The paper focuses on the efficient preparation of matrix-product states (MPS) in constant depth using measurements and classical communication, a process referred to as MPS fusion. The authors establish that MPS fusion is possible only for states with a flat entanglement spectrum, such as the Affleck-Kennedy-Lieb-Tasaki (AKLT) state. They introduce a more general method called split-index MPS (SIMPS) fusion, which can prepare states with non-onsite symmetries, including those protected by non-onsite symmetries and serving as resources for long-range quantum teleportation. SIMPS fusion is shown to be more efficient than MPS fusion in certain cases, reducing resource overhead. The paper also discusses the limitations of MPS fusion and provides a detailed comparison between MPS and SIMPS fusion, highlighting the advantages of SIMPS fusion in handling states with non-onsite symmetries.The paper focuses on the efficient preparation of matrix-product states (MPS) in constant depth using measurements and classical communication, a process referred to as MPS fusion. The authors establish that MPS fusion is possible only for states with a flat entanglement spectrum, such as the Affleck-Kennedy-Lieb-Tasaki (AKLT) state. They introduce a more general method called split-index MPS (SIMPS) fusion, which can prepare states with non-onsite symmetries, including those protected by non-onsite symmetries and serving as resources for long-range quantum teleportation. SIMPS fusion is shown to be more efficient than MPS fusion in certain cases, reducing resource overhead. The paper also discusses the limitations of MPS fusion and provides a detailed comparison between MPS and SIMPS fusion, highlighting the advantages of SIMPS fusion in handling states with non-onsite symmetries.