Sector Lengths: Quantum, Landau & Dark Sectors

• Sector lengths are quantitative measures that partition physical parameters—ranging from multi-qubit correlations to spatial and decay scales—into distinct length-resolved sectors.
• In quantum information, sector lengths (Aₖ) are computed from grouped Pauli correlators and decompose state purity into k-partite contributions, thereby revealing entanglement properties.
• In Landau theory and dark-sector studies, sector lengths govern the stability of externally written fields and determine observable decay distances of exotic particles in experiments.

Searching arXiv for the specified papers to ground the article in current sources. Recent arXiv literature uses the phrase sector lengths in distinct technical senses. In multi-qubit quantum information, sector lengths $A_k$ quantify the amount of $k$-partite correlations in a basis-independent manner and organize the Pauli/Fano–Bloch expansion of an $n$-qubit state (Wyderka et al., 2019). In the nonintrinsic sector of Landau theory, the relevant lengths are the correlation length $\xi$, the writing length $\ell_D$, and the frustration length $\ell_{\rm fr}$, whose hierarchy determines whether externally written microscale fields survive coarse graining as coefficient fields in the continuum free energy (Li, 23 Mar 2026). In flavoured dark-sector phenomenology, the operative lengths are proper decay lengths $c\tau$ of dark pions and the corresponding emergence length scales of dark jets in detectors (Renner et al., 2018). The common feature is structural: each usage packages physically distinct information into length-resolved sectors, but the mathematical objects, observables, and applications differ substantially across fields.

1. Distinct technical meanings of sector lengths

The phrase appears in at least three sharply different frameworks. In quantum information, sector lengths are quadratic invariants of correlators. In Landau theory, the phrase refers to a hierarchy of spatial scales that delineates a nonintrinsic regime of coarse-grained free energies. In dark-sector phenomenology, it denotes experimentally relevant propagation distances.

Context	Meaning of sector lengths	Primary objects
Multi-qubit states	$A_k$, the squared norm of weight-$k$ Pauli correlators	Density operators, Pauli strings, reduced-state purities
Nonintrinsic Landau theory	$\xi$, $k$0, $k$1	Coarse-grained coefficient fields, written disorder or doping patterns
Flavoured dark sector	$k$2 and emergence lengths	Dark pions, mediator-induced decay widths, detector-scale decay profiles

This terminological multiplicity matters because the same phrase can otherwise suggest a single invariant formalism. The literature instead supports a field-dependent usage: in one case sector lengths are algebraic correlation measures, in another they are control and stability scales of a continuum theory, and in a third they are macroscopic decay distances.

2. Sector lengths as correlation invariants in multi-qubit systems

For an $k$3-qubit state $k$4, the relevant starting point is the Pauli/Fano–Bloch expansion in the $k$5-qubit Pauli basis. Grouping the expansion by the number of nontrivial local Pauli matrices gives

$k$6

The $k$7-partite sector length, denoted $k$8 in the paper, is defined by

$k$9

where the sum runs over all Pauli strings $n$0 acting nontrivially on exactly $n$1 subsystems. Equivalently, $n$2, and $n$3 by normalization (Wyderka et al., 2019).

These quantities are invariant under local unitaries because local basis changes rotate local Pauli frames orthogonally and therefore preserve the squared Euclidean norm of each correlation tensor. They are also directly linked to purity: $n$4 For pure states, $n$5. This makes sector lengths a decomposition of total purity into correlation sectors indexed by weight.

For pure states, the sector lengths are further constrained by MacWilliams-type identities,

$n$6

for all integers $n$7. A subset of $n$8 of these equations is linearly independent. Together with shadow inequalities, these relations supply the main analytic machinery for deriving bounds on allowed correlation distributions.

3. Global constraints, low-party characterizations, and structural bounds

The basic individual-sector bounds are explicit. One has $n$9, with equality for pure product states such as $\xi$0, and $\xi$1, tight for all $\xi$2 (Wyderka et al., 2019). For the three-body sector,

$\xi$3

The equality cases are also explicit: for $\xi$4, the GHZ state has $\xi$5; for $\xi$6, the state

$\xi$7

has $\xi$8; and for $\xi$9, product states attain $\ell_D$0. For the full-body sector, the paper proves $\ell_D$1 for odd $\ell_D$2, tight for odd-$\ell_D$3 GHZ states, and $\ell_D$4 for even $\ell_D$5, tight for even-$\ell_D$6 GHZ states.

For two qubits, the feasible pairs $\ell_D$7 are completely characterized by

$\ell_D$8

The extremal points include pure product states with $\ell_D$9, Bell states with $\ell_{\rm fr}$0, and the maximally mixed state with $\ell_{\rm fr}$1. For three qubits, the feasible triples $\ell_{\rm fr}$2 form a convex polytope defined by

$\ell_{\rm fr}$3

$\ell_{\rm fr}$4

$\ell_{\rm fr}$5

and

$\ell_{\rm fr}$6

Pure three-qubit states lie on the line segment connecting $\ell_{\rm fr}$7 and $\ell_{\rm fr}$8.

A notable structural point is that a complete characterization for $\ell_{\rm fr}$9 remains open. The paper states a conjecture that for each $c\tau$0 there exists $c\tau$1 such that for all $c\tau$2, $c\tau$3, and reports numerics suggesting this for $c\tau$4 at about $c\tau$5. It also remarks that why the feasible region for $c\tau$6 forms a polytope with few linear constraints is not yet understood.

4. Entanglement detection, monogamy, entropy relations, and estimation

Sector lengths support several concrete applications. For two qubits, separable states satisfy $c\tau$7, so any state with $c\tau$8 is entangled (Wyderka et al., 2019). For three qubits, biseparable states satisfy $c\tau$9, while all states satisfy $A_k$0; hence $A_k$1 certifies genuine multipartite entanglement. For four qubits, highest-order correlations alone cannot detect genuine multipartite entanglement because both biseparable and general states satisfy $A_k$2. The next-to-highest sector does distinguish them: biseparable states obey $A_k$3, while all states obey $A_k$4, and the state $A_k$5 with $A_k$6 is detected as not biseparable by $A_k$7.

The formalism also yields monogamy-type bounds. The non-symmetric relation

$A_k$8

limits the total amount of pairwise two-body sector length that one qubit can share with the others. This is closely related to Osborne–Verstraete-type monogamy. In addition, the paper proves a symmetrized strong subadditivity statement for linear entropy in three-qubit systems: $A_k$9 Equivalently, in terms of mutual linear entropies,

$k$0

and by induction,

$k$1

for $k$2.

Experimentally and computationally, the quantities are accessible because

$k$3

The number of required $k$4-body correlators is $k$5. The paper notes that randomized measurement schemes can estimate sector lengths from sampled Pauli settings. Depolarizing noise acts simply: for

$k$6

one has $k$7. The formalism is therefore experimentally useful precisely because sector lengths are in one-to-one correspondence with purities of reduced states.

5. Sector lengths in the nonintrinsic sector of Landau theory

In Landau theory, the relevant sector lengths are not correlation invariants but a hierarchy of spatial scales defining when an externally written microscale field survives coarse graining as a coefficient field in the continuum free energy. The paper distinguishes conventional intrinsic Landau theory, where coefficients are fixed and spatially uniform, from a nonintrinsic sector in which a written field $k$8 enters the free-energy functional as a spatially prescribed coefficient field (Li, 23 Mar 2026). A representative Landau–Ginzburg functional is

$k$9

For the quadratic coefficient,

$\xi$0

and, more explicitly,

$\xi$1

The three key lengths are the correlation length $\xi$2, the writing length $\xi$3, and the frustration length $\xi$4. The correlation length sets the width of the coarse-graining kernel and is the resolution limit of effective coefficient fields; in Landau mean-field settings, $\xi$5, and near a continuous transition $\xi$6, with mean-field $\xi$7. The writing length $\xi$8 is the spatial variation scale of the externally written field and is fixed by the lithography, irradiation, or doping protocol. The frustration length $\xi$9 is the emergent scale beyond which long-range elastic, dipolar, or electrostatic back-action reconstructs the imposed pattern.

The nonintrinsic sector requires the strict hierarchy

$k$00

The first inequality ensures that the convolution with $k$01 does not smear out the written pattern; the second ensures that the pattern is not rebuilt by long-range forces. The failure modes are explicit. If $k$02, coarse graining washes out the pattern and the coefficients revert to effectively uniform intrinsic parameters. If $k$03, long-range back-action reconstructs the landscape, removing metastability and writable control.

This framework has a concrete dynamic consequence. For relaxational dynamics

$k$04

gradients in the quadratic coefficient bias interface motion according to

$k$05

The paper identifies ion-patterned FeRh as a plausible realization. Focused He$k$06 irradiation can locally increase chemical disorder at tens-of-nanometers resolution, shifting the transition temperature $k$07 and hence $k$08. In FeRh, $k$09 is described as short compared with the patterning scale near the metamagnetic transition, and the AF–FM transition involves a small volume change of $k$10–$k$11, which supports a relatively large $k$12.

6. Proper decay lengths and emergence scales in a flavoured dark sector

In flavoured dark-sector phenomenology, the lengths of interest are physical distances traversed by unstable dark hadrons before visible decay. The paper studies a QCD-like dark sector with gauge group SU$k$13, $k$14, and $k$15 light dark quark flavours, coupled to right-handed down-type Standard Model quarks through a heavy scalar mediator $k$16 (Renner et al., 2018). The lightest composite mesons are dark pions $k$17, and their proper decay lengths span a broad range, from millimetres to hundreds of meters. The reason is helicity-suppressed flavour dependence: the decay width is proportional to squared visible-quark masses, so dark pions connected to $k$18-quarks decay much faster than species connected only to light or strange quarks.

The proper decay length is

$k$19

In the partonic regime $k$20 GeV, the inclusive width for $k$21 is

$k$22

Accordingly,

$k$23

The paper gives an explicit benchmark: for $k$24, $k$25, $k$26, final state $k$27, and $k$28, one finds $k$29; for $k$30, $k$31; for $k$32, $k$33. In the chiral regime near $k$34 GeV, CP forbids $k$35 decays at leading order for real $k$36, so $k$37 widths are suppressed and very long lifetimes result.

These decay lengths determine fixed-target and collider observables. For a lab-frame decay length $k$38, the probability to decay inside a vessel extending from $k$39 to $k$40 is

$k$41

The paper uses $k$42 m and $k$43 m for SHiP, and $k$44 m and $k$45 m for NA62 dump mode. With

$k$46

it finds sizeable regions in which the “13” and “23” flavour scenarios yield $k$47 expected events at SHiP and NA62.

At the LHC, the same multi-scale lifetime structure produces emerging jets. The decay probability before radius $k$48 is

$k$49

The paper emphasizes that different dark-pion species generate different emergence scales: $k$50-connected species with $k$51 in the mm–cm range decay in the tracker, while metre-scale species can first deposit visible energy in calorimeters or even in the muon system. This yields a two-scale emergence pattern, with a fast rise from $k$52-decays and a slower tail from light-flavour decays. By contrast, dark hadronization occurs at distances of order $k$53–$k$54 fm and is negligible on detector scales; the experimentally relevant emergence length is controlled by $k$55, not by hadronization.