Inferring the density matrix for a system of an unknown Hamiltonian
A. Rigo,
1
M. Casas,
1
and A. Plastino
2,3
1
Departament de Fı ´sica, Universitat de les Illes Balears, 07071 Palma de Mallorca, Spain
2
National University La Plata, Casilla de Correo 727, 1900 La Plata, Argentina
3
Argentine National Council (CONICET), Casilla de Correo 727, 1990 La Plata, Argentina
Received 7 July 1997; revised manuscript received 4 November 1997
An approximate inference approach is advanced in order to determine the density matrix in cases in which
the pertinent Hamiltonian is unknown, and the available expectation values pertain to noncommuting observ-
ables. The inference of both pure and mixed states is addressed, and on an equal footing, without facing
infinities in the pure state limit. Information-theoretical tools are employed. S1050-29479802604-3
PACS numbers: 03.65.Ca
I. INTRODUCTION
In order to attempt a quantum description of a system so
as to obtain the concomitant wave function or density matrix,
knowledge of the pertinent Hamiltonian is an all important
requirement, as otherwise Schro
¨
dinger’s equation or Von
Neumann’s cannot be written down. However, cases in
which the Hamiltonian is unknown abound, and indeed much
effort, from both theoretical and experimental viewpoints,
has been expended in trying to determine, at the very least,
appropriate effective Hamiltonians in order to remedy such
informational failure.
Since in statistical mechanics SM inference is the over-
all purpose, because incomplete information is always pre-
supposed, SM methods would seem to be called for in the
case of an unknown Hamiltonian. Indeed, the main idea un-
derlying statistical mechanics is that of describing the most
salient characteristics of a given system by recourse to a
small set of relevant expectation values just H in the case
of Gibb’s canonical distribution1. But once again, one is
assumed to know the system’s Hamiltonian in order to pro-
ceed to build up canonical, grand-canonical, or other types of
ensembles. What is to be done if such energetic information
is not available?
Several types of hardship are to be faced in numerically
dealing with quantal problems by recourse to methods of
statistical mechanics. Among them we can list the following:
for pure states, the thermodynamical entropy is identically
zero; the Hamiltonian may not be known; pure states and
mixed states are often dealt with on an unequal footing, due
to technical difficulties; some Lagrange multipliers may di-
verge; and the Kubo transform must be implemented in deal-
ing with non-commuting observables 2, which is some-
times a rather difficult task.
Recourse to information theory IT3, as employed by
Jaynes 4 in his celebrated reformulation of SM see also
Refs. 4–8, allows one to conveniently tackle the first two
of the above problems, but subject to the following restric-
tions: i the pertinent methodology can be applied only in
the case of pure states, and ii the a priori input information
must refer only to commuting observables. For details, the
reader is referred to Refs. 9–19. The purpose of the present
effort is that of addressing the last three of the hardships
listed above, overcoming also the aforementioned limita-
tions.
Concerning the fourth of the above-listed difficulties, if
the associated scenario makes it possible to infer a statistical
operator for mixed states a
`
la Jaynes, as one approaches the
pure state limit the associated Lagrange multipliers diverge,
to either plus or minus infinity 20, which seems to erect a
formidable barrier to the workings of numerical techniques.
As for the last item in the lists of problems to be overcome,
we mention that, if the available information refers to expec-
tation values of noncommuting observables, recourse to the
Kubo transform KT2 becomes mandatory. Its beauty and
elegance notwithstanding, the KT confronts one, in general,
with involved integrals that are not always easy to deal with.
In what follows we show how to apply Jaynes’ method-
ology in quantum-mechanical scenarios for which, the
Hamiltonian not being available, just a small set of expecta-
tion values of noncommuting operators constitutes the only
prior information. Both pure and mixed states will be tackled
on an equal footing, without infinities in the Lagrange mul-
tipliers, and no recourse to the KT needs to be made.
One would envision utilizing this formalism in situations
in which scarce information is available concerning the de-
tails of the interactions governing the physics of the system
to be described. Inference would then be the name of the
game. A number of such scenarios found, for example, in
Refs. 21–47, is by no means an exhaustive list. The for-
malism is introduced in Sec. II, some illustrations are dis-
cussed in Sec. III, and conclusions are drawn in Sec. IV.
II. PRESENT FORMALISM
We assume that M expectation values are at our disposal,
A
r
=d
r
, r =1, . . . , M , 1
and we do not assume that A
ˆ
r
are commuting operators.
In the arbitrary basis | i the statistical operator is of the
customary form
ˆ
=
i , j
N
| i f
ij
j | , 2
and our input-information can be cast in the fashion
PHYSICAL REVIEW A APRIL 1998 VOLUME 57, NUMBER 4
57 1050-2947/98/574/23196/$15.00 2319 © 1998 The American Physical Society