Identifying local unitary equivalence based on reduction of quantum states

Source: arXiv:2607.20266 · Published 2026-07-22 · By Yanjun Chu, Chenyang Cui, Yuhang Xie, Mengli Liu, Shao-Ming Fei

TL;DR

This paper tackles the challenging problem of determining local unitary (LU) equivalence of a special but important class of degenerate quantum states characterized by having one highly degenerate eigenvalue and the remaining eigenvalues being simple and non-degenerate. LU equivalence is fundamental in quantum information for classifying entanglement, but existing criteria struggle with degeneracies due to eigenvector non-uniqueness. The authors introduce a novel “reduction” procedure that constructs "reduced states" by nullifying the eigenvalue of maximal multiplicity. They prove that LU equivalence between the original degenerate states is exactly equivalent to LU equivalence of their corresponding reduced states. Since the reduced states are either pure or non-degenerate on their supports, standard LU invariants and fixed-point subgroup criteria can then be applied to determine equivalence. This effectively reduces a difficult degenerate problem to tractable non-degenerate or pure-state problems.

They demonstrate this approach with explicit examples involving bipartite qutrit and multipartite qubit states and show how their method can verify LU equivalence where prior methods could not directly apply. The methodology is also used to establish LU equivalence between two families of single-parameterized multipartite mixed states constructed from perturbations of absolutely maximally entangled (AME) states of distinct combinatorial origins (orthogonal Latin squares versus orthogonal arrays). This application highlights the approach’s generality and utility for quantum resource theory. Overall, it provides a systematic framework to handle LU equivalence for a class of degenerate quantum states that are pivotal in quantum information.

Key findings

• Proposed a reduction procedure where the eigenvalue of the highest multiplicity is nullified, producing reduced states whose LU equivalence coincides exactly with the original states' LU equivalence.
• For states with two distinct non-zero eigenvalues λ1 > λ2, the reduced states are pure states simplifying LU equivalence verification using standard invariants (Eqs. 4 and 5).
• In multipartite examples, verifying invariants for constructed matrices Ai showed that the reduced pure states are LU equivalent, thus proving LU equivalence of original degenerate states.
• For states with multiple distinct eigenvalues and one degenerate eigenvalue, the reduced states become non-degenerate on their support, enabling use of known LU invariants [38] for classification.
• Applied the method to two single-parameter families of 4-partite AME(4,3) states (derived from orthogonal Latin squares and orthogonal arrays), proving full LU equivalence for all parameters except singular points.
• Demonstrated that for any p,q in (0, 1/81)∪(1/81, 1), families ρ1(p) and ρ2(q) are LU equivalent, highlighting usefulness in studying structured multipartite mixed states.
• Reduction idea generalizes the LU equivalence problem from complicated degenerate density matrices to simpler problems on reduced states that can be handled with existing invariant-based techniques.

Methodology — deep read

1. Threat Model & Assumptions: The adversary here is conceptualized in the quantum information context as attempting to distinguish quantum states under local unitary operations, with the problem being purely mathematical — no explicit attacker capabilities are modeled beyond the standard LU equivalence definitions. The focus is on states with one eigenvalue exhibiting high degeneracy and all other eigenvalues simple and non-degenerate.

2. Data: The "data" are multipartite quantum states represented as density matrices on finite-dimensional Hilbert spaces. They consider general bipartite and multipartite states with spectral decompositions fitting the eigenvalue multiplicity pattern. Examples focus on qutrit (dimension 3) and qubit systems, including families of states constructed by perturbations around absolutely maximally entangled states.

3. Architecture / Algorithm: The core innovation is a reduction operation: given a state ρ with eigenvalues λ1,...,λs where one eigenvalue λk has multiplicity D−s+1 (large), define ( \hat{\rho} = \frac{1}{\lambda_k D - 1} (\lambda_k I_D - \rho) ). This reduced state has zero for the degenerate eigenvalue and transformed eigenvalues for the others. The authors prove LU equivalence of ρ and ρ' is equivalent to that of ( \hat{\rho} ) and ( \hat{\rho}' ). Two cases arise: ( \hat{\rho} ) is either a pure state (rank 1) or a non-degenerate mixed state on its support. In either case, known LU invariants and criteria (e.g., fixed-point subgroup tests, polynomial trace invariants from [38]) become applicable.

4. Training Regime: Not applicable; this is a theoretical mathematical framework not dependent on training.

5. Evaluation Protocol: The method is tested on concrete examples demonstrating computations of invariants Eqs. (4) and (5), construction of matrices A_i corresponding to eigenvectors, and checking the equality of these under tensor-decomposable unitary transformations. Key examples include bipartite qutrit states and multipartite three-qubit Werner-type states. Additional application is to the two single-parameter families of AME-derived states to verify equivalence over continuous parameter ranges.

6. Reproducibility: The paper gives explicit formulas, examples, and theoretical proofs of equivalence but no code. The examples are sufficiently detailed that a practitioner with knowledge of quantum information theory and linear algebra could reproduce the calculations manually or with standard computational tools. The datasets (quantum states) are fully specified.

Example End-to-End: For two qutrit states ρ1, ρ2 with eigenvalues λ1=3/25 (multiplicity 8) and λ2=1/25 (multiplicity 1), their reductions ( \hat{\rho}_i = \frac{1}{\lambda_1 - \lambda_2}(\lambda_1 I - \rho_i) ) are rank-1 pure states ( |\phi\rangle\langle\phi| ) and ( |\psi\rangle\langle\psi| ). The respective matrices A1 and A2 derived from eigenvectors are explicitly constructed and satisfy the same polynomial invariants (Eq. 5) showing the pure states are LU equivalent. By the theorem, this proves ρ1 and ρ2 are LU equivalent, addressing a problem that direct degeneracy-based tests cannot handle.

Technical innovations

• Introduction of a reduction procedure nullifying the eigenvalue of maximal multiplicity to map degenerate states to reduced states, preserving LU equivalence.
• Proof that LU equivalence of original degenerate states is equivalent to LU equivalence of their reduced counterparts, enabling application of existing invariants.
• Use of block diagonalization and eigenvector matrix invariants on reduced pure or non-degenerate states to fully characterize LU equivalence in this class.
• Application of reduction method to systematically verify LU equivalence between single-parameter families of multipartite AME-derived mixed states.

Datasets

• Bipartite qutrit states — 9x9 density matrices from explicit construction in examples — specified in the paper
• Multipartite three-qubit states — 8x8 density matrices with given spectral structure — specified in examples
• Families of AME(4,3) states parameterized by p ∈ (0,1) — 81x81 density matrices from orthogonal Latin squares and orthogonal arrays constructions

Baselines vs proposed

• Standard polynomial LU invariants and fixed point subgroup criteria from Ref. [38]: applicable only to non-degenerate states.
• Prior methods [33, 38]: cannot directly resolve LU equivalence for degenerate states with a large multiplicity eigenvalue.
• Proposed reduction + existing invariants: enables LU equivalence testing of degenerate states via reduced states, with explicit examples demonstrating successful discrimination where prior methods failed.

Limitations

• The reduction method applies specifically to states with exactly one highly degenerate eigenvalue and all other eigenvalues non-degenerate; general degenerate states with multiple degenerate eigenvalues remain unaddressed.
• Positivity constraints require the maximal eigenvalue to be well defined and positive for the affine transformation; this limits the direct application to certain state classes.
• No adversarial robustness or operational complexity analysis is provided; computational cost of verifying invariants for large-dimensional states can be high.
• The method depends on spectral decomposition and assumes knowledge of eigenvalues and eigenvectors, which may be challenging for experimentally obtained states with noise.
• No consideration of noise, experimental imperfections, or approximate LU equivalences.
• Code or software tools for automating these checks are not released, impacting immediate reproducibility.

Open questions / follow-ons

• Can the reduction approach be extended to states with multiple degenerate eigenvalues beyond a single highly degenerate eigenvalue?
• How does this method scale computationally for larger multipartite systems, and can the invariant computations be made more efficient or automated?
• Is it possible to develop analogous reduction or invariant-based criteria that tolerate noise or approximate LU equivalence for experimental applications?
• Can this framework be generalized to incorporate stochastic local operations and classical communication (SLOCC) equivalence classification?

Why it matters for bot defense

Although not directly related to CAPTCHA or bot-defense, this paper’s insights into discerning equivalence classes under local unitary transformations can inspire techniques for symmetry and invariance detection in high-dimensional data representations. The approach of reducing a complicated equivalence verification problem to a simpler canonical form and then applying well-understood invariants echoes the methodology behind many bot-detection and CAPTCHA analysis algorithms where complex behavior is mapped to invariant signatures. For quantum-inspired or physics-guided machine learning models used in bot-defense, understanding how to efficiently characterize equivalence under transformations could inform robust feature extraction. Additionally, the paper demonstrates techniques for handling degeneracies and structural symmetries in data representations—concepts broadly applicable in bot detection frameworks needing to distinguish genuinely distinct from spurious states or actions.

Cite

@article{arxiv2607_20266,
  title={ Identifying local unitary equivalence based on reduction of quantum states },
  author={ Yanjun Chu and Chenyang Cui and Yuhang Xie and Mengli Liu and Shao-Ming Fei },
  journal={arXiv preprint arXiv:2607.20266},
  year={ 2026 },
  url={https://arxiv.org/abs/2607.20266}
}

Read the full paper

• arXiv:2607.20266 (abstract)
• arXiv:2607.20266 (PDF)