This work gives a detailed investigation of matrix product state (MPS) representations for pure multipartite quantum states. We determine the freedom in representations with and without translation symmetry, derive respective canonical forms and provide efficient methods for obtaining them. Results on frustration free Hamiltonians and the generation of MPS are extended, and the use of the MPS-representation for classical simulations of quantum systems is discussed.

We give a classification of gapped quantum phases of one-dimensional systems in the framework of Matrix Product States (MPS) and their associated parent Hamiltonians, for systems with unique as well as degenerate ground states, and both in the absence and presence of symmetries. We find that without symmetries, all systems are in the same phase, up to accidental ground state degeneracies. If symmetries are imposed, phases without symmetry breaking (i.e., with unique ground states) are classified by the cohomology classes of the symmetry group, this is, the equivalence classes of its projective representations, a result first derived in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011); arXiv:1008.3745]. For phases with symmetry breaking (i.e., degenerate ground states), we find that the symmetry consists of two parts, one of which acts by permuting the ground states, while the other acts on individual ground states, and phases are labelled by both the permutation action of the former and the cohomology class of the latter. Using Projected Entangled Pair States (PEPS), we subsequently extend our framework to the classification of twodimensional phases in the neighborhood of a number of important cases, in particular systems with unique ground states, degenerate ground states with a local order parameter, and topological order. We also show that in two dimensions, imposing symmetries does not constrain the phase diagram in the same way it does in one dimension. As a central tool, we introduce the isometric form, a normal form for MPS and PEPS which is a renormalization fixed point. Transforming a state to its isometric form does not change the phase, and thus, we can focus on to the classification of isometric forms.

The projected entangled pair state (PEPS) representation of quantum states on two-dimensional lattices induces an entanglement based hierarchy in state space. We show that the lowest levels of this hierarchy exhibit a very rich structure including states with critical and topological properties. We prove, in particular, that coherent versions of thermal states of any local 2D classical spin model correspond to such PEPS, which are in turn ground states of local 2D quantum Hamiltonians. This correspondence maps thermal onto quantum fluctuations, and it allows us to analytically construct critical quantum models exhibiting a strict area law scaling of the entanglement entropy in the face of power law decaying correlations. Moreover, it enables us to show that there exist PEPS which can serve as computational resources for the solution of NP-hard problems. The concept of entanglement plays a central role in both fields of quantum information theory and of strongly correlated systems. In quantum information theory it lies at the heart of many applications and it is viewed as a resource for various information processing tasks. In condensed matter theory, entanglement is one of the roots for the notorious complexity of quantum many-body systems: its presence necessitates a description within an exponentially growing Hilbert space and it is intimately connected with many of the fascinating properties which quantum matter can exhibit at small temperatures.Many fundamental questions arise at the crossing of these fields: how is entanglement related to the power of quantum computation on the one hand, and the difficulties of classical simulations on the other? What is the scaling of the entanglement entropy in spin systems, its relation to criticality, and the appearance of topological quantum order? All these questions can be addressed very easily within the framework of so-called projected entangled pair states (PEPS)-this is the intention of this Letter. We will see, in particular, that all the above-mentioned properties emerge naturally already within the simplest classes of PEPS which include cluster, toric code, and resonating valence bond states. This will enable us to settle a recent debate about the relation between criticality and entropy scaling, and it allows us to find computational resources for the solution of NP-hard problems. The central tool of the Letter is a general correspondence between thermal states of classical 2D spin models and 2D quantum states with a simple PEPS representation. This correspondence substitutes thermal by quantum fluctuations while preserving the correlations, and it thus allows to map temperature driven classical phase transitions to zero-temperature quantum phase transitions.We begin by recalling the PEPS formalism, which was introduced in the context of numerical renormalization group methods for simulating strongly correlated quantum spin systems [1,2]. PEPS can be viewed as generalizations of the Affleck, Kennedy, Lieb, and Tasaki (AKLT) valence bond solids [3] to arbitrar...

We introduce a framework for characterizing Matrix Product States (MPS) and Projected Entangled Pair States (PEPS) in terms of symmetries. This allows us to understand how PEPS appear as ground states of local Hamiltonians with finitely degenerate ground states and to characterize the ground state subspace. Subsequently, we apply our framework to show how the topological properties of these ground states can be explained solely from the symmetry: We prove that ground states are locally indistinguishable and can be transformed into each other by acting on a restricted region, we explain the origin of the topological entropy, and we discuss how to renormalize these states based on their symmetries. Finally, we show how the anyonic character of excitations can be understood as a consequence of the underlying symmetries.