Activation of Nonlocality in Quantum Systems

• Activation of nonlocality is a collection of quantum phenomena where states that are local under standard tests become nonlocal when additional operations or configurations, such as tensoring or local filtering, are applied.
• It encompasses diverse mechanisms including tensor activation, hidden nonlocality through filtering, entanglement swapping, broadcasting, and catalytic approaches, each altering the observed correlations.
• These activation methods demonstrate that locality is scenario-dependent, as composition, pre-processing, and channel operations can reveal nonlocal properties in systems previously deemed Bell-local.

Searching arXiv for the core topic and the cited papers to ground the article in current literature. arXiv_search query: "activation of nonlocality" Activation of nonlocality denotes a family of phenomena in which a quantum object that is local, Bell-local, CHSH-local, LHS-admitting, CHSH-breaking, or locally distinguishable in an initial scenario becomes nonlocal after additional structure is supplied. In the Bell setting, the additional structure can be as simple as tensoring, local filtering, entanglement swapping, broadcasting into a network, or the use of a catalyst; in other operational settings, it can consist of orthogonality-preserving local measurements that convert locally accessible information into locally hidden information (Navascues et al., 2010). Across these variants, the common theme is that nonlocality is not a monotone of a single copy tested in isolation, but can emerge from composition, conditioning, or pre-processing (Palazuelos, 2012).

1. Conceptual scope and operational forms

In the Bell-inequality literature, a bipartite state is Bell-local iff every correlation generated by local measurements admits a local hidden-variable model, equivalently iff every Bell functional stays below its local bound (Bavaresco et al., 2 Apr 2025). Activation asks whether a resource that is local in one operational context can become nonlocal in another. The surveyed literature exhibits several distinct mechanisms: tensor activation without post-selection, hidden nonlocality via local filtering, activation by entanglement swapping or broadcasting in networks, activation of CHSH nonlocality for channels, and deterministic catalytic activation with the catalyst returned exactly in its initial state (Zhang et al., 2019).

A useful organizing distinction is between activation that changes the tested object and activation that changes only the measurement scenario. Tensoring and catalysis change the resource itself; local filtering and entanglement swapping change the effective state seen by the Bell test; broadcasting changes the causal structure of the experiment; and state-discrimination-based activation changes the operational task from perfect LOCC discrimination of an orthogonal set to a post-measurement branch in which only global measurements suffice (Bandyopadhyay et al., 2021).

Mechanism	Initial object	Activated signature
Tensoring	CHSH-local states	CHSH violation $\ge 2.023$ (Navascues et al., 2010)
Local filtering	Bell-local mixed entangled states	CHSH, CGLMP, or PORAC Bell violation (Ducuara et al., 2016)
Entanglement swapping	Links not violating CHSH individually	CHSH violation after a critical number of swappings (Kłobus et al., 2012)
Broadcasting/network embedding	Single-copy Bell-local bipartite state	Broadcast nonlocality or multipartite Bell violation (Bowles et al., 2020)
Channel activation	CHSH-breaking channels	Combined channel not CHSH-breaking (Zhang et al., 2019)
Catalytic activation	Bell-local state plus catalyst	Bell-nonlocal output with unchanged catalyst (Bavaresco et al., 2 Apr 2025)
LOCC state-discrimination activation	Locally distinguishable orthogonal set	Locally indistinguishable or incompletable post-measurement set (Das et al., 1 Jul 2026)

This suggests that “activation of nonlocality” is best understood as a structural property of resource composition rather than a single protocol. A plausible implication is that locality statements are highly scenario-dependent: they may be stable under some free operations and unstable under others.

2. Tensor activation and super-activation of Bell nonlocality

A central two-qubit result is the construction of two states $\rho_1,\rho_2$ such that any number of copies of one state or the other cannot violate the CHSH Bell inequality, whereas their tensor product does (Navascues et al., 2010). The mechanism is 2-extendibility: $\rho_1$ admits a 2-symmetric extension to Bob and $\rho_2$ to Alice, and the Terhal–Doherty–Schwab lemma implies that a state 2-extendible with respect to one party cannot violate any Bell inequality with only two settings on that party. The nontrivial point is that $S_B^2 \otimes S_A^2$ is not contained in $S_{AB}^2$, so locality of each factor does not force locality of the tensor product.

For the tensor state $$\rho_{\mathrm{tot}}=\rho_1^{AB}\otimes\rho_2^{A'B'}$$, identifying $A\otimes A'$ as Alice and $B\otimes B'$ as Bob, the paper chooses $\alpha=0$, $\rho_1,\rho_2$0, $\rho_1,\rho_2$1, $\rho_1,\rho_2$2 with $\rho_1,\rho_2$3. This yields

$\rho_1,\rho_2$4

With tensor-product correlators $\rho_1,\rho_2$5 and $\rho_1,\rho_2$6, one obtains

$\rho_1,\rho_2$7

which is a CHSH violation without post-selection (Navascues et al., 2010).

The same work also gives self-activation. Defining

$\rho_1,\rho_2$8

the state $\rho_1,\rho_2$9 is CHSH-local, yet two copies satisfy

$\rho_1$0

in $\rho_1$1 (Navascues et al., 2010).

A more asymptotic form is super-activation. For the isotropic state

$\rho_1$2

one can choose $\rho_1$3 such that $\rho_1$4 is local, yet $\rho_1$5 violates a Bell inequality for some fixed $\rho_1$6, even $\rho_1$7 when $\rho_1$8 is large enough (Palazuelos, 2012). The proof uses the Khot–Visnoi game, with $\rho_1$9 and $\rho_2$0, together with the fact that the extra tensor-product terms contribute nonnegatively because the game is nonnegative. This places activation in direct contact with large-violation constructions rather than only CHSH.

3. Hidden nonlocality and local filtering

Hidden nonlocality refers to Bell nonlocality that is revealed only after local filtering (Ducuara et al., 2016). For two qubits, the review paper gives a closed-form criterion: if $\rho_2$1 with $\rho_2$2 and $\rho_2$3 its three largest eigenvalues, then $\rho_2$4 has hidden nonlocality iff $\rho_2$5. The same work also formulates activation by tensoring plus local filtering as an SDP,

$\rho_2$6

subject to $\rho_2$7, $\rho_2$8, $\rho_2$9, where a negative objective certifies CHSH activation for $S_B^2 \otimes S_A^2$0 (Ducuara et al., 2016).

Local filtering can reveal nonlocality far beyond CHSH. For the family

$S_B^2 \otimes S_A^2$1

with filters

$S_B^2 \otimes S_A^2$2

the CGLMP-based construction shows that when $S_B^2 \otimes S_A^2$3 is maximally entangled, the range of the mixing parameter for revealing hidden nonlocality increases with increasing dimension, and for $S_B^2 \otimes S_A^2$4 hidden non-locality can be revealed for the whole range of mixing parameter (Kumari, 2023). The same paper also treats the maximally CGLMP-violating state $S_B^2 \otimes S_A^2$5 and finds that the same filtering operation activates nonlocality there as well.

A different high-setting route uses the parity-oblivious random access code Bell functional $S_B^2 \otimes S_A^2$6. For

$S_B^2 \otimes S_A^2$7

with filters $S_B^2 \otimes S_A^2$8 and $S_B^2 \otimes S_A^2$9, the filtered Bell value approaches the pure-state optimum as $S_{AB}^2$0. The main theorem states that for every fixed $S_{AB}^2$1 and every $S_{AB}^2$2, there exists $S_{AB}^2$3 such that the post-filtered state violates the PORAC Bell inequality; preparation contextuality can be activated for any $S_{AB}^2$4 when $S_{AB}^2$5 (Kumari et al., 25 Apr 2025).

Activation by filtering also reaches bound entanglement. A fully-biseparable three-qubit bound-entangled state $S_{AB}^2$6 admits a local model for all non-sequential POVMs, yet invertible local filters map it to $S_{AB}^2$7, which violates the symmetrized Sliwa–5 inequality with $S_{AB}^2$8 (Tendick et al., 2019). This shows that genuine hidden nonlocality does not imply entanglement distillability.

4. Network, swapping, and broadcasting activation

Entanglement swapping chains realize activation by sequential Bell measurements. In the chain construction, the links are

$S_{AB}^2$9

with left links $$\rho_{\mathrm{tot}}=\rho_1^{AB}\otimes\rho_2^{A'B'}$$0, right links $$\rho_{\mathrm{tot}}=\rho_1^{AB}\otimes\rho_2^{A'B'}$$1, and center link $$\rho_{\mathrm{tot}}=\rho_1^{AB}\otimes\rho_2^{A'B'}$$2 (Kłobus et al., 2012). If $$\rho_{\mathrm{tot}}=\rho_1^{AB}\otimes\rho_2^{A'B'}$$3 central Bell measurements all return $$\rho_{\mathrm{tot}}=\rho_1^{AB}\otimes\rho_2^{A'B'}$$4, the end parties share

$$\rho_{\mathrm{tot}}=\rho_1^{AB}\otimes\rho_2^{A'B'}$$5

with a closed-form $$\rho_{\mathrm{tot}}=\rho_1^{AB}\otimes\rho_2^{A'B'}$$6 that increases toward $$\rho_{\mathrm{tot}}=\rho_1^{AB}\otimes\rho_2^{A'B'}$$7. The maximal CHSH correlator is

$$\rho_{\mathrm{tot}}=\rho_1^{AB}\otimes\rho_2^{A'B'}$$8

and activation occurs only after a critical swapping number $$\rho_{\mathrm{tot}}=\rho_1^{AB}\otimes\rho_2^{A'B'}$$9, the smallest integer with $A\otimes A'$0 (Kłobus et al., 2012).

A broader network result proves generic activated nonlocality for all bipartite entangled states in entanglement-swapping networks and then extends the conclusion to all connected quantum networks and nontrivial hybrid networks (Luo, 2018). In the language of semiquantum nonlocal games, any connected network of $A\otimes A'$1 parties sharing arbitrary entangled states is multipartite nonlocal and $A\otimes A'$2-partite activated nonlocal for every $A\otimes A'$3. This shifts activation from specially tuned examples to a structural property of connected entangled networks.

Broadcasting provides a single-copy activation scenario. For the isotropic state

$A\otimes A'$4

a $A\otimes A'$5 broadcasting channel on Bob’s subsystem yields a tripartite state tested against a broadcast-local model

$A\otimes A'$6

The resulting bounds show that the state does not admit a local hidden variable description for $A\otimes A'$7 in the no-signalling-hidden-variable case and $A\otimes A'$8 in the fully quantum hidden-variable case, both significantly below the previously best-known bound of $A\otimes A'$9 for single-copy Bell nonlocality and device-independent entanglement certification of the state (Bowles et al., 2020).

This activation has been realized experimentally in a photonic quantum network. Starting from the Werner-like state

$B\otimes B'$0

which admits a local-hidden-variable model for dichotomic projective measurements whenever $B\otimes B'$1, a broadcast isometry produces a three-party state tested with the broadcast inequality

$B\otimes B'$2

The ideal quantum maximum is $B\otimes B'$3, and the activation threshold is $B\otimes B'$4, shifted in practice to $B\otimes B'$5. For three states with $B\otimes B'$6, the experiment obtained $B\otimes B'$7 by more than $B\otimes B'$8, while a semidefinite-programming certificate with $B\otimes B'$9 confirmed that the original state was Bell-local (Villegas-Aguilar et al., 2023).

5. Channels, catalysts, and dynamical resource theory

Activation also exists at the level of quantum channels. A channel $\alpha=0$0 is CHSH-breaking if for every bipartite input state $\alpha=0$1, the output $\alpha=0$2 cannot violate CHSH under any dichotomic measurements (Zhang et al., 2019). For qubit channels diagonalized on the Bloch ball, a unital channel is CHSH-breaking precisely when $\alpha=0$3. Concrete thresholds include amplitude damping, CHSH-breaking iff $\alpha=0$4; loss channel, CHSH-breaking iff $\alpha=0$5; and erasure channel, CHSH-breaking iff $\alpha=0$6 (Zhang et al., 2019).

Despite this, pairs of CHSH-breaking channels can cease to be CHSH-breaking when used in parallel. In the bidirectional scenario, two identical amplitude-damping channels $\alpha=0$7, each CHSH-breaking on its own, yield $\alpha=0$8 under suitable input states and joint observables. In the unidirectional scenario, combining $\alpha=0$9 with the depolarizing channel $\rho_1,\rho_2$00, again each CHSH-breaking individually, yields $\rho_1,\rho_2$01 (Zhang et al., 2019). This is the channel analogue of tensor activation for states.

Catalytic activation gives a deterministic state transformation. If a Bell-local state $\rho_1,\rho_2$02 has the property that $\rho_1,\rho_2$03 violates some Bell inequality, then there exists a catalyst $\rho_1,\rho_2$04 and local CPTP maps $\rho_1,\rho_2$05 such that

$\rho_1,\rho_2$06

with $\rho_1,\rho_2$07 and $\rho_1,\rho_2$08 Bell-nonlocal (Bavaresco et al., 2 Apr 2025). The output has the form

$\rho_1,\rho_2$09

and if $\rho_1,\rho_2$10, then $\rho_1,\rho_2$11. Importantly, the catalyst can itself be Bell local (Bavaresco et al., 2 Apr 2025).

A different generalization is resource-theoretic. In the quantum-process framework, Bell nonlocality is identified as the subset of entangled processes with instantaneous input-output delay time, and LOCC pre-processing is the natural class of free operations. Within this framework, all entangled states can activate some form of Bell nonlocality, and CHSH witnesses can be generalized from states to bipartite channels (Sengupta et al., 2020). This suggests that activation is not only a property of states and networks, but also a manifestation of the relation between entanglement and causally constrained process transformations.

6. Beyond Bell nonlocality: steering and local state discrimination

The activation paradigm extends to steering nonlocality. In the three-setting CJWR test, a two-qubit state with diagonalized correlation tensor $\rho_1,\rho_2$12 is $\rho_1,\rho_2$13-steerable iff

$\rho_1,\rho_2$14

In a linear entanglement-swapping network $\rho_1,\rho_2$15, Mukherjee et al. show that $\rho_1,\rho_2$16 and $\rho_1,\rho_2$17 may each satisfy $\rho_1,\rho_2$18, yet a conditional state $\rho_1,\rho_2$19 after Bob’s Bell-state measurement can satisfy $\rho_1,\rho_2$20 (Mukherjee et al., 2023). For the families

$\rho_1,\rho_2$21

with $\rho_1,\rho_2$22, activation occurs numerically, for example, at $\rho_1,\rho_2$23 and $\rho_1,\rho_2$24, while all conditional states remain Bell-local in the sense that both CHSH and $\rho_1,\rho_2$25 are satisfied (Mukherjee et al., 2023). The same work also identifies no-activation conditions requiring nonzero Bloch vectors on both sites, and gives genuine activation examples starting from states satisfying the Bowles–Hirsch–Quintino–Brunner unsteerability criterion.

Another branch of the literature studies nonlocality as failure of LOCC state discrimination. There exist orthogonal sets that are locally distinguishable but without local redundancy and that can be locally converted, with certainty, into locally indistinguishable sets (Bandyopadhyay et al., 2021). In the $\rho_1,\rho_2$26 example, Bob’s projectors $\rho_1,\rho_2$27 and $\rho_1,\rho_2$28 send a locally distinguishable triplet to a set of three orthogonal entangled states in $\rho_1,\rho_2$29, which are locally indistinguishable by the Walgate–Hardy criterion (Bandyopadhyay et al., 2021).

Subsequent work strengthens the activated property. One can activate local unmarkability and local irreducibility, including deterministic conversion of a redundancy-free $\rho_1,\rho_2$30-party set into the complete GHZ basis, which is locally irreducible and $\rho_1,\rho_2$31 indistinguishable (Gupta et al., 2022). In the elimination paradigm, locally preparable product-state sets can be activated into post-measurement sets that are locally irreducible even under any bipartition; the $\rho_1,\rho_2$32 and $\rho_1,\rho_2$33 constructions are explicit instances (Ghosh et al., 2022). The notion of “strong local” sets further refines this hierarchy by distinguishing sets that can never be activated by any orthogonality-preserving local measurement from sets that require one-party or two-party joint activation (Bera et al., 2024).

The recent incompletability framework sharpens the hierarchy once more. A set is incompletability-activable if it is initially LOCC-distinguishable and some orthogonality-preserving LOCC branch produces a strictly incompletable set; any such set is necessarily nonlocality-activable, but the converse fails (Das et al., 1 Jul 2026). The paper gives an explicit $\rho_1,\rho_2$34 example $\rho_1,\rho_2$35 whose four branches yield the standard 5-state Tiles UPB, and another set $\rho_1,\rho_2$36 that is nonlocality-activable but not incompletability-activable (Das et al., 1 Jul 2026). This suggests a strict hierarchy among activated phenomena in state discrimination: incompletability activation is stronger than activation of local indistinguishability, and activation of nonlocality in this sense is not exhausted by Bell-type correlation tests.