Undecidability of MM-QFAs Language Equivalence Problem

Let $L_{>\lambda }(\mathcal{A})$ and $L_{\ge \lambda }(\mathcal{A})$ be the languages recognized by {\em measure many 1-way quantum finite automata (MM-QFA)} (or,{\em enhanced 1-way quantum finite automata(EQFA)}) $\mathcal{A}$ with strict and non-strict cut-point $\lambda$, respectively. We consider the language equivalence problem and show the following

1. both strict and non-strict language equivalence are undecidable;
2. we provide an another proof of the undecidability of non-strict and strict emptiness of MM-QFA and EQFA, and then reducing the language equivalence problem to emptiness problem;
3. lastly, we obtain some other properties which can be derived from the above results.