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arXiv:2609.34255

A threshold for maximal Schmidt number from spectrum

Junhyeong An, Soojoon Lee

quant-phSubmitted Sep 28, 2026• Announced Sep 29, 2026• 0 citations• Revised Oct 3, 2026 (v2)
v1 · Sep 28, 2026v2 · Oct 3, 2026
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Abstract

The Schmidt number quantifies the dimensionality of entanglement in bipartite quantum states. We investigate when the spectrum of a state on $\mathbb{C}^d \otimes \mathbb{C}^d$ alone guarantees that its Schmidt number is not maximal. By deriving spectral bounds for $(d-1)$-block positive operators, we prove that $\mathrm{LS}_p\subseteq \mathrm{ASN}_{d-1}$ for $p\leq \lfloor d^2/2\rfloor$, where $\mathrm{LS}_p$ denotes the set of states whose largest eigenvalue does not exceed the sum of their $p$ smallest eigenvalues, and $\mathrm{ASN}_{d-1}$ denotes the set of states whose Schmidt number remains at most $d-1$ under all global unitaries. This also yields a simple sufficient condition involving only the largest eigenvalue. As an application, we show that states satisfying the reduction criterion under arbitrary global unitaries belong to $\mathrm{ASN}_{d-1}$ in every dimension. This subsumes the earlier result that absolutely positive partial transpose states belong to $\mathrm{ASN}_{d-1}$. The global unitary requirement is nevertheless essential: for every $d\geq3$, we construct states of maximal Schmidt number satisfying the reduction criterion on both subsystems.

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The Schmidt number quantifies the dimensionality of entanglement in bipartite quantum states. We investigate when the spectrum of a state on Cd⊗Cd\mathbb{C}^d \otimes \mathbb{C}^dCd⊗Cd alone guarantees that its Schmidt number is not maximal. By deriving spectral bounds for (d−1)(d-1)(d−1)-block positive operators, we prove that LSp⊆ASNd−1\mathrm{LS}_p\subseteq \mathrm{ASN}_{d-1}LSp​⊆ASNd−1​ for p≤⌊d2/2⌋p\leq \lfloor d^2/2\rfloorp≤⌊d2/2⌋, where LSp\mathrm{LS}_pLSp​ denotes the set of states whose largest eigenvalue does not exceed the sum of their ppp smallest eigenvalues, and ASNd−1\mathrm{ASN}_{d-1}ASNd−1​ denotes the set of states whose Schmidt number remains at most d−1d-1d−1 under all global unitaries. This also yields a simple sufficient condition involving only the largest eigenvalue. As an application, we show that states satisfying the reduction criterion under arbitrary global unitaries belong to ASNd−1\mathrm{ASN}_{d-1}ASNd−1​ in every dimension. This subsumes the earlier result that absolutely positive partial transpose states belong to ASNd−1\mathrm{ASN}_{d-1}ASNd−1​. The global unitary requirement is nevertheless essential: for every d≥3d\geq3d≥3, we construct states of maximal Schmidt number satisfying the reduction criterion on both subsystems.
We introduce a framework to upper bound the entanglement content of a bipartite quantum state from its spectrum alone. Using linear maps and their inverses, we derive rigorous constraints on the maximal entanglement that can be activated under global unitary transformations. We use as entanglement quantifiers the negativity and the Schmidt number; however, our framework is general and applies to any other entanglement measure. Our approach yields compact analytical sufficient criteria for bounding the entanglement of full-rank states in arbitrary dimensions and reveals new spectral constraints on Schmidt number witnesses.
The dimensionality of entanglement, quantified by the Schmidt number, is a valuable resource for a wide range of quantum information processing tasks. In this work, we introduce the notion of the absolute Schmidt number, referring to states whose Schmidt number cannot be increased by any global unitary transformation. We provide a characterization of the set of arbitrary-dimensional states whose Schmidt number is invariant under all global unitaries. Our approach enables us to develop both witness-based and moment-based techniques to detect nonabsolute Schmidt number states which could provide significant operational advantages through Schmidt number enhancement by global unitaries. We next formulate two resource-theoretic measures of nonabsolute Schmidt number states, based respectively on Schmidt number witness and robustness, and demonstrate an operational utility of the latter in a channel discrimination task. Finally, we extend our analysis to quantum channels by introducing a new class of channels that possess the absolute Schmidt number property. We derive a necessary and sufficient condition for identifying when a channel has the absolute Schmidt number property, confining our analysis to the class of covariant channels.
Genuine high-dimensional entanglement, i.e., the property of having a high Schmidt number, constitutes an instrumental resource in quantum communication, overcoming limitations of low-dimensional systems. States with a positive partial transpose (PPT) are generally considered weakly entangled, as they can never be distilled into pure entangled states. This naturally raises the question of whether high Schmidt numbers are possible for PPT states. Volume estimates suggest that optimal, i.e., linear, scaling in the local dimension should be possible, albeit without providing insight into the possible slope. We provide the first explicit construction of a family of PPT states that achieves linear scaling in the local dimension and we prove that random PPT states typically share this feature. Our construction also allows us to prove a recent conjecture of Chen et al. on the existence of PPT states whose Schmidt number increases by an arbitrarily large amount upon partial transposition. Finally, we link the Schmidt number to entangled sub-block matrices of a quantum state. We use this connection to prove that quantum states that are either (i) invariant under partial transposition on the smallest of their two subsystems, or (ii) absolutely PPT cannot have a maximal Schmidt number. Overall, our findings shed new light on some fundamental problems in entanglement theory.
A deep understanding of quantum entanglement is vital for advancing quantum technologies. The strength of entanglement can be quantified by counting the degrees of freedom that are entangled, which results in a quantity called the Schmidt number. A particular challenge is to identify the strength of entanglement in quantum states that remain positive under partial transpose (PPT), otherwise recognized as undistillable states. Finding PPT states with high Schmidt numbers has become a mathematical and computational challenge. In this Letter, we introduce efficient analytical tools for calculating the Schmidt number for a class of bipartite states called grid states. Our methods improve the best-known bounds for PPT states with high Schmidt numbers. Most notably, we construct a Schmidt number 3 PPT state in five-dimensional systems and a family of states with a Schmidt number of (d+1)/2 for odd d-dimensional systems, representing the best-known scaling of the Schmidt number in a local dimension. Additionally, these states possess intriguing geometrical properties, which we utilize to construct indecomposable entanglement witnesses.
The main topic of this compilation thesis is the investigation of multipartite entanglement of finite dimensional systems. We developed a numerical algorithm that detects if a multipartite state is entangled or separable in a finite number of steps of a semi-definite optimization task. This method is an extension of previously known semi-definite methods, which are inconclusive when the state is separable. In our case, if the state is separable, an explicit decomposition into a mixture of separable states can be extracted. This was achieved by mapping the entanglement problem onto the mathematically well studied truncated moment problem. Additionally, a new way of writing the partially transposed state for symmetric multi-qubit states was developed which simplifies many results previously known in entanglement theory. This new way of writing the partial transpose criterion unifies different interpretations and alternative formulations of the partial transpose criterion and it is also a part in the aforementioned semi-definite algorithm. The geometric properties of entangled symmetric states of two qubits were studied in detail: We could answer the question of how far a given pure state is from the convex hull of symmetric separable states, as measured by the Hilbert-Schmidt distance, by giving an explicit formula for the closest separable symmetric state. For mixed states we could provide a numerical upper and analytical lower bound for this distance. For a larger number of qubits we investigated the ball of absolutely classical states, i.e. symmetric multi-qubit states that stay separable under any unitary transformation. We found an analytical lower bound for the radius of this ball around the maximally mixed symmetric state and gave a numerical upper bound on this radius, by searching for an entangled state as close as possible to the maximally mixed symmetric state. The tensor representation of a symmetric multi-qubit state, or spin-j state, allowed us to study entanglement properties based on the spectrum of the tensor via tensor eigenvalues. The definiteness of this tensor relates to the entanglement of the state it represents and, hence, the smallest tensor eigenvalue can be used to detect entanglement. However, the tensor eigenvalues are more difficult to determine than the familiar matrix eigenvalues which made the investigation computationally more challenging. The relationship between the value of the smallest tensor eigenvalue and the amount of entanglement in the state was also investigated. It turned out that they are strongly correlated for small system sizes, i.e. for up to six qubits. However, to investigate this correlation we needed an independent way to gauge the amount of entanglement of a state and in order to do so we improved existing numerical methods to determine the distance of a state to the set of separable symmetric states, using a combination of linear and quadratic programming. The tensor representation of symmetric multi-qubit states was also used to formally define a new tensor class of regularly decomposable tensors that corresponds to the set of separable symmetric multi-qubit states.
High-dimensional bipartite entanglement depends on how probability is distributed across the Schmidt spectrum, whereas a single reference-state fidelity resolves only one spectral scale. We develop multiscale Schmidt-spectrum bounds that connect geometric measures with quantitative Schmidt-number witnesses. For any partition of a pure-state Schmidt spectrum, several nested Vidal tails determine block masses. We derive the sharp upper boundary of the associated normalized nuclear-norm coordinate and show that equality holds if and only if the spectrum is uniform within each block. Refining the partition gives a monotone hierarchy of tighter bounds whenever the added tail data distinguish unequal block means. We also solve a relaxed weighted multiscale optimization globally: one scalar parameter specifies its unique full-support optimizer and explicit value on the nontrivial branch. Using the established single-tail fidelity--resource curve as a baseline, we obtain exact-fidelity equality refinements, convex-roof lower bounds for mixed states, and quantitative calibrations of Schmidt-number witnesses. A higher-tail relation further bounds convex-roof extended negativity in terms of Vidal tails and identifies the pure-state equality spectra. Leakage-aware and joint-confidence formulations state how these bounds can be used with incomplete data. Multistep tails require block-resolved or independently certified spectral information; they are not determined by one projector expectation.