Entanglement does not in general imply Einstein-Podolsky-Rosen steering. We identify a boundary-geometric mechanism that closes this gap on product-null boundary strata of two-qubit state space, where Bob's conditional states touch the boundary of the Bloch ball. The key obstruction is local: if a projective assemblage approaches a Bloch-sphere boundary contact with a first-order tangential displacement but only a second-order inward defect, then no finite-measure local-hidden-state model can reproduce it. For two-qubit states with a product vector in the kernel, this boundary contact is exactly the tangency of Bob's steering ellipsoid to the Bloch sphere. At such a product-null tangency, a single tangential coherence controls both partial-transpose negativity and the boundary-contact scaling obstruction. The same boundary minor gives a compact experimental witness: once the product-null contact is verified or guaranteed, the tangential coherence supplies the steering signal. Consequently, every entangled two-qubit rank-two state, and every entangled rank-three state whose null space is spanned by a product vector, is two-way projectively steerable. The same boundary idea extends to arbitrary steering cuts: the Bloch-sphere contact is replaced by a rank-deficient trusted conditional state, and support-kernel first-order coherence implies both NPT entanglement and projective steering.
Entanglement does not in general imply Einstein-Podolsky-Rosen steering. We identify a boundary-geometric mechanism that closes this gap on product-null boundary strata of two-qubit state space, where Bob's conditional states touch the boundary of the Bloch ball. The key obstruction is local: if a projective assemblage approaches a Bloch-sphere boundary contact with a first-order tangential displacement but only a second-order inward defect, then no finite-measure local-hidden-state model can reproduce it. For two-qubit states with a product vector in the kernel, this boundary contact is exactly the tangency of Bob's steering ellipsoid to the Bloch sphere. At such a product-null tangency, a single tangential coherence controls both partial-transpose negativity and the boundary-contact scaling obstruction. The same boundary minor gives a compact experimental witness: once the product-null contact is verified or guaranteed, the tangential coherence supplies the steering signal. Consequently, every entangled two-qubit rank-two state, and every entangled rank-three state whose null space is spanned by a product vector, is two-way projectively steerable. The same boundary idea extends to arbitrary steering cuts: the Bloch-sphere contact is replaced by a rank-deficient trusted conditional state, and support-kernel first-order coherence implies both NPT entanglement and projective steering.
We prove that every rank-2 bipartite entangled state is already steerable with projective measurements, closing the first possible gap between entanglement and Einstein-Podolsky-Rosen (EPR) steering. In arbitrary finite local dimensions, any rank-2 entangled state is projectively steerable in at least one direction; when the effective dimensions are equal, steering is two-way. The argument is geometric: a rank obstruction forces some projective outcome on the larger local system to reach the boundary of the trusted-state space. From this boundary contact, a neighboring measurement produces a linear displacement that cannot be reproduced by a local-hidden-state model, because positivity only permits a quadratic filling into a kernel direction. This same block witnesses partial-transpose entanglement. If the contact is degenerate, a Schur complement removes one product layer and reduces the problem to a lower-rank entangled residual; each such reduction lowers the rank, so the recursion terminates. More broadly, we derive a directional rank criterion: if an m⊗n entangled state ρ satisfies $\rankρ\le 1+\lfloor(m-1)/(n-1)\rfloor$, then it is projectively steerable from A to B. Since counterexamples already occur at rank three, rank two is the unique mixed rank for which entanglement guarantees projective steering, and the proof provides an operational certificate based solely on the support and kernel of the density matrix.
Abstract For general mixed states, entanglement does not guarantee steerability. But this separation disappears when the density matrix has sufficiently low rank. We establish a unifying theorem: every bipartite entangled state whose rank does not exceed the larger local dimension is steerable in both directions. At every higher rank, there exist entangled states that are unsteerable in both directions under arbitrary measurements. This bound covers all rank-2 entangled states. To prove this, we construct a finite steering inequality based on the relation between coherence and measurement probabilities, and show that every entangled state satisfying the rank bound can violate it in both directions. Testing this inequality requires only two fixed measurement bases on the trusted side, while the other party can use either projective measurements or a fixed interferometer with controlled attenuation.
Quantum nonlocality is an essential resource in quantum information and is commonly classified into three distinct forms: quantum entanglement, Einstein-Podolsky-Rosen (EPR) steering, and Bell's nonlocality. Gisin's theorem shows that pure-state entanglement implies Bell nonlocality, and it motivates the question of how entanglement and EPR steering are related beyond pure states. Here we prove that every entangled rank-2 two-qubit state is EPR steerable, with rank-1 states included as a special case. The proof combines a local-unitary parametrization of rank-2 two-qubit states, a necessary and sufficient separability condition, and a state-dependent nonlinear steering inequality. This gives an analytical characterization of EPR steerability for rank-2 two-qubit entangled states. Thus, rank-2 two-qubit entangled states provide certifiable EPR-steering resources in quantum information.
We show that the spectral data of a two-qubit state does not determine whether it isEinstein-Podolsky-Rosen steerable, already at rank three. On the slice of states withmaximally mixed marginal on the untrusted side - the normalized Choi states of qubitchannels - the two-qubit reduction/partial-transpose relation forces the partial-transposespectrum to be an affine image of the ordinary spectrum, so that isospectral pairs on thisslice are automatically iso-partial-transpose and have equal negativity. We then exhibit two exact pairs. The first is a one-parameter family of pairs with identicalspectra, identical partial-transpose spectra and equal negativity, one member of which isunsteerable for arbitrary POVMs while the other is two-way projectively steerable; thedifference lies entirely in whether the kernel is spanned by an entangled or a product vector.The second pair is stronger: two states sharing not merely the kernel type but the identicalkernel projector, the same spectrum, the same partial-transpose spectrum and the samenegativity, of which one carries an explicit local-hidden-state model for all projectivemeasurements with exact rational slack 7653/985000, while the other violates a three-settinglinear steering inequality with exact value sqrt(145156269/141610000). Consequently no function of the eigenvalues of rho and rho^{T_B} - including all moments,purity, entropies, rank and negativity - and no such function augmented by the full kernelprojector, can classify steering at rank three. As an application we exhibit an exactcounterexample, valid on a continuous interval of width 0.0116713828..., to the unrestrictedhigher-rank extension of a recently proposed Peres-type steering criterion; the provedWerner, pure-state and rank-two results of that criterion are untouched. The operativemissing coordinate is the orientation of the positive eigenspaces within the support relativeto the tensor-product structure. All stated constants are reproduced in exact arithmetic by the verification script includedin the deposit. The deposit also contains a claim ledger mapping every claim to its status,evidence and external dependency, and an explicit DOES_NOT_PROVE statement of scope limits. Certification note. Results were developed and independently re-derived across two separateautomated reasoning venues under a preregistration and dual-verification protocol, with everystated constant replayed in exact arithmetic in a second venue before inclusion. The depositincludes the replay script, its captured output, a machine-readable claim/dependency ledger,a clean build log, and SHA-256 checksums for every file. Scope. Theorem 2 and its corollary are unconditional, resting only on peer-reviewed criteria(Bowles-Hirsch-Quintino-Brunner, Phys. Rev. A 93, 022121 (2016); Cavalcanti-Jones-Wiseman-Reid,Phys. Rev. A 80, 032112 (2009)). Theorem 1 is conditional on two declared 2026 preprints(arXiv:2607.27757, arXiv:2605.21245) and is labelled as such in the manuscript.
Bound-entangled states with positive partial transpose (PPT) can exhibit quantum steering and Bell nonlocality, disproving the long-standing Peres conjecture and its stronger steering counterpart. These counterexamples, obtained through explicit constructions or numerical searches, raise the question of whether they share a common structural origin. We address this question in the low-rank regime by asking whether known counterexamples can be obtained, up to invertible local filtering, from the complement of an unextendible product basis (UPB). We find that the answer depends crucially on the local dimension. In 3⊗3, the rank-four structure of PPT entangled states implies that both the steerable PPT state of Moroder \textit{et al.} and the Bell-nonlocal PPT state of Vértesi and Brunner can be obtained from UPB-complement states under invertible local filtering. For three qubits, in contrast, rank-four states that are PPT across every bipartition are separable across each cut. Moreover, comparison of the product-vector structure of the relevant kernels shows that the known three-qubit Bell-nonlocal PPT states cannot be obtained by invertible local filtering from the complement of any three-qubit UPB. Going beyond rank four, we further show that a known rank-eight three-qutrit state, supported on the complement of a UPB and PPT across every bipartition, is genuinely multipartite entangled. These results show that UPBs provide a unifying structural origin for important low-rank PPT entangled states in 3⊗3, but that nonlocal bound entanglement is not, in general, a UPB phenomenon.
Quantum inseparabilities can be classified into three inequivalent forms: entanglement, Einstein-Podolsky-Rosen (EPR) steering, and Bell's nonlocality. Bell-nonlocal states form a strict subset of EPR steerable states which also form a strict subset of entangled states. Recently, EPR steerable states have been shown to be fundamental resources for one-sided device-independent quantum information processing tasks and, hence, identification of EPR steerable states becomes important from foundational as well as informational theoretic perspectives. In the present paper, we propose a criterion to detect whether a given two-qubit state is EPR steerable. From an arbitrary given two-qubit state, another two-qubit state is constructed in such a way that the given state is EPR steerable if the new constructed state is entangled. Hence, EPR steerability of an arbitrary two-qubit state can be detected by detecting entanglement of the newly constructed state. Apart from providing a distinctive way to detect EPR steering without using any steering inequality, the present paper also provides an alternative way to avoid locality loopholes in EPR steering tests and to reduce the ``complexity cost'' present in experimentally detecting EPR steering. We also generalize our criteria to detect EPR steering of higher dimensional quantum states. Finally, we illustrate our result by using our proposed technique to detect EPR steerability of various families of mixed states.