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Entropy

Analysis of Quantum Correlations Obtained Using Local Optimal Universal Asymmetric Cloners

Cătălina Cîrneci, Iulia Ghiu

Published Dec 23, 2022• 0 citations
Publisher

Abstract

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.

DOI
10.3390/e25010029

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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.
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.
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.
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.
In this work, we exhaustively investigate 1→21 \rightarrow 21→2 local and non local broadcasting of entanglement and correlations (optimal and non-optimal) using asymmetric Pauli cloners, with the most general two qubit state as a resource. We exemplify the broadcasting of entanglement using Maximally Entangled Mixed States and demostrate the variation of broadcasting range with the amount of entanglement in resource state as well as the asymmetry in the cloner. We show that it is impossible to optimally broadcast quantum correlations that go beyond entanglement (Geometric Discord) with the help of these asymmetric Pauli cloning machines. We introduce the concept of asymmetry in correlations to quantify the degree of variation in the amount of correlation present in the output pairs obtained after broadcasting. We also study the problem of 1→31 \rightarrow 31→3 broadcasting of entanglement using non-maximally entangled state (NME) as a resource. We adopt two different strategies : successive application of 1→21 \rightarrow 21→2 optimal cloning machines and application of 1→31 \rightarrow 31→3 optimal cloning machines. Interestingly, we show that 1→31 \rightarrow 31→3 optimal cloner does a better job at broadcasting than the successive application of 1→21 \rightarrow 21→2 cloners. Finally, we create an example to show that there are local unitaries, which when applied on non-maximally entangled state locally give us a better range of broadcasting, that surpasses the best possible range obtained using cloning machines. This result opens up a new direction in exploration of methods to perform broadcasting which may outperform standard broadcasting strategies implemented through cloning transformations.
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.
Szab`o et al. [4] show that universal quantum cloning machines may be used to manipulate entanglement between bipartite states. Entanglement is measured with concurrence, C, and it is shown that as the fidelity of the cloner improves, more entanglement transfers from the reference to the cloned state. This thesis characterizes the nature of this entanglement transfer by analyzing the entanglement distribution of the cloner. In this analysis we will show that entanglement is generally destroyed after cloning except in cases where there is no cloning, two identical qubits are created, or there is complete qubit transfer. Along with a proof of this claim this thesis covers entanglement measures, a class of functions which quantify entanglement and entanglement distribution, a scheme for analyzing how quantum circuits transfer entanglement.