Discrete Pauli Pairs: Sampling & Structure

• Discrete Pauli pairs are defined as pairs of functions or quantum operators whose amplitudes match on prescribed discrete spatial and Fourier sampling sets, extending classical phase retrieval.
• They reveal sharp density thresholds that determine when discrete sampling forces global modulus equality, distinguishing between full and weak recovery under Gaussian decay conditions.
• In finite quantum systems, discrete Pauli pairs characterize commutation and anticommutation relations among Pauli operators, with implications for stabilizer theory, finite geometry, and efficient algorithmic methods.

Discrete Pauli pairs denote a family of phase-retrieval and Pauli-structure problems in which only discrete or finitely encoded data are retained. In the harmonic-analytic sense now standard in recent work, a discrete Pauli pair consists of two functions $f,g$ whose moduli agree on a discrete sampling set $\Lambda$ and whose Fourier-transform moduli agree on a discrete sampling set $M$; the central question is when such sampled equalities already force the global modulus equalities of a classical Pauli pair (Lysen, 21 May 2026). In a distinct but related quantum-information usage, the term is applied to finite pairs of generalized Pauli operators or Pauli strings, classified by whether they commute or anticommute via a symplectic commutation law, with consequences for Pauli graphs, finite geometry, stabilizer theory, and fast classical subroutines (Sarkar et al., 2023).

1. Historical origin and basic definitions

Pauli’s original reconstruction question asked whether a quantum state is determined, up to a constant phase, by its position and momentum densities. In $L^2(\mathbb R^n)$, a classical Pauli pair is a pair $f,g$ such that

$$|f|=|g| \quad \text{and} \quad |\widehat f|=|\widehat g|$$

almost everywhere, while $f$ and $g$ are not related by a trivial global phase (Shkarin, 2012). Shkarin’s work made this failure of uniqueness explicit through new constructions, including spatially localized examples and step-function examples governed by finite autocorrelation constraints (Shkarin, 2012).

Recent work recasts this as a sampled or discrete problem. For discrete sets $\Lambda,M\subset\mathbb R$, $f,g$ form a discrete Pauli pair for $\Lambda$0 when

$\Lambda$1

The corresponding global notion is still the ordinary Pauli pair, and a weak Pauli pair is defined by the weaker alternative that either $\Lambda$2 globally or $\Lambda$3 globally (Lysen, 21 May 2026).

The modern sharp theory is formulated for

$\Lambda$4

and for the Gaussian-decay class

$\Lambda$5

with the symmetric case $\Lambda$6, $\Lambda$7, playing the decisive role (Lysen, 21 May 2026). In this setting, “Gaussian decay” means

$\Lambda$8

A persistent misconception is that sampled modulus agreement should determine a state on arbitrary discrete sets. It does not: without extra decay assumptions, there exist $\Lambda$9 that form a discrete Pauli pair on arbitrary discrete $M$0 while neither global modulus agrees (Lysen, 21 May 2026).

2. Sharp density thresholds for sampled Pauli uniqueness

The principal recent advance is the determination of sharp density thresholds for when a discrete Pauli pair with Gaussian decay must already be a classical Pauli pair (Lysen, 21 May 2026). The relevant density is encoded through the asymptotic spacings

$M$1

for sequences unbounded in both directions.

A first threshold concerns one-sided modulus recovery. Define

$M$2

If $M$3 and

$M$4

then sampled equality $M$5 on $M$6 forces

$M$7

Conversely, if the corresponding $M$8 is strictly above $M$9, there exist $L^2(\mathbb R^n)$0 with sampled equality on $L^2(\mathbb R^n)$1 but not global modulus equality (Lysen, 21 May 2026).

Combining this with the decay-transfer theorem gives the sharp full Pauli threshold. If

$L^2(\mathbb R^n)$2

and $L^2(\mathbb R^n)$3, $L^2(\mathbb R^n)$4, then

$L^2(\mathbb R^n)$5

(Lysen, 21 May 2026). The optimal constant in the Ramos–Sousa theorem is therefore

$L^2(\mathbb R^n)$6

A second threshold governs weak Pauli recovery. Define

$L^2(\mathbb R^n)$7

If

$L^2(\mathbb R^n)$8

and $L^2(\mathbb R^n)$9, $f,g$0 form a discrete Pauli pair, then they form a weak Pauli pair: $f,g$1 Conversely, if the corresponding $f,g$2 condition is strictly above $f,g$3, then there exist discrete Pauli pairs that are not even weak Pauli pairs (Lysen, 21 May 2026).

These two thresholds are genuinely different. For $f,g$4, the full and weak thresholds coincide; for smaller $f,g$5, they separate, and the weak threshold isolates a specifically phaseless phenomenon that does not appear in ordinary Fourier uniqueness theory (Lysen, 21 May 2026).

3. Proof architecture, nonuniqueness mechanisms, and Hardy-type consequences

The harmonic-analytic theory proceeds by converting phaseless sampling into a zero-set problem for entire functions. The standard auxiliary objects are

$f,g$6

so that sampled modulus equality on $f,g$7 becomes $f,g$8 for all $f,g$9. On the Fourier side one similarly considers

$$|f|=|g| \quad \text{and} \quad |\widehat f|=|\widehat g|$$0

Indicator lower bounds derived from dense real zeros are then compared against Gaussian growth bounds obtained from the decay class $$|f|=|g| \quad \text{and} \quad |\widehat f|=|\widehat g|$$1 (Lysen, 21 May 2026).

For the weak-Pauli threshold, the argument is refined by introducing

$$|f|=|g| \quad \text{and} \quad |\widehat f|=|\widehat g|$$2

which yields

$$|f|=|g| \quad \text{and} \quad |\widehat f|=|\widehat g|$$3

and an analogous expression for $$|f|=|g| \quad \text{and} \quad |\widehat f|=|\widehat g|$$4. A nontrivial optimization over decay parameters attached to $$|f|=|g| \quad \text{and} \quad |\widehat f|=|\widehat g|$$5 and $$|f|=|g| \quad \text{and} \quad |\widehat f|=|\widehat g|$$6 produces the exact piecewise formula for $$|f|=|g| \quad \text{and} \quad |\widehat f|=|\widehat g|$$7 (Lysen, 21 May 2026).

The sharpness constructions use the standard phaseless ansatz

$$|f|=|g| \quad \text{and} \quad |\widehat f|=|\widehat g|$$8

If $$|f|=|g| \quad \text{and} \quad |\widehat f|=|\widehat g|$$9 and $f$0 are engineered to vanish on complementary parts of the sampling sets, sampled modulus equalities follow automatically. The sharp constructions combine Hardy-class Fourier nonuniqueness, canonical products with prescribed indicators, lower bounds on derivatives at zeros, and weighted simultaneous interpolation (Lysen, 21 May 2026).

The pre-sharp stage of the theory was established earlier. It was shown that if one function has Gaussian decay in space and frequency and the sampling sets accumulate like suitable small multiples of $f$1 at infinity, then sampled modulus equality implies global modulus equality; conversely, arbitrary discrete sets or insufficient accumulation admit counterexamples (Ramos et al., 2024). The same work also established a discrete Hardy principle: for $f$2, sufficiently dense discrete sampling together with

$f$3

forces $f$4, matching the classical Hardy threshold $f$5 (Ramos et al., 2024).

Historically, the nonuniqueness side reaches back to explicit continuous constructions. Shkarin exhibited step-function Pauli pairs whose coefficient vectors satisfy equal modulus and equal autocorrelation constraints, and also constructed countably infinite UZD-sets in $f$6 and finite UZD-sets in $f$7, all of which generate large families of Pauli states by phase coding (Shkarin, 2012). This suggests that the sampled theory is best viewed not as an isolated uniqueness theorem, but as a phaseless nonlinear analogue of Fourier uniqueness with a large underlying nonuniqueness landscape.

4. Discrete Pauli pairs in finite Pauli systems

A separate literature uses essentially the same phrase for finite pairs of generalized Pauli operators, especially when the pair relation is commuting versus anticommuting. For phase-free $f$8-qubit Pauli strings,

$f$9

the support and weight are

$g$0

For two strings $g$1, the conflict set is

$g$2

and the commutation sign is determined by its parity: $g$3 Hence $g$4 and $g$5 commute iff $g$6 is even, and anticommute iff it is odd (Cha et al., 10 May 2026).

In the binary symplectic representation,

$g$7

an $g$8-qubit string becomes $g$9, and

$\Lambda,M\subset\mathbb R$0

This is the standard stabilizer-theoretic criterion, and it makes discrete Pauli pairs a symplectic parity problem (Cha et al., 10 May 2026).

For qudits of dimension $\Lambda,M\subset\mathbb R$1, the corresponding phase-stripped operators are encoded by exponent vectors $\Lambda,M\subset\mathbb R$2, and commutation is governed by the alternating bilinear form

$\Lambda,M\subset\mathbb R$3

Two Pauli operators commute exactly iff

$\Lambda,M\subset\mathbb R$4

Failure to commute is measured by the central phase

$\Lambda,M\subset\mathbb R$5

(Sarkar et al., 2023).

For a single qudit of dimension $\Lambda,M\subset\mathbb R$6, the same structure is often written with representatives $\Lambda,M\subset\mathbb R$7, in which case

$\Lambda,M\subset\mathbb R$8

so two operators commute iff

$\Lambda,M\subset\mathbb R$9

(Planat et al., 2011). A common misconception here is to identify “commuting up to phase” with exact commutation. The module-theoretic literature is explicit that exact commutation means commutator $f,g$0, while a nontrivial central phase is a genuine failure to commute (Sarkar et al., 2023).

5. Graphs, finite geometry, and contextual structures

Once the pair relation is fixed by the symplectic form, it can be organized globally in several ways. For a single qudit of dimension $f,g$1, the Pauli graph $f,g$2 has as vertices the $f,g$3 nontrivial phase classes $f,g$4, with adjacency defined by commuting pairs. Maximal cliques are maximal mutually commuting sets, each of size $f,g$5, and the total number of such maximal commuting sets is $f,g$6, while the projective-line subset associated with free cyclic submodules has size

$f,g$7

(Planat et al., 2011). The distinction between square-free and non-square-free $f,g$8 is structural: when $f,g$9 contains a square, $\Lambda$00, producing extra isotropic lines and more intricate intersection patterns (Planat et al., 2011).

For qubit–qu$\Lambda$01dit systems, the organization of maximal commuting sets can be lifted to an incidence geometry in which points are maximal commuting sets and collinearity means intersection in exactly $\Lambda$02 elements. Explicit computations for $\Lambda$03 show a nested configuration of

$\Lambda$04

with exceptional maximal commuting sets forming a recursive core; the authors stressed that these generic properties were inferred from purely computer-handled cases of $\Lambda$05, and a rigorous proof for $\Lambda$06 remained open (Saniga et al., 2011).

For $\Lambda$07-qubit systems, commuting Pauli pairs are encoded in the symplectic polar space $\Lambda$08: two Pauli observables commute iff their associated points are symplectically orthogonal. Maximal pairwise commuting sets correspond to generators $\Lambda$09, and these generators are parametrized by the binary Lagrangian Grassmannian $\Lambda$10. Over $\Lambda$11, principal-minor coordinates yield a bijection between maximal commuting classes of the $\Lambda$12-qubit Pauli group and a distinguished subset of observables of the $\Lambda$13-qubit Pauli group (Holweck et al., 2013).

These same pairwise structures also control contextuality and hidden-variable obstructions. In the $\Lambda$14-qubit setting, the poset $\Lambda$15 of isotropic subspaces of $\Lambda$16 models Pauli measurements, with maximal contexts given by $\Lambda$17. Spectral-graph arguments show

$\Lambda$18

for the maximum fraction of maximal contexts admitting a consistent partial hidden-variable assignment, while for even $\Lambda$19,

$\Lambda$20

for the minimum inconsistency of complete contextual assignments (Bankston, 2022). This makes the pairwise overlap structure of Pauli contexts not merely combinatorial, but a quantitatively rigid obstruction to noncontextual models.

6. Algorithmic and algebraic developments

The most concrete algorithmic advance concerns counting anticommuting unordered pairs in large sparse collections of Pauli strings. Given $\Lambda$21 phase-free strings $\Lambda$22, the target quantity is

$\Lambda$23

The standard method checks all unordered pairs in quadratic time via the binary symplectic test. In the bounded-locality regime, however, an incremental pattern-counting data structure yields an $\Lambda$24 algorithm in $\Lambda$25 for fixed locality $\Lambda$26 (Cha et al., 10 May 2026).

The data structure stores counts $\Lambda$27 of labeled subpatterns $\Lambda$28 appearing in previously inserted strings. For a query string $\Lambda$29 with support $\Lambda$30, one defines

$\Lambda$31

and the number of prior strings anticommting with $\Lambda$32 is

$\Lambda$33

The key identity is

$\Lambda$34

which turns parity of conflict sets into a subset-zeta transform (Cha et al., 10 May 2026).

If the input weights are $\Lambda$35, the total expected time is

$\Lambda$36

with additional dictionary space

$\Lambda$37

For $\Lambda$38 uniformly, this simplifies to

$\Lambda$39

which is linear in $\Lambda$40 for fixed $\Lambda$41 (Cha et al., 10 May 2026). The same framework supports exact counting, certification of pairwise commutativity, and witness finding.

At a more structural level, arbitrary prescribed commutation relations can be realized through module-theoretic normal forms. For an alternating commutation matrix $\Lambda$42, the minimal number of qudits needed to realize $\Lambda$43 is $\Lambda$44, where $\Lambda$45 is the $\Lambda$46-submodule generated by the columns of $\Lambda$47. The alternating Smith normal form decomposes $\Lambda$48 into canonical $\Lambda$49 blocks

$\Lambda$50

which are precisely canonical non-commuting Pauli pairs (Sarkar et al., 2023). The same paper proves that the maximum size of a collection of non-commuting pairs on $\Lambda$51 qudits is $\Lambda$52, where $\Lambda$53 is the number of distinct prime factors of $\Lambda$54, and that the maximum size of a pairwise non-commuting set on one qudit is

$\Lambda$55

(Sarkar et al., 2023).

Pairwise anticommutation also drives Lie generation. Restricting generators to Pauli strings, the minimal set generating $\Lambda$56 has size

$\Lambda$57

and the lower-bound proof splits according to pairwise commutation structure: a $\Lambda$58-element set fails whether all distinct pairs anticommute or whether at least one commuting pair exists (Smith et al., 2024). The corresponding compiler constructs nested commutator expressions of optimal $\Lambda$59 depth and $\Lambda$60 runtime by repeatedly engineering useful anticommuting pairs (Smith et al., 2024).

A final, more representational development identifies finite phase-space pairs $\Lambda$61 with Heisenberg–Weyl or Pauli basis elements, and shows that discrete Wigner kernels are inverse Fourier transforms of these operators. For qubits,

$\Lambda$62

so a “Pauli pair” may also mean a phase-space coordinate pair indexing a Pauli operator (Cai et al., 2018). This suggests that the coexistence of harmonic-analytic and operator-theoretic usages is not accidental: both are ultimately organized by duality, symmetry, and sparse discrete structure.