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arXiv:2507.17279 · Physical Review A

Virtual cloning of quantum states

Zhi-Hao Bi, Jing-Tao Qiu, Xiao‐Dong Yu

Published Jul 7, 2025• 0 citations
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Abstract

The inherent limitations of physical processes prevent the copying of arbitrary quantum states. Furthermore, even if we only aim to clone two distinct quantum states, it remains impossible unless they are mutually orthogonal. To overcome this limitation, we propose a virtual-cloning protocol that bypasses the restrictions imposed by the quantum no-cloning theorem. Specifically, we begin by outlining the general framework for virtual cloning and deriving a necessary and sufficient criterion for the existence of a virtual operation capable of simultaneously cloning a set of states. Subsequently, through an analysis of the simulation cost of the virtual-cloning process, we demonstrate that the problem of identifying an optimal virtual-cloning protocol can be cast as a semidefinite programming problem. Finally, we establish a connection between virtual cloning and state discrimination, from which universal bounds on the optimal cloning cost are derived.

DOI
10.1103/4y3n-p641

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The inherent limitations of physical processes prevent the copying of arbitrary quantum states. Furthermore, even if we only aim to clone two distinct quantum states, it remains impossible unless they are mutually orthogonal. To overcome this limitation, we propose a virtual-cloning protocol that bypasses the restrictions imposed by the quantum no-cloning theorem. Specifically, we begin by outlining the general framework for virtual cloning and deriving a necessary and sufficient criterion for the existence of a virtual operation capable of simultaneously cloning a set of states. Subsequently, through an analysis of the simulation cost of the virtual-cloning process, we demonstrate that the problem of identifying an optimal virtual-cloning protocol can be cast as a semidefinite programming problem. Finally, we establish a connection between virtual cloning and state discrimination, from which universal bounds on the optimal cloning cost are derived.
The quantum no-broadcasting theorem states that it is impossible to produce perfect copies of an arbitrary quantum state, even if the copies are allowed to be correlated. Here we show that, although quantum broadcasting cannot be achieved by any physical process, it can be achieved by a virtual process, described by a Hermitian-preserving trace-preserving map. This virtual process is canonical: it is the only map that broadcasts all quantum states, is covariant under unitary evolution, is invariant under permutations of the copies, and reduces to the classical broadcasting map when subjected to decoherence. We show that the optimal physical approximation to the canonical broadcasting map is the optimal universal quantum cloning, and we also show that virtual broadcasting can be achieved by a virtual measure-and-prepare protocol, where a virtual measurement is performed, and, depending on the outcomes, two copies of a virtual quantum state are generated. Finally, we use canonical virtual broadcasting to prove a uniqueness result for quantum states over time.
It is a well-known fact that the no-cloning theorem forbids the creation of identical copies of an arbitrary unknown quantum state.In other words,there does not exist a quantum cloning machine that can clone all quantum states.However,it is possible to clone given quantum states under certain conditions,for instance,k distinct pure states|ψ1,|ψ2,...,|ψk can be cloned simultaneously if and only if they are orthogonal.This paper discusses the existence and construction of simultaneous cloning machines for mixed states.It is proved that k distinct mixed statesρ1,ρ2,...,ρkof the n-dimensional quantum system Cncan be cloned simultaneously,that is,there exists a quantum channel Φ on MnMn and a state Σ in Mn,such that Φ(ρiΣ)=ρiρi for all i,if and only ifρiρj=0(i j).Also,the constructing procedure of the desired simultaneous cloning machine is given.
Quantum broadcasting is central to quantum information processing and characterizes the correlations within quantum states. Nonetheless, traditional quantum broadcasting encounters inherent limitations dictated by the principles of quantum mechanics. In a previous study, Parzygnat et al. [Phys. Rev. Lett. 132, 110203 (2024)] introduced a canonical broadcasting quantum map that goes beyond the quantum no-broadcasting theorem through a virtual process. In this work, we generalize the concept of virtual broadcasting to unilocal broadcasting by incorporating a reference system and introduce protocols that can be approximated using physical operations with minimal cost. First, we propose a universal unilocal protocol enabling multiple parties to share the correlations of a target bipartite state, which is encoded in the expectation value for any observable. Second, we formalize the simulation cost of a virtual quantum broadcasting protocol into a semidefinite programming problem. Notably, we propose a specific protocol with optimal simulation cost for the two-broadcasting scenario, revealing an explicit relationship between simulation cost and the quantum system's dimension. Moreover, we establish upper and lower bounds on the simulation cost of the virtual nnn-broadcasting protocol and demonstrate the convergence of the lower bound to the upper bound as the quantum system's dimension increases.
Due to the no-cloning theorem, the unknown quantum state can only be cloned approximately or exactly with some probability. There are two types of cloners: universal and state-dependent cloner. The optimal universal cloner has been found and can be viewed as a special state-dependent quantum cloner that has no information about the states. In this paper, we investigate the state-dependent cloning when the state set contains more than two states. We get some bounds of the global fidelity for these processes. This method is not dependent on the number of the states contained in the state set. It is also independent of the numbers of copying.
The no-broadcasting theorem, a fundamental limitation on the communication of quantum information, holds that a physical process cannot broadcast copies of an unknown quantum state to two or more receivers. Recent work has explored ways of circumventing this limitation using "virtual" implementations of non-physical processes using measurement and data-processing on statistical samples of the unknown input. However, the statistical fluctuations of this data degrades the virtual copies so much that the protocol effectively depletes, rather than proliferate, the sample size -- thereby rendering it worse than the "naive" approach of splitting the given sample and sending a subsample to each receiver. In this work, we circumvent this flaw by allowing a small amount of systematic bias in the broadcast data, resulting in approximate virtual copies. We provide efficient semidefinite programs (SDP's) to determine the minimum sample size required to keep the approximation error below a desired threshold and vice versa. For reasonably small error values, we find approximate virtual broadcasting to be viable with sample sizes smaller than naive sample-splitting would demand. Along the way, we prove several symmetry-based simplifications to the problem, allowing optimal approximate broadcasting to be characterized in terms of the simple class of depolarizing channels.