Symmetry-protected topological entanglement

We propose an order parameter for the symmetry-protected topological (SPT) phases which are protected by Abelian on-site symmetries. This order parameter, called the SPT entanglement, is defined as the entanglement between $A$ and $B$, two distant regions of the system, given that the total charge (associated with the symmetry) in a third region $C$ is measured and known, where $C$ is a connected region surrounded by $A, B$, and the boundaries of the system. In the case of one-dimensional systems we prove that in the limit where $A$ and $B$ are large and far from each other compared to the correlation length, the SPT entanglement remains constant throughout a SPT phase, and furthermore, it is zero for the trivial phase while it is nonzero for all the nontrivial phases. Moreover, we show that the SPT entanglement is invariant under the low-depth quantum circuits which respect the symmetry, and hence it remains constant throughout a SPT phase in the higher dimensions as well. Also, we show that there is an intriguing connection between SPT entanglement and the Fourier transform of the string order parameters, which are the traditional tool for detecting SPT phases. This leads to an algorithm for extracting the relevant information about the SPT phase of the system from the string order parameters. Finally, we discuss implications of our results in the context of measurement-based quantum computation.

Figure 1

The light cone corresponding to a low-depth circuit. After applying the unitary, information which is initially localized in region $A$, will stay inside region $A^{\prime }$, which is only slightly larger than $A$.

Figure 2

Region $C$ is surrounded by regions $A$ and $B$ and the boundaries of the system.

Figure 3

State ${\mathrm{\Omega }}^{\left(AB\right|C)}\left(\rho \right)$ describes the joint state of the local degrees of freedom at regions $A$ and $B$, and register $K_C$, which corresponds to the total charge in region $C$. The SPT entanglement is defined as the bipartite entanglement of this state relative to the partitions $A|K_{C}B$ or $A{K}_C|B$.

Figure 4

Under the symmetric low-depth circuit $V$ quantum information and charge from region $A$ can leak to region $\mathrm{\Delta }A$, that is a subset of sites in region $C$ which are close to the sites in region $A$. Applying the symmetric unitary $U_A$ on the region $A^{\prime }=A\cup \mathrm{\Delta }A$, Alice can restore the leaked charge and quantum information from region $A$ back to this region.

Figure 5

Isometry $S$ maps virtual subsystems $i_L$ and $i_R$ to the physical site $i$.

Figure 6

Top: We look at the entanglement of regions $A$ and $B$, given the value of charge in region $C$ is known. Here each isometry $S$ maps two virtual systems to a physical system. Middle: Applying local isometry $S$ does not change entanglement, or charge. Therefore, we can ignore the isometries and look at the charge and entanglement at the level of the virtual systems. Bottom: For each pair of virtual systems in state $|\lambda \rangle$ the total charge is zero. Thus, if both pairs are inside region $C$ then they do not contribute in the total charge in this region, and therefore we can ignore them. Furthermore, all the virtual subsystems in $A$ or $B$, except the ones at the boundary, are uncorrelated with the virtual systems in the other regions. Therefore, since we are only interested in the entanglement between the two regions, we can ignore them. The resulting state is state ${\sigma }^{\left(ab\right)}$ defined in proposition 7, where $a$ and $b$ correspond, respectively, to the virtual systems $a_R$ and $b_L$ in this figure.

Figure 7

By consuming the maximally entangled state ${|\mathrm{\Theta }\rangle }_{Z_a{Z}_b}$, Alice and Bob can measure the total charge in systems $a^{\prime }$ and $b^{\prime }$, and thereby prepare state ${\sigma }^{\left(ab\right)}$, which is equivalent to state ${\mathrm{\Omega }}^{\left(AB\right|C)}\left(\mathrm{\Psi }\right)$ up to local operations. The idea is that by adding systems $Z_a$ to the Alice's side and $Z_b$ to the Bob's side, the representations of symmetry $G$ on both systems $a^{\prime }{Z}_a$ and $b^{\prime }{Z}_b$ become nonprojective. Therefore, Alice and Bob can locally measure the charges in $a^{\prime }{Z}_a$ and $b^{\prime }{Z}_b$. But, because the total charge in ${|\mathrm{\Theta }\rangle }_{Z_a{Z}_b}$ is zero, the total charge in $a^{\prime }{b}^{\prime }$ is equal to the sum of the charges in $a^{\prime }{Z}_a$ and $b^{\prime }{Z}_b$, denoted by ${\kappa }_A$ and ${\kappa }_B$ in the above figure.

Figure 8

Region $A^{\prime }$ is partitioned to region $A$ and region $\mathrm{\Delta }A$. Similarly, region $B^{\prime }$ is partitioned to region $B$ and region $\mathrm{\Delta }B$. Region $C^{\prime }\equiv C\setminus \left(A^{\prime }\cup B^{\prime }\right)$ is the subregion of $C$ surrounded by $A^{\prime }, B^{\prime }$, and the boundaries of the system.