Nonprojective Bell-state measurements

The Bell-state measurement (BSM) is the projection of two qubits onto four orthogonal maximally entangled states. Here we first propose how to appropriately define more general BSMs, which have more than four possible outcomes, and then study whether they exist in quantum theory. We observe that nonprojective BSMs can be defined in a systematic way in terms of equiangular tight frames of maximally entangled states, i.e., a set of maximally entangled states, where every pair is equally, and in a sense maximally, distinguishable. We show that there exists a five-outcome BSM through an explicit construction and find that it admits a simple geometric representation. Then we prove that there exists no larger BSM on two qubits by showing that no six-outcome BSM is possible. We also determine the most distinguishable set of six equiangular maximally entangled states and show that it falls only somewhat short of forming a valid quantum measurement. Finally, we study the nonprojective BSM in the contexts of both local state discrimination and entanglement-assisted quantum communication. Our results put forward natural forms of nonprojective joint measurements and provide insight into the geometry of entangled quantum states. Published by the American Physical Society 2024