Group-Symmetric Data Hiding States

• The paper's main contribution is constructing quantum state ensembles using group symmetries to hide classical data from LOCC operations while remaining globally accessible.
• It employs methodologies such as group twirling, projector-based constructions, and quantification of asymmetry to establish secure protocols with strict distinguishability metrics.
• The study has practical implications for secret sharing and quantum key distribution by providing rigorous bounds on LOCC bias via convex optimization and numerical techniques.

Group-symmetric data hiding states are a class of multipartite quantum states designed so that classical information encoded in them is perfectly accessible to global quantum measurements yet remains nearly (or completely) inaccessible to parties limited to locally restricted operations—most notably, local operations and classical communication (LOCC), separable operations, or more generally, operations covariant with respect to a symmetry group. The essential feature is to exploit group symmetries to enforce indistinguishability, thus constructing instances of “nonlocality without entanglement,” and enabling secure protocols such as data hiding, secret sharing, and reference frame alignment.

1. Mathematical Foundations: Symmetry, Stabilizers, and State Construction

Group-symmetric data hiding states are constructed using quantum symmetries, most commonly under global actions of Lie groups such as $\mathrm{U}(d)$, permutation groups, or finite subgroups. The core methodology is to exploit invariance under collective group actions (e.g., $g^{\otimes n}$ for $g\in G$) so that the encoded information is accessible globally but hidden from any subset of parties unable to jointly break the symmetry.

For symmetric pure states of $n$ qubits, the Majorana representation provides a canonical geometric characterization: any symmetric state is determined, up to phase, by a “Majorana configuration” of $n$ points on the Bloch sphere. All local unitary (LU)-equivalence classes are characterized by orbits of these configurations under global SU(2) action; thus, for symmetric mixed or pure states,

$$\rho' = (g^{\otimes n})\, \rho\, (g^{\dagger})^{\otimes n}$$

for some $g \in \mathrm{U}(2)$ exhausts the equivalence class (Cenci et al., 2010).

In the context of LOCC-restricted state discrimination, group twirling (averaging over $G$) or similar symmetrization operations typically ensure that group symmetry is preserved, locking the classical data away from locally accessible degrees of freedom. This is a central aspect in both explicit constructions (e.g., via projectors on group-invariant subspaces (Mele et al., 3 Oct 2025)) and random coding arguments.

2. Information Hiding: Resource Theory of Asymmetry and Indistinguishability

The operational principle at the heart of group-symmetric data hiding is that, under symmetric (i.e., $G$-covariant) operations, the set of distinguishable features of a state is limited to its “asymmetry” relative to $G$. Formally, the asymmetry or frameness is quantified by the characteristic function,

$\chi_\psi(g) = \langle \psi | U(g) | \psi \rangle$

for $g \in G$. The state’s ability to encode “unspeakable” information about group elements is precisely determined by this function and its reduction onto irreducible representations (Marvian et al., 2011).

When information is encoded into a group orbit (via global rotations or group averaging), LOCC or G-covariant operations cannot distinguish data corresponding to different group elements because the measurement statistics are group-invariant. More generally, the resource theory of asymmetry provides necessary and sufficient conditions for state conversion under symmetric operations, characterizing when information hiding is possible, and under which protocols.

3. Distinguishability Under Restricted Measurements

A central technical framework is the analysis of distinguishability norms defined with respect to sets of restricted measurements such as LO, LOCC, separable (SEP), or positive partial transpose (PPT) protocols (Aubrun et al., 2014). Given states $\rho$ and $\sigma$, the measurement distinguishability norm for a set $\mathcal{M}$ is

$$\|\rho-\sigma\|_{\mathcal{M}} = \sup_{M\in\mathcal{M}} \sum_i |\operatorname{Tr}(M_i (\rho-\sigma))|$$

and quantifies the bias achievable when distinguishing $\rho$ from $\sigma$ using measurements from $\mathcal{M}$.

Group-symmetric data hiding states are engineered so that

• $\|\rho-\sigma\|_{\text{ALL}} \approx 2$ (perfect global distinguishability)
• $\|\rho-\sigma\|_{\text{LOCC}}\ll 1$ (LOCC indistinguishability, can be made exponentially small).

For high-dimensional systems, these gaps are generic, with the data hiding ratio scaling as $\Theta(\min\{n_A,n_B\})$ for a bipartite $n_A\times n_B$ system (Lami et al., 2017). For multipartite or continuous-variable systems, the role of dimension is replaced by energy or photon number (Lami, 2021).

4. Explicit Constructions and Group-Invariant Examples

a. Projector-Based Construction (Orthogonal, Separable, PPT States)

An explicit method utilizes four mutually orthogonal projectors (e.g., onto maximally entangled, maximally correlated, and swap-invariant subspaces) to construct states

$$\sigma_0^{(d)} = \frac{1}{d}\,\Theta_0 + \frac{2}{d^2}\,\Theta_2,\quad \sigma_1^{(d)} = \frac{1}{2(d-1)}\,\Theta_1 + \frac{1}{d(d-1)}\,\Theta_3$$

and applies a group-twirling map $T_G$ to ensure full symmetry. These states are:

• separable,
• orthogonal,
• positive under partial transpose.

By combining $k$ independent copies and encoding data via parity (i.e., “even” or “odd” number of $\sigma_1^{(d)}$ factors), one constructs states whose LOCC bias decays exponentially, i.e.,

$$\frac{1}{2}\|\rho_1^{(k,d)} - \rho_0^{(k,d)}\|_{\text{LOCC}} \leq 2 \mu_d^k$$

with $\mu_d<1$ for $d \geq 3$ (Mele et al., 3 Oct 2025).

b. Randomized and Symmetrized Protocols

Random coding with Haar-random unitaries or group symmetrization is often used to build hiding codes for data-hiding protocols over noisy quantum channels. The symmetry ensures average output states for different classical messages are nearly invariant under all LOCC measurements, thus indistinguishable (Lupo et al., 2015).

c. Continuous-Variable (Gaussian) Constructions

In continuous-variable systems, Gaussian displacement of two-mode squeezed states, or weak two-mode thermal states with phase encodings, are employed. Here, hiding is achieved due to the inability of local Gaussian operations plus classical communication (GLOCC) to jointly access required phase-space correlations (Wang et al., 2 Feb 2025).

d. Separable State Ensembles

Constructing hiding schemes with orthogonal separable states (rather than just entangled states) is now possible by carefully choosing ensembles in low-dimensional systems and analyzing PPT- or LOCC-constrained success probabilities via iterative polynomial bounds (Ha et al., 25 Feb 2025).

5. Applications: Secret Sharing, Data Hiding Capacity, and Limitations

Group-symmetric data hiding is foundational for quantum cryptographic primitives:

• Secret Sharing: Classical bits are encoded in such a way that only global (collaborative) measurements can recover the secret (Ha et al., 8 Mar 2024).
• Distributed Data Hiding: Information is perfectly recoverable only by bringing all subsystems together, while arbitrary local coalitions are rendered mathematically incapable of decoding the secret (1804.01982).
• Quantum Key Distribution Repeater Chains: Private states leveraging group-symmetry-based hiding allow the extension of cryptographic distances via optimal entanglement swapping (Christandl et al., 2016).

The data-hiding capacity of a quantum channel quantifies how much classical information can be hidden per channel use, and is upper bounded by the difference between Holevo information and accessible information under LOCC (Lupo et al., 2015). In Gaussian and CV protocols, the energy requirement for information hiding reflects a tradeoff analogous to dimension in the finite case.

A practical implication is that only global resources—such as full entanglement, high squeezing, or total photon number access—unlock the hidden data. Even in the absence of entanglement, as in certain separable or PPT states, nonlocality without entanglement is manifested via perfect global but negligible local distinguishability.

6. Connections, Extensions, and Theoretical Impact

The design of group-symmetric data hiding states connects the theory of tensor norms (projective vs. injective) in Banach spaces (Lami et al., 2017), resource theory of asymmetry (Marvian et al., 2011), and quantum state verification (QSV) via dualities in sample complexity and security levels (Akibue et al., 1 Sep 2025).

Group-symmetric constructions generalize to arbitrary probabilistic theories (GPTs), where the maximum achievable data hiding ratio for any bipartite system is shown to scale linearly with the minimal local dimension, and can be achieved in models with suitable symmetry (Lami et al., 2017).

In multipartite settings, encoding protocols partition receiver sets and exploit subgroup symmetries so that unless all parties collaborate, hidden data remains unrecoverable even for unrestricted global quantum operations within any proper subset (Ha et al., 8 Mar 2024). Protocols incorporate features such as remote deletion (“burning” the data before revealing), concatenation of subprotocols, and unextendible product bases.

7. Methodological and Numerical Advances

The paper and optimization of group-symmetric data hiding states integrate methods from convex optimization, linear and semidefinite programming, and numerical analysis tools such as Tikhonov-regularized least squares and probabilistic tail bounds (e.g., via Sanov’s theorem) to analyze LOCC norms and convergence rates (Mele et al., 3 Oct 2025). These advances facilitate explicit quantitative analysis, providing rigorous and computable security and indistinguishability guarantees.

Optimization over PPT relaxations enables analytically tractable bounds on the LOCC distinguishability bias for composite protocols, yielding resource-efficient and implementation-friendly blueprints for experimental hiding schemes in low-dimensional, multipartite, or continuous-variable platforms.

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Group-symmetric data hiding states thus embody a fundamental paradigm in quantum information science: the deliberate deployment of symmetry to control—and restrict—information accessibility, yielding protocols with strong nonclassicality, operational significance in secure multi-party computation, and deep connections to the structure of measurement, entanglement, and quantum theory itself.