Approximate cloning of structured pure states

2026-10-05

AI Summary

This study addresses the exponential sample complexity bottleneck in approximately cloning structured pure states, a fundamental limitation imposed by the quantum no-cloning theorem. To overcome this challenge, the authors establish a unified representation-theoretic framework tailored for structured state families, including fermionic and bosonic states. By integrating saddle-point evaluation with controlled frame potential analysis techniques, they design efficient quantum cloning channels. The proposed approach achieves optimal or near-optimal cloning protocols, reducing the sample complexity from exponential to polynomial scaling. Furthermore, this work reveals an intrinsic scaling law demonstrating that both cloning and tomography complexities scale linearly with the manifold dimension, thereby significantly lowering quantum resource overhead.

Abstract

The no-cloning theorem is a cornerstone result in quantum mechanics that forbids copying of general quantum information. More quantitatively, even given $N$ copies of an unknown state, any channel producing an $(N+1)$-copy state incurs a nonzero trace-distance error on some inputs. Turning this around, one can ask about the sample complexity of approximate $N\rightarrow N+1$ cloning: for a desired small error $\epsilon$, what $N$ suffices? For arbitrary pure states on a Hilbert space $H$, the answer is $N\simeq \dim H/\epsilon$ - astronomically large for most $H$ of interest. What if the input is promised to lie in a structured family? We examine a range of families fundamental to many-body physics and quantum information: $n$-mode fermionic Gaussian and Slater states, $n$-mode bosonic Gaussian states, and qudit phase states. For fermions, we construct optimal cloning channels, reducing the complexity from exponential to polynomial in $n$. For bosonic Gaussian states, we construct an explicit cloner that certifies polynomial complexity, without any constraint on the energy of the state; this channel, however, is not optimal in general. The fermionic and bosonic results follow from a single representation-theoretic framework, which generalizes previous work by Werner and by Chiribella and Yang; it also yields the cloners'unitary implementation. Furthermore, we explain why for many families the cloning complexity is linear in the dimension of the family manifold, via a controlled saddle-point evaluation of the frame potential at high $N$. The same calculation accounts for the analogous scaling of tomography complexity. The above framework does not cover phase states; for these we give an independent construction of a channel with optimal cloning fidelity. Our work complements recent results on cloning stabilizer states.

Problem

• approximate cloning
• sample complexity
• structured pure states
• no-cloning theorem
• Gaussian states

Innovation

• approximate quantum cloning
• structured pure states
• sample complexity
• representation theory
• frame potential