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arXiv:2606.08189

Every Rank-2 Bipartite Entangled State is Projectively Steerable

Yu-Xuan Zhang, Jing-Ling Chen

quant-phSubmitted Jun 6, 2026• Announced Jun 9, 2026• 0 citations• Revised Aug 21, 2026 (v3)
v1 · Jun 6, 2026v2 · Jun 29, 2026v3 · Aug 21, 2026
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Abstract

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 $m\otimes n$ 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.

Comments
Main 5 pages + SM 12 pages, 0 figure. Second revised version

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Quantum nonlocality manifests in multipartite systems through entanglement, Bell's nonlocality, and Einstein-Podolsky-Rosen (EPR) steering. While Peres's positive-partial-transpose criterion provides a simple and powerful test for entanglement, a comparably elegant spectral criterion for detecting EPR steering remains an open challenge. In this work, we systematically explore whether a Peres-type criterion can be established for EPR steering in the two-qubit system. Focusing on rank-2 (including rank-1) states and the two-qubit Werner state, we analyze the eigenvalues of their partially transposed density matrices and construct a significant steering criterion based on symmetric combinations of these eigenvalues. We prove that this criterion serves as a necessary and sufficient condition for steerability for the Werner state, all two-qubit pure states, all two-qubit rank-2 states. Furthermore, we validate the criterion for higher-rank states (rank-3 and rank-4) and show that the results align with known steering inequalities. Our findings suggest a more unified framework for detecting quantum nonlocality via partial transposition and open avenues for further theoretical and numerical investigations into steering detection.

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Spectra and kernels do not determine quantum steering

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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.

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Necessary and sufficient criterion of steering for two-qubit T states

Xiao‐Gang Fan, Huan Yang, Fei Ming +3 more

Einstein-Podolsky-Rosen (EPR) steering is the ability that an observer persuades a distant observer to share entanglement by making local measurements. Determining a quantum state is steerable or unsteerable remains an open problem. Here, we derive a new steering inequality with infinite measurements corresponding to an arbitrary two-qubit T state, from consideration of EPR steering inequalities with N projective measurement settings for each side. In fact, the steering inequality is also a sufficient criterion for guaranteering that the T state is unsteerable. Hence, the steering inequality can be viewed as a necessary and sufficient criterion to distinguish whether the T state is steerable or unsteerable. In order to reveal the fact that the set composed of steerable states is the strict subset of the set made up of entangled states, we prove theoretically that all separable T states can not violate the steering inequality. Moreover, we put forward a method to estimate the maximum violation from concurrence for arbitrary two-qubit T states, which indicates that the T state is steerable if its concurrence exceeds 1/4.

arXiv:2103.04280•Mar 7, 2021•0 citations
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⊗nm\otimes nm⊗n entangled state ρρρ satisfies \rankρ≤1+⌊(m−1)/(n−1)⌋\rankρ\le 1+\lfloor(m-1)/(n-1)\rfloor\rankρ≤1+⌊(m−1)/(n−1)⌋, then it is projectively steerable from AAA to BBB. 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 entanglement does not necessarily imply Einstein-Podolsky-Rosen steering. We identify a \emph{boundary mechanism} that closes this gap when an entangled state meets the boundary of the trusted state space in a nondegenerate way. The mechanism is local: a projective assemblage that approaches a boundary contact with first-order tangential coherence but only second-order inward defect cannot be reproduced by any finite-measure local-hidden-state model. In finite dimensions this yields a support-kernel criterion for an arbitrary fixed steering cut: if a trusted conditional state is rank-deficient and its first-order block couples its support to its kernel, then the state is both NPT (negative partial transpose) and projectively steerable. The two-qubit product-null sector provides the minimal transparent realization. A product vector in the kernel gives the pure boundary contact, and a single coherence entry simultaneously serves as the tangential displacement, the NPT minor, and the steering obstruction. Hence, on this boundary, entanglement collapses to two-way projective steering -- including the genuinely mixed rank-three branch and the rank-two case as a low-dimensional instance. We also provide a filtered standard form, an explicit rank-three Cholesky parametrization, and a compact boundary certificate.
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.
Quantum nonlocality manifests in multipartite systems through entanglement, Bell's nonlocality, and Einstein-Podolsky-Rosen (EPR) steering. While Peres's positive-partial-transpose criterion provides a simple and powerful test for entanglement, a comparably elegant spectral criterion for detecting EPR steering remains an open challenge. In this work, we systematically explore whether a Peres-type criterion can be established for EPR steering in the two-qubit system. Focusing on rank-2 (including rank-1) states and the two-qubit Werner state, we analyze the eigenvalues of their partially transposed density matrices and construct a significant steering criterion based on symmetric combinations of these eigenvalues. We prove that this criterion serves as a necessary and sufficient condition for steerability for the Werner state, all two-qubit pure states, all two-qubit rank-2 states. Furthermore, we validate the criterion for higher-rank states (rank-3 and rank-4) and show that the results align with known steering inequalities. Our findings suggest a more unified framework for detecting quantum nonlocality via partial transposition and open avenues for further theoretical and numerical investigations into steering detection.
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.
Einstein-Podolsky-Rosen (EPR) steering is the ability that an observer persuades a distant observer to share entanglement by making local measurements. Determining a quantum state is steerable or unsteerable remains an open problem. Here, we derive a new steering inequality with infinite measurements corresponding to an arbitrary two-qubit T state, from consideration of EPR steering inequalities with N projective measurement settings for each side. In fact, the steering inequality is also a sufficient criterion for guaranteering that the T state is unsteerable. Hence, the steering inequality can be viewed as a necessary and sufficient criterion to distinguish whether the T state is steerable or unsteerable. In order to reveal the fact that the set composed of steerable states is the strict subset of the set made up of entangled states, we prove theoretically that all separable T states can not violate the steering inequality. Moreover, we put forward a method to estimate the maximum violation from concurrence for arbitrary two-qubit T states, which indicates that the T state is steerable if its concurrence exceeds 1/4.