Quantum Inf Process (2011) 10:257–269
Multipartite quantum systems and symplectic
Received: 18 February 2010 / Accepted: 5 August 2010 / Published online: 21 August 2010
© Springer Science+Business Media, LLC 2010
Abstract In this paper we study the geometrical structures of multi-qubit states
based on symplectic toric manifolds. After a short review of symplectic toric man-
ifolds, we discuss the space of a single quantum state in terms of these manifolds.
We also investigate entangled multipartite states based on moment map and Delzant’s
construction of toric manifolds and algebraic toric varieties.
Keywords Multipartite quantum systems · Symplectic toric manifolds ·
Quantum entanglement · Quantum information
During recent years the geometrical, topological, and combinatorial structures of
multipartite quantum systems have been parts of ongoing research in the ﬁelds of
foundations of quantum theory, quantum information, and quantum computing [1–4].
These mathematical methods are also very important in solving complex problems
and visualizing the difﬁcult physical concepts in other branches of physics such as
general relativity, gauge theory, and string theory .
Recently we have investigated the combinatorial and geometrical structures of
quantum systems using complex projective toric varieties [6,7]. In this paper we will
establish a relation between multipartite quantum states and symplectic toric mani-
folds. These manifolds of dimension 2n are compact connected symplectic manifolds
which have effective hamiltonian actions of n-torus with corresponding moment maps.
In particular, in Sect. 2 we give a short introduction to complex projective varieties.
In Sect. 3 we will review the construction of symplectic toric manifolds. The
H. Heydari (
Physics Department, Stockholm university, 10691 Stockholm, Sweden