Localizable Bipartite Rank-One Ideal Measurements Have a Block-Replicated Nice-Bell Structure

Abstract

We prove a complete structural characterization of finite-dimensional bipartite rank-one ideal projective measurements that are localizable without communication in the sense of Akibue and Miyazaki. Up to local-unitary equivalence, every such measurement basis is obtained by replicating a single nice Bell basis across equal-dimensional local subspace blocks. The necessity proof combines the causal block structure of complete measurements established by Beckman, Gottesman, Nielsen, and Preskill with their eigenstate-composition theorem for localizable superoperators. A reference block is first forced to be a nice Bell basis; the same theorem then propagates that basis consistently along every row, column, and interior block. The converse follows from the explicit localization protocol of Akibue and Miyazaki. This establishes their Conjecture 1 and gives a protocol-independent classification of bipartite rank-one ideal measurements in this setting.