Encrypted clones can leak: Classification of informative subsets in Quantum Encrypted Cloning

Abstract: Encrypted cloning enables the redundant storage of an unknown qubit while remaining compatible with the no-cloning theorem, since only one clone can later be recovered through key-consuming decryption. Because encryption in this protocol is introduced to enable cloning-compatible redundancy rather than to guarantee confidentiality by design, its secrecy properties must be assessed explicitly. Here we classify the subsets of the encrypted-clone storage register into authorized, completely non-informative, and partially informative sets. We show that intermediate non-authorized subsets may retain only a restricted residual dependence on the input state, and we characterize exactly when this dependence occurs. The resulting leakage pattern is parity-dependent, revealing a structural confidentiality limitation of encrypted cloning.

• The paper introduces a classification of leakage in quantum encrypted cloning, showing that some unauthorized subsets reveal partial information due to parity effects.
• It systematically analyzes n=1, 2, and 3 cases to confirm that missing a complete signal-noise pair results in non-informativeness, while aligned subsets follow parity rules.
• The study underscores implications for secure quantum storage, with unauthorized subsets having an odd number of signal qubits leaking the y-component of the input state.

Information Leakage Structures in Quantum Encrypted Cloning: Parity-Dependent Patterns

Introduction

The paper "Encrypted clones can leak: Classification of informative subsets in Quantum Encrypted Cloning" (2604.10155) critically examines the confidentiality properties of quantum encrypted cloning, a protocol originally motivated by the need for redundant—but non-copyable—storage of unknown quantum states. The authors emphasize that while the protocol is compliant with the no-cloning theorem and ensures recoverability only for authorized subsets, the encryption mechanism does not guarantee all-or-nothing secrecy. Through systematic analysis, the paper provides an exhaustive classification of information leakage across all possible subsets of the storage register, uncovering subtle parity-based dependencies in the leakage pattern.

Encrypted Cloning: Protocol and Problem Statement

Encrypted cloning leverages a Pauli-based unitary transformation acting on an input qubit and a set of Bell pairs. The goal is to encode the information redundantly such that only a specific subset ("authorized") of the storage register—including one complete signal-noise pair and at least one qubit from all remaining pairs—allows perfect recovery. Notably, this redundancy is made compatible with the no-cloning theorem via quantum encryption; however, encryption's primary function here is not confidentiality per se.

The central question is whether all unauthorized subsets (i.e., those not satisfying the authorization threshold for recovery) are equally uninformative, and to what degree certain subsets may retain partial information about the input state. This distinction is not merely theoretical: it impacts the practical deployment and architectural choices for quantum storage systems based on encrypted cloning.

Formal Criteria for Informativeness

Subsets of the storage register are evaluated in terms of their dependence on the input state $\ket{\psi}$. A subset is called completely uninformative if its reduced state is independent of $\ket{\psi}$. Partial informativeness (leakage) occurs if the reduced state retains a residual, but nontrivial, dependence on $\ket{\psi}$, even if full recovery is impossible.

A key proposition established is that any subset missing an entire signal-noise pair is always completely uninformative, regardless of the subset's size or composition. For other subsets, particularly those containing exactly one qubit from each signal-noise pair, a more nuanced, parity-based analysis is required.

Explicit Analysis in Low-Dimensional Cases

The authors work through the cases of $n=1,2,3$ encrypted clones to expose the mechanism of leakage:

• $n=1$: The single signal qubit subset $\{S_1\}$ is partially informative—it reveals the $y$-component of the input state's Bloch vector, while the noise qubit $\{N_1\}$ is completely uninformative.
• $n=2$: All unauthorized subsets with one qubit from each pair ($|B|=2$) are completely uninformative; all Bloch-dependent terms cancel.
• $\ket{\psi}$0: Subsets containing an odd number of signal qubits (and none missing a complete pair) exhibit partial informativeness, leaking a function of the $\ket{\psi}$1-Bloch component. Subsets with an even number of signal qubits remain uninformative.

This pattern suggests leakage is not generic but is instead governed by global parity conditions.

General Classification: Parity-Induced Leakage Patterns

The main result generalizes the observations to arbitrary $\ket{\psi}$2 using algebraic decomposition of the encoded state's reduced density matrices. The analysis demonstrates:

• Missing a complete pair: Any subset missing even a single signal-noise pair reveals no information (completely non-informative).
• Aligned subsets with $\ket{\psi}$3 (one qubit per pair):
  • If $\ket{\psi}$4 is even, all such subsets are completely uninformative.
  • If $\ket{\psi}$5 is odd, only those subsets with an odd number of signal qubits are partially informative; their reduced state encodes the $\ket{\psi}$6-component of the input state, explicitly:

  $\ket{\psi}$7

  for $\ket{\psi}$8 the $\ket{\psi}$9-component of the Bloch vector, and $\ket{\psi}$0 the Pauli matrix.

• Authorized subsets ($\ket{\psi}$1 meeting the protocol's pair overlap condition): These allow full recovery.

The emergence of parity-dependent leakage constitutes a structural limitation on confidentiality: for some specific, unauthorized subsets, information proportional to the $\ket{\psi}$2-Bloch component is available, rather than secrecy being uniformly maximal.

Confidentiality and Architectural Implications

The findings highlight that encrypted cloning cannot be assessed solely based on recoverability or the restriction on single-clone decryption. Instead, the finer structure of leakage means that confidentiality may fail in a sharply delineated, subset-dependent fashion. For practical quantum storage, this result implies the physical or logical co-localization of certain qubit types (signal or noise) and their arrangement may impact the risk profile for confidentiality leakage, particularly in architectures where qubit subsets are selectively exposed.

Theoretical and Future Directions

The presence of parity-dependent residual leakage suggests several directions:

• Informativeness with Source Qubit Inclusion: The authors propose extending the classification to subsets involving the input register $\ket{\psi}$3 to complete the analysis of possible leaks.
• Higher-Dimensional and Generalized Protocols: A deeper investigation into the structure of leakage in the setting of generalized (e.g., $\ket{\psi}$4-dimensional) quantum states as developed in (Ceară, 6 Apr 2026) could reveal whether the observed patterns persist or whether additional structures emerge.
• Connection to Quantum Secret Sharing: The access structure observed has analogies to quantum secret sharing, but also distinct features due to the non-binary classification (fully, partially, or non-informative) and the source of leakage.

Conclusion

This work provides a rigorous, explicit classification of information leakage in quantum encrypted cloning as a function of subset structure and parity, establishing that confidentiality is not absolute and exhibits parity-dependent exceptions. The analysis exposes the necessity of considering not just recoverability but also partial informativeness when evaluating the security of quantum memory protocols. The results have direct implications for secure quantum storage architecture and motivate further exploration into multidimensional extensions and related quantum cryptographic primitives.