Every Rank-Two Entangled State is Projectively Steerable

Abstract:Pure entangled states are already steerable by suitable projective measurements within their Schmidt supports, whereas rank two is the first genuinely mixed rank at which entanglement and Einstein--Podolsky--Rosen steering could bifurcate. We prove that this bifurcation does not occur even under the restricted measurement class of projective measurements: every rank-two bipartite entangled state in arbitrary finite local dimensions is projectively steerable in at least one direction, and is two-way projectively steerable when the effective local dimensions are equal. The proof is a boundary theorem rather than a steering-inequality optimization or a two-qubit reduction. A dimension--rank obstruction forces a projective outcome on the larger effective party to hit a pure boundary point of the trusted state cone. At such a contact, a nonzero support--kernel tangent block is simultaneously an NPT minor and a projective-steering certificate; if the first contact is degenerate, the rank-two constraint closes that branch and forces a second, active contact. Thus the entire first mixed-state stratum is a complete projective-steering sector, and the same support--kernel geometry gives a practical low-rank certificate whenever the relevant boundary contact is nondegenerate.

arXiv:2606.08189 [quant-ph]

(or arXiv:2606.08189v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2606.08189 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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