Nonlocality Without Entanglement

• Nonlocality without entanglement is defined by orthogonal product states that, despite being unentangled, cannot be perfectly distinguished using local operations and classical communication.
• Mathematical constructions and explicit OPB examples demonstrate that measurement limitations under LOCC can create operational gaps compared to global strategies.
• The phenomenon challenges the traditional link between entanglement and nonlocality, providing actionable insights for quantum data hiding, secret sharing, and resource theories.

Nonlocality without Entanglement refers to a set of quantum phenomena in which unentangled—specifically orthogonal product—states or separable measurements exhibit nonlocal characteristics, such as local indistinguishability or operational gaps between local operations and classical communication (LOCC) and global operations. This class of nonclassical effects is evidenced by ensembles, measurements, or protocols in which the nonlocality does not originate from entanglement, but rather emerges from the global structure of quantum state sets and the limitations of local quantum operations. The concept fundamentally challenges the association between nonlocality and entanglement and provides a more nuanced understanding of quantum resources and their operational behaviors.

1. Conceptual Foundations and Historical Motivation

The concept originated from the realization that certain sets of orthogonal product states in composite Hilbert spaces are locally indistinguishable by LOCC, although they are globally orthogonal and perfectly distinguishable if all parties cooperate. The seminal demonstration by Bennett et al. (1999) established that even in the absence of entanglement, ensembles of product states could display nonlocality in the sense that “the whole is more than the sum of its parts.” This type of quantum nonlocality is fundamentally different from Bell-nonlocality, which is operationally defined via the violation of Bell inequalities and is a direct consequence of entanglement. In subsequent work, this phenomenon was termed “quantum nonlocality without entanglement” (NWE or NLWE) and explored in various operational, foundational, and information-theoretic settings (Bhattacharya et al., 2019, Mal et al., 2020, Li et al., 2020).

An early operational framework, inspired by resource theory, asserts that whereas entanglement cannot be generated by LOCC, nonlocal correlations—here modeled as joint probability distributions—cannot be created by a broader class of operations, including pre-input classical communication and wirings (WCCPI) (Gallego et al., 2011). The paper of NWE focuses on the inability of LOCC protocols to simulate or discriminate between certain unentangled resources optimally, extending the exploration of quantum nonclassicality beyond the paradigm of Bell inequalities.

2. Taxonomy and Formal Criteria

A central task in the paper of NWE is to delineate precise notions and classifications:

• Locally Indistinguishable Product Bases: An orthogonal product basis (OPB) is said to be locally indistinguishable if no sequence of LOCC can perfectly discriminate its elements. The original Bennett et al. construction, and the later “domino” and “twisted” product bases, fall within this category (Halder et al., 2018).
• Local Irreducibility: A stronger property concerns “locally irreducible” sets: OPSs for which no subset can be locally eliminated via an orthogonality-preserving measurement, even sequentially. Such sets are called “strongly nonlocal” or “locally irreducible in all bipartitions,” indicating that local discrimination is not only impossible for the entire set, but remains so for all proper subsets after any nontrivial measurement (Halder et al., 2018, Yuan et al., 2020, Zhou et al., 3 Jul 2024).
• Genuine Nonlocality: Encompasses sets that are locally indistinguishable in every bipartition of the parties, thus guaranteeing that nonlocal features persist regardless of how the system is grouped (Li et al., 2020, Zhen et al., 2022).
• Genuine Hidden Nonlocality: Describes sets whose indistinguishability (and hence nonlocality) can only be activated by specific sequences of orthogonality-preserving local measurements, emphasizing that nonlocal characteristics may be “hidden” until triggered by a suitable protocol (Li et al., 2021).
• Measure-Dependence: Whether an OPS exhibits NWE can depend on the operational measure or discrimination strategy—minimum-error versus unambiguous discrimination, etc.—since sometimes LOCC gaps exist only for certain discrimination tasks (Murshid et al., 25 Jun 2025).

This taxonomy is operationalized mathematically. For a discrimination task involving an ensemble $\mathcal{E}_\psi = \{(\eta_i, |\psi_i\rangle)\}$, nonlocality without entanglement is evidenced if the optimal figure of merit $$f_\mathrm{L}(\mathcal{E}_\psi) < f_\mathrm{G}(\mathcal{E}_\psi)$$, where the subscript “L” restricts to LOCC and “G” allows global operations.

3. Mathematical and Structural Results

Numerous mathematical results underpin the manifestations and limitations of NWE:

• Orthogonal Product Base Constructions: Explicit constructions for locally indistinguishable product sets have been achieved in bipartite, multipartite, and high-dimensional systems (Zhen et al., 2022, Zhou et al., 3 Jul 2024). For instance, in $(\mathbb{C}^d)^{\otimes n}$, sets of $n(d-1)+1$ orthogonal product states that are LOCC-indistinguishable were defined using systematic basis manipulations and “stopper” states to enforce triviality of local measurement operators.
• Reduction of Measurement Operators: Lemmas concerning the structure of orthogonality-preserving POVMs demonstrate that off-diagonal terms must be zero and diagonal entries must be equal, forcing all such measurements to be proportional to the identity in key constructions (Zhen et al., 2022).
• Strong Nonlocality via Local Irreducibility: Recent advances constructed strongly nonlocal OPSs—locally irreducible in all bipartitions or in every $(n-1)$-partition—for arbitrary multipartite systems. These constructions are significantly resource-efficient, using fewer states than previous extremal sets, yet still exhibit robust nonlocality in the sense that no nontrivial measurement by any party (or any subset of parties) can commence a successful discrimination protocol (Zhou et al., 3 Jul 2024, Yuan et al., 2020).
• Measure-Dependence: Distinct discrimination measures can affect whether a given OPS set shows NWE. Example constructions exist wherein for minimum-error discrimination, LOCC cannot attain the global optimum, yet for unambiguous discrimination, global and local strategies are equally effective (Murshid et al., 25 Jun 2025). This highlights the operational task’s centrality in ascribing nonlocality to a given state ensemble.
• Generalized Probabilistic Theories: The phenomena are not limited to quantum mechanics; analogous NWE behaviors arise in a wide class of generalized probabilistic theories (GPTs), and the “strength” of NWE can be quantitatively compared via the gap $\Delta=1-P_L$ in optimal discrimination (Bhattacharya et al., 2019).

4. Distinction from Entanglement and Bell Nonlocality

Nonlocality without entanglement is distinct from both entanglement and Bell-nonlocality:

• Inequivalence to Entanglement: There exist genuinely multipartite entangled (GME) states that do not display genuine multipartite nonlocality—they admit local hidden variable models even for generalized measurements (Augusiak et al., 2014). Conversely, product states in NWE scenarios can manifest strongly nonlocal features operationally inaccessible to entangled states in some instances.
• Non-Monotonicity: The degree of nonlocality, e.g., the level of violation of a Bell inequality, is not necessarily a monotonic function of entanglement. Notably, in the I$_{3322}$ inequality, the maximal quantum violation is attained not by maximally entangled states but by non-maximally entangled ones (or even, in principle, with embezzlement states that are not maximally entangled) (Vidick et al., 2010).
• Unified Criterion: Structural analysis reveals that LOCC-indistinguishability (hence, NWE) of a product set is closely linked to the irreducibility of the set under global unitary operations. The action of, e.g., a CNOT transformation can “reveal” entanglement only in the presence of NWE (Mal et al., 2020). This provides a unified framework relating different forms of quantum nonlocality based on measurement and state structure.
• Generalized Framework for Nonlocality: The operational resource theory of nonlocality, based on allowed WCCPI operations and associated equivalence classes (e.g., TOBL, NSBL), formalizes the distinction from entanglement resource theory, showing that nonlocal and entangled resources are operationally and physically distinct notions (Gallego et al., 2011).

5. Applications and Operational Impact

NWE has direct and consequential implications in quantum information science:

• Data Hiding and Secret Sharing: The resistance of certain OPS sets to LOCC discrimination allows for protocols in which classical information is hidden from spatially separated parties, unless joint global measurements are performed. This underpins quantum data hiding and secret sharing schemes where security is guaranteed even in the absence of entanglement (Li et al., 2021).
• Resource Constraints in Distributed Computing: Strongly nonlocal OPSs have been constructed whose discrimination requires substantial entanglement resources—sometimes a genuinely multipartite entangled state (e.g., a GHZ state) rather than bipartite Bell resources—if all parties are to collaborate via LOCC (Bhunia et al., 2021). This has operational consequences for distributed quantum computing, quantum communication complexity, and controlled information sharing.
• Cryptographic Protocols Tunable by Prior Information: The occurrence and even the very presence of NWE can be “locked” or “unlocked” by availability of post-measurement information or by the choice of prior probabilities in state-preparation. This suggests that cryptographic schemes may exploit such tunability to design unified data hiding or secret sharing protocols by varying priors or public information (Ha et al., 2021, Ha et al., 2021, Ha et al., 2021).
• Device-Independent Certification: Self-testing protocols have been developed to device-independently certify the occurrence of NWE in quantum networks, bridging network nonlocality and measurement nonlocality (Šupić et al., 2022). This advances the applicability of NWE as a diagnostic and benchmarking tool in experimental quantum networks.

6. Network and Resource Scaling Considerations

Recent research has investigated the interplay between entanglement and network nonlocality, especially in networks with multiple sources:

• Network Nonlocality without All Sources Entangled: In star-topology networks, it is possible to achieve non-n-locality (i.e., observe violation of n-local inequalities) even when not all sources are entangled, provided that the product of the concurrence of all sources exceeds ½. In contrast, in linear-topology networks, every source must be entangled for nonlocality to be observed (Mukherjee et al., 19 Oct 2024). This distinction points to a nuanced picture in which topology and entanglement interact to determine nonlocal behavior, and nonlocality can arise in networks with partial entanglement, subject to threshold conditions captured by explicit inequalities, e.g., for star networks:

$$B_{n\text{-star}} \leq \sqrt{1 + \prod_{i=1}^n C_i^{2/n}}$$

where $C_i$ is the concurrence of the $i$th source.

• Scaling of Strongly Nonlocal Sets: Constructions of strongly nonlocal OPSs with quantum states scaling only as $\Theta\left(d^{\lceil (n+1)/2 \rceil}\right)$ for $n$-partite $d$-dimensional systems allow for experimental and theoretical efficiency when implementing NWE in high-dimensional settings (Zhou et al., 3 Jul 2024).

7. Theoretical and Foundational Implications

NWE phenomena constrain and elucidate foundational aspects of quantum mechanics:

• Quantum-Classic Divide and GPTs: Since analogous behavior arises in general probabilistic or polygonal theories, and quantum theory admits only limited “strength” of NWE compared to certain discrete theories, the phenomenon acts as a diagnostic for the topology and structure of state spaces, potentially informing axiomatic reconstructions (Bhattacharya et al., 2019).
• Nonlocality Activation and Genuine Hidden Structures: The possibility to “activate” or “hide” nonlocality in ensembles without entanglement, sometimes requiring elaborate measurement sequences or the nontrivial structure of product states, demonstrates the subtleties in quantum resource theory beyond canonical entanglement.
• Resource Theory Distinctions: The operational distinction between what cannot be created by LOCC (entanglement) and by WCCPI or LOCC in discrimination settings (nonlocality) reflects a broader stratification of quantum resources (Gallego et al., 2011, Augusiak et al., 2014), with NWE being a salient signature of this stratification.

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In summary, nonlocality without entanglement characterizes a variety of quantum behaviors wherein nonentangled states or measurements manifest global nonclassical features—such as the local indistinguishability of certain ensembles, operational gaps between LOCC and global measurement, or resource-obstructed protocols—arising from the structure of quantum mechanics, sometimes shared by even broader operational theories. Its paper has clarified fundamental distinctions between entanglement and nonlocality, enabled the design of efficient quantum information protocols, and motivated deeper exploration into the hierarchy of quantum resources and the architecture of quantum theory itself.