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Information geometry and reduced quantum description
Vol. 38 (1996)
REPORTS
ON
MATHEMATICAL
No. 3
PHYSICS
INFORMATION GEOMETRY REDUCED QUANTUM DESCRIPTION
AND
R. F. STREATER Department of Mathematics, King’s College, Strand, London WC2R 2LS, UK e-mail: ray. streaterQkc1. ac .uk (Received
April
10, 1996)
We consider the quantum dynamics of a system with two time-scales, having discrete time and unbounded slow variables, under a useful set of axioms. We argue that the Kubo susceptibility tensor is the appropriate generalisation of the Fisher information metric on the set of density matrices. We define the Amari connections V(*‘) in the unbounded case. The geometric picture of reduced quantum dynamics, described by Balian, Alhassid and Reinhardt [Phys. Rep. 131, 2-1461, is extended to some unbounded cases. We show that the reduced dynamics is a contraction in the Bogoliubov norm. Keywords:
Fisher
information
metric,
Kubo
susceptibility,
Kato-bounded
perturba-
tions.
1.
Introduction
and summary
Non-equilibrium
statistical
t,ical dynamics
[I].
is implemented
by following
ables, Xi, are random necessarily
mechanics
We here develop
is best studied within the framework
the projection
the time-evolution
. . . , X, say, called the “slow” variables. variables,
while in the quantum
bounded,
on a Hilbert
case, and the technical the C*-algebra
method
version
they are self-adjoint
space ‘H. In this paper we shall consider
operators
of statis-
in [2, 3, 41. This
of the means of a selected list of observIn classical statistical mechanics these
issues arising when some of the Xj
of all bounded
described
are not bounded.
operators,
not
the quantum Let A denote
on N, and let C be the set of normal
states
on
A. The real linear span X= should
contain
the unit operator,
very slow variables!
We can assume
SpanR{Xi,...,X,} Xi
= 1, and the energy without
(1)
Ho of the system;
loss in generality
that
these
are
dim X = n.
When our information about the system consists of the means of X E X, Jaynes [5] and Ingarden [6] advocate that the system should be assigned the state of maximum entropy, among all states p E C with the given means. According to these authors. this
420
R.F.STREATER
is the best
estimate
for the state
define the map Q, written domain
of states,
of the system,
given our limited
on the right of the state
as follows:
the state
knowledge.
thus p t+ pQ, acting
pQ E C is the state of maximum
Let us
on a suitable
entropy
obeying
the n equations
Tr(pQ) = p&.X, Here, for a state
=
p and observable
Trp~p.l=
To make sense, If dimti
wherepi,
(3)
X we write
= Tr(pX).
we must limit the states < 0~) this is automatically
j=l,...
(2)
j=2,...,n.
p.Xj E qj,
p.x
X E X.
1,
, n, are determined
(4) for which pX is of trace-class
to those true,
and it is easy to establish
by the n - 1 equations
for all
that
(3) and /3’ = log 2, with
Z = Tr exp(-$p’Xj)
denoting
the partition
of C consisting
say.
V,-1 dim V,_l
Both
function
of states
for the “Hamiltonian”
of the form
and the non-linear
PH = cz
,@Xj.
The submanifold
(5) is said to form an exponential map
Q are determined
family,
by the choice
of X.
V,_l In
= n - 1, since 71 = 1 due to normalisation. For the same reason, p1 = log 2 is not a free coordinate. The parameters ~2, . . , qn act as coordinates for Vn-l, and are known as “mixture” or “expectation” coordinates. We might equally use P2,...,pn as coordinates for s/n-l; these are known as “canonical” coordinates. In chemistry, 712,. . . , vn are usually extensive variables and p2,. . , /?” are intensive variables. If dim7-f < co, the geometry of Vn-l can be completely worked out [4]. general
We follow the methods expressing
our ignorance
but
not the philosophy
of the state
of the system,
of [5, 61; we do not regard an ignorance
caused
Q as
by our having
measured the means only of X E X and not more. Instead we make use of Q to represent the complete thermalising of the fast variables. We take it that this thermalising takes place
in one time-step,
whether
or not we know the means
of all the slow variables,
and, indeed, takes place even if we make no measurements at all. The time-step, then, represents the relaxation time of the fast variables, and in the spirit of [l, 41 we regard it as one of the adjustable parameters of the model. The fast variables will arise as elements of the orthogonal complement to X in the Bogoliubov scalar product. Quite often, the slow variables are the integrals of the conserved currents over mesoscopic regions; their time-variation is proportional to the area of the surface of the region, and this is small compared with the size of the operator, which is proportional to the volume of the region.
INFORMATION
This explains expressed most
GEOMETRY
why the dynamics
in terms
general
of densities
formulation
fast under the linear
AND REDUCED
QUANTUM
of many macroscopic of conserved
of statistical
theories,
quantities.
dynamics,
part of the dynamics
like hydrodynamics,
It should
in the model;
By reduced
dynamics
we mean the construction
and let {VA} be a family of unitary t,he index
set) represents
and the scattering
as follows.
operators
of the fast variables.
is called a random the whole system bistochastic, Td : C +
quantum
dynamics.
in one time-step.
and that
=
projection
on the states,
Td is also called a random by
entropy.
quantum
The reduced
UxAU,-l .I .J7
VA (for X E J.
measure
on J;
then
dJ
(7)
T is completely
positive
of H is a fixed point of T. We denote
defined on the state for all
dynamics,
dynamics
the
during the time step,
It gives the linear part of the time-evolution
pTd.A = p.TA
to increase
law obeying
with He. Each
It is easy to show that
any spectral
E the dual action
the fast but unknown
of the slow variables
TA
are not all visibly
[I] of a dynamical
Let dX be a probability
T:d-A,
are in the
Let He be the energy of the system
commuting
the time-evolution
that
they are made fast by
the action of Q, which thermalises them, and which can replace dynamics which is omitted from the reduced description. first and second laws of thermodynamics
be noted
the fast variables
chosen
421
DESCRIPTION
of and by
p by
AEA.
(8)
and is seen to conserve
mean energy and
is a map T : C + C given by Td followed
Q: p-pTdQ=pr
where
Q is associated
with
the given
X.
(9)
It is then
clear
that
T obeys
both
laws of
thermodynamics
where S. the entropy
PT.&
=
p.H”,
(10)
S(P7)
2
S(p) 1
(11)
(P 1% PI .
(12)
of the whole system,
is
S(P) = -Tr
It is possible to identify C with the density matrices, a subset of A. In that case we can drop the “d” in Td, and regard A as a bimodule under the semigroup of time-evolution. This is the way we treated the classical version [7]. In the present paper, we shall concentrate on the geometrica. description of t,he map Q, in the case when dim% = 00. For the classical case (71, we have described pQ as the foot of the straight
line from p to IL,_,
which cuts the latter
at right angles
in the
Fisher metric [8]; a similar description including also the quantum case was given earlier by Kossakowski, by Ingarden et al., by Grabert and by Balian et al. [2, 9: 3, 41. The ‘straight line” used in the projection is a geodesic relative to V(-‘), this being one of a
422
R. F. STREATER
pair of dual affine connections described in the classical case in Amari’s notable treatise [lo]. In the quantum case, Hasagawa and independently Nagaoka [ll] have defined quantum analogues of the Amari family V ca) of affine structures, at least if dim3-1 < oo. We shall need only V(*‘), which are the only members of the family that are dually flat: V(*l) have zero curvature and are torsion free [ll]. They therefore have dual affine coordinates (pj, r]k), and V(*l) are the covariant derivatives for which the geodesics are straight lines in the coordinates ,@ (for V(+l)) or rlj (for V(-l)). If dim% < 00 it is known that these are smooth coordinates for V,_l. The coordinates run over convex sets, namely lR”-’ in the case of the canonical coodinates, and a convex subset of the hyperinterval aj 5 qj 5 bj for the mixture coordinates, where for each j, [aj, bj] is the convex hull of the spectrum of Xj. Let & = a/a,@ and di = a/aqi. These are vector fields on the manifold K-1. The metric is a scalar product between vector fields, denoted by (m,l), and the Fisher metric turns out to be gii = (&,a,)
= -$
3’
2 5 i,j 5 72.
(13)
This is called the susceptibility, since it gives the response of an extensive quantity, such as spin, to a small change in an intensive variable such as temperature. This was actually suggested by Weinhold in equilibrium thermodynamics [12], which does not distinguish between classical and quantum probability. In finite dimensions it is easy to show that gii is nondegenerate and smooth in the differential structure given by the coordinates {rlj} or the {pi}. B y using the Matsubara interaction picture in imaginary time [13], otherwise known as the Duhamel method, one shows [14] that the metric is given by the celebrated Kubo formula [15, 161 gii(p)
=
2-l
Il
Ch Tr C(1-“‘4H(X~
- p.Xi)e-“‘H(Xj
- p.Xj) ,
(14)
0
where2
E vn-l.
In finite dimensions, the entropy S is a smooth function on the manifold V,_l
(15) and
(16) From this point the problem is one of information geometry, and Amari’s theory shows that (vi, pi) are dual coordinates, and that the divergence of the structure is the relative information S(p,a)=Trp(logp-log(T). 07) To give a geometrical picture of the map Q, we note that if p E Vn-l then the state after one time-step, pi given in eq. (9) will not in general be in &I, but will be in a larger manifold V&l involving fast variables as well as X. We shall assume that m is finite; this is in the spirit of the subject. Then If,_, can be coordinatised by
INFORMATION
(7/2>... ,qnrqn.+_l,.
GEOMETRY
AND REDUCED
or by (p2,.
. . ,vm),
QUANTUM
. . r/3n,pn+1!.
In terms
. . ,pm).
423
DESCRIPTION
Vn-l
of the latter,
is the set
Q from l&-l
The projection relative
to the connection
onto V,_l in which 772, . . , qn are held constant, it is a path in V,_,
V(-‘1;
whose tangent
is a geodesic
must, be a sum of
8”. with k = 71f 1.. . . , m. On the other hand, any tangent to V,_l is a sum of &, with i=2 3. . . , n. The scalar product between such a tangent vector and the V(-‘)-geodesic is therefore a sum of (a”, &) = Sf , with Ic > n and i 1. n, and so vanishes. Hence the map Q can be described
by the V(-‘)-geodesic
the same as in the classical
V,_l
cutting
The case with dim ‘H = co is not so easy. In Section the operator
He and the set X of slow variables
and can be continued is that
it is positive
at right angles;
and that
2 we offer some natural
exp{-,f3He}
be of trace
manifold.
:Xiz, X,],
should
be He-bounded.
made in the Rellich-Kato differentiable
function
is our metric
The first is a common
In statistical as generating states
dynamics,
as p = Z,l
state
of the system
In Section
with derivative
equal to (minus)
operator
of X.
This
fields described
X2))
=
which
with X E X,
non-equilibrium to the well-known
as arising
as the equilibrium
by X.
of the Bogoliubov (l--aWxl
z-’
is enough
of this metric.
Ho+X,
is in contrast
it is
that Z is a
the Kubo tensor,
continuity
it is just a way to parametrise
by elements
in external
X we
for example
that on the commutators,
[15, 161, in which p is regarded
3 we derive some properties
((Xl,
assumption;
the proof of the Lipschitz
of the system;
exp{-/?H} response
On the space
[17], and is enough to establish
we do not regard the perturbed
the dynamics
t,heory of linear
theory
of p J. The second condition,
we also sketch
gjk:
Sufficient
less than E
of eigenvalues
situation.
16),
on Ho
with bound less than 1, and that each commutator,
perturbation
to show that qj is differentiable,
The condition
class for every ,0 > 0.
and that the number
grow at most as a power of E. This is quite a common be He-bounded,
axioms on
which allow us to arrive at eqs. (13)-(
Vn_-l is a smooth
to show that
for this is that Ho have point spectrum, impose that each Xj
this is entirely
case [7].
scalar product
e-aOH~2
(19)
e-
and in particular Bogoliubov
In Section stochastic
2.
the isothermal
4 we ask whether
evolution
this is true if generated
we show that
dynamics
leads to a contraction
in the
norm.
A
through
is an algebra,
the
A, but
reduced
dynamics
with a time-dependent
but that
in general
can be expressed generator.
it seems that
as a linear
We show that
the natural
dynamics
by the dual to the map r is not positive.
The Fisher
metric
in quantum
mechanics
In this section we give some conditions on HO the energy operator, and the set X of slow variables, which together are sufficient for the existence of the manifold V,_, of exponential
states
and the smooth
metric
given by eqs. (13)-(16).
In order for the map
424
R. F. STREATER
Q to be defined, we just need to assume that the fast variables involved in one time-step r obey the same conditions as we have assumed for the elements of X. We express these conditions as axioms. AXIOM
exp{ -pH,*}
I.
Ho is essentially self-adjoint is of trace-class.
on a domain D and for
each ,B >
0,
This axiom enables us to define the canonical state Pp = z-le-PHo
z = Tre-flHo .
with
(20)
(I trust that no confusion will arise by dropping the closure symbol * on Ho here and later). More, by the spectral theorem, Hoe --OH0 has a bounded closure. It then follows that for each p > 0 the operator Ho exp{-PHo} h as a closure of trace-class, since Hoe-@&
=
(
Hoe-@&/2 ) (e--PH0I2)
is the product of a bounded operator by a trace-class operator. Thus the equilibrium state has finite energy. In the same way, all the moments of HO are finite in the equilibrium state. It follows from Axiom I that HO is bounded below; without loss of generality we may assume that HO 2 0. The next axiom concerns the set X of slow variables. Without loss in generality we may assume that dim X = n, that is, 1, HO, X3, . . . , X, are linearly independent. AXIOM II. There exists a basis 1 = X1, HO = X2, XJ, . . . ,X, Xi is defined on D and for i = 2,. . . , n,
\I(1 + Ho)-~X~II
< 1.
of X such that each
(21)
It follows from Axiom II that Xi and HO + Xi are essentially self-adjoint on V, using the theorem of Rellich [17, Theorem V 4.41, and that HO + Xi is bounded below [17, Theorem V 4.111. It is also easy to see that Axiom II is a convex condition: if 0 < X 2 1 then /I(1 + HO)-‘(Xxi + (1 - X)Xj)ll < 1. Let us denote by V the set of X E X such that X is Ho-bounded with bound less than 1, and such that there exists S > 0 with HO + X 2 61; then V is open, convex and of dimension n. It follows that exp{ -,L?H} can be defined as a bounded operator, where H = HO + X, X E V. As a simple example of Axiom II, suppose that n = 3 and HO and X3 = N are the Hamiltonian and the number operator of the free Boson field: Ho =
dkw(kW(k), SW JO3 N=
0
dk N(k),
0
where N(k) = a*(lc)a(k). If w(k) > m > 0, then p(Ho - ,uN) is bounded below in the open convex region p > 0, p < m. It is clear that for each i, Xi exp{ -pHo} is of trace-class, since when written as (X,(1 +
HO)-‘)
((I+
Ho)
exp{-P&j)
INFORMATION
GEOMETRY
it is seen to be the product the relevant
means
The important our purposes.
of a bounded of Golden
and Grumm
We give Ruskai’s 1 (Ruskai).
THEOREM
and A + B
is of trace-class,
THEOREM
Since
was established
in the unbounded
[ 191. either of which is sufficient
for
are essentially
self-adjoint
on 2,
and
2 Tr (e-“e-“)
.
= Tr e--B(Ho+X)
as a function
of pj, with logarithmic
neighbourhood
m < Tre-P(Ho+X) Proof:
all
then
For all X E V, exp{-$(Ho+X)}
2.
Thus
from I and II, we show that Zx
and small enollgh compact c CC such that
operator.
version:
It. is now easy to get some results;
and is differentiable
by a trace-class
and Thompson
Tr (e--(A+B))
exists
425
DESCRIPTION
state.
[18] and by Ruskai
If A, B
bounded below, and exp{ -B}
QUANTUM
operator
are finite in the canonical inequality
case by Breitenecker
AND REDUCED
derivative
is of trace-class,
-r13
and for each X0 E V
K of X,J there exist m, 4.4, with 0 < m c M
5 Al
V is open, there exists
X E K.
(22)
(1 + 6)X
E V for all X E I<.
for all
6 > 0 such that
Put
A = /3(1+ S)-l(Ha
B = /?S(l + S)plHO.
+ (1 + 6)X),
Then A+B=@H=/3(Ho+X). Since (1+6)X Then
Tr e-OH
on K.
E?=/?(Ho+X+SX)
inequality, =
Tre-(A+B)
5
IIe-AII.IJe-BII1
=
M(AK)
2 n
(e-Ae-B)
5 eF~(le~~HoII1
The lower bound is proved similarly:
(23)
we put A = /3(Ho + X - SX) and
toget Tr e--2,3(&+x)
This enables
below, and as X roams over the compact
set K, the lower bound can be chosen to be C, the same for all X E K.
by the Golden-Thompson
uniformly
is bounded
E V, Ho + (1+6)X
finite-dimensional
5 e-C’Tr
us to define the exponential px = Zgl
e-OH,
X E V,
e&j%+X+6X)
0
family of states H = Ho + X,
2~ = TreeoH
>
.
(24)
426
R. F. STREATER
An element P2,
@3r..
. , pm.
of this family is determined by the values of the n - 1 variables p = We shall see that the dimension of the family is n - 1. This will fol-
low from the non-degeneracy
of the metric
gjk which
will be defined
on the tangent
space. We now show that
THEOREM
H inherits
an important
3. Fork = 2,3,...
of Ho.
property
,nandXEV,
H=Ho+X,wehave
= cx < co.
11(1+H)-%&
Moreover for any X0 E V there is a compact neighbourhood K of X0 such that sup
cx=c
XEKO
Proof :
We see that
(1 + H)-‘(l+ is uniformly
bounded
H,,) = [l+
on some compact
(1 + H,,-1X]-1 K of Xe, since
neighbourhood
Ho)-~XII,= dx
]](l+
1
as X runs over K. Then
Ho)llmI Cl- 4-l
]I(1 + H)-r(l+
uniformly
in K .
Therefore ]](l+
uniformly
H)-lXk]]
H)-r(l + Ho)]]co]](I
5
]](I +
1.
(1 - d)-‘]I(1
+ H&lXk]]
+ Ho)-lXk]], = c
0
in K.
COROLLARY. For k = 2,. . . , n and any /? > 0, XkepPH is of trace class, with tracenorm bounded in some compact neighbourhood of any X0 E V. (25) Indeed, ]]Xke-PH]]r
=
+ H)e-PH/2e-BH/211
]]Xk(l
+ H)-I(1
<
IIxk(l
+ H)-‘llmll(l
< -
C.(2/p)e-1+0/2.M(/3/2)
the first factor is given by Theorem the third factor by Theorem 2. As a result of this corollary
+ H)e-PH’211~Ile-PH’21(,
3, the second
;
factor
by the spectral
we can define the expectation
r]k = zilTrXkePpH
>
k=2,...,n.
(or mixture)
theorem,
and
coordinates (26)
In the next two proofs, we estimate the trace norm of the integrand given by Matsubara’s interaction picture [ 131, and show that it is bounded by an integrable function of imaginary time. An application of Lebesgue’s dominated convergence theorem and Fubini’s theorem for traces then shows that the integrals in question converge in trace-norm.
INFOR.MATION GEOMETRY THEOREM
AND REDUCED QUANTUM DESCRIPTION
log ZX is differentiable in ,&, . . . , pn and
4.
a log 2x
ap
Proof :
427
= -qj.
We first remark that Z is bounded
hood of X E V, so it is enough
to discuss
(27)
away from zero in a compact
the differentiability
of 2.
neighbour-
Let 6X
= h,flJ_X,
and put H’ = H + SX. Then (Spj)-‘[Z(X We use Duhamel’s
+6X)
- Z(X)]
= (S,@-lTr(e”H’
- epaH).
formula 1 ,-Off’
_
e-OH
=
da ,-d3H’
_
(pQj)
,-(l-~VflH
(28)
s0 where strong operator
convergence
with respect
to (Y, noting
the integrand
is understood. w(o)
that the domains
of the trace
1
difference
of H* and H’* include
of two operators
of trace
e-(lma)fiH;I). class.
In fact,
Similarly
we can
and write
rlj =
the desired
the unitary
= e-aflH’e-(i-n)OH
in eq. (28) is the product
use the cyclicity
Then
To prove eq. (28) we differentiate
Tr
da &HX
s0
’
.c-(i-a)PH 3
(29)
qj, with
Z(X + SW- Z(X)
q’ = 3
Spj
+
Z(X)%
obeys the relation
SJ 1
=
-S@T’r
1
dcr
0
shorten
To
d,j e-XaPH’Xje-(l-X)aBHXje-(l-a)BH,
0
some of the formulae in the proof, let us denote the resolvent (1 + H)-l by Similarly, Ro, R’ and X’ are defined using Ho, H’ and X. In the
R, and 1 - Q by o’. region
0 5 Q 5 l/2 the integrand
using Holder’s
n I
inequality
is controlled
by exp(-cY’PH},
II = n (1 +H')~-"Q?H'/~ e-~WI/2~l~j II aXPH”211~llR’XjllMIJX~e-a’BH’211111e-X’aOHlloo x 5 Ile-
,-~X~H’X~~-(~-X)OPHX~-(~-~)SH
[
and we can estimate
for traces:
e-h0H~je+3H
I
=
K
>(
x ]]e--a’RH’2(1
+ H)]],]]R(l
O(1)
+ 0.
as spj
>
+ H’)e-UX4H’/211m
428
R. F.STREATER
In the region f < o 5 1 we split JdX into 0 2 X < l/2 and l/2 < X < 1. In the first, the integrand is controlled by e-(l-X)aPH (n [ (e-MH’)
(X,e(i-+BV)
5
IIPAPH
=
O(1)
and we write
(e-(l-+P~lzXj)
(e-(l-~)P)]
(
IloollX,e-(‘-X)“gHIJ1J(e-(l-~)~~HX,/I~I/,-_(l-~)~HII, as S/P -+ 0.
i < A 5 1, the integrand is controlled by the factor ePaXpH’, Intheregioni
[(
e -aA{H’Xj)
(e-X’@H)
(Xj~)
(e-“‘P”)
((I + H)eMaXgH’)]
,
which is O(1) since the first and last factors are of trace-class, and the middle factors are bounded, all uniformly over the range of integration and for 6X E K for some compact set K. For example, Ile-aXPH’12XjJI < Ile-~(“x-f)H’llmlle-~H’X3111
5 IlePfH’Xjlli 0
and this is bounded uniformly in SX by the corollary to Theorem 3. To prove the next theorem we need a further axiom. AXIOM III. For each X, Y E X, (1 + Ho)-l[X,
Y] is bounded.
Then we have the Kubo formula for the susceptibility. THEOREM
5. For any j, k > 1, q
is differentiable
(xk - p.xk)e-
in ,B”, and
(l - “‘PH(Xj - P.Xj).
On differentiating the Z-’ in eq. (26) we get -Z-2dZ/dflk = Z-$k by Proof : Theorem 4, and this cancels the cross moment in eq. (30). So it is enough to show that is d ifferentiable in /3”, with derivative Tr(exp{-PC& +X)1X,) 1
da (e- @HXk)
-Tr
(,-(I-“)“HXj
So we must look at e-PH’ _ e-PH Wk
1
xj -
dae -~YPHXke-(l-WHXj
J0
1 .
We use Duhamel’s formula for the difference of the exponentials, and find that FjI, = fip” L’dal’dhTr
[e-
Xa4H’xke-(l-X)@Hx
ke-(l--a)PH~j
1 .
INFORMATION
GEOMETRY
AND REDUCED
QUANTUM
This is of order S/3” if the integrand is bounded
DESCRIPTION
(as SD” -+ 0).
429
We now prove this
boundedness .
In the region 0 < LY5 $ the trace of the integrand is written as the trace of
All the factors are obviously bounded in norm uniformly in S/3 (using previous results) except the last. For this, we write e-(i-&HXJ(I
+ H’) = e- (l-Q)gHxj(l
+ H)(1+
H)-1(1+
H’).
We may drop the factor (1+ H)_i(l which is uniformly norm-bounded ,-c’4Hxj(I
+ H)
+ H’)eCX”OH
as SD” + 0. We are then left with
=
e-(i-4:H(I
=
-(i-&H(I Ce
+ H)X, + N,)
+ (,+a)+(1
+ e-(l-4$H[H,
X,]
(&4Qj)
+
+ Ho)) ((1+
H&i[H,
Xj])
Both terms on the right side contain a factor of trace-class, multiplied by bounded operators. Hence we have a bound on the trace-norm in the region Q < i. In the region $ 5 (Y 5 1, 0 2 X 2 $,the exponential exp{-(1 - X)aPH) dominates, so we group them as (
(I + H’)Xke-X’agH
>(
e-“a$HX
k) (f~-~“~>
(XjR'e-X"bH')
All the factors are bounded uniformly in norm in the given range, and the worst part of the first factor, the quadratic term jy’xke-X’a$H
=
(H'R)([H,Xk]t-X'a~H+X,&'a~H(l
+ u))
is the sum of two trace-class operators; for, the first term is (H’(1 + H)-‘)
([H,Xk](I
+ He)-‘)
((I + ~s)e-(l-x)~‘H)
;
all factors are bounded and the last is of trace class. The other term is even easier; we omit the details. In the region $ 5 (Y 5 1, $ < X < 1 the dominant exponential is exp{-XapH’} so we write the integrand as (I + H’)XjewXngH’)
(e-xOqW’Xk)
(e-(‘-“)-““)
(Xk(I + H’))‘)
430
R.F.STREATER
The factors are all bounded, uniformly as Q, A and S,B” vary over the given regions. The proof is completed by the remark that the first factor is of trace-class, as is seen by the identity (1 +
fp)q+4~’
=
([H’,Xj](l +
COROLLARY.
+ Ho)-l)
((l+
Ho)e-““$“‘)
(Xje-AaaH’)((I+ f$‘)e-‘“ZH’)
gj,kisLipschitz continuous.
t q
(31)
For by the estimates just given, gj,k(X + 6X) - gj,k(X) can be written as the sum of two integrals of the above form, and are similarly shown to be O(S,B”). In fact, we can continue and show that the manifold is infinitely smooth. is positive definite, the dimension of the manifold V,_l is n - 1. 3.
Since gj,k
Rival norms on Z A number of papers [20] a dvacate either the KMS scalar product
(A, a,
= Tr w*q
(32)
or, for Hermitian operators, its real part Re(A, B)P = f Tr p (AB + BA)
(33)
as the metric to use on the space of observables. Let us suppose that p is a faithful state, as happens when it is the canonical state of the system. Then eq. (32) defines a norm on the algebra. The dual norm is then
IIUII = {Tr
(p-ln2)}1’2
(34)
giving a metric on the space of states. In [l] the author adopted the metric eq. (34); it seemed a good idea at the time, because of a number of useful properties: 1. It reduces to the Fisher metric [8] if the algebra is abelian; 2. The unique state of minimal dual norm is p itself; that is, Tr
(p-la2)
2 1, for all states cr, with equality only if u = p;
3. (Xj, qp = vj; 4. Any isothermal dynamics is a contraction in this norm. More generally, any linear dynamics obeying detailed balance (in the general form given in [l]) is a contraction.
INFOR.MATION
Property
GEOMETRY
AND REDUCED
4. was first established
by Majewski
QUANTUM
[21]; recently
with a simple proof has been found [l, 221. Because
431
DESCRIPTION
a discrete-time
of this, we can picture
version
the approach
to equilibrium under isothermal conditions as being along an orbit in state-space whose metric distance to the fixed point p, the canonical state at the given temperature, is reduced
at each step, until it reaches
is an alternative with time. complex
statement
However,
scalar
p, the state with norm 1, the global minimum.
to the decrease
as we now show, the properties
products
defined on the algebra
= Tr
(A, Bjp+ which have been studied Zegarlinski
[23].
real vector
of the non-linear
before
(
Our real interest
are also true of the family of
>
(35)
,
the case (Y = i was used by Majewski
is in the Bogoliubov
operators
This
the free energy,
by
paA*p(‘-a)B
(141. Recently
space of Hermitian
l.-4.
function,
scalar
product,
defined
on
Fisher
of its close relation metric.
with Kubo’s
Indeed,
susceptibility
(36)
tensor
We now show that all these obey l.-4.
the norm of the scalar product
and thus to the quantum
We start
with the case of ]( l ]]p.Cr,
eq. (35).
1. is clear if all operators
first find a formula
commute,
and 3. is obvious.
Before
proving
2. we
for the dual norm, defined by
((ullp,_a = sup{(u.Al : (IAll,,, = 11. We expect
the
by eq. (19)
((A, B)) = .I’ Tr[p”Ap’-Y?]da, because
and
(37)
that the answer is
THEOREM 6. ll~ll&
and even for some unbounded
for all states. certainly
So let u be such that
finite,
namely
defined by the spectrum
matrices
operators,
the right-hand
consider
the dual norm
of p, and put
IIAllp,a = 1, so putting (]c$,,-~
> ITroAl
1’2 .
this A in eq. (37) gives
= ll’~p-%p-~+“IN-~
not be finite
in Theorem
in the orthonormal
the operator
One checks that
might
side of the equality
of finite rank when expressed
N = (Tr(p-%p-l+“a)) Then
.
We note that whereas the norm coming from eq. (35) is finite for all bounded
Proof: operators
= Tr (p-“crp-lfao)
= N.
6 is basis
432
R. F. STREATER
For the inequality in the other direction, Ic.A12
=
/n [(p(“-1)/20p-n/2) (pa/2Ap(1-“)/2)]I2
<
n (p-Sqf--lgp-%
= N2 IIAII;,a = N2 if I14La = 1. Hence the supremum over all such A is also bounded by N2, and we get 0 IlallP,-a 5 N. Combining with the other inequality gives the theorem. We now return to the proof of 2. for these norms. We have
I14Jp,-a=
s;~{l~.Al
: IJAIl,,, = 1) L a.1 = 1 = ll~ll~,-a
(since lllllP,~ = 1 we can put A = 1 and get a lower bound for the supremum). equality only if o = p. We note that in 0 5 (Y 5 $ the norms are decreasing:
II4lP,a5 ll4,~
if
Qf>P.
(33) We get
(39)
This is implicit in [14]. The corresponding dual norms go the other way:
II4lP,-a 2 ll4-0
if
QZP,
(40)
which can be regarded as the extension of the same inequality for negative values of the parameters. To prove 4., we first prove that any isothermal dynamics obeys detailed balance relative to the scalar product eq. (35). We need a definition. DEFINITION. Let A be a W*-algebra with a scalar product (0, l ), and let T be a stochastic map on A. We say that T obeys detailed balance relative to the given scalar product if the adjoint map T* (relative to the scalar product) is also a stochastic map.
Then there is the following result [l, 221: THEOREM 7. If T, a stochastic map, obeys detailed balance relative to a scalar product, then it is a contraction in the corresponding norm.
In statistical dynamics we must distinguish between isolated dynamics, in which no heat is exchanged with the surroundings and the total energy is constant, and isothermal dynamics, in which the temperature is kept constant by removing, or adding, heat as necessary. In isolated dynamics the entropy increases, and in isothermal dynamics the free energy decreases. Isothermal dynamics is obtained from a special case of the general scheme described in Section 1, and is linear. It is this linear map that obeys detailed balance; we showed this for the scalar product eq. (32) in [l]. We now show it for the family eq. (35). Recall that one time-step in isothermal dynamics is obtained as follows.
INFORMATION
The W*-algebra and
d,
GEOMETRY
is a tensor
is the algebra
AND REDUCED
product
d = A, @ 4,
of the heat-particle
as a slow variable.
and the map Q is the
the tensor
of its partial
heat)
product
traces
[l].
QUANTUM
where
A, is
Any Hermitian
“stoss map”,
the algebra element
of the system
of
which replaces
over the two factors.
433
DESCRIPTION
-4,is
a state
The total energy
regarded on A by
(including
is a sum H = H, @ 1+
TJ commutes
reversible
of such automorphisms. dynamics
motion
with H. Random
7j3 at beta
is given by an inner automorphism
dynamics
Then
U
l7
l
‘. where
is given by a mixture
T is bistochastic
/3 is defined
(41)
1@ HA,
and completely
to be the predual
positive.
of the following
The isothermal
linear
dynamics
on
induced
by
C(A):
Here, cc is any state
of the system,
T and pr,o is the canonical stochastic and obeys defined by
state
detailed
balance
Tr,. (P,,@(AP) is also stochastic. dynamics 11.12
Indeed,
immediately eq. (35),
guarantees
and that
LEMMA.
canonical
~(0)
We proved
to the canonical
dynamics
of
A
[I] that
state
for all
~(0) is the isothermal l
such a T[, is
~~,a, that
A, B E
our result
given in Theorem
7,: is therefore
a contraction
of-AC; that
is, r(@+)
in the norm
11l ((P,-O.
of rO relative to (A, B)p,a.
where p = pr,,j zs the
obeys
= Tr, (p”Ap’-O~~(l?))
for all
A, B E
d,
Then 7(35n) _ 7(3).
(+a)(A),
Let A, B be Hermitian Bjp,n
random
= 7-i. The next lemma generalises this result to all CY. and that 7. obeys detailed balance relative to the scalar products of
Tr, (pa&~)(A)pl-V)
Proof:
is. .(“)
A,
for the time-reversed
U. Let us call this -r,*, so that
Let ~(fi>~) be the adjoint state
relative
= Tr, (p,,&p(B))
T” = C &U-l
of [I] is that
Td is the dual map on the states
of the heat-particle.
=
(-4 ~,@))~,a
=
Tr,
=
%,,
(
elements
T%P*
A,,and
put po = p @ p_,,g; then
- Tr, (P%-~~,(B))
~,d(p~Apl-~)B (
of
>
= Tr, (Tr,[Td(paApl-a
@v;,/dL-
Wpl-
@ P$3V(B
6~ P,,~)]B)
@ 1))
434
R. F. STREATER
=
Tr c,y
(p,“Td(A @~)P;-“(B8 1,)
since Vi commutes with pp = p @ pr,fl =
‘I’r,,,
(Td(A 8 l)p;-(B
@1)~;)
= nc,-,(Td(A@~)(P~-~BP~) @pr,o) = nc,y((A8 W[(P~-~~P~) @~y,pl) = nc (A& (Tdd(~l-“B~a @P,,P,)) =
Tr, (A~;~(p~-‘V3p”))
=
($4 Bjp,a.
= Tr, ($(A)p’-“BP”)
It follows that r(P@) = r(p) for all cz and is a stochastic map, being the isothermal dynamics for the time-reversed dynamics. Therefore @%a) obeys detailed balance, and rP is a contraction in the norm ]Il ]]p,a and therefore that its dual r: acting on the states 0 is a contraction in the dual norm. Since we have just proved that rO obeys
we see that rO is also a contraction seems to be new. 4.
Non-linear
in the Bogoliubov norm. In this generality, this result
stochastic processes
So far our (non-linear) dynamics of an isolated system is given by an orbit through the family V,_l of states, one time-step being p -+ pTdQ. This can be regarded as a non-linear version of the von Neumann dynamics, which is the Schrodinger picture for density matrices. In the case of isothermal dynamics, and where the slow variables form an algebra A,, we can reduce to a linear stochastic process obeying detailed balance. This can be transferred to a quantum stochastic process on the algebra, by duality; thus in the linear case there is no problem in setting up the corresponding Heisenberg picture. As a result, a meaning can be given to the multitime correlation functions PO
(xl(h),
. ..,xk(tk))
,
(42)
where po is the initial state. In [l] we discuss whether the dynamics can be thrown onto the observables also in the non-linear case. Such a procedure is desirable, since the multitime correlation functions and, equivalently, the Wightman functions, exist in the reversible case, and should retain some meaning at least if the dissipation is small. We impose the constraint that the change in the expectation of each X E X should be the
INFORMATION
GEOMETRY
same in both pictures. on t#hestates obeys
AND REDUCED
QUANTUM
Thus we seek a linear map Q,(t)
pT”Qf.A
for all
= pTdQ.A
on
DESCRIPTION
435
A such that its dual Q:(t)
AEd.
(43)
Then we could define one time-step of the Heisenberg dynamics at time t to be A(t) H
Q,(t)TA(t) THEOREM
of
= A(t + 1). 8.
Let Q*(t)
be the orthogonal
projection
onto X (as a subvector
space
d) in the KMS scalar product at pTdQ; that is
(4 B)pTdQ =
pTdQ.A*B.
(44)
Then eq. (43) holds. Proof:
WriteA=Ae+Al, pTdDlf.A
where A0 = Q,A.
Then
=
pTd.Q,A
= pTd.A,,
=
PT~Q-&
= (1, Ao)~T~Q
=
(17-4,~d~
=
pTdQ.A.
as 1 L A’ 0
This proof is more transparent than the one given in [l], p. 256. We see from the proof that instead of the KMS scalar product, we can use for Q* the projection onto X using any scalar product obeying (3) of the last section. In particular, we could use the Bogoliubov scalar product. We obtain in this way a linear map A H Q,TA, but the (discrete time) dynamics got from this is not stationary, since Q* depends on the current state. In fact we must solve the non-linear dynamics for p(t) in the Schrodinger picture before we can find Q* at each time. If the initial state is an equilibrium state, then p(t) is independent of time, and this problem does not arise. It is not clear that the map Q* is positive, let alone completely positive. This is true whether we use the KMS or the Bogoliubov scalar product. This would be needed for a satisfactory interpretation of the many-time correlation functions. Positivity does hold, however, in one special but common situation; this is when X includes the Hermitian part of an algebra A,,and A is a tensor product A, @ A,. Positivity then holds on d,. Indeed, for any pr the map
which gives one time-step, is linear, and takes states to states; it is therefore the dual of a linear positive map on the algebra. This is time-dependent, since the construction depends on the current state, pr, of the heat-particle. This gives a non-linear stochastic process, in which the temperature varies with time. It has been worked out in detail for a Brownian particle in a potential [24].
436
R. F. STREATER
Acknowledgements The author is indebted to E. B. Davies and Y. Safarov for bringing to his attention some useful parts of the literature. REFERENCES [l] R. F. Streater: Statistical Dynamics, Imperial College Press, London 1995. [2] A. Kossakowski: Bull. acad. polonaise sci. 1’7 (1969), 263. [3] H. Grabert: Projection operator techniques in non-equilibrium statistical mechanics, Springer Tracts in Modern Physics, 95, Springer, Berlin 1982. [4] R. Balian, Y. Alhassid and H. Reinhardt: Physics Reports 131 (1986), 2, North Holland. [5] E. T. Jaynes: Phys. Rev. 106, 620, and ditto II, ibid, 108, 171 (1957). [6] R. S. Ingarden: Open Syst. Inform. Dyn. 1 (1992), 75, and early references therein. [7] R. F. Streater: Statistical dynamics and information geometry, to appear in Geometrical and Topological Methods in Physics, Eds. P. Combe and H. Nencka, Amer. Math. Sot. 1996. [S] R. A. Fisher: Proc. Camb. Phil. Sot. 22 (1925), 700. [9] R. S. Ingarden, Y. Sato, K. Sagura, and T. Kawaguchi: Tensor 33 (1979), 347; R. S. Ingarden, H. Janyszek, A. Kossakowski, and T. Kawaguchi: Tensor 37 (1982), 105. [lo] S.-i. Amari: Differential Geometric Methods in Statistics, Lecture Notes in Statistics, 28, Springer, Berlin 1985. [ll]
H. Hasagawa: Non-commutative extension of information geometry; pp. 327-337 in Quantum Communication and Measurement, Eds. V. P. Belavkin, 0. Hirota and R. L. Hudson, Plenum Press, 1995. Also, H. Nagaoka: Differential geometrical aspects of quantum state estimation and relative entropy, ibid, pp. 449-452. F. Weinhold: J. Chem. Phys. 63 (1975), 2479; ibid 65 (1975), 559. T. Matsubara: Prog. Theor. Phys. 14 (1955), 351. G. Roepstorfi Path-integral Approach to Quantum Physics, Springer, Berlin 1994. R. Kubo: J. Phys. Sot. Japan 12 (1957), 570. J. Naudts, A. Verbeure and R. Weder: Commun. Math. Phys. 44 (1975), 87. T. Kato: Perturbation Theory for Linear Operators, Springer, Berlin 1966. M. Breitenecker and H. R. Griimm: Commun. Math. Phys. 26 (1972), 276.
[12] [13] [14] [15] [16] [17] [18] [19] M.-B. Ruskai: Commun. Math. Phys. 26 (1972), 280. [20] D. J. C. Bures: Trans. Amer., Math. Sot. 135 (1969), 199. A. Uhlmann: Rep. Math. Phys. 9 (1976), 273. W. K. Wootters: Phys. Rev. D23 (1981), 357. [al] W. A. Majewski: J. Math. Phys. 25 (1984), 614. [22] W. A. Majewski and R. F. Streater: Detailed balance and quantum dynamical semigroups, to appear. [23] W. A. Majewski and B. Zegarlinski: On quantum stochastic dynamics and noncommutative L, spaces, to appear. [24] R. F. Streater: The Brownian particle in statistical dynamics, to appear.
REPORTS
ON
MATHEMATICAL
No. 3
PHYSICS
INFORMATION GEOMETRY REDUCED QUANTUM DESCRIPTION
AND
R. F. STREATER Department of Mathematics, King’s College, Strand, London WC2R 2LS, UK e-mail: ray. streaterQkc1. ac .uk (Received
April
10, 1996)
We consider the quantum dynamics of a system with two time-scales, having discrete time and unbounded slow variables, under a useful set of axioms. We argue that the Kubo susceptibility tensor is the appropriate generalisation of the Fisher information metric on the set of density matrices. We define the Amari connections V(*‘) in the unbounded case. The geometric picture of reduced quantum dynamics, described by Balian, Alhassid and Reinhardt [Phys. Rep. 131, 2-1461, is extended to some unbounded cases. We show that the reduced dynamics is a contraction in the Bogoliubov norm. Keywords:
Fisher
information
metric,
Kubo
susceptibility,
Kato-bounded
perturba-
tions.
1.
Introduction
and summary
Non-equilibrium
statistical
t,ical dynamics
[I].
is implemented
by following
ables, Xi, are random necessarily
mechanics
We here develop
is best studied within the framework
the projection
the time-evolution
. . . , X, say, called the “slow” variables. variables,
while in the quantum
bounded,
on a Hilbert
case, and the technical the C*-algebra
method
version
they are self-adjoint
space ‘H. In this paper we shall consider
operators
of statis-
in [2, 3, 41. This
of the means of a selected list of observIn classical statistical mechanics these
issues arising when some of the Xj
of all bounded
described
are not bounded.
operators,
not
the quantum Let A denote
on N, and let C be the set of normal
states
on
A. The real linear span X= should
contain
the unit operator,
very slow variables!
We can assume
SpanR{Xi,...,X,} Xi
= 1, and the energy without
(1)
Ho of the system;
loss in generality
that
these
are
dim X = n.
When our information about the system consists of the means of X E X, Jaynes [5] and Ingarden [6] advocate that the system should be assigned the state of maximum entropy, among all states p E C with the given means. According to these authors. this
420
R.F.STREATER
is the best
estimate
for the state
define the map Q, written domain
of states,
of the system,
given our limited
on the right of the state
as follows:
the state
knowledge.
thus p t+ pQ, acting
pQ E C is the state of maximum
Let us
on a suitable
entropy
obeying
the n equations
Tr(pQ) = p&.X, Here, for a state
=
p and observable
Trp~p.l=
To make sense, If dimti
wherepi,
(3)
X we write
= Tr(pX).
we must limit the states < 0~) this is automatically
j=l,...
(2)
j=2,...,n.
p.Xj E qj,
p.x
X E X.
1,
, n, are determined
(4) for which pX is of trace-class
to those true,
and it is easy to establish
by the n - 1 equations
for all
that
(3) and /3’ = log 2, with
Z = Tr exp(-$p’Xj)
denoting
the partition
of C consisting
say.
V,-1 dim V,_l
Both
function
of states
for the “Hamiltonian”
of the form
and the non-linear
PH = cz
,@Xj.
The submanifold
(5) is said to form an exponential map
Q are determined
family,
by the choice
of X.
V,_l In
= n - 1, since 71 = 1 due to normalisation. For the same reason, p1 = log 2 is not a free coordinate. The parameters ~2, . . , qn act as coordinates for Vn-l, and are known as “mixture” or “expectation” coordinates. We might equally use P2,...,pn as coordinates for s/n-l; these are known as “canonical” coordinates. In chemistry, 712,. . . , vn are usually extensive variables and p2,. . , /?” are intensive variables. If dim7-f < co, the geometry of Vn-l can be completely worked out [4]. general
We follow the methods expressing
our ignorance
but
not the philosophy
of the state
of the system,
of [5, 61; we do not regard an ignorance
caused
Q as
by our having
measured the means only of X E X and not more. Instead we make use of Q to represent the complete thermalising of the fast variables. We take it that this thermalising takes place
in one time-step,
whether
or not we know the means
of all the slow variables,
and, indeed, takes place even if we make no measurements at all. The time-step, then, represents the relaxation time of the fast variables, and in the spirit of [l, 41 we regard it as one of the adjustable parameters of the model. The fast variables will arise as elements of the orthogonal complement to X in the Bogoliubov scalar product. Quite often, the slow variables are the integrals of the conserved currents over mesoscopic regions; their time-variation is proportional to the area of the surface of the region, and this is small compared with the size of the operator, which is proportional to the volume of the region.
INFORMATION
This explains expressed most
GEOMETRY
why the dynamics
in terms
general
of densities
formulation
fast under the linear
AND REDUCED
QUANTUM
of many macroscopic of conserved
of statistical
theories,
quantities.
dynamics,
part of the dynamics
like hydrodynamics,
It should
in the model;
By reduced
dynamics
we mean the construction
and let {VA} be a family of unitary t,he index
set) represents
and the scattering
as follows.
operators
of the fast variables.
is called a random the whole system bistochastic, Td : C +
quantum
dynamics.
in one time-step.
and that
=
projection
on the states,
Td is also called a random by
entropy.
quantum
The reduced
UxAU,-l .I .J7
VA (for X E J.
measure
on J;
then
dJ
(7)
T is completely
positive
of H is a fixed point of T. We denote
defined on the state for all
dynamics,
dynamics
the
during the time step,
It gives the linear part of the time-evolution
pTd.A = p.TA
to increase
law obeying
with He. Each
It is easy to show that
any spectral
E the dual action
the fast but unknown
of the slow variables
TA
are not all visibly
[I] of a dynamical
Let dX be a probability
T:d-A,
are in the
Let He be the energy of the system
commuting
the time-evolution
that
they are made fast by
the action of Q, which thermalises them, and which can replace dynamics which is omitted from the reduced description. first and second laws of thermodynamics
be noted
the fast variables
chosen
421
DESCRIPTION
of and by
p by
AEA.
(8)
and is seen to conserve
mean energy and
is a map T : C + C given by Td followed
Q: p-pTdQ=pr
where
Q is associated
with
the given
X.
(9)
It is then
clear
that
T obeys
both
laws of
thermodynamics
where S. the entropy
PT.&
=
p.H”,
(10)
S(P7)
2
S(p) 1
(11)
(P 1% PI .
(12)
of the whole system,
is
S(P) = -Tr
It is possible to identify C with the density matrices, a subset of A. In that case we can drop the “d” in Td, and regard A as a bimodule under the semigroup of time-evolution. This is the way we treated the classical version [7]. In the present paper, we shall concentrate on the geometrica. description of t,he map Q, in the case when dim% = 00. For the classical case (71, we have described pQ as the foot of the straight
line from p to IL,_,
which cuts the latter
at right angles
in the
Fisher metric [8]; a similar description including also the quantum case was given earlier by Kossakowski, by Ingarden et al., by Grabert and by Balian et al. [2, 9: 3, 41. The ‘straight line” used in the projection is a geodesic relative to V(-‘), this being one of a
422
R. F. STREATER
pair of dual affine connections described in the classical case in Amari’s notable treatise [lo]. In the quantum case, Hasagawa and independently Nagaoka [ll] have defined quantum analogues of the Amari family V ca) of affine structures, at least if dim3-1 < oo. We shall need only V(*‘), which are the only members of the family that are dually flat: V(*l) have zero curvature and are torsion free [ll]. They therefore have dual affine coordinates (pj, r]k), and V(*l) are the covariant derivatives for which the geodesics are straight lines in the coordinates ,@ (for V(+l)) or rlj (for V(-l)). If dim% < 00 it is known that these are smooth coordinates for V,_l. The coordinates run over convex sets, namely lR”-’ in the case of the canonical coodinates, and a convex subset of the hyperinterval aj 5 qj 5 bj for the mixture coordinates, where for each j, [aj, bj] is the convex hull of the spectrum of Xj. Let & = a/a,@ and di = a/aqi. These are vector fields on the manifold K-1. The metric is a scalar product between vector fields, denoted by (m,l), and the Fisher metric turns out to be gii = (&,a,)
= -$
3’
2 5 i,j 5 72.
(13)
This is called the susceptibility, since it gives the response of an extensive quantity, such as spin, to a small change in an intensive variable such as temperature. This was actually suggested by Weinhold in equilibrium thermodynamics [12], which does not distinguish between classical and quantum probability. In finite dimensions it is easy to show that gii is nondegenerate and smooth in the differential structure given by the coordinates {rlj} or the {pi}. B y using the Matsubara interaction picture in imaginary time [13], otherwise known as the Duhamel method, one shows [14] that the metric is given by the celebrated Kubo formula [15, 161 gii(p)
=
2-l
Il
Ch Tr C(1-“‘4H(X~
- p.Xi)e-“‘H(Xj
- p.Xj) ,
(14)
0
where2
E vn-l.
In finite dimensions, the entropy S is a smooth function on the manifold V,_l
(15) and
(16) From this point the problem is one of information geometry, and Amari’s theory shows that (vi, pi) are dual coordinates, and that the divergence of the structure is the relative information S(p,a)=Trp(logp-log(T). 07) To give a geometrical picture of the map Q, we note that if p E Vn-l then the state after one time-step, pi given in eq. (9) will not in general be in &I, but will be in a larger manifold V&l involving fast variables as well as X. We shall assume that m is finite; this is in the spirit of the subject. Then If,_, can be coordinatised by
INFORMATION
(7/2>... ,qnrqn.+_l,.
GEOMETRY
AND REDUCED
or by (p2,.
. . ,vm),
QUANTUM
. . r/3n,pn+1!.
In terms
. . ,pm).
423
DESCRIPTION
Vn-l
of the latter,
is the set
Q from l&-l
The projection relative
to the connection
onto V,_l in which 772, . . , qn are held constant, it is a path in V,_,
V(-‘1;
whose tangent
is a geodesic
must, be a sum of
8”. with k = 71f 1.. . . , m. On the other hand, any tangent to V,_l is a sum of &, with i=2 3. . . , n. The scalar product between such a tangent vector and the V(-‘)-geodesic is therefore a sum of (a”, &) = Sf , with Ic > n and i 1. n, and so vanishes. Hence the map Q can be described
by the V(-‘)-geodesic
the same as in the classical
V,_l
cutting
The case with dim ‘H = co is not so easy. In Section the operator
He and the set X of slow variables
and can be continued is that
it is positive
at right angles;
and that
2 we offer some natural
exp{-,f3He}
be of trace
manifold.
:Xiz, X,],
should
be He-bounded.
made in the Rellich-Kato differentiable
function
is our metric
The first is a common
In statistical as generating states
dynamics,
as p = Z,l
state
of the system
In Section
with derivative
equal to (minus)
operator
of X.
This
fields described
X2))
=
which
with X E X,
non-equilibrium to the well-known
as arising
as the equilibrium
by X.
of the Bogoliubov (l--aWxl
z-’
is enough
of this metric.
Ho+X,
is in contrast
it is
that Z is a
the Kubo tensor,
continuity
it is just a way to parametrise
by elements
in external
X we
for example
that on the commutators,
[15, 161, in which p is regarded
3 we derive some properties
((Xl,
assumption;
the proof of the Lipschitz
of the system;
exp{-/?H} response
On the space
[17], and is enough to establish
we do not regard the perturbed
the dynamics
t,heory of linear
theory
of p J. The second condition,
we also sketch
gjk:
Sufficient
less than E
of eigenvalues
situation.
16),
on Ho
with bound less than 1, and that each commutator,
perturbation
to show that qj is differentiable,
The condition
class for every ,0 > 0.
and that the number
grow at most as a power of E. This is quite a common be He-bounded,
axioms on
which allow us to arrive at eqs. (13)-(
Vn_-l is a smooth
to show that
for this is that Ho have point spectrum, impose that each Xj
this is entirely
case [7].
scalar product
e-aOH~2
(19)
e-
and in particular Bogoliubov
In Section stochastic
2.
the isothermal
4 we ask whether
evolution
this is true if generated
we show that
dynamics
leads to a contraction
in the
norm.
A
through
is an algebra,
the
A, but
reduced
dynamics
with a time-dependent
but that
in general
can be expressed generator.
it seems that
as a linear
We show that
the natural
dynamics
by the dual to the map r is not positive.
The Fisher
metric
in quantum
mechanics
In this section we give some conditions on HO the energy operator, and the set X of slow variables, which together are sufficient for the existence of the manifold V,_, of exponential
states
and the smooth
metric
given by eqs. (13)-(16).
In order for the map
424
R. F. STREATER
Q to be defined, we just need to assume that the fast variables involved in one time-step r obey the same conditions as we have assumed for the elements of X. We express these conditions as axioms. AXIOM
exp{ -pH,*}
I.
Ho is essentially self-adjoint is of trace-class.
on a domain D and for
each ,B >
0,
This axiom enables us to define the canonical state Pp = z-le-PHo
z = Tre-flHo .
with
(20)
(I trust that no confusion will arise by dropping the closure symbol * on Ho here and later). More, by the spectral theorem, Hoe --OH0 has a bounded closure. It then follows that for each p > 0 the operator Ho exp{-PHo} h as a closure of trace-class, since Hoe-@&
=
(
Hoe-@&/2 ) (e--PH0I2)
is the product of a bounded operator by a trace-class operator. Thus the equilibrium state has finite energy. In the same way, all the moments of HO are finite in the equilibrium state. It follows from Axiom I that HO is bounded below; without loss of generality we may assume that HO 2 0. The next axiom concerns the set X of slow variables. Without loss in generality we may assume that dim X = n, that is, 1, HO, X3, . . . , X, are linearly independent. AXIOM II. There exists a basis 1 = X1, HO = X2, XJ, . . . ,X, Xi is defined on D and for i = 2,. . . , n,
\I(1 + Ho)-~X~II
< 1.
of X such that each
(21)
It follows from Axiom II that Xi and HO + Xi are essentially self-adjoint on V, using the theorem of Rellich [17, Theorem V 4.41, and that HO + Xi is bounded below [17, Theorem V 4.111. It is also easy to see that Axiom II is a convex condition: if 0 < X 2 1 then /I(1 + HO)-‘(Xxi + (1 - X)Xj)ll < 1. Let us denote by V the set of X E X such that X is Ho-bounded with bound less than 1, and such that there exists S > 0 with HO + X 2 61; then V is open, convex and of dimension n. It follows that exp{ -,L?H} can be defined as a bounded operator, where H = HO + X, X E V. As a simple example of Axiom II, suppose that n = 3 and HO and X3 = N are the Hamiltonian and the number operator of the free Boson field: Ho =
dkw(kW(k), SW JO3 N=
0
dk N(k),
0
where N(k) = a*(lc)a(k). If w(k) > m > 0, then p(Ho - ,uN) is bounded below in the open convex region p > 0, p < m. It is clear that for each i, Xi exp{ -pHo} is of trace-class, since when written as (X,(1 +
HO)-‘)
((I+
Ho)
exp{-P&j)
INFORMATION
GEOMETRY
it is seen to be the product the relevant
means
The important our purposes.
of a bounded of Golden
and Grumm
We give Ruskai’s 1 (Ruskai).
THEOREM
and A + B
is of trace-class,
THEOREM
Since
was established
in the unbounded
[ 191. either of which is sufficient
for
are essentially
self-adjoint
on 2,
and
2 Tr (e-“e-“)
.
= Tr e--B(Ho+X)
as a function
of pj, with logarithmic
neighbourhood
m < Tre-P(Ho+X) Proof:
all
then
For all X E V, exp{-$(Ho+X)}
2.
Thus
from I and II, we show that Zx
and small enollgh compact c CC such that
operator.
version:
It. is now easy to get some results;
and is differentiable
by a trace-class
and Thompson
Tr (e--(A+B))
exists
425
DESCRIPTION
state.
[18] and by Ruskai
If A, B
bounded below, and exp{ -B}
QUANTUM
operator
are finite in the canonical inequality
case by Breitenecker
AND REDUCED
derivative
is of trace-class,
-r13
and for each X0 E V
K of X,J there exist m, 4.4, with 0 < m c M
5 Al
V is open, there exists
X E K.
(22)
(1 + 6)X
E V for all X E I<.
for all
6 > 0 such that
Put
A = /3(1+ S)-l(Ha
B = /?S(l + S)plHO.
+ (1 + 6)X),
Then A+B=@H=/3(Ho+X). Since (1+6)X Then
Tr e-OH
on K.
E?=/?(Ho+X+SX)
inequality, =
Tre-(A+B)
5
IIe-AII.IJe-BII1
=
M(AK)
2 n
(e-Ae-B)
5 eF~(le~~HoII1
The lower bound is proved similarly:
(23)
we put A = /3(Ho + X - SX) and
toget Tr e--2,3(&+x)
This enables
below, and as X roams over the compact
set K, the lower bound can be chosen to be C, the same for all X E K.
by the Golden-Thompson
uniformly
is bounded
E V, Ho + (1+6)X
finite-dimensional
5 e-C’Tr
us to define the exponential px = Zgl
e-OH,
X E V,
e&j%+X+6X)
0
family of states H = Ho + X,
2~ = TreeoH
>
.
(24)
426
R. F. STREATER
An element P2,
@3r..
. , pm.
of this family is determined by the values of the n - 1 variables p = We shall see that the dimension of the family is n - 1. This will fol-
low from the non-degeneracy
of the metric
gjk which
will be defined
on the tangent
space. We now show that
THEOREM
H inherits
an important
3. Fork = 2,3,...
of Ho.
property
,nandXEV,
H=Ho+X,wehave
= cx < co.
11(1+H)-%&
Moreover for any X0 E V there is a compact neighbourhood K of X0 such that sup
cx=c
XEKO
Proof :
We see that
(1 + H)-‘(l+ is uniformly
bounded
H,,) = [l+
on some compact
(1 + H,,-1X]-1 K of Xe, since
neighbourhood
Ho)-~XII,= dx
]](l+
1
as X runs over K. Then
Ho)llmI Cl- 4-l
]I(1 + H)-r(l+
uniformly
in K .
Therefore ]](l+
uniformly
H)-lXk]]
H)-r(l + Ho)]]co]](I
5
]](I +
1.
(1 - d)-‘]I(1
+ H&lXk]]
+ Ho)-lXk]], = c
0
in K.
COROLLARY. For k = 2,. . . , n and any /? > 0, XkepPH is of trace class, with tracenorm bounded in some compact neighbourhood of any X0 E V. (25) Indeed, ]]Xke-PH]]r
=
+ H)e-PH/2e-BH/211
]]Xk(l
+ H)-I(1
<
IIxk(l
+ H)-‘llmll(l
< -
C.(2/p)e-1+0/2.M(/3/2)
the first factor is given by Theorem the third factor by Theorem 2. As a result of this corollary
+ H)e-PH’211~Ile-PH’21(,
3, the second
;
factor
by the spectral
we can define the expectation
r]k = zilTrXkePpH
>
k=2,...,n.
(or mixture)
theorem,
and
coordinates (26)
In the next two proofs, we estimate the trace norm of the integrand given by Matsubara’s interaction picture [ 131, and show that it is bounded by an integrable function of imaginary time. An application of Lebesgue’s dominated convergence theorem and Fubini’s theorem for traces then shows that the integrals in question converge in trace-norm.
INFOR.MATION GEOMETRY THEOREM
AND REDUCED QUANTUM DESCRIPTION
log ZX is differentiable in ,&, . . . , pn and
4.
a log 2x
ap
Proof :
427
= -qj.
We first remark that Z is bounded
hood of X E V, so it is enough
to discuss
(27)
away from zero in a compact
the differentiability
of 2.
neighbour-
Let 6X
= h,flJ_X,
and put H’ = H + SX. Then (Spj)-‘[Z(X We use Duhamel’s
+6X)
- Z(X)]
= (S,@-lTr(e”H’
- epaH).
formula 1 ,-Off’
_
e-OH
=
da ,-d3H’
_
(pQj)
,-(l-~VflH
(28)
s0 where strong operator
convergence
with respect
to (Y, noting
the integrand
is understood. w(o)
that the domains
of the trace
1
difference
of H* and H’* include
of two operators
of trace
e-(lma)fiH;I). class.
In fact,
Similarly
we can
and write
rlj =
the desired
the unitary
= e-aflH’e-(i-n)OH
in eq. (28) is the product
use the cyclicity
Then
To prove eq. (28) we differentiate
Tr
da &HX
s0
’
.c-(i-a)PH 3
(29)
qj, with
Z(X + SW- Z(X)
q’ = 3
Spj
+
Z(X)%
obeys the relation
SJ 1
=
-S@T’r
1
dcr
0
shorten
To
d,j e-XaPH’Xje-(l-X)aBHXje-(l-a)BH,
0
some of the formulae in the proof, let us denote the resolvent (1 + H)-l by Similarly, Ro, R’ and X’ are defined using Ho, H’ and X. In the
R, and 1 - Q by o’. region
0 5 Q 5 l/2 the integrand
using Holder’s
n I
inequality
is controlled
by exp(-cY’PH},
II = n (1 +H')~-"Q?H'/~ e-~WI/2~l~j II aXPH”211~llR’XjllMIJX~e-a’BH’211111e-X’aOHlloo x 5 Ile-
,-~X~H’X~~-(~-X)OPHX~-(~-~)SH
[
and we can estimate
for traces:
e-h0H~je+3H
I
=
K
>(
x ]]e--a’RH’2(1
+ H)]],]]R(l
O(1)
+ 0.
as spj
>
+ H’)e-UX4H’/211m
428
R. F.STREATER
In the region f < o 5 1 we split JdX into 0 2 X < l/2 and l/2 < X < 1. In the first, the integrand is controlled by e-(l-X)aPH (n [ (e-MH’)
(X,e(i-+BV)
5
IIPAPH
=
O(1)
and we write
(e-(l-+P~lzXj)
(e-(l-~)P)]
(
IloollX,e-(‘-X)“gHIJ1J(e-(l-~)~~HX,/I~I/,-_(l-~)~HII, as S/P -+ 0.
i < A 5 1, the integrand is controlled by the factor ePaXpH’, Intheregioni
[(
e -aA{H’Xj)
(e-X’@H)
(Xj~)
(e-“‘P”)
((I + H)eMaXgH’)]
,
which is O(1) since the first and last factors are of trace-class, and the middle factors are bounded, all uniformly over the range of integration and for 6X E K for some compact set K. For example, Ile-aXPH’12XjJI < Ile-~(“x-f)H’llmlle-~H’X3111
5 IlePfH’Xjlli 0
and this is bounded uniformly in SX by the corollary to Theorem 3. To prove the next theorem we need a further axiom. AXIOM III. For each X, Y E X, (1 + Ho)-l[X,
Y] is bounded.
Then we have the Kubo formula for the susceptibility. THEOREM
5. For any j, k > 1, q
is differentiable
(xk - p.xk)e-
in ,B”, and
(l - “‘PH(Xj - P.Xj).
On differentiating the Z-’ in eq. (26) we get -Z-2dZ/dflk = Z-$k by Proof : Theorem 4, and this cancels the cross moment in eq. (30). So it is enough to show that is d ifferentiable in /3”, with derivative Tr(exp{-PC& +X)1X,) 1
da (e- @HXk)
-Tr
(,-(I-“)“HXj
So we must look at e-PH’ _ e-PH Wk
1
xj -
dae -~YPHXke-(l-WHXj
J0
1 .
We use Duhamel’s formula for the difference of the exponentials, and find that FjI, = fip” L’dal’dhTr
[e-
Xa4H’xke-(l-X)@Hx
ke-(l--a)PH~j
1 .
INFORMATION
GEOMETRY
AND REDUCED
QUANTUM
This is of order S/3” if the integrand is bounded
DESCRIPTION
(as SD” -+ 0).
429
We now prove this
boundedness .
In the region 0 < LY5 $ the trace of the integrand is written as the trace of
All the factors are obviously bounded in norm uniformly in S/3 (using previous results) except the last. For this, we write e-(i-&HXJ(I
+ H’) = e- (l-Q)gHxj(l
+ H)(1+
H)-1(1+
H’).
We may drop the factor (1+ H)_i(l which is uniformly norm-bounded ,-c’4Hxj(I
+ H)
+ H’)eCX”OH
as SD” + 0. We are then left with
=
e-(i-4:H(I
=
-(i-&H(I Ce
+ H)X, + N,)
+ (,+a)+(1
+ e-(l-4$H[H,
X,]
(&4Qj)
+
+ Ho)) ((1+
H&i[H,
Xj])
Both terms on the right side contain a factor of trace-class, multiplied by bounded operators. Hence we have a bound on the trace-norm in the region Q < i. In the region $ 5 (Y 5 1, 0 2 X 2 $,the exponential exp{-(1 - X)aPH) dominates, so we group them as (
(I + H’)Xke-X’agH
>(
e-“a$HX
k) (f~-~“~>
(XjR'e-X"bH')
All the factors are bounded uniformly in norm in the given range, and the worst part of the first factor, the quadratic term jy’xke-X’a$H
=
(H'R)([H,Xk]t-X'a~H+X,&'a~H(l
+ u))
is the sum of two trace-class operators; for, the first term is (H’(1 + H)-‘)
([H,Xk](I
+ He)-‘)
((I + ~s)e-(l-x)~‘H)
;
all factors are bounded and the last is of trace class. The other term is even easier; we omit the details. In the region $ 5 (Y 5 1, $ < X < 1 the dominant exponential is exp{-XapH’} so we write the integrand as (I + H’)XjewXngH’)
(e-xOqW’Xk)
(e-(‘-“)-““)
(Xk(I + H’))‘)
430
R.F.STREATER
The factors are all bounded, uniformly as Q, A and S,B” vary over the given regions. The proof is completed by the remark that the first factor is of trace-class, as is seen by the identity (1 +
fp)q+4~’
=
([H’,Xj](l +
COROLLARY.
+ Ho)-l)
((l+
Ho)e-““$“‘)
(Xje-AaaH’)((I+ f$‘)e-‘“ZH’)
gj,kisLipschitz continuous.
t q
(31)
For by the estimates just given, gj,k(X + 6X) - gj,k(X) can be written as the sum of two integrals of the above form, and are similarly shown to be O(S,B”). In fact, we can continue and show that the manifold is infinitely smooth. is positive definite, the dimension of the manifold V,_l is n - 1. 3.
Since gj,k
Rival norms on Z A number of papers [20] a dvacate either the KMS scalar product
(A, a,
= Tr w*q
(32)
or, for Hermitian operators, its real part Re(A, B)P = f Tr p (AB + BA)
(33)
as the metric to use on the space of observables. Let us suppose that p is a faithful state, as happens when it is the canonical state of the system. Then eq. (32) defines a norm on the algebra. The dual norm is then
IIUII = {Tr
(p-ln2)}1’2
(34)
giving a metric on the space of states. In [l] the author adopted the metric eq. (34); it seemed a good idea at the time, because of a number of useful properties: 1. It reduces to the Fisher metric [8] if the algebra is abelian; 2. The unique state of minimal dual norm is p itself; that is, Tr
(p-la2)
2 1, for all states cr, with equality only if u = p;
3. (Xj, qp = vj; 4. Any isothermal dynamics is a contraction in this norm. More generally, any linear dynamics obeying detailed balance (in the general form given in [l]) is a contraction.
INFOR.MATION
Property
GEOMETRY
AND REDUCED
4. was first established
by Majewski
QUANTUM
[21]; recently
with a simple proof has been found [l, 221. Because
431
DESCRIPTION
a discrete-time
of this, we can picture
version
the approach
to equilibrium under isothermal conditions as being along an orbit in state-space whose metric distance to the fixed point p, the canonical state at the given temperature, is reduced
at each step, until it reaches
is an alternative with time. complex
statement
However,
scalar
p, the state with norm 1, the global minimum.
to the decrease
as we now show, the properties
products
defined on the algebra
= Tr
(A, Bjp+ which have been studied Zegarlinski
[23].
real vector
of the non-linear
before
(
Our real interest
are also true of the family of
>
(35)
,
the case (Y = i was used by Majewski
is in the Bogoliubov
operators
This
the free energy,
by
paA*p(‘-a)B
(141. Recently
space of Hermitian
l.-4.
function,
scalar
product,
defined
on
Fisher
of its close relation metric.
with Kubo’s
Indeed,
susceptibility
(36)
tensor
We now show that all these obey l.-4.
the norm of the scalar product
and thus to the quantum
We start
with the case of ]( l ]]p.Cr,
eq. (35).
1. is clear if all operators
first find a formula
commute,
and 3. is obvious.
Before
proving
2. we
for the dual norm, defined by
((ullp,_a = sup{(u.Al : (IAll,,, = 11. We expect
the
by eq. (19)
((A, B)) = .I’ Tr[p”Ap’-Y?]da, because
and
(37)
that the answer is
THEOREM 6. ll~ll&
and even for some unbounded
for all states. certainly
So let u be such that
finite,
namely
defined by the spectrum
matrices
operators,
the right-hand
consider
the dual norm
of p, and put
IIAllp,a = 1, so putting (]c$,,-~
> ITroAl
1’2 .
this A in eq. (37) gives
= ll’~p-%p-~+“IN-~
not be finite
in Theorem
in the orthonormal
the operator
One checks that
might
side of the equality
of finite rank when expressed
N = (Tr(p-%p-l+“a)) Then
.
We note that whereas the norm coming from eq. (35) is finite for all bounded
Proof: operators
= Tr (p-“crp-lfao)
= N.
6 is basis
432
R. F. STREATER
For the inequality in the other direction, Ic.A12
=
/n [(p(“-1)/20p-n/2) (pa/2Ap(1-“)/2)]I2
<
n (p-Sqf--lgp-%
= N2 IIAII;,a = N2 if I14La = 1. Hence the supremum over all such A is also bounded by N2, and we get 0 IlallP,-a 5 N. Combining with the other inequality gives the theorem. We now return to the proof of 2. for these norms. We have
I14Jp,-a=
s;~{l~.Al
: IJAIl,,, = 1) L a.1 = 1 = ll~ll~,-a
(since lllllP,~ = 1 we can put A = 1 and get a lower bound for the supremum). equality only if o = p. We note that in 0 5 (Y 5 $ the norms are decreasing:
II4lP,a5 ll4,~
if
Qf>P.
(33) We get
(39)
This is implicit in [14]. The corresponding dual norms go the other way:
II4lP,-a 2 ll4-0
if
QZP,
(40)
which can be regarded as the extension of the same inequality for negative values of the parameters. To prove 4., we first prove that any isothermal dynamics obeys detailed balance relative to the scalar product eq. (35). We need a definition. DEFINITION. Let A be a W*-algebra with a scalar product (0, l ), and let T be a stochastic map on A. We say that T obeys detailed balance relative to the given scalar product if the adjoint map T* (relative to the scalar product) is also a stochastic map.
Then there is the following result [l, 221: THEOREM 7. If T, a stochastic map, obeys detailed balance relative to a scalar product, then it is a contraction in the corresponding norm.
In statistical dynamics we must distinguish between isolated dynamics, in which no heat is exchanged with the surroundings and the total energy is constant, and isothermal dynamics, in which the temperature is kept constant by removing, or adding, heat as necessary. In isolated dynamics the entropy increases, and in isothermal dynamics the free energy decreases. Isothermal dynamics is obtained from a special case of the general scheme described in Section 1, and is linear. It is this linear map that obeys detailed balance; we showed this for the scalar product eq. (32) in [l]. We now show it for the family eq. (35). Recall that one time-step in isothermal dynamics is obtained as follows.
INFORMATION
The W*-algebra and
d,
GEOMETRY
is a tensor
is the algebra
AND REDUCED
product
d = A, @ 4,
of the heat-particle
as a slow variable.
and the map Q is the
the tensor
of its partial
heat)
product
traces
[l].
QUANTUM
where
A, is
Any Hermitian
“stoss map”,
the algebra element
of the system
of
which replaces
over the two factors.
433
DESCRIPTION
-4,is
a state
The total energy
regarded on A by
(including
is a sum H = H, @ 1+
TJ commutes
reversible
of such automorphisms. dynamics
motion
with H. Random
7j3 at beta
is given by an inner automorphism
dynamics
Then
U
l7
l
‘. where
is given by a mixture
T is bistochastic
/3 is defined
(41)
1@ HA,
and completely
to be the predual
positive.
of the following
The isothermal
linear
dynamics
on
induced
by
C(A):
Here, cc is any state
of the system,
T and pr,o is the canonical stochastic and obeys defined by
state
detailed
balance
Tr,. (P,,@(AP) is also stochastic. dynamics 11.12
Indeed,
immediately eq. (35),
guarantees
and that
LEMMA.
canonical
~(0)
We proved
to the canonical
dynamics
of
A
[I] that
state
for all
~(0) is the isothermal l
such a T[, is
~~,a, that
A, B E
our result
given in Theorem
7,: is therefore
a contraction
of-AC; that
is, r(@+)
in the norm
11l ((P,-O.
of rO relative to (A, B)p,a.
where p = pr,,j zs the
obeys
= Tr, (p”Ap’-O~~(l?))
for all
A, B E
d,
Then 7(35n) _ 7(3).
(+a)(A),
Let A, B be Hermitian Bjp,n
random
= 7-i. The next lemma generalises this result to all CY. and that 7. obeys detailed balance relative to the scalar products of
Tr, (pa&~)(A)pl-V)
Proof:
is. .(“)
A,
for the time-reversed
U. Let us call this -r,*, so that
Let ~(fi>~) be the adjoint state
relative
= Tr, (p,,&p(B))
T” = C &U-l
of [I] is that
Td is the dual map on the states
of the heat-particle.
=
(-4 ~,@))~,a
=
Tr,
=
%,,
(
elements
T%P*
A,,and
put po = p @ p_,,g; then
- Tr, (P%-~~,(B))
~,d(p~Apl-~)B (
of
>
= Tr, (Tr,[Td(paApl-a
@v;,/dL-
Wpl-
@ P$3V(B
6~ P,,~)]B)
@ 1))
434
R. F. STREATER
=
Tr c,y
(p,“Td(A @~)P;-“(B8 1,)
since Vi commutes with pp = p @ pr,fl =
‘I’r,,,
(Td(A 8 l)p;-(B
@1)~;)
= nc,-,(Td(A@~)(P~-~BP~) @pr,o) = nc,y((A8 W[(P~-~~P~) @~y,pl) = nc (A& (Tdd(~l-“B~a @P,,P,)) =
Tr, (A~;~(p~-‘V3p”))
=
($4 Bjp,a.
= Tr, ($(A)p’-“BP”)
It follows that r(P@) = r(p) for all cz and is a stochastic map, being the isothermal dynamics for the time-reversed dynamics. Therefore @%a) obeys detailed balance, and rP is a contraction in the norm ]Il ]]p,a and therefore that its dual r: acting on the states 0 is a contraction in the dual norm. Since we have just proved that rO obeys
we see that rO is also a contraction seems to be new. 4.
Non-linear
in the Bogoliubov norm. In this generality, this result
stochastic processes
So far our (non-linear) dynamics of an isolated system is given by an orbit through the family V,_l of states, one time-step being p -+ pTdQ. This can be regarded as a non-linear version of the von Neumann dynamics, which is the Schrodinger picture for density matrices. In the case of isothermal dynamics, and where the slow variables form an algebra A,, we can reduce to a linear stochastic process obeying detailed balance. This can be transferred to a quantum stochastic process on the algebra, by duality; thus in the linear case there is no problem in setting up the corresponding Heisenberg picture. As a result, a meaning can be given to the multitime correlation functions PO
(xl(h),
. ..,xk(tk))
,
(42)
where po is the initial state. In [l] we discuss whether the dynamics can be thrown onto the observables also in the non-linear case. Such a procedure is desirable, since the multitime correlation functions and, equivalently, the Wightman functions, exist in the reversible case, and should retain some meaning at least if the dissipation is small. We impose the constraint that the change in the expectation of each X E X should be the
INFORMATION
GEOMETRY
same in both pictures. on t#hestates obeys
AND REDUCED
QUANTUM
Thus we seek a linear map Q,(t)
pT”Qf.A
for all
= pTdQ.A
on
DESCRIPTION
435
A such that its dual Q:(t)
AEd.
(43)
Then we could define one time-step of the Heisenberg dynamics at time t to be A(t) H
Q,(t)TA(t) THEOREM
of
= A(t + 1). 8.
Let Q*(t)
be the orthogonal
projection
onto X (as a subvector
space
d) in the KMS scalar product at pTdQ; that is
(4 B)pTdQ =
pTdQ.A*B.
(44)
Then eq. (43) holds. Proof:
WriteA=Ae+Al, pTdDlf.A
where A0 = Q,A.
Then
=
pTd.Q,A
= pTd.A,,
=
PT~Q-&
= (1, Ao)~T~Q
=
(17-4,~d~
=
pTdQ.A.
as 1 L A’ 0
This proof is more transparent than the one given in [l], p. 256. We see from the proof that instead of the KMS scalar product, we can use for Q* the projection onto X using any scalar product obeying (3) of the last section. In particular, we could use the Bogoliubov scalar product. We obtain in this way a linear map A H Q,TA, but the (discrete time) dynamics got from this is not stationary, since Q* depends on the current state. In fact we must solve the non-linear dynamics for p(t) in the Schrodinger picture before we can find Q* at each time. If the initial state is an equilibrium state, then p(t) is independent of time, and this problem does not arise. It is not clear that the map Q* is positive, let alone completely positive. This is true whether we use the KMS or the Bogoliubov scalar product. This would be needed for a satisfactory interpretation of the many-time correlation functions. Positivity does hold, however, in one special but common situation; this is when X includes the Hermitian part of an algebra A,,and A is a tensor product A, @ A,. Positivity then holds on d,. Indeed, for any pr the map
which gives one time-step, is linear, and takes states to states; it is therefore the dual of a linear positive map on the algebra. This is time-dependent, since the construction depends on the current state, pr, of the heat-particle. This gives a non-linear stochastic process, in which the temperature varies with time. It has been worked out in detail for a Brownian particle in a potential [24].
436
R. F. STREATER
Acknowledgements The author is indebted to E. B. Davies and Y. Safarov for bringing to his attention some useful parts of the literature. REFERENCES [l] R. F. Streater: Statistical Dynamics, Imperial College Press, London 1995. [2] A. Kossakowski: Bull. acad. polonaise sci. 1’7 (1969), 263. [3] H. Grabert: Projection operator techniques in non-equilibrium statistical mechanics, Springer Tracts in Modern Physics, 95, Springer, Berlin 1982. [4] R. Balian, Y. Alhassid and H. Reinhardt: Physics Reports 131 (1986), 2, North Holland. [5] E. T. Jaynes: Phys. Rev. 106, 620, and ditto II, ibid, 108, 171 (1957). [6] R. S. Ingarden: Open Syst. Inform. Dyn. 1 (1992), 75, and early references therein. [7] R. F. Streater: Statistical dynamics and information geometry, to appear in Geometrical and Topological Methods in Physics, Eds. P. Combe and H. Nencka, Amer. Math. Sot. 1996. [S] R. A. Fisher: Proc. Camb. Phil. Sot. 22 (1925), 700. [9] R. S. Ingarden, Y. Sato, K. Sagura, and T. Kawaguchi: Tensor 33 (1979), 347; R. S. Ingarden, H. Janyszek, A. Kossakowski, and T. Kawaguchi: Tensor 37 (1982), 105. [lo] S.-i. Amari: Differential Geometric Methods in Statistics, Lecture Notes in Statistics, 28, Springer, Berlin 1985. [ll]
H. Hasagawa: Non-commutative extension of information geometry; pp. 327-337 in Quantum Communication and Measurement, Eds. V. P. Belavkin, 0. Hirota and R. L. Hudson, Plenum Press, 1995. Also, H. Nagaoka: Differential geometrical aspects of quantum state estimation and relative entropy, ibid, pp. 449-452. F. Weinhold: J. Chem. Phys. 63 (1975), 2479; ibid 65 (1975), 559. T. Matsubara: Prog. Theor. Phys. 14 (1955), 351. G. Roepstorfi Path-integral Approach to Quantum Physics, Springer, Berlin 1994. R. Kubo: J. Phys. Sot. Japan 12 (1957), 570. J. Naudts, A. Verbeure and R. Weder: Commun. Math. Phys. 44 (1975), 87. T. Kato: Perturbation Theory for Linear Operators, Springer, Berlin 1966. M. Breitenecker and H. R. Griimm: Commun. Math. Phys. 26 (1972), 276.
[12] [13] [14] [15] [16] [17] [18] [19] M.-B. Ruskai: Commun. Math. Phys. 26 (1972), 280. [20] D. J. C. Bures: Trans. Amer., Math. Sot. 135 (1969), 199. A. Uhlmann: Rep. Math. Phys. 9 (1976), 273. W. K. Wootters: Phys. Rev. D23 (1981), 357. [al] W. A. Majewski: J. Math. Phys. 25 (1984), 614. [22] W. A. Majewski and R. F. Streater: Detailed balance and quantum dynamical semigroups, to appear. [23] W. A. Majewski and B. Zegarlinski: On quantum stochastic dynamics and noncommutative L, spaces, to appear. [24] R. F. Streater: The Brownian particle in statistical dynamics, to appear.