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arXiv:2609.28241

Beyond Maximal Entanglement: Exact Resources for Multiparty Encrypted Quantum Cloning

Pritam Roy, Shashank Gupta

quant-phSubmitted Sep 23, 2026• Announced Sep 24, 2026• 0 citations
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Abstract

Encrypted quantum cloning distributes an unknown $k$-qubit state among $m$ encrypted clones so that no individual clone reveals the input, yet the state can be recovered from any one clone together with a common quantum key. We ask the inverse question: for a fixed encoding architecture, which multipartite pure states can serve as exact resources for this task? For $m\ge2$, we completely characterize the pure resources compatible with a sector-wise two-Pauli encoder, with necessity holding for arbitrary completely positive trace-preserving (CPTP) recovery maps. For even $m$, exact recovery requires maximal entanglement across the signal--noise cut. For odd $m$, less entanglement can suffice, provided that the surviving signal correlations have the structure selected by the encoder. We further show, without fixing the encoder, that exact recovery from every authorized subsystem already implies perfect concealment of each individual signal and requires at least $(m-1)k$ ebits of signal--noise entanglement. For graph states, the resource classification reduces to an exact condition on the kernel of the signal--noise cut matrix, leading to binary certification and constructive Clifford recovery. We identify rank-deficient graph resources that attain the architecture-independent entanglement bound and prove that the entire Dicke family, including $W$ states, is excluded for every sector-wise two-Pauli encoder. Our results show that encrypted recovery depends not only on how much entanglement a resource contains, but also on how its correlations are organized relative to the encoder.

Comments
17 pages (double column) + 28 pages (single column), 7 figure; Comments are welcome

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Encrypted quantum cloning distributes an unknown kkk-qubit state among mmm encrypted clones so that no individual clone reveals the input, yet the state can be recovered from any one clone together with a common quantum key. We ask the inverse question: for a fixed encoding architecture, which multipartite pure states can serve as exact resources for this task? For m≥2m\ge2m≥2, we completely characterize the pure resources compatible with a sector-wise two-Pauli encoder, with necessity holding for arbitrary completely positive trace-preserving (CPTP) recovery maps. For even mmm, exact recovery requires maximal entanglement across the signal--noise cut. For odd mmm, less entanglement can suffice, provided that the surviving signal correlations have the structure selected by the encoder. We further show, without fixing the encoder, that exact recovery from every authorized subsystem already implies perfect concealment of each individual signal and requires at least (m−1)k(m-1)k(m−1)k ebits of signal--noise entanglement. For graph states, the resource classification reduces to an exact condition on the kernel of the signal--noise cut matrix, leading to binary certification and constructive Clifford recovery. We identify rank-deficient graph resources that attain the architecture-independent entanglement bound and prove that the entire Dicke family, including WWW states, is excluded for every sector-wise two-Pauli encoder. Our results show that encrypted recovery depends not only on how much entanglement a resource contains, but also on how its correlations are organized relative to the encoder.
We investigate noise propagation in encrypted quantum cloning by formulating the protocol in terms of effective recovery channels. For local single-qubit completely positive trace-preserving noise, a Pauli-basis expansion of the Kraus operators reveals a strong dependence on the location of the physical channel. Noise on an auxiliary signal qubit leaves the reduced recovered state unchanged, whereas noise on an auxiliary key qubit induces a Pauli-diagonal effective channel on the recovered qubit. For the core qubits, the error propagation additionally depends on the parity of the number of entangled pairs. In particular, for noise on the core key qubit with odd parity, distinct physical Pauli sectors merge into the same effective sector, converting isotropic depolarizing noise into an anisotropic recovery channel. We also relate Werner-type resource imperfections to local depolarizing noise and distinguish distributed product depolarization from genuine global depolarization. The results demonstrate that the effective recovery noise is governed not only by noise strength, but also by its physical location and by the structure of the encryption-decryption protocol.
Quantum encrypted cloning shows that an unknown quantum state can be distributed into multiple encrypted copies without contradicting the no-cloning theorem: each copy is unusable on its own, but can be redeemed together with a suitable quantum key. Recent work has related canonical encrypted-cloning protocols to particular forms of quantum secret sharing. Here we take the converse perspective: instead of mapping a given encrypted-cloning protocol into QSS, we use QSS access structures as a design library from which encrypted-cloning schemes can be extracted. The criterion is access-structural. A QSS scheme supports a quantum encrypted-cloning structure whenever it contains a family of qualified sets with a non-qualified common intersection. The common subsystem is interpreted as the key, while the non-common parts are interpreted as encrypted clones relative to that key. Thus quantum encrypted cloning does not require a new notion of recoverability beyond QSS; what changes is the operational reading of QSS constituents as a mechanism for delayed and alternative redemption opportunities. This viewpoint separates redemption from perfect secrecy. Perfect QSS yields encrypted-cloning schemes with forbidden non-qualified subsystems, whereas ramp QSS naturally allows intermediate, partially informative non-redeeming subsystems. The resulting framework broadens quantum encrypted cloning from a specific protocol to a general access-structure primitive. We illustrate the extraction principle with threshold-like, ramp, hierarchical, and compartmented architectures, showing how encrypted clones may be symmetric or asymmetric, individual or composite, perfectly hidden or leaky. Equivalently, these constructions can be viewed as overlapping erasure-recovery regions of an isometric quantum code. This establishes secret sharing as a systematic design language for encrypted quantum redundancy.
Quantum encrypted cloning, introduced by Yamaguchi and Kempf, is a Pauli-based protocol that distributes an unknown input qubit into multiple encrypted signal-noise pairs in such a way that redundancy is created without violating the no-cloning theorem, since at most one clone can later be perfectly recovered through an appropriate decoding procedure. In previous work we showed that unauthorized subsets of the storage register are not, in general, completely uninformative, and we identified a parity-dependent leakage pattern. In the present work we extend the analysis to subsets that also include the transformed source qubit A. Exploiting the purity of the global encoded state and the complementarity between storage-only subsets and subsets containing A, we derive a full classification of the informativeness of all sets of the form H={A}∪CH=\{A\}\cup CH={A}∪C. We show that these subsets are fully informative in the generic case. Two exceptions arise. First, if all pairs are incomplete and |C|<n, then the reduced state is completely uninformative. Second, if |C|=n, n is odd, and the number q of signal qubits in C is even, then the reduced state is partially informative. In this latter case, the residual dependence on the input state is confined to the y-component of the Bloch vector. These results provide a complete parity-based characterization of leakage for subsets containing the transformed input qubit.
The no-cloning theorem prohibits the creation of identical copies of quantum information, imposing fundamental constraints on quantum technologies. A recently proposed protocol, encrypted cloning, introduced by Yamaguchi and Kempf, showed that perfect qubit clones can be produced if they are simultaneously encrypted with a single-use key. They also observed a connection between this scheme and quantum secret sharing (QSS). However, it remained an open question whether encrypted cloning could be generalised to arbitrary dimensions, and the broader relationship between the two schemes had not been formally established. In this work, we address both questions by framing encrypted clones as Absolutely Maximally Entangled (AME) states. In parallel with recent work by Ceará that utilises Zadoff-Chu sequences, we independently develop a complementary framework for arbitrary dimensions based on Weyl-Heisenberg displacement operators, both tracing back to the original qubit construction by Yamaguchi and Kempf. We analytically compute the encrypted state and prove that an encrypted qudit system comprising two signal-noise qudit pairs is equivalent to a five-party AME state in any dimension, provided the input state is uniform. We then formalise the connection to QSS by proving that a threshold QSS scheme can achieve the fundamental objectives of encrypted cloning, establishing QSS as the natural general framework within which encrypted cloning can be contextualised.
Quantum encrypted cloning encodes an unknown state into several encrypted components, each offering an alternative way to recover it later. We show that this redundancy can also serve for one-shot fault diagnosis: instead of inspecting the encrypted state, we measure relational Pauli observables testing consistency conditions imposed by the encoding. The canonical Yamaguchi--Kempf scheme encodes the input into nnn signal--key pairs, all carrying the same coherent Bell label, plus a transformed copy of the input qubit. Comparing the labels of two pairs yields a deterministic check that localizes an anomalous pair and identifies its Pauli-error class without revealing the label. Such checks cannot tell whether the signal or the key of that pair is faulty, and no measurement on the pairs can (we prove that they already generate the whole group of deterministic state-blind observables supported there, of rank 2(n−1)2(n-1)2(n−1)). This ambiguity matters: a faulty signal costs one redemption path, a faulty key threatens them all. However, retaining also the transformed input qubit contributes exactly two further independent checks, at any multiplicity, and these suffice to identify any single fault drawn from the single-qubit Pauli set. For three clones this gives six checks in all, and we prove that no smaller set of state-blind observables achieves the same resolution. Across clone multiplicities, parity governs how many components must be measured jointly to reach this resolution, not the resolution itself. These findings are then abstracted into a general framework --- state-blind observables, deterministic healthy references, syndrome-induced fault partitions, redemption-oriented sufficiency --- yielding \textit{relational diagnosability}: a syndrome need not identify every fault, only enough of it to select a safe redemption path.
We determine the preshared entanglement required for spatially separated parties, restricted to local operations and classical communication, to attain globally optimal probabilistic two-copy purification of arbitrary bipartite pure states under depolarizing noise. In every local dimension d≥2d\ge 2d≥2, one shared maximally entangled qubit pair suffices: local controlled-SWAP operations exactly reproduce the globally optimal successful transformation. Conversely, any finite-dimensional preshared resource state that attains the same benchmark, even under a positive-partial-transpose relaxation, must have entanglement of formation at least one ebit. For pure resources with one ebit, or for two-qubit resources including mixed states, attaining the benchmark is possible only for states with exactly two nonzero Schmidt weights, both equal to 1/21/21/2, namely those locally unitarily equivalent to a maximally entangled qubit pair. These findings provide a benchmark for evaluating the entanglement demands of noise management in quantum networks and modular quantum computers.