Magic, a key quantum resource beyond entanglement, remains poorly understood in terms of its structure and classification. In this paper, we demonstrate a striking connection between high-dimensional symmetric lattices and quantum magic states. By mapping vectors from the $E_8, B W_{16}$, and $E_6$ lattices into Hilbert space, we construct and classify stabiliser and maximal magic states for two-qubit, three-qubit and one-qutrit systems. In particular, this geometric approach gives closed-form lattice representatives of maximal magic states and motivates a conjecture for the number of three-qubit maximal states. In the three-qubit case, we further classify the extremal magic states according to their entanglement structure. We also show that the 45 one-qutrit maximal magic states selected by the second shell of $E_6$ split into two Clifford orbits. Our findings suggest that deep algebraic and geometric symmetries underlie the structure of extremal magic states.
Magic, a key quantum resource beyond entanglement, remains poorly understood in terms of its structure and classification. In this paper, we demonstrate a striking connection between high-dimensional symmetric lattices and quantum magic states. By mapping vectors from the E8,BW16, and E6 lattices into Hilbert space, we construct and classify stabiliser and maximal magic states for two-qubit, three-qubit and one-qutrit systems. In particular, this geometric approach gives closed-form lattice representatives of maximal magic states and motivates a conjecture for the number of three-qubit maximal states. In the three-qubit case, we further classify the extremal magic states according to their entanglement structure. We also show that the 45 one-qutrit maximal magic states selected by the second shell of E6 split into two Clifford orbits. Our findings suggest that deep algebraic and geometric symmetries underlie the structure of extremal magic states.
Achieving a genuine quantum advantage relies on two distinct non-classical resources that restrict efficient classical simulation: entanglement and magic (nonstabilizerness). We investigate the interplay between these resources by characterizing the Pareto frontiers of extreme magic at fixed entanglement for systems of two qutrits (d=3) and two ququints (d=5). Unlike the case of two qubits, the Schmidt spectrum for two qutrits features two independent entanglement parameters, resulting in two-dimensional Pareto surfaces. For the lower frontier, we recast the minimal magic as a compact function of concurrence and negativity, with a maximal value of ln2. For the upper frontier, we determine the maximal stabilizer Rényi entropy to be M2=ln(81/17)≈1.561, which tightens the previous theoretical bound of ln5≈1.609 and improves on earlier numerical estimates. The maximum magic is achieved at eighteen distinct maxima categorized into three families of six permutation-equivalent spectra. We provide analytical expressions for the maximal magic in the neighborhood of each maximum and for the corresponding maximally magical states which turn out to be Weyl-Heisenberg-covariant fiducial states for mutually unbiased bases. Finally, numerical analysis of two ququints (d=5) reveals six permutation-inequivalent maxima with a peak magic value of M2=ln(625/49)≈2.546. Based on these findings, we conjecture that the maximal magic for a bipartite system of two qudits with prime dimension d is given by ln[d4/(2d2−1)], which reproduces the previously known value for qubits, as well as the values derived here for qutrits and ququints.
Abstract Magic, a key quantum resource beyond entanglement, remains poorly understood in terms of its structure and classification. In this paper, we demonstrate a striking connection between high-dimensional symmetric lattices and quantum magic states. By mapping vectors from the E8,BW16, and E6 lattices into Hilbert space, we construct and classify stabiliser and maximal magic states for two-qubit, three-qubit and one-qutrit systems. In particular, this geometric approach allows us to construct, for the first time, closed-form expressions for the maximal magic states in the three-qubit and one-qutrit systems, and to conjecture their total counts. In the three-qubit case, we further classify the extremal magic states according to their entanglement structure. We also examine the distinctive behaviour of one-qutrit maximal magic states with respect to Clifford orbits. Our findings suggest that deep algebraic and geometric symmetries underlie the structure of extremal magic states.
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 G-stabilizer spaces, states, and codes for arbitrary finite subgroups G⊂U(H). We define analytic families of G-covariant functionals and prove that any G-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 G-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.
Quantum magic is a necessary resource for fault-tolerant quantum computing, capturing the departure of quantum states from the classically simulable stabiliser set. Motivated by its role in quantum information processing, we investigate the correlation structure of states that maximize the Rényi stabiliser entropies, focusing on the T-type and Hoggar states. Inspired by their distinctive properties, we explore the notion of Pauli flat states, defined as N-qubit states whose expectation values on all Pauli strings are equal in absolute value. These states reproduce the correlation structure of highly magical states, which could also be interpreted as a mixed variant of Pauli covariant N-qubit symmetric informationally complete (SIC) Weyl-Heisenberg fiducial states, while not necessarily possessing nontrivial quantum magic. We investigate their existence and possible constructions, including examples both inside and outside of the stabiliser polytope. To identify the latter, we examine a magic witness tailored to the introduced family of states. Finally, we discuss their entanglement and Bell nonlocality, providing a broader characterization of their quantum properties. For completeness, we find pure states with a relaxed condition on the Pauli expectations, which forces unequal elements to zero, and discuss their non-stabiliserness. Our results offer a new perspective on the relation between quantum magic and Pauli correlation structures. They hold relevance to geometrical approaches to quantum mechanics, entanglement and magic detection, as well as estimation tasks.
Distinct Clifford orbits of magic states can exhibit different stabilizer ranks at small tensor powers. We establish this for qutrits, where the single-qutrit Clifford group has four inequivalent orbits of magic states: Strange, Norrell, Hadamard-eigenstate, and the qutrit T-state, but a nontrivial upper bound on the asymptotic exponent had been pinned down for only the qutrit T-state. For the other three orbits we give explicit stabilizer decompositions, yielding upper bounds on the per-copy asymptotic stabilizer-rank exponent: γS≤log3(2)/2≈0.316 for the Strange state, and γH3,γN≤log3(4)/3≈0.421 for the Hadamard-eigenstate and Norrell orbits, all strictly below the prior γT3≤1/2 baseline. We also prove the first nontrivial Ω(m/logm) asymptotic lower bounds for the Hadamard-eigenstate and Norrell orbits, and exhibit two-qutrit Clifford circuits that convert two copies of these states into an injectable phase state with constant success probability, enabling constant-overhead injection of one non-Clifford diagonal gate per orbit. In the case of qubits, we give a closed-form decomposition of the qubit T-type orbit at four copies matching the existing γT≤log2(3)/4≈0.396 exponent via a direct algebraic identity rather than an entangled cat-state construction. An open-source library stabrank accompanies the paper, with Lean 4 proof formalizations of all the decompositions.
We develop a Clifford-orbit framework for studying magic-protected entanglement, which we refer to as magical entanglement: the part of bipartite entanglement that remains after optimal stabilizer simplification. This construction leverages residual entanglement under Clifford reduction as a state-level organizing principle for magic state space. It defines canonical representatives, spectra, and ranks that characterize the Clifford-irreducible structure of a state. We identify two regimes of the magic-entanglement interplay. In the T-magic regime, local nonstabilizer resources can coexist with entanglement, but the protected component remains weak and state-dependent. In the W-magic regime, by contrast, entanglement is Clifford-irreducibly tied to nonstabilizerness, producing typical, strongly self-averaging behavior. Analytical examples and random-circuit numerics support a crossover from broad T-magic fluctuations to concentrated, Haar-like W-magic behavior. These results identify magical entanglement as an orbit-level diagnostic of how nonstabilizerness protects quantum correlations against Clifford reduction.