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arXiv:2610.36076

An algebraic characterisation of nnn-qubit pure-state invariants

Markus Frembs

quant-phmath-phSubmitted Oct 26, 2026• Announced Oct 28, 2026• 0 citations
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Abstract

We study the set of $n$-qubit pure quantum states. Specifically, we consider their incarnation as density matrices in Bloch representation. Building on prior work, we define a family of quadratic functions in Pauli expectation values, and fully characterise all instances that result in state-independent purity constraints. The resulting family of pure-state invariants is surprisingly rich in structure, with strong ties to the geometry of symplectic polar spaces. After charting the general landscape, we define a sub-family of pure-state invariants that satisfies the dynamical constraint of being stable under state update. We prove that it corresponds with (different embeddings of) symplectic Lie algebras $\mathfrak{sp}(2^{n-1})\subset\mathfrak{su}(2^n)$, which admit a generating set of Pauli operators. We carefully distinguish the symplectic case from its orthogonal counterpart, using the Arf invariant of quadratic refinements of the symplectic form on the projective Pauli group, which we further identify with quaternionic structures on $\mathbb{C}^{2^n}$ that are both generated by and restricted to Pauli operators. As a consequence of this analysis, we reveal previously largely overlooked structure in the $n$-qubit Pauli group. We highlight this in a number of examples. We characterise all state-independent uncertainty relations in variance for Pauli observables, as well as all Pauli channels for which the output fidelity and output purity are state-independent. We further show that the orbits of the projective stabiliser groups of pure-state invariants, which for $n\geq2$ are proper subgroups of the Clifford group, form complex projective state $2$-designs, while their unitary lifts form symplectic $2$-designs.

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44+6 pages; comments welcome

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We study the set of nnn-qubit pure quantum states. Specifically, we consider their incarnation as density matrices in Bloch representation. Building on prior work, we define a family of quadratic functions in Pauli expectation values, and fully characterise all instances that result in state-independent purity constraints. The resulting family of pure-state invariants is surprisingly rich in structure, with strong ties to the geometry of symplectic polar spaces. After charting the general landscape, we define a sub-family of pure-state invariants that satisfies the dynamical constraint of being stable under state update. We prove that it corresponds with (different embeddings of) symplectic Lie algebras sp(2n−1)⊂su(2n)\mathfrak{sp}(2^{n-1})\subset\mathfrak{su}(2^n)sp(2n−1)⊂su(2n), which admit a generating set of Pauli operators. We carefully distinguish the symplectic case from its orthogonal counterpart, using the Arf invariant of quadratic refinements of the symplectic form on the projective Pauli group, which we further identify with quaternionic structures on C2n\mathbb{C}^{2^n}C2n that are both generated by and restricted to Pauli operators. As a consequence of this analysis, we reveal previously largely overlooked structure in the nnn-qubit Pauli group. We highlight this in a number of examples. We characterise all state-independent uncertainty relations in variance for Pauli observables, as well as all Pauli channels for which the output fidelity and output purity are state-independent. We further show that the orbits of the projective stabiliser groups of pure-state invariants, which for n≥2n\geq2n≥2 are proper subgroups of the Clifford group, form complex projective state 222-designs, while their unitary lifts form symplectic 222-designs.
We present a family of quadratics in Pauli expectation values, and prove that they constitute state-independent invariants for all n-qubit pure states. This family generalises the two-qubit `pentagon identities', discovered in the reconstruction programme of [P. A. Höhn, Quantum 1, 38 (2017), P. A. Höhn and C. S. P. Wever, Phys. Rev. A 95, 012102 (2017)], where they characterise the space of pure states, as well as the unitary group, and are interpreted as complementarity equalities in the Brukner-Zeilinger information measure. The generalisation to arbitrarily many qubits is nontrivial as it requires new tools which in turn reveal novel structural properties that are absent in the two-qubit case. A thorough analysis of these properties, and their relation with mutual unbiasedness and complementarity in the n-qubit Pauli group, can be found in two companion papers.
The nnn-qubit Pauli group is an essential ingredient to most quantum applications, from computing and error correction to benchmarking and simulation. Despite comprising merely a discrete set of operators, it exhibits many quintessential features of quantum theory, including contextuality, which has been identified as a key resource to quantum advantage in a variety of different flavours. Here, we extend this analysis, introducing the notion of a `noncontextual property' whose nonexistence proves the Kochen-Specker theorem, similarly to and generalising common arguments based on the nonexistence of valuations. We relate this notion formally to the existence of Boolean-valued frame functions, and characterise all such frame functions in the case of the nnn-qubit Pauli group. For two qubits, we show that the Pauli group admits noncontextual properties, despite admitting no valuations. We then establish this as the only nontrivial such case with n≥2n\geq 2n≥2, by proving that any Boolean-valued frame function on stabiliser states is constant for more than two qubits. We also perform a similar analysis for the symplectic theory underlying the nnn-qubit Pauli group, for which nonconstant Boolean-valued frame functions exist for all nnn, yet only in restricted form. By comparison, this shows that contextuality in the nnn-qubit Pauli group is not only a consequence of the projective nature of the Pauli group as a representation of its underlying symplectic vector space, but of the geometry of symplectic polar spaces itself. In geometric terms, our result determines all Cameron-Liebler sets of maximal totally isotropic flats in the binary affine-symplectic space.
Understanding the dynamical properties of quantum systems is an essential task in quantum computing, quantum control, and many-body physics. Tools such as representation theory and Lie theory provide crucial information on reachability and computational power. However, this information can be difficult to access exactly or compute efficiently for arbitrary generating sets. Here we focus on the setting of Pauli strings, which satisfy numerous exceptional properties that simplify the problem. We find deep connections between Pauli Lie algebras and certain subgroups of the Clifford group generated by transvections, through the symplectic properties of the Pauli strings. This allows us to give an invariant-based perspective on these objects and their reachability, in the language of Pauli orbits, symmetries, and invariant subspaces. The invariant-based approach provides efficient algorithms for identifying Lie algebras and orbits, as well as a simple framework for analyzing structured Pauli generating sets. We also show in an elementary way that Clifford subgroups generated by transvections provide 3-designs for the corresponding Pauli Lie groups. We illustrate the framework through structured examples from variational quantum algorithms, restricted quantum computation, many-body systems, and random circuits.
Magic states enable universal, fault-tolerant quantum computation within the stabilizer framework. Their non-stabilizerness supplies the resource needed to bypass the Eastin-Knill theorem while allowing fault-tolerant distillation. Although many measures of magic exist, not every nonzero-magic state is known to be distillable, and many of currently known distillable states are special cases of Clifford-stabilizer states, defined as pure states uniquely stabilized by finite Clifford subgroups. We develop a general framework for group-covariant functionals on Hermitian operators, introducing the notions of GGG-stabilizer spaces, states, and codes for arbitrary finite subgroups G⊂U(H)G \subset \mathrm{U}(\mathcal{H})G⊂U(H). We define analytic families of GGG-covariant functionals and prove that any GGG-invariant pure state is extremal for a broad class of derived functionals, including symmetric, max-type, and Rényi-type functionals, whenever the underlying family is GGG-covariant. This extremality holds for purity-preserving variations orthogonal to the stabilized subspace. Specializing to the Pauli and Clifford groups, we recover the extremality structure of canonical magic measures such as mana, stabilizer Rényi entropies, generalized Rényi entropies, and stabilizer fidelity, and show that Clifford-stabilizer states extremize them. We classify such states for qubits, qutrits, ququints, and two-qubit systems, identifying new candidates for magic distillation. We further propose an inefficient distillation protocol for a two-qubit magic state with stabilizer fidelity exceeding standard benchmarks, and conjecture that SIC-POVM fiducial states are Clifford-stabilizer states.
Abstract Parametrized and random unitary (or orthogonal) n -qubit circuits play a central role in quantum information. As such, one could naturally assume that circuits implementing symplectic transformations would attract similar attention. However, this is not the case, as S P ( d / 2 ) —the group of d × d unitary symplectic matrices—has thus far been overlooked. In this work, we aim at starting to fill this gap. We begin by presenting a universal set of generators G for the symplectic algebra sp ( d / 2 ) , consisting of one- and two-qubit Pauli operators acting on neighboring sites in a one-dimensional lattice. Here, we uncover two critical differences between such set, and equivalent ones for unitary and orthogonal circuits. Namely, we find that the operators in G cannot generate arbitrary local symplectic unitaries and that they are not translationally invariant. We then review the Schur–Weyl duality between the symplectic group and the Brauer algebra, and use tools from Weingarten calculus to prove that Pauli measurements at the output of Haar random symplectic circuits can converge to Gaussian processes. As a by-product, such analysis provides us with concentration bounds for Pauli measurements in circuits that form t -designs over S P ( d / 2 ) . To finish, we present tensor-network tools to analyze shallow random symplectic circuits, and we use these to numerically show that computational-basis measurements anti-concentrate at logarithmic depth.
Lie groups, and therefore Lie algebras, are fundamental structures in quantum physics that determine the space of possible trajectories of evolving systems. However, classification and characterization methods for these structures are often impractical for larger systems. In this work, we provide a comprehensive classification of Lie algebras generated by an arbitrary set of Pauli operators, from which an efficient method to characterize them follows. By mapping the problem to a graph setting, we identify a reduced set of equivalence classes: the free-fermionic Lie algebra, the set of all anti-symmetric Paulis on n qubits, the Lie algebra of symplectic Paulis on n qubits, and the space of all Pauli operators on n qubits, as well as controlled versions thereof. Moreover, out of these, we distinguish 6 Clifford inequivalent cases and find a simple set of canonical operators for each, which allow us to give a physical interpretation of the dynamics of each class. Our findings reveal a no-go result for the existence of small Lie algebras beyond the free-fermionic case in the Pauli setting and offer efficiently computable criteria for universality and extendibility of gate sets. These results bear significant impact in ideas in a number of fields like quantum control, quantum machine learning, or classical simulation of quantum circuits.