Researchers prove that a single bit of classical information can be encoded into a quantum ciphertext so that two non‑communicating attackers cannot both learn the bit, even if each attacker is later given the decryption key. In this work the authors give a concrete encryption construction and a mathematical proof that the best coordinated strategy for the attackers cannot do better than random guessing. The failure probability of any attack drops exponentially with a security parameter, and the security claim requires no computational assumptions.

What the team did is give a fully quantum construction of “uncloneable encryption” and a detailed security proof. They model the threat as two parties who share whatever quantum resources they like before the ciphertext is produced, but who are not allowed to communicate afterwards. The proof shows that, for their encoding (related to a Clifford 2‑design and a dual “six‑state” game), the probability that both attackers simultaneously guess the encrypted bit correctly approaches the value for random guessing (one half) at a rate that is exponentially small in the security parameter (written exp(−λ)).

At a high level the security rests on two quantum facts. The first is decoupling, which certifies that parts of a quantum system can be made statistically independent and so supports extraction of secure random bits. The second is the monogamy of entanglement, the physical principle that a quantum system cannot be strongly entangled with two separate systems at the same time. The authors formalise that monogamy by using strong subadditivity of quantum entropy, and they bring in several technical tools from quantum information theory (smooth entropy methods, the quantum de Finetti theorem, and the asymptotic equipartition property) to turn those principles into a full, quantitative security bound.