Two new ways to define 'almost identical' quantum states

Most quantum information protocols assume sources produce perfectly independent, identical states — an idealization that rarely holds in practice. This work introduces two alternative ways to quantify how close a multi-party quantum state is to being i.i.d.: one using the normalized quantum Wasserstein distance, another by averaging over k-body marginals. The three competing definitions, including a recent one by Mazzola et al., are shown to form a strict hierarchy from most to least demanding, with explicit examples confirming the separations. The framework clarifies which physical scenarios each definition captures and lays groundwork for more robust quantum information theorems.