Combinatorics

Discrete mathematics

Quantum entanglement

Physics

Pure mathematics

Dimension (graph theory)

Mathematics

LOCC

Quantum nonlocality

Geometry

Converse

Quantum mechanics

Product (mathematics)

Quantum

Physical Review Letters

Quantum nonlocality is usually associated with entangled states by their violations of Bell-type inequalities. However, even unentangled systems, whose parts may have been prepared separately, can show nonlocal properties. In particular, a set of product states is said to exhibit ``quantum nonlocality without entanglement'' if the states are locally indistinguishable; i.e., it is not possible to optimally distinguish the states by any sequence of local operations and classical communication. Here, we present a stronger manifestation of this kind of nonlocality in multiparty systems through the notion of local irreducibility. A set of multiparty orthogonal quantum states is defined to be locally irreducible if it is not possible to locally eliminate one or more states from the set while preserving orthogonality of the postmeasurement states. Such a set, by definition, is locally indistinguishable, but we show that the converse does not always hold. We provide the first examples of orthogonal product bases on ${\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}\ensuremath{\bigotimes}{\mathbb{C}}^{d}$ for $d=3$, 4 that are locally irreducible in all bipartitions, where the construction for $d=3$ achieves the minimum dimension necessary for such product states to exist. The existence of such product bases implies that local implementation of a multiparty separable measurement may require entangled resources across all bipartitions.

Strong Quantum Nonlocality without Entanglement