Connections and metrics respecting purification
of quantum states
J. Dittmann
a)
Mathematisches Institut, Universita
¨
t Leipzig,
Augustusplatz 10/11, 04109 Leipzig, Germany
A. Uhlmann
Institut fur Theoretische Physik, Universita
¨
t Leipzig, Leipzig, Germany
͑Received 5 January 1999; accepted for publication 5 April 1999͒
Standard purification interlaces Hermitian and Riemannian metrics on the space of
density operators with metrics and connections on the purifying Hilbert–Schmidt
space. We discuss connections and metrics which are well adopted to purification,
and present a selected set of relations between them. A connection, as well as a
metric on state space, can be obtained from a metric on the purification space. We
include a condition, with which this correspondence becomes one to one. Our
methods are borrowed from elementary
*
-representation and fiber space theory. We
lift, as an example, solutions of a von Neumann equation, write down holonomy
invariants for cyclic ones, and ‘‘add noise’’ to a curve of pure states. © 1999
American Institute of Physics. ͓S0022-2488͑99͒02107-6͔
I. INTRODUCTION
In Ref. 1, see also Ref. 2, the monotone Hermitian and Riemannian metrics in the ͑finite
dimensional͒ spaces of all density operators are classified. Based on the theory of operator means
3
they are indexed by a real function, f, operator monotone
4
on (0,ϱ). These metrics play an
important role in domains like quantum information geometry, quantum versions of statistical
estimation, and decision rules.
5–7
D. Petz communicated his main results to us prior to publication, and about that time we
started to ask for the effect of a purifying lift to these metrics. There are clear reasons for this. One
of the present authors ͑A.U.͒ had defined 1986 in Ref. 8 an extension of the geometric phase,
9,10
see also Refs. 11 and 12, to curves of density operators by the help of a ‘‘parallelity condition.’’
The condition singles out, up to a global gauge ͑or a global partial isometry͒, a distinguished
‘‘parallel lift’’ within all purifying lifts of a curve of density operators. It turns out
13
that a
connection form ͑a gauge potential͒, here called a
geo
, is governing the transport of the purifying
vectors, such that the parallelity condition results from the request for horizontality. In 1992 G.
Rudolph and one of the authors ͑J.D.͒ considered a large class of gauge potentials, including a
geo
,
which rests on a purification scheme and which enables variants of the geometric phase along
curves of density operators. It seems natural to ask for a link between these objects: ͑a͒ the
connection forms just mentioned, ͑b͒ certain Hermitian ͑Riemannian͒ metrics on the purification
space, and, if respecting the symmetry of the scheme, ͑c͒ metrics induced from ͑b͒ on the space of
density operators.
Purification is essentially representation theory of observables and of the algebra in which
they are contained. Principally one may use any unital
*
-representation of the ‘‘algebra of observ-
ables’’ over which the states can be defined. Its Hilbert representation space should only be large
enough to allow for a representation of the states by vectors. If this condition is fulfilled, transport
mechanism, its noncommutative phases, metrics, and other geometric objects can be constructed
by relying on their form and appearance in the pure state case.
a͒
Electronic mail: dittmann@mathematik.uni-leipzig.de
JOURNAL OF MATHEMATICAL PHYSICS VOLUME 40, NUMBER 7 JULY 1999
32460022-2488/99/40(7)/3246/22/$15.00 © 1999 American Institute of Physics