Canonical Form and Separability of PPT States in 2xMxN Composite Quantum Systems.pdf
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Canonical Form and Separability of PPT States in
C2 ? CM ? CN Composite Quantum Systems
Xiao-Hong Wang1, Shao-Ming Fei1, 2, Zhi-Xi Wang1, and Ke Wu1
1 Department of Mathematics, Capital Normal University, Beijing, China.
2 Institute of Applied Mathematics, University of Bonn, 53115 Bonn, Germany
Abstract
We investigate the canonical forms of positive partial transposition (PPT)
density matrices in C2 ?CM ?CN composite quantum systems with rank
N . A general expression for these PPT states are explicitly obtained.
From this canonical form a sufficient separability condition is presented.
Key words: Separability, Quantum entanglement
PACS number(s): 03.67.Hk, 03.65.Ta, 89.70.+c
Quantum entangled states have become one of the key resources in the rapidly expanding
field of quantum information processing and computation [1, 2, 3, 4]. Nevertheless, the study
of physical character and mathematical structure of the quantum entanglement is far from
being satisfied. One even does not have a general criterion to judge if a quantum (mixed)
state is entangled or not. For bipartite states a number of entanglement measures such as
entanglement of formation and distillation [5, 6], negativity [7], von Neumann entropy and
relative entropy [5, 8] have been proposed. However most proposed measures of entanglement
involve extremizations which are difficult to handle analytically. For instance, explicit analytic
formulae for entanglement of formation [9] have been found only for a pair of qubits system [10],
and for some symmetric states [11] and a class of special states [12]. For multipartite systems
there is no well defined measure of entanglement yet.
The separability problem for pure states is quite well understood [13]. Nevertheless, in real
conditions, due to the interactions with environment, one encounters mixed states rather than
pure ones. The manifestations of mixed-state entanglement can be very subtle [1
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