Quantum Information Processing, Vol. 4, No. 3, August 2005 (© 2005)
Derivation of the Lewenstein–Sanpera Decomposition
via Semideﬁnite Programming
M. A. Jafarizadeh,
and M. Rezaee
Received September 1, 2004; accepted April 27, 2005
In many theoretical proposals appeared recently, semideﬁnite programming was
considered as a way to express quantum entanglement. Using semideﬁnite opti-
mization method, we prove the Lewenstein–Sanpera lemma in a simple elegant
manner. Particularly, using this method we obtain Lewenstein–Sanpera decompo-
sition for some examples such as: generic two qubit state in Wootters’s basis,
Iso-concurrence state, Bell decomposable state and 2⊗ 3 Bell decomposable state.
KEY WORDS: Lewenstein–Sanpera decomposition; semideﬁnite programming;
Bell decomposable states; Wootters’s basis and Iso-concurrence states.
Entanglement is one of the most striking features of quantum mechan-
. In the case of pure states it is easy to check whether a given state
is, or is not, entangled. For mixed states, however, the statistical properties
of the mixture can hide the quantum correlations embodied in the system,
making thus the distinction between separable and entangled states enor-
mously difﬁcult. In the pioneering parer
a very interesting description of
entanglement was achieved by deﬁning the best separable approximation
(BSA) of a mixed state. In the case of 2-qubit system, it consists of a
decomposition of the state into a linear combination of mixed separa-
ble part and a pure entangled one. In this way, the whole non-separabil-
ity properties are concentrated in the pure part. In Ref. 3, the numerical
Department of Theoretical Physics and Astrophysics, Tabriz University, Tabriz 51664, Iran.
Institute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795, Iran.
Research Institute for Fundamental Sciences, Tabriz 51664, Iran.
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