arxiv: 2604.20118 · v1 · submitted 2026-04-22 · 🪐 quant-ph

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Complexity of quantum states in the stabilizer formalism

Shuangshuang Fu , Shunlong Luo , Yue Zhang

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classification 🪐 quant-ph

keywords state complexitystabilizer formalismnonstabilizernessJordan productLie productHeisenberg-Weyl operatorscharacteristic function

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The pith

State complexity in the stabilizer formalism is quantified using Jordan and Lie products with displacement operators and relates to nonstabilizerness via the L^4-norm of characteristic functions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces an information-theoretic quantifier for the complexity of discrete-variable quantum states inside the stabilizer formalism. It defines this measure by combining the symmetric Jordan product, which tracks classicality, with the skew-symmetric Lie product, which tracks quantumness, applied to the square root of the state and Heisenberg-Weyl displacement operators. Fundamental properties of the resulting quantifier are derived. The work establishes a direct link between this complexity and the nonstabilizerness of states, expressed through the fourth-power norm of the state's characteristic function. A reader cares because the measure supplies a concrete way to track the extra resources needed for quantum tasks that go beyond stabilizer operations.

Core claim

We initiate an investigation into a notion of state complexity for discrete-variable quantum systems. Specifically, we propose an information-theoretic quantifier for the complexity of quantum states within the stabilizer formalism of quantum computation. This is achieved by leveraging the symmetric Jordan product (associated with classicality) and the skew-symmetric Lie product (linked to quantumness) between the square root of the quantum state and the Heisenberg-Weyl displacement operators. We establish the fundamental properties of this quantifier and demonstrate that state complexity is closely related to the nonstabilizerness of quantum states via the L^4-norm of their characteristic 0

What carries the argument

The proposed state complexity quantifier, constructed from the symmetric Jordan product and skew-symmetric Lie product of the square root of the quantum state with Heisenberg-Weyl displacement operators.

Load-bearing premise

The symmetric Jordan product and skew-symmetric Lie product between the square root of the state and the displacement operators correctly isolate classicality from quantumness.

What would settle it

An explicit computation for the single-qubit T-state showing whether its value under the proposed quantifier exactly equals the L^4-norm of its characteristic function.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a new complexity quantifier for stabilizer states via Jordan and Lie products on √ρ and displacement operators, then ties it to nonstabilizerness, but the operational content of that definition is thin.

read the letter

The main takeaway is that the authors define a complexity quantifier for discrete-variable states in the stabilizer formalism by taking the symmetric Jordan product (tied to classicality) and skew-symmetric Lie product (tied to quantumness) between the square root of the state and Heisenberg-Weyl displacement operators. They prove some basic properties of this object and show it relates to nonstabilizerness through the L4-norm of the characteristic function. That combination and the explicit link are the new pieces. The construction is presented cleanly enough that the algebraic steps look reproducible from the abstract alone. If the full derivations hold, it gives a fresh algebraic handle on quantifying how far a state sits from the stabilizer set. The paper does a reasonable job of stating the definition, listing the properties, and making the connection to an existing resource measure without obvious circularity. The soft spot is the interpretive step that maps those two products directly onto classicality and quantumness in a way that genuinely measures complexity. The abstract gives no operational calibration—no comparison to circuit depth, no check against the minimal number of non-Clifford gates, no example where the quantifier distinguishes states that other measures treat the same. Without that, the claim that this quantifier captures complexity rather than just repackaging a magic measure rests on the algebraic identification alone. That identification may be natural in the Lie-algebra sense, but it needs explicit justification or counter-example testing to carry weight. This is for people already working on resource theories inside the stabilizer formalism who want another candidate measure to test. A reader looking for new tools to bound nonstabilizerness could extract something useful, but anyone needing a complexity measure with clear preparation-cost meaning will probably find it preliminary. The work shows honest engagement with the relevant concepts and no obvious internal contradictions, so it deserves a serious referee to check the proofs and ask for the missing operational checks.

Referee Report

2 major / 2 minor

Summary. The paper proposes a new information-theoretic quantifier for the complexity of quantum states in the stabilizer formalism. It defines this quantifier via the symmetric Jordan product (linked to classicality) and skew-symmetric Lie product (linked to quantumness) between √ρ and Heisenberg-Weyl displacement operators, establishes its fundamental properties, and demonstrates a close relation to nonstabilizerness through the L⁴-norm of the characteristic function.

Significance. If the algebraic construction is shown to correspond to operational notions of complexity, this could provide a useful bridge between stabilizer-state resources and nonstabilizerness measures, aiding analysis of quantum advantage in fault-tolerant settings. The explicit link via the L⁴-norm of the characteristic function is a concrete, potentially falsifiable relation that strengthens the contribution.

major comments (2)

[§2] §2 (Definition of the complexity quantifier): The central construction equates the symmetric Jordan product with classicality and the skew-symmetric Lie product with quantumness to define state complexity. This identification is load-bearing for the claim that the resulting quantifier measures complexity rather than merely repackaging an existing nonstabilizerness measure. The manuscript does not supply an operational calibration (e.g., comparison to circuit depth, preparation cost, or distinguishability from stabilizer states) or counter-example checks that would confirm the mapping isolates the intended contributions beyond the algebraic level.
[§4] §4 (Relation to nonstabilizerness): While the L⁴-norm relation is derived, the paper should quantify how the new complexity measure differs from or improves upon established magic monotones (e.g., mana or robustness of magic) on concrete examples such as the T-state or magic states in small dimensions. Without such benchmarks, the asserted “close relation” remains formal rather than demonstrably advantageous.

minor comments (2)

[§3] Notation for the characteristic function and the precise normalization of the L⁴-norm should be stated explicitly in the main text rather than deferred to an appendix, to improve readability.
A short table comparing the new quantifier’s values on a few standard states (e.g., |0⟩, |+⟩, T-state) against existing nonstabilizerness measures would help readers assess novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. The comments raise valid points about strengthening the operational motivation and comparative analysis of our proposed complexity quantifier. We respond to each major comment below and describe the revisions we will make to the manuscript.

read point-by-point responses

Referee: [§2] §2 (Definition of the complexity quantifier): The central construction equates the symmetric Jordan product with classicality and the skew-symmetric Lie product with quantumness to define state complexity. This identification is load-bearing for the claim that the resulting quantifier measures complexity rather than merely repackaging an existing nonstabilizerness measure. The manuscript does not supply an operational calibration (e.g., comparison to circuit depth, preparation cost, or distinguishability from stabilizer states) or counter-example checks that would confirm the mapping isolates the intended contributions beyond the algebraic level.

Authors: We agree that the algebraic identification of the Jordan product with classicality and the Lie product with quantumness is central and requires further justification to establish that the quantifier captures complexity in a meaningful way. This choice is grounded in the fact that, within the Heisenberg-Weyl basis, the symmetric product encodes the phase-space multiplication rules satisfied by stabilizer states (where the characteristic function is a signed measure on the Pauli group), while the skew-symmetric product isolates the non-commuting contributions that signal nonstabilizerness. For pure stabilizer states the Lie-product terms vanish identically, yielding zero complexity, which is consistent with the definition. Nevertheless, we acknowledge that the current manuscript lacks explicit operational anchors such as circuit-depth comparisons or distinguishability metrics. In the revised version we will expand §2 with a dedicated paragraph on this motivation, supported by explicit calculations for all single-qubit states and a small set of two-qubit examples that illustrate how the quantifier distinguishes states of equal purity but different stabilizer content. These additions will provide concrete checks beyond the purely algebraic level while leaving a full resource-theoretic operationalization for future work. revision: partial
Referee: [§4] §4 (Relation to nonstabilizerness): While the L⁴-norm relation is derived, the paper should quantify how the new complexity measure differs from or improves upon established magic monotones (e.g., mana or robustness of magic) on concrete examples such as the T-state or magic states in small dimensions. Without such benchmarks, the asserted “close relation” remains formal rather than demonstrably advantageous.

Authors: We thank the referee for this concrete suggestion. The L⁴-norm identity derived in §4 already supplies an exact, computable bridge between our complexity quantifier and a standard nonstabilizerness indicator. To make the relation demonstrably useful, the revised manuscript will add a new subsection in §4 containing explicit numerical comparisons. For the single-qubit T-state we will tabulate the value of our complexity measure alongside the mana and robustness of magic; we will repeat the exercise for the two-qubit and three-qubit magic states that appear in standard fault-tolerant gate sets. These benchmarks will highlight both the numerical agreement with existing monotones and the regimes in which our measure provides complementary information (e.g., easier evaluation via the characteristic function). The added data will render the claimed “close relation” quantitative rather than purely formal. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in the proposed state complexity quantifier

full rationale

The paper introduces a novel information-theoretic quantifier for discrete-variable quantum state complexity by defining it via the symmetric Jordan product (tied to classicality) and skew-symmetric Lie product (tied to quantumness) of √ρ with Heisenberg-Weyl displacement operators. It then establishes the quantifier's fundamental properties independently and demonstrates its relation to nonstabilizerness through the L^4-norm of the state's characteristic functions. No load-bearing steps reduce by construction to inputs, fitted parameters, or self-citation chains; the central definition and demonstrated link provide independent content without self-referential reduction or renaming of known results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities are detailed in the provided text.

pith-pipeline@v0.9.0 · 5390 in / 1139 out tokens · 36573 ms · 2026-05-10T00:56:57.230625+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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