Approximate Quantum Information Masking

• Approximate quantum information masking is a quantitative framework that relaxes strict state hiding by requiring high, but not perfect, similarity in local reduced states.
• The methodology uses fidelity and trace distance metrics along with random isometries to gauge and control the unavoidable leakage in bipartite and multipartite systems.
• Practical implications include improved designs for approximate quantum error correction codes and experimental protocols that achieve high retrieval fidelities in complex quantum architectures.

Approximate quantum information masking is the relaxation of quantum information masking in which an encoding no longer requires the local reduced states to be exactly identical for all allowed inputs, but only close according to specified figures of merit. In the standard masking task, a physical process encodes a quantum state into a composite system so that no local subsystem reveals the input and the information is present only in correlations. Exact universal masking is impossible in bipartite systems, and approximate formulations quantify how much local leakage is unavoidable, how this depends on dimension and architecture, and when multipartite encodings or random isometries can nevertheless realize high-quality masking (Modi et al., 2016, Li et al., 2019, Li et al., 25 Jul 2025).

1. Formal definitions and approximation criteria

In the exact bipartite setting, a masker is a physical process $S$ that maps input states $\{|\phi_k\rangle_A\}$ to bipartite states $\{|\Psi_k\rangle_{AB}\}$ such that the reduced states of each subsystem are independent of the label $k$: $$\rho_A^{(k)} = \operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right), \qquad \rho_B^{(k)} = \operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right).$$ Operationally, the information about the input must be hidden entirely in the correlations of the joint state rather than in either local marginal (Modi et al., 2016).

A standard approximate relaxation is $\epsilon$-approximate masking. In that formulation, one replaces exact equality of marginals by a fidelity requirement: $$F(\rho_{A|k},\rho_{A|k'}) \ge 1-\epsilon,\qquad F(\rho_{B|k},\rho_{B|k'}) \ge 1-\epsilon,$$ for all $k,k'$. Using the standard relation between fidelity and trace distance, this implies

$$\|\rho_{X|k}-\rho_{X|k'}\|_1 \le 2\sqrt{2}\,\epsilon,\qquad X\in\{A,B\}.$$

Thus approximate masking is a controlled leakage model rather than a categorical hiding condition (Li et al., 2019).

A more recent formulation introduces several notions of approximate quantum information masking (AQIM) for a code subspace $C\subset H_{B_1\cdots B_m}$. For subsystem $\{|\phi_k\rangle_A\}$0, the maximal subsystem variation is

$\{|\phi_k\rangle_A\}$1

and the order-$\{|\phi_k\rangle_A\}$2 quantity is

$\{|\phi_k\rangle_A\}$3

The same work also defines inaccuracy relative to the average reduced state $\{|\phi_k\rangle_A\}$4, and proves that variation and inaccuracy are equivalent up to factors of $\{|\phi_k\rangle_A\}$5. In particular, the maximal-version notion implies the average-version notion, and the distance to the average reduced state controls pairwise distinguishability, and vice versa (Li et al., 25 Jul 2025).

2. Exact no-go theorems and the necessity of approximation

The no-masking theorem establishes that no masker can mask all states of a qubit, and more generally that an arbitrary finite-dimensional quantum state cannot be masked in the exact bipartite sense. The obstruction is linearity: if two basis states are mapped to bipartite outputs with identical marginals, then arbitrary superpositions generate cross terms in the partial traces that cannot vanish for all coefficients without forcing trivial or locally swappable encodings (Modi et al., 2016).

Allowing postselected success does not remove this obstruction. A probabilistic masker, modeled by a completely positive trace-decreasing map that succeeds with probabilities $\{|\phi_k\rangle_A\}$6, is still impossible in the universal setting: conditioned on success, the normalized outputs would have to mask perfectly, and the same superposition constraints reappear. The result is that a probabilistic masker that can mask all states in $\{|\phi_k\rangle_A\}$7 is impossible (Li et al., 2019).

Approximation weakens the theorem, but only partially. Universal approximate masking is subject to a dimension-dependent lower bound on the achievable masking error $\{|\phi_k\rangle_A\}$8, so the leakage cannot be made arbitrarily small. The same lower bound remains valid for probabilistic approximate masking. For qubits, the bound implies that the local fidelity cannot exceed about $\{|\phi_k\rangle_A\}$9, which quantifies how much information is inevitably visible in the marginals of any universal masker (Li et al., 2019).

These impossibility statements are sharpened by span-based results. Any informationally complete set of quantum states is antiscrambling and not maskable, and any set of states with nonzero measure is not hideable or maskable. In the qubit case, a set is maskable iff its Bloch vectors lie in a disk; equivalently, a qubit set is informationally complete iff its Bloch vectors are not contained in any disk (Zhu, 2020).

3. Exact maskable families as the baseline for AQIM

Approximate masking is most naturally understood against the exact geometry of restricted maskable sets. A canonical exact masker $\{|\Psi_k\rangle_{AB}\}$0 sends basis states to $\{|\Psi_k\rangle_{AB}\}$1 and masks families of pure states with fixed amplitudes and variable phases. Representative examples lie on a “great hyper-disk,” and more generally states of the form

$\{|\Psi_k\rangle_{AB}\}$2

with fixed amplitudes $\{|\Psi_k\rangle_{AB}\}$3 are maskable. The reduced states then have fixed spectra, so the phase information is hidden while the amplitude information remains visible locally (Modi et al., 2016, Cao et al., 2020).

This hyperdisk geometry was developed further into an exact characterization of maskable sets. For qubits, the set of maskable states is either a two-dimensional hyperdisk or exactly two pure states. For higher-dimensional systems, the set of maskable states can consist of two or more hyperdisks, and in fully degenerate cases even infinitely many hyperdisks. The maskable set is therefore geometrically thin but structurally nontrivial (Ding et al., 2019, Liu et al., 2020).

Mixed-state masking enlarges the exact landscape without restoring universality. Commuting mixed states can be simultaneously masked by an isometry, and for specific maskers the maximal maskable mixed-state sets are characterized by fixed diagonals in a preferred basis or in a Fourier-transformed basis (Cao et al., 2020). The same line of work on photonic masking states that commuting mixed states can be simultaneously masked, that broadcastable states are a proper subset of maskable states, and that the maximal maskable set for an arbitrary $\{|\Psi_k\rangle_{AB}\}$4-dimensional qudit state lies on a hyperdisk in $\{|\Psi_k\rangle_{AB}\}$5-dimensional Euclidean space (Liu et al., 2020).

A particularly important exact family is the set of real quantum states. Real density matrices are maskable, $\{|\Psi_k\rangle_{AB}\}$6 is a maximal maskable set in quantum theory, and real pure input states are mapped to maximally entangled outputs. The construction uses Hurwitz-Radon matrices satisfying Clifford-type anticommutation relations, but the required local output dimension grows exponentially with the input dimension. This gives a concrete maximal exact subtheory against which approximate relaxations are often interpreted (Zhu, 2020). Experimentally, a masking protocol for the real ququart was realized with a photonic quantum walk, and the hidden information was retrieved from correlation measurements with a fidelity of about $\{|\Psi_k\rangle_{AB}\}$7 (Zhang et al., 2021).

4. Quantitative AQIM and random isometries

Recent AQIM work reformulates masking as a quantitative property of code subspaces rather than of isolated state families. In multipartite systems, exact $\{|\Psi_k\rangle_{AB}\}$8-uniform masking requires

$\{|\Psi_k\rangle_{AB}\}$9

so no measurement on any $k$0-party subsystem reveals information about which codeword was encoded. AQIM relaxes this by bounding subsystem variation or inaccuracy rather than demanding exact equality (Li et al., 25 Jul 2025).

The random-isometry perspective leads to a sharp bipartite–multipartite dichotomy. In bipartite systems, for a random subspace $k$1, the expected masking variation obeys a nonzero lower bound: $k$2 Moreover,

$k$3

so deviations below the lower bound are exponentially unlikely. The resulting “no-random-AQIM theorem” states that if $k$4 is a random isometry from $k$5 to a bipartite Hilbert space, then the probability that $k$6 is a $k$7-approximate masker with $k$8 is exponentially small, both for the maximum and for the average subsystem variation (Li et al., 25 Jul 2025).

The multipartite case is the opposite. For a random subspace $k$9 with equal local dimensions $$\rho_A^{(k)} = \operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right), \qquad \rho_B^{(k)} = \operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right).$$0, the paper proves concentration bounds showing that all $$\rho_A^{(k)} = \operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right), \qquad \rho_B^{(k)} = \operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right).$$1-party marginals are close to maximally mixed with high probability. Under the stated scaling regimes, a random isometry is a good approximate $$\rho_A^{(k)} = \operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right), \qquad \rho_B^{(k)} = \operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right).$$2-uniform masker with exponentially high probability. For qubits, the number of physical qubits required to randomly mask $$\rho_A^{(k)} = \operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right), \qquad \rho_B^{(k)} = \operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right).$$3 logical qubits grows only linearly in $$\rho_A^{(k)} = \operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right), \qquad \rho_B^{(k)} = \operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right).$$4, not exponentially (Li et al., 25 Jul 2025).

This establishes a modern AQIM picture: randomization does not circumvent no-masking in bipartite systems, but it does so approximately in multipartite systems. The bipartite obstruction appears as a quantitative tradeoff between the two marginals, whereas multipartite success is a concentration-of-measure phenomenon on random subspaces (Li et al., 25 Jul 2025).

5. Randomness cost, information conservation, and the AQECC connection

Masking can also be studied as a resource-theoretic task requiring randomness. In that formulation, a universal masker is written as

$$\rho_A^{(k)} = \operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right), \qquad \rho_B^{(k)} = \operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right).$$5

where the “safe state” $$\rho_A^{(k)} = \operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right), \qquad \rho_B^{(k)} = \operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right).$$6 supplies the randomness, and the randomness cost is

$$\rho_A^{(k)} = \operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right), \qquad \rho_B^{(k)} = \operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right).$$7

A central identity is the information conservation law

$$\rho_A^{(k)} = \operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right), \qquad \rho_B^{(k)} = \operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right).$$8

which implies that if one output system has no information about the input, then the other system must carry all of it. The no-hiding and no-masking theorems follow immediately from this identity (Lie et al., 2019).

The same framework yields lower bounds that are robust to incomplete masking. A principal inequality is

$$\rho_A^{(k)} = \operatorname{Tr}_B\!\left(|\Psi_k\rangle\langle\Psi_k|\right), \qquad \rho_B^{(k)} = \operatorname{Tr}_A\!\left(|\Psi_k\rangle\langle\Psi_k|\right).$$9

which shows that the randomness cost is controlled not only by mean leakage but by how unevenly information is distributed between the two parties. The approximate-setting content is explicit: the channel-mixing theorem is stated for an “almost-erasure” channel with entanglement-assisted classical capacity $\epsilon$0, and the bounds shift by $\epsilon$1. The paper states that the results are robust to incompleteness of quantum masking (Lie et al., 2019).

AQIM is also connected directly to approximate quantum error correction. For a code space $\epsilon$2, the intrinsic subsystem variance

$\epsilon$3

is essentially the same as the maximum masking inaccuracy. For replacement errors on subsystems of size at most $\epsilon$4, the inaccuracy and the approximate quantum error-correction error satisfy

$\epsilon$5

Under the stated conditions, $\epsilon$6-uniform AQIM is therefore equivalent to approximate quantum error correction up to constants and square roots. Combined with the random multipartite results, this yields approximate quantum error-correcting codes with constant code rates and exponentially small correction inaccuracies (Li et al., 25 Jul 2025).

6. Multipartite architectures, noise, and experimental realizations

Exact masking becomes broadly possible once the bipartite restriction is removed. Multipartite constructions based on Fourier-phase encodings and mutually orthogonal Latin squares can mask arbitrary $\epsilon$7-level states into $\epsilon$8 parties or, when combinatorial designs exist, into tripartite systems. These are exact deterministic maskers, not approximate ones, but they define the principal architectures through which approximate and noisy variants are analyzed (Li et al., 2019).

Teleportation has also been reinterpreted as masking when the measurement apparatus is included as a quantum subsystem. In that picture, occupation probabilities and coherence are masked in two steps, and exact tripartite and four-partite maskers follow from controlled operations and entangled channels (Shang et al., 2021). Related multipartite schemes show that an arbitrary $\epsilon$9-level quantum state can be deterministically masked into an $$F(\rho_{A|k},\rho_{A|k'}) \ge 1-\epsilon,\qquad F(\rho_{B|k},\rho_{B|k'}) \ge 1-\epsilon,$$0-qudit system with local dimension $$F(\rho_{A|k},\rho_{A|k'}) \ge 1-\epsilon,\qquad F(\rho_{B|k},\rho_{B|k'}) \ge 1-\epsilon,$$1, provided $$F(\rho_{A|k},\rho_{A|k'}) \ge 1-\epsilon,\qquad F(\rho_{B|k},\rho_{B|k'}) \ge 1-\epsilon,$$2, and with input dimension bounded by

$$F(\rho_{A|k},\rho_{A|k'}) \ge 1-\epsilon,\qquad F(\rho_{B|k},\rho_{B|k'}) \ge 1-\epsilon,$$3

The paper emphasizes that this bound is tighter than the quantum Singleton bound for the corresponding quantum error-correcting-code setting (Shang et al., 2022).

Noise robustness has been studied explicitly in non-Hermitian and probabilistic masking models. Under the Pauli/Weyl and global depolarizing channels analyzed there, the identical-marginal property survives, so deterministic and probabilistic masking remain effective even though the composite state becomes less pure or less ideal. The same work introduces $$F(\rho_{A|k},\rho_{A|k'}) \ge 1-\epsilon,\qquad F(\rho_{B|k},\rho_{B|k'}) \ge 1-\epsilon,$$4-uniform probabilistic masking in multipartite systems (Lv et al., 2022).

Laboratory realizations have generally implemented exact protocols on restricted sets while producing approximately ideal data because of device imperfections. In the photonic masking machine based on a fusion gate, the experiment reported a mean retrieval fidelity of $$F(\rho_{A|k},\rho_{A|k'}) \ge 1-\epsilon,\qquad F(\rho_{B|k},\rho_{B|k'}) \ge 1-\epsilon,$$5, an average total absolute spectra error of $$F(\rho_{A|k},\rho_{A|k'}) \ge 1-\epsilon,\qquad F(\rho_{B|k},\rho_{B|k'}) \ge 1-\epsilon,$$6, average trace distances of the marginals $$F(\rho_{A|k},\rho_{A|k'}) \ge 1-\epsilon,\qquad F(\rho_{B|k},\rho_{B|k'}) \ge 1-\epsilon,$$7 for Alice and $$F(\rho_{A|k},\rho_{A|k'}) \ge 1-\epsilon,\qquad F(\rho_{B|k},\rho_{B|k'}) \ge 1-\epsilon,$$8 for Bob, and an average fidelity of the masked bipartite states with theory of $$F(\rho_{A|k},\rho_{A|k'}) \ge 1-\epsilon,\qquad F(\rho_{B|k},\rho_{B|k'}) \ge 1-\epsilon,$$9 (Liu et al., 2020). The real-ququart experiment similarly reported faithful retrieval with a fidelity of about $k,k'$0 from correlation measurements (Zhang et al., 2021). These results are best read as approximate physical realizations of exact masking protocols.

7. Scope, boundaries, and recurrent misconceptions

Approximate quantum information masking is distinct from several neighboring tasks that also use the word “masking.” One such task is classical state masking over a quantum channel, where the hidden object is the channel-state system $k,k'$1 together with the encoder’s CSI measurement outcome $k,k'$2, and the leakage criterion is

$k,k'$3

That problem concerns reliable classical communication with bounded leakage about a quantum channel state and its measurement record, not masking of arbitrary input quantum states (Pereg et al., 2021).

Another recurrent source of confusion is the masking of a known state via its classical description. The comment on the no-masking theorem argues that any known finite-dimensional quantum state can be masked by masking its classical description, for example by encoding a classical bit as

$k,k'$4

That paper explicitly does not develop an approximate masking theory, does not give trace-distance or fidelity bounds, and describes its exception as conditional or operational rather than approximate in the technical sense (He, 2023).

The modern AQIM literature therefore occupies a specific conceptual niche. It is not a repeal of the no-masking theorem, nor merely an operational hiding of classical descriptions, nor a synonym for channel-state privacy. It is a quantitative theory of how closely local marginals can be equalized while preserving recoverability of quantum information, with exact bipartite no-go results as its background, random multipartite subspaces as a major constructive mechanism, and approximate quantum error correction as one of its principal structural correspondences (Li et al., 2019, Li et al., 25 Jul 2025).