Strongly Local Sets: Insights & Constructions

• Strongly local sets are defined by robust local constraints that yield a small global difference set while ensuring each subset maintains expansive local properties.
• They appear in diverse fields—from additive combinatorics to quantum state discrimination and geometric approximation—with each domain tailoring the local criteria to its specific operational framework.
• Recursive constructions and precise local approximations underpin these sets, providing key insights into distinct-distance problems, activation hierarchies, and efficient localized algorithms.

“Strongly local sets” is a field-dependent term rather than a single canonical notion. In additive combinatorics, it denotes finite sets $A\subset \mathbb R$ whose global difference set is small while every subset $A'\subseteq A$ still has a large difference set, uniformly in $|A'|$ (Fish et al., 2018). In multipartite quantum state discrimination, a “strong-local” set is a locally distinguishable orthogonal set of pure states that cannot be converted by any orthogonality-preserving local measurement into a locally indistinguishable set, even after allowing coalitions of parties (Bera et al., 2024). Related research also uses the adjective “strongly-local” for algorithms whose runtime depends only on seed-set parameters rather than the ambient instance size (Huang et al., 2023), and studies sets that are “strongly” locally well approximable by model classes through tangent-set methods (Badger et al., 2014). The common theme is stringent control of local behavior, but the formal object and the mathematical stakes vary sharply across domains.

1. Terminological scope and domain-specific meanings

Domain	Underlying object	Meaning of “strongly local”
Additive combinatorics	Finite $A\subset\mathbb R$	Small $|A-A|$ globally, but every $k$-subset has $|A'-A'|\ge k^\alpha$ (Fish et al., 2018)
Quantum state discrimination	Orthogonal multipartite pure states	No OPLM can turn a locally distinguishable set into a locally indistinguishable one (Bera et al., 2024)
Quantum elimination paradigm	Orthogonal multipartite states	“Strongly local in the strongest possible sense” means elimination-based nonlocality across every bipartition (Ghosh et al., 2022)
Geometric measure theory	Closed $A\subset\mathbb R^n$	Sets characterized by strong local approximation to a model class via $\Theta$, $\beta$, and tangent sets (Badger et al., 2014)
Localized hypergraph optimization	Seeded densest-subhypergraph problems	A strongly-local algorithm explores only a seed-dependent local region (Huang et al., 2023)

Taken together, these uses show that “strong locality” is a structural label whose exact content is supplied by the ambient theory. In some settings it constrains subset growth, in others it forbids activation of nonlocality, and in still others it limits the spatial or combinatorial footprint of approximation or optimization procedures.

A further point is that the terminology is not uniform even within quantum information. One paper places strong-local sets at the “most local” end of a hierarchy of locally distinguishable ensembles (Bera et al., 2024), while another uses “strongly local in the strongest possible sense” for sets that remain elimination-based nonlocal in every bipartition (Ghosh et al., 2022). This suggests that the phrase should always be read relative to the paper-specific operational framework.

2. Strongly local difference sets in additive combinatorics

For $A'\subseteq A$0, the difference set is

$A'\subseteq A$1

A set $A'\subseteq A$2 of size $A'\subseteq A$3 is called strongly local, or said to have the strong local difference-set property, with exponent $A'\subseteq A$4 if

$A'\subseteq A$5

for every $A'\subseteq A$6 of size $A'\subseteq A$7 (Fish et al., 2018).

The principal existence theorem states that when $A'\subseteq A$8 and $A'\subseteq A$9 is a power of $|A'|$0, there exists $|A'|$1 with $|A'|$2 such that

$|A'|$3

and every subset $|A'|$4 of size $|A'|$5 satisfies

$|A'|$6

Asymptotically, this is written

$|A'|$7

If $|A'|$8 is not a power of two, the same proof gives

$|A'|$9

The same construction yields a planar distinct-distances corollary: $A\subset\mathbb R$0 which is described as the first nontrivial upper bound for the distinct-distances-with-local-properties problem in the plane where the local threshold $A\subset\mathbb R$1 grows polynomially in $A\subset\mathbb R$2 (Fish et al., 2018).

The significance of the construction is twofold. Globally, the full set has difference-set size $A\subset\mathbb R$3, well below the trivial $A\subset\mathbb R$4 scale. Locally, no $A\subset\mathbb R$5-point subset can collapse to substantially fewer than $A\subset\mathbb R$6 differences. In that precise sense, the construction is simultaneously sparse in its total difference structure and uniformly expansive on every scale.

3. Recursive construction and proof architecture

The construction proceeds recursively through sets $A\subset\mathbb R$7 with $A\subset\mathbb R$8, where $A\subset\mathbb R$9 (Fish et al., 2018). One first chooses positive real numbers $|A-A|$0 that are linearly independent over $|A-A|$1, meaning that the only integer solution to

$|A-A|$2

is $|A-A|$3 for all $|A-A|$4. The base case is $|A-A|$5, and for $|A-A|$6,

$|A-A|$7

The final set is $|A-A|$8, with $|A-A|$9.

The global bound follows from a complete description of the differences. Every element of $k$0 has the form

$k$1

By the $k$2-independence of the $k$3, all such sums are distinct. Since there are $k$4 coefficient choices,

$k$5

The local bound is proved by induction on the recursive depth. For a fixed $k$6 of size $k$7, the proof sets

$k$8

so that $k$9. Elements of $|A'-A'|\ge k^\alpha$0 are grouped by the highest stage at which they first appear, and one chooses two translates $|A'-A'|\ge k^\alpha$1 and $|A'-A'|\ge k^\alpha$2 with $|A'-A'|\ge k^\alpha$3 minimal so that

$|A'-A'|\ge k^\alpha$4

are both nonempty. The key recursive estimate is

$|A'-A'|\ge k^\alpha$5

Writing $|A'-A'|\ge k^\alpha$6, each $|A'-A'|\ge k^\alpha$7 splits into two pieces, and the proof derives

$|A'-A'|\ge k^\alpha$8

Induction reduces the problem to a numerical inequality,

$|A'-A'|\ge k^\alpha$9

which is justified by Claim 2.2 in the paper together with a calculus check.

At the top level $A\subset\mathbb R^n$0, this yields

$A\subset\mathbb R^n$1

and hence $A\subset\mathbb R^n$2. Small cases already display the pattern: for $A\subset\mathbb R^n$3, $A\subset\mathbb R^n$4 has size $A\subset\mathbb R^n$5; for $A\subset\mathbb R^n$6, $A\subset\mathbb R^n$7 has size $A\subset\mathbb R^n$8. The recursive doubling and ternary difference-counting are the essential mechanism behind the exponent $A\subset\mathbb R^n$9.

4. Strong-local sets in quantum state discrimination

In multipartite state discrimination, let $\Theta$0 be a set of mutually orthogonal $\Theta$1-partite pure states. An orthogonality-preserving local measurement is a collection of local POVM or PVM elements $\Theta$2 such that in each branch the states $\Theta$3 remain mutually orthogonal, up to elimination of some states. A locally distinguishable set $\Theta$4 is called strong-local if there exists no OPLM, even allowing any subset of the parties to combine into a single lab, that transforms $\Theta$5 into a locally indistinguishable set in any induced partition (Bera et al., 2024).

An equivalent formulation is given in terms of joint projective measurements. For every choice of parties $\Theta$6 performing a joint projective measurement $\Theta$7, each post-measurement branch

$\Theta$8

must remain locally distinguishable in the resulting $\Theta$9-partite scenario. In the hierarchy adopted in that work, strong-local sets occupy the extreme “most local” position.

Several characterization results isolate cases where activation is impossible. Any orthogonal product basis in $\beta$0 is completely distinguishable under local projective measurements and classical communication. Consequently, any set of orthogonal product states in $\beta$1 or $\beta$2 is strong-local. There is also a tripartite theorem for $\beta$3: if a product basis is LPCC-distinguishable in the $\beta$4 cut, then whenever either $\beta$5 or $\beta$6 goes first with any nontrivial PVM, the post-measurement branches remain product bases in $\beta$7 or $\beta$8, and thus remain LPCC-distinguishable. No activation of nonlocality is possible unless $\beta$9 performs a nontrivial measurement.

The same paper introduces a hierarchy of activability. An LOCC-distinguishable $A'\subseteq A$00-partite set is $A'\subseteq A$01-activable if, in some grouping into $A'\subseteq A$02 labs, there is an OPLM that renders all post-measurement branches locally irreducible within that $A'\subseteq A$03-partition. If those branches remain locally irreducible in at least one $A'\subseteq A$04-partition, the set is strong-$A'\subseteq A$05-activable. Non-activable sets are then called strong local, meaning “not 2-activable.” The resulting scale is

$A'\subseteq A$06

This hierarchy formalizes locality as resistance to operational activation.

5. Explicit quantum constructions, activation, and terminological divergence

Two infinite families in $A'\subseteq A$07 illustrate the distinction between different degrees of locality (Bera et al., 2024). The family $A'\subseteq A$08 is “Type-I activable”: a single party, Bob, can apply an OPLM so that each post-measurement branch is locally indistinguishable in the $A'\subseteq A$09 cut. The family $A'\subseteq A$10 is “Type-II activable”: no single party suffices, but a pair such as $A'\subseteq A$11 and $A'\subseteq A$12 acting jointly can produce local indistinguishability in $A'\subseteq A$13.

For $A'\subseteq A$14, the paper gives explicit unnormalized state vectors $A'\subseteq A$15, indexed by $A'\subseteq A$16 and $A'\subseteq A$17, with the $A'\subseteq A$18 choices arranged so that the total count is exactly $A'\subseteq A$19 orthogonal states. The set is LPCC-distinguishable. Yet when Bob measures in the computational basis $A'\subseteq A$20, each outcome isolates a sub-block exactly isomorphic to Bennett’s $A'\subseteq A$21 product basis in the $A'\subseteq A$22 cut, which is locally indistinguishable there.

For $A'\subseteq A$23, the construction begins with a central seed

$A'\subseteq A$24

and then adds eight further classes of states through block prescriptions indexed by $A'\subseteq A$25 and by $A'\subseteq A$26 or $A'\subseteq A$27. No single party’s PVM can produce a locally indistinguishable branch. However, if $A'\subseteq A$28 and $A'\subseteq A$29 join into one lab and measure the three-dimensional subspaces

$A'\subseteq A$30

each branch again yields one of Bennett’s $A'\subseteq A$31 product bases in the $A'\subseteq A$32 cut. The paper uses this distinction to argue that $A'\subseteq A$33 is “more local” than $A'\subseteq A$34, because it requires nonlocal operations to exhibit nonlocality.

The same work gives a data-hiding interpretation. A Type-I activable set can be made locally inaccessible if the party performing the OPLM withholds the classical measurement record; by contrast, strong local sets cannot be turned into data-hiding resources by any proper subset of the parties.

A different quantum usage appears in the elimination paradigm (Ghosh et al., 2022). There, a set is elimination-based nonlocal if no party or group can perform a nontrivial orthogonality-preserving local measurement that eliminates one of the states outright; equivalently, every such local operator must be proportional to the identity on its subsystem. The paper then says that a set is “strongly local in the strongest possible sense” if no party alone can eliminate any state and no coalition of all but one party can eliminate any state, so the set remains elimination-based nonlocal in every bipartition. Activation theorems show that a locally distinguishable set $A'\subseteq A$35 can be mapped, by Bob’s two-outcome OPM

$A'\subseteq A$36

to Bennett–DiVincenzo–Mor–Shor–Smolin–Terhal’s $A'\subseteq A$37 UPB in either branch. Likewise, a 27-state set $A'\subseteq A$38 becomes, after each party applies the same local OPM, the 27-state strongly nonlocal set of Halder–Banik–Agrawal–Bandyopadhyay in every one of the $A'\subseteq A$39 outcome branches.

These two quantum papers therefore assign the phrase “strong local” to different operational regimes. One reserves it for locally distinguishable sets that are never activable (Bera et al., 2024); the other uses it in a setting where the target property is elimination-based nonlocality under every bipartition (Ghosh et al., 2022). The divergence is terminological rather than mathematical: both papers formulate precise operational criteria, but the label itself is not standardized.

6. Related notions of strong locality in geometry and localized optimization

In geometric measure theory, the relevant object is not a specially named “strongly local set” but a closed set $A'\subseteq A$40 that is locally uniformly approximated by a cone $A'\subseteq A$41 of model sets (Badger et al., 2014). The basic bilateral approximation quantity is the Walkup–Wets distance

$A'\subseteq A$42

and the unilateral analogue is

$A'\subseteq A$43

Tangent sets $A'\subseteq A$44 and pseudotangents $A'\subseteq A$45, defined through Attouch–Wets limits of blow-ups, characterize pointwise and local approximability. The paper proves bilateral and unilateral characterization theorems, a pointwise decomposition

$A'\subseteq A$46

and dimension bounds derived from covering profiles. In this setting, “strong” locality refers to local approximation quality and tangent-cone organization, not to subset combinatorics or LOCC behavior.

In localized dense-substructure discovery, “strongly-local” instead describes the computational footprint of an algorithm (Huang et al., 2023). For a hypergraph $A'\subseteq A$47 with seed set $A'\subseteq A$48, the Anchored Densest Subhypergraph objective is

$A'\subseteq A$49

with locality parameter $A'\subseteq A$50. A strongly-local algorithm is one whose runtime depends only on seed-set parameters and the returned set, not on $A'\subseteq A$51; formally, it explores only $A'\subseteq A$52 vertices and hyperedges and runs in time $A'\subseteq A$53. The paper proves a sharp threshold: when $A'\subseteq A$54, there is a strongly-local flow-based algorithm that explores at most

$A'\subseteq A$55

hyperedges and

$A'\subseteq A$56

vertices, where $A'\subseteq A$57 and $A'\subseteq A$58 is the hypergraph rank; when $A'\subseteq A$59, no strongly-local algorithm exists for ADSH or ADSH-F, because one can construct instances where the optimizer $A'\subseteq A$60 is arbitrarily large compared to $A'\subseteq A$61.

These adjacent literatures broaden the semantic range of strong locality. In geometric approximation it governs how a set looks under infinitesimal blow-up; in hypergraph optimization it governs how far computation must propagate from a seed. A plausible implication is that “strong locality” functions less as a universal definition than as a recurrent research pattern: local constraints are made uniform, quantitative, and resistant to degeneration under scale change, coalition formation, or expansion of the ambient space.