We prove that, for an $n$ -qubit system of dimension $d=2^n$ , every state satisfying $\operatorname{Tr}(ρ^2)\le 1/(d-a_\ast)$ , with $a_\ast=0.458327\cdots$ , lies inside the stabilizer polytope and is therefore magic-free. Combining this result with general geometric properties of high-dimensional polytopes, we establish quantitative estimates for the Hilbert--Schmidt inradius and volume radius of the stabilizer polytope, and use them to characterize the typicality of magic in random induced states obtained by tracing out a $k$ -dimensional subsystem from a $d\times k$ -dimensional Haar-random pure state. We prove a sharp phase transition in the probability of such states having magic, whose transition dimension $k_\star$ is bounded between $Ω(d^2/\log^2d)$ and $\mathcal{O}(d^2)$ . We further prove that the number of facets of the stabilizer polytope lies between $\exp[Ω(d^2/\log^2 d)]$ and $\exp[\mathcal{O}(d^2\log^2 d)]$ employing a result of Bourgain and Milman in convex geometry, substantially improving upon the previous quasipolynomial lower bound and implying that doubly-exponentially many linear inequalities in the number of qubits are required for an exact description of the magic-free region. Overall, our results reveal the near-extremal geometry of the stabilizer polytope and provide a quantitative foundation for understanding the typicality, robustness, and detectability of magic.We prove that, for an $n$-qubit system of dimension $d=2^n$, every state satisfying $\operatorname{Tr}(ρ^2)\le 1/(d-a_\ast)$, with $a_\ast=0.458327\cdots$, lies inside the stabilizer polytope and is therefore magic-free. Combining this result with general geometric properties of high-dimensional polytopes, we establish quantitative estimates for the Hilbert--Schmidt inradius and volume radius of the stabilizer polytope, and use them to characterize the typicality of magic in random induced states obtained by tracing out a $k$-dimensional subsystem from a $d\times k$-dimensional Haar-random pure state. We prove a sharp phase transition in the probability of such states having magic, whose transition dimension $k_\star$ is bounded between $Ω(d^2/\log^2d)$ and $\mathcal{O}(d^2)$. We further prove that the number of facets of the stabilizer polytope lies between $\exp[Ω(d^2/\log^2 d)]$ and $\exp[\mathcal{O}(d^2\log^2 d)]$ employing a result of Bourgain and Milman in convex geometry, substantially improving upon the previous quasipolynomial lower bound and implying that doubly-exponentially many linear inequalities in the number of qubits are required for an exact description of the magic-free region. Overall, our results reveal the near-extremal geometry of the stabilizer polytope and provide a quantitative foundation for understanding the typicality, robustness, and detectability of magic.

quant-pharXiv:2609.03944Sep 4, 20260 citations

The Stabilizer Dimension of n-Qubit Symmetric States

Bo Li, LI Jiao-jiao, Zhi‐Xi Wang

International Journal of Theoretical PhysicsOct 16, 20130 citations

Entropy Stability and Spectral Concentration under Convex Block Constraints

Hassan Nasreddine

We study quadratic stability of entropy minimization under convex constraints with block-separable structure. For constraint sets of the formC = { ⊕ᵢ pᵢ ρᵢ : p ∈ Π, ρᵢ ∈ Cᵢ },we establish explicit stability estimates showing that, under a confining (fixed-support) hypothesis, the entropy gap controls the squared trace-norm distance to the set of entropy minimizers, with constants determined by the geometry of the constraint. The stability constant admits a natural decomposition into marginal and conditional components. The marginal contribution is governed by the curvature of Shannon entropy on the marginal polytope Π at extreme points, while the conditional contribution is governed by the curvature of von Neumann entropy at entropy-minimizing states within each block. We prove that the quadratic exponent is optimal by constructing explicit examples for which no linear stability bound holds uniformly. Exploiting the block-separable structure, the analysis reduces the global stability problem to independent marginal and conditional subproblems, yielding a geometric characterization of the stability constants in terms of constraint curvature. This stability phenomenon cannot be derived from Pinsker-type inequalities or standard entropy continuity bounds, since no reference state is fixed and the minimizer emerges intrinsically from the constraint geometry.We study quadratic stability of entropy minimization under convex constraints with block-separable structure. For constraint sets of the formC = { ⊕ᵢ pᵢ ρᵢ : p ∈ Π, ρᵢ ∈ Cᵢ },we establish explicit stability estimates showing that, under a confining (fixed-support) hypothesis, the entropy gap controls the squared trace-norm distance to the set of entropy minimizers, with constants determined by the geometry of the constraint. The stability constant admits a natural decomposition into marginal and conditional components. The marginal contribution is governed by the curvature of Shannon entropy on the marginal polytope Π at extreme points, while the conditional contribution is governed by the curvature of von Neumann entropy at entropy-minimizing states within each block. We prove that the quadratic exponent is optimal by constructing explicit examples for which no linear stability bound holds uniformly. Exploiting the block-separable structure, the analysis reduces the global stability problem to independent marginal and conditional subproblems, yielding a geometric characterization of the stability constants in terms of constraint curvature. This stability phenomenon cannot be derived from Pinsker-type inequalities or standard entropy continuity bounds, since no reference state is fixed and the minimizer emerges intrinsically from the constraint geometry.

Dec 17, 20250 citations

Predicting Magic from Very Few Measurements

J. M. Varela, L. L. Keller, A. de Oliveira Junior +3 more

The nonstabilizerness of quantum states is a necessary resource for universal quantum computation, yet its characterization is notoriously demanding. Quantifying nonstabilizerness typically requires an exponential number of measurements and a doubly exponential classical post-processing cost to evaluate its standard monotones. In this work, we show that nonstabilizerness is, to a large extent, in the eyes of the beholder: it can be witnessed and quantified using any set of $m$ $n$ -qubit Pauli measurements, provided the set contains anti-commuting pairs. We introduce a general framework that projects the stabilizer polytope onto the subspace defined by these observables and provide an algorithm that estimates magic from Pauli expectation values with runtime exponential in the number of measurements $m$ and polynomial in the number of qubits $n$ . By relating the problem to a stabilizer-restricted variant of the quantum marginal problem, we also prove that deciding membership in the corresponding reduced stabilizer polytope is NP-hard. In particular, unless $\mathrm{P} = \mathrm{NP}$ , no algorithm polynomial in $m$ can solve the problem in full generality, thus establishing fundamental complexity-theoretic limitations. Finally, we employ our framework to compute nonstabilizerness in different Hamiltonian ground states, demonstrating the practical performance of our method in regimes beyond the reach of existing techniques.The nonstabilizerness of quantum states is a necessary resource for universal quantum computation, yet its characterization is notoriously demanding. Quantifying nonstabilizerness typically requires an exponential number of measurements and a doubly exponential classical post-processing cost to evaluate its standard monotones. In this work, we show that nonstabilizerness is, to a large extent, in the eyes of the beholder: it can be witnessed and quantified using any set of $m$ $n$-qubit Pauli measurements, provided the set contains anti-commuting pairs. We introduce a general framework that projects the stabilizer polytope onto the subspace defined by these observables and provide an algorithm that estimates magic from Pauli expectation values with runtime exponential in the number of measurements $m$ and polynomial in the number of qubits $n$. By relating the problem to a stabilizer-restricted variant of the quantum marginal problem, we also prove that deciding membership in the corresponding reduced stabilizer polytope is NP-hard. In particular, unless $\mathrm{P} = \mathrm{NP}$, no algorithm polynomial in $m$ can solve the problem in full generality, thus establishing fundamental complexity-theoretic limitations. Finally, we employ our framework to compute nonstabilizerness in different Hamiltonian ground states, demonstrating the practical performance of our method in regimes beyond the reach of existing techniques.

quant-pharXiv:2602.18939Feb 24, 20260 citations

Uncertainty characterization of stabilizer states and magic states for qubit systems

Bowen Wang, Jiayu He, Shuangshuang Fu

Quantum Information ProcessingApr 30, 20252 citations

distanceStudying the geometric phase for a two-qubit model in terms of

Mehdi Amniat-Talab, Hossein Rangani Jahromi, Leyla Abbasi

انجمن اپتیک و فوتونیک ایرانMar 15, 20150 citations

We prove that, for an

n

-qubit system of dimension

d=2^n

, every state satisfying

\operatorname{Tr}(ρ^2)\le 1/(d-a_\ast)

, with

a_\ast=0.458327\cdots

, lies inside the stabilizer polytope and is therefore magic-free. Combining this result with general geometric properties of high-dimensional polytopes, we establish quantitative estimates for the Hilbert--Schmidt inradius and volume radius of the stabilizer polytope, and use them to characterize the typicality of magic in random induced states obtained by tracing out a

k

-dimensional subsystem from a

d\times k

-dimensional Haar-random pure state. We prove a sharp phase transition in the probability of such states having magic, whose transition dimension

k_\star

is bounded between

Ω(d^2/\log^2d)

and

\mathcal{O}(d^2)

. We further prove that the number of facets of the stabilizer polytope lies between

\exp[Ω(d^2/\log^2 d)]

and

\exp[\mathcal{O}(d^2\log^2 d)]

employing a result of Bourgain and Milman in convex geometry, substantially improving upon the previous quasipolynomial lower bound and implying that doubly-exponentially many linear inequalities in the number of qubits are required for an exact description of the magic-free region. Overall, our results reveal the near-extremal geometry of the stabilizer polytope and provide a quantitative foundation for understanding the typicality, robustness, and detectability of magic.

m

n

-qubit Pauli measurements, provided the set contains anti-commuting pairs. We introduce a general framework that projects the stabilizer polytope onto the subspace defined by these observables and provide an algorithm that estimates magic from Pauli expectation values with runtime exponential in the number of measurements

m

and polynomial in the number of qubits

n

. By relating the problem to a stabilizer-restricted variant of the quantum marginal problem, we also prove that deciding membership in the corresponding reduced stabilizer polytope is NP-hard. In particular, unless

\mathrm{P} = \mathrm{NP}

, no algorithm polynomial in

m

can solve the problem in full generality, thus establishing fundamental complexity-theoretic limitations. Finally, we employ our framework to compute nonstabilizerness in different Hamiltonian ground states, demonstrating the practical performance of our method in regimes beyond the reach of existing techniques.