Classical Pauli Pairs: Constructions & Counterexamples

• Classical Pauli pairs are pairs of quantum states in L²(ℝᵈ) that, despite having identical position and momentum moduli, are not related by a unimodular scalar, highlighting a fundamental nonuniqueness in quantum state determination.
• They arise from both classical symmetry-based constructions—such as reflection and conjugation—and nonclassical mechanisms involving periodic phase factors and support disjointness, expanding the landscape of counterexamples.
• Recent studies demonstrate that under specific Gaussian decay and dense sampling conditions, discrete modulus data can force global modulus equivalence, thus clarifying regimes where Pauli’s conjecture fails.

Searching arXiv for recent and foundational papers on classical Pauli pairs. Classical Pauli pairs are pairs of quantum states, represented by functions $f,g\in L^2(\mathbb{R}^d)$, that have identical position and momentum moduli while remaining distinct as states. In the standard formulation, this means

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

yet $f$ and $g$ are not identified by a unimodular scalar factor. The notion originates in the failure of Pauli’s 1958 conjecture that position and momentum distributions determine a quantum state up to global phase. Within the modern literature, “classical Pauli pair” typically designates this global, continuous modulus-equality setting, in contrast to discrete or sampled analogues (Shkarin, 2012, Ramos et al., 2024, Lysen, 21 May 2026).

1. Definition and conceptual setting

A Pauli pair in $L^2(\mathbb R^n)$ is a pair of linearly independent functions $f,g\in L^2(\mathbb R^n)$ such that

$$|f|=|g| \quad\text{a.e.},\qquad |\widehat f|=|\widehat g| \quad\text{a.e.}$$

This is the central definition in the modern treatment of the Pauli problem (Shkarin, 2012). In the language used in more recent work, this is precisely what is meant by a classical Pauli pair: global modulus agreement in physical space and Fourier space (Ramos et al., 2024, Lysen, 21 May 2026).

The underlying quantum-mechanical interpretation is standard. For a wave function $f$, the position density is $|f|^2$, while the momentum density is $(2\pi)^{-n}|\widehat f|^2$ in the normalization of Shkarin (Shkarin, 2012). Thus Pauli pairs are pairs of states that are indistinguishable by measuring position and momentum distributions alone. The original Pauli conjecture asserted that such indistinguishability should imply equality up to a global phase; the conjecture is false (Shkarin, 2012).

Recent papers distinguish three levels of phaseless equivalence. A Pauli pair is the full global equality above; a weak Pauli pair requires only one of the two global modulus equalities; and a discrete Pauli pair requires equality of the sampled moduli on prescribed discrete sets in time and frequency (Lysen, 21 May 2026). This taxonomy clarifies that the classical notion is the strongest of the three.

2. Historical problem and the failure of uniqueness

The classical Pauli problem asks whether knowledge of $$|f(x)|=|g(x)| \quad\text{a.e. on }\mathbb R^d, \qquad |\widehat f(\xi)|=|\widehat g(\xi)| \quad\text{a.e. on }\mathbb R^d,$$0 and $$|f(x)|=|g(x)| \quad\text{a.e. on }\mathbb R^d, \qquad |\widehat f(\xi)|=|\widehat g(\xi)| \quad\text{a.e. on }\mathbb R^d,$$1 determines $$|f(x)|=|g(x)| \quad\text{a.e. on }\mathbb R^d, \qquad |\widehat f(\xi)|=|\widehat g(\xi)| \quad\text{a.e. on }\mathbb R^d,$$2 up to multiplication by a unimodular constant. Shkarin formulates the conjecture in $$|f(x)|=|g(x)| \quad\text{a.e. on }\mathbb R^d, \qquad |\widehat f(\xi)|=|\widehat g(\xi)| \quad\text{a.e. on }\mathbb R^d,$$3 and emphasizes that it fails already in one dimension (Shkarin, 2012). The failure is not confined to isolated pathologies: the paper constructs new families of counterexamples and organizes them through several structural mechanisms.

One basic source of counterexamples comes from symmetry. If $$|f(x)|=|g(x)| \quad\text{a.e. on }\mathbb R^d, \qquad |\widehat f(\xi)|=|\widehat g(\xi)| \quad\text{a.e. on }\mathbb R^d,$$4 is even, then $$|f(x)|=|g(x)| \quad\text{a.e. on }\mathbb R^d, \qquad |\widehat f(\xi)|=|\widehat g(\xi)| \quad\text{a.e. on }\mathbb R^d,$$5 and $$|f(x)|=|g(x)| \quad\text{a.e. on }\mathbb R^d, \qquad |\widehat f(\xi)|=|\widehat g(\xi)| \quad\text{a.e. on }\mathbb R^d,$$6 satisfy

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

so $$|f(x)|=|g(x)| \quad\text{a.e. on }\mathbb R^d, \qquad |\widehat f(\xi)|=|\widehat g(\xi)| \quad\text{a.e. on }\mathbb R^d,$$8 is a Pauli pair whenever $$|f(x)|=|g(x)| \quad\text{a.e. on }\mathbb R^d, \qquad |\widehat f(\xi)|=|\widehat g(\xi)| \quad\text{a.e. on }\mathbb R^d,$$9 is not a scalar multiple of a real-valued function (Shkarin, 2012). This is the prototype of what the literature informally treats as a “classical” or symmetry-generated example.

The 2012 paper also recalls the Moroz–Perelomov construction. If $f$0 is symmetric with respect to a vertical line $f$1, and

$f$2

then

$f$3

When the reflected phase is genuinely nontrivial, this yields a Pauli pair (Shkarin, 2012). Moroz and Perelomov conjectured that all one-dimensional Pauli pairs arise by this mechanism. Shkarin shows that this conjecture is false (Shkarin, 2012).

This negative result is significant because it separates the general nonuniqueness phenomenon from the visible symmetry heuristics that first motivated it. A plausible implication is that classical Pauli pairs cannot be understood purely through simple involutive symmetries such as conjugation and reflection; additional mechanisms are essential.

3. Classical constructions and nonclassical constructions

The literature draws a sharp distinction between symmetry-based constructions and genuinely nonclassical ones. In Shkarin’s treatment, the symmetry-based examples include conjugation, reflection, linear phase twists, and especially the Moroz–Perelomov reflected-phase construction (Shkarin, 2012).

A more flexible nonclassical construction, recalled from Ismagilov, uses periodic phase factors. For a $f$4-periodic measurable $f$5 and a nonzero $f$6 supported on an interval of length $f$7, define

$f$8

Then both $f$9 and $g$0 are independent of $g$1, and for sufficiently small $g$2, the pair $g$3 is a Pauli pair provided $g$4 is not of the exponential form

$g$5

This produces a one-parameter family of Pauli pairs (Shkarin, 2012).

The construction is important for two reasons. First, it already lies outside the Moroz–Perelomov mechanism. Second, the resulting functions necessarily have unbounded support in both physical and Fourier space (Shkarin, 2012). This prompted the question of whether strong localization might restore the older “classical” picture. Shkarin shows that it does not.

The paper’s main structural mechanism for generating large families is the notion of a UZD-set (ultimate zero divisor set): a set $g$6 with at least two nonzero elements such that for distinct $g$7,

$g$8

Thus distinct functions have disjoint supports both in physical space and Fourier space (Shkarin, 2012). If $g$9 is a UZD-set and $L^2(\mathbb R^n)$0 with $L^2(\mathbb R^n)$1, then different phase choices yield distinct functions with identical position and momentum moduli. This converts strong support disjointness into large Pauli families (Shkarin, 2012).

Theorem 1.3 of Shkarin states that there exists a countably infinite UZD-set $L^2(\mathbb R^n)$2 (Shkarin, 2012). From this, the paper derives the existence of a closed infinite-dimensional subspace $L^2(\mathbb R^n)$3 such that every nonzero vector in $L^2(\mathbb R^n)$4 is a Pauli state (Shkarin, 2012). This result decisively broadens the scope of the nonuniqueness phenomenon: Pauli states are not isolated exceptions but can occupy an infinite-dimensional closed subspace.

4. Spatial localization and bounded-support examples

One of the central developments in the theory is the demonstration that compact support does not force classicality. Before Shkarin’s work, many known constructions either relied on symmetry or produced functions with unbounded support. The possibility remained that bounded support might exclude genuinely new Pauli pairs. The paper disproves this (Shkarin, 2012).

The relevant model class is that of step functions. For $L^2(\mathbb R^n)$5, define

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

Then $L^2(\mathbb R^n)$7 has support $L^2(\mathbb R^n)$8, and its Fourier transform satisfies

$L^2(\mathbb R^n)$9

where $f,g\in L^2(\mathbb R^n)$0 is the indicator of $f,g\in L^2(\mathbb R^n)$1 (Shkarin, 2012).

The resulting characterization is algebraic. Lemma 4.1 states that

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

if and only if

$f,g\in L^2(\mathbb R^n)$3

and

$f,g\in L^2(\mathbb R^n)$4

Thus the compactly supported Pauli problem for step functions reduces to a system of homogeneous quadratic algebraic equations (Shkarin, 2012).

The first genuinely new phenomenon occurs at $f,g\in L^2(\mathbb R^n)$5. Shkarin provides an explicit pair of compactly supported step functions $f,g\in L^2(\mathbb R^n)$6 on $f,g\in L^2(\mathbb R^n)$7,

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

and

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

and proves that $$|f|=|g| \quad\text{a.e.},\qquad |\widehat f|=|\widehat g| \quad\text{a.e.}$$0 is a Pauli pair (Shkarin, 2012).

The significance is explicit in the paper: both functions have bounded support, both are step functions, and the pair is neither MP1 nor MP2, the two classes encoding the Moroz–Perelomov-type “classical” compact-support mechanisms (Shkarin, 2012). This shows that even strong spatial localization does not restore uniqueness from position and momentum distributions.

5. Classification of classical versus nonclassical bounded-support pairs

To formalize which compactly supported examples come from the older symmetry framework, Shkarin introduces two notions. A Pauli pair $$|f|=|g| \quad\text{a.e.},\qquad |\widehat f|=|\widehat g| \quad\text{a.e.}$$1 is an MP1-pair if $$|f|=|g| \quad\text{a.e.},\qquad |\widehat f|=|\widehat g| \quad\text{a.e.}$$2 has a vertical symmetry line,

$$|f|=|g| \quad\text{a.e.},\qquad |\widehat f|=|\widehat g| \quad\text{a.e.}$$3

and an MP2-pair if there exist $$|f|=|g| \quad\text{a.e.},\qquad |\widehat f|=|\widehat g| \quad\text{a.e.}$$4 such that

$$|f|=|g| \quad\text{a.e.},\qquad |\widehat f|=|\widehat g| \quad\text{a.e.}$$5

If a pair comes from the Moroz–Perelomov proposition, then it must be of one of these two types (Shkarin, 2012).

For step functions $$|f|=|g| \quad\text{a.e.},\qquad |\widehat f|=|\widehat g| \quad\text{a.e.}$$6, the paper provides concrete criteria. In the MP1 case the modulus pattern must be symmetric,

$$|f|=|g| \quad\text{a.e.},\qquad |\widehat f|=|\widehat g| \quad\text{a.e.}$$7

while the MP2 criterion is expressed through the corresponding reflected/conjugated coefficient relation described in Remark 1.8 (Shkarin, 2012). These criteria make the distinction between “classical” and genuinely new localized examples operational.

The most complete result is the classification of 4-step Pauli pairs. Proposition 4.2 states that all solutions with $$|f|=|g| \quad\text{a.e.},\qquad |\widehat f|=|\widehat g| \quad\text{a.e.}$$8 and $$|f|=|g| \quad\text{a.e.},\qquad |\widehat f|=|\widehat g| \quad\text{a.e.}$$9 fall into three families: the trivial family $f$0, a first nontrivial 4-parameter family identified exactly with the MP1-type solutions, and a second nontrivial 4-parameter family containing the genuinely new examples (Shkarin, 2012).

The second family is given by

$f$1

$f$2

with

$f$3

The MP2-pairs appear inside this family only under the additional constraint

$f$4

Hence much of this family lies genuinely beyond the classical Moroz–Perelomov pattern (Shkarin, 2012).

This classification is valuable because it isolates, in a minimal compact-support model, where the old symmetry heuristics end and the broader nonuniqueness phenomenon begins.

6. Discrete Pauli pairs and when they become classical

Recent work has shifted part of the focus from constructing classical Pauli pairs to understanding when sampled phaseless data force the classical conclusion. This discrete theory does not classify all classical Pauli pairs, but it gives rigorous criteria under which discrete modulus equalities imply the global equalities that define them (Ramos et al., 2024, Lysen, 21 May 2026).

In the formulation of Ramos–Sousa and its successors, a discrete Pauli pair for discrete sets $f$5 satisfies

$f$6

The question is when such sampled equalities force the full global identities

$f$7

that is, when a discrete Pauli pair must be a classical Pauli pair (Ramos et al., 2024, Lysen, 21 May 2026).

The 2024 paper proves a substantial sufficient condition. If $f$8, one of the functions satisfies Gaussian space-frequency decay,

$f$9

and the sample sets are dense enough at infinity in the sense

$|f|^2$0

then discrete modulus agreement implies

$|f|^2$1

on all of $|f|^2$2 (Ramos et al., 2024). The asymptotic spacing condition corresponds heuristically to nodes accumulating like suitable small multiples of $|f|^2$3 at infinity (Ramos et al., 2024).

The mechanism is analytic. Gaussian control propagates from the nodes to global Gaussian decay; then Paley–Wiener-type arguments give entire extensions of order $|f|^2$4; finally the sampled modulus equalities become zero sets of auxiliary entire functions such as

$|f|^2$5

and zero-density arguments force $|f|^2$6, hence $|f|^2$7 globally (Ramos et al., 2024). This shows that the classical Pauli conclusion can be recovered from discrete phaseless data under strong regularity and density assumptions.

The 2026 paper sharpens this picture. It introduces the Hardy class

$|f|^2$8

and determines sharp density thresholds (Lysen, 21 May 2026). For $|f|^2$9, it defines

$(2\pi)^{-n}|\widehat f|^2$0

and proves that sufficiently dense time-side samples force global time-side modulus equality (Lysen, 21 May 2026). Combining time and frequency yields the sharp classical-Pauli threshold

$(2\pi)^{-n}|\widehat f|^2$1

in the sense that, under

$(2\pi)^{-n}|\widehat f|^2$2

a discrete Pauli pair with $(2\pi)^{-n}|\widehat f|^2$3, $(2\pi)^{-n}|\widehat f|^2$4 must be a classical Pauli pair (Lysen, 21 May 2026).

These results do not classify classical Pauli pairs themselves. Rather, they characterize a regime in which sparse phaseless observations already force the full classical equalities. This suggests that, in Gaussian-decay classes, the nonuniqueness of the Pauli problem is constrained not only by symmetry and support geometry but also by sampling density.

7. Significance, limitations, and related misconceptions

The theory of classical Pauli pairs establishes a precise failure of phase retrieval from position and momentum distributions. Several misconceptions are corrected by the cited work.

First, compact support does not imply that every Pauli pair must come from reflection or conjugation symmetries. Shkarin’s 4-step example is compactly supported and neither MP1 nor MP2 (Shkarin, 2012).

Second, the nonuniqueness is not confined to isolated examples. Countably infinite UZD-sets in $(2\pi)^{-n}|\widehat f|^2$5 and an infinite-dimensional closed subspace of Pauli states show that the phenomenon can be structurally large (Shkarin, 2012).

Third, discrete modulus agreement is not automatically informative. Without Gaussian decay, there exist $(2\pi)^{-n}|\widehat f|^2$6 that agree in sampled modulus on arbitrary discrete sets in time and frequency while failing both global modulus equalities (Ramos et al., 2024, Lysen, 21 May 2026). Thus the discrete-to-classical implication is genuinely conditional.

Fourth, even within the discrete theory there are distinct rigidity levels. A pair may be a discrete Pauli pair without being a weak Pauli pair, or a weak Pauli pair without being a classical Pauli pair (Lysen, 21 May 2026). The 2026 paper determines a separate sharp threshold $(2\pi)^{-n}|\widehat f|^2$7 for when sampled data force at least one of the two global modulus identities (Lysen, 21 May 2026). This separates full classical recovery from one-sided recovery.

From a broader perspective, the modern theory shows that “classical Pauli pairs” are best understood not as a single family of symmetry-generated counterexamples but as a multi-layered phase-retrieval phenomenon. The 2012 work develops explicit constructions and localized nonclassical examples (Shkarin, 2012). The 2024 and 2026 works place the same concept into a Fourier-uniqueness framework, identifying when discrete phaseless data compel the full classical conclusion (Ramos et al., 2024, Lysen, 21 May 2026). Taken together, these results show that the failure of Pauli’s conjecture is both robust and highly structured: it persists under smoothness, under compact support, and across infinite-dimensional families, but can be suppressed under sufficiently strong Gaussian decay and sufficiently dense phaseless sampling.