Homogeneous multiscale-entanglement-renormalization-ansatz states: An information theoretical analysis

Homogeneous multiscale entanglement renormalization ansatz states have been recently introduced to describe quantum critical systems. Here we present an extensive analysis of the properties of such states by clarifying the definition of their transfer superoperator whose structure is studied within an informational theoretical approach. Explicit expressions for computing the expectation values of symmetric observables are given both in the case of finite size systems and in the thermodynamic limit of infinitely many particles.

Figure 1

(Color online) (a) Graphical representation of a typical one-dimensional MERA tensor network [9] for a many-body system of $N=16$ sites. The red (light gray) elements correspond to the disentanglers tensors $\chi$, the blue (dark gray) elements are the isometry tensors $\lambda$, and the green (upper) element $\mathcal{C}$ is the hat of the MERA. Any two joined legs from any two distinct nodes indicate saturation of the associated indices of the corresponding tensor [9]. The dashed line indicates periodic boundary conditions (i.e., the rightmost $\chi$ re-emerge on the left of the graph). Alternative MERA decompositions can be obtained by reordering the links of the graph, e.g., see Ref. [15]. (b) Representation of the contraction rules of Eqs. (2, 3) imposed on $\chi$ and $\lambda$ (here the inverted elements represent their adjoints counterparts).

Figure 2

(Color online) (a) The black elements of the graph represent the tensor ${\mathcal{Q}}_k$ of the causal cone associated with the triple formed by the fifth to seventh sites of the system [see Eq. (7)]. It can be identify by connecting the triple with the MERA’s hat via percolation starting from the bottom of the graph. Thanks to contraction rules (2, 3) the remaining $\chi$’s and $\lambda$’s (in gray) do not contribute when evaluating the expectation values of observables which act locally on the triple. The light box underlines the tensor $\mathcal{M}$ of Eq. (6). Here $N=16$. (b) The two alternative ways in which a tensor $\mathcal{M}$ can enter in ${\mathcal{Q}}_k$: the empty circle represent the links that connect with the neighboring elements of the cone.

Figure 3

(Color online) The black and green (dark gray) elements represent the causal cone structure associated to the product ${\widehat{A}}_k\otimes {\widehat{B}}_{k^{\prime }}$ (here $N=32$, $k=9$, and $k^{\prime }=24$): it is formed by merging the CC of the $k$ triple (black elements on the left) with the CC of the $k^{\prime }$ triple (green elements on the right) which intercept at the $\left(\overline{m}+1\right)\text{th}$ MERA level. The empty circles describe the six-qudit quantum state on which the two single-triple CCs operate on. It is associated with the tensor ${\mathcal{X}}_{k{k}^{\prime }}$ of Eq. (9) and with the density matrix ${\widehat{\sigma }}_{k,k^{\prime }}$ of Eq. (30).

Figure 4

(Color online) (a) The black link represents an example of first neighboring causal shadow of depth 2 associated with three consecutive links of the second layer of an $N=32$-MERA. (b) Tensor of a causal shadow of depth 2 associated with a triple. The insets show its decomposition in terms of products [Eq. (7)]: starting from the top-left corner and moving clockwise we have ${\mathcal{M}}_L\cdot {\mathcal{M}}_L$, ${\mathcal{M}}_R\cdot {\mathcal{M}}_R$, ${\mathcal{M}}_L\cdot {\mathcal{M}}_R$, and ${\mathcal{M}}_R\cdot {\mathcal{M}}_L$.

Figure 5

(Color online) The black and green (dark gray) elements represent the causal shadows ${\text{CS}}_A$ and ${\text{CS}}_B$, respectively. The arrows indicate their leftmost sites $k_A$ and $k_B$ (here $N=32$, $k_A=8$, and $k_B=20$). The empty circles describe the six sites described by the density matrix ${\widehat{\sigma }}^{\left(k_A,{\Delta }_k\right)}$ of Eq. (37).