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arXiv:2412.21083

Quantum States with Maximal Magic

Gianluca Cuffaro, Christopher A. Fuchs

Published Dec 30, 2024• 1 citation
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Abstract

Finding ways to quantify magic is an important problem in quantum information theory. Recently Leone, Oliviero and Hamma introduced a class of magic measures for qubits, the stabilizer entropies of order $α$, to aid in studying nonstabilizer resource theory. This suggests a way to search for those states that are as distinct as possible from the stabilizer states. In this work we explore the problem in any finite dimension $d$ and characterize the states that saturate an upper bound on stabilizer entropies of order $α\geq2$. Particularly, we show that if a Weyl-Heisenberg (WH) covariant Symmetric Informationally Complete (SIC) quantum measurement exists, its states uniquely maximize the stabilizer entropies by saturating the bound. No other states can reach so high. This result is surprising, as the initial motivation for studying SICs was a purely quantum-foundational concern in QBism. Yet our result may have implications for quantum computation at a practical level, as it demonstrates that this notion of maximal magic inherits all the difficulties of the 25-year-old SIC existence problem, along with the deep questions in number theory associated with it.

DOI
10.48550/arxiv.2412.21083

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Finding ways to quantify magic is an important problem in quantum information theory. Recently Leone, Oliviero and Hamma introduced a class of magic measures for qubits, the stabilizer entropies of order ααα, to aid in studying nonstabilizer resource theory. This suggests a way to search for those states that are as distinct as possible from the stabilizer states. In this work we explore the problem in any finite dimension ddd and characterize the states that saturate an upper bound on stabilizer entropies of order α≥2α\geq2α≥2. Particularly, we show that if a Weyl-Heisenberg (WH) covariant Symmetric Informationally Complete (SIC) quantum measurement exists, its states uniquely maximize the stabilizer entropies by saturating the bound. No other states can reach so high. This result is surprising, as the initial motivation for studying SICs was a purely quantum-foundational concern in QBism. Yet our result may have implications for quantum computation at a practical level, as it demonstrates that this notion of maximal magic inherits all the difficulties of the 25-year-old SIC existence problem, along with the deep questions in number theory associated with it.
Abstract Magic is a quantum resource essential for universal quantum computation and represents the deviation of quantum states from those that can be simulated efficiently using classical algorithms. Using the stabilizer Rényi entropy (SRE), we investigate two-qubit states with maximal magic, which are most distinct from classical simulability, and provide strong numerical evidence that the maximal second order SRE is ln ⁡ ( 16 / 7 ) ≈ 0.827 , establishing a tighter bound than the prior ln ⁡ ( 5 / 2 ) ≈ 0.916 . We identify 480 states saturating the new bound, which turn out to be the fiducial states for the mutually unbiased bases (MUBs) generated by the orbits of the Weyl–Heisenberg (WH) group, and conjecture that WH-MUBs are the maximal magic states for n -qubit, when n ≠ 1 and 3. We also reveal a striking interplay between magic and entanglement: the entanglement of maximal magic states is restricted to two possible values, 1 / 2 and 1 / 2 , as quantified by the concurrence; none is maximally entangled.
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.
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.
In the stabilizer formalism of quantum computation, the Gottesman–Knill theorem shows that universal fault-tolerant quantum computation requires the resource called magic (nonstabilizerness). Thus stabilizer states serve as “classical states,” and states beyond them are necessary for genuine quantum computation. Characterization, detection, and quantification of magic states are basic issues in this context. In the paradigm of quantum measurement, symmetric informationally complete positive operator valued measures (SIC-POVMs, further abbreviated as SICs) play a prominent role due to their structural symmetry and remarkable features. However, their existence in all dimensions, although strongly supported by extensive theoretical and numerical evidence, remains an elusive open problem (Zauner’s conjecture). A standard method for constructing SICs is via the orbit of the Heisenberg–Weyl group on a fiducial state, and most known SICs arise in this way. A natural question arises regarding the relation between stabilizer states and fiducial states. In this paper, we connect them by showing that they are on two extremes with respect to the ppp -norms of characteristic functions of quantum states. This not only reveals a simple path from stabilizer states to SIC fiducial states, showing quantitatively that they are as far away as possible from each other, but also provides a simple reformulation of Zauner’s conjecture in terms of extremals for the ppp -norms of characteristic functions. A convenient criterion for magic states and some interesting open problems are also presented.
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=3d=3d=3) and two ququints (d=5d=5d=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 ln⁡2\ln 2ln2. For the upper frontier, we determine the maximal stabilizer Rényi entropy to be M2=ln⁡(81/17)≈1.561M_2 = \ln(81/17) \approx 1.561M2​=ln(81/17)≈1.561, which tightens the previous theoretical bound of ln⁡5≈1.609\ln 5\approx 1.609ln5≈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=5d=5d=5) reveals six permutation-inequivalent maxima with a peak magic value of M2=ln⁡(625/49)≈2.546M_2 = \ln(625/49) \approx 2.546M2​=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 ddd is given by ln⁡[d4/(2d2−1)]\ln [ d^4 / (2d^2 - 1) ]ln[d4/(2d2−1)], which reproduces the previously known value for qubits, as well as the values derived here for qutrits and ququints.
In most stabilizer-based quantum computing schemes, so-called magic states are a necessary resource for implementing non-transversal quantum gates. With the resource theory of magic, it is possible to analyze and quantify the generation of the non-stabilizer states. The relative entropy is a measure used in various resource theories. For single qubits, we characterize magic states and their closest stabilizer states by applying analytical results known from the relative entropy of entanglement and show that the magic states and their closest stabilizer states are arranged symmetrically around the states at the centers of the faces of the stabilizer octahedron. For tensor products of single-qubit states, we prove analytically that the relative entropy of magic is nonadditive in almost all cases.