Quantum Information Processing, Vol. 4, No. 3, August 2005 (© 2005)
Further Results on the Cross Norm Criterion
Received February 13, 2005; accepted April 27, 2005
In the present paper we develop and investigate a novel approach that aims to
characterize quantum entanglement by using cross norms. In the ﬁrst part of the
paper we further develop the mathematical theory by determining the value of the
greatest cross norm for Werner states, for isotropic states and for Bell diagonal
states. In the second part we show that our techniques induce a novel powerful
analytical and computable separability criterion for bipartite systems. This new
criterion complements the well-known Peres positive partial transpose criterion in
several aspects. It is a necessary but in general not a sufﬁcient criterion for sepa-
rability. We demonstrate the power of the new criterion by evaluating the criterion
for a number of important examples. We also demonstrate that the new criterion
is able to detect bound entangled states.
KEY WORDS: Quantum entanglement; separability criteria; tensor norm;
bound entangled states; greatest cross norm.
PACS: 03.67.Mn, 03.67.a, 02.30.Tb.
Quantum entanglement is one of the most fascinating features of quantum
theory. Its study is an important area of research that has received much
attention in the development of quantum information theory. Mathemat-
ically entanglement is a complex property of quantum states deﬁned on
tensor product Hilbert spaces. Physically the role of entanglement in quan-
tum information processing is manifold. Quantum entanglement has been
identiﬁed as an essential resource in many applications of quantum infor-
mation processing. In recent years considerable progress has been made
towards developing a general theory of quantum entanglement,
references therein. In particular criteria to decide whether or not a given
Dipartimento di Fisica “A. Volta,” Universit
a degli Studi di Pavia, via Bassi 6, I-27100
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1570-0755/05/0800-0219/0 © 2005 Springer Science+Business Media, Inc.