Localizable Quantumness: Frameworks & Applications

• Localizable quantumness is the property whereby quantum features like entanglement and coherence are concentrated in localized observables, subsystems, or spacetime regions rather than being inherently global.
• In quantum field theory, it replaces traditional particle localization with local algebras and smeared operator-valued distributions to maintain causality and Lorentz covariance.
• In quantum information, localizable resources enable precise local measurements and operations that enhance teleportation fidelity, entanglement certification, and the study of measurement-induced phase transitions.

Localizable quantumness denotes a class of notions in which quantum structure is not treated as an irreducibly global feature, but as something that can be represented, concentrated, preserved, or operationally accessed within localized observables, spacetime regions, subsystems, subspaces, or projected ensembles. In relativistic quantum field theory, the relevant objects are local observables and strictly localizable operator-valued distributions rather than sharply localized particle wave functions (Khoze et al., 2018, Balachandran, 2016, Kubicki et al., 2016). In quantum information, the same phrase and nearby constructions refer to entanglement, concurrence, coherence, and measurement structure that can be concentrated onto selected parties by local measurements or by non-adaptive local operations assisted by shared entanglement (Akibue et al., 6 Jan 2026, Consiglio et al., 2021, Hamma et al., 2020, Sadhukhan et al., 2015). In monitored circuits, certification theory, and localization-based many-body protocols, it refers to the persistence of usable quantum properties under projection, measurement, or disorder-driven localization (Manna et al., 20 Jan 2026, Du et al., 22 Sep 2025, Sahoo et al., 2023).

1. Relativistic localization: observables rather than particles

In relativistic quantum field theory, Born localization of a single-particle wave function does not survive unchanged. In nonrelativistic quantum mechanics one may define

$$P_K=\int_K d^dx\, |\vec{x}\rangle\langle \vec{x}|, \qquad \psi_K=P_K|\psi\rangle,$$

so that wave functions supported in disjoint regions are orthogonal. The review by Schroer emphasizes that this reciprocity between localized states and localized observables breaks down in relativistic physics: the Newton–Wigner position operator is not Lorentz covariant, and the Currie–Jordan–Sudarshan no-interaction theorem excludes a covariant position operator for interacting relativistic point particles. What replaces particle localization is symplectic localization of observables (Balachandran, 2016).

For a free scalar field, the localized object is the smeared field

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$

and the associated Weyl algebra is generated by

$W(f)=e^{i\varphi(f)/\sqrt{2}},$

with

$$W(f)W(g)=W(f+g)e^{-\frac{i}{2}\operatorname{Im}(f,g)},\qquad W(f)^\ast=W(-f).$$

Localization is therefore encoded in a net of local algebras $\mathcal W(K)$, not in sharply localized particle states. The net is isotonic,

$$K_1\subseteq K_2 \;\Rightarrow\; \mathcal{W}(K_1)\subseteq \mathcal{W}(K_2),$$

and for the causal complement $K'$ one requires Haag duality,

$\mathcal{W}(K')=\mathcal{W}'(K).$

In modular localization for a wedge $W$, one introduces

$$S_W=J_W\Delta_W^{1/2}, \qquad \mathrm{Re}\,\mathcal{H}(W)=\{\zeta_W\in\mathcal{H}\;|\;S_W\zeta_W=\zeta_W\},$$

so that localization is carried by real standard subspaces of the one-particle Hilbert space and the corresponding Weyl algebras. This same framework yields the Unruh effect through the KMS property of the vacuum restricted to a wedge algebra, and encodes spin-statistics through the modified involution $\varphi(f)=\int d^4x\, f(x)\varphi(x),$0 for spin-$\varphi(f)=\int d^4x\, f(x)\varphi(x),$1 (Balachandran, 2016).

A related but distinct formulation appears in the Heisenberg-picture locality program of Deutsch and Hayden and its generalization to arbitrary states. There, local observables of a composite system are $\varphi(f)=\int d^4x\, f(x)\varphi(x),$2 and $\varphi(f)=\int d^4x\, f(x)\varphi(x),$3, and the claim is that complete quantum information can be represented by state-dependent values of such local observables, either as Deutsch–Hayden matrix values

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$4

or as noncommutative values

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$5

The locality condition for operations on $\varphi(f)=\int d^4x\, f(x)\varphi(x),$6 is

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$7

This formalism localizes quantum information in local observables, while explicitly leaving open how the full noncommutative value can be retrieved by local processes in practice (Kong, 2022).

2. Strict localizability in local quantum field theory

The mathematically sharp QFT version of localizable quantumness is strict localizability in the sense of Jaffe. The basic object is an operator-valued distribution $\varphi(f)=\int d^4x\, f(x)\varphi(x),$8, and actual operators are defined only after smearing,

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$9

For strict localizability, $W(f)=e^{i\varphi(f)/\sqrt{2}},$0 belongs to the space $W(f)=e^{i\varphi(f)/\sqrt{2}},$1 of smooth test functions with compact support in spacetime. The inclusion relations are

$W(f)=e^{i\varphi(f)/\sqrt{2}},$2

so tempered distributions form a smaller class than strictly localizable distributions. Compact support is the ingredient that preserves locality and causality: if the supports of $W(f)=e^{i\varphi(f)/\sqrt{2}},$3 and $W(f)=e^{i\varphi(f)/\sqrt{2}},$4 are spacelike separated, then

$W(f)=e^{i\varphi(f)/\sqrt{2}},$5

This is the precise sense in which compactly supported smearing “ensures the causality property of QFT” (Khoze et al., 2018).

The same paper places semiclassical Higgsplosion rates inside this broader framework. Starting from the Källén–Lehmann representation,

$W(f)=e^{i\varphi(f)/\sqrt{2}},$6

and similarly for the $W(f)=e^{i\varphi(f)/\sqrt{2}},$7PI two-point function and time-ordered correlator, the point is that rapid spectral growth need not violate locality provided the field is treated as a smeared distribution. Jaffe’s criterion allows

$W(f)=e^{i\varphi(f)/\sqrt{2}},$8

equivalently

$W(f)=e^{i\varphi(f)/\sqrt{2}},$9

with $$W(f)W(g)=W(f+g)e^{-\frac{i}{2}\operatorname{Im}(f,g)},\qquad W(f)^\ast=W(-f).$$0 polynomial. The paper contrasts three regimes: $$W(f)W(g)=W(f+g)e^{-\frac{i}{2}\operatorname{Im}(f,g)},\qquad W(f)^\ast=W(-f).$$1 corresponds to tempered distributions, $$W(f)W(g)=W(f+g)e^{-\frac{i}{2}\operatorname{Im}(f,g)},\qquad W(f)^\ast=W(-f).$$2 to strictly localizable fields, $$W(f)W(g)=W(f+g)e^{-\frac{i}{2}\operatorname{Im}(f,g)},\qquad W(f)^\ast=W(-f).$$3 to a quasi-localizable boundary case, and $$W(f)W(g)=W(f+g)e^{-\frac{i}{2}\operatorname{Im}(f,g)},\qquad W(f)^\ast=W(-f).$$4 to non-localizable distributions for which compactly supported test functions do not exist (Khoze et al., 2018).

In the Higgsplosion application, the smeared semiclassical rate is

$$W(f)W(g)=W(f+g)e^{-\frac{i}{2}\operatorname{Im}(f,g)},\qquad W(f)^\ast=W(-f).$$5

and the smeared self-energy takes the form

$$W(f)W(g)=W(f+g)e^{-\frac{i}{2}\operatorname{Im}(f,g)},\qquad W(f)^\ast=W(-f).$$6

The central admissibility statement is therefore not that the theory is tempered, but that exponential growth consistent with $$W(f)W(g)=W(f+g)e^{-\frac{i}{2}\operatorname{Im}(f,g)},\qquad W(f)^\ast=W(-f).$$7 remains compatible with local QFT once strict localizability replaces the narrower tempered-distribution framework (Khoze et al., 2018).

A complementary construction appears for the $$W(f)W(g)=W(f+g)e^{-\frac{i}{2}\operatorname{Im}(f,g)},\qquad W(f)^\ast=W(-f).$$8-dimensional Dirac field in a cavity. Instead of momentum-like global quanta, one quantizes modes localized in a finite interval at an initial time, obtaining a local algebra generated by operators such as $$W(f)W(g)=W(f+g)e^{-\frac{i}{2}\operatorname{Im}(f,g)},\qquad W(f)^\ast=W(-f).$$9 and a local vacuum annihilated by all of them. The resulting one-particle states satisfy Knight’s strict localization condition relative to the complementary region, but the representation is unitarily inequivalent to the standard global Fock representation because the Bogoliubov $\mathcal W(K)$0-coefficients are not Hilbert–Schmidt. The paper identifies the global vacuum’s spatial correlations as the obstruction to “common sense localization” in the standard Fock picture (Kubicki et al., 2016).

3. Localizable resources in quantum information

In finite-dimensional quantum information, localizable quantumness is usually an operational resource obtained by acting locally on part of a multipartite system. One precise example is the theory of localizable measurements. A bipartite POVM $\mathcal W(K)$1 is localizable by a resource state $\mathcal W(K)$2 if Alice and Bob can implement it using only non-adaptive local operations on the target system plus their shares of $\mathcal W(K)$3, with no communication before or during the measurement and only a final classical post-processing $\mathcal W(K)$4. For rank-$\mathcal W(K)$5 PVMs on $\mathcal W(K)$6, the algebraic criterion reduces to the existence of rank-$\mathcal W(K)$7 non-redundant local POVMs and a pattern function $\mathcal W(K)$8 such that

$\mathcal W(K)$9

Theorem 3 states that a rank-$$K_1\subseteq K_2 \;\Rightarrow\; \mathcal{W}(K_1)\subseteq \mathcal{W}(K_2),$$0 PVM containing an element of maximal Schmidt rank $$K_1\subseteq K_2 \;\Rightarrow\; \mathcal{W}(K_1)\subseteq \mathcal{W}(K_2),$$1 can be localized by a resource state with Schmidt number at most $$K_1\subseteq K_2 \;\Rightarrow\; \mathcal{W}(K_1)\subseteq \mathcal{W}(K_2),$$2 iff it is a maximally entangled basis and the associated operators form a nice unitary error basis. For two qubits, Theorem 4 gives a complete classification: the localizable rank-$$K_1\subseteq K_2 \;\Rightarrow\; \mathcal{W}(K_1)\subseteq \mathcal{W}(K_2),$$3 PVMs are exactly those LU-equivalent to the computational basis, the Bell basis, or a BB84 basis. For ideal two-qudit measurements containing a maximal-Schmidt-rank element, Theorem 5 further sharpens the result to nice Bell bases (Akibue et al., 6 Jan 2026).

The teleportation literature isolates a different localizable resource. For protocols using a Bell measurement inside an arbitrary multipartite channel, the relevant quantity is localizable concurrence, defined as the maximum average concurrence that can be concentrated onto two chosen qubits by optimal local measurements on the remaining qubits. For GHZ-symmetric three-qubit states,

$$K_1\subseteq K_2 \;\Rightarrow\; \mathcal{W}(K_1)\subseteq \mathcal{W}(K_2),$$4

The paper shows that the teleportation fidelity obeys

$$K_1\subseteq K_2 \;\Rightarrow\; \mathcal{W}(K_1)\subseteq \mathcal{W}(K_2),$$5

so a protocol beats the classical threshold $$K_1\subseteq K_2 \;\Rightarrow\; \mathcal{W}(K_1)\subseteq \mathcal{W}(K_2),$$6 iff $$K_1\subseteq K_2 \;\Rightarrow\; \mathcal{W}(K_1)\subseteq \mathcal{W}(K_2),$$7 is nonzero. In the depolarized GHZ example, three-tangle, GME concurrence, and negativity can all vanish while the fidelity remains above $$K_1\subseteq K_2 \;\Rightarrow\; \mathcal{W}(K_1)\subseteq \mathcal{W}(K_2),$$8; the operational resource persists exactly until localizable concurrence vanishes (Consiglio et al., 2021).

Localizable quantum coherence extends the same logic to basis-dependent coherence stored in a designated subsystem $$K_1\subseteq K_2 \;\Rightarrow\; \mathcal{W}(K_1)\subseteq \mathcal{W}(K_2),$$9. If $K'$0, one may define coherence by tracing out the ancilla,

$K'$1

or by measuring $K'$2 in a preferred basis and averaging the coherence of the post-selected states,

$K'$3

For the $K'$4-norm coherence measure,

$K'$5

The paper shows that the measurement-assisted version can preserve much more local coherence than simple tracing, and that the resulting quantities reveal real-space structure in localized, ETH-like, MBL-like, and topological states (Hamma et al., 2020).

A parallel construction for genuine multipartite entanglement is the localizable generalized geometric measure. Given an $K'$6-party pure state $K'$7, local measurements on $K'$8 parties generate post-measurement $K'$9 party states $\mathcal{W}(K')=\mathcal{W}'(K).$0, and the LGGM is

$\mathcal{W}(K')=\mathcal{W}'(K).$1

where $\mathcal{W}(K')=\mathcal{W}'(K).$2 is the generalized geometric measure

$\mathcal{W}(K')=\mathcal{W}'(K).$3

For generalized GHZ states, $\mathcal{W}(K')=\mathcal{W}'(K).$4; for generalized $\mathcal{W}(K')=\mathcal{W}'(K).$5 states with single-qubit measurement, $\mathcal{W}(K')=\mathcal{W}'(K).$6; and for Dicke states the relation can be $\mathcal{W}(K')=\mathcal{W}'(K).$7 or $\mathcal{W}(K')=\mathcal{W}'(K).$8 depending on $\mathcal{W}(K')=\mathcal{W}'(K).$9 and the excitation number. The paper also shows that measuring multiple parties can increase the localizable multipartite entanglement in some families (Sadhukhan et al., 2015).

4. Order parameters, projected ensembles, and subspace weights

A particularly sharp use of localizable quantumness appears in measurement-induced phase transitions. In random Clifford brickwall circuits, localizable entanglement between qubits $W$0 and $W$1 is defined as

$W$2

where $W$3 is the ensemble generated by local measurements on all other qubits. The proposed order parameter is

$W$4

In the area-law phase, $W$5; in the volume-law phase, $W$6. The spatially averaged correlation function

$W$7

decays as

$W$8

for $W$9, saturates to a constant for $$S_W=J_W\Delta_W^{1/2}, \qquad \mathrm{Re}\,\mathcal{H}(W)=\{\zeta_W\in\mathcal{H}\;|\;S_W\zeta_W=\zeta_W\},$$0, and yields an entanglement length diverging as

$$S_W=J_W\Delta_W^{1/2}, \qquad \mathrm{Re}\,\mathcal{H}(W)=\{\zeta_W\in\mathcal{H}\;|\;S_W\zeta_W=\zeta_W\},$$1

with the crossing located near $$S_W=J_W\Delta_W^{1/2}, \qquad \mathrm{Re}\,\mathcal{H}(W)=\{\zeta_W\in\mathcal{H}\;|\;S_W\zeta_W=\zeta_W\},$$2. The paper interprets this localizable entanglement both as a teleportation resource and as a quantum analogue of percolation connectivity, and proposes a two-ancilla protocol in which a final CNOT yields $$S_W=J_W\Delta_W^{1/2}, \qquad \mathrm{Re}\,\mathcal{H}(W)=\{\zeta_W\in\mathcal{H}\;|\;S_W\zeta_W=\zeta_W\},$$3 in the area-law phase after averaging over input stabilizer states (Manna et al., 20 Jan 2026).

A broader projected-ensemble framework turns localizable quantumness into a certification primitive. Fix a bipartition into a small accessible subsystem $$S_W=J_W\Delta_W^{1/2}, \qquad \mathrm{Re}\,\mathcal{H}(W)=\{\zeta_W\in\mathcal{H}\;|\;S_W\zeta_W=\zeta_W\},$$4 and its complement $$S_W=J_W\Delta_W^{1/2}, \qquad \mathrm{Re}\,\mathcal{H}(W)=\{\zeta_W\in\mathcal{H}\;|\;S_W\zeta_W=\zeta_W\},$$5. Measuring $$S_W=J_W\Delta_W^{1/2}, \qquad \mathrm{Re}\,\mathcal{H}(W)=\{\zeta_W\in\mathcal{H}\;|\;S_W\zeta_W=\zeta_W\},$$6 in a local basis produces the ensemble

$$S_W=J_W\Delta_W^{1/2}, \qquad \mathrm{Re}\,\mathcal{H}(W)=\{\zeta_W\in\mathcal{H}\;|\;S_W\zeta_W=\zeta_W\},$$7

and for an experimental state $$S_W=J_W\Delta_W^{1/2}, \qquad \mathrm{Re}\,\mathcal{H}(W)=\{\zeta_W\in\mathcal{H}\;|\;S_W\zeta_W=\zeta_W\},$$8,

$$S_W=J_W\Delta_W^{1/2}, \qquad \mathrm{Re}\,\mathcal{H}(W)=\{\zeta_W\in\mathcal{H}\;|\;S_W\zeta_W=\zeta_W\},$$9

For a property $\varphi(f)=\int d^4x\, f(x)\varphi(x),$00 with free set $\varphi(f)=\int d^4x\, f(x)\varphi(x),$01, the localizable-quantumness metric is

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$02

and the conditional-fidelity witness is

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$03

Free states satisfy $\varphi(f)=\int d^4x\, f(x)\varphi(x),$04, while $\varphi(f)=\int d^4x\, f(x)\varphi(x),$05. The sample complexity is

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$06

so if $\varphi(f)=\int d^4x\, f(x)\varphi(x),$07 and $\varphi(f)=\int d^4x\, f(x)\varphi(x),$08, then $\varphi(f)=\int d^4x\, f(x)\varphi(x),$09. The paper applies the framework to entanglement, circuit complexity, measurement-assisted circuit complexity, quantum magic, and fidelity certification, with soundness for mixed states and a random-basis variant based on

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$10

and spectral gap $\varphi(f)=\int d^4x\, f(x)\varphi(x),$11 (Du et al., 22 Sep 2025).

A distinct but formally precise notion is localization of a quantum state within a subspace $\varphi(f)=\int d^4x\, f(x)\varphi(x),$12. For a positive operator $\varphi(f)=\int d^4x\, f(x)\varphi(x),$13, one has the unique decomposition

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$14

with

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$15

For a density operator $\varphi(f)=\int d^4x\, f(x)\varphi(x),$16,

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$17

and the localization probability is

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$18

This $\varphi(f)=\int d^4x\, f(x)\varphi(x),$19 is not the ordinary overlap $\varphi(f)=\int d^4x\, f(x)\varphi(x),$20 with the projector onto $\varphi(f)=\int d^4x\, f(x)\varphi(x),$21; rather, it is the largest fraction of the state that can be written as a state fully supported in $\varphi(f)=\int d^4x\, f(x)\varphi(x),$22, with $\varphi(f)=\int d^4x\, f(x)\varphi(x),$23, equality only when $\varphi(f)=\int d^4x\, f(x)\varphi(x),$24. In the rank-one case $\varphi(f)=\int d^4x\, f(x)\varphi(x),$25,

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$26

The map $\varphi(f)=\int d^4x\, f(x)\varphi(x),$27 is positively homogeneous, super-additive, and concave, and the corresponding $\varphi(f)=\int d^4x\, f(x)\varphi(x),$28 is concave in $\varphi(f)=\int d^4x\, f(x)\varphi(x),$29 (Salcedo, 14 Jan 2026).

5. Localization as an operational resource

Several works treat localization itself as the quantum resource. In the one-dimensional fermionic Aubry–André–Harper lattice,

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$30

with localization transition at $\varphi(f)=\int d^4x\, f(x)\varphi(x),$31, the parameter $\varphi(f)=\int d^4x\, f(x)\varphi(x),$32 can be sensed by exploiting the fragility of states near the delocalization-localization transition. The quantum Fisher information satisfies $\varphi(f)=\int d^4x\, f(x)\varphi(x),$33, and in the adiabatic scheme the peak scales as

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$34

for the single-particle and half-filled noninteracting cases. For the directly measurable charge-density-wave operator

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$35

the observable Fisher information scales as

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$36

In the interacting half-filled case, the natural observable

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$37

gives

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$38

while the full QFI remains close to Heisenberg scaling,

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$39

In dynamical sensing, short-time transients exhibit

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$40

for suitable initial states (Sahoo et al., 2023).

Localization of correlations in QFT ground states can also be converted into a circuit principle. For a lattice scalar field theory with Gaussian ground state covariance matrix $\varphi(f)=\int d^4x\, f(x)\varphi(x),$41, the classical two-point function decays exponentially,

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$42

The state-preparation circuit can be reorganized from exact $\varphi(f)=\int d^4x\, f(x)\varphi(x),$43-rotations into systematically localizable $\varphi(f)=\int d^4x\, f(x)\varphi(x),$44-rotations ordered by control distance $\varphi(f)=\int d^4x\, f(x)\varphi(x),$45, with

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$46

These angles decay with separation like the two-point function rather than the more localized mutual information or the hyper-localized negativity,

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$47

and truncating the circuit beyond a control distance $\varphi(f)=\int d^4x\, f(x)\varphi(x),$48 yields exponentially convergent fidelity,

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$49

The result is a digital state-preparation scheme aligned with the physical correlation length of the QFT ground state (Klco et al., 2019).

In open disordered systems, localization can be selected and stabilized dissipatively rather than destroyed. For the Anderson Hamiltonian

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$50

with Lindblad evolution

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$51

the local pairwise dissipators

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$52

act as phase-selective mode filters. For $\varphi(f)=\int d^4x\, f(x)\varphi(x),$53, $\varphi(f)=\int d^4x\, f(x)\varphi(x),$54 favors lower-band-edge Anderson modes, $\varphi(f)=\int d^4x\, f(x)\varphi(x),$55 favors upper-band-edge modes, and $\varphi(f)=\int d^4x\, f(x)\varphi(x),$56 makes the dissipators Hermitian and yields the completely mixed asymptotic state $\varphi(f)=\int d^4x\, f(x)\varphi(x),$57. In the quantum-jump picture, the localized steady regime shows intermittent dynamics consisting of long trapping events near selected modes interrupted by jumps between them (Yusipov et al., 2016).

Other engineered settings use wave-function localization directly. A two-dimensional alternate quantum walk with step-dependent phase gates,

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$58

displays a strong localization-like effect when $\varphi(f)=\int d^4x\, f(x)\varphi(x),$59 and $\varphi(f)=\int d^4x\, f(x)\varphi(x),$60, with particularly strong accumulation at the origin around $\varphi(f)=\int d^4x\, f(x)\varphi(x),$61; the coherence norm

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$62

tracks the distinction between strong localization, weak localization, and no localization (Franco et al., 2013). In high-dimensional optical communication, the localization/delocalization contrast of twin-photon wave functions in the correct and wrong basis underlies a data-basis-shuffling protocol, where the wrong-basis disagreement probability approaches $\varphi(f)=\int d^4x\, f(x)\varphi(x),$63 and the error ratio scales as

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$64

The protocol uses localization as a basis-consistency certificate encoded in the same photons that carry the data (Santagati et al., 2019).

6. Localization-protected order, distinctions, and limits

In many-body localized systems, localization can protect order rather than merely freeze transport. For disordered Ising, Majorana, and Dirac chains, the claim is that individual highly excited many-body eigenstates can break the global $\varphi(f)=\int d^4x\, f(x)\varphi(x),$65 symmetry or support topological order because localized defects cannot propagate and melt the order. In the ordered MBL phase, each eigenstate comes in a nearly degenerate “feline” pair with splitting

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$66

and the system can pass between a localized ordered phase and a localized disordered phase through a non-thermodynamic eigenstate transition, possibly governed by a localized infinite-randomness critical point with

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$67

The same work identifies a spectral transition inside the ordered MBL phase, between paired Poisson and ordinary Poisson statistics, diagnosed by the consecutive-gap ratio

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$68

Localization is therefore presented as a mechanism that protects symmetry breaking and topological order at energy densities where equilibrium thermodynamics would predict disorder (Huse et al., 2013).

Pretko and Nandkishore generalize this logic from point particles to extended objects. A string or membrane moves only by propagating lower-dimensional internal disturbances along its worldsheet or worldvolume. If those internal modes many-body localize, the entire extended object localizes. For a string described by $\varphi(f)=\int d^4x\, f(x)\varphi(x),$69,

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$70

localization of the internal modes yields a convergent “string locator expansion,” and the long-time diagnostic becomes an out-of-time-order string correlator built from string $\varphi(f)=\int d^4x\, f(x)\varphi(x),$71-bits. The same hierarchical construction extends to membranes, domain walls, loop excitations, and flux lines in type-II superconductors (Pretko et al., 2017).

Several recurrent distinctions prevent the phrase from collapsing into a single universal resource theory. First, localizable does not mean tempered in QFT: strictly localizable fields may grow like $\varphi(f)=\int d^4x\, f(x)\varphi(x),$72 with $\varphi(f)=\int d^4x\, f(x)\varphi(x),$73, whereas $\varphi(f)=\int d^4x\, f(x)\varphi(x),$74 marks the loss of spacetime localizability (Khoze et al., 2018). Second, in relativistic QFT the localized entities are observables or local algebras, not sharply localized particle wave packets in the nonrelativistic sense (Balachandran, 2016). Third, in distributed measurement theory, localizable measurements implemented by non-adaptive local operations plus shared entanglement are strictly weaker than adaptive LOCC; maximal Schmidt rank is not enough, and the underlying measurement algebra must be that of a nice unitary error basis (Akibue et al., 6 Jan 2026). Fourth, nonzero discord need not signal nonlocal quantumness: it can also arise from local superposition. In the two-qubit framework based on $\varphi(f)=\int d^4x\, f(x)\varphi(x),$75 and directional discords $\varphi(f)=\int d^4x\, f(x)\varphi(x),$76, a state has only local quantumness iff

$\varphi(f)=\int d^4x\, f(x)\varphi(x),$77

so discord conflates local and nonlocal sources of nonclassicality (Agrawal et al., 2015).

These distinctions suggest that “localizable quantumness” is best treated as a structured family of notions unified by a common operational motif: quantum properties that appear global in one description can, under the appropriate localization framework, be concentrated in local observables, localized subsystems, projected ensembles, or disorder-pinned excitations. The precise meaning depends on whether the relevant locality is that of compactly supported test functions, local operator algebras, non-adaptive distributed measurements, measurement-conditioned entanglement, projected-state witnesses, or spatially localized many-body dynamics.