On a Parametrization of Puriﬁcations of a Qubit
and Viswanath Ramakrishna
Received November 29, 2002; accepted February 1, 2003
This work provides, constructively, explicit one–one parametrizations of all pur-
iﬁcations of a mixed state in dimension 2 and all joint puriﬁcations, if any, of two
mixed states in the same dimension. The former is parametrized by SOð3; RÞ,while
the latter is parametrized by SOð2; RÞ, except when the state being puriﬁed is already
pure. These parametrizations are derived without any passage to the spectral
decompositions of the state(s) being puriﬁed. Using this, we show how to calculate
certain measures of quantum information. The appendix considers an alternative one–
one parametrization of mixed states in C
, which provides a diﬀerent explicit
construction of joint puriﬁcations.
KEY WORDS: Puriﬁcations; parametrizations; maximal entangled fraction.
PACS: 03.67-a; 03.67-Hk; 03.67-Lx.
The notion of puriﬁcation of a mixed state plays an important role in several
It provides insight into the question of decoherence. It is
important in quantum information theory from several points of view. For
instance, quantitative measures such as the fully entangled fraction,
be explicitly deﬁned in terms of puriﬁcations.
The purpose of this note is to provide an explicit one–one para-
metrization of all possible puriﬁcations of a mixed state in two dimensions,
and joint puriﬁcations (if any) of two mixed states in a two dimensional
Hilbert space. This parametrization does not rely on the spectral decom-
position of the mixed states being puriﬁed, and in particular does not
involve Schmidt decompositions of pure states in C
. This circum-
1570-0755/02/1000-0409/0 # 2003 Plenum Publishing Corporation
Department of Mathematical Sciences and Center for Signals, Systems and Communications,
University of Texas at Dallas, P. O. Box 830688, Richardson, Texas 75083.
Author to whom correspondence should be addressed. E-mail: email@example.com
Quantum Information Processing, Vol. 1, No. 5, October 2002 (# 2003)