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arXiv:2603.06393

Continuous-variable approximate unitary 2-design, with applications to unclonable encryption

Arpan Akash Ray, Boris Skoric

quant-phSubmitted Mar 6, 2026• Announced Mar 9, 2026• 0 citations
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Abstract

We introduce an $\varepsilon$-approximate unitary 2-design that is compatible with the structure of p- and q-quadratures in continuous-variable (CV) quantum systems. The design unitaries are defined on a finite-dimensional discretisation of the CV space and can be physically implemented as operations on the full CV space. This establishes the first approximate unitary design for CV systems. The design alternatingly acts with unitaries based on the quadrature operators $\hat q$ and $\hat p$. We prove that the parameter $\varepsilon$ is given by $1/d^\ell$, where $d$ is the dimension of the truncated Hilbert space and $\ell$ is the number of iterations. We propose an Unclonable Encryption scheme in which the encryption operators are given by the unitaries which constitute the approximate unitary design. We prove its security using recent results on decoupling. This establishes unclonable-indistinguishable security for a CV encryption for the first time.

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We introduce an ε\varepsilonε-approximate unitary 2-design that is compatible with the structure of p- and q-quadratures in continuous-variable (CV) quantum systems. The design unitaries are defined on a finite-dimensional discretisation of the CV space and can be physically implemented as operations on the full CV space. This establishes the first approximate unitary design for CV systems. The design alternatingly acts with unitaries based on the quadrature operators q^\hat qq^​ and p^\hat pp^​. We prove that the parameter ε\varepsilonε is given by 1/dℓ1/d^\ell1/dℓ, where ddd is the dimension of the truncated Hilbert space and ℓ\ellℓ is the number of iterations.
We propose an Unclonable Encryption scheme in which the encryption operators are given by the unitaries which constitute the approximate unitary design. We prove its security using recent results on decoupling. This establishes unclonable-indistinguishable security for a CV encryption for the first time.
We propose the first continuous-variable (CV) unclonable encryption scheme, extending the paradigm of quantum encryption of classical messages to CV systems. In our construction, a classical message is first encrypted classically and then encoded using an error-correcting code. Each bit of the codeword is mapped to a CV mode by creating a coherent state which is squeezed in the q or p quadrature direction, with a small displacement that encodes the bit. The squeezing directions are part of the encryption key. We prove unclonability in the framework introduced by Broadbent and Lord, via a reduction of the cloning game to a CV monogamy-of-entanglement game. Furthermore, we demonstrate that our scheme can be readily implemented with current technology. By incorporating realistic imperfections such as channel noise and detector inefficiencies, we show that the protocol remains robust under these conditions.
We generalize the notion of quantum state designs to infinite-dimensional spaces. We first prove that, under the definition of continuous-variable (CV) state t -designs from [Blume-Kohout , ], no state designs exist for t≥2 . Similarly, we prove that no CV unitary t -designs exist for t≥2 . We propose an alternative definition for CV state designs, which we call rigged t -designs, and provide explicit constructions for t=2 . As an application of rigged designs, we develop a design-based shadow-tomography protocol for CV states. Using energy-constrained versions of rigged designs, we define an average fidelity for CV quantum channels and relate this fidelity to the CV entanglement fidelity. As an additional result of independent interest, we establish a connection between torus 2-designs and complete sets of mutually unbiased bases. Published by the American Physical Society 2024
We give an unconditional construction of information-theoretically secure one-time private-key unclonable encryption for one-bit messages, with efficient encryption and decryption and exponentially small unclonable-indistinguishability advantage. Previous work by Bhattacharyya and Culf (Nature Physics, 2026) and Bhattacharyya, Broadbent, and Culf (arXiv:2603.08916) either only showed inverse-polynomial security loss or required inefficient encryption/decryption operations; we avoid both of these caveats. Our scheme's key is a uniformly random non-identity phase-free Pauli on nnn qubits, and bit aaa is encrypted as a random (−1)a(-1)^a(−1)a eigenstate of that Pauli. We prove that the probability that both receivers recover the bit is at most 1/2+O(2−n/2).1/2 + O\left(2^{-n/2}\right).1/2+O(2−n/2). By a lower bound due to Broadbent, Culf, and Rochette (arXiv:2510.00903), this is the best probability bound achievable with nnn-qubit ciphertexts while ensuring perfect correctness (up to the constant hidden in the O(⋅)O(\cdot)O(⋅)).
Unitary t -designs are the bread and butter of quantum information theory and beyond. An important issue in practice is that of efficiently constructing good approximations of such unitary t -designs. Building on results by Aubrun (Comm. Math. Phys. 2009), we prove that sampling dtpoly(t,log⁡d,1/ϵ) unitaries from an exact t -design provides with positive probability an ϵ -approximate t -design, if the error is measured in one-to-one norm. As an application, we give a randomized construction of a quantum encryption scheme that has roughly the same key size and security as the quantum one-time pad, but possesses the additional property of being non-malleable against adversaries without quantum side information.
We investigate the notion of untelegraphable encryption (UTE), a quantum encryption primitive that is a special case of uncloneable encryption (UE), where the adversary’s capabilities are restricted to producing purely classical information rather than arbitrary quantum states. We present an unconditionally secure construction of UTE that achieves untelegraphable-indistinguishability security, together with natural multi-ciphertext and bounded collusion-resistant extensions, without requiring any additional assumptions. We also extend this to the unbounded case, assuming pseudo-random unitaries, yielding everlasting security. Furthermore, we derive results on UE using approaches from UTE in the following ways: first, we provide new lower bounds on UTE, which give new lower bounds on UE; second, we prove an asymptotic equivalence between UTE and UE in the regime where the number of adversaries in UE grows. These results suggest that UTE may provide a new path toward achieving a central open problem in the area: indistinguishability security for UE in the plain model.
The unclonable bit question asks whether quantum encryption can prevent an adversary from producing two systems that both reveal the plaintext once the key is disclosed. We introduce and study Pauli Encodings, a simple class of one-bit encryption schemes whose ciphertexts are normalized eigenspace projectors of Pauli strings. For every Pauli Encoding with K Pauli strings, we prove a universal lower bound 1/2+1/(2K)1/2+1/(2\sqrt{K})1/2+1/(2K​) on the optimal monogamy-of-entanglement winning probability, together with sharper bounds for several structured families. We then establish two limitations of natural approaches to unclonable security. First, if the Pauli strings are restricted to strings of X and Z of length n, the encoding is not secure. Second, we identify a universal 3/4 obstruction showing that arguments based only on pairwise guessing marginals cannot establish unclonable-indistinguishable security. When the Pauli strings all pairwise anticommute, the protocol becomes the one studied in [Quantum 10, 2157 (2026)]. We exploit the symmetry of this protocol to solve the third level of the natural semidefinite programming relaxation obtaining an asymptotic upper bound of approximately 0.5556 on the winning probability. Finally, we prove strong unclonable-indistinguishable security against bounded-local-dimension adversaries and strong indistinguishability security for several efficient Pauli families. First-level NPA computations provide additional numerical evidence towards the strong unclonable-indistinguishable security.