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Spectra and kernels do not determine quantum steering

Brent W. Jonah

Published Sep 2, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.22260681

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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.
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 $\rankρ\le 1+\lfloor(m-1)/(n-1)\rfloor$, 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.
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.
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.
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⊗33\otimes33⊗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⊗33\otimes33⊗3, but that nonlocal bound entanglement is not, in general, a UPB phenomenon.
Bargmann invariants constructed from a bipartite density operator and its two lifted marginals are polynomial invariants of local-unitary conjugation. We determine the precise information encoded in these invariants. Whenever one subsystem is a qubit, the ordinary marginal-word family determines the full spectrum of the partial transpose and hence decides whether the state has the positive-partial-transpose property. In 2⊗22\otimes22⊗2 and 2⊗32\otimes32⊗3 systems, this yields complete separability criteria. For the two-qubit system, a finite subfamily additionally separates local-unitary orbits, and a finite extension generates the polynomial invariant ring. These three tasks already diverge for qubit-qutrit states: we exhibit full-rank, locally maximally mixed states that agree on all ordinary marginal-word invariants yet have different operator-Schmidt ranks, together with a quartic correlation invariant that separates them. When both local dimensions are at least three, the analogous collapse on the locally maximally mixed sector produces isospectral pairs consisting of one separable state and one entangled state with negative partial transpose. The ordinary Bargmann algebra therefore coincides with the full local unitary invariant ring if and only if both subsystems are qubits. The missing data are geometric: they encode the placement of global eigenspaces relative to the tensor-product decomposition
We propose a sufficient criterion S=λ1+λ2−(λ1−λ2)2<0{S}=λ_1+λ_2-(λ_1-λ_2)^2<0S=λ1​+λ2​−(λ1​−λ2​)2<0 to detect Einstein-Podolsky-Rosen steering for arbitrary two-qubit density matrix ρABρ_{AB}ρAB​. Here λ1,λ2λ_1,λ_2λ1​,λ2​ are respectively the minimal and the second minimal eigenvalues of ρABTBρ^{T_B}_{AB}ρABTB​​, which is the partial transpose of ρABρ_{AB}ρAB​. By investigating several typical two-qubit states such as the isotropic state, Bell-diagonal state, maximally entangled mixed state, etc., we show this criterion works efficiently and can make reasonable predictions for steerability. We also present a mixed state of which steerability always exists, and compare the result with the violation of steering inequalities.