Non-local Information Encoding

• Non-local information encoding is a method that distributes quantum data across multipartite systems so that recovery requires coordinated actions.
• The approach leverages entanglement and delocalized correlations to achieve error robustness, secure secret sharing, and efficient network coding.
• Applications span quantum networks, error correction, and gravitational holography, linking advanced encoding techniques with practical quantum protocols.

A non-local information encoding scheme encodes logical data such that recovery or manipulation of the information requires coordinated actions or access to distributed quantum subsystems, with individual parties or locations lacking full access through any local operation. In quantum settings, such encodings leverage entanglement or delocalized correlations—often across a network—to achieve error robustness, security against node failures, and functionalities such as quantum secret sharing, distributed error correction, and network coding. In gravitational and holographic contexts, non-local encoding arises through mechanisms such as massive graviton dynamics, leading to phenomena like entanglement islands where the information about certain spacetime regions is encoded in spatially disjoint exterior systems.

1. Foundational Notions of Non-local Information Encoding

At the core, non-local information encoding operates by distributing the logical state of interest across a multipartite quantum system so that no single subsystem or region holds the complete information. This principle is realized in several domains:

• Entanglement-based codes: Logical qubits are encoded in large-scale entangled states (graph states, stabilizer codes, Dicke states, MPS/correlation space wires) spread across network nodes, rendering the logical data robust to loss and inaccessible to local measurement or manipulation by a subset of nodes (Miguel-Ramiro et al., 2019).
• Encoding in gravitational systems: In models with entanglement islands, the interior information of a black hole is encoded non-locally in an external bath, challenging classical locality notions (Geng, 12 Feb 2025).
• Locally inaccessible subspaces: Quantum systems can possess subspaces such that the information encoded within is inaccessible to any sequence of local operations and classical communication (LOCC), except in the special case of dimension two (Croke, 2022).
• Network coding with non-local resources: In communication protocols, non-signaling or quantum-correlated resources enable higher rates than possible classically, by distributing the encoding operation across multiple parties without signaling (Yun et al., 2020).

2. Quantum Network and Error Correction Schemes

Quantum communication networks and distributed circuits provide canonical settings for non-local encoding. Two emblematic classes are:

• Stabilizer and Graph-State Encodings: Logical qubits are represented by stabilizer codewords, often in graph states. For an $n$-node network, the logical subspace is spanned by states such as $|0_L\rangle = |G\rangle$, $|1_L\rangle = Z^{\otimes n}|G\rangle$, with robust logical operations $\bar{X}, \bar{Z}$ acting nontrivially across all nodes. Decoding back to a physical node typically requires coordinated measurements and corrections based on global measurement outcomes (Miguel-Ramiro et al., 2019).
• Dicke-State and Correlation-Space Encodings: Dicke states $|n,k\rangle$ and matrix-product states enable delocalized encodings with tunable resilience: for Dicke states, logical codewords can be chosen to maximize survival fidelity under partial loss; in MPS, the correlation space hosts the logical state, with physical realization achieved by distributed local operations and nearest-neighbor entanglement. Localization ("downloading") of logical information typically requires open-destination or heralded protocols based on measurement-induced filtration, where full recovery is probabilistic but can be made arbitrarily close to deterministic by adjusting circuit depth (Miguel-Ramiro et al., 2019).

The resource cost, entanglement distribution, and operational locality are quantitatively codified: e.g., the required entanglement per network edge for exact encoding is given by $$E_{\rm enc}(e)=\log_2{\rm rank}(\tilde\Phi_{D,e}^+)$$, linked to tree-like network structures and the sequential application of state-splitting and state-merging protocols (Yamasaki et al., 2018).

3. Locally Inaccessible Information and Measurement Non-locality

The concept of locally inaccessible information formalizes when an encoded quantum state cannot be fully observed or manipulated by LOCC, a property distinguished by the dimension of the encoding subspace:

• Two-dimensional subspaces (qubit case): All measurements (POVMs) can be implemented by a recursive one-way LOCC protocol. Any two-outcome instrument on a logical qubit subspace $S$ decomposes into a sequence of local Kraus operators with classical feed-forward, leveraging the Walgate decomposition for bipartite and multipartite systems. Thus, $d=2$ encodings are LOCC-trivial (Croke, 2022).
• Higher-dimensional subspaces ($d\geq 3$): Genuinely non-local information becomes possible. There exist subspaces and (orthonormal) bases that cannot be distinguished by any LOCC protocol, including the construction of unextendible product bases (UPBs) and "nonlocality without entanglement" settings (Croke, 2022). This threshold underpins fundamental limitations and possibilities for secret sharing and error correction, with precise characterization of which subspaces are "LOCC-nontrivial" remaining an open problem.

4. Physical Mechanisms in Gravitational and Holographic Systems

In gravitational contexts, non-local encoding of information manifests through mechanisms fundamentally distinct from conventional multipartite entanglement:

• Entanglement Islands and Graviton Mass: The island rule in AdS black hole physics leads to an apparent non-local mapping: all quantum operators supported inside the "island" region $\mathcal{I}$ are encoded in an external region $R$. The emergence of this non-local encoding from a local theory is attributed to the graviton acquiring a small mass, which breaks the global Gauss law constraints and enables the support of local operators in $\mathcal{I}$ that commute with operators in $R^c$ (Geng, 12 Feb 2025).
• Stückelberg Fields and Encoded Observables: After the graviton becomes massive, operators in the island must be "dressed" with a Stückelberg field. The physical encoding map for an operator $O_{\mathcal{I}}(x)$ in $R$ involves an exponential (Yukawa-type) nonlocal kernel $G_m(x, x')$:

$$O_R(x) = \int_{\mathcal{I}} d^d x' \, G_m(x, x')\, O_{\mathcal{I}}(x') + \dots$$

with $G_m(x, x') \sim e^{-m|x-x'|}/|x-x'|^{d-2}$. This mechanism is concretely illustrated in the Karch–Randall braneworld, where the encoding arises from the holographic geometry and bulk Wilson-line dressings, providing a direct link to ER=EPR dualities (Geng, 12 Feb 2025).

• Operational Consequences: This encoding mechanism underlies the recovery of the Page curve and the apparent monogamy violation for boundary-reconstructable regions in gauge/gravity duality, rooting all non-local features in the spontaneous breaking of boundary diffeomorphisms by bath coupling.

5. Applications to Quantum Communication and Network Coding

Non-local encoding principles extend to quantum communication and classical network coding scenarios, often yielding strict separations in capacity:

• Non-Local Network Coding: In interference-channel scenarios, encoding and decoding by exploiting non-signaling (PR-box-type) or quantum resources outperform any classical (local) coding scheme. Bell-type boxes ($P(a,b|x,y)$), parameterized to enforce the no-signaling constraint, form the building blocks for such codes (Yun et al., 2020).
• Capacity Hierarchies: For a fixed interference channel, classical, quantum, and non-signaling resources induce distinct sum-capacities $C^{(L)}(N)<C^{(Q)}(N)<C^{(\rm NS)}(N)$. Explicit formulas show, for example, that using a PR-box can double the classical capacity. Critically, the degree of non-locality (as measured by Bell violation) does not monotonically determine channel capacity; some more non-local boxes have strictly worse capacity, indicating the nuanced resource theory required for non-local encoding utility (Yun et al., 2020).

6. Operational Asymmetries and Resource Trade-offs

Non-local encoding schemes exhibit distinct operational asymmetries, quantified resource trade-offs, and protocols sharply tied to their physical context:

• Encoding vs. Decoding Costs: Distributed encoding (spreading) and decoding (concentrating) are inverse isometries, but the one-shot entanglement cost for encoding along a network can exceed that of decoding. This arises from asymmetries in state-splitting (which must actively distribute entanglement) versus state-merging (which can exploit pre-existing classical correlations) (Yamasaki et al., 2018).
• Robustness and Security: Non-local encodings ensure that no single node or region can fully access the logical information, offering intrinsic resistance to node failure, local decoherence, or eavesdropping. Dicke and correlation-space encodings in quantum networks allow graceful degradation in fidelity under partial loss, and secret-sharing protocols guarantee that only joint action permits full measurement or recovery (Miguel-Ramiro et al., 2019, Croke, 2022).
• Protocol Design: In multipartite networks, choosing between stabilizer, Dicke, or MPS-based encodings involves balancing deterministic localization, ease of preparation, passive/active error protection, and entanglement overhead (Miguel-Ramiro et al., 2019). In network coding, optimizing non-local resources necessitates fine-grained analysis beyond simple non-locality metrics.

7. Outlook and Open Problems

Non-local information encoding remains central to both foundational quantum theory and practical quantum information science. Key challenges and directions include:

• Classification of LOCC-inaccessible subspaces for $d \geq 3$: While $d=2$ is universally LOCC-accessible, the structure of higher-dimensional subspaces supporting genuine non-locality is not fully characterized (Croke, 2022).
• Resource theory for network coding: Precise operational metrics for non-locality in communication capacity, beyond Bell inequalities, invite further study (Yun et al., 2020).
• Gravitational embedding: The role of graviton mass and Stückelberg fields in holographic non-local encoding suggests broader classes of quantum-gravity models with similar behavior (Geng, 12 Feb 2025).

Future developments will likely clarify the taxonomy of non-local encodings, deepen the links between gravitational and quantum information paradigms, and advance robust quantum network applications.