of the
following form
dX = Ω∇ < H > dt+∇(∆2L)dt+ 1√
∇ < L > dW (25)
where ∇ and Ω∇ are the gradient and the skew gradient on S2n−1/S1, and X
denotes the set of 2n− 2 coordinates on the reduced phase space S2n−1/S1.
Equations like (25) have been analyzed as candidates for a description of the
spontaneous state reduction in [13], or in the case L̂ = Ĥ in [12].
3 QSD metric and qualitative properties of
dynamics
Application of formula (16) gives for the case (5) of the QSD equation an
explicit procedure for calculation of the diffusion metric coefficients gij, in
terms of the coefficients of the Linblad operators and the coefficients of the
state ψ > in some bases |ψ >= ∑i ci|i >. The components of the diffusion
matrix G = BB† are given by
Bkk′(c, c̄) =
(Ll− < Ll > 1)kj(Ll†− < Ll† > 1)k′j′cj c̄′j (26)
where: < Ll >=
ss′ Lss′c
sc̄s. Expressing ci, c̄i in terms of x1 . . . x2n
xi = (c̄i + ci)/
2 i = 1, . . . n
−1(c̄i − ci)/
2 i = n+ 1, . . . 2n, (27)
separating of GR and GI and substituting in (16) finally gives the 4n2 entries
of the real matrix G.
We shall study the diffusion metric for the following three types of envi-
ronments: (a) dephasing environment; (b) the environment corresponding to
measurement of an observable and (c) thermal environment. The first two
are represented by Hermitian and the third one by a non-Hermitian Linblad
operators. The main geometrical object which we shall study are the diffusion
metric norm of a state vector and its scalar curvature. In order to illustrate
how these objects depend on the environment we shall use the simplest but
important quantum system, namely a single qubit. The system operators
can be expressed as combinations of the Pauli sigma matrices σ̂x, σ̂y, σ̂z, a
state |ψ > of unit norm is determined by < σ̂x >,< σ̂y >,< σ̂z > or by the
spherical angles (θ, φ) given by
< σ̂z > = cos(θ)
< σ̂x > = sin(θ) cos(φ)
< σ̂y > = sin(θ) sin(φ), (28)
The environment operators are [20],[3]
L̂ = µσ̂+σ̂− (29)
for the dephasing and
L̂ = µ1σ̂+ + µ2σ̂− (30)
for the thermal environment, with µ1 and µ2 proportional to the temperature,
and finally for the measurement of, say, σ̂z the Linblad operator is just
L̂ = µσ̂z. (31)
The formulas for the entries gij of the diffusion metrics in terms of the
coordinates x1, x2, x3, x4 in the three considered cases can be conveniently
written using the following notation:
s = x1x2 + x3x4, a = x1x4 − x2x3. (32)
Because many of the metric entries are repeated, it is more convenient to
present them in a list rather than to write down the corresponding matrices.
Using the notation (32), the entries of the metrics in the three considered
cases are: For dephasing:
g11 = 1/2 + (µ
2/16)d2
(2 + d2
)2, g12 = (µ
2/16)sd2
(2 + d2
), g13 = 0
g14 = (µ
2/16)ad2
(2 + d2
), g22 = 1/2 + (µ
2/16)d4
, g23 = −g14
g24 = 0, g33 = g11, g34 = g21, g44 = g22; (33)
For the thermal environment:
g11 = 1/2 + d
+ (2 + d2
]/16,
g12 = s[d
(2 + d2
)µ1 + (2 + d
µ2]/16, g1,3 = 0
g14 = a[d
(2 + d2
)µ1 + (2 + d
µ2]/16,
g22 = 1/2 + d
+ (2 + d2
g13 = 0, g23 = −g14, g24 = 0, g33 = g11, g34 = g21, g44 = g22; (34)
3,0 1
3,0 1
3,0 1
3,0 1
3,0 1
3,0 1
Figure 1: Poincaré sections for the separability constrained non-symmetric
quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a) µ = 1.3,
(b) µ = 1.7
and for the measurement of σ̂z
g11 = 1/2 + (g
2/16)d12(2 + d2
g12 = (g
2/16)s(d2
− 2)(d2
+ 2), g13 = 0
g14 = a(d
), g22 = 1/2 + (g
2/16)d2
(2 + d2
g23 = −g14, g24 = 0, g33 = g11, g34 = g21, g44 = g22. (35)
These formulas are used to compute the diffusion metric norm and the
scalar curvature as functions of the state parameters θ and φ. We shall first
consider the dependence of the stated properties of the diffusion metric on
the type of the environment and the coupling strengths µ, µ1, µ2 and then
analyze the relation between these properties and the stochastic dynamics of
the state vectors.
0,0 0,5 1,0 1,5 2,0 2,5 3,0
0,0 0,5 1,0 1,5 2,0 2,5 3,0
D (c)
0,0 0,5 1,0 1,5 2,0 2,5 3,0
0,0 0,2 0,4 0,6 0,8 1,0
0 0.5 1 1.5 2 2.5 3
0 0.5 1.0 1.5 2.0 2.5 3.0
Figure 2: Poincaré sections for the separability constrained non-symmetric
quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a) µ = 1.3,
(b) µ = 1.7
In Figures 1 and 2 we illustrate the diffusion metric norm and curva-
ture considered as functions on the sphere of states fixed by the type of
environment and the value of the corresponding coupling µ, µ1, µ2. Consider
first Figure 1. The first row (fig. (a),(c),(e)) represent the diffusion metric
norm and the second row (fig. (b),(d),(f)) the curvature for the three types
of the environments and for some typical fixed values of the corresponding
coupling strengths. The curvature is not constant, and can be positive or
negative depending on the state vector and on the coupling strength. The
maxima of the curvature can be sharp like in the cases of the dephasing and
measurement of σz. On the other hand, in the thermal case the maxima is
surrounded by a large neighborhood of states with almost maximal value of
the curvature. Thus, the curvature has a sharp maxima only at the states
which are clearly favored by the environment. If there are no such states
the curvature maximum differs very little from the neighboring values. The
curvature minima are at the states that are like repellers for the environment
dominated dynamics.
Dependence of the curvature maxima and the norm on the coupling
strength is illustrated in Figure 2 for the dephasing and the thermal en-
vironments. The most important information from these Figures is that in
the dephasing and measurement ( not shown) cases there are clearly sharp
values of corresponding coupling strength where the curvature maxima goes
from negative to positive values. Also, we see that the curvature minima are
negative for all values of the coupling strength.
We shall now study the relation between the sign of the curvature maxima
and a stability of the stochastic dynamics of the state vector. The relation
will not be analyzed in a mathematically rigorous way using an appropriate
notion of the stochastic stability and considering the evolution of the met-
ric as a stochastic process governed by the process |ψ(t) >. Instead, our
strategy is to compute the curvature along different sample paths and see
if the path remains near the state corresponding to the curvature maxima.
We do such computations for the evolution governed by the environment and
an additional fixed small hamiltonian, and we pay special attention to the
case when the Linblad operators and the hamiltonian do not commute. The
computations are repeated for the values of the coupling to the environment
slightly above and below the critical value when the curvature maxima is
zero. If the Hamiltonian perturbation is zero the sample paths that started
near a maximum of the curvature remain near this maximum. For very
small added Hamiltonian part and for a fixed value of the coupling to the
environment, the sample paths of the system could wonder away from the
maximum or could remain near it. In the former case we shall say that the
stochastic dynamics is unstable and in the later case it is stable. The relevant
computations are illustrated in Figures 3 and 4.
In the case of the dephasing environment (or the measurement of σ̂z),
when the maxima of the curvature are sharply picked, Figures 3 clearly illus-
trate that positive curvature maxima correspond to the stability and negative
to instability in the above mentioned sense. On the other hand, in the ther-
mal case, the dynamics is always unstable even if there is no Hamiltonian
perturbation. This is illustrated in Figure 4. We can conclude that the dif-
fusion metric curvature provides us with a clear picture of the qualitative
properties of the system’s dynamics under strong influence of the environ-
ment.
It is well known that if the Linblad operators are Hermitian and commute
with the hamiltonian, than the attractors of the stochastic QSD dynamics
are the common eigenstates of the Hamiltonian and the Linblad operators
[6],[13]. The curvature maxima coincide with the eigenstates of the Linblad
1,0 -1,0
1,0-0,9
0 400 800 1200
0 400 800 1200
1,0 -1,0
1,0-0,9
σ yσx
Figure 3: Poincaré sections for the separability constrained non-symmetric
quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a) µ = 1.3,
(b) µ = 1.7
1.0 -1.0
1.0-0.9
0 200 400 600 800 100012001400
Figure 4: Poincaré sections for the separability constrained non-symmetric
quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a) µ = 1.3,
(b) µ = 1.7
operators, and consequently with the eigenstates of the hamiltonian. The
probability of convergence to one of the attractors is, in this case, determined
solely by the distance of the initial state from the attractor eigenstate, that
is by the quantum mechanical transition probability, and does not depend
on the parameters of the Hamiltonian and stochastic terms. The sign of the
curvature maxima has no effect on this probability. This is the reason why
we expected that the relevance of the sign of the curvature maxima on the
stochastic stability is manifested if the Linblad operator and the Hamiltonian
perturbation do not commute. This expectation is qualitatively confirmed,
as we described and illustrated in Figures 3 and 4, by numerical computa-
tions. Observations of numerical sample paths, when the Linblad and the
Hamiltonian operators do not commute, are enough to establish the quali-
tative connection between the maxima of the curvature and the stability of
small domains near the maxima.
Finally, our treatment of the relation between the geometry of the dif-
fusion and the stability of the stochastic dynamics is rather heuristic. We
treated the diffusion metric as a given field on R2n (determined by the Lin-
blad operators), and we numerically studied the paths of the stochastic pro-
cess |ψ(t) > in relation to the sign of the curvature maxima. However, the
problem of stability versus the properties of the diffusion metric should be
formulated and studied using the appropriate notions of stochastic stability
[21],[22]. Nevertheless, we think that the numerical evidence strongly indi-
cates that there is a clear relation between the sign and the shape of the
curvature maxima and the systems dynamical stability.
4 Summary and discussion
According to the view of QSD theory, evolution of a state of an open quan-
tum system is a diffusion process governed by a complex stochastic differen-
tial equation on the Hilbert space of the system. The diffusion term of the
QSD evolution equation explicitly depends on the operators modelling the
environment and on the current state vector of the system. We have stud-
ied the Riemannian metric associated with the diffusion term in the QSD
equation. The metric is defined on the real 2n-dimensional space (here n is
the complex dimension of the Hilbert space) and is directly related to the
properties of the Linblad operators of the environment. We have shown that
the scalar curvature of the metric has local maxima at states that are favored
by the corresponding environment. The curvature at different points, and
in particular its local maxima, can be be negative or positive depending on
the strength of the coupling to the environment. Also, the sharpness of the
curvature maxima reflects the type of the environment. We have shown that
there is a sense in which the sign of the curvature maxima is related to the
stability of the corresponding state under the addition of a small perturba-
tion that does not commute with the considered Linblad operator. If the
environment type and the coupling strength are such that the curvature has
sharp positive maxima, than the corresponding state is likely to attract the
states of the system whose evolution is governed by the environment and a
Hamiltonian that do not necessarily commute. On the other hand, if the
curvature maxima are negative, the corresponding states are dynamically
unstable under a small Hamiltonian perturbation that does not commute
with the Linblad operators. In conclusion, the curvature of the diffusion
metric is a relatively easy to calculate, and a very good indicator of what the
environment dominated dynamics of the system would look like.
The QSD equation describes the evolution of a pure quantum state using
the Hilbert space of the quantum system, but, because it is norm-preserving,
it gives also an equation on the state space, namely on the space of rays of
the Hilbert space. The Riemannian metric associated with the diffusion on
R2n gives a Hermitian modification of the Fubini-Study metric on CP n−1. It
is common to consider the complex projective manifold with the associated
Fubini-Study metric as the proper framework for the geometry of quantum
states [23],[24], so the modification of the metric due to the diffusion should
also be formulated within this framework.
The examples that we have analyzed in this paper are restricted on a
single qubit under the influence of various types of environments. It would
be interesting to analyze the properties of the diffusion metric in the case of
coupled gubits, and in particular to see what is the curvature at the entangled
states. Probably the proper framework for such analyzes is the formulation
on CP n−1, mentioned in the previous paragraph, because the entangled states
then have characteristic geometric interpretation [23].
Acknowledgements This work is partly supported by the Serbian Min-
istry of Science contract No. 141003. I should also like to acknowledge the
support and hospitality of the Abdus Salam ICTP.
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[23] Brody D C and Hughston L P 2001 J.Geom. Phys. 38 19
[24] Bengtsson I and Žyczkowski K 2006 Geometry of Quantum States (Cam-
bridge: Cambridge University Press)
FIGURE CAPTIONS
Figure 1 Diffusion metric norm (a,c,e) and curvature (b,d,f) as functions
of state parametrized by (θ, φ), for dephasing environment with µ = 0.6
(a,b); measurement of σ̂z with µ = 1. (c,d) and thermal environment with
µ1 = 2, µ2 = 1 (e,f).
Figure 2 Diffusion metric norm (a,c) and curvature (b,d) as functions of
θ for different values of the parameters µ or µ1, µ2, and the maximum over
(θ, φ) of the curvature as a function of µ (e) or µ1 = 2µ2 = 2µ(f) . Figures
a,b,e corespond to the dephasing and c, d, f to the thermal environment.
Figure 3Diffusion metric curvature (a,c) along the corresponding stochas-
tic path illustrated in b,d for the dephasing environment and µ = 0.3 when
maxR < 0 (a,b), and µ = 0.5 when maxR > 0(c,d). The small Hamiltonian
perturbation is 0.01σ̂x.
Figure 4 Diffusion metric curvature (a) along a stochastic path (b) for
the thermal environment and µ1 = 2µ2 = 1.6. The Hamiltonian part is zero.
	Introduction
	Riemannian metric of a complex diffusion
	 QSD metric and qualitative properties of dynamics
	Summary and discussion

  Riemannian metric on real 2n-dimensional space associated with the equation
governing complex diffusion of pure states of an open quantum system is
introduced and studied. Examples of a qubit under the influence of dephasing
and thermal environments are used to show that the curvature of the diffusion
metric is a good indicator of the properties of the environment dominated
evolution and its stability.

Introduction
The states of an open quantum system are commonly described by a density
matrix ρ̂. In many cases, the evolution of ρ̂(t) is governed by a master
equation of the Linblad form [1],[2] for the density matrix ρ̂(t)
dρ̂(t)
= −i[Ĥ, ρ̂] + 1
[L̂lρ̂, L̂
l ] + [L̂l, ρ̂L̂
l ], (1)
where the Linblad operators L̂l describe the influence of the environment.
The equation (1) represents the general form of an evolution equation for a
quantum system which satisfies Markov property.
However, this theoretical approach to the dynamics of open quantum
systems is not unique. In real experiments it is often useful to understand and
model the dynamics of pure quantum states [3],[4],[5]. Indeed, the evolution
of an open system can be described directly in terms of the dynamics of the
system’s pure state. The corresponding evolution equation is a stochastic
modification of the unitary Schroedinger equation. In fact, the density matrix
ρ̂ can be written, in different but equivalent ways, as a convex combination
of pure states. Each of these results in a stochastic differential equation for
|ψ(t) > in the Hilbert spaceH. Such stochastic Schroedinger equations (SSE)
are called stochastic unravelling [6],[7],[2] of the Linblad master equation for
the reduced density matrix ρ̂(t). There are many different forms of nonlinear
and linear SSE that have been used in the context of open systems [8],[3],
[2],[6], [7] or suggested as fundamental modifications of the Schroedinger
equation [9] [10],[11],[6] [12],[13]. They are all consistent with the requirement
that the solutions of (1) and of SSE satisfy
ρ̂(t) = E[|ψ(t) >< ψ(t)|]. (2)
where E[|ψ(t) >< ψ(t)|] is the expectation with respect to the distribution
of the stochastic process |ψ(t) >. The advantages of the description in terms
of the pure states and SSE over the description by ρ̂ are twofold. On the
practical side, the computations are much more practical, as soon as the size
of the Hilbert space is moderate or large [14]. On the theoretical side, the
stochastic evolution of pure states provides valuable insides which can not
be inferred from the density matrix approach [15],[16],[6],[2],[5],[17].
There are two main approaches to the unravelling of the Linblad master
equation: the method of quantum state diffusion [6] and the relative state
method [3], [2], with specific advantages associated with each of the methods.
The relative state method is usually used do describe the situations when the
measurement is the dominant interaction with the environment. The method
offers particular flexibility in that the master equation can be unravelled into
different stochastic equations conditioned on the results of measurement.
On the other hand the correspondence between the QSD equations and the
Linblad master equations is unique, and is not related to a particular mea-
surement scheme, or the form of the Markov environment. The resulting
SSE is always of the form of a diffusion process on the Hilbert space of pure
states, which is its main property to be explored in this paper.
We shall concentrate on the unique unravelling of the master equation
given by the quantum state diffusion equation, and explore the fact that it
represents a diffusion process. QSD equation is the unique unravelling of (1)
which preserves the norm of the state vector and has the same invariance as
(1) under the unitary transformations of the environment operators {L̂l}, [6].
The equation is given by the following formula:
|dψ > = −iĤ|ψ > dt
2 < L̂
l > L̂l − L̂
l L̂l− < L̂
l >< L̂l >
|ψ(t) > dt
(L̂l− < Ll >)|ψ(t) > dWl (3)
where <> denotes the quantum expectation in the state |ψ(t) > and dWl are
independent increments (indexed by l) of complex Wiener c-number processes
Wl(t).
The equation (3) represent a diffusion process on a complex vector space.
We shall utilize the diffusion matrix of this process to define a Riemannian
metric on the corresponding real space. We shall then study the properties of
this diffusion metric as a field fixed by the environment and in relation to the
stochastic evolution of the state vector, for different types of the environment.
It will be shown, using examples of the dephasing and thermal environments
and the measurement of an observable, that the curvature of the diffusion
metric is a good indicator of the properties of the environment dominated
evolution and its stability. We shall see that the curvature maxima of the
diffusion metric coincide with the states that are preferred by the particular
type of the environment. Furthermore, if the maxima are sharp and positive
the stochastic dynamics governed by the environment and a Hamiltonian
perturbation that does not commute with L̂l, is likely to be attracted to the
state with the maximal (positive) curvature. On the other hand the states
that correspond to the negative values of the curvature are unstable. Our
analyzes of the QSD equation, and the results, are strictly related to the
fact that the equation represent a norm-preserving diffusion process, and in
this sense are applicable to the stochastic modifications of the Schroedinger
equation that describe a norm-preserving diffusion on the Hilbert space of
pure states, like the QSD equation and, for example, the equations of the
spontaneous collapse models [13].
The structure of the paper is as follows. We shall first discuss, in the next
section, a way to relate a Riemannian metric on a real space R2n to a complex
diffusion process on Cn. Then, in section 3, we shall apply this procedure to
define the Riemannian metric associated with QSD, and than the properties
of this metric for various types of environments will be studied. Finally, in
section 4, we shall summarize and discuss our results.
2 Riemannian metric of a complex diffusion
Using the following notation
f(|ψ >) = −iĤ |ψ > (4)
2 < ψ|L̂†l |ψ > L̂l − L̂
l L̂l− < ψ|L̂
l |ψ >< ψ|L̂l|ψ >
|ψ >,
B(|ψ >)dW =
(L̂l− < ψ|Ll|ψ >)|ψ > dWl. (5)
the QSD equation (3) assumes the standard form of a stochastic differen-
tial equation (SDE) for an n-dimensional autonomous (stationary) complex
diffusion process:
d|ψ >= f(|ψ >)dt+B(|ψ >)dW. (6)
|ψ(t) > and f(|ψ(t) >) are complex vectors of complex dimension n, and dW
are differential increments of an m-dimensional complex Wiener process:
E[dWl] = E[dWldWl′ ] = 0,
E[dWldW̄ l′ ] = δl,l′dt,
l = 1, 2 . . .m, (7)
where E[·] denotes the expectation with respect to the probability distribu-
tion given by the (m-dimensional) process W , and W̄l is the complex conju-
gate of Wl. B(|ψ >) is n × m matrix, where m is at most n2 − 1, and the
diffusion matrix is
G = BB†. (8)
Thus, G(|ψ >) is Hermitian and nonnegative-definite. Notice that, unlike the
case of a general SDE, the dissipative part of the drift (4) and the diffusion
term (5) are determined by the same operators L̂l, and related in such a way
that the diffusion equation preserves the norm of the state vector.
The complex n-dimensional equation (3) generates 2n-dimensional real
diffusion. Let us introduce the following real n dimensional vectors
(ψ̄ − ψ), q =
(ψ̄ + ψ)
(q + ip), ψ̄ =
(q − ip), (9)
and a 2n dimensional vector X = (q, p). Similarly, we introduce real and
imaginary parts of the vector f and order them as components of a 2n real
vector F = (fR, f I), and introduce real and imaginary parts of the incre-
ments of the complex m-dim Wiener process dW by
dWi = (dW
i + idW
2, i = 1, 2, . . .m (10)
It is easily checked that the real and the imaginary parts are increments
of a real 2m-dimensional process, i.e.
E(dWRi dW
j ) = E(dW
j ) = δi,jdt, E(dW
j ) = 0. (11)
With this notation we have
dψ̄ + dψ
dψ̄ − dψ
. (12)
Substitution of the complex equation (3) and its complex conjugate, leads to
the following 2n dimensional real SDE:
fR(p, g)
f I(p, q)
BR −BI
BI BR
, (13)
The matrix B of dimension 2n× 2m
B = 1√
BR −BI
BI BR
, (14)
where
(B)ij = (B
R)ij + i(B
I)ij (15)
gives the diffusion matrix G for the real 2n dimensional diffusion described
by the process (13)
G = BBT = 1
(BR)(BR)T + (BI)(BI)T (BR)(BI)T − (BI)(BR)T
(BI)(BR)T − (BR)(BI)T (BI)(BI)T + (BR)(BR)T
We can write the matrix G in terms of real and imaginary components of
the n× n complex matrix G = BB† as follows
G = 1
GR GI
−GI GR
where −GI = (GI)T , since the matrix G is Hermitian. Furthermore, one
can see that, besides the equalities between the entries corresponding to the
symmetry of the matrix, there are other equalities
(G)i,j = (G)i+n,j+n, i, j = 1, 2 . . . n (18)
The matrix G is symmetric and nonnegative, but it could be singular.
However, the matrix Diag{1/2, 1/2, . . . , 1/2}+ G gives a Riemannian metric
on the real 2n dimensional vector space. The factor 1/2 of the Euclidian
part is chosen in order that the Euclidian norm of a vector corresponding to
a complex n-vector of unit norm is also unity.
Once the diffusion metric Diag{1/2, 1/2, . . . , 1/2} + G is calculated the
standard formulas [18] give the connection coefficients Γkµν of the Levi-Civita
connection for this metric in terms of the coefficients gµν = δµν/2 + (G)µν
Γkµν =
gkλ(∂µgλν + ∂νgλµ − ∂λgµν) (19)
Curvature tensor, Ricci tensor and the scalar curvature of the diffusion metric
are also given by the standard formulas [18]:
Rkλµν = ∂µΓ
νλ − ∂νΓkµλ + Γ
µη − Γ
νη, (20)
Ricµν = R
µλν , R = gµνRicµν . (21)
Before we present the results of calculations of the diffusion metric and
its curvature for different types of environments, we would like to consider
briefly real representation of the QSD equation in the case when the Linblad
operators are Hermitian. This includes, for example, the dephasing environ-
ment or measurement, or the primary QSD [19], [6] and other fundamental
stochastic modifications of the Schroedinger equation [9],[10],[12]. The goal
of this digression is to point out to the connection between the general QSD
equation (3) and some other stochastic modifications of the Schroedinger
equation that have the form of a norm-preserving diffusion equation, and
that consequently the construction of the diffusion metrics and its proper-
ties are applicable to these equations also. In the case of Hermitian Linblad
operators the real representation of (3) assumes a specially simple and illumi-
nating form. Applying the same derivation as from equation (9) to equation
(13) one obtains the following:
dpi = −Hijqjdt+ (2 < L > Lij − (L2)ij− < L2 > δij)pjdt
(Lij− < L > δij)pjdWR +
(Lij− < L > δij)qjdW I , (22)
where we have, for reasons of simplicity, included only one Linblad operator
and the summation over repeated indexes is assumed. Noticing that for an
arbitrary linear operator B
Bijqj = δij
∂ < B >
, Bijpi = δij
∂ < B >
equation (23) becomes
dpi = −
∂ < H >
∂ < L >
dWR +
∂ < L >
, (24)
where ∆2L =< L2 > − < L >2. There is an analogous equation for dqi. The
two sets of equations represent a diffusion process on R2n, consisting of the
drift given by a Hamiltonian dynamical system on R2n with the Hamilton’s
function < H > and the dissipative part determined by ∆2L =< L2 > − <
L >2 and the diffusion term determined by< L >. The drift and the diffusion
are such that the norm of the vectors in R2n is preserved. Furthermore, the
equations are invariant under a global gauge transformation corresponding
to the multiplication of vectors |ψ > by a phase factor. Takeing into the
account the norm invariance and the global phase symmetry the equations
can be written as a diffusion equation on the phase space S2n−1/S1 of the
following form
dX = Ω∇ < H > dt+∇(∆2L)dt+ 1√
∇ < L > dW (25)
where ∇ and Ω∇ are the gradient and the skew gradient on S2n−1/S1, and X
denotes the set of 2n− 2 coordinates on the reduced phase space S2n−1/S1.
Equations like (25) have been analyzed as candidates for a description of the
spontaneous state reduction in [13], or in the case L̂ = Ĥ in [12].
3 QSD metric and qualitative properties of
dynamics
Application of formula (16) gives for the case (5) of the QSD equation an
explicit procedure for calculation of the diffusion metric coefficients gij, in
terms of the coefficients of the Linblad operators and the coefficients of the
state ψ > in some bases |ψ >= ∑i ci|i >. The components of the diffusion
matrix G = BB† are given by
Bkk′(c, c̄) =
(Ll− < Ll > 1)kj(Ll†− < Ll† > 1)k′j′cj c̄′j (26)
where: < Ll >=
ss′ Lss′c
sc̄s. Expressing ci, c̄i in terms of x1 . . . x2n
xi = (c̄i + ci)/
2 i = 1, . . . n
−1(c̄i − ci)/
2 i = n+ 1, . . . 2n, (27)
separating of GR and GI and substituting in (16) finally gives the 4n2 entries
of the real matrix G.
We shall study the diffusion metric for the following three types of envi-
ronments: (a) dephasing environment; (b) the environment corresponding to
measurement of an observable and (c) thermal environment. The first two
are represented by Hermitian and the third one by a non-Hermitian Linblad
operators. The main geometrical object which we shall study are the diffusion
metric norm of a state vector and its scalar curvature. In order to illustrate
how these objects depend on the environment we shall use the simplest but
important quantum system, namely a single qubit. The system operators
can be expressed as combinations of the Pauli sigma matrices σ̂x, σ̂y, σ̂z, a
state |ψ > of unit norm is determined by < σ̂x >,< σ̂y >,< σ̂z > or by the
spherical angles (θ, φ) given by
< σ̂z > = cos(θ)
< σ̂x > = sin(θ) cos(φ)
< σ̂y > = sin(θ) sin(φ), (28)
The environment operators are [20],[3]
L̂ = µσ̂+σ̂− (29)
for the dephasing and
L̂ = µ1σ̂+ + µ2σ̂− (30)
for the thermal environment, with µ1 and µ2 proportional to the temperature,
and finally for the measurement of, say, σ̂z the Linblad operator is just
L̂ = µσ̂z. (31)
The formulas for the entries gij of the diffusion metrics in terms of the
coordinates x1, x2, x3, x4 in the three considered cases can be conveniently
written using the following notation:
s = x1x2 + x3x4, a = x1x4 − x2x3. (32)
Because many of the metric entries are repeated, it is more convenient to
present them in a list rather than to write down the corresponding matrices.
Using the notation (32), the entries of the metrics in the three considered
cases are: For dephasing:
g11 = 1/2 + (µ
2/16)d2
(2 + d2
)2, g12 = (µ
2/16)sd2
(2 + d2
), g13 = 0
g14 = (µ
2/16)ad2
(2 + d2
), g22 = 1/2 + (µ
2/16)d4
, g23 = −g14
g24 = 0, g33 = g11, g34 = g21, g44 = g22; (33)
For the thermal environment:
g11 = 1/2 + d
+ (2 + d2
]/16,
g12 = s[d
(2 + d2
)µ1 + (2 + d
µ2]/16, g1,3 = 0
g14 = a[d
(2 + d2
)µ1 + (2 + d
µ2]/16,
g22 = 1/2 + d
+ (2 + d2
g13 = 0, g23 = −g14, g24 = 0, g33 = g11, g34 = g21, g44 = g22; (34)
3,0 1
3,0 1
3,0 1
3,0 1
3,0 1
3,0 1
Figure 1: Poincaré sections for the separability constrained non-symmetric
quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a) µ = 1.3,
(b) µ = 1.7
and for the measurement of σ̂z
g11 = 1/2 + (g
2/16)d12(2 + d2
g12 = (g
2/16)s(d2
− 2)(d2
+ 2), g13 = 0
g14 = a(d
), g22 = 1/2 + (g
2/16)d2
(2 + d2
g23 = −g14, g24 = 0, g33 = g11, g34 = g21, g44 = g22. (35)
These formulas are used to compute the diffusion metric norm and the
scalar curvature as functions of the state parameters θ and φ. We shall first
consider the dependence of the stated properties of the diffusion metric on
the type of the environment and the coupling strengths µ, µ1, µ2 and then
analyze the relation between these properties and the stochastic dynamics of
the state vectors.
0,0 0,5 1,0 1,5 2,0 2,5 3,0
0,0 0,5 1,0 1,5 2,0 2,5 3,0
D (c)
0,0 0,5 1,0 1,5 2,0 2,5 3,0
0,0 0,2 0,4 0,6 0,8 1,0
0 0.5 1 1.5 2 2.5 3
0 0.5 1.0 1.5 2.0 2.5 3.0
Figure 2: Poincaré sections for the separability constrained non-symmetric
quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a) µ = 1.3,
(b) µ = 1.7
In Figures 1 and 2 we illustrate the diffusion metric norm and curva-
ture considered as functions on the sphere of states fixed by the type of
environment and the value of the corresponding coupling µ, µ1, µ2. Consider
first Figure 1. The first row (fig. (a),(c),(e)) represent the diffusion metric
norm and the second row (fig. (b),(d),(f)) the curvature for the three types
of the environments and for some typical fixed values of the corresponding
coupling strengths. The curvature is not constant, and can be positive or
negative depending on the state vector and on the coupling strength. The
maxima of the curvature can be sharp like in the cases of the dephasing and
measurement of σz. On the other hand, in the thermal case the maxima is
surrounded by a large neighborhood of states with almost maximal value of
the curvature. Thus, the curvature has a sharp maxima only at the states
which are clearly favored by the environment. If there are no such states
the curvature maximum differs very little from the neighboring values. The
curvature minima are at the states that are like repellers for the environment
dominated dynamics.
Dependence of the curvature maxima and the norm on the coupling
strength is illustrated in Figure 2 for the dephasing and the thermal en-
vironments. The most important information from these Figures is that in
the dephasing and measurement ( not shown) cases there are clearly sharp
values of corresponding coupling strength where the curvature maxima goes
from negative to positive values. Also, we see that the curvature minima are
negative for all values of the coupling strength.
We shall now study the relation between the sign of the curvature maxima
and a stability of the stochastic dynamics of the state vector. The relation
will not be analyzed in a mathematically rigorous way using an appropriate
notion of the stochastic stability and considering the evolution of the met-
ric as a stochastic process governed by the process |ψ(t) >. Instead, our
strategy is to compute the curvature along different sample paths and see
if the path remains near the state corresponding to the curvature maxima.
We do such computations for the evolution governed by the environment and
an additional fixed small hamiltonian, and we pay special attention to the
case when the Linblad operators and the hamiltonian do not commute. The
computations are repeated for the values of the coupling to the environment
slightly above and below the critical value when the curvature maxima is
zero. If the Hamiltonian perturbation is zero the sample paths that started
near a maximum of the curvature remain near this maximum. For very
small added Hamiltonian part and for a fixed value of the coupling to the
environment, the sample paths of the system could wonder away from the
maximum or could remain near it. In the former case we shall say that the
stochastic dynamics is unstable and in the later case it is stable. The relevant
computations are illustrated in Figures 3 and 4.
In the case of the dephasing environment (or the measurement of σ̂z),
when the maxima of the curvature are sharply picked, Figures 3 clearly illus-
trate that positive curvature maxima correspond to the stability and negative
to instability in the above mentioned sense. On the other hand, in the ther-
mal case, the dynamics is always unstable even if there is no Hamiltonian
perturbation. This is illustrated in Figure 4. We can conclude that the dif-
fusion metric curvature provides us with a clear picture of the qualitative
properties of the system’s dynamics under strong influence of the environ-
ment.
It is well known that if the Linblad operators are Hermitian and commute
with the hamiltonian, than the attractors of the stochastic QSD dynamics
are the common eigenstates of the Hamiltonian and the Linblad operators
[6],[13]. The curvature maxima coincide with the eigenstates of the Linblad
1,0 -1,0
1,0-0,9
0 400 800 1200
0 400 800 1200
1,0 -1,0
1,0-0,9
σ yσx
Figure 3: Poincaré sections for the separability constrained non-symmetric
quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a) µ = 1.3,
(b) µ = 1.7
1.0 -1.0
1.0-0.9
0 200 400 600 800 100012001400
Figure 4: Poincaré sections for the separability constrained non-symmetric
quantum dynamics (39). The parameters are ω = 1, h = 1.5 and (a) µ = 1.3,
(b) µ = 1.7
operators, and consequently with the eigenstates of the hamiltonian. The
probability of convergence to one of the attractors is, in this case, determined
solely by the distance of the initial state from the attractor eigenstate, that
is by the quantum mechanical transition probability, and does not depend
on the parameters of the Hamiltonian and stochastic terms. The sign of the
curvature maxima has no effect on this probability. This is the reason why
we expected that the relevance of the sign of the curvature maxima on the
stochastic stability is manifested if the Linblad operator and the Hamiltonian
perturbation do not commute. This expectation is qualitatively confirmed,
as we described and illustrated in Figures 3 and 4, by numerical computa-
tions. Observations of numerical sample paths, when the Linblad and the
Hamiltonian operators do not commute, are enough to establish the quali-
tative connection between the maxima of the curvature and the stability of
small domains near the maxima.
Finally, our treatment of the relation between the geometry of the dif-
fusion and the stability of the stochastic dynamics is rather heuristic. We
treated the diffusion metric as a given field on R2n (determined by the Lin-
blad operators), and we numerically studied the paths of the stochastic pro-
cess |ψ(t) > in relation to the sign of the curvature maxima. However, the
problem of stability versus the properties of the diffusion metric should be
formulated and studied using the appropriate notions of stochastic stability
[21],[22]. Nevertheless, we think that the numerical evidence strongly indi-
cates that there is a clear relation between the sign and the shape of the
curvature maxima and the systems dynamical stability.
4 Summary and discussion
According to the view of QSD theory, evolution of a state of an open quan-
tum system is a diffusion process governed by a complex stochastic differen-
tial equation on the Hilbert space of the system. The diffusion term of the
QSD evolution equation explicitly depends on the operators modelling the
environment and on the current state vector of the system. We have stud-
ied the Riemannian metric associated with the diffusion term in the QSD
equation. The metric is defined on the real 2n-dimensional space (here n is
the complex dimension of the Hilbert space) and is directly related to the
properties of the Linblad operators of the environment. We have shown that
the scalar curvature of the metric has local maxima at states that are favored
by the corresponding environment. The curvature at different points, and
in particular its local maxima, can be be negative or positive depending on
the strength of the coupling to the environment. Also, the sharpness of the
curvature maxima reflects the type of the environment. We have shown that
there is a sense in which the sign of the curvature maxima is related to the
stability of the corresponding state under the addition of a small perturba-
tion that does not commute with the considered Linblad operator. If the
environment type and the coupling strength are such that the curvature has
sharp positive maxima, than the corresponding state is likely to attract the
states of the system whose evolution is governed by the environment and a
Hamiltonian that do not necessarily commute. On the other hand, if the
curvature maxima are negative, the corresponding states are dynamically
unstable under a small Hamiltonian perturbation that does not commute
with the Linblad operators. In conclusion, the curvature of the diffusion
metric is a relatively easy to calculate, and a very good indicator of what the
environment dominated dynamics of the system would look like.
The QSD equation describes the evolution of a pure quantum state using
the Hilbert space of the quantum system, but, because it is norm-preserving,
it gives also an equation on the state space, namely on the space of rays of
the Hilbert space. The Riemannian metric associated with the diffusion on
R2n gives a Hermitian modification of the Fubini-Study metric on CP n−1. It
is common to consider the complex projective manifold with the associated
Fubini-Study metric as the proper framework for the geometry of quantum
states [23],[24], so the modification of the metric due to the diffusion should
also be formulated within this framework.
The examples that we have analyzed in this paper are restricted on a
single qubit under the influence of various types of environments. It would
be interesting to analyze the properties of the diffusion metric in the case of
coupled gubits, and in particular to see what is the curvature at the entangled
states. Probably the proper framework for such analyzes is the formulation
on CP n−1, mentioned in the previous paragraph, because the entangled states
then have characteristic geometric interpretation [23].
Acknowledgements This work is partly supported by the Serbian Min-
istry of Science contract No. 141003. I should also like to acknowledge the
support and hospitality of the Abdus Salam ICTP.
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FIGURE CAPTIONS
Figure 1 Diffusion metric norm (a,c,e) and curvature (b,d,f) as functions
of state parametrized by (θ, φ), for dephasing environment with µ = 0.6
(a,b); measurement of σ̂z with µ = 1. (c,d) and thermal environment with
µ1 = 2, µ2 = 1 (e,f).
Figure 2 Diffusion metric norm (a,c) and curvature (b,d) as functions of
θ for different values of the parameters µ or µ1, µ2, and the maximum over
(θ, φ) of the curvature as a function of µ (e) or µ1 = 2µ2 = 2µ(f) . Figures
a,b,e corespond to the dephasing and c, d, f to the thermal environment.
Figure 3Diffusion metric curvature (a,c) along the corresponding stochas-
tic path illustrated in b,d for the dephasing environment and µ = 0.3 when
maxR < 0 (a,b), and µ = 0.5 when maxR > 0(c,d). The small Hamiltonian
perturbation is 0.01σ̂x.
Figure 4 Diffusion metric curvature (a) along a stochastic path (b) for
the thermal environment and µ1 = 2µ2 = 1.6. The Hamiltonian part is zero.
	Introduction
	Riemannian metric of a complex diffusion
	 QSD metric and qualitative properties of dynamics
	Summary and discussion
