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arXiv:2608.21526

Asymptotic entanglement in circle stabilizer states and states forbidding arbitrary vertex-minors

Kenneth Goodenough, Marcelo Sales

quant-phmath.COSubmitted Aug 21, 2026• Announced Aug 25, 2026• 0 citations
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Abstract

Stabilizer states play a central role in quantum information theory, and understanding their entanglement has motivated a large body of work. A well-studied question in particular is when a stabilizer state $|ψ\rangle$ can be transformed into another stabilizer state $|φ\rangle$ using only single-qubit Clifford operations and Pauli measurements. If this is possible, we say that $|φ\rangle$ is a vertex-minor of $|ψ\rangle$. Assuming Geelen's weak structural conjecture on vertex-minors, we establish the following general statement. For any fixed stabilizer state $|φ\rangle$, the entanglement in stabilizer states $|ψ\rangle$ that do not contain $|φ\rangle$ as a vertex-minor is asymptotically constrained. More concretely, we show that the distance of any sufficiently rank-connected $|ψ\rangle$ not containing $|φ\rangle$ as a vertex-minor grows as $O(\log n)$, and prove similar results for the so-called locally accessible information, a quantity that captures the amount of information that can be learned through single-qubit Pauli measurements. Our results rely on (i) connecting the above two entanglement measures to rank functions of multimatroids, (ii) connecting the rank functions of circle stabilizer states to rank functions on $4$-regular multigraphs, which asymptotically constrains the entanglement of circle stabilizer states, and (iii) using Geelen's weak structural conjecture on vertex-minors to `lift' the previous result to sufficiently connected states in proper vertex-minor-closed families of stabilizer states. Our results establish a connection between asymptotic stabilizer entanglement and forbidden vertex-minors, with direct implications for the entanglement that can be generated in quantum devices.

Comments
29 pages, 7 figures, mathematical details in appendix

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Stabilizer states play a central role in quantum information theory, and understanding their entanglement has motivated a large body of work. A well-studied question in particular is when a stabilizer state ∣ψ⟩|ψ\rangle∣ψ⟩ can be transformed into another stabilizer state ∣φ⟩|φ\rangle∣φ⟩ using only single-qubit Clifford operations and Pauli measurements. If this is possible, we say that ∣φ⟩|φ\rangle∣φ⟩ is a vertex-minor of ∣ψ⟩|ψ\rangle∣ψ⟩. Assuming Geelen's weak structural conjecture on vertex-minors, we establish the following general statement. For any fixed stabilizer state ∣φ⟩|φ\rangle∣φ⟩, the entanglement in stabilizer states ∣ψ⟩|ψ\rangle∣ψ⟩ that do not contain ∣φ⟩|φ\rangle∣φ⟩ as a vertex-minor is asymptotically constrained. More concretely, we show that the distance of any sufficiently rank-connected ∣ψ⟩|ψ\rangle∣ψ⟩ not containing ∣φ⟩|φ\rangle∣φ⟩ as a vertex-minor grows as O(log⁡n)O(\log n)O(logn), and prove similar results for the so-called locally accessible information, a quantity that captures the amount of information that can be learned through single-qubit Pauli measurements. Our results rely on (i) connecting the above two entanglement measures to rank functions of multimatroids, (ii) connecting the rank functions of circle stabilizer states to rank functions on 444-regular multigraphs, which asymptotically constrains the entanglement of circle stabilizer states, and (iii) using Geelen's weak structural conjecture on vertex-minors to `lift' the previous result to sufficiently connected states in proper vertex-minor-closed families of stabilizer states. Our results establish a connection between asymptotic stabilizer entanglement and forbidden vertex-minors, with direct implications for the entanglement that can be generated in quantum devices.
Stabilizer states form an important class of states in quantum information, and are of central importance in quantum error correction. Here, we provide an algorithm for deciding whether one stabilizer (target) state can be obtained from another stabilizer (source) state by single-qubit Clifford operations (LC), single-qubit Pauli measurements (LPM) and classical communication (CC) between sites holding the individual qubits. What is more, we provide a recipe to obtain the sequence of LC+LPM+CC operations which prepare the desired target state from the source state, and show how these operations can be applied in parallel to reach the target state in constant time. Our algorithm has applications in quantum networks, quantum computing, and can also serve as a design tool—for example, to find transformations between quantum error correcting codes. We provide a software implementation of our algorithm that makes this tool easier to apply. A key insight leading to our algorithm is to show that the problem is equivalent to one in graph theory, which is to decide whether some graph G ′ is a vertex-minor of another graph G . The vertex-minor problem is, in general, -Complete, but can be solved efficiently on graphs which are not too complex. A measure of the complexity of a graph is the rank-width which equals the Schmidt-rank width of a subclass of stabilizer states called graph states, and thus intuitively is a measure of entanglement. Here, we show that the vertex-minor problem can be solved in time O (| G | 3 ), where | G | is the size of the graph G , whenever the rank-width of G and the size of G ′ are bounded. Our algorithm is based on techniques by Courcelle for solving fixed parameter tractable problems, where here the relevant fixed parameter is the rank width. The second half of this paper serves as an accessible but far from exhausting introduction to these concepts, that could be useful for many other problems in quantum information. This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.
We study the notion of kkk-stabilizer universal quantum state, that is, an nnn-qubit quantum state, such that it is possible to induce any stabilizer state on any kkk qubits, by using only local operations and classical communications. These states generalize the notion of kkk-pairable states introduced by Bravyi et al., and can be studied from a combinatorial perspective using graph states and kkk-vertex-minor universal graphs. First, we demonstrate the existence of kkk-stabilizer universal graph states that are optimal in size with n=Θ(k2)n=Θ(k^2)n=Θ(k2) qubits. We also provide parameters for which a random graph state on Θ(k2)Θ(k^2)Θ(k2) qubits is kkk-stabilizer universal with high probability. Our second contribution consists of two explicit constructions of kkk-stabilizer universal graph states on n=O(k4)n = O(k^4)n=O(k4) qubits. Both rely upon the incidence graph of the projective plane over a finite field Fq\mathbb{F}_qFq​. This provides a major improvement over the previously known explicit construction of kkk-pairable graph states with n=O(23k)n = O(2^{3k})n=O(23k), bringing forth a new and potentially powerful family of multipartite quantum resources.
Entanglement is central to quantum information processing, while stabilizer operations underpin fault-tolerant quantum computation. We ask how much entanglement remains visible or distillable under stabilizer restrictions. We quantify stabilizer-visible entanglement by restricting the measured relative entropy of entanglement to stabilizer measurements, thereby obtaining converse bounds on entanglement distillation under stabilizer operations. We demonstrate magic-free asymptotic entanglement hiding: we construct explicit convex mixtures of pure stabilizer states on NNN qutrits per party whose unrestricted visible entanglement and LOCC-distillable entanglement both grow as Ω(N/log⁡N)Ω(N/\log N)Ω(N/logN), while their stabilizer-visible and stabilizer-distillable entanglement vanish as N→∞N\to\inftyN→∞. Thus, an unbounded amount of LOCC-distillable entanglement carried by stabilizer states can become asymptotically invisible and undistillable under stabilizer restrictions. We further prove that stabilizer-visible entanglement is O(1)O(1)O(1) with high probability for Haar-random pure states despite extensive unrestricted visibility, and vanishes uniformly over entangled Werner states as the local dimension grows through odd primes. These results reveal a fundamental separation between entanglement and magic as resources, exposing intrinsic limits on entanglement extraction using stabilizer operations.
We prove that every pure state of stabilizer rank at most kkk has stabilizer extent at most 2O(klog⁡(k+1))2^{O(\sqrt{k\log(k+1)})}2O(klog(k+1)​), and establish the analogous bound for the squared Clifford coefficient norm of Clifford-rank-kkk operators. This implies stabilizer fidelity at least 2−O(klog⁡(k+1))2^{-O(\sqrt{k\log(k+1)})}2−O(klog(k+1)​), resolving the quantitative conjecture of (Mehraban-Tamasbi, STOC, 2025), and proves an Ω(n2/log⁡n)Ω(n^2/\log n)Ω(n2/logn) lower bound for the approximate stabilizer rank of tensor powers of any non-stabilizer qubit state. The latter result generalizes the best-known lower bound for tensor powers of TTT-states (Mehraban-Tamasbi, STOC, 2024) to arbitrary non-stabilizer qubit states, including magic states. As a consequence of the Clifford rank--norm inequality, we obtain an Ω(n2/log⁡n)Ω(n^2/\log n)Ω(n2/logn) lower bound for exact representations of nnn-bit AND function by quadratic phases, improving the previous best-known linear bound. Further consequences rule out pseudorandom state and unitary ensembles with approximate stabilizer and Clifford rank O((log⁡n)2/log⁡log⁡n)O((\log n)^2/\log\log n)O((logn)2/loglogn), respectively, a log⁡n\log nlogn improvement over prior work (Kalra-Sinha, Quantum, 2026). We also obtain tomography algorithms for states of stabilizer rank at most kkk, with poly(n)2O(klog⁡(k+1))poly(n)2^{O(\sqrt{k\log(k+1)})}poly(n)2O(klog(k+1)​) time and copy complexity, a nearly square-root improvement in the exponent over the best-known algorithm.
Stabilizer states and graph states find application in quantum error correction, measurement-based quantum computation and various other concepts in quantum information theory. In this work, we study party-local Clifford (PLC) transformations among stabilizer states. These transformations arise as a physically motivated extension of local operations in quantum networks with access to bipartite entanglement between some of the nodes of the network. First, we show that PLC transformations among graph states are equivalent to a generalization of the well-known local complementation, which describes local Clifford transformations among graph states. Then, we introduce a mathematical framework to study PLC equivalence of stabilizer states, relating it to the classification of tuples of bilinear forms. This framework allows us to study decompositions of stabilizer states into tensor products of indecomposable ones, that is, decompositions into states from the entanglement generating set (EGS). While the EGS is finite up to 3 parties [Bravyi et al., J. Math. Phys. 47 , 062106 (2006)], we show that for 4 and more parties it is an infinite set, even when considering party-local unitary transformations. Moreover, we explicitly compute the EGS for 4 parties up to 10 qubits. Finally, we generalize the framework to qudit stabilizer states in prime dimensions not equal to 2 , which allows us to show that the decomposition of qudit stabilizer states into states from the EGS is unique.
The stabilizer rank of a quantum state ψ is the minimal r such that |ψ⟩=∑j=1rcj|φj⟩ for cj∈C and stabilizer states φj . The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the n -th tensor power of single-qubit magic states.We prove a lower bound of Ω(n) on the stabilizer rank of such states, improving a previous lower bound of Ω(n) of Bravyi, Smith and Smolin \cite{BSS16}. Further, we prove that for a sufficiently small constant δ , the stabilizer rank of any state which is δ -close to those states is Ω(n/log⁡n) . This is the first non-trivial lower bound for approximate stabilizer rank.Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of F2n , and we use tools from analysis of boolean functions and complexity theory. The proof of the first result involves a careful analysis of directional derivatives of quadratic polynomials, whereas the proof of the second result uses Razborov-Smolensky low degree polynomial approximations and correlation bounds against the majority function.