Majorana Sampling: Methods in Neutrino, Spin & Devices

• Majorana sampling is an umbrella term for various probabilistic methods that use randomization to extract hidden information from Majorana phases, observables, or spectra.
• These methodologies span neutrino phenomenology, fermionic quantum measurements, symmetric spin-state analysis, and device spectroscopy to yield statistically informed predictions.
• Practical frameworks include Monte Carlo sampling for unknown phases, randomized joint measurements in fermionic systems, geometric sampling in spin constellations, and ensemble spectroscopy for distinguishing Majorana modes.

In the literature considered here, Majorana sampling does not denote a single standardized procedure. It refers instead to several sampling-based methodologies in which the sampled objects are tied to Majorana structures: unknown Majorana phases in neutrino phenomenology, even Majorana monomials in fermionic measurement theory, random Majorana constellations in symmetric spin systems, and ensembles of Majorana-sensitive spectra in condensed-matter devices. A common feature is the replacement of deterministic envelopes, single-shot signatures, or exhaustive separate measurements by explicitly probabilistic, randomized, or ensemble-based constructions (Benato, 2015, McNulty et al., 2024, Goldberg et al., 2021, Ziesen et al., 2022).

1. Scope and principal usages

The phrase is best understood as an umbrella label for several technically distinct practices rather than as a single formalism. In each case, the “sampling” step changes what can be inferred: probabilities replace extremal bounds in neutrino mass studies, parent POVMs replace families of incompatible sharp measurements in fermionic tomography, random point sets on the sphere induce non-Haar ensembles of symmetric quantum states, and repeated spectroscopy under parameter variation converts ambiguous zero-bias features into statistically testable patterns.

Usage	Sampled object	Representative aim
Neutrino phenomenology	Majorana phases and oscillation inputs	Probability distribution of $|m_{\beta\beta}|$
Fermionic quantum measurement	Random Majorana products, Gaussian settings, occupation outcomes	Joint estimation of many non-commuting Majorana observables
Symmetric spin-state theory	i.i.d. Majorana constellation points on $\mathcal S_2$	Typical quantumness and metrological structure
Majorana-device spectroscopy	Parameter-varied subgap spectra	Statistical discrimination of MBS from ABS

A recurring misconception is to equate Majorana sampling with a single condensed-matter protocol for Majorana zero modes. The literature instead supports a broader interpretation. Some works sample unknown phases, some sample measurement settings, some sample geometric constellations, and some sample ensembles of spectra. A plausible implication is that the unifying content lies less in a particular hardware platform than in the use of sampling to access otherwise hidden Majorana-dependent structure.

2. Majorana-phase sampling in neutrino phenomenology

In neutrino physics, Majorana sampling refers to Monte Carlo sampling of the unknown Majorana phases entering the effective mass for neutrinoless double beta decay, together with the measured oscillation parameters (Benato, 2015). The central observable is

$$|m_{\beta\beta}|=\left|\sum_{i=1}^3 U_{ei}^2 m_i\right|,$$

written in the convention

$$| m_{\beta\beta} | = \bigl| c_{12}^2c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 e^{i\alpha} + s_{13}^2 m_3 e^{i\beta} \bigr|,$$

with real-imaginary decomposition

$$| m_{\beta\beta} | = \sqrt{ \bigl( c_{12}^2 c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 \cos{\alpha} + s_{13}^2 m_3 \cos{\beta} \bigr)^2 + \bigl( s_{12}^2 c_{13}^2 m_2 \sin{\alpha} + s_{13}^2 m_3 \sin{\beta} \bigr)^2 }.$$

The analysis is performed in the standard three-neutrino scheme for both Normal Hierarchy and Inverted Hierarchy, parameterized by the lightest neutrino mass $m_{\min}$. The sampled oscillation inputs are $\delta m_{\odot}^2$, $\Delta m_{atm}^2$, $s_{12}^2$, and $s_{13}^2$, using Gaussian distributions centered on the quoted best-fit values. The two Majorana phases $\mathcal S_2$0 and $\mathcal S_2$1 are sampled independently from a flat distribution on $\mathcal S_2$2, which is the defining prior assumption of the analysis. Correlations among oscillation parameters are explicitly neglected.

For each bin in $\mathcal S_2$3, $\mathcal S_2$4 random parameter combinations are generated. Rather than plotting only deterministic allowed bands obtained by extremizing over phases, the study displays coverage regions labeled $\mathcal S_2$5. The main conceptual shift is from asking what values of $\mathcal S_2$6 are merely allowed to asking what values are probable under explicit priors. High values of $\mathcal S_2$7 are favored within the allowed bands for both orderings, and the low-$\mathcal S_2$8 part of the Normal Hierarchy is statistically disfavored because it requires special phase alignments.

The most explicit numerical statement concerns the Normal Hierarchy cancellation region. In the “most unfortunate case” $\mathcal S_2$9, the probability of having $$|m_{\beta\beta}|=\left|\sum_{i=1}^3 U_{ei}^2 m_i\right|,$$0 is only about $$|m_{\beta\beta}|=\left|\sum_{i=1}^3 U_{ei}^2 m_i\right|,$$1, equivalently there is “at least $$|m_{\beta\beta}|=\left|\sum_{i=1}^3 U_{ei}^2 m_i\right|,$$2 probability” that an experiment with $$|m_{\beta\beta}|=\left|\sum_{i=1}^3 U_{ei}^2 m_i\right|,$$3 sensitivity to the effective mass would detect a signal. Small values of $$|m_{\beta\beta}|=\left|\sum_{i=1}^3 U_{ei}^2 m_i\right|,$$4 arise only for correlated phases, roughly $$|m_{\beta\beta}|=\left|\sum_{i=1}^3 U_{ei}^2 m_i\right|,$$5; more geometrically, $$|m_{\beta\beta}|=\left|\sum_{i=1}^3 U_{ei}^2 m_i\right|,$$6 and $$|m_{\beta\beta}|=\left|\sum_{i=1}^3 U_{ei}^2 m_i\right|,$$7 must lie in opposite quadrants so that both sine and cosine contributions cancel.

The same framework is extended by replacing $$|m_{\beta\beta}|=\left|\sum_{i=1}^3 U_{ei}^2 m_i\right|,$$8 with the total neutrino mass

$$|m_{\beta\beta}|=\left|\sum_{i=1}^3 U_{ei}^2 m_i\right|,$$9

sampled from the Gaussian input

$$| m_{\beta\beta} | = \bigl| c_{12}^2c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 e^{i\alpha} + s_{13}^2 m_3 e^{i\beta} \bigr|,$$0

with quoted $$| m_{\beta\beta} | = \bigl| c_{12}^2c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 e^{i\alpha} + s_{13}^2 m_3 e^{i\beta} \bigr|,$$1 CL limit $$| m_{\beta\beta} | = \bigl| c_{12}^2c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 e^{i\alpha} + s_{13}^2 m_3 e^{i\beta} \bigr|,$$2. This cosmology-inclusive study uses $$| m_{\beta\beta} | = \bigl| c_{12}^2c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 e^{i\alpha} + s_{13}^2 m_3 e^{i\beta} \bigr|,$$3 sampled points. The method gives conditional distributions within each ordering, not a posterior probability for Normal versus Inverted Hierarchy. Under the adopted cosmological assumption, the study finds a “weak preference” for values of $$| m_{\beta\beta} | = \bigl| c_{12}^2c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 e^{i\alpha} + s_{13}^2 m_3 e^{i\beta} \bigr|,$$4 closer to the quasi-degenerate region, but also stresses that this preference is highly sensitive to the assumed distribution for $$| m_{\beta\beta} | = \bigl| c_{12}^2c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 e^{i\alpha} + s_{13}^2 m_3 e^{i\beta} \bigr|,$$5.

The probabilistic output is then mapped to isotope-dependent half-life expectations through

$$| m_{\beta\beta} | = \bigl| c_{12}^2c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 e^{i\alpha} + s_{13}^2 m_3 e^{i\beta} \bigr|,$$6

For $$| m_{\beta\beta} | = \bigl| c_{12}^2c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 e^{i\alpha} + s_{13}^2 m_3 e^{i\beta} \bigr|,$$7, $$| m_{\beta\beta} | = \bigl| c_{12}^2c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 e^{i\alpha} + s_{13}^2 m_3 e^{i\beta} \bigr|,$$8, and $$| m_{\beta\beta} | = \bigl| c_{12}^2c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 e^{i\alpha} + s_{13}^2 m_3 e^{i\beta} \bigr|,$$9, the adopted phase-space factors and NME ranges generate wide half-life bands, and the study identifies NME calculation as “the bottleneck” in converting half life to $$| m_{\beta\beta} | = \sqrt{ \bigl( c_{12}^2 c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 \cos{\alpha} + s_{13}^2 m_3 \cos{\beta} \bigr)^2 + \bigl( s_{12}^2 c_{13}^2 m_2 \sin{\alpha} + s_{13}^2 m_3 \sin{\beta} \bigr)^2 }.$$0. With the cosmological prior included, the reported $$| m_{\beta\beta} | = \sqrt{ \bigl( c_{12}^2 c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 \cos{\alpha} + s_{13}^2 m_3 \cos{\beta} \bigr)^2 + \bigl( s_{12}^2 c_{13}^2 m_2 \sin{\alpha} + s_{13}^2 m_3 \sin{\beta} \bigr)^2 }.$$1-coverage discovery thresholds are

$$| m_{\beta\beta} | = \sqrt{ \bigl( c_{12}^2 c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 \cos{\alpha} + s_{13}^2 m_3 \cos{\beta} \bigr)^2 + \bigl( s_{12}^2 c_{13}^2 m_2 \sin{\alpha} + s_{13}^2 m_3 \sin{\beta} \bigr)^2 }.$$2

The paper is explicit that these are not model-independent truths: they depend on flat priors for $$| m_{\beta\beta} | = \sqrt{ \bigl( c_{12}^2 c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 \cos{\alpha} + s_{13}^2 m_3 \cos{\beta} \bigr)^2 + \bigl( s_{12}^2 c_{13}^2 m_2 \sin{\alpha} + s_{13}^2 m_3 \sin{\beta} \bigr)^2 }.$$3 and on the adopted Gaussian prior for $$| m_{\beta\beta} | = \sqrt{ \bigl( c_{12}^2 c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 \cos{\alpha} + s_{13}^2 m_3 \cos{\beta} \bigr)^2 + \bigl( s_{12}^2 c_{13}^2 m_2 \sin{\alpha} + s_{13}^2 m_3 \sin{\beta} \bigr)^2 }.$$4.

3. Randomized joint measurements of Majorana observables

In fermionic quantum-information settings, Majorana sampling denotes randomized measurement schemes whose single-shot outcomes can be post-processed into unbiased estimators for many incompatible Majorana observables simultaneously (Majsak et al., 2024, McNulty et al., 2024). The relevant system is an $$| m_{\beta\beta} | = \sqrt{ \bigl( c_{12}^2 c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 \cos{\alpha} + s_{13}^2 m_3 \cos{\beta} \bigr)^2 + \bigl( s_{12}^2 c_{13}^2 m_2 \sin{\alpha} + s_{13}^2 m_3 \sin{\beta} \bigr)^2 }.$$5- or $$| m_{\beta\beta} | = \sqrt{ \bigl( c_{12}^2 c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 \cos{\alpha} + s_{13}^2 m_3 \cos{\beta} \bigr)^2 + \bigl( s_{12}^2 c_{13}^2 m_2 \sin{\alpha} + s_{13}^2 m_3 \sin{\beta} \bigr)^2 }.$$6-mode fermionic Fock space with Majorana operators

$$| m_{\beta\beta} | = \sqrt{ \bigl( c_{12}^2 c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 \cos{\alpha} + s_{13}^2 m_3 \cos{\beta} \bigr)^2 + \bigl( s_{12}^2 c_{13}^2 m_2 \sin{\alpha} + s_{13}^2 m_3 \sin{\beta} \bigr)^2 }.$$7

satisfying

$$| m_{\beta\beta} | = \sqrt{ \bigl( c_{12}^2 c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 \cos{\alpha} + s_{13}^2 m_3 \cos{\beta} \bigr)^2 + \bigl( s_{12}^2 c_{13}^2 m_2 \sin{\alpha} + s_{13}^2 m_3 \sin{\beta} \bigr)^2 }.$$8

For any even subset $$| m_{\beta\beta} | = \sqrt{ \bigl( c_{12}^2 c_{13}^2 m_1 + s_{12}^2 c_{13}^2 m_2 \cos{\alpha} + s_{13}^2 m_3 \cos{\beta} \bigr)^2 + \bigl( s_{12}^2 c_{13}^2 m_2 \sin{\alpha} + s_{13}^2 m_3 \sin{\beta} \bigr)^2 }.$$9, one defines

$m_{\min}$0

or, in the alternative notation,

$m_{\min}$1

for degree-$m_{\min}$2 monomials. These are the observables entering fermionic partial tomography and quadratic-plus-quartic chemistry Hamiltonians.

The simplest protocol proceeds by randomizing over a Majorana product $m_{\min}$3, applying a fermionic Gaussian unitary, measuring fermionic occupation numbers or disjoint Majorana pairs, and then post-processing the raw data into a binary variable $m_{\min}$4. For a target observable $m_{\min}$5, the output is distributed exactly as the noisy measurement

$m_{\min}$6

with visibility $m_{\min}$7 in one construction. This is not hardware noise: it is unsharpness induced by implementing a parent POVM and then coarse-graining it. The resulting unbiased estimator is

$m_{\min}$8

A central design principle is randomization over only a small number of fermionic Gaussian settings. One construction achieves all Majorana pairs with $m_{\min}$9 settings, all quartics with $\delta m_{\odot}^2$0 settings in general, and chemistry Hamiltonians with only $\delta m_{\odot}^2$1 deterministic settings. The visibility scales as

$\delta m_{\odot}^2$2

yielding simultaneous-estimation sample complexities

$\delta m_{\odot}^2$3

In a rectangular lattice of qubits encoding an $\delta m_{\odot}^2$4-mode fermionic system via Jordan–Wigner, the same approach can be implemented in circuit depth $\delta m_{\odot}^2$5 with $\delta m_{\odot}^2$6 two-qubit gates.

A complementary formulation develops the same idea in joint-measurement language for all even-degree Majorana observables up to degree $\delta m_{\odot}^2$7. The noisy marginals are written as

$\delta m_{\odot}^2$8

and the optimal common sharpness is characterized by the incompatibility robustness

$\delta m_{\odot}^2$9

By exploiting braid-group symmetry, this robustness is related to the operator norm of an SYK-type Hamiltonian,

$\Delta m_{atm}^2$0

For $\Delta m_{atm}^2$1, the paper shows

$\Delta m_{atm}^2$2

and presents a joint measurement scheme achieving the asymptotically optimal noise using a small number of fermionic Gaussian unitaries and uniform sampling from the set of all Majorana monomials. The corresponding all-observable estimation complexity is

$\Delta m_{atm}^2$3

These protocols are closely analogous to fermionic classical shadows but are framed operationally as parent POVMs plus classical post-processing rather than as explicit shadow-state reconstruction. The practical significance is that one experimental shot amortizes over a large non-commuting family of observables.

4. Random Majorana constellations in symmetric spin systems

A different usage of Majorana sampling concerns sampling the Majorana constellation itself for a spin-$\Delta m_{atm}^2$4 state (Goldberg et al., 2021). A pure spin-$\Delta m_{atm}^2$5 state can be viewed as a fully symmetrized state of $\Delta m_{atm}^2$6 spin-$\Delta m_{atm}^2$7 particles and represented by $\Delta m_{atm}^2$8 points on the sphere. The state is written as

$\Delta m_{atm}^2$9

with rotated creation operators

$s_{12}^2$0

The constellation $s_{12}^2$1 defines the state, and the coherent-state wavefunction

$s_{12}^2$2

has zeros precisely at the constellation points.

The random ensemble is defined by taking the $s_{12}^2$3 points independently and uniformly on $s_{12}^2$4 with Haar measure

$s_{12}^2$5

This is not the same as drawing a Haar-random pure state in the $s_{12}^2$6-dimensional symmetric subspace. The paper emphasizes that random Majorana constellations and Haar-random symmetric states are distinct ensembles for all but $s_{12}^2$7, because Haar-random symmetric states induce correlations between the Majorana points.

The principal diagnostics are multipoles and the quantumness measure

$s_{12}^2$8

together with cumulative multipole content

$s_{12}^2$9

Random Majorana states are found to be markedly nonclassical: their multipole weight is concentrated mostly at small $s_{13}^2$0, but with a much heavier high-$s_{13}^2$1 tail than coherent states. Numerically, the most contributing multipole scales as

$s_{13}^2$2

The hierarchy drawn in the paper is precise. Coherent states are the least quantum in this multipole sense; random Majorana states are more quantum than coherent states; Haar-random coefficient states are more quantum still. For the CUE/Haar ensemble on the symmetric Hilbert space, the average cumulative multipoles satisfy

$s_{13}^2$3

and the average quantumness is

$s_{13}^2$4

very close to the maximum possible single pure-state value $s_{13}^2$5. Random Majorana states are therefore atypical relative to both classical coherent states and Haar-random symmetric states.

The metrological interpretation in the paper is qualitative rather than a full sensing theory. Random Majorana constellations retain substantial higher-order multipole content even in the large-$s_{13}^2$6 limit, which suggests metrological usefulness. The text provided does not supply explicit QFI formulas, channel models, or analytical robustness bounds, so stronger metrological claims require additional sources.

5. Statistical spectroscopy and real-space screening of Majorana zero modes

In Majorana-device physics, one sampling strategy is to generate an ensemble of spectroscopy traces by varying external control parameters and then infer Majorana content from the statistics of the resulting spectra (Ziesen et al., 2022). The central observable is the averaged subgap spectral density

$s_{13}^2$7

with $s_{13}^2$8 when a true MBS is present and $s_{13}^2$9 when it is absent. Both cases can display a center peak near zero bias, so the decisive signature is the sign of the side-lobe oscillations: with an MBS the first side lobe is a maximum, without an MBS it is a minimum. The protocol requires sufficiently strong level mixing, $\mathcal S_2$00, sufficiently good resolution, $\mathcal S_2$01, and enough independent samples. The paper estimates

$\mathcal S_2$02

and for $\mathcal S_2$03 concludes that around $\mathcal S_2$04 samples are enough for practical discrimination.

A complementary, state-resolved line of work evaluates Majorana polarization as a local or nonlocal classifier of zero-energy states (Bena, 2017, Awoga et al., 2024). For a lattice BdG eigenstate with site amplitudes $\mathcal S_2$05 and $\mathcal S_2$06, the local Majorana-polarization vector is

$\mathcal S_2$07

For a region $\mathcal S_2$08 containing one candidate Majorana, the integrated criterion is

$\mathcal S_2$09

The interpretation is explicit: $\mathcal S_2$10 for a true Majorana state, $\mathcal S_2$11 for a regular electron/hole state in the ideal trivial limit, and $\mathcal S_2$12 for quasi-Majorana states with locally strong but non-aligned Majorana texture.

The newer nonlocal construction defines, for the lowest-energy state,

$\mathcal S_2$13

then integrates over left and right halves,

$\mathcal S_2$14

and forms the real scalar

$\mathcal S_2$15

According to the paper, $\mathcal S_2$16, with $\mathcal S_2$17 for ideal non-overlapping Majorana zero modes on opposite ends, $\mathcal S_2$18 but still negative for overlapping Majoranas, and $\mathcal S_2$19 for trivial states. In uniform wires, $\mathcal S_2$20 reproduces the topological transition at

$\mathcal S_2$21

In NS and SNS junctions it separates topological Majorana modes from confinement-induced trivial zero modes, and under disorder it remains informative even when spectra develop many accidental zero crossings.

Taken together, these methods show two distinct notions of sampling in Majorana-device studies. One is ensemble sampling of spectra across parameter settings; the other is parameter-space or disorder-realization screening using real-space Majorana-sensitive observables. Both aim to discriminate genuine Majorana structure from spectrally similar trivial states.

6. Readout, remote imaging, and control primitives

Several related works do not define sampling procedures in the same probabilistic sense, but they provide operational primitives on which Majorana-sampling architectures may depend. Continuous charge sensing of a tunnel-coupled quantum dot, for example, can implement a projective measurement of a Majorana operator product through a conserved combined parity rather than through the bare Majorana product itself (Steiner et al., 2020). In the single-dot tetron setting,

$\mathcal S_2$22

The readout is symmetry-protected provided $\mathcal S_2$23 is conserved and the two eigenspaces generate distinguishable detector outputs. The detector current takes the diffusive form

$\mathcal S_2$24

but the paper shows that the average current generically does not distinguish the parity sectors at long times. Instead, the information can reside in current-noise correlations. The measurement is valid only if the sectors are distinguishable, which in the single-dot model requires

$\mathcal S_2$25

A different form of remote Majorana-sensitive probing is the mirage of a Majorana mode in an elliptical corral of free electrons (Sacramento, 2018). The wire-end Majorana is coupled to a two-dimensional electron gas confined in an ellipse, and the relevant distinction is between ordinary fermionic tunneling and Majorana tunneling with equal electron and hole components. In favorable cases—wire end at a focus and chemical potential tuned to a corral eigenstate with strong weight at both foci—the Majorana case produces an enhanced mirage effect with spectral weight mainly confined around the foci. The self-conjugacy of the induced state is monitored through

$\mathcal S_2$26

which vanishes for a self-conjugate Majorana-like image.

Gate-controlled tunneling also provides a transport and braiding primitive for Majorana networks (Sau et al., 2010). Two coupled Majoranas are described by

$\mathcal S_2$27

and adiabatic switching in a three-Majorana configuration,

$\mathcal S_2$28

moves an unpaired Majorana according to

$\mathcal S_2$29

An exchange of two Majoranas is implemented by the braid unitary

$\mathcal S_2$30

with $\mathcal S_2$31. In a triangular junction, the sign is fixed by a chirality

$\mathcal S_2$32

leading to

$\mathcal S_2$33

These control primitives are not themselves sampling algorithms, but they specify how Majorana operators can be measured, imaged, and transformed in hardware.

Across these strands, Majorana sampling emerges as a family of probabilistic and randomized methodologies attached to Majorana structure at very different levels: unknown phases, observable algebras, geometric state representations, spectral ensembles, and device-level readout. The literature therefore supports a precise but plural definition: Majorana sampling is any framework in which sampling, randomization, or ensemble generation is used to infer physically relevant information carried by Majorana phases, Majorana operators, Majorana constellations, or Majorana zero modes.