Matrix product states and projected entangled pair states: Concepts, symmetries, theorems

The theory of entanglement provides a fundamentally new language for describing interactions and correlations in many-body systems. Its vocabulary consists of qubits and entangled pairs, and the syntax is provided by tensor networks. How matrix product states and projected entangled pair states describe many-body wave functions in terms of local tensors is reviewed. These tensors express how the entanglement is routed, act as a novel type of nonlocal order parameter, and the manner in which their symmetries are reflections of the global entanglement patterns in the full system is described. The manner in which tensor networks enable the construction of real-space renormalization group flows and fixed points is discussed, and the entanglement structure of states exhibiting topological quantum order is examined. Finally, a summary of the mathematical results of matrix product states and projected entangled pair states, highlighting the fundamental theorem of matrix product vectors and its applications, is provided.

Figure 1

Construction of projected entangled pair states on a 2D lattice.

Figure 2

Tensor network description of (a) a MPS, (b) a PEPS on a square lattice, and their corresponding marginals.

Figure 3

Definitions of (a) matrix product operators (MPOs) and (b) projected entangled pair operators (PEPOs).

Figure 4

Sufficient condition for a translationally invariant MPO $\rho$ to be positive.

Figure 5

(a) Tensors generating a MPU after blocking. (b) The MPU can be described as a quantum circuit, with alternating layers of unitary operators $u$ and $v$ acting on nearest neighbors with even-odd indices and odd-even indices, respectively.

Figure 6

(a) MPU representation of the left-moving shift operator. (b) MPU made of two shift operators, one right moving and another left moving.

Figure 7

A sufficient condition for two PEPSs to be equal to each other is the existence of a MPO satisfying the pulling through equation.

Figure 8

(a) Tensor network description of the reduced density matrix of $n$ spins in an infinite translationally invariant MPS. $|{\rho }^{R,L}⟩$ correspond to the right and left fixed points, respectively, of the transfer matrix; see the main text. (b) Argument showing that the eigenvalues of the reduced density matrix of $n$ sites in a MPS coincide with those of ${({\rho }^L{\rho }^R)}^{\otimes 2}$ in the limit of large $n$.

Figure 9

(a) Staircase quantum circuit representing a MPS. (b) Tree tensor network.

Figure 10

Construction of a translation invariant MPO approximation to infinite translation invariant ground states by truncating the bond dimension in an intermediate region ([308]). The resulting MPO ${\rho }_k$ for $k$ sites is then patched together and superimposed with its translations.

Figure 11

For a given MPS, one considers the two possible canonical forms, given by tensors $A_L$ and $A_R$, where the right and left fixed points of the transfer operator, respectively, are the identities; see Sec. 4 for more details. They are distinguished by the direction of the triangle. They allow one to express the tangent space projector $P_{T(A)}$ as a sum of MPOs. The variable $i$ in the sum corresponds to the position of the central (blue) index.

Figure 12

Bulk-boundary correspondence. For any given region $R$, ${\rho }_R$ can be isometrically mapped to a density operator ${\sigma }_{\partial R}$ that is defined in the auxiliary indices of the boundary of $R$ via a polar decomposition; see the main text.

Figure 13

Examples of different possibilities for RGFPs in a 2D square configuration, where (a) a unitary operation is applied to the physical indices of four spins and invertible operations are applied to the auxiliary ones. (b) One can disentangle several spins, i.e., obtain the original tensor and other ones as product states. (c) One can obtain the original tensor and other generating MPSs. (d) One can obtain two copies of the original tensor.

Figure 14

Inverse renormalization procedure. (a) For MPSs, if $E$ is divisible, one can rewrite the tensor in terms of two tensors of the same bond dimension and iterate the procedure. (b) For PEPSs in two dimensions, the same step would imply that the bond dimension of the new tensor has to be square rooted.