Absolutely maximally entangled states in tripartite heterogeneous systems

Absolutely maximally entangled (AME) states are closely related to quantum error correction codes. They are typically defined in homogeneous systems. However, the heterogeneous system is very common in a practical setup. In this work, we focus on the AME states in tripartite heterogeneous systems. We first introduce irreducible AME states as the basic elements to construct AME states in the systems with large local dimensions. Then, we introduce an array called magic solution array and show it is related to the AME states in the systems whose local dimensions are l, m, n with 3 ≤ l< m< n≤ m+ l- 1. Furthermore, we identify in which kinds of heterogeneous systems the AME states are irreducible. We also indicate some applications of our results. First, we propose a protocol to prepare a kind of tripartite heterogeneous AME states. Second, we present a method to construct k-uniform states of more parties from two AME states. In addition, we establish the connection between heterogeneous AME states and multi-isometry matrices, and apply heterogeneous AME states to realize quantum steering.