Researching the Link Between the Geometric and Rènyi Discord for Special Canonical Initial States Based on Neural Network Method

Xiaoyu LiQinsheng Zhu Qingyu MengCaishu YouMingzheng ZhuYong HuYiming Huang Hao Wu and Desheng Zheng

Abstract:Quantum correlation which is different to the entanglement and classical correlation plays important role in quantum information field.In our setup,neural network method is adopted to simulate the link between the Rènyi discord (α = 2) and the geometric discord (Bures distance) for special canonical initial states in order to show the consistency of physical results for different quantification m ethods.Our results are useful for studying the differences and commonalities of different quantizing methods of quantum correlation.Keywords:Neural network method,quantum correlation,rènyi discord,geometric discord.

1 Introduction

As part of both artificial intelligence and statistics,machine learning come from the computer science field in which the goal is to learn the potential patterns from prior given data sets.It can make a decision or prediction for future unknown situation based on this learned patterns.Recently,some quantum problems has been study using machine learning method,such as quantum state tomography [Giacomo,Guglielmo,Juan et al.(2018)],and quantum many-body problem [Giuseppe and Matthias (2017)].The results of the ese works suggest that machine learning can be a new platform for solving some problems of quantum physics.

Quantum correlation which plays important role in quantum information field is firstly quantified by the concept of "quantum discord" which is introduced by Harold et al.[Harold and Wo jciech (2001)] and Henderson et al.[Henderson and Vedral (2001)] in about ten years ago.It show us that there is an universal consensus that entanglement entirely captures quantum correlation only for a global pure state[Marco,Thomas,Rosario et al.(2015)],namely the entangle mentdoesno taccount foralln onclassi calcorrelations and thateven the states with zero entangle mentusually contain quantum correlations[Harold and Wojciech(2001);Kavan,Aharon,Hu et al.(2012)].So,many related works have been presented[Xu,Xu,Li et al.(2010);Claudia,Fabrizio,Paolo et al.(2013);Zhu,Ding,Wu et al.(2016);Zhu,Ding,Wu etal.(2015);Zhu,Fu and Lai(2013);Zhu,Ding,Wu etal.(2015);Huang(2014);Li,Zhu,Zhu et al.(2018);Davide,A lexandre,Vittorio et al.(2014);Manabendra(2014);Benjam in,Rosario,Giuseppe et al.(2013);Qu,Zhu,Wang et al.(2018);Liu,Chen,Liu et al.(2018);Qu,Wu,Wang et al.(2017);Qu,Cheng,Liu et al.(2018)].In general,theseworks are classified two class for the different quantification methods.One is theentropy style,such as,quantum discord and the Rènyientropy discord(RED)[Mario,Kaushik and Mark(2015);Kaushik,Mario and Mark(2015)].The other is geometric quantizationmethods,such as Hilber-Schmidt(DHS),Bures distance[Marco,Thomas,Wojciech,Rosario et al.(2015);Davide,A lexandre,Vittorio et al.(2014);Manabendra(2014)](DBr),trace-norm and Hellinger[Marco,Thomas,Wojciech et al.(2015);Benjam in,Rosario,Giuseppe etal.(2013)](DHL).

From the point of view of invariance of physical laws,even for differentmethods,the same physical problem should have the same result.So,finding the relation between different quantification methods will help us better study the properties of quantum correlation.Unfortunately,it need face the complex nonlinear mathematical forms of differentquantificationmethodswhenwewish to resolve thisproblem from physicalview.Looking at this problem from data processing perspective,this problem can be solved by machine learningmethod.In thiswork,we extentourworks[Zhu,Li,Zhu etal.(2018);Ding,Zhu,Wu etal.(2017)]and further construct the link betweenDBrand the RED ofα=2 by theuse of machine learningmethod for specialcanonical initialstates(SCI).

2 The link between the D B r and RED(α =2)for SCI

2.1 The brief of D B r and RED

Cianciaruso etal.discussed thegeometricmeasure of dicord-type correlationsbased on the Buresdistance(dBu)[Marco,Thomas,Rosario etal.(2015)],which isdefined as follows:

where the set of classical-quantum statesis a probability distribution,{|i＞A}denotes an orthogonal bas′is for subsystem A,is an arbitrary ensemble of states for subsystem B,anddBu(ρ,χ)is the Buresdistance.

Because it is difficult to obtain mathematically analytic form of Eq.(6)for general models,some numerical calculation methods were proposed in Davide etal.[Davide,Alexandre,Vittorio et al.(2014);Manabendra(2014)]which are also adopted in this work to studyDBrbased on the relation between quantum Fisher information and the Bures distance.

The Buresdistance can be rew ritten

whereFdenotes the quantum Fisher information,F(ρAB;)=withqi,|ψi〉denoting respectively the eigenvaluesand eigenvectors ofρAB,and them inimum is taken over the set of all local Hamiltonians

The Rènyiquantum discord ofρABis an extension of quantum discord and is defined forα∈(0,1)∪(1,2]as follows[Mario,Kaushik and Mark(2015);Kaushik,Mario and Mark(2015)]:

where the Rènyiconditionalmutual informationIα(E;B|X)τX(jué)EBsatisfies:

where the the classicaloutputXdenotes the measure mentacting on systemAandEisan environment for the measure mentmap[Kaushik,Mario and Mark(2015)].In thispaper,we choose thevon Neumannm easurementΠi′=|i′〉〈i′|(i=0,1) with two angular parametersθandφ:|0′〉=cos(θ/2)|0〉+eiφsin(θ/2)|1〉and|1′〉=sin(θ/2)|0〉-eiφcos(θ/2)|1〉(0≤θ≤π/2;0≤φ≤π).The properties of the e Rènyiquantum discord are shown in Kaushik etal.[Kaushik,Mario and Mark(2015)].

2.2 The sample of SCIstates

For the class of canonical initial(CI)states[Mazzola,Piilo and Maniscalco(2010)] of the densitymatrix

The SCIstatesneeds to be satisfied further[Titas,Amit,Anindyaetal.(2015)]:

In order to generatemore SCIstates,we consider the bit-flip(BF)noise channel[Titas,Amit,Anindya et al.(2015)].In this scenario,it is easy to show that during the evolution ofρ(t),c11,c10,andc01remain unchanged,whereas the correlationscaaand themagnetizationc0aandca0(a=2,3)decay withγas(1-γ)2and(1-γ),respectively.Here,γ=1-exp(-Γ)andΓ＞0.Simultaneously,it is also easy to check theρs(t)satisfy Eq.(2).Finally,forevery sample of SCIstates,the values ofDBrand RED can be obtained by the define ofDBrand RED.

3 Neural network model

Considering the complex nonlinear mathematical forms ofDBr and RED, the link betweenthem maybe nonlinear function. So we need to find a method which simulates this nonlinearfunction. Inspired by the biological neural network we build artificial neural network tostudy the data. Artificial neural networks is a multi-layer perception model, for each layerwe have input data and output data which is the next layer’s input. The neural network isused to construct the link between DBr and RED because the multi-layer neural networkcan represent any function no matter how complex it is. We train the neural network toadjusted the the weight and bias parameters and use the well-trained neural network topredict the output according to the input we give [Simon (2008)]. Fig.1 shows a structuregraph of our neural network. It has input data xi, hidden layer neural ali and output data y,satisfying

where thelthlayern euralcells denoteand for

The activation functionf(z)is used to realize the nonlinear relation between input and output of each neural node,

We adjust the parameters of neural network to minimize the cost function by using back-propagation algorithm and gradient descent method.

The summation for all the training data(training samples).y′andydenote the predicted value and realvalue ofDBr.

For our problem,the 4 layer neural network is constructed.The number of neurons per layer is7,8,1,1.The Tab.1 shows the learning process of neural network.

Figure1:The structure graph of our neural network.b l is the biasunit of lth layer, is the ith neural node of l layer and the input layer satisfies l =1.W (l )is the weight matrix,the elements is the weight of connection between and

Table1:Learning process of neural network

Figure2:The figure of the mean-squareerror(MSE)which is equal to the expectations of cost change with epoch.The solid(+)line shows the behavior of test(train)data

In this paper,we generate 200193 samples under the bit-flip(BF)noise channel(Here,C11∈[-1,1],C10,C01,C22andC33∈[-0.3,0.3],t∈[0,2],with step 0.1.Γ=[1,2,3,4,5]).The to talnumber of samples are more than one hundred and twenty thousand with the repetition rate less than 1%.Considering the calculation process of entropy,the important characteristic parameters of the e density matrix are the eigenvalues and the optimized selection of measurement in RED calculation,seven parameters,including the foureigenvalues ofρs(t)andθandφwhich are introduced in theREDcalculation process,and the value ofRED,are chosen as the input features of neural network.In Fig.2,the+line shows that at the end of training,the mean-square error(MSE)which isequal to the expectations ofcostrapidly decreases at first hundreds epochs and eventually converges after hundred thousand epochs.The MSE is less than 0.0042.This means that a good link is constructed based on ourmodel.

3.1 Over fitting

As we apply themachine learningmethods,the overfitting need to be carefully avoided in the training process.A regularization technique named dropout is applied to reduce overfitting in neural networks by preventing complex co-adaptations on training data.Dropoutcan effectively preventoverfitting,which means that we temporarily and randomly remove some units from the network,along with all its incoming and outgoing connections[Nitish,Ge of frey,Alex et al.(2014)].In Fig.2,it is shown that after 10000 epoch of training the MSE of training data has reduced to 0.0042 and the MSE of test data is close to that(the magnitudes of the e distance between two lines is 10-4).We can claim that the neural network has constructed without over fitting.It also further demonstrates that we obtain the link betweenRED(α=2)andDBrfor SCIstates.Our result spave a way for the further study of the physicalnature of quantum correlation.Finally,as a conclusion of this section,our neural network model is appropriate and successfully simulated the relationship between geometry and entropy style discord for SCI.

4 Conclusion

In this paper,we calculate the values ofDBrand RED which are used to quantify the quantum correlation for SCI,and the link between geometric(DBr)and entropy(RED)style discord is successfully constructed by our neural network model forSCI.From the physical perspective,the quantum correlation shows the different characteristics of the e quantum states contrasting with the classical states or the changing degree of the quantum states when it suffers the local disturb.So,the system information presented by different discord like definitions will be different.Searching the link between these defines,it will not only help us to understand the differences and commonalities of systematic information obtained by different definitions,but also help us to understand the total properties of quantum states,such as coherence,and the properties of entanglement.

Acknowledgement:This work was supported by the National Natural Science Foundation of China[61502082].