ISSN 1063-7397, Russian Microelectronics, 2009, Vol. 38, No. 3, pp. 199–205. © Pleiades Publishing, Ltd., 2009.
Original Russian Text © A.Yu. Chernyavskiy, 2009, published in Mikroelektronika, 2009, Vol. 38, No.3, pp. 217–223.
Numerical characterization of quantum entangle-
ment is one of the priority problems in quantum infor-
mation theory. A complete study and classiﬁcation has
been performed for the entanglement of pure states of
two qudits. Among other things, the criterion for deriv-
ing one such state from the other by local operations
and classical communications (LOCC)  has been
found. However, to date, only a few results have been
obtained for a larger number of subsystems. These
results are brieﬂy listed below.
The classiﬁcation of pure three-qubit states is com-
pletely described in terms of stochastic LOCC
For mixed states, the conditions of monotony of the
measures of entanglement with respect to LOCC are
well understood (e.g., [3, 4])
In different studies (e.g., [6–8]), some measures of
entanglement of pure and mixed states and some crite-
ria of success of states for quantum computations are
In this study, we present the measure of entangle-
ment of pure multiparticle states. The main advantages
of this measure of entanglement are as follows:
(1) This measure is based on the representation of a
state that is fully in agreement with the Schmidt expan-
sion for the case of two subsystems.
(2) The algorithm of calculation of this measure has
It should be noted that ﬁnding adequate measures of
entanglement for pure multiparticle states is probably
one of the most intricate problems of the theory of
quantum entanglement, since this problem is practi-
cally equivalent to the search for counterparts of vari-
ous matrix expansions for high-order tensors.
2. ENTANGLEMENT OF TWO-PARTICLE STATES
The entanglement of pure states of two particles has
been much studied. It is determined by the Schmidt
coefﬁcients of this state [1, 5].
be the pure state of the com-
posed system AB. Then, there exist such orthonormalized
of the system A and
of the system B that
> 0, and = 1.
are referred to as the Schmidt coefﬁcients.
For the measure of entanglement, the entropy of the
Schmidt coefﬁcients squared (the von Neumann
entropy of the states of the subsystem
The Computable Measure of Quantum Entanglement
of Multiqubit States
A. Yu. Chernyavskiy
Institute of Physics and Technology, Russian Academy of Sciences, Moscow, 117218 Russia
Received September 23, 2008
—The search for adequate measures of the entanglement of pure quantum states for an arbitrary num-
ber of subsystems is one of the priority problems of quantum information science. In this paper, I present the
computable measure of entanglement of an arbitrary quantum state. The measure is the generalization of the
von Neumann entropy to an arbitrary number of subsystems.