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Results in Physics

Asymmetric cloning of steering, quantum discord, and coherence

Irina Ion, Iulia Ghiu

Published Jun 15, 2025• 1 citation
Publisher

Abstract

Cloning of steering, quantum discord , and coherence is investigated by using two types of cloners: local and nonlocal 1 → 2 optimal universal asymmetric cloning machines. We prove the impossibility of generating two steerable output states when local cloners are used. On the other hand, when nonlocal cloning machines are employed, we show that there are ranges of the parameters which characterize the initial state and the cloning machines, such that the two output mixed states are steerable. Moreover, we obtain analytical expressions of the quantum discord and the coherence of the two output states generated by local and nonlocal asymmetric cloners. Further we make a detailed comparison of steering, quantum discord, and coherence of the output states obtained by local versus nonlocal cloning machines.

DOI
10.1016/j.rinp.2025.108323

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arXiv:1702.02123•Physical Review A•Feb 13, 2019•10 citations
Cloning of steering, quantum discord , and coherence is investigated by using two types of cloners: local and nonlocal 1 → 2 optimal universal asymmetric cloning machines. We prove the impossibility of generating two steerable output states when local cloners are used. On the other hand, when nonlocal cloning machines are employed, we show that there are ranges of the parameters which characterize the initial state and the cloning machines, such that the two output mixed states are steerable. Moreover, we obtain analytical expressions of the quantum discord and the coherence of the two output states generated by local and nonlocal asymmetric cloners. Further we make a detailed comparison of steering, quantum discord, and coherence of the output states obtained by local versus nonlocal cloning machines.
We analyze the output mixed states generated when the asymmetric cloners are applied on a pure two-qubit state. We investigate two kinds of cloners: (a) local [Formula: see text] optimal universal asymmetric cloning machines and (b) nonlocal [Formula: see text] optimal universal asymmetric cloning machines. In both cases we find analytical formulae of the geometric quantum discord of the two clones. Finally, we prove that the geometric quantum discord generated by the nonlocal asymmetric cloners is not smaller than the geometric quantum discord given by the local asymmetric cloners.
We apply the local optimal universal asymmetric cloning machine on an initially pure entangled state of two qubits. As output, we obtain two final states which present quantum correlations. We analyze three types of quantum correlations among the final states, namely, concurrence, quantum discord, and consonance. A detailed comparison between concurrence, quantum discord, and consonance is made, and we find that consonance is greater than quantum discord, which is in turn greater than concurrence.
Due to the axioms of quantum mechanics, perfect cloning of an unknown quantum state is impossible. But since imperfect cloning is still possible, a question arises: ``Is there an optimal quantum cloning machine?'' Buzek and Hillery answered this question and constructed their famous B-H quantum cloning machine. The B-H machine clones the state of an arbitrary single qubit in an optimal manner and hence it is universal. Generalizing this machine for a two-qubit system is straightforward, but during this procedure, except for product states, this machine loses its universality and becomes a state-dependent cloning machine. In this paper, we propose some classes of optimal universal local quantum state cloners for a particular class of two-qubit systems, more precisely, for a class of states with known Schmidt basis. We then extend our machine to the case that the Schmidt basis of the input state is deviated from the local computational basis of the machine. We show that more local quantum coherence existing in the input state corresponds to less fidelity between the input and output states. Also we present two classes of a state-dependent local quantum copying machine. Furthermore, we investigate local broadcasting of two aspects of quantum correlations, i.e., quantum entanglement and quantum discord, defined, respectively, within the entanglement-separability paradigm and from an information-theoretic perspective. The results show that although quantum correlation is, in general, very fragile during the broadcasting procedure, quantum discord is broadcasted more robustly than quantum entanglement.
Quantum steering in a global state allows an observer to remotely steer a subsystem into different ensembles by performing different local measurements on the other part. We show that, in general, this property cannot be perfectly cloned by any joint operation between a steered subsystem and a third system. Perfect cloning is viable if and only if the initial state is of zero discord. We also investigate the process of cloning the steered qubit of a Bell state using a universal cloning machine. Einstein–Podolsky–Rosen (EPR) steering, which is a type of quantum correlation existing in the states without a local-hidden-state model, is observed in the two copy subsystems. This contradicts the conclusion of no-cloning of quantum steering (EPR steering) [C. Y. Chiu et al., npj Quantum Inf. 2, 16020 (2016)] based on a mutual information criterion for EPR steering.
Quantum mechanical properties like entanglement, discord, and coherence act as fundamental resources in various quantum information processing tasks. Consequently, the technique of generating more resources from a few, typically termed as broadcasting, serves as a promising candidate for the design of quantum networks. One way to broadcast resources could be using a cloning operation. In this article, broadcasting of quantum resources beyond $2\ensuremath{\bigotimes}2$ systems is investigated. In particular, in $2\ensuremath{\bigotimes}3$ dimensions, a class of states not useful for broadcasting of entanglement is characterized considering an optimal universal Heisenberg cloning machine. The broadcasting ranges for maximally entangled mixed states and two-parameter class of states are obtained to exemplify our protocol. A significant derivative of our protocol is that the cloning operation generates a qutrit ($3\ensuremath{\bigotimes}3$) entangled pair with positive partial transpose on one of the local sides, and an absolutely separable qubit ($2\ensuremath{\bigotimes}2$) pair on the other side of the input bipartite $2\ensuremath{\bigotimes}3$-dimensional resource state. Moving beyond entanglement, in $2\ensuremath{\bigotimes}d$ dimensions, the impossibility to optimally broadcast quantum discord and quantum coherence (l1{l}_{1}l1​ norm) is established. However, some significant illustrations are provided to highlight that nonoptimal broadcasting of discord and coherence is still possible.
In this work, we exhaustively investigate $1\ensuremath{\rightarrow}2$ local and nonlocal broadcasting of entanglement as well as correlations beyond entanglement (geometric discord) using asymmetric Pauli cloners with the most general two-qubit state as the resource. We exemplify asymmetric broadcasting of entanglement using ``maximally entangled mixed states.'' We demonstrate the variation of broadcasting range with the amount of entanglement present in the resource state as well as with the asymmetry in the cloner. We show that it is impossible to optimally broadcast geometric discord with the help of these asymmetric Pauli cloning machines. We also study the problem of $1\ensuremath{\rightarrow}3$ broadcasting of entanglement using nonmaximally entangled state as the resource. For this task, we introduce a method we call successive broadcasting which involves application of $1\ensuremath{\rightarrow}2$ optimal cloning machines multiple times. We compare and contrast the performance of this method with the application of direct $1\ensuremath{\rightarrow}3$ optimal cloning machines. We show that $1\ensuremath{\rightarrow}3$ optimal cloner does a better job at broadcasting than the successive application of $1\ensuremath{\rightarrow}2$ cloners and the successive method can be beneficial in the absence of $1\ensuremath{\rightarrow}3$ cloners. We also bring out the fundamental difference between the tasks of cloning and broadcasting in the final part of the paper. We create examples to show that there exist local unitaries which can be employed to give a better range for broadcasting. Such unitary operations are not only economical, but also surpass the best possible range obtained using existing cloning machines enabling broadcasting of lesser entangled states. This result opens up a direction in exploration of methods to facilitate broadcasting which may outperform the standard strategies implemented through cloning transformations.