Approximate Quantum Information Masking

• AQIM is the approximate relaxation of quantum masking, replacing exact identical marginals with near-indistinguishability measured by fidelity or trace-norm criteria.
• Key methodologies involve analyzing subsystem variations, randomness cost, and the redistribution of information into global correlations through entanglement.
• AQIM establishes links between no-go theorems and approximate quantum error correction, influencing both bipartite and multipartite quantum system designs.

Approximate Quantum Information Masking (AQIM) is the approximate relaxation of quantum information masking: an encoding task in which information about an input quantum state is hidden from local subsystems and retained only in global correlations. In the exact bipartite setting, masking requires the reduced states of each subsystem to be independent of the encoded input, but universal exact masking of arbitrary quantum states is impossible; AQIM replaces exact equality of marginals by approximate indistinguishability, typically through fidelity-, trace-norm-, or inaccuracy-based criteria (Modi et al., 2016, Li et al., 2019, Li et al., 25 Jul 2025). The subject sits at the intersection of no-go theorems, entanglement structure, randomness cost, multipartite correlation theory, and approximate quantum error correction, with recent work showing that AQIM is not a single definition but a family of closely related notions (Li et al., 25 Jul 2025).

1. Formal definition and operational criteria

In the standard exact formulation, a masker maps a family of input states $\{|a_k\rangle\}$ to bipartite output states $\{|\Psi_k\rangle_{AB}\}$ such that neither local subsystem reveals the input label. Operationally, this means

$$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right),\qquad \rho_B=\operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right),$$

with $\rho_A$ and $\rho_B$ independent of $k$ (Modi et al., 2016). The information is therefore not destroyed; it is redistributed into correlations.

AQIM relaxes exact equality. A standard bipartite definition requires the local marginals for different inputs to be approximately indistinguishable, for example through

$$F(\rho_{A|k},\rho_{A|k'})\ge 1-\epsilon,\qquad F(\rho_{B|k},\rho_{B|k'})\ge 1-\epsilon,$$

for all $k,k'$, where $F$ is quantum fidelity (Li et al., 2019). In this sense, AQIM measures how much local leakage remains when exact masking is unattainable.

More recent work organizes AQIM through several figures of merit rather than a single $\epsilon$-criterion. For a code or image subspace $\{|\Psi_k\rangle_{AB}\}$0, the literature introduces maximum and average subsystem variation, along with masking inaccuracy relative either to the average encoded state $\{|\Psi_k\rangle_{AB}\}$1 or to the maximally mixed state (Li et al., 25 Jul 2025). The same work shows that these notions are equivalent up to constants: small inaccuracy implies small variation, and small variation implies small inaccuracy up to a factor of $\{|\Psi_k\rangle_{AB}\}$2 (Li et al., 25 Jul 2025). In multipartite language, the exact target becomes $\{|\Psi_k\rangle_{AB}\}$3-uniform masking, where every $\{|\Psi_k\rangle_{AB}\}$4-party reduction is input-independent, and the approximate version is an $\{|\Psi_k\rangle_{AB}\}$5-approximate $\{|\Psi_k\rangle_{AB}\}$6-uniform state or code (Li et al., 25 Jul 2025).

2. Exact masking as the baseline for AQIM

AQIM is defined against a sharply constrained exact theory. The no-masking theorem states that an arbitrary quantum state cannot be masked in bipartite systems; this holds first for qubits and then for all finite dimensions (Modi et al., 2016). At the same time, exact masking is not empty. Certain restricted continuous families of nonorthogonal pure states can be masked, and for the canonical isometry $\{|\Psi_k\rangle_{AB}\}$7 the maximal maskable family consists of states with a fixed amplitude profile and arbitrary phases. The same exact literature shows that commuting subsets of mixed states can be masked by an isometry $\{|\Psi_k\rangle_{AB}\}$8, whereas all mixed states cannot be masked by any operator (Cao et al., 2020).

The boundary of exact maskability is particularly clear for real states. Real ququart states can be completely hidden in bipartite correlations of two-qubit hybrid entangled states, and the set of real density matrices is a maximal maskable set: any superset of the real density matrices cannot be masked (Zhang et al., 2021). Mixed-state masking work further showed that simultaneously maskable states lie on hyperdisks in the state hypersphere and strictly contain the broadcastable states (Liu et al., 2020). These exact results supply the ideal benchmark for AQIM: local states should be nearly input-independent, while the full state should retain recoverable information in correlations.

Exact masking also has a multipartite escape route. All $\{|\Psi_k\rangle_{AB}\}$9-level states can be masked into tripartite systems of local dimension $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right),\qquad \rho_B=\operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right),$$0 or $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right),\qquad \rho_B=\operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right),$$1 using mutually orthogonal Latin squares (Li et al., 2019). A later construction proved that all $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right),\qquad \rho_B=\operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right),$$2-level quantum states can be masked into $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right),\qquad \rho_B=\operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right),$$3-qudit systems with $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right),\qquad \rho_B=\operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right),$$4, local dimension $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right),\qquad \rho_B=\operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right),$$5, and an upper bound tighter than the quantum Singleton bound (Shang et al., 2022). This makes AQIM especially significant in bipartite or dimension-constrained regimes, where exact masking is impossible or structurally too restrictive.

3. Approximate and probabilistic relaxations

The first systematic AQIM bounds show that allowing approximation does not remove the no-go structure. Universal approximate masking has a nonzero error floor: the local marginals cannot be made arbitrarily close for all input states, so some information necessarily leaks into the subsystems (Li et al., 2019). In the qubit case, the best local fidelity cannot exceed about $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right),\qquad \rho_B=\operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right),$$6 (Li et al., 2019). The same paper proves that probabilistic $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right),\qquad \rho_B=\operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right),$$7-approximate masking obeys the same lower bound, so permitting failure does not improve the fundamental universal limit (Li et al., 2019).

A nearby operational relaxation is probabilistic exact masking. Deterministic masking is possible for mutually orthogonal states, while probabilistic masking is possible for linearly independent states by a general unitary-reduction operation followed by postselection (Li et al., 2019). For two initial states, the maximal successful probability can be explicitly bounded in terms of input and target overlaps (Li et al., 2019). This is not AQIM in the strict quantitative sense, because successful runs remain exactly masked, but it broadens the set of maskable inputs by giving up certainty rather than precision.

Related robustness questions have also been pursued outside the standard Hermitian setting. In non-Hermitian quantum systems, mutually orthogonal states can be deterministically masked, arbitrary states still cannot be universally masked, and deterministic and probabilistic masking were analyzed under Pauli, Weyl, and global depolarizing noise channels; the same work proposed $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right),\qquad \rho_B=\operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right),$$8-uniform probabilistic quantum information masking in multipartite systems (Lv et al., 2022). This is not a core AQIM framework, but it situates approximate or imperfect masking within a broader family of noisy and generalized masking problems.

4. Geometry, entanglement, and randomness cost

The geometry of exact maskability strongly informs AQIM. For qubit input spaces, the set of maskable states is either a two-dimensional hyperdisk or a set of two states (Ding et al., 2019). In higher dimensions, maskable sets can be unions of multiple hyperdisks, and in completely degenerate cases even infinitely many hyperdisks can arise (Ding et al., 2019). At the same time, a stronger general geometric conjecture about unitarily maskable states was later disproved, and the algebraic analysis shifted attention from simple “disk” pictures to information-theoretic structure (Lie et al., 2019). This suggests that AQIM should not be understood as a mere thickening of a single geometric object, although the exact hyperdisk picture remains a useful local guide.

Entanglement is not incidental in this structure. For arbitrary two-state masking in the qubit setting, whether the states are commuting or non-commuting and whether they are pure or mixed, the masked states remain entangled unless the input is an equal mixture of the two pure states (Saha et al., 2023). In the formulations studied there, entanglement is indispensable for masking an arbitrary set of two single-qubit states (Saha et al., 2023). For AQIM, a plausible implication is that approximation can relax exact equal-marginal constraints, but it does not remove the central role of entanglement-mediated correlations.

Masking also has a resource cost. The randomness-cost analysis models a universal masker as an isometry fed with a safe state whose entropy is the randomness cost, and proves that masking quantum information is impossible without randomness as a resource (Lie et al., 2019). The same work derives lower bounds controlled by the evenness of information distribution across output subchannels and shows that more informative subchannels must be more strongly suppressed in the random mixture (Lie et al., 2019). Importantly for AQIM, these bounds are robust to incompleteness of quantum masking (Lie et al., 2019).

5. Random AQIM, multipartite asymmetry, and approximate QEC

The most explicit AQIM framework to date studies random isometries as approximate maskers. It introduces several AQIM notions, proves profound intrinsic connections among them, and uses figures of merit to quantify deviation from exact masking (Li et al., 25 Jul 2025). The central result is asymmetric. In bipartite systems, a fundamental lower bound exists for a key figure of merit, and almost all random isometries fail to realize AQIM; this is formulated as a no-random-AQIM theorem for bipartite systems (Li et al., 25 Jul 2025). In particular, the expected variation of a random bipartite masking subspace is bounded below by $$\rho_A=\operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right),\qquad \rho_B=\operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right),$$9, and the probability of obtaining a $\rho_A$0-approximate masker with $\rho_A$1 is exponentially small (Li et al., 25 Jul 2025).

In multipartite systems the conclusion reverses. Almost all random isometries can realize AQIM, and the failure probability is exponentially small in the ambient dimension (Li et al., 25 Jul 2025). The number of physical qubits required to randomly mask a single logical qubit scales only linearly (Li et al., 25 Jul 2025). This probabilistic success in the multipartite regime mirrors the exact multipartite constructions of Latin-square and maximum-entangled-basis type, but now in an approximate and high-dimensional random setting.

A further consequence is the link to approximate quantum error correction. Under certain conditions, approximate quantum error correction is equivalent to AQIM (Li et al., 25 Jul 2025). For replacement noise on up to $\rho_A$2 parties, the relationship is quantified by

$\rho_A$3

where $\rho_A$4 is the subsystem variance or masking inaccuracy of the code and $\rho_A$5 is the QEC inaccuracy (Li et al., 25 Jul 2025). Consequently, AQIM gives rise to approximate quantum error correction codes with constant code rates and exponentially small correction inaccuracies (Li et al., 25 Jul 2025). This places AQIM within the same operational family as approximate decoupling and local-noise correctability.

6. Experiments, applications, and interpretive issues

Experimental work predates the recent formal AQIM framework and provides practical benchmarks rather than AQIM definitions. A photonic implementation extending masking to mixed states reported a mean fidelity of reconstruction of $\rho_A$6, an average total absolute spectra error of $\rho_A$7, and trace distances between Alice’s and Bob’s marginals around $\rho_A$8 and $\rho_A$9 (Liu et al., 2020). These are experimental errors and finite-precision deviations from ideal exact masking, not a foundational AQIM criterion (Liu et al., 2020).

A photonic quantum-walk realization of masking for real ququart states reported an experimental fidelity of about $\rho_B$0, while the hidden information could be faithfully retrieved with a fidelity of about $\rho_B$1 from correlation measurements (Zhang et al., 2021). The same experiment showed that introducing a genuinely complex component causes the observed correlations to deteriorate, illustrating in practice that arbitrary quantum states cannot be perfectly masked (Zhang et al., 2021). On IBM hardware, restricted bipartite masking of selected states was demonstrated with fidelities $\rho_B$2, $\rho_B$3, and $\rho_B$4 for local-state comparisons, whereas arbitrary bipartite states failed with $\rho_B$5 and $\rho_B$6; a tripartite implementation yielded subsystem fidelities $\rho_B$7, $\rho_B$8, and $\rho_B$9 (Ghosh et al., 2019). These experiments function as empirical approximations to exact protocols and therefore as operational reference points for AQIM.

A recurrent interpretive issue concerns what is being masked. The no-masking theorem applies to arbitrary unknown quantum states, but a later comment argued that this does not imply qubit commitment is impossible in general, because commitment protocols concern known states whose descriptions are classical data (He, 2023). In that view, one can mask the classical description of a known state by encoding bits in Bell-state correlations, even though this is not masking the quantum amplitudes themselves (He, 2023). The distinction matters for AQIM as well: task statements depend on whether the object to be hidden is the state itself, a preparation description, or a hybrid classical-quantum record.

The broader masking program continues to expand beyond strict AQIM. Exact masking states have been analyzed under entanglement swapping for cryptographic redistribution of hidden correlations (Ji et al., 2021), and multipartite exact maskers have been constructed in the Kitaev model using Abelian and Ising anyons, with braiding and circling acting as masking-preserving operations on extended hyperdisks in anyonic space (Shen et al., 2024). These works are not AQIM in the strict technical sense, but they reinforce the same core theme: locality can be made approximately or exactly uninformative only by transferring information into a sufficiently structured pattern of nonlocal correlations.