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arXiv:2610.17454

Norm properties of quantum uniformity measures

Berke Ünal

math-phmath.FAquant-phSubmitted Oct 12, 2026• Announced Oct 14, 2026• Revised Oct 20, 2026 (v2)
v1 · Oct 12, 2026v2 · Oct 20, 2026
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Abstract

We give a complete classification of the norm properties of quantum uniformity measures, introduced by Bu, Gu, and Jaffe to analytically characterize the Clifford hierarchy. Resolving an open question they posed, we show that quantum uniformity measures define norms precisely for $k=2,3,4$. In particular, we establish the triangle inequality for $k=4$, whereas for every $k\geq5$ it fails in some dimension. The higher-order result is proved using the structure theory of norming systems of linear equations over finite fields. We also study the geometry of quantum uniformity measures, establishing explicit bounds on the modulus of convexity whenever they define norms. Finally, we give a complete classification of the norm properties of matrix-valued Gowers uniformity functionals, showing that only $k=2$ defines a norm.

Comments
19 pages; added results on the uniform convexity of quantum uniformity measures and made explicit the classification of norm properties of matrix-valued uniformity functionals

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We give a complete classification of the norm properties of quantum uniformity measures, introduced by Bu, Gu, and Jaffe to analytically characterize the Clifford hierarchy. Resolving an open question they posed, we show that quantum uniformity measures define norms precisely for k=2,3,4k=2,3,4k=2,3,4. In particular, we establish the triangle inequality for k=4k=4k=4, whereas for every k≥5k\geq5k≥5 it fails in some dimension. The higher-order result is proved using the structure theory of norming systems of linear equations over finite fields. We also study the geometry of quantum uniformity measures, establishing explicit bounds on the modulus of convexity whenever they define norms. Finally, we give a complete classification of the norm properties of matrix-valued Gowers uniformity functionals, showing that only k=2k=2k=2 defines a norm.
We consider the problem of testing whether an unknown unitary is close to a specified level of the Clifford hierarchy. Bu, Gu, and Jaffe proposed a candidate tester for this task based on a connection with noncommutative analogues of the Gowers uniformity norms. The complexity of this tester -- whose analysis depends on a robust characterization of the near-extremizers of these norms -- was left open. We establish such a characterization for the fourth noncommutative uniformity norm and, as a consequence, obtain an efficient tester for the third level of the Clifford hierarchy. We further discuss possible routes toward resolving the problem of testing for all higher levels, highlighting the main barriers that remain.
We propose a mathematical framework that we call quantum, higher-order Fourier analysis. This generalizes the classical theory of higher-order Fourier analysis, which led to many recent advances in number theory and combinatorics. We define a family of "quantum measures" on linear transformations on a Hilbert space, that reduce in the case of diagonal matrices to the uniformity norms introduced by Timothy Gowers. We show that our quantum measures and our related theory of quantum higher-order Fourier analysis characterize the Clifford hierarchy, an important notion of complexity in quantum computation. In particular, we give a necessary and sufficient analytic condition that a unitary is an element of the [Formula: see text] level of the Clifford hierarchy.
Abstract We propose a family of quantum algorithms for estimating Gowers uniformity norms Uk U^k Uk U k over finite abelian groups, extending earlier quantum methods for the U2 U^2 U2 U 2 -norm to arbitrary prime fields and higher-order uniformity norms. Our algorithms prepare quantum states encoding higher-order finite differences and apply Fourier sampling together with amplitude estimation to obtain estimates of Gowers norms. As a central application, we study algebraic property testing problems of distinguishing whether a bounded function f:Fpn→C f: \mathbb {F}_p^n \rightarrow \mathbb {C} f:Fpn​→C f : F p n → C is a low-degree phase polynomial or is far from any such structure. We show that whenever an inverse theorem for the Ud+1 U^{d+1} Ud+1 U d + 1 -norm is available, our quantum framework yields a corresponding structure-testing algorithm whose query and measurement complexity depends explicitly on the quantitative bounds of that inverse theorem. In particular, using the recent quasipolynomial inverse theorem for the U4 U^4 U4 U 4 -norm over Fpn \mathbb {F}_p^n Fpn​ F p n , we obtain quasipolynomial-time quantum algorithms for detecting cubic phase polynomials. For higher-order norms such as U5 U^5 U5 U 5 and U6 U^6 U6 U 6 over F2n \mathbb {F}_2^n F2n​ F 2 n , the best known inverse theorems provide only tower-type quantitative bounds; accordingly, our detection algorithms remain correct but inherit complexity corresponding to tower-type quantitative bounds. We also present a quantum method for estimating the number of 3-term arithmetic progressions in Boolean functions via the U2 U^2 U2 U 2 -norm. Although not query-optimal compared to Grover-style counting, this approach is sensitive to additive structure and naturally aligned with tools from higher-order Fourier analysis. Finally, we observe that Gowers norms are invariant under certain classes of shift-type noise, implying that our algorithms retain robustness under natural quantum noise models. This suggests that Gowers norm-based quantum procedures may serve as stable primitives for quantum property testing, learning theory, and the analysis of pseudorandomness in the NISQ regime.
Abstract We introduce a framework unifying the mathematical characterisation of different measures of general quantum resources and allowing for a systematic way to define a variety of faithful quantifiers for any given convex quantum resource theory. The approach allows us to describe many commonly used measures such as matrix norm-based quantifiers, robustness measures, convex roof-based measures, and witness-based quantifiers together in a common formalism based on the convex geometry of the underlying sets of resource-free states. We establish easily verifiable criteria for a measure to possess desirable properties such as faithfulness and strong monotonicity under relevant free operations, and show that many quantifiers obtained in this framework indeed satisfy them for any considered quantum resource. We derive various bounds and relations between the measures, generalising and providing significantly simplified proofs of results found in the resource theories of quantum entanglement and coherence. We also prove that the quantification of resources in this framework simplifies for pure states, allowing us to obtain more easily computable forms of the considered measures, and show that many of them are in fact equal on pure states. Further, we investigate the dual formulation of resource quantifiers, which provide a characterisation of the sets of resource witnesses. We present an explicit application of the results to the resource theories of multi-level coherence, entanglement of Schmidt number k , multipartite entanglement, as well as magic states, providing insight into the quantification of the four resources by establishing novel quantitative relations and introducing new quantifiers, such as a measure of entanglement of Schmidt number k which generalises the convex roof–extended negativity, a measure of k -coherence which generalises the ℓ 1 norm of coherence, and a hierarchy of norm-based quantifiers of k -partite entanglement generalising the greatest cross norm.
Abstract. We consider a family of vector and operator norms which we refer to as Schmidt norms. We show that these norms have several uses in quantum information theory – they can be used to help classify k-positive linear maps (and hence entanglement witnesses), they are useful for approaching the problem of finding non-positive partial transpose bound entangled Werner states, they are related to the quantum fidelity and trace distance measures, and they are connected to the recently-defined local numerical range. We show that the vector norms can be explicitly calculated, and we derive several inequalities in order to bound the operator norms and compute them in special cases. We show that one particular entangled Werner state is bound entangled if and only if a certain norm inequality holds on a given family of projections, and we use our inequalities to study that family of projections. We also develop a family of semidefinite programs that can be used to further bound the operator norms. We extend these norms to arbitrary convex mapping cones and explore their implications with positive partial transpose states. 1.
There are various notions of positivity for matrices and linear matrix-valued maps that play important roles in quantum information theory. The cones of positive semidefinite matrices and completely positive linear maps, which represent quantum states and quantum channels respectively, are the most ubiquitous positive cones. There are also many natural cones that can been regarded as "more" or "less" positive than these standard examples. In particular, entanglement theory deals with the cones of separable operators and entanglement witnesses, which satisfy very strong and weak positivity properties respectively. Rather complementary to the various cones that arise in entanglement theory are norms. The trace and operator norms for operators and the diamond and completely bounded norms for superoperators are the typical norms that are seen throughout quantum information theory. In this work our main goal is to develop a family of norms that play a role analogous to the cone of entanglement witnesses. We investigate the basic mathematical properties of these norms, including their relationships with other well-known norms, their isometry groups, and their dual norms. We also make the place of these norms in entanglement theory rigorous by showing that entanglement witnesses arise from minimal operator systems, and analogously our norms arise from minimal operator spaces. Finally, we connect the various cones and norms considered here to several seemingly unrelated problems from other areas. We characterize the problem of whether or not non-positive partial transpose bound entangled states exist in terms of one of our norms, and provide evidence in favour of their existence. We also characterize the minimum gate fidelity of a quantum channel, the maximum output purity and its completely bounded counterpart, and the geometric measure of entanglement in terms of these norms.