"- INTRODUCTION

Quantum systems are notoriously fragile: small losses or weak interactions with the outside world usually destroy quantum coherence. Since quantum information cannot be copied, any leakage of information leads to its destruction in the original system. To fully retrieve it, one usually needs full control over the environment. The loss of coherence is at the heart of quantum information, whether we want to fight it or impose it to an adversary, but it plays an important role in a broader area of physics, including thermodynamics, quantum control, and black hole physics [1]. Among the strategies devised to try and overcome this fragility are quantum error correcting (QEC) codes and quantum secret sharing (QSS) schemes. Secret sharing is an important primitive for mutlipartite cryptography, used for example in electronic voting, byzantine agreement and secure multiparty computation. The original classical protocol was introduced by Shamir in the seventies [2], and the first quantum versions appeared in a series of works about twenty years ago [3, 4]. In QSS schemes, a dealer delocalizes the information between several players, so that only authorized subsets of them (access parties) can fully reconstruct the original information without the shares of the other players. Unuathorized sets on the other hand get in principle no information about the secret. It was shown for qubits that QSS schemes are equivalent to erasure correcting codes [5], protecting against loss of part of the system (the complement of any authorized set). As most ECC codes, QSS schemes are highly structured. However, random codes exist that have been proven to optimally protect the state of a set of qubits from erasure errors [6]. Their randomness makes them a natural model in a variety of physical scenarios where information is lost. Alternative to qubits, information can be encoded in the state of infinite dimensional quantum systems, known as continuousvariable (CV) systems. The name comes from the existence of observables with continuous spectra, such as position and momentum, here referred to as quadratures, as is customary in quantum optics. CV systems are of great practical importance in quantum technologies: the possibility to experimentally generate entanglement in a deterministic fashion makes them interesting candidates for the realization of quantum communication and computation protocols. Several CV generalizations of QSS [7] and erasure-correcting codes [8] have been proposed, and some have been experimentally demonstrated [9]. Each of these schemes, however, requires encoding the secret in carefully chosen states. No CV random code has been proposed to date. This gap poses serious limitations to the experimental realization of CV-QSS. For example, unless the experimental setup is specifically tailored for the task, CV-QSS could not be carried out, or experimental imperfections might hinder its implementation.

- RESULTS

We fill this gap by introducing a form of random coding for CV. Namely, we show that QSS can be implemented in bosonic systems mixing a secret state with squeezed states, the workhorse of CV quantum information, through almost any energy preserving transformation. The latter correspond to passive interferometers in the optical setting. Squeezed states are obtained from vacuum by reducing the noise in one quadrature (e.g. momentum) while simultaneously increasing the noise in the conjugated one (e.g. position). We show that for almost any passive transformation there exists a decoding scheme,that each authorized set can construct efficiently, such that the secret can be recovered to arbitrary precision, provided the initial squeezing is high enough. The decoding only requires Gaussian resources, considered relatively easy to implement experimentally. We show that in the optical case, decoding can be achieved by interferometry, homodyne detection and a of single mode squeezers independent of the number of players. We stress that our results follow from simple linear algebra and general considerations on the number of modes. Our approach also generalizes earlier proposals by allowing the secret to be an arbitrary multimode state, as long as enough players are considered.

These results have immediate experimental and technological applications. Indeed, they imply that almost any experimental setup involving squeezed states can beused for QSS. Moreover, small deviations of the setup from a theoretical target one are not important, as long as they can be characterized. This opens the possibility to share resource states securely over a network of CV systems with arbitrarily distributed entanglement links, which may pave the way to server-client architectures for CV quantum computation. But the relevance of CVrandom codes is not limited to their practicality. The optimality of random erasure correcting codes for qubits was used in a seminal paper to estimate the rate of information leakage from black holes through Hawking radiation [1]. The most relevant objects in this setting are however fields, namely CV systems. This stimulated work applying CV techniques, notably related to QSS, in relativistic contexts [10]. The existence of efficient CV random EC codes may open new avenues for tackling the black hole information puzzle and related fundamental questions.

- REFERENCES

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[4] R. Cleve, D. Gottesman, and H.-K. Lo, Phys. Rev. Lett. 83, 648 (1999).

[5] D. Gottesman, Phys. Rev. A 61, 042311 (2000).

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[7] T. Tyc and B. C. Sanders, Phys. Rev. A 65, 042310 (2002).

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[10] P. Hayden, S. Nezami, G. Salton, and B. C. Sanders, New Journal of Physics 18, 083043 (2016)."