# Disentanglement of Qubits in Classical Limit of Interaction

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## Abstract

Two qubits coupled by integral spin object are studied in the semi-classical limit of interaction intermediary. It is shown that initial entanglement of qubits becomes more robust when mediated by semi-classical interaction and does not decay below certain value at a given time. The statements are supported by numerical averaging with respect to a set of randomly chosen initial preparations. There are evidences that such a robustness holds true also for different types of quantum correlations.

## Keywords

Entanglement Quantum discord Semi-classical regime Quantum open systems## 1 Introduction

Entanglement [1] is one of these phenomena which are peculiar for quantum world and seem to be absent in classical limit. There are various, often highly sophisticated, techniques of producing entangled states of bi-partite systems. Entanglement can be created via non-local interaction of two parties. Such interaction can be direct, typically described by suitable tensor product term of operators acting on states of parties incorporated in Hamiltonian. Such a modelling is suitable for systems which are ‘not very far away’ from each other. In the case when entanglement of separate (non-proximal) parties is desired one can adapt indirect interaction, caused e.g. by an environment mediating information between parties or by other systems such as cavity mode or wave guide in quantum optics [2, 3, 4, 5, 6]. Here we focus our attention on a case of indirect interaction via quantum object (agent) which serves as a intermediary.

We consider here tri-partite setting: two qubits, which entanglement is going to be studied, coupled via intermediary which properties affect entanglement of qubits. It seems to be intuitively clear, that in the classical limit of an intermediary, the qubits in initially separable state are less likely to entangle in time in comparison with deeply quantum regime of the intermediary. In this paper we analyze opposite and probably less intuitive problem: does initial entanglement become more robust against interaction caused by semi-classical intermediary? We present affirmative answer to that question. Entangled systems, when coupled via semi-classical agent, disentangle less likely in comparison with the case of deeply quantum interaction. We show that entanglement of two qubits does not decay below certain value for a given time of running evolution provided that the intermediary is ‘sufficiently classical’. In other words the qubits does not disentangle via semi-classical interaction. We also present evidences that such a property holds true for different types of quantum correlations. Results of this paper shed light on the general properties of quantum-classical hybrid systems [7, 8, 9, 10, 11, 12] which has recently been studied also in the context of quantum correlations [13, 14].

The paper is organized as follows: first two section we devote to the entanglement dynamics in (semi) classical limit of interaction mediating agent. Next we consider different type of correlations qualified by quantum discord. We attempt to unify and generalize the results based upon numerical evidences in conjectural statements concerning properties of quantum correlations in classical regime of interaction.

## 2 Entanglement Dynamics

*ħ*=1)

*A*,

*B*interacting via quantum agent labelled by

*M*. In this work we also assume that the intermediary is given by integral

*j*-spin system:

*j*→∞ of such systems studied e.g. in the context of quantum chaos [15, 16] or kicked entanglement dynamics [17]. It is known that in the limit of large (integral)

*j*dynamics of spin systems can be effectively described by classical maps and hence observables of such systems become classical [15]. In that sense large

*j*spins, although described in fully quantum mechanical fashion, can mimic behavior of classical system and hence become natural candidates for components of composites mimicking quantum-classical hybrids.

*ϱ*(

*t*) with respect to intermediary i.e.

*ρ*

_{ AB }(

*t*)=Tr

_{ M }

*ϱ*(

*t*). Its entanglement can be quantified by the concurrence [1]

*λ*

_{ i }are eigenvalues (

*λ*

_{1}is the largest) of a matrix

*QuTip*, Python-based computational toolbox [19, 20]. The results are presented in Fig. 1. With increasing

*j*(what corresponds to classical limit [16, 17]) entanglement of qubits appears less likely. This result is intuitively acceptable: quantum entanglement does not emerge in quantum objects placed in (coupled by) classical world.

However, the aim of our work is to consider a problem which is less intuitive: what happens if the qubits are *initially entangled* and the intermediary becomes (semi) classical? Does quantum entanglement decay more or less likely for the interaction mediating agent with larger *j*? A motivation for our work is the following: we investigate how the ‘classicality’ of the mediating agent affects the drain of correlations shared by composite systems.

*E*

_{ i }〉

_{ AB }:

*j*the more entangled remain the qubits. Let us notice that for sufficiently large

*j*characteristics of the concurrence corresponding to different

*j*become ‘ordered’ with respect to increasing

*j*and the entanglement does not decay below certain value for a given time of evolution. Let us notice that this time increases with increasing

*j*. The time of evolution in all figures is chosen approximately an order of magnitude larger than characteristic time related to qubit energy in Eq. (2). Let us also notice that as long as the time evolution of the total system is unitary one expects periodic behavior typical for finite quantum systems. However, one can also expect that with increasing

*j*periodicity occurs for larger and larger time scales. Qubit–qubit

*A*−

*B*system given by Eq. (2) can formally be considered as an open system coupled to (non-thermal) ‘environment’

*M*. As this environment is finite it cannot cause truly irreversible properties of qubits’ pair. Nevertheless, for sufficiently large

*M*(with

*j*→∞) one can consider certain qubit–qubit characteristics, inferred form

*A*−

*B*reduced dynamics, as effectively decaying.

^{3}randomly chosen initial states of the form:

*ξ*∈[0,

*π*/2] and

*ϵ*

_{1,2}∈[0,2

*π*) are independent uniformly distributed random numbers. In other words concurrence becomes a (classical) random variable with a given uniform probability distribution. In Fig. 3 we present two quantifiers of this random variable: mean (expected) value \(\langle\!\langle\mathcal{C}\rangle\!\rangle\) (upper panel) and its variance \(\langle\!\langle\mathcal{C}^{2}\rangle\!\rangle-\langle\!\langle\mathcal{C}\rangle\!\rangle^{2}\) (lower panel) indicating relation of exceptional to typical states. All the features of ‘expected value’ of the entanglement in Fig. 3 qualitatively agree with these presented in Fig. 2.

## 3 Dissipation

*C*and the damping amplitude

*ε*:

*C*=

*C*

_{1}acts on the intermediary

*M*, in the second case (for

*C*=

*C*

_{2}) dissipation affects directly one of the qubits:

*A*−

*M*−

*B*triple. Such a dissipation, contrary to many known examples [26, 27, 28], does not play any constructive role in entanglement dynamics. The model of dissipation considered here is general enough to incorporate energy exchange between

*A*−

*M*−

*B*system and the environment and hence, contrary to e.g. pure dephasing [18] or driven non-equilibrium systems, entanglement disappears for asymptotically large time. Although physical origin for

*C*

_{1}and

*C*

_{2}can be different an impact of both types of dissipation on entanglement of qubits

*A*,

*B*is, as presented in Fig. 4, qualitatively the same. For weak dissipation conclusions of previous section remain valid. In the presence of Markovian dissipation entanglement of qubits becomes more robust with increasing

*j*but the time scale when the concurrence does not decay below certain value becomes shorter for

*ε*>0. Let us notice that, as presented in Fig. 5, qualitatively similar behavior occurs when the single-qubit terms in the system Hamiltonian Eq. (2) are different e.g. when we replace

*σ*

^{ z }→

*σ*

^{ x }for either

*A*or

*B*. As an example we consider Hamiltonian

## 4 Beyond Entanglement: Quantum Discord

Quantum entanglement, although best known, is not the only (effective for applications) quantifier of quantum correlations. There are other types of correlations which are essentially non-classical and which can exist even in the absence of entanglement [29]. It is natural to ask if robustness of the entanglement in the classical limit of intermediary reported here holds also for other types of quantum correlation between qubits. Unfortunately, mathematical setting for entanglement (incorporating e.g. tensor products of state spaces) has not been uniquely established so far for general quantum correlations which remain, to some extent, ‘definition-dependent’. We limit our consideration to a single type of correlation: the one qualified by quantum discord. It is a very specific but also very popular measure of correlations. Its properties are summarized in the review paper [29] (equipped there with a comprehensive list of references). Let us stress that, in general, quantum discord is neither the only nor always the best quantifier of quantum correlation [29]. Nevertheless, quantum correlations quantified by the quantum discord can open new avenues for quantum computations and quantum communication schemes [30, 31, 32, 33].

*A*and

*B*. Total correlation, encoded in mutual entropy of

*A*and

*B*, is formulated in terms of a difference between entropies:

*ρ*

_{ A }=Tr

_{ B }

*ρ*

_{ AB }and

*ρ*

_{ B }=Tr

_{ A }

*ρ*

_{ AB }. That what is known about

*A*, under the condition that on

*B*was performed a measurement \(\varPi_{j}^{B}\), can be quantified in terms of conditional entropy

*θ*,

*ϕ*is a standard parameterization of a single qubit Bloch sphere.

*j*quantum discord never decays below certain value for a given time of evolution. We can only conjecture that such a characteristics hold true for any ‘reasonable choice’ of quantum correlation quantifier. Let us notice that such a direct relation between different types of quantum correlations is not always generic even is a simplest case of quantum entanglement and quantum discord [35, 36].

## 5 Summary

In this paper we analyze entanglement dynamics of qubits coupled via integral *j*-spin intermediary. We confirm an intuitive expectation that for large *j*, corresponding to interaction agent operating in semi-classical regime, the concurrence of initially non-entangled qubits remains small in time. We also consider less intuitive problem of initially maximally entangled qubits. We show that initial entanglement is more robust for intermediaries with large *j*. For sufficiently large *j* entanglement never decays below certain value for a given time of evolution i.e. the qubits does not disentangle via semi-classical interaction. This is central result of present work. Such behavior is typical in the sense that it holds true for the ‘expected value’ of the concurrence obtained by averaging with respect to a set of randomly chosen initial preparations. It also holds true in the presence of weak Markovian dissipation locally affecting any of the components of the system. Numerical studies of quantum discord suggest a conclusion that this feature is generic for a broad class of different and non-equivalent types of quantum correlations.

In other words, one can say that quantum information qualified here by two types of quantum correlations, shared initially by qubits *A* and *B* drain into the system *M* coupled to qubits slower provided that *M* is (semi)classical. The coupling via classical mediating agent causes not only slower correlation but also slower discorrelation of the qubits in comparison to the case when the mediating agent is fully quantum mechanical.

Finally, let us mention that results of this paper not only touch general properties of quantum-classical hybrid systems [7, 8, 9, 10, 11, 12, 13, 14] but also can be of potential practical value for nanoscience operating on the border of classical and quantum world [37, 38, 39] motivated by the role played by quantum entanglement in quantum information processing [40].

## Notes

### Acknowledgements

The work has been supported by the NCN Grant N202 052940.

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