Weak Pauli Pairs: Thresholds and Analysis

• Weak Pauli pairs are defined as function pairs that agree in modulus on either the spatial or Fourier side, representing a relaxed uniqueness condition.
• Recent discrete threshold theories establish sharp density criteria that determine when sampled data forces one-sided modulus agreement.
• The analytic framework employs entire function zero-set analysis to link Fourier uniqueness with phase retrieval challenges in both mathematical and quantum contexts.

Searching arXiv for papers on Pauli pairs and weak Pauli pairs. Weak Pauli pairs arise in two related but distinct strands of recent work. In the mathematical Pauli problem, they are a weakened phaseless equivalence notion for two functions $f,g:\mathbb R\to\mathbb C$: instead of requiring agreement of both spatial and Fourier moduli, one requires global agreement on at least one side. In the quantum-measurement literature, the phrase is not formalized in the same way, but sequential weak measurements of ordered pairs of non-commuting Pauli observables address an operationally adjacent question: how weakly accessed ordered Pauli products encode information not available from ordinary simultaneous measurement. The formal definition of weak Pauli pairs is given in recent discrete Pauli-pair theory, while earlier work on Pauli pairs and on weak measurements supplies the conceptual and technical background (Lysen, 21 May 2026).

1. Formal definitions and conceptual boundaries

For $f,g:\mathbb R\to\mathbb C$, with Fourier transform normalized by

$$\hat f(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i x\xi}\,dx,$$

a Pauli pair satisfies

$$|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R, \qquad |\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$

A weak Pauli pair satisfies only one of these two global identities: $|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R$ or

$$|\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$

A discrete Pauli pair for $(\Lambda,M)$, where $\Lambda,M\subset\mathbb R$ are discrete, satisfies only sampled modulus equalities,

$$|f(\lambda)|=|g(\lambda)| \quad \text{for all }\lambda\in\Lambda, \qquad |\hat f(\mu)|=|\hat g(\mu)| \quad \text{for all }\mu\in M.$$

The weak-Pauli question is then whether such sampled phaseless data force at least one global modulus identity. The 2026 paper makes this notion explicit and treats it as a phaseless analogue of Fourier uniqueness/nonuniqueness pairs (Lysen, 21 May 2026).

Earlier literature used only the stronger notion. In particular, Shkarin defined Pauli pairs as linearly independent $f,g\in L^2(\mathbb R^n)$ with

$f,g:\mathbb R\to\mathbb C$0

almost everywhere, thereby formalizing nontrivial failure of uniqueness in reconstructing a quantum state from position and momentum distributions (Shkarin, 2012). By contrast, the 2024 discrete theory studies sampled modulus equalities on $f,g:\mathbb R\to\mathbb C$1 and asks when they imply the full global Pauli conclusion, but it does not explicitly introduce the phrase “weak Pauli pair” (Ramos et al., 2024).

A recurrent source of confusion is terminological. In harmonic analysis, weak Pauli pairs concern global modulus agreement on one side only. In quantum measurement, “weak” refers instead to weak system-pointer coupling. The latter context studies weak values and sequential weak values of Pauli observables, including non-commuting pairs, but does not formalize “weak Pauli pair” as a definition (Chen et al., 2018).

2. Sharp threshold theory for discrete weak Pauli pairs

The central recent theorem identifies a sharp density threshold for when a discrete Pauli pair must be a weak Pauli pair. The ambient classes are

$f,g:\mathbb R\to\mathbb C$2

and, for $f,g:\mathbb R\to\mathbb C$3,

$f,g:\mathbb R\to\mathbb C$4

The theory mostly considers $f,g:\mathbb R\to\mathbb C$5 with $f,g:\mathbb R\to\mathbb C$6 (Lysen, 21 May 2026).

For $f,g:\mathbb R\to\mathbb C$7, define

$f,g:\mathbb R\to\mathbb C$8

If $f,g:\mathbb R\to\mathbb C$9 satisfy

$$\hat f(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i x\xi}\,dx,$$0

and if $$\hat f(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i x\xi}\,dx,$$1, $$\hat f(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i x\xi}\,dx,$$2, and $$\hat f(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i x\xi}\,dx,$$3 form a discrete Pauli pair for $$\hat f(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i x\xi}\,dx,$$4, then $$\hat f(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i x\xi}\,dx,$$5 form a weak Pauli pair. Conversely, if

$$\hat f(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i x\xi}\,dx,$$6

then there exist $$\hat f(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i x\xi}\,dx,$$7, $$\hat f(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i x\xi}\,dx,$$8 forming a discrete Pauli pair for $$\hat f(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i x\xi}\,dx,$$9 such that they do not form a weak Pauli pair, i.e.

$$|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R, \qquad |\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$0

This theorem is sharp in the precise sense that $$|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R, \qquad |\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$1 is the boundary between unavoidable one-sided global agreement and the existence of counterexamples with global disagreement on both sides (Lysen, 21 May 2026).

The theorem should be distinguished from the stronger threshold for forcing a full Pauli pair. That threshold is $$|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R, \qquad |\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$2, where

$$|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R, \qquad |\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$3

Below the stronger threshold, a discrete Pauli pair must be a full Pauli pair; below $$|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R, \qquad |\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$4, one knows only that it must be a weak Pauli pair; above $$|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R, \qquad |\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$5, both global modulus failures can coexist. This makes weak Pauli pairs an intermediate rigidity notion between full phaseless uniqueness and total nonuniqueness (Lysen, 21 May 2026).

3. Analytic mechanism and connection to Fourier uniqueness

The modern theory converts sampled modulus equalities into zero-set problems for entire functions. Writing

$$|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R, \qquad |\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$6

one defines

$$|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R, \qquad |\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$7

and

$$|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R, \qquad |\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$8

If $$|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R, \qquad |\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$9 for $|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R$0, then $|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R$1 on $|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R$2; if $|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R$3 for $|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R$4, then $|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R$5 on $|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R$6. The weak-Pauli question is thereby reframed as whether both quadratic entire functions can remain nonzero under the given density and decay assumptions (Lysen, 21 May 2026).

The decisive input is the interaction between zero density and growth indicators. For a $|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R$7-smooth sequence $|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R$8, density $|f(x)|=|g(x)| \quad \text{a.e. on }\mathbb R$9 means

$$|\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$0

together with

$$|\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$1

In the present setting $$|\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$2. Real zero density yields lower bounds on indicators, while Gaussian decay yields upper bounds through Phragmén–Lindelöf control. The proof then derives coupled inequalities for the half-line densities of $$|\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$3 and $$|\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$4, optimizes them, and recovers the piecewise constant $$|\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$5 as the sharp threshold separating forced weak-Pauli behavior from true two-sided nonuniqueness (Lysen, 21 May 2026).

This is explicitly presented as a phaseless version of Fourier uniqueness theory. A pair $$|\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$6 is a Fourier uniqueness pair for a space $$|\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$7 if

$$|\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$8

Kulikov–Nazarov–Sodin introduced the corresponding linear supercritical/subcritical theory in terms of asymptotic spacings, and the recent Pauli literature adapts that philosophy to modulus data by replacing linear vanishing with quadratic identities such as $$|\hat f(\xi)|=|\hat g(\xi)| \quad \text{a.e. on }\mathbb R.$$9 and $(\Lambda,M)$0 (Lysen, 21 May 2026).

The 2024 paper already framed the sampled Pauli problem as a nonlinear analogue of Fourier uniqueness. There the key analytic objects are

$(\Lambda,M)$1

so that on the real line $(\Lambda,M)$2 and $(\Lambda,M)$3. Sampled modulus equality becomes vanishing of $(\Lambda,M)$4 on the sampling set, and the argument proceeds through propagation of Gaussian decay, entire extension, and zero-density theorems for entire functions of order $(\Lambda,M)$5 (Ramos et al., 2024).

4. Earlier Pauli-pair constructions and structural mechanisms

Before the formal introduction of weak Pauli pairs, the theory of ordinary Pauli pairs had already produced a rich catalogue of nonuniqueness mechanisms. Shkarin’s 2012 paper gave new examples of Pauli pairs and in particular constructed spatially localized Pauli pairs (Shkarin, 2012).

One structural device is the ultimate zero divisor set (UZD-set). A set $(\Lambda,M)$6 is a UZD-set if

$(\Lambda,M)$7

The paper proves that there is a countable infinite UZD-set $(\Lambda,M)$8. If $(\Lambda,M)$9 is such a set and

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

then

$\Lambda,M\subset\mathbb R$1

This shows that changing phases across mutually disjoint components can leave both moduli unchanged. A plausible implication is that later weak-Pauli theory inherits its nonuniqueness intuition from precisely this sort of phase hiding (Shkarin, 2012).

Another important mechanism is the Moroz–Perelomov symmetry construction. If $\Lambda,M\subset\mathbb R$2 lies in $\Lambda,M\subset\mathbb R$3, $\Lambda,M\subset\mathbb R$4 is Borel measurable, and

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

then

$\Lambda,M\subset\mathbb R$6

satisfy $\Lambda,M\subset\mathbb R$7 and $\Lambda,M\subset\mathbb R$8. If $\Lambda,M\subset\mathbb R$9 is non-constant, $$|f(\lambda)|=|g(\lambda)| \quad \text{for all }\lambda\in\Lambda, \qquad |\hat f(\mu)|=|\hat g(\mu)| \quad \text{for all }\mu\in M.$$0 is a Pauli pair. This gave a standard symmetry-based source of counterexamples prior to more elaborate constructions (Shkarin, 2012).

Shkarin also developed a finite-dimensional step-function criterion. For $$|f(\lambda)|=|g(\lambda)| \quad \text{for all }\lambda\in\Lambda, \qquad |\hat f(\mu)|=|\hat g(\mu)| \quad \text{for all }\mu\in M.$$1, let $$|f(\lambda)|=|g(\lambda)| \quad \text{for all }\lambda\in\Lambda, \qquad |\hat f(\mu)|=|\hat g(\mu)| \quad \text{for all }\mu\in M.$$2 be the $$|f(\lambda)|=|g(\lambda)| \quad \text{for all }\lambda\in\Lambda, \qquad |\hat f(\mu)|=|\hat g(\mu)| \quad \text{for all }\mu\in M.$$3-step function on $$|f(\lambda)|=|g(\lambda)| \quad \text{for all }\lambda\in\Lambda, \qquad |\hat f(\mu)|=|\hat g(\mu)| \quad \text{for all }\mu\in M.$$4. Then $$|f(\lambda)|=|g(\lambda)| \quad \text{for all }\lambda\in\Lambda, \qquad |\hat f(\mu)|=|\hat g(\mu)| \quad \text{for all }\mu\in M.$$5 and $$|f(\lambda)|=|g(\lambda)| \quad \text{for all }\lambda\in\Lambda, \qquad |\hat f(\mu)|=|\hat g(\mu)| \quad \text{for all }\mu\in M.$$6 hold precisely when

$$|f(\lambda)|=|g(\lambda)| \quad \text{for all }\lambda\in\Lambda, \qquad |\hat f(\mu)|=|\hat g(\mu)| \quad \text{for all }\mu\in M.$$7

and

$$|f(\lambda)|=|g(\lambda)| \quad \text{for all }\lambda\in\Lambda, \qquad |\hat f(\mu)|=|\hat g(\mu)| \quad \text{for all }\mu\in M.$$8

For $$|f(\lambda)|=|g(\lambda)| \quad \text{for all }\lambda\in\Lambda, \qquad |\hat f(\mu)|=|\hat g(\mu)| \quad \text{for all }\mu\in M.$$9, every Pauli pair of step functions comes from the Moroz–Perelomov construction; for $f,g\in L^2(\mathbb R^n)$0, the paper gives a complete classification under a normalization and exhibits a bounded-support Pauli pair that is neither MP1 nor MP2. This suggests that compact support alone does not collapse Pauli nonuniqueness to the classical symmetry mechanism (Shkarin, 2012).

5. Discrete partial-data rigidity before the sharp threshold

The 2024 paper established a broad partial-data framework that anticipates the later weak-Pauli threshold. It asks whether sampled equalities

$f,g\in L^2(\mathbb R^n)$1

force

$f,g\in L^2(\mathbb R^n)$2

The main positive theorem states that for each $f,g\in L^2(\mathbb R^n)$3 there exists $f,g\in L^2(\mathbb R^n)$4 such that if $f,g\in L^2(\mathbb R^n)$5, if $f,g\in L^2(\mathbb R^n)$6 satisfies

$f,g\in L^2(\mathbb R^n)$7

and if the discrete sets satisfy

$f,g\in L^2(\mathbb R^n)$8

then sampled modulus equality implies

$f,g\in L^2(\mathbb R^n)$9

The paper interprets the spacing law as the regime of sets accumulating like suitable small multiples of $f,g:\mathbb R\to\mathbb C$00 at infinity (Ramos et al., 2024).

This positive result is paired with two negative statements. First, without additional structural assumptions, arbitrary discrete sets $f,g:\mathbb R\to\mathbb C$01 admit counterexamples $f,g:\mathbb R\to\mathbb C$02 with

$f,g:\mathbb R\to\mathbb C$03

while still satisfying sampled modulus equalities on $f,g:\mathbb R\to\mathbb C$04 and $f,g:\mathbb R\to\mathbb C$05. Second, if the sets are too sparse, specifically if

$f,g:\mathbb R\to\mathbb C$06

then there exists an infinite-dimensional space $f,g:\mathbb R\to\mathbb C$07 for small $f,g:\mathbb R\to\mathbb C$08 such that distinct $f,g:\mathbb R\to\mathbb C$09 have identical sampled moduli on $f,g:\mathbb R\to\mathbb C$10 but no global modulus equality on either side (Ramos et al., 2024).

The same paper also proves a sharp discrete version of Hardy’s uncertainty principle. For $f,g:\mathbb R\to\mathbb C$11, there exists $f,g:\mathbb R\to\mathbb C$12 such that under the corresponding dense-sampling condition, if $f,g:\mathbb R\to\mathbb C$13 satisfies

$f,g:\mathbb R\to\mathbb C$14

then $f,g:\mathbb R\to\mathbb C$15. In context, this linear theorem supplies part of the uniqueness backbone behind the nonlinear Pauli results (Ramos et al., 2024).

6. Weakly measured Pauli pairs in quantum measurement

A different but adjacent use of “weak” appears in the study of sequential weak measurements of Pauli observables. For an observable $f,g:\mathbb R\to\mathbb C$16, pre-selected state $f,g:\mathbb R\to\mathbb C$17, and post-selected state $f,g:\mathbb R\to\mathbb C$18, the weak value is

$f,g:\mathbb R\to\mathbb C$19

For a sequence of observables $f,g:\mathbb R\to\mathbb C$20, the sequential weak value is

$f,g:\mathbb R\to\mathbb C$21

For a pair,

$f,g:\mathbb R\to\mathbb C$22

Because the numerator contains an ordered product, non-commuting Pauli pairs are intrinsically order dependent (Chen et al., 2018).

For Pauli observables $f,g:\mathbb R\to\mathbb C$23, the interaction unitary satisfies

$f,g:\mathbb R\to\mathbb C$24

and the pointer averages obey exact finite-strength formulas,

$f,g:\mathbb R\to\mathbb C$25

The paper explicitly states that for Pauli-type observables, both weak values and sequential weak values are independent of measurement strength. This makes ordered Pauli pairs unusually tractable in weak measurement (Chen et al., 2018).

The Pauli algebra clarifies the ordered-pair structure: $f,g:\mathbb R\to\mathbb C$26 Hence, for distinct Pauli matrices,

$f,g:\mathbb R\to\mathbb C$27

while reversing the order changes the sign. With the convention

$f,g:\mathbb R\to\mathbb C$28

one obtains, for example,

$f,g:\mathbb R\to\mathbb C$29

Thus sequential weak measurement of a non-commuting Pauli pair probes ordered products and commutator structure rather than a classical joint value. A plausible implication is that this operational notion of “weak Pauli pairs” is conceptually parallel, though not identical, to the mathematical weak-Pauli theory: both concern what survives when direct simultaneous reconstruction is replaced by constrained or partial data (Chen et al., 2018).

Experimentally, the 2018 photonic work demonstrated sequential weak measurements of three non-commuting Pauli observables using heralded single photons, with the system qubit encoded in polarization and a modular design permitting ordered weak measurements of $f,g:\mathbb R\to\mathbb C$30, $f,g:\mathbb R\to\mathbb C$31, and $f,g:\mathbb R\to\mathbb C$32. The figures were reported to show measured sequential weak values of arbitrary two observables and three observables for different post-selected states, with agreement between theory and data for coupling strengths $f,g:\mathbb R\to\mathbb C$33 and $f,g:\mathbb R\to\mathbb C$34, thereby verifying the claimed measurement-strength independence for Pauli observables (Chen et al., 2018).