Multipartite quantum systems and symplectic toric manifolds

A toric varieties approach to geometrical structure of multipartite states

We investigate the geometrical structures of multipartite states based on construction of toric varieties. In particular, we describe pure quantum systems in terms of affine toric varieties and projective embedding of these varieties in complex projective spaces. We show that a quantum system can be corresponds to a toric variety of a fan which is constructed by gluing together affine toric varieties of polytopes. Moreover, we show that the projective toric varieties are the spaces of separable multipartite quantum states. The construction is a generalization of the complex multi-projective Se…

复曲面簇：多体量子系统的简单组合和几何结构

Toric varieties: Simple combinatorial and geometrical structure of multipartite quantum systems

我们基于复曲面簇的构造研究了多体态的几何结构。我们证明了复曲面簇表示一般纯态空间，而投射复曲面簇定义了多比特态的可分集空间。我们还详细讨论了单比特、双比特、三比特和多比特态的构造.这种构造给出了多体量子系统的几何结构的非常简单和优雅的视觉表示。

Multipartite entanglement of the topologically ordered state in a perturbed toric code

We demonstrate that multipartite entanglement, witnessed by the quantum Fisher information (QFI), can characterize topological quantum phase transitions in the spin-$\frac{1}{2}$ toric code model on a square lattice with external fields. We show that the QFI density of the ground state can be written in terms of the expectation values of gauge-invariant Wilson loops for different sizes of square regions and identify $\mathbb{Z}_2$ topological order by its scaling behavior. Furthermore, we use this multipartite entanglement witness to investigate thermalization and disorder-assisted stabilizati…

Geometrical structures of multipartite quantum systems

In this paper I will investigate geometrical structures of multipartite quantum systems based on complex projective varieties. These varieties are important in characterization of quantum entangled states. In particular I will establish relation between multi-projective Segre varieties and multip-qubit quantum states. I also will discuss other geometrical approaches such as toric varieties to visualize complex multipartite quantum systems.

Quantum entangled state, resolution of conifold singularity, and toric variety

We investigate the geometrical and combinatorial structures of multipartite quantum systems based on conifold and toric variety. In particular, we study the relations between resolution of conifold, toric variety, a separable state, and a quantum entangled state for bipartite and multi-qubit states. For example we show that the resolved or deformed conifold is equivalent with the space of a pure entangled two-qubit state. We also generalize this result into multi-qubit states.

Multipartite quantum correlations: symplectic and algebraic geometry approach

We review a geometric approach to classification and examination of quantum correlations in composite systems. Since quantum information tasks are usually achieved by manipulating spin and alike systems or, in general, systems with a finite number of energy levels, classification problems are usually treated in frames of linear algebra. We proposed to shift the attention to a geometric description. Treating consistently quantum states as points of a projective space rather than as vectors in a Hilbert space we were able to apply powerful methods of differential, symplectic and algebraic geomet…