Measurement-Induced Nonlocality and Geometric Discord in the Spin-Boson Model

Dynamics of measurement-induced-nonlocality (MIN) and geometric measure of discord (GD) in the spin-boson model are studied. Analytical results show that for two large classes of initial states, MINs are equal but GDs are different. At the end of evolution, MIN and GD initially stored in the spin system transfer completely to reservoirs. The quantum beats for MIN and GD are also found, which are the results of quantum interference between two local non-Markovian dynamics via quantum correlation.


Introduction
Quantum correlation arises from noncommutativity of operators representing states, observables, and measurements [1].Quantum entanglement, as the earliest known quantum correlation, has acquired extensive research and is found to be an useful resource in quantum communication and quantum computation [2].However, entanglement is not the unique kind of quantum correlation.A more general concept, quantum discord [3], was found, which is regarded as the measure of all nonclassical correlations in a bipartite system, being the entanglement of a particular case of it.It was shown that there exists separable states with nonzero discord which can be used to perform quantum computation [4,5].Quantum discord is also useful in the studies of quantum phase transition [6,7] and estimation of quantum correlations in Grover search algorithm [8].Unfortunately, evaluation of quantum discord in general requires considerable numerical minimization and analytical results that are known only for certain classes of states [9][10][11].More recently, two measures out of geometric perspective in measurements, GD [12] and MIN [13], were proposed, which as the authors shown have the merit of easier evaluation.In fact, Luo and Fu [14,15] have evaluated the GD for some typical classes of states and found the tight lower bound.The dissecting about the meaning of GD was also done [16,17].
Realistic quantum systems cannot avoid interactions with their environments, leading to the change of quantum correlation.In the last decade, the influences of en-vironments to quantum entanglement [18] and quantum discord [19] have been investigated extensively.An interesting phenomenon, named as "entanglement sudden death" (ESD) [20,21] for a pair of entangled qubits exposed to local Markovian environments was found.In contrast, quantum discord in similar conditions decays only in asymptotic time [22], which signifies that quantum discord is more robust against Markovian noise than entanglement.There are also many works involved in the evolutions of quantum entanglement and discord in the non-markovian environments [23][24][25][26][27][28][29].Especially, trapping [30] and quantum beats [31] for quantum entanglement and discord for a pair of qubits in local structured environments were found.
In this paper, we investigate the dynamics of GD and MIN in a system that consists of two independent spins (qubits) coupled respectively to their local environments.Our motivation is to find the evolutional properties of GD and MIN and the difference between their evolutions and make a comparison between the evolutions of quantum entanglement or/and discord.The paper is constructed as follows.In Section 2, we introduce the original definitions of GD and MIN.In Section 3, we first introduce the interaction model, and then study the evolution of GD and MIN of different partitions for two classes of initial states.Section 4 is devoted to the study of quantum beats for GD and MIN.And the conclusion is arranged in Section 5.

GD and MIN
Let  be a bipartite state shared by parties A and B. The GD of  is defined as [12], where the min is taken over all von Neumann measurements on subsystem A, and the Hilbert Schmidt . On the other hand, the MIN is defined as [13]     here the max is taken over the von Neumann measurements which do not disturb A  locally, that is, For a two-qubit system, the general state may be written as, .
Here i  are the Pauli operators, and  are the components of the correlation tensor      .For this state, we have [12,13] and where

Model and Dynamics of MIN and GD
The model we consider consists of two independent spins interacting respectively with their local boson reservoirs.The total Hamiltonian is, .
Here z i  and i k g .For simplicity, we assume the two spins have equal Zeeman splitting 0  , and the reservoirs are initially in vacuum states and not correlated to the spins.Under these conditions, the model can be solved exactly for any initial state of the spins and any form of spectral reservoirs.In this paper, we will mainly discuss two types of reservoirs-unstructured flat reservoirs and structured Lorentzian reservoirs, and also two large types of initial states of spins-double-excitation and one-excitation states.

Double-Excitation Dynamics of MIN and GD
Let us first study the double-excitation case, i.e., the joint initial state for the whole system is, where Here we introduce the collective state to denote the one-excitation state of the reservoir i [19,32].The coefficients r


, which are determined by the quantum dynamical equation, satisfy and 1.
Here the kernel function is defined by The reduced density matrix for the two spins reads From Equations ( 4) and ( 5), the analytical expressions for MIN and GD can be written as and Similarly, the MIN and GD for the reservoirs reads, In order to further demonstrate the dynamical features of MIN and GD, we specify our study to two exemplary reservoir spectra widely used in the literature, flat spectrum and Lorentzian spectrum.First, we consider the case that both spins are respectively embedded into two equal but independent flat spectral reservoirs, where  is a constant that is commonly used as the Markov approximation with the interval of the spectral density much broader than the corresponding energy scale of the system.For this set of spectra, we obtain the time-dependent coefficients,   We plot the time evolution of and for different partitions as in Figure 1, where we take two initial states for spins: Bell state with and Belllike state with 1 8, 7 8     .From the figure along with the analytical expressions of Equations ( 12)-( 15), we can find the following features.Firstly, there is clearly correlation transference between the spins and reservoirs [Figure 1(a

 
And in the equilibrium, , i.e., the transference is complete.Secondly, both and of the spins deplete gradually and no sudden death occurs in the process of transference, which is different from the concurrence [20].Indeed, the concurrence of the spin system , the Bell-like initial state in the figure), it will occur sudden death.Thirdly, due to the interaction of the spins with their own reservoirs, any spin   r [Figure 1(d)] due to the initial correlation of the spins.However, at the end of evolution there is no correlation between any one of the spins and reservoirs.Therefore, the spin system can finally be discarded without any effect, and all correlations transfer to the reservoirs.Lastly, we find that the  13) and (15)].
Up to now, we have mainly concentrated on the ca m flat reservoirs where the dynamics is Markovian.There will be different features for structured reservoirs which will lead to obvious non-Markovian dynamics.Assume that the two spins are now embedded in the local Lorentzian reservoirs with spectra, This spectrum is commonly used to describe a twolevel atom in an imperfect cavity.Where 0  is the Bohr frequency of the spins, and    is the frequency detuning between spin avity mode.The quantity i and i  is the photon-leakage rate of the cavity whose inverse notes the reservoir correlation time.The ideal cavity limit is obtained for 0 de   , where with The time evolution of and es , we obtain, N D for different partitions in this case is presented in ure 2. For clarity, we only present the evolution of N and D for the initial state of the spins to be in Bell te.The istinctive property compared with the case of flat reservoirs is the oscillation in the evolution of N and D , which is the result of non-Markovian effe Other alike properties include: complete correlation transference between the spins and reservoirs, no sudden death of correlations for the spins in the transferring process, the vanishing correlations between spins and reservoirs at the end of evolution, and the roughly overlapping evolutional curves for N and D.
For the case of non-r onance also exist , b s ut the amplitude of oscillation becomes aller compared with the resonant case.From Equatween the spin and its reservoir.The steady values of sm tions ( 12)-( 15), we can also find that the correlation transference between the spins and reservoirs is also complete, but the transferring time becomes longer compared with the corresponding resonant case.This is because the non-resonant effect decreases the coupling be-

Single-Excitation Dynamics of MIN and GD
We w stud the correlation evolution no y for another type of initial state which has only one excitation in the spin system, The dynamical evolution of the overall system in this case can be solved as, It is interesting to find that and .Copyright © 2013 SciRes.

Quant Beat for MIN and GD
Quantum beat is a very inte resting phenomenon in quantum optics.We discussed the entanglement and discord n two-level sys-quantum beats in detail early [31] in ope tems.Here, we find that for the time evolution of N and D a similar phenomenon also appears.Let us still assume that the two spins are plugged into their own Lorentz reservoirs with spectral density given by Equation (18).For simplicity, we only discuss the non-resonant and two-excitation case.For resonant or/and oneexcitation case, similar phenomenon of quantum beat for N and D also exists.The quantum correlations N and D for the spins and the reservoirs are described respectively by Equations ( 12)-( 15) with parameters i  and i  given by Equation (20).The corresponding time evolutions are depicted in Figure 7, where  and 2  is the demand of observing quantum beat.Only in this way can we induce two harmonic oscillations with tiny different frequencies, whose interfering superpo ion forms quantum beat.These quantum beats originate from non-Markovian effect, as no any direct or mediated interaction exists between the two spins or the two reservoirs.It is the result of both non-Markovian effect and quantum interference.The detailed mathematical analysis may be consulted in Reference [31].

Conclusions
We have studied al rese voirs.We cons dered o large types of initial states of the spins, i , one-d o-excitation states.We found that N of differ t spectively couple to their loc r i tw .e. an en tw partitions is identical for the two types of initial states, while D depends on the initial states.In the situations of flat and Lorentz spectral reservoirs, we have both analytically and numerically simulated the time evolutions of N and D .We found that there exist complete correlation transference between the system and reservoirs, all N and D initially stored in the spins completely transfer to reservoirs at the end of evolution.There is no sudden death of correlations for the spin system in the transferring process, which forms bright contrast to the quantum entanglement.For flat spectral reservoirs, there is no oscillation for N and D in the transferring process.While for memory Lorentz reservoirs, oscillation appears which is the symbol of non-Markovian effect.In particular, when the detunings of the spins with their reservoirs have tiny difference, the quantum beats for N and D are observed which signify the quantum interference between the two dynamics of the spins through quantum correlations.For the Lorentz reservoirs and in the case of resonance between the spins and their reservoirs, the oscillation amplitudes of N and D are larger, and the time of the correlation transference is shorter.While in the detunings (particularly for large detunings), the oscillation amplitudes become smaller and the transferring time becomes longer.Finally, we found that though the different definitions, the evolutions of N and D are very close for most cases under consideration.This is an astonishing result which highlights in some sense the relation between the two definitions.Also note that it may happen sudden change for the evolution f D .
Note that a similar work for the evolutions of the quantum and classical correlations was done [29].The transference of N  owever it also appears many differences, for example, N is the same for one-and two-excitation cases, and D may happen sudd change in the evolutional process.Especially, N and D have very coincident evolutions in most cases under considerations.
Quantum correlation is a kind of unique characteristics quantum system which could be a new resource in quantum information tech ology.Exploring the relation between various quantum correlations an en namical rules of them in practical environments can not only contribute to a better understanding of the concepts, but also offer possible references for applications.

r
Due to the conservation of excitation number under the evolution of Jaynes-Cummings model, the dynamical state of the whole system has the form, ) and 1(b)].Both and initially stored in the spins run into reservoirs gradually.N D Figure 1.(Color online) Evolution of and N D among different partitions with flat spectral density and for the double-excitation initial spin state   0 

8 .. 2 rs 1 with s 2 ; (b) Reservoirs r 1 with r 2 ; (c) Spin s 1 with reservoir r 1 ; (d) Spin s 1 with reservo r 2 .
Figure 2. (Color online) Dynamics of and N D for the double-excitation initial Bell state of s and e Lor pins in th entz spectrum with ,    W 1 2 20 0  


are also zero [not show completely in Figure3(c) and (d)].Note that the evolutions of N and D also roughly coincide.

s 1 with s 2 ; (b) Reser- voirs r 1 with r 2 ; ( Spin s 1 with reservoir r 1 ;
Figure 4. (Color online) Dynamics of and N D for the single-excitation initial spin state   0  Figure 6.(Color online) Evolution of and N D among different partitions for non-resonan nt ectrum t Lore z sp      1 2 20 , 20 W     and for the sin le-exc initia g s d

r 2 .
Figure 7. (Color online) Dynamics for and N D for the double-excitation initial Bell state of spi nonresonant Lorentz spectrum ns and for and D studied here has some similar properties to at