Absolutely maximally entangled states in tripartite heterogeneous systems

Absolutely maximally entangled (AME) states are typically defined in homogeneous systems. However, the quantum system is more likely to be heterogeneous in a practical setup. In this work we pay attention to the construction of AME states in tripartite heterogeneous systems. We first introduce the concept of irreducible AME states as the basic elements to construct AME states with high local dimensions. Then we investigate the tripartite heterogeneous systems whose local dimensions are $l,m,n$, with $3\leq l<m<n\leq m+l-1$. We show the existence of AME states in such heterogeneous systems is related to a kind of arrays called magic solution array. We further identify the AME states in which kinds of heterogeneous systems are irreducible. In addition, we propose a method to construct $k$-uniform states of more parties using two existing AME states. We also build the connection between heterogeneous AME states and multi-isometry matrices, and indicate an application in quantum steering.