How Simple Spectral Inequalities Enable Practical Detection of Absolute PPT Without Full State Reconstruction

Study conducted by Prof. Raymond Nung-sing SZEand his research team

Entanglement sits at the centre of quantum information science: it enables tasks like teleportation, secure communication and certain speedups in computation. Yet entanglement is also difficult to diagnose in realistic laboratory settings because most states are mixed, they contain noise and imperfections, and mixed-state entanglement is notoriously hard to characterise.

In recent work published in the Journal of Mathematical Physics [1], Prof. Raymond Nung-sing SZE, Associate Dean of the Faculty of Computer and Mathematical Sciences and Associate Professor of the Department of Applied Mathematics at The Hong Kong Polytechnic University, and his research team addressed a practical and conceptually appealing idea: can we decide useful entanglement-related properties using only the eigenvalues of the density matrix? Eigenvalues are attractive because they do not depend on the choice of basis, and in some scenarios, they may be estimated more directly or more robustly than the full state description. This line of research is often described as “separability (or entanglement) from spectrum.”

The specific target in the work is a strong robustness property called absolute positive partial transpose (PPT). It is built on the well-known PPT test, which is a standard tool in entanglement detection. The PPT test is computationally manageable and, in some low-dimensional cases, exactly characterises separability. However, PPT is not stable under arbitrary transformations of the whole system: a state that is PPT in one basis may fail the PPT test after a global unitary transformation.

That observation leads naturally to the “absolute” version: a state is absolutely PPT if it remains PPT even after any global unitary change of basis. This is important in scenarios where the state might undergo uncontrolled or unknown global dynamics, or where guarantees are wanted that entanglement (as witnessed by PPT violation) cannot be “activated” by global operations.

At first glance, absolute PPT looks extremely hard to test. It quantifies over all global unitary, an infinite and high-dimensional set. The clever point, emphasised throughout the literature and systematically applied here, is that global unitary preserves the eigenvalues. Therefore, while the eigenvectors can change dramatically, the spectrum stays fixed. This makes it plausible, though not obvious, that absolute PPT might be captured by inequalities involving only eigenvalues.

Earlier work had already achieved complete spectral characterisations in certain cases, notably for qubit–qudit and qutrit–qudit systems. The combinatorics and geometry become more complicated, however, when both subsystems are higher-dimensional, for example when one side is a 4-level system (“ququart”).

This work pushes the boundary in two key ways:

1. An exact spectral test is provided for ququart–qudit systems (4 × n, n ≥ 4), though the test is more elaborate than in the 3 × n case.

2. Simple, easy-to-apply sufficient rules are extracted for general m × n systems, rules that depend only on a small handful of eigenvalues at the top and bottom of the spectrum.

This second contribution is particularly relevant for experimentalists and for researchers doing large numerical surveys, because it provides a fast-screening method without requiring heavy optimisation or large semidefinite programs.

The technical engine behind the exact results is a framework developed in earlier work that links absolute PPT to certain structured checks built from the eigenvalues. Roughly speaking, how the partial transpose behaves in pure states (where structure comes from the Schmidt decomposition) is studied, then that structure is used to derive worst-case constraints for mixed states. In the ququart–qudit setting, this produces a finite list of 12 checks that are necessary and sufficient.

The research team is careful to acknowledge a practical limitation: even though twelve checks might not sound large number, the process is still somewhat involved for routine use. Therefore, they focus significant effort on deriving simpler sufficient condition: if the spectrum satisfies a compact inequality, then absolute PPT is guaranteed. This sufficient condition, along with the other sufficient conditions established in the paper, are not meant to replace the exact characterisation, but to make the theory usable when only a quick and reliable certificate is required.

The notion is similar to many “engineering-friendly” criteria in mathematical physics: instead of performing the full optimal test, we use a strong condition that is fast to verify and works well in practice.

A simple eigenvalue rule for general m × n

For a bipartite system with dimensions m and n (with n ≥ m ≥ 2), the eigenvalues of the density matrix are ordered from largest to smallest: λ₁ ≥ λ₂ ≥ ⋯ ≥ λ_mn. The research team provides the following sufficient condition ensuring that the state is absolutely PPT:

                                   2λ_mn  + λ_mn–1 + ⋯ + λ_m(n–1)+1  ≥  λ₁ + λ₂ + ⋯ + λ_m–1

This inequality can be explained in plain terms: On the right, one adds up the largest m – 1 eigenvalues. On the left, one adds up the smallest m eigenvalues, with the very smallest one counted twice. If the “weight in the bottom tail” (left side) is large enough compared with the “weight in the top head” (right side), then no global unitary can push the state into a configuration whose partial transpose becomes negative. In other words, the spectrum is “spread out” sufficiently to prevent PPT violation under any global rotation. Several additional related inequalities are derived in the paper; interested readers are referred to the original article for full details.

A practical advantage of the team’s approach is immediately clear: to check the inequality, we do not need the full matrix, and we do not need all mn eigenvalues. We only need a small number of extremes, roughly the first m – 1 and the last m. That is a major reduction when n is large.

Absolute PPT is easiest to violate when a state has a highly uneven spectrum, one or a few eigenvalues dominate while many others are tiny. In such cases, a global unitary can, in effect, “repackage” the dominant weight into directions that are unfriendly to partial transposition, potentially producing negative eigenvalues after partial transpose. The inequality is a spectral balance condition. It requires that the smallest eigenvalues are not too small relative to the largest ones. When the bottom of the spectrum has sufficient mass, the state behaves more like a well-mixed object, and extreme basis changes cannot create the sharp structures that typically cause PPT violation.

This is also consistent with the broader theme in the separability-from-spectrum literature: states close to maximally mixed are robustly separable/PPT, while highly pure or sharply peaked spectra allow more room for entanglement signatures to emerge under suitable transformations.

Researchers often describe “how mixed” a state is by using purity, which provides a single-number summary of spectral concentration. Highly pure states have spectra dominated by a few large eigenvalues; highly mixed states have spectra that are more evenly distributed. From the perspective of Absolute PPT, this matters because spectral concentration is exactly what makes worst-case global rotations dangerous: the more weight sits in a few eigen-directions, the more freedom a unitary has to rearrange that weight into patterns that can trigger partial-transpose negativity. The inequality condition can be viewed as a local, targeted purity control: instead of computing a global mixedness score, it directly compares the top end of the spectrum to the bottom end. In practice, this is useful because two states can have similar overall mixedness yet differ in how thin their smallest eigenvalues are, and those smallest eigenvalues can be decisive for robustness under arbitrary global transformations.

Prof. Sze’s work reinforces a central idea in modern quantum information: for certain robust properties, the spectrum carries surprisingly strong information. Absolute PPT is defined through a worst-case quantification over all global unitary transformations, yet the results show that, in many cases, a single eigenvalue inequality can already certify the property.

Prof. Sze is currently the Associate Dean of the Faculty of Applied Mathematics and an Associate Professor in the Department of Applied Mathematics at The Hong Kong Polytechnic University. He has a solid foundation and extensive experience in research on linear algebra, matrix theory and operator theory, with a particular focus on quantum computing and quantum information science.

[1] Xiong, L., & Sze. N. S. (2026). Spectral criteria for absolute positive partial transpose (PPT) in qudit-qudit state systems, Journal of Mathematical Physics 67, 012201 (2026), https://doi.org/10.1063/5.0273779

Prof. Raymond Nung-sing SZE Associate Dean, Faculty of Computer and Mathematical Sciences Associate Professor, Department of Applied Mathematics