Symmetries and local transformations of translationally invariant matrix product states

We determine the local symmetries and local transformation properties of certain many-body states called translationally invariant matrix product states (TIMPSs). We focus on physical dimension $d=2$ of the local Hilbert spaces and bond dimension $D=3$ and use the procedure introduced in Sauerwein et al. [Phys. Rev. Lett. 123, 170504 (2019)] to determine all (including nonglobal) symmetries of those states. We identify and classify the stochastic local operations assisted by classical communication (SLOCC) that are allowed among TIMPSs. We scrutinize two very distinct sets of TIMPSs and show the big diversity (also compared to the case $D=2$) occurring in both their symmetries and the possible SLOCC transformations. These results reflect the variety of local properties of MPSs, even if restricted to translationally invariant states with low bond dimension. Finally, we show that states with nontrivial local symmetries are of measure zero for $d=2$ and $D>3$.

Figure 1

Illustration of the group structure of the cycles for normal MPSs generated by the fiducial states $\mathbb{1}\otimes b\otimes \mathbb{1}\left|M\right(\omega )\rangle$. The depicted labels correspond to the cycles as given in Table 2. A line connecting two labels indicates that the symmetry group associated to the higher elevated cycle is a subgroup of the symmetry group associated to the cycle below.

Figure 2

(a) Set-theoretic sketch (Euler diagram) of $b$'s leading to nontrivial cycles in $G_b$ [considering the fiducial state $\mathbb{1}\otimes b\otimes \mathbb{1}\left|M\right(\omega )\rangle ]$. The sketch illustrates the inclusion relations and intersections between the different sets of $b$'s leading to certain cycles as labeled in Table 2. Let us point out a few of them. For instance, one sees that the set $b\left(C_3\right)$ is a union of the (pairwise) nonintersecting sets $b\left(T_6^{\tau }\right), b\left(T_6^{\epsilon }\right)$, and $b\left(T_6^{\kappa }\right)$, and each of $b\left(T_6^{\sigma }\right)$ is a subset of $b\left(T_0^{\sigma }\right)$ for $\sigma \in \{\tau ,\epsilon ,\kappa \}$. This illustrates that, e.g., there does exist a $b$ s.t. $G_b$ exhibits the three cycles $T_6^{\tau }, T_0^{\tau }$, and $C_3$, but there does not exist any $b$ leading to the cycle $C_3$ only. We find that $b\left(T_0^{\tau }\right), b\left(T_0^{\epsilon }\right), b\left(T_0^{\kappa }\right)$, and $b\left(C_0\right)$ intersect. However, the three sets $b\left(T_i^{\tau }\right), b\left(T_j^{\epsilon }\right)$, and $b\left(T_k^{\kappa }\right)$ are disjoint for any $i,j,k\in \{1,2,3,6,7\}$. In fact, $b\left(T_3^{\tau ,\epsilon ,\sigma }\right)$ are completely isolated in the sense that they do not intersect with any other family. The set $b\left(T_1^{\sigma }\right)$ is a subset of $b\left(T_0^{\sigma }\right)$ (for $\sigma \in \{\tau ,\epsilon ,\kappa \}$), but does not intersect with any other family. Note that the sketch does not correctly represent the geometry or sizes of the sets. As explained in the paper, the diagram gives as a complete characterization of symmetries of all normal TIMPSs generated by fiducial states $\mathbb{1}\otimes b\otimes \mathbb{1}\left|M\right(\omega )\rangle$. While some of the displayed inclusion relations follow immediately considering the group structure of the cycles as in Fig. 1, additional relations reveal themselves only considering the full characterization of $b$ as in Table 3. An instance of the latter case would be one of the example mentioned above—the fact that there exists no $b$ s.t. $G_b$ possesses the cycle $C_3$, but does not possess any of the cycles $T_6^{\tau }, T_6^{\epsilon }$, and $T_6^{\kappa }$. (b) Detailed view of an aspect of (a).

Figure 3

Sketch of the domain of ${\chi }_k, \mathcal{D}$. $\mathcal{D}$ comprises complex numbers in the upper half of the complex plane which have absolute value larger than or equal to one, excluding the negative real axis and, moreover, excluding numbers with negative real part whose absolute value equals 1. The imaginary unit $i$ is included in $\mathcal{D}$.

Figure 4

Flowchart showing the characterization of the symmetries of all normal MPSs generated by $\mathbb{1}\otimes b\otimes \mathbb{1}|LLT\rangle$, i.e., for any $b$ with $b_{20}\ne 0$ (w.l.o.g. $b_{20}=1$) and $\mathbf{v}\ne \mathbf{0}$ (shaded rectangles with solid contour); cf. Observation t14. For any such $b$, the symmetries of the corresponding MPS may be determined by calculating $T$ and $\mathbf{v}$ as in Eqs. (22) and (23) and then following the procedure described in the paper, which is shown in the flowchart. Additionally, the flowchart also shows the cycles in $G_b$ obtained for nonnormal MPSs generated by $b$ such that $b_{20}=1$ and $\mathbf{v}=\mathbf{0}$ (shaded rectangles with dashed contour). Note that for nonnormal MPSs the symmetry group might be larger than displayed, as the utilized methods may fail to identify the full symmetry group (and yield a subgroup instead). Here “no sym.” indicates that the corresponding MPS possesses only the trivial symmetry. Generic $b$ belong to box VI, as indicated in the flowchart.

Figure 5

Summary of the SLOCC classification of all normal MPSs generated by $\mathbb{1}\otimes b\otimes \mathbb{1}|LLT\rangle$ (shaded rectangles with solid contour) plus partial results on SLOCC classes of some nonnormal MPSs (shaded rectangles with dashed contour). We display the number of SLOCC classes corresponding to each type and give representatives for every SLOCC class (we count complex parameters). To simplify the presentation not all redundancies in the representatives of the continuous SLOCC families are avoided. Redundancy may be removed straightforwardly though. The flowchart is following the same structure as the one in Fig. 4, in particular, the displayed type labels agree. Note that additional nonnormal MPSs that have not been identified as SLOCC equivalent might in fact be equivalent. The displayed nonnormal MPSs vanish in case of odd $N$.