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Closed-form autonomous optimal cloning on spin networks beyond the star: fractional revival, perfect transfer and balanced beam splitters in symmetric sectors

Hikaru Wakaura

Published Oct 19, 2026• 0 citations
Publisher

Abstract

Abstract Copying quantum information is forbidden by the no-cloning theorem, yet approximate copies with the highest fidelity allowed by quantum mechanics are a basic resource for quantum networks and for assessing the security of quantum key distribution. Such cloners are usually gate sequences, but a spin network with fixed couplings can perform the transformation autonomously: the unknown qubit and blank qubits sit on the vertices of a graph and the free evolution does the rest. Until now, the only spin graph known to reach the optimal phase-covariant fidelity for every number of clones $M$ was the star with a tuned anisotropy. Which other graphs can clone optimally, and why, has remained open. Here we show that the complete bipartite graph $K_{2,M-1}$---the unknown qubit and one target on one side, the remaining targets on the other---reaches the optimal $1\to M$ phase-covariant fidelity for every $M\ge 3$, with all couplings given in closed form. For odd $M$ a staggered Ising anisotropy suffices, with the parameters fixed by two integers; for even $M$ an additional Ising coupling among the middle targets is required, with its value an explicit algebraic number. Both families reduce, in the permutation-symmetric sectors, to two four-level problems: a perfect-state-transfer ladder on a four-cycle and a freeze block whose characteristic polynomial we obtain analytically. We verify the formulas with no free parameter up to $M=10$ and classify exactness for all four-vertex graphs with a least-squares test of the exact revival and transfer equations. The results connect the fractional-revival engineering of spin chains to optimal cloning and identify the hidden involution behind both. Beyond cloning, optimal quantum machines can thus be read off from the spectrum of small symmetric blocks, suggesting autonomous implementations of other covariant transformations on spin and Rydberg arrays.

DOI
10.21203/rs.3.rs-11396207/v1

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Abstract Copying quantum information is forbidden by the no-cloning theorem, yet approximate copies with the highest fidelity allowed by quantum mechanics are a basic resource for quantum networks and for assessing the security of quantum key distribution. Such cloners are usually gate sequences, but a spin network with fixed couplings can perform the transformation autonomously: the unknown qubit and blank qubits sit on the vertices of a graph and the free evolution does the rest. Until now, the only spin graph known to reach the optimal phase-covariant fidelity for every number of clones MMM was the star with a tuned anisotropy. Which other graphs can clone optimally, and why, has remained open. Here we show that the complete bipartite graph K2,M−1K_{2,M-1}K2,M−1​---the unknown qubit and one target on one side, the remaining targets on the other---reaches the optimal 1→M1\to M1→M phase-covariant fidelity for every M≥3M\ge 3M≥3, with all couplings given in closed form. For odd MMM a staggered Ising anisotropy suffices, with the parameters fixed by two integers; for even MMM an additional Ising coupling among the middle targets is required, with its value an explicit algebraic number. Both families reduce, in the permutation-symmetric sectors, to two four-level problems: a perfect-state-transfer ladder on a four-cycle and a freeze block whose characteristic polynomial we obtain analytically. We verify the formulas with no free parameter up to M=10M=10M=10 and classify exactness for all four-vertex graphs with a least-squares test of the exact revival and transfer equations. The results connect the fractional-revival engineering of spin chains to optimal cloning and identify the hidden involution behind both. Beyond cloning, optimal quantum machines can thus be read off from the spectrum of small symmetric blocks, suggesting autonomous implementations of other covariant transformations on spin and Rydberg arrays.
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We present a scheme which transforms a one-input qubit to MMM identical copies with optimal fidelity via free dynamical evolution of spin star networks. We demonstrate that the Heisenberg XXZXXZXXZ coupling can meet the challenge. The initial state of the copying machine and the parameters of the spin Hamiltonian are discussed in detail. Furthermore we have proposed a feasible method to prepare the initial state of the copying machine.
While the no-cloning theorem, which forbids the perfect copying of quantum states, is well-known as one of the defining features of quantum mechanics, the question of how well the theory allows a state to be cloned is yet to be completely solved. In this paper, rigorous solutions to the problem of M→NM\rightarrow NM→N asymmetric cloning of qudits are obtained in a number of interesting cases. The central result is the solution to the 1→N1 \rightarrow N1→N universal asymmetric qudit cloning problem for which the exact trade-off in the fidelities of the clones for every NNN and ddd is derived. Analogous results are proven for qubits when M=N−1M=N-1M=N−1. We also consider state-dependent 1→N1 \rightarrow N1→N qubit cloning, providing a general parametrization in terms of a Heisenberg star Hamiltonian. In all instances, we determine the feasibility of implementing the cloning economically, i.e., without an ancilla, and determine the dimension of the ancilla when an economic implementation is not possible.