Fisher information of two superconducting charge qubits under dephasing noisy
aa r X i v : . [ qu a n t - ph ] D ec Fisher information of two superconducting charge qubits underdephasing noisy
Hamada El-Shiekh a,b , Nour Zidan a and Nasser Metwally c,da Mathematics Department, Faculty of Science, Sohag University, Sohag, Egypt. b Mathematics Department, College of Science, Al-Taif University, Saudi Arabia. c Department of Mathematics, College of Science, Bahrain University, Bahrain. d Department of Mathematics, Faculty of Science, Aswan University, Aswan, Egypt.
Abstract
The dynamics of a charged two-qubit system prepared initially in a maximum entangledstate is discussed, where each qubit interacts independently with a dephasing channel. TheFisher information is used to estimate the channel and the energy parameters. Moreover,the contribution from different parts of Fisher information is discussed. We show that,the degree of estimation of any parameter depends on the initial value of this parameter.It turns out that the upper bounds of Fisher information with respect to the dephasingparameter increase and that corresponding to coupling parameter decreases as the initialenergy of each qubit increases.
Quantum Fisher information is considered as one of the important measurements in the esti-mation theory [1, 2], which characterizes the sensitivity of a given system with respect to thechanges in one of its parameters. Recently, in the context of quantum estimation theory Fisherinformation has paid some attentions. For example, Z. Qiang et. al [3] examined the problem ofparameter estimation for two initially entangled qubits subject to decoherence . Xiao et. al.[4]showed that the partial measurements can greatly enhance the quantum Fisher information tele-portation under decoherence. F. Fr¨owis et. al proposed an experimentally accessible scheme todetermine lower bounds on the quantum Fisher information [5]. Quantum Fisher informationof the Greenberger-Horne-Zeilinger state in decoherence channels was discussed by Ma et.al [6].The dynamic of quantum Fisher information of W state in the three basic decoherence channelswas studied by Ozaydin [7]. The properties of quantum Fisher information in a general super-position of a 3-qubit GHZ state and two W states were investigated by Yi et. al [9]. The timeevolution of the quantum Fisher information of a system whose the dynamics was described bythe phase-damped model is discussed by Obada et. al [10]. The effect of the Unruh parameter onthe precision of estimating the channel parameter was discussed in [11]. Recently, Metwally[12]employed the Fisher information to estimate the teleported and the gained parameters by usingan accelerated channel. 1owever, investigating the Fisher information for the charged qubits has not paid muchattention. Therefore, we are motivated to use the Fisher information to estimate the initialparameters of the charged qubits system; the Josephon energies parameters and the mutualcoupling energy between the two qubits and the channel noise parameter.This paper is organized as follows: In Sec. 2, we present the model and its solution underdephasing noisy. In Sec. 3, we estimate the charged system’s parameters and the noisy channelparameter by means of Fisher information, where analytical forms are introduced. We summarizeour results and conclusion in Sec. 4.
Charged qubit represents one of the most promising particles in quantum computation andinformation. There are several studies have been introduced to investigate the properties ofentangled charged qubits. For example, Liao et. al, [13] investigated the entanglement betweentwo Josephson charge qubits. The dynamics of charge qubits coupled to a nonmechanical res-onator under the influence a phonon bath is discussed by Abdel-Aty et.al [14]. Dynamics of thequantum deficit of two charged qubit is investigated in [15]. The possibility of using chargedqubit to perform quantum teleportation is investigated by Metwally [16].However, the Hamiltonian which can be used to describe a two charged qubit system isdefined as [17, 18, 19]: H = − n κ σ (1) z ⊗ I + κ I ⊗ σ (2) z + E J σ (1) x ⊗ I + E J I ⊗ σ (2) x − E m σ (1) z ⊗ σ (2) z o (1)where κ = 2 E c (1 − n g ) + E m (1 − n g ), κ = 2 E c (1 − n g ) + E m (1 − n g ), I , I are theunit operators for the first and the second qubit, respectively and E c i , E J i are the charging andJosephon energies for the first and the second qubit, respectively. The mutual coupling energybetween the two qubits is defined by the parameter E m . The operators σ ( i ) z,x , i = 1 , ρ ψ + = (cid:12)(cid:12) ψ + (cid:11)(cid:10) ψ + (cid:12)(cid:12) , (cid:12)(cid:12) ψ + (cid:11) = √ ( (cid:12)(cid:12) (cid:11) + (cid:12)(cid:12) (cid:11) ). In our investigation, we consider the following assumptions:(i) The two charged qubits are identical, namely, E J = E J = E J .(ii) The system is assumed to be is in the degenerate point, namely, n g = n g = 0 . H = − n E J σ (1) x ⊗ I + E J I ⊗ σ (2) x − E m σ (1) z ⊗ σ (2) z o . (2)The master equation which governed the system is given by:2 ρ ( t ) dt = − i [ H , ρ ] + Γ8 X j =1 , (cid:0) σ ( j ) z ρσ ( j ) z − σ ( j ) z σ ( j ) z ρ − ρσ ( j ) z σ ( j ) z (cid:1) (3)where, Γ is the dephasing parameter. By solving this master equation, one obtains ρ ( t ), whichdescribes the evolution of the charged system. This state is described by a square matrix oforder 4 with elements are given by: ̺ = ̺ = Υ n λ exp[2Γ t ] − λ µ − R +1 − λ µ + R +2 o ,̺ = ̺ = Υ n λ exp[2Γ t ] + λ µ − R +1 + λ µ + R +2 o ,̺ = ̺ = Υ n λ exp[ − t ] + λ µ − R − + λ µ + R − o ,̺ = ̺ = Υ n λ exp[ − t ] − λ µ − R − − λ µ + R − o ,̺ = ̺ = ̺ = ̺ = Υ n E m h cosh (cid:16) √ λ t (cid:17) − cos (cid:16) √ λ t (cid:17)i + i √ h λ sinh (cid:16) √ λ t (cid:17) + λ sin (cid:16) √ λ t (cid:17)io ,̺ = ̺ = ̺ = ̺ = ̺ ∗ (4)where, Υ = exp( − t ) / λ , Υ = E j exp( − t )8 λ , λ = λ λ λ ,µ ± = λ ± (Γ + E m − E J ) R ± = √
2Γ sin( √ λ t ) ± λ cos( √ λ t ) ,R ± = √
2Γ sinh( √ λ t ) ± λ cosh( √ λ t ) ,λ , = q λ ± (cid:0) E m + E j − Γ (cid:1) ,λ = q E m + (cid:0) E j − Γ (cid:1) + 2 E m (cid:0) E j + Γ (cid:1) . (5)The eigenvalues of the density operator (4) are given by: ǫ = ̺ − ̺ , ǫ = ̺ − ̺ , ǫ , = − p α + β µ ± (6)and the corresponding eigenvectors are given by: | V i = 1 √ − , , , , | V i = 1 √ , − , , , | V i = 1 p µ − ) (1 , µ − exp( − iθ ) , µ − exp( − iθ ) , , | V i = 1 p µ ) (1 , µ + exp( − iθ ) , µ + exp( − iθ ) , , (7)where, 3 = tan − (cid:18) βα (cid:19) ,α = E j E m exp( − t )4 λ h cosh (cid:16) √ λ t (cid:17) − cos (cid:16) √ λ t (cid:17)i ,β = √ E j exp( − t )8 λ h λ sinh (cid:16) √ λ t (cid:17) + λ sin (cid:16) √ λ t (cid:17)i µ ± = 14 p α + β (cid:16) − ̺ − ̺ + ̺ + ̺ ± p ( ̺ + ̺ + ̺ + ̺ ) + 16 ̺ ̺ (cid:17) . (8) In the estimation theory, the quantum Fisher information plays a central role. There are someparameters cannot be quantified directly, so quantum Fisher information can be used to estimatethese parameters [20]. Let η represents the parameter to be estimated and the density opera-tor of the spectral decomposition is given as ̺ η = P ni =1 E i |V i i hV i | where, E i and |V i i are theeigenvalues and eigenvectors of of the density operator ρ η , respectively. The Fisher informationcorresponding to the parameter η is given by (see [1, 11, 21, 22]): F η = F ηC + F ηP − F ηM , (9)where, F ηC = n X i =1 E i (cid:18) ∂ E i ∂η (cid:19) , E i = 0 (10) F ηP = 4 n X i =1 E i (cid:18)(cid:28) ∂ V i ∂η (cid:12)(cid:12)(cid:12)(cid:12) ∂ V i ∂η (cid:29) − (cid:28) V i (cid:12)(cid:12)(cid:12) ∂ V i ∂η (cid:29)(cid:19) , F ηM = 8 n X i = j E i E j E i + E j (cid:12)(cid:12)(cid:12) (cid:28) V i (cid:12)(cid:12)(cid:12) ∂ V i ∂η (cid:29) (cid:12)(cid:12)(cid:12) , E i + E j = 0Using the eigenvalues ǫ i and the eigenvectors (cid:12)(cid:12) V i (cid:11) of the density operator (4), one obtainsthe explicit form of Fisher information for arbitrary parameter η as: F ηC = X i =1 ǫ i (cid:18) ∂ǫ i ∂η (cid:19) , (11) F ηP = 4 ǫ (cid:2) µ ′− + µ − θ ′ (cid:3) (1 + µ − ) + 4 ǫ (cid:2) µ ′ +2 + µ θ ′ (cid:3) (1 + µ ) , F ηM = 8 ǫ ǫ ( ǫ + ǫ ) (1 + µ ) (1 + µ − ) n ( µ + − µ − ) h(cid:0) µ − (cid:1) µ ′ + (cid:0) µ (cid:1) µ ′ − i +2 µ µ − (cid:0) µ (cid:1) (cid:0) µ − (cid:1) θ ′ o , (12)4 F Γ ( a ) t t ( b ) F Γ Figure 1: Fisher information F Γ as a function of time t . In ( a ), we set Γ = 0 . E j = E m = 0 . , . , .
2, respectively. For( b ), we set E j = E m = 0 . . , . , .
5, respectively.where µ ′± = ∂µ ± ∂η and θ ′ = ∂θ∂η . In the following subsection, we investigate the initial parametersof the charged qubit E j and E m and the channel noisy parameter Γ. Γ In this subsection, we estimate the parameter Γ by evaluating the corresponding Fisher infor-mation, namely F Γ . In Fig.(1a), we investigate the effect of E i on the dynamics of F Γ . It isclear that, at t = 0, F Γ is zero. However, as soon as the interaction is switched on, the Fisherinformation increases suddenly to reach its maximum value. For further values of t , F Γ decreasesfast to reach its minimum value. The minimum values depend on the values of E i , where theseminimum values are larger for larger values of E i . On the other hand, this behavior is unchangedfor further values of t .In Fig.(1b), we investigate the effect of different initial values of Γ on the F Γ , where we set E j = E m = 0 .
1. It is evident that, for small values of t , a similar behavior is depicted as thatshown in Fig.(1a), i.e., the sudden increasing behavior is displayed as soon as the interactionis switched on. Fisher information reaches its upper bound as t increases. However, the upperbounds depend on the initial values of Γ. It is clear that, as the initial values of Γ increases,the upper bounds of F Γ increases. For further values of t , the Fisher information decreases fastto reach its lower bounds at different values of t depending on the initial values Γ. Also, F Γ behaves continently as the interaction time increases.From Fig.(1), one concludes that, for small range of the interaction time, one can estimatethe parameter Γ, with high probability. One can increase, the degree of estimation by startingwith a larger values of the parameter Γ or E i parameters. The larger values of interaction timehas no effect on the degree of estimation. 5 F G a nd F i G ( a ) t F G a nd F i G t ( b )Figure 2: Fisher information F Γ and its components F Γ i , i = C, P, M as a function of time t whenΓ = 0 .
4. The black-solid, red-dotted, blue-dashed and green-dotdashed curves for F Γ , F Γ C , F Γ P and F Γ M , respectively. We set in ( a ) E j = E m = 0 . b ) E j = E m = 0 . F Γ ; the classical Fisherinformation F C , the pure Fisher information F P and the mixture of pure state F M on thebehavior of F Γ . It is clear that, the for small values of t the F C has the the large contribution.However, as t increases, the pure part F P and the mixture part F M increase, therefor the Fisherinformation F Γ decreases. The effect of the energy parameters E j , and the coupling parameter, E m can be seen by comparing Fig.(2a) and Fig.(2b), where the the rate of increasing F P and F M in Fig.(2a) is smaller than that depicted in Fig.(2b). This shows that as the energy and thecoupling parameters increase, the entangled charged state turns into a product pure states andthe possibility of construct the state of the charged qubit as a mixture of pure states increases.From Fig.(2), we can see that as soon as the interaction is switched on, the classical part F C increases, namely, the entangled qubits lose their quantum correlation. For larger time there aresome pure states are generated and consequently the pure Fisher information increases. Also,the initial state behaves as a mixture of pure state, namely F M increases. E j Fig.(3a) displays the behavior of Fisher information corresponding to the energy parameter E i ,where we set E = E = E . In Fig.(3a), we fix the value of the depshing parameter Γ anddifferent initial energies are considered. It is clear that, F E j increases as soon as the interactionis switched on. The upper bounds depend on the initial values of E , where for small initialvalues of E , the upper bounds of Fisher information F E j are larger than those depicted for smallvalues of E .The effect of different values of the depshing parameter Γ is displayed in Fig.(3b), wheredifferent values of Γ are considered. It is clear that, as Γ increases, the Fisher information F E j decreases. Moreover, for small values of the depashing parameter, Fisher information increases6 F E j ( a ) t F E j ( b ) t Figure 3: Fisher information F E j as a function of time t . The same parameters and propertiesin Fig. 1 are used F E j a nd F i E j ( a ) t F E j a nd F i E j ( b ) t Figure 4: Fisher information F E j and its components F E j i , i = C, P, M as a function of time t ,with the same parameters in Fig. 2as t increases. However as Γ increases further, the Fisher information decreases.The effect of the different part of the Fisher information are displayed in Fig.(4a), wherethe classical part has the larger contribution, while the pure and mixed parts have a smalleffect. This behavior is changed dramatically as one increases the energy parameter E , whereas the interaction time t increases, the pure and the mixed parts of Fisher information rapidlyincreasing . This explain that why the upper bounds of the Fisher information at E = 0 , E = 0 .
2. Also, as it is shown in Fig.(4b), the charged qubit turnsinto product pure state and the possibility of representing it as a mixture of pure states increasesas the interaction time increases. E m To estimate the coupling parameter E m , we evaluate the corresponding Fisher information F E m .The effect of different initial coupling on the Fisher information F E m is displayed in Fig.(5a),7 F E m ( a ) t F E m ( b ) t Figure 5: Fisher information F E m as a function of time t . The same parameters and propertiesin Fig. 1 are used F E m a nd F i E m ( a ) t F E m a nd F i E m ( b ) t Figure 6: Fisher information F E m and its components F E m i , i = C, P, M as a function of time t , with the same parameters in Fig. 2where we fixed the values of the dephasing end energy parameters. It is evident that, as soonas the interaction is switched on, the Fisher information increases F E m increases as t increases.However, the upper bounds of F E m , depend on the coupling parameter, where its upper boundsare small for larger values the coupling parameter. For larger values of the energy parameter( E = 0 . F P has the largest contribution, while the classical part F C has the smallest contribution. However as one increases the energy and coupling parameters,the classical part and the mixed part increases also for larger interaction time, the pure partdecreases. This explain the decreases of the upper bounds of the Fisher information in Fig.(6b).8rom Fig.(6) one can conclude that one can increase the possibility of estimating the couplingparameter either by starting with small values of this parameter with small energy, or by largerinitial value of the coupling and energy parameters. Also, the possibility of behaving the singletstate as pure increases for smaller values of the energy and the coupling parameters. In this contribution, we investigated the dynamics of a charged two-qubit system initially pre-pared in a maximum entangled state. Each particle interacts independently with a noisy de-phasing channel. The final state of the charged qubits was obtained analytically. We estimatedthe channel parameter as well as the energy parameters by means of Fisher information.We showed that, Fisher information with respect to the energy increases suddenly as soon asthe interaction is switched on to reach its upper bounds. The upper bounds of Fisher informationdepend on the interaction time, the initial values of the qubits’ energy each the charged qubit.For further values of the interaction time, the Fisher information decreases gradually, where thedecay rate increases as the energy increases. The long-lived behavior Fisher information withrespect to the noisy parameter is clearly depicted for larger values of interaction time. Meanwhile,Fisher information with respect to the energies of the charged qubits increases suddenly as theinteraction time increases with larger values of the noisy strength. On the other hand thebehavior of Fisher information with respect to the mutual coupling constant increases graduallyas the noisy strength increases.The effect of the different parts of the Fisher information is discussed for all cases. We showedthat classical part has the larger contribution on the total Fisher information with respect to thenosy and energy parameters, while for the mutual coupling the pure part has the largest effect.This result is changed for larger values of the initial energy, where the classical has a largesteffect on the mutual parameter’s information.From the behavior of the differen parts of Fisher information, one deduced when the finalstate turns into a separable pure state. It is clear that, the initial charged qubit system is robustagainst the larger energy, since there still a contribution from the pure and mixed parts.
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