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Stabilizer entropies and nonstabilizerness monotones
Tobias Haug, Lorenzo Piroli
TL;DR
The paper asks whether stabilizer entropies are reliable nonstabilizerness monotones and how they relate to established magic measures. It analyzes explicit protocols and measurement conditions, compares SEs with other monotones, and develops MPS methods for computation. SEs fail general monotonicity for 0 ≤ n < 2 and strong monotonicity for every Rényi index, while remaining useful for comparisons and efficient numerical evaluation.
Problem
The paper addresses whether stabilizer entropies are monotones under stabilizer protocols and how they compare with known nonstabilizerness measures.
Method
The authors use explicit counterexamples, analytical and numerical comparisons with magic monotones, and tensor-network methods including perfect MPS sampling.
Results
SEs are not monotones for 0 ≤ n < 2, even for pure-state protocols, and do not satisfy strong monotonicity under computational-basis measurements for any Rényi index.
Takeaways & Limitations
Despite these failures, SEs can support comparisons and bounds involving genuine monotones and can be computed efficiently for important MPS classes.
Takeaways & Limitations
Whether SE monotonicity holds for Rényi index n ≥ 2 remains unresolved, and the paper identifies the need for an N-independent SE lower bound in terms of min-relative entropy.
Abstract
from arXiv · showhide
We study different aspects of the stabilizer entropies (SEs) and compare them against known nonstabilizerness monotones such as the min-relative entropy and the robustness of magic. First, by means of explicit examples, we show that, for Rényi index $0\leq n<2$, the SEs are not monotones with respect to stabilizer protocols which include computational-basis measurements, not even when restricting to pure states (while the question remains open for $n\geq 2$). Next, we show that, for any Rényi index, the SEs do not satisfy a strong monotonicity condition with respect to computational-basis measurements. We further study SEs in different classes of many-body states. We compare the SEs with other measures, either proving or providing numerical evidence for inequalities between them. Finally, we discuss exact or efficient tensor-network numerical methods to compute SEs of matrix-product states (MPSs) for large numbers of qubits. In addition to previously developed exact methods to compute the Rényi SEs, we also put forward a scheme based on perfect MPS sampling, allowing us to compute efficiently the von Neumann SE for large bond dimensions.
1 Introduction
The paper examines stabilizer entropies as measures of nonstabilizerness, focusing on their monotonicity, relations to other magic measures, and efficient computation for many-body states. It finds failures of monotonicity in important settings while developing methods for evaluating SEs in MPSs.
- Motivation: Stabilizer entropies quantify nonstabilizerness using Pauli-string expectation values without minimization procedures.Although typical-state computation is exponentially costly, SEs can be computed efficiently for matrix-product states.
- Stabilizer entropies: For pure states, the SE vanishes exactly on stabilizer states, is Clifford-invariant, and is additive under tensor products.The paper restricts its analysis to pure states because the zero-characterization does not generally hold for mixed stabilizer states.
- Open questions: Whether SEs are monotones under stabilizer protocols was an open resource-theory question, including protocols with measurements, discarding, and conditioned operations.The paper also asks how SEs compare with established magic monotones such as min-relative entropy and robustness of magic.
- Main results: For Rényi index 0 ≤ n < 2, the paper gives counterexamples showing that SEs are not monotones under general stabilizer protocols, even for pure states.It also shows that SEs fail strong monotonicity under computational-basis measurements for every Rényi index, contradicting an earlier claim for Rényi-1/2 SE.
- Further contributions: The study compares SEs with known magic monotones, proves or numerically supports inequalities, and develops tensor-network methods for MPS computation.A perfect-MPS-sampling scheme enables efficient computation of the von Neumann SE for large bond dimensions, beyond the reach of an earlier exact method.
- Applications: The numerical techniques are motivated by applications of stabilizer entropies to quantum certification, purity estimation, many-body quantum chaos, and entanglement spectra.These applications are presented as reasons the computational methods may be practically useful despite the monotonicity results.
2 Stabilizer Protocols and monotonicity
This section defines stabilizer protocols, stabilizer entropies, and two established magic monotones, then formulates ordinary and strong monotonicity conditions. The paper’s central result is that SEs fail these conditions in specified settings despite remaining potentially useful for comparison and computation.
- Stabilizer protocols: Stabilizer protocols comprise Clifford operations, stabilizer-state composition, computational-basis measurements, discarding, and operations conditioned on measurement outcomes.These operations form a minimal commonly used set of free operations in the resource theory of nonstabilizerness.
- Monotonicity conditions: Strong monotonicity requires that average nonstabilizerness does not increase after computational-basis measurements.This condition is desirable in many-body settings but is not required for a function to qualify as a magic monotone.
- Stabilizer entropies: The Rényi stabilizer entropy is defined for pure states through the distribution of squared Pauli expectation values, with the von Neumann SE obtained at n → 1.The Pauli-weight distribution sums to one and gives the SE as a Rényi entropy up to an offset −N log 2.
- Known monotones: Robustness of magic and min-relative entropy are established nonstabilizerness monotones, but only robustness satisfies strong monotonicity.The min-relative entropy measures the distance from a pure state to its nearest stabilizer state through stabilizer fidelity.
- Main conclusion: For 0 ≤ n < 2, SEs fail monotonicity under stabilizer protocols, and for every n they fail strong monotonicity under computational-basis measurements.The paper notes that these failures do not eliminate the possibility that SEs provide useful bounds through order relations with genuine monotones.
3 Counterexamples to monotonicity
The paper constructs explicit pure-state counterexamples showing that stabilizer entropies violate monotonicity for 0 ≤ n < 2 and strong monotonicity for every Rényi index under computational-basis measurements.
- Violation of monotonicity for Rényi index 0 ≤ n < 2: For 0 ≤ n < 2, a four-qubit stabilizer protocol deterministically transforms |φ∗⟩ into |ψ∗⟩ while increasing the stabilizer entropy.The protocol measures the first qubit and applies outcome-dependent Clifford feedback; the feedback makes the output independent of the measurement outcome for this input.
- Violation of monotonicity for Rényi index 0 ≤ n < 2: For n ≥ 2, the four-qubit construction finds no monotonicity violation, while the question remains open even for pure states.Increasing the system size yields strong-monotonicity violations beyond n = 2, but not the deterministic monotonicity violation required here.
- Violation of strong monotonicity: For 0 ≤ n < 2, the same counterexample violates strong monotonicity because the output entropy exceeds the input entropy.Additivity and Mn(|1⟩) = 0 reduce the comparison to the previously established entropy difference.
- Violation of strong monotonicity: For sufficiently large N and n > 1, a single-qubit measurement violates strong monotonicity, including all n ≥ 2.The relevant outcome probability remains finite as N → ∞, and the nonstabilizerness can increase from O(1) to O(N) with finite probability.
- Violation of strong monotonicity: The results contradict earlier claims that the Rényi-1/2 stabilizer entropy satisfies strong monotonicity.The contradiction follows because the Rényi-1/2 stabilizer entropy coincides with a measure previously claimed to obey that condition.
4 Relations to other monotones
The paper compares stabilizer entropies with the min-relative entropy and robustness of magic, deriving upper bounds and showing that some N-independent lower bounds cannot exist.
- The SEs are compared with the min-relative entropy and robustness of magic through analytical inequalities and numerical evidence.
- For n > 1, an upper bound on the SE is derived in terms of the min-relative entropy.
- An N-independent lower bound for the SE in terms of the log-robustness is impossible for n > 1/2.
- Numerical minimization provides evidence that the ratio Mn(|ψ⟩)/Dmin(|ψ⟩) approaches a constant depending only on n for small n and up to four qubits.
5 Numerical methods for the SE
The paper develops tensor-network methods for computing stabilizer entropies of matrix-product states, including exact Rényi methods and perfect sampling for the von Neumann SE. These methods support efficient calculations for large systems, with sampling accuracy controlled by the sample count.
- For MPSs, the exact integer-Rényi method replaces an exponential sum with an MPS norm of bond dimension χ^2n, costing O(Nχ^6n).
- Perfect MPS sampling computes the von Neumann SE by exactly sampling the probability distribution ΞP(|ΨN⟩).
- The conditional Pauli-string probabilities can be evaluated for MPSs at cost O(χ^3N), improving favorably on exact integer-Rényi evaluation in χ.
- In the XXZ ground state at half-filling, von Neumann and Rényi-2 SE densities vary similarly with anisotropy, while their decrease near Δ = 1 becomes sharper at larger N.
- Extending perfect sampling to arbitrary Rényi indices may require a number of samples that scales exponentially with N for fixed accuracy.
6 Outlook
The paper concludes that stabilizer entropies fail to be genuine monotones in the studied regimes, while remaining computationally useful. It identifies efficient monotones and broader lower-bound questions as open problems.
- For 0 ≤ n < 2, SEs are not monotones under general stabilizer protocols, and they are not strong monotones for any Rényi index.
- Whether monotonicity can be established or disproved for n ≥ 2 remains open.
- The paper leaves open whether an N-independent lower bound relates an SE to the min-relative entropy for some Rényi index.
- The results motivate finding genuine nonstabilizerness monotones that remain efficiently computable for restricted pure-state classes such as MPSs.
A Details on the violation of monotonicity
The appendix details numerical searches and asymptotic checks behind the monotonicity counterexamples, including a four-qubit state that violates both strong and ordinary monotonicity.
- Gradient descent found four-qubit states violating strong monotonicity for 0 ≤ n < 2, while no violation was found for N ≤ 3.
- The state obtained from the strong-monotonicity search is also a counterexample to ordinary monotonicity because it is an eigenstate of a Clifford operator.
- For N = 5, the numerical minimization becomes cumbersome and did not produce states with the same useful Clifford-eigenstate properties.
- For the constructed state |ψε⟩ and n > 1, the relevant SE remains bounded in N, with corrections that are exponentially small.
- Exact evaluation verifies the asymptotic prediction (72), with numerical data showing excellent agreement for ε = 0.5.
B Additional numerical results
Additional numerical results validate the perfect-MPS sampling method and examine how stabilizer-entropy estimates depend on bond dimension. Sampling errors follow the expected inverse-square-root scaling, while both m1 and m2 converge as χ increases.
- The von Neumann SE estimate m1 converges as the MPS bond dimension χ increases.The estimate uses the method presented in Sec. 5 and is evaluated for the half-filled Heisenberg-model ground state.
- The Rényi-2 SE density m2 decreases and converges to a constant as χ increases.m2 is computed using the method of Ref. [25] for the half-filled Heisenberg-model ground state.
- Figure 5 jointly compares m2, the estimated m1, and the fidelity of the MPS approximation with the true ground state as functions of χ.The true ground state is estimated using a very large bond dimension, approximately χ ≃ 200.