Information-theoretical analysis of topological entanglement entropy and multipartite correlations

A special feature of the ground state in a topologically ordered phase is the existence of large-scale correlations depending only on the topology of the regions. These correlations can be detected by the topological entanglement entropy or by a measure called irreducible correlation. We show that these two measures coincide for states obeying an area law and having zero correlation length. Moreover, we provide an operational meaning for these measures by proving its equivalence to the optimal rate of a particular class of secret sharing protocols. This establishes an information-theoretical approach to multipartite correlations in topologically ordered systems.

Figure 1

The illustration shows the regions $A,B,C$ used for the calculation of the TEE and the irreducible correlation. Region $\left(\text{a}\right)$ and $\left(\text{b}\right)$ correspond to the Kitaev-Preskill–type version [5] and $\left(\text{c}\right)$ to the Levin-Wen–type version [6] of the TEE. Due to the difference of the topology of the regions, the value of the TEE for $\left(\text{a}\right)$ is half of that for $\left(\text{b}\right)$ and $\left(\text{c}\right)$.

Figure 2

A geometrical illustration of the functions $D^{\left(k\right)}$ and $C^{\left(k\right)}$. $D^{\left(k\right)}\left(\rho \right)$ is the distance from the set of $k$-correlated states, and $C^{\left(k\right)}$ is the difference between crossing points in ${\overline{Q}}_k$ and ${\overline{Q}}_{k-1}$ measured by the quantum relative entropy.

Figure 3

A graphical illustration of how to divide the regions $A,B,C$ from Fig. 1 in the proof of Theorem t1. Note that in any configuration $\left(a\right),\left(b\right),$ or $\left(c\right)$, the RDMs of three consecutive subregions are QMSs according to assumption (II).

Figure 4

The upper illustration shows the merge of two local QMSs into one global QMS, which is consistent with the local original QMSs. The lower illustration shows the merge of six local QMSs in a cyclic way as used in the proof of Theorem t1 for regions as in Figs. 1 and 1.