Genuine Activation of Hidden Nonlocality

• The paper demonstrates that quantum states, initially local under standard measurements, reveal nonlocality after applying local filtering or orthogonality-preserving operations.
• It shows that genuine hidden nonlocality occurs when states remain local for all POVMs, requiring sequential measurements or ancillary assistance to violate Bell or Svetlichny inequalities.
• The studies extend the phenomenon to multipartite systems and state discrimination scenarios, outlining practical conditions and resource-theoretic implications in high-dimensional settings.

Genuine activation of hidden nonlocality denotes a family of phenomena in which a quantum object that is initially operationally local becomes nonlocal only after an additional local pre-processing stage. In the Bell-inequality literature, the standard mechanism is local filtering followed by a Bell test; in the local-discrimination literature, the mechanism is an orthogonality-preserving local measurement that converts a locally distinguishable set into a locally indistinguishable one. The qualifier “genuine” does not have a single universal meaning across these settings. In bipartite Bell scenarios it typically means that the initial state is local even for all POVMs, so its nonlocality is revealed only by a sequence of measurements rather than by any non-sequential test (Hirsch et al., 2013, Guerini, 2014). In multipartite Bell scenarios it can mean that the activated correlations are genuinely multipartite nonlocal, e.g. by violating Svetlichny-type inequalities after filtering (Bowles et al., 2015, Sun et al., 2021). In the state-discrimination setting it refers to activation that is not attributable to local redundancy, so that locally available information is converted into locally hidden information in a structurally nontrivial way (Bandyopadhyay et al., 2021, Li et al., 2021).

1. Conceptual scope and basic definitions

Hidden nonlocality, in the Bell sense, is the phenomenon that a state can be Bell-local in a standard Bell test but Bell-nonlocal after local preprocessing. The relevant local hidden-variable factorization is

$$p(a,b|x,y)=\int_{\Lambda}\pi(\lambda)\,p_A(a|x,\lambda)\,p_B(b|y,\lambda)\,d\lambda,$$

and the canonical Bell test is often CHSH (Guerini, 2014). In this framework, local filters are stochastic local operations applied before the Bell test, producing

$$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$

or, in multipartite form,

$$\rho = \dfrac{F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} } {\mathrm{Tr}(F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} )}.$$

If the post-filtered state violates a Bell inequality, then the original state has hidden nonlocality (Camalet, 2017, Tendick et al., 2019).

A stronger notion, commonly called genuine hidden nonlocality, arises when the original state is local not merely for projective measurements but for all POVMs. In that case, even the most general single-shot measurement scenario cannot reveal nonlocality directly; only a sequence of local operations can do so (Hirsch et al., 2013, Guerini, 2014). In the thesis literature this distinction is stated explicitly as the difference between ordinary hidden nonlocality, where the state is local for projective measurements, and genuine hidden nonlocality, where it is local even for POVMs (Guerini, 2014).

A separate usage appears in state discrimination. There, a set of orthogonal states is called local if it is perfectly distinguishable by LOCC, and nonlocal if perfect discrimination requires a global measurement. Hidden nonlocality is activated when a locally distinguishable set is transformed by an orthogonality-preserving local measurement into sets that are locally indistinguishable. In this setting, “genuine” means that the effect cannot be reduced to discarding redundant subsystems; the initial set is locally irredundant (Bandyopadhyay et al., 2021, Li et al., 2021).

2. Bipartite Bell hidden nonlocality and the POVM-local criterion

The bipartite Bell formulation is centered on CHSH. In one convention used in the literature,

$$E_{00}+E_{01}+E_{10}-E_{11}\le 2, \qquad E_{xy}=P(a=b|x,y)-P(a\neq b|x,y),$$

while another writes

$$\langle \mathcal{B}_{\mathrm{CHSH}} \rangle=\langle QS\rangle+\langle RS\rangle+\langle RT\rangle-\langle QT\rangle \le 2.$$

For two-qubit states, the Horodecki criterion is used repeatedly: $$\max \langle \mathcal{B}_{\mathrm{CHSH}} \rangle_\rho = 2\sqrt{M(\rho)}.$$ This criterion underlies explicit filtering-based demonstrations of hidden nonlocality (Guerini, 2014).

A central construction for genuine hidden nonlocality is the lifting from a state $\rho_0$ that is local for dichotomic projective measurements to a state

$$\rho'=\frac{1}{d^2}\Big[ \rho_0+(d-1)(\rho_A\otimes \sigma_B+\sigma_A\otimes \rho_B) +(d-1)^2\sigma_A\otimes \sigma_B \Big],$$

where $\rho_A=\mathrm{tr}_B(\rho_0)$ and $\rho_B=\mathrm{tr}_A(\rho_0)$. If $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$0 is local for dichotomic projective measurements, then $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$1 is local for all POVMs (Hirsch et al., 2013, Guerini, 2014). The simulation protocol decomposes POVM elements as $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$2, chooses $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$3 with probability $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$4, simulates a dichotomic projective measurement $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$5, and uses $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$6 only when the first branch fails. This provides an explicit operational meaning of POVM-locality (Guerini, 2014).

The standard two-qubit example begins with

$$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$7

which is local for dichotomic projective measurements for $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$8. Using $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$9, one obtains

$$\rho = \dfrac{F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} } {\mathrm{Tr}(F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} )}.$$0

which is local for POVMs (Guerini, 2014). The local filters are

$$\rho = \dfrac{F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} } {\mathrm{Tr}(F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} )}.$$1

After filtering,

$$\rho = \dfrac{F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} } {\mathrm{Tr}(F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} )}.$$2

and the CHSH value reaches

$$\rho = \dfrac{F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} } {\mathrm{Tr}(F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} )}.$$3

in the $$\rho = \dfrac{F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} } {\mathrm{Tr}(F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} )}.$$4 limit, which is strictly greater than $$\rho = \dfrac{F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} } {\mathrm{Tr}(F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} )}.$$5 whenever $$\rho = \dfrac{F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} } {\mathrm{Tr}(F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} )}.$$6. This is the prototypical demonstration that a POVM-local entangled state can nevertheless violate CHSH after local filtering (Guerini, 2014).

The earlier paper on “genuine hidden quantum nonlocality” establishes the same logic directly and emphasizes that there exist entangled states “the nonlocality of which can be revealed only by using a sequence of measurements” (Hirsch et al., 2013). It also presents maximal genuine hidden nonlocality through erasure-type constructions: an initial state local for all POVMs can be filtered exactly to the singlet $$\rho = \dfrac{F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} } {\mathrm{Tr}(F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} )}.$$7, yielding the Tsirelson value $$\rho = \dfrac{F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} } {\mathrm{Tr}(F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} )}.$$8 (Hirsch et al., 2013).

3. Activation theorems and the equivalence-with-entanglement viewpoint

The strongest general activation theorem in the supplied literature states that every entangled finite-dimensional multipartite state can be made to violate CHSH when assisted by an additional ancillary state that by itself does not violate CHSH, after suitable local filtering (Liang et al., 2012). More precisely, for an $$\rho = \dfrac{F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} } {\mathrm{Tr}(F_A \otimes F_B \otimes F_C \ \rho_L \ F_A^{\dagger} \otimes F_B^{\dagger} \otimes F_C^{\dagger} )}.$$9-partite state $$E_{00}+E_{01}+E_{10}-E_{11}\le 2, \qquad E_{xy}=P(a=b|x,y)-P(a\neq b|x,y),$$0,

$$E_{00}+E_{01}+E_{10}-E_{11}\le 2, \qquad E_{xy}=P(a=b|x,y)-P(a\neq b|x,y),$$1

where $$E_{00}+E_{01}+E_{10}-E_{11}\le 2, \qquad E_{xy}=P(a=b|x,y)-P(a\neq b|x,y),$$2 is the set of CHSH-preprocessed-1-local states (Liang et al., 2012). Activation here means

$$E_{00}+E_{01}+E_{10}-E_{11}\le 2, \qquad E_{xy}=P(a=b|x,y)-P(a\neq b|x,y),$$3

The Bell test is a generalized CHSH experiment applied only to parties 1 and 2 after all parties may apply local filters. A separable SLO map has Kraus form

$$E_{00}+E_{01}+E_{10}-E_{11}\le 2, \qquad E_{xy}=P(a=b|x,y)-P(a\neq b|x,y),$$4

The set $$E_{00}+E_{01}+E_{10}-E_{11}\le 2, \qquad E_{xy}=P(a=b|x,y)-P(a\neq b|x,y),$$5 admits the characterization

$$E_{00}+E_{01}+E_{10}-E_{11}\le 2, \qquad E_{xy}=P(a=b|x,y)-P(a\neq b|x,y),$$6

for all $$E_{00}+E_{01}+E_{10}-E_{11}\le 2, \qquad E_{xy}=P(a=b|x,y)-P(a\neq b|x,y),$$7 and all $$E_{00}+E_{01}+E_{10}-E_{11}\le 2, \qquad E_{xy}=P(a=b|x,y)-P(a\neq b|x,y),$$8, with

$$E_{00}+E_{01}+E_{10}-E_{11}\le 2, \qquad E_{xy}=P(a=b|x,y)-P(a\neq b|x,y),$$9

The proof uses a dual-cone/Farkas lemma argument. A particular filter projects each local factor onto a maximally entangled state between $$\langle \mathcal{B}_{\mathrm{CHSH}} \rangle=\langle QS\rangle+\langle RS\rangle+\langle RT\rangle-\langle QT\rangle \le 2.$$0 and an ancillary copy $$\langle \mathcal{B}_{\mathrm{CHSH}} \rangle=\langle QS\rangle+\langle RS\rangle+\langle RT\rangle-\langle QT\rangle \le 2.$$1, reducing the CHSH test to the witness condition

$$\langle \mathcal{B}_{\mathrm{CHSH}} \rangle=\langle QS\rangle+\langle RS\rangle+\langle RT\rangle-\langle QT\rangle \le 2.$$2

Assuming no such $$\langle \mathcal{B}_{\mathrm{CHSH}} \rangle=\langle QS\rangle+\langle RS\rangle+\langle RT\rangle-\langle QT\rangle \le 2.$$3 exists places $$\langle \mathcal{B}_{\mathrm{CHSH}} \rangle=\langle QS\rangle+\langle RS\rangle+\langle RT\rangle-\langle QT\rangle \le 2.$$4 in the dual cone $$\langle \mathcal{B}_{\mathrm{CHSH}} \rangle=\langle QS\rangle+\langle RS\rangle+\langle RT\rangle-\langle QT\rangle \le 2.$$5, and a technical lemma then forces $$\langle \mathcal{B}_{\mathrm{CHSH}} \rangle=\langle QS\rangle+\langle RS\rangle+\langle RT\rangle-\langle QT\rangle \le 2.$$6 to be fully separable, contradicting entanglement (Liang et al., 2012).

The paper’s interpretation is explicit: entanglement is sufficient for hidden CHSH nonlocality once one allows activation by an auxiliary state, and entanglement and nonlocality become equivalent in this activated sense (Liang et al., 2012). A plausible implication is that “genuine activation” in many later works can be read as a refinement of this general principle, with additional constraints on the initial notion of locality, the Bell scenario, or the target form of nonlocality.

An explicit bipartite example is provided by the two-qubit Werner state

$$\langle \mathcal{B}_{\mathrm{CHSH}} \rangle=\langle QS\rangle+\langle RS\rangle+\langle RT\rangle-\langle QT\rangle \le 2.$$7

which is entangled iff $$\langle \mathcal{B}_{\mathrm{CHSH}} \rangle=\langle QS\rangle+\langle RS\rangle+\langle RT\rangle-\langle QT\rangle \le 2.$$8. For $$\langle \mathcal{B}_{\mathrm{CHSH}} \rangle=\langle QS\rangle+\langle RS\rangle+\langle RT\rangle-\langle QT\rangle \le 2.$$9, the ancillary state

$$\max \langle \mathcal{B}_{\mathrm{CHSH}} \rangle_\rho = 2\sqrt{M(\rho)}.$$0

with the displayed matrix $$\max \langle \mathcal{B}_{\mathrm{CHSH}} \rangle_\rho = 2\sqrt{M(\rho)}.$$1 in the source is PPT, hence bound entangled, and belongs to $$\max \langle \mathcal{B}_{\mathrm{CHSH}} \rangle_\rho = 2\sqrt{M(\rho)}.$$2. Yet

$$\max \langle \mathcal{B}_{\mathrm{CHSH}} \rangle_\rho = 2\sqrt{M(\rho)}.$$3

which is negative for $$\max \langle \mathcal{B}_{\mathrm{CHSH}} \rangle_\rho = 2\sqrt{M(\rho)}.$$4. This is an explicit superactivation of hidden CHSH nonlocality (Liang et al., 2012).

4. Multipartite genuine activation under local filtering

In multipartite Bell scenarios, “genuine” often refers to genuinely multipartite nonlocality rather than merely to POVM-locality. One construction shows that for any number of parties $$\max \langle \mathcal{B}_{\mathrm{CHSH}} \rangle_\rho = 2\sqrt{M(\rho)}.$$5, there exist genuinely multipartite entangled states that admit a fully local hidden-variable model for all non-sequential local measurements, yet whose nonlocality can be activated using sequences of local measurements, specifically local filtering operations, leading to genuinely multipartite nonlocality (Bowles et al., 2015). The starting point is a bipartite unsteerable state

$$\max \langle \mathcal{B}_{\mathrm{CHSH}} \rangle_\rho = 2\sqrt{M(\rho)}.$$6

which is assembled in a star network and postprocessed by a GHZ projection

$$\max \langle \mathcal{B}_{\mathrm{CHSH}} \rangle_\rho = 2\sqrt{M(\rho)}.$$7

For suitable parameters, the resulting $$\max \langle \mathcal{B}_{\mathrm{CHSH}} \rangle_\rho = 2\sqrt{M(\rho)}.$$8-party state is fully local but genuinely multipartite entangled (Bowles et al., 2015).

To obtain locality for all POVMs, that construction uses the symmetrized embedding

$$\max \langle \mathcal{B}_{\mathrm{CHSH}} \rangle_\rho = 2\sqrt{M(\rho)}.$$9

Each party then applies the same filter

$\rho_0$0

with $\rho_0$1. The filtered state becomes very close to the $\rho_0$2-qubit GHZ state, with fidelity

$\rho_0$3

which tends to $\rho_0$4 when $\rho_0$5. Since GHZ states exhibit genuinely multipartite nonlocality, the filtered state does so for suitable parameters (Bowles et al., 2015).

Tripartite and four-partite analyses formulate this activation through Svetlichny-type inequalities. For three qubits, the Svetlichny operator is

$\rho_0$6

and bi-local models satisfy $\rho_0$7 (Sun et al., 2021). The paper derives the bound

$\rho_0$8

where $\rho_0$9 is the maximal singular value of the filtered correlation tensor. For

$$\rho'=\frac{1}{d^2}\Big[ \rho_0+(d-1)(\rho_A\otimes \sigma_B+\sigma_A\otimes \rho_B) +(d-1)^2\sigma_A\otimes \sigma_B \Big],$$0

with $$\rho'=\frac{1}{d^2}\Big[ \rho_0+(d-1)(\rho_A\otimes \sigma_B+\sigma_A\otimes \rho_B) +(d-1)^2\sigma_A\otimes \sigma_B \Big],$$1, the hidden genuine nonlocality region is

$$\rho'=\frac{1}{d^2}\Big[ \rho_0+(d-1)(\rho_A\otimes \sigma_B+\sigma_A\otimes \rho_B) +(d-1)^2\sigma_A\otimes \sigma_B \Big],$$2

For the GHZ state with colored noise,

$$\rho'=\frac{1}{d^2}\Big[ \rho_0+(d-1)(\rho_A\otimes \sigma_B+\sigma_A\otimes \rho_B) +(d-1)^2\sigma_A\otimes \sigma_B \Big],$$3

the hidden genuine nonlocality region is

$$\rho'=\frac{1}{d^2}\Big[ \rho_0+(d-1)(\rho_A\otimes \sigma_B+\sigma_A\otimes \rho_B) +(d-1)^2\sigma_A\otimes \sigma_B \Big],$$4

In both examples, the initial state does not violate Svetlichny, but the filtered state does (Sun et al., 2021).

For four qubits, the Seevinck-Svetlichny operator $$\rho'=\frac{1}{d^2}\Big[ \rho_0+(d-1)(\rho_A\otimes \sigma_B+\sigma_A\otimes \rho_B) +(d-1)^2\sigma_A\otimes \sigma_B \Big],$$5 satisfies $$\rho'=\frac{1}{d^2}\Big[ \rho_0+(d-1)(\rho_A\otimes \sigma_B+\sigma_A\otimes \rho_B) +(d-1)^2\sigma_A\otimes \sigma_B \Big],$$6 for partially separable states. The paper proves

$$\rho'=\frac{1}{d^2}\Big[ \rho_0+(d-1)(\rho_A\otimes \sigma_B+\sigma_A\otimes \rho_B) +(d-1)^2\sigma_A\otimes \sigma_B \Big],$$7

with $$\rho'=\frac{1}{d^2}\Big[ \rho_0+(d-1)(\rho_A\otimes \sigma_B+\sigma_A\otimes \rho_B) +(d-1)^2\sigma_A\otimes \sigma_B \Big],$$8 and $$\rho'=\frac{1}{d^2}\Big[ \rho_0+(d-1)(\rho_A\otimes \sigma_B+\sigma_A\otimes \rho_B) +(d-1)^2\sigma_A\otimes \sigma_B \Big],$$9 the largest singular values of the original and filtered four-body correlation matrices (Hossain et al., 2023). For the noisy GHZ-class state

$\rho_A=\mathrm{tr}_B(\rho_0)$0

the original violation threshold at $\rho_A=\mathrm{tr}_B(\rho_0)$1 is $\rho_A=\mathrm{tr}_B(\rho_0)$2, whereas after filtering the paper reports violation already for

$\rho_A=\mathrm{tr}_B(\rho_0)$3

so genuine hidden nonlocality is revealed in

$\rho_A=\mathrm{tr}_B(\rho_0)$4

This is the paper’s main concrete example of genuine hidden nonlocality activated by local filtering (Hossain et al., 2023).

5. Bound entanglement, higher-dimensional inequalities, and resource-theoretic consequences

The relation between hidden nonlocality and weak forms of entanglement is sharpened by the three-qubit bound-entangled example of a fully-biseparable state $\rho_A=\mathrm{tr}_B(\rho_0)$5 that is local for the most general non-sequential measurements, yet becomes Bell-nonlocal after suitable local filters (Tendick et al., 2019). The state is generated from a nonlocal state $\rho_A=\mathrm{tr}_B(\rho_0)$6 by invertible local filters $\rho_A=\mathrm{tr}_B(\rho_0)$7: $\rho_A=\mathrm{tr}_B(\rho_0)$8 The explicit $\rho_A=\mathrm{tr}_B(\rho_0)$9 is invariant under partial transpose with respect to any party and invariant under permutation of parties; hence it is PPT and bi-separable with respect to any bipartite cut (Tendick et al., 2019).

Nonlocality is witnessed by Sliwa’s inequality number 5,

$\rho_B=\mathrm{tr}_A(\rho_0)$0

with

$\rho_B=\mathrm{tr}_A(\rho_0)$1

and $\rho_B=\mathrm{tr}_A(\rho_0)$2. The resulting quantum value is

$\rho_B=\mathrm{tr}_A(\rho_0)$3

The authors then prove, by an SDP-based method, that $\rho_B=\mathrm{tr}_A(\rho_0)$4 itself admits a local model for POVMs with optimization outcome

$\rho_B=\mathrm{tr}_A(\rho_0)$5

Applying the explicit local filters $\rho_B=\mathrm{tr}_A(\rho_0)$6 recovers $\rho_B=\mathrm{tr}_A(\rho_0)$7, so $\rho_B=\mathrm{tr}_A(\rho_0)$8 has genuine hidden nonlocality (Tendick et al., 2019). The paper’s conceptual conclusion is equally explicit: $\rho_B=\mathrm{tr}_A(\rho_0)$9 Since invertible local filters do not change the entanglement class, a bound-entangled state remains bound entangled after filtering (Tendick et al., 2019).

A related resource-theoretic result states that if a bipartite state can be transformed into a nonlocal state by local filtering, then it can be deterministically transformed into a nonlocal state by LOCC using one bit from Alice to Bob and one bit from Bob to Alice (Camalet, 2017). The deterministic state takes the block form

$$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$00

with

$$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$01

This yields a communication-based characterization of hidden nonlocality and a measure-independent “anomaly of nonlocality”: a Bell-local state with hidden nonlocality is more, or equally, entangled than some nonlocal state (Camalet, 2017).

Higher-dimensional Bell inequalities widen the activation landscape. For the qudit family

$$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$02

the CGLMP inequality is used instead of CHSH. With

$$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$03

the range of the mixing parameter revealing hidden nonlocality increases with dimension, and for $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$04 hidden nonlocality can be revealed for the whole range of the mixing parameter (Kumari, 2023). A related PORAC-based construction studies the Bell functional

$$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$05

with local bound

$$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$06

and preparation-noncontextual bound

$$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$07

After filtering, the paper shows nonlocality for any nonzero $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$08 provided $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$09 for even $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$10, and preparation contextuality for any nonzero $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$11 provided $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$12 for even $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$13 (Kumari et al., 25 Apr 2025). This suggests that activation may become easier in families where the Bell functional scales with dimension or input size.

6. Genuine activation in local discrimination and the strong-locality boundary

In the local-discrimination literature, genuine activation of hidden nonlocality concerns orthogonal-state ensembles rather than Bell inequalities. An orthogonal set $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$14 is locally distinguishable if LOCC can identify the state with certainty, and locally indistinguishable otherwise. An orthogonality-preserving local measurement (OPLM) is a local measurement after which the post-measurement states remain pairwise orthogonal (Bandyopadhyay et al., 2021, Bera et al., 2024). Activation is the transformation

$$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$15

The adjective “genuine” excludes cases caused by local redundancy: the original set must become nonorthogonal upon discarding any proper local part (Bandyopadhyay et al., 2021).

The first systematic construction shows that there exist orthogonal sets that are locally distinguishable but without local redundancy whose nonlocality can be activated by local measurements with certainty (Bandyopadhyay et al., 2021). In $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$16, for example, the three unnormalized states

$$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$17

are locally distinguishable, have no local redundancy, and are converted by Bob’s OPLM $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$18 into triplets that are locally indistinguishable (Bandyopadhyay et al., 2021). The paper develops analogous examples in $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$19, $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$20, and $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$21 (Bandyopadhyay et al., 2021).

The question whether such genuine activation can occur without any entanglement is answered affirmatively by explicit product-state constructions (Li et al., 2021). That work separates two types. Type I preserves the number of states in every branch: $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$22 whereas Type II allows cardinality decrease in at least one branch: $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$23 for some $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$24 (Li et al., 2021). For odd $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$25, there exist orthogonal product-state sets in $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$26 with genuine hidden nonlocality of Type I, and for all $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$27, there exist orthogonal product-state sets in $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$28 with genuine hidden nonlocality of Type II (Li et al., 2021). A key lemma used to certify local irredundancy states that if $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$29 is a quantum channel and $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$30, then

$$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$31

so partial trace cannot turn nonorthogonal states into orthogonal ones (Li et al., 2021).

Later work organizes these phenomena into a hierarchy of locality. In $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$32 systems, TYPE-I activable sets admit activation by a single party’s local measurement, whereas TYPE-II activable sets cannot be activated by any single party acting alone but can be activated if two parties perform a joint PVM (Bera et al., 2024). This yields a relative comparison: $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$33 is more local than $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$34 because more cooperation is needed to reveal its hidden nonlocality (Bera et al., 2024). The same work introduces “strong local” sets as non-activable extremes (Bera et al., 2024).

The strongest boundary result concerns complete orthogonal product bases. For a complete orthonormal product basis $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$35, any nontrivial orthogonality-preserving local projective measurement can only keep or eliminate members of $$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$36; it cannot generate a genuinely new orthogonal product structure. The post-measurement states take the form

$$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$37

and orthogonality preservation requires

$$\rho'=\frac{M\otimes N\, \rho\, M^\dagger\otimes N^\dagger}{p}, \qquad p=\operatorname{tr}(M^\dagger M\otimes N^\dagger N\, \rho),$$38

From this, the paper concludes that a complete orthogonal product basis that is initially distinguishable by LOCC cannot be made locally indistinguishable via any LPCC protocol (Bhunia et al., 28 May 2026). It further defines strongly local sets as locally distinguishable sets that remain non-activable under all bipartitions (Bhunia et al., 28 May 2026). This establishes a structural boundary: incompleteness is necessary for activation of nonlocality without entanglement under the orthogonality-preserving projective protocols studied there (Bhunia et al., 28 May 2026).