Based-nonequilibrium-environment non-Markovianity, quantum Fisher information and quantum coherence
BBased-nonequilibrium-environment non-Markovianity, quantum Fisher informationand quantum coherence
Danping Lin, Hong-Mei Zou, ∗ and Jianhe Yang Synergetic Innovation Center for Quantum Effects and Application,Key Laboratory of Low-dimensional Quantum Structures and Quantum Control of Ministry of Education,School of Physics and Electronics, Hunan Normal University, Changsha, 410081, P.R. China.
In this work, we investigate the non-Markovianity, quantum Fisher information (QFI) and quan-tum coherence of a qubit in a nonequilibrium environment and have obtained the expressions ofQFI and quantum coherence as well as their relationship. We have also discussed in detail the influ-ences of the different noise parameters on these quantum sffects. The results show that the suitableparameters of the nonequilibrium environment can retard the QFI and quantum coherence in bothMarkovian and non-Markovian regions. In addition, the smaller memory effects and the larger thejumping rate, the greater the QFI and quantum coherence. And a larger QFI naturally correspondsto a larger quantum coherence, which indicates that the quantum coherence can enlarge the QFIand can effectively enhance the quantum metrology.
PACS numbers:
Keywords:
Non-Markovianity; Quantum Fisher information; Quantum Coherence; Nonequilibrium environment
I. INTRODUCTION
The parameter estimation is a very basic and impor-tant content in quantum information theory. Quantummetrology can attain a measurement precision that sur-passes the classically achievable limit by using quan-tum effects. As we know, the metrology precisioncan be raised by increasing quantum Fisher informa-tion (QFI)[1]. And the Braunstein-Caves theorem provedthat the QFI basing on the symmetric logarithmic deriva-tive(SLD) contains all possible information[2]. On theother hand, quantum coherence(QC) isAs we all knows, the memory effect of the environ-ment is caused by the long time correlation betweenthe system and the environment and the environmentcan be divided into Markovian and non-Markovian typesaccording to the memory effects. Many methods forquantifying non-Markovian features have been proposed,such as Rivas-Huelga-Plenio (RHP) measure[10], Breuer-Laine-Piilo (BLP) measure[11] and Luo-Fu-Song (LFS)measure[12]. Besides, Zhi He et al.[13] has used theBLP measurement to calculate the non-Markovianity forsingle-channel open systems, and they obtain a very tightlower bound of non-Markovianity. Yu Liu et al.[14] hascalculated the non-Markovianity in the dissipative cavityand used it to explain the dynamical behavior of quan-tum coherence.Recently, the influence of environments on the QFIand quantum coherence of quantum systems has becomean important research topic. ...... one of our authorshave obtained the analytical solution of QFI through theSLD method and have enhanced or retarded the QFI bythe dissipation cavity[3]. Very recently, more and moreattention has been paid to study the relationship be- ∗ [email protected] tween different quantum effects. For examples, Zou andFang have discussed that the discord and entanglementin non-Markovian environments at finite temperatures[4];Zou has discussed the influence of the non-Markovian ef-fect and detuning on the lower bound of the quantumentropic uncertainty relation and entanglement witnessin detail[5]; Hou has studied the relationship betweenquantum discord and coherence[6]. Feng has verifiedthat the QFI can be utilized to quantify the quantumcoherence[7]. Wang has shown that quantum coherencealso increases the precision of parameter estimation[8].Although many important progresses have been ac-quired in experimental and theoretical researches on thequantum Fisher information and quantum coherence ofopen quantum systems, these investigations mentionedabove are mainly focused on the Lorentzian and Ohmicenvironments. Actually, the nonequilibrium environ-ment of dichotomic nature has been observed in someexperiments[25–27], and the steady-state entanglementand coherence of two coupled qubits[15] and the decoher-ence induced by non-Markovian noise[18] in a nonequilib-rium environment are investigated. In our work, we willstudy the non-Markovianity, QFI and quantum coher-ence of a qubit in the nonequilibrium environment. Ourmain purpose are to understand whether these quantumeffects of a qubit in a nonequilibrium environment aredifferent from those in a Lorentzian and an Ohmic en-vironments, and to understand how nonequilibrium en-vironment parameters affect these quantum effects. Inaddition, we also hope to get the relationship betweenthe QFI and quantum coherence. Hence, we believe thatthe study of the non-Markovianity, QFI and quantum co-herence of a qubit in the nonequilibrium environment isalso important and meaningfulThe outline of the paper is organized as follows. InSection II, we give the model of a qubit in a nonequilib-rium environment. In Section III, we calculate the non-Markovianty, QFI and quantum coherence of the qubit in a r X i v : . [ qu a n t - ph ] D ec a nonequilibrium environment. In Section IV, we analyzethe influence of the nonequilibrium parameter, the mem-ory effects and the jumping rate on the non-Markovianity,QFI and coherence. Finally, we end with a brief summaryof important results in Section V. II. PHYSICAL MODEL
We consider a qubit coupled to a nonequilibrium envi-ronment. The Hamiltonian is written as[16] H = (cid:126) ω + ξ ( t )] σ z , (1)where σ z and ω are the Pauli matrix and the intrinsictransition frequency for the qubit, respectively. ξ ( t ) de-notes the environmental noise caused by environmentaleffects, which is originally introduced by A. Fuli´nski[17].For simplify, we suppose that ξ ( t ) is subject to a non-stationary and non-Markovian stochastic process in thispaper.The time evolution of the system is described by theLiouville equation ∂∂t ρ ( t ; ξ ( t )) = − i (cid:126) [ H ( t ) , ρ ( t ; ξ ( t ))] , (2)where the notation ρ ( t ; ξ ( t )) indicates the density opera-tor under the environmental noise ξ ( t ). The reduced den-sity operator of the qubit can be derived by taking an av-erage over the environmental noise as ρ ( t ) = (cid:104) ρ ( t ; ξ ( t )) (cid:105) .By solving Eq. (2), the density matrix in the basis {| e (cid:105) , | g (cid:105)} can be given in the following from ρ ( t ) = (cid:18) ρ ee (0) ρ eg (0) e − iω t G ∗ ( t ) ρ ge (0) e iω t G ( t ) ρ gg (0) (cid:19) , (3)where the G ( t ) = (cid:104) e i (cid:82) t dt (cid:48) ξ ( t ) (cid:105) is a complex time-dependent function.According to Ref.[18], the environmental noise ξ ( t )is subject to a nonstationary random telegram processthat randomly jumps back and forth between − ν and+ ν at an average rate of λ . The noise process satis-fies the conditional probability such as ∂∂t P ( ± ν, t | ξ (cid:48) , t (cid:48) ) = ∓ (cid:82) tt (cid:48) dt (cid:48) K ( t − τ )[ λP ( ν, τ | ξ (cid:48) , t (cid:48) ) − λP ( − ν | ξ (cid:48) , t (cid:48) )], where K ( t − τ ) represents a generalized memory kernel. By de-riving the calculation, the initial condition of this work isset to P ( ξ ) = 2 − [(1 ± a ) δ ξ , ± ν ]. And the memory kernel K satisfies K ( t − τ ) = κe − κ ( t − τ ) . The environment noisecan be characterized by the nonequilibrium parameter a ( | a |≤
1) and the memory decay rate κ ( κ >
0) as well asthe amplitude switches randomly with the jumping rate λ between the values ± ν ( ν > a = 0 indicates thatthe environment is equilibrium (and a (cid:54) = 0 vice versa).The smaller κ and bigger ν represent the stronger non-Markovianity. Under nonequilibrium environment, G ( t ) in Eq. (3)can be analytically solved as[18] G ( t ) = L − [ G ( s )] , G ( s ) = s + ( κ + iaν ) s + κ (2 λ + iaν ) s + κs + (2 κλ + ν ) s + κν , (4)where L − indicates the inverse Laplace transform andthe initial condition is G (0) = 1.Using G (0) = 1, one can obtain easily G ( s ) = ( s − u + )( s − u − )( s − s )( s − s )( s − s ) where u ± are roots of s + ( κ + iaν ) s + κ (2 λ + iaν ) = 0, and s j ( j = 1 , ,
3) are roots of s + κs + (2 κλ + ν ) s + κν = 0We can acquire the exact solution of G ( t ) as[19], i.e. G ( t ) = (cid:88) j =1 G ( s ) e s j t . (5)Let an arbitrary initial state is | ψ (cid:105) = cos θ | e (cid:105) + e iφ sin θ | g (cid:105) . The reduced density matrix ρ ( t ) in Eq. (3)can write as ρ ( t ) = (cid:18) ρ ee ( t ) ρ eg ( t ) ρ ge ( t ) ρ gg ( t ) (cid:19) , (6)with the elements ρ ee ( t ) = 12 (1 + cos θ ) ,ρ eg ( t ) = 12 sin θe − i ( φ + ω t ) G ∗ ( t ) ,ρ ge ( t ) = ρ ∗ eg ( t ) ,ρ gg ( t ) =1 − ρ ee ( t ) . (7) III. NON-MARKOVIANITY, QFI ANDQUANTUM COHERENCE
In the three subsections, we briefly review the mea-surements of the non-Markovianity, QFI and quantumcoherence.
A. non-Markovianity
In this part, we introduce the non-Markovianity of thequbit by using the BLP measure, which based on thetrace distance between two quantum states ρ and ρ [11].The non-Markovianity N is defined as N = max ρ (0) ,ρ (0) (cid:82) σ> σ [ t, ρ (0) , ρ (0)] dt , (8)where σ [ t, ρ (0) , ρ (0)] is the change rate of the tracedistance defined as σ [ t, ρ (0) , ρ (0)] = ddt D [ ρ ( t ) , ρ ( t )],and the trace distance is defined as D [ ρ ( t ) , ρ ( t )] = T r | ρ ( t ) − ρ ( t ) | with | A | = √ A † A and 0 ≤ D ≤ D = 0, the two state are the same, and if D = 1,the two state are totaly distinguishable. That is, when ρ (0) = | e (cid:105)(cid:104) e | and ρ (0) = | g (cid:105)(cid:104) g | , the trace distance ismaximum. B. Quantum Fisher Information
Based on thee classical Fisher information (CFI)[20–22], the quantum Cram´er-Rao (QCR) theorem of theQFI has also be proposed, which is (cid:104) ( (cid:52) ˆ λ ) (cid:105) λ ≥ N F λ ,where ˆ λ is the unbiased estimator of parameter λ . TheQFI theorem gives the highest accuracy of parameter es-timation in the case of quantum mechanics. The smallerthe (cid:104) ( (cid:52) ˆ λ ) (cid:105) λ , the larger the F λ . That is, we can get thehigher the quantum metrology precision.Using positive-operator valued measurement (POVM), F λ can be written as F λ = T r ( ρ λ L λ ) = T r ( ∂ λ ρ λ L λ ) , (9)this is quantum Fisher information (QFI), where L λ issymmetric logarithmic derivatives(SLD) for the param-eter λ , which is a Hermitian operator determined by ∂ λ ρ λ = { ρ λ , L λ } where ∂ λ ≡ ∂∂λ and {· , ·} denotes theanticommutator. If the density operator ρ λ satisfies thespectral decomposition ρ λ = (cid:80) i p i | ψ i (cid:105)(cid:104) ψ i | and withoutconsidering the existence of zero eigenvalues, the F λ canbe expressed as follows F λ = (cid:80) i (cid:48) ( ∂ λ p i (cid:48) ) p i (cid:48) + 2 (cid:80) i (cid:54) = j ( p i − p j ) p i + p j |(cid:104) ψ i | ∂ λ ψ j (cid:105)| , (10)Using Eq. (9) to calculate the QFIs of parameter θ and φ . Inserting Eq. (6) into Eq. (9), the QFIs are obtainedas F θ = 1 F φ = | G ( t ) | sin θ , (11)From Eq. (11), it is known that F θ is always 1 and the F φ is determined by two factors, G ( t ) and θ . C. Quantum Coherence
A reasonable measure to quantify quantum coher-ence should fulfill the following condition: non-egativity,monotonicity, strong monotonicity, convexity, uniquenessfor pure states and additivity[23]. According to a set ofproperties of quantum coherence measure, some coher-ence measures are proposed. Here, we focus on the l norm of coherence[24]. From Eq. (6) it can be expressedas C l ( t ) = (cid:88) i,j ( i (cid:54) = j ) | ρ ij ( t ) | = | G ( t ) || sin θ | , (12)where | ρ ij ( t ) | ( i (cid:54) = j ) is the absolute value of the non-diagonal elements ρ ij ( t ) of the density matrix ρ ( t ).Comparing Eq. (11) and Eq. (12), we can obtain theanalytical relationship between the F φ and the C l ( t ),which is F φ = C l ( t ) , (13)From Eq. (12), we know that the QFI will enlarge withthe quantum coherence increasing, which indicates thatthe quantum coherence can augment the QFI and caneffectively enhance the quantum metrology. IV. DISCUSSION AND RESULTS
In this section, we illustrate the influence of envi-ronmental parameters (the nonequilibrium parameter a ,the memory decay rate κ and the jumping rate λ ) onnon-Markovianity, QFI and quantum coherence. Thequantum system exhibits a Markovian behavior when( ν/λ < ν/λ > N as func-tion of the dimensionless quantity ν/λ , which shows that N is dependent on ν/λ . However, due to the differ-ence between the selected parameters and the calculationmethod, the curve change of N is different from that ofthe Ref.[16]. In short, N increases as ν/λ increases. Forthe same equilibrium parameter a , the larger the valueof ν is, the larger N is. For the same the value of ν/λ , the smaller the equilibrium parameter a , the larger N .As is plotted in Fig. 2, N associated to the memory de-cay rate κ/λ . N decreases as κ/λ increases, and N willeventually disappear when κ → ∞ . When ν = 0 . λ and κ/λ = 0, the maximum value of N can reach 4.0, indicat-ing that the non-Markovian effect is very obvious. When ν = 0 . λ and κ/λ = 10, the non-Markovianity is zero.The dynamic behavior of the qubit presents Markoviancharacteristics. For the same ν , the smaller κ , the larger N . And, the non-Markovianity has the similar dynamicbehaviors in the cases of ν = 0 . λ and ν = 4 λ , while the N is larger in the former than in the latter. / λ a = = = FIG. 1: (Color online) Non-Markovianity N as afunction of the dimensionless quantity ν/λ . Here θ = π , φ = 0. The memory decay rate κ = 8 λ. Fig. 3 is plotted F φ and C l as a function of λt underdifferent values of κ and ν for a = 0 .
5. Fig. 3 (a) showsthat the F φ has the periodic oscillation when κ = 0 and ν = 0 . λ . However, when κ = 10 λ and ν = 0 . λ , the F φ decays monotonically, which implies that the F φ dynam-ics of the qubit becomes Markovian. Therefore, with κ decreasing, the evolution of the qubit will change fromMarkovian to non-Markovian, corresponding to the ten-dency of F φ from monotonical decay to oscillation. Asshown in Fig. 3 (b), when κ = 10 λ and ν = 4 λ , the F φ could also take the behavior of oscillations. In addition,the maximum value of oscillation recovery of F φ increases κ / λ ν = λν = λ FIG. 2: (Color online) Non-Markovianity N as afunction of the dimensionless quantity κ/λ . Here θ = π , φ = 0, a = 0 . ( a ) λ t ℱ ϕ ( b ) λ t ℱ ϕ ( c ) λ t ℓ ( t ) ( d ) λ t ℓ ( t ) FIG. 3: (Color online) QFI F φ and quantum coherence C l as a function of λt for a = 0 . κ : κ = 0 (red dot dash line), κ = λ (blue dotted line), κ = 10 λ (green line). (a) and(c) ν = 0 . λ ; (b) and (d) ν = 4 λ . The other parameterare θ = π and φ = 0. FIG. 4: (Color online) QFI F φ as a function of thedimensionless quantity λt and the memory decay rate κ for a = 0 .
5. Here θ = π and φ = 0. (a) ν = 0 . λ ;(b) ν = 4 λ .with the decreasing of κ , which implies that the smallerthe value of κ/λ , the more obvious the non-Markovianeffects. In Fig. 3 (c) and (d), C l is plotted as a functionof λt for a = 0 . κ and ν . Fig. 3(c) displays an oscillatory behavior of C l when κ is verysmall and ν = 0 . λ . However, C l decays monotonicallyfor larger κ , which shows that the dynamical evolutionbecomes Markovian. Similar to the case of Fig. 3 (c),as κ increase, the C l rapidly oscillations to zero when ν = 4 λ , as shown in the Fig. 3 (d).In order to observe comprehensively the dependence of F φ on κ and λt , we can plot a three-dimensional diagram.Fig. 4 is plotted F φ as a function of the dimensionlessquantity λt and the memory decay rate κ for a = 0 . F φ appears the phenomenon ofrapidly revivals when κ/λ = 10 λ and ν = 0 . λ . How-ever, when κ/λ (cid:29) ν = 0 . λ , F φ appears decayphenomenon. Fig. 4(b) shows that no matter what κ is, F φ appears the phenomenon of oscillations. The mem-ory effect is the reason for the revival of F φ . Namely, thememory effect can enhance the non-Markovian behaviorand the value of QFI. But, the oscillatory behavior ofthe F φ does not disappear in the memoryless limit. Thethree-dimensional picture of C l ( t ) is similar to the F φ , weomit it in order to reduce the space.In Fig.5, we plot the F φ as a functions of the dimen-sionless quantity λt and the memory decay rate κ/λ for a = 0 .
5. From this figure, it can be seen that the smallerthe value of κ/λ and λt , the larger the value of F φ . Mean-while, the evolution behavior of C l ( t ) is similar to the F φ , we omit it in order to reduce the space. Further-more, as is shown above, the evolution behavior of C l issimilar qualitatively to the F φ in both Markovian andnon-Markovian regions. That is, the smaller κ is benefi-cial to the F φ and C l . Comparing Fig. 3 (a) and (c) (or ( a ) λ t κ / λ ( b ) λ t κ / λ FIG. 5: (Color online) F φ as a functions of thedimensionless quantity λt and the memory decay rate κ for a = 0 .
5. Here θ = π and φ = 0. (a) ν = 0 . λ ;(b) ν = 4 λ .Fig. 3 (b) and (d)), we can see the tendency of F φ and C l with ν = 0 . λ and ν = 4 λ is coincident.In Fig. 6(a), F φ is plotted as a function of the dimen-sionless quantity λt for the different nonequilibrium pa-rameter a in the Markovian dynamic region ( ν = 0 . λ ).As is shown above, there is a little difference among thefive lines and F φ decreases monotonically with time t andvanishes in the asymptotic limit t → ∞ , which impliesthe accuracy of the parameter estimation is smaller. Ifthe environment is out of equilibrium (0 < | a | ≤ | a | , the greater the initial valueof F φ . But they all close to zero when time t increases.And the curves of F φ are coincident for a = 0 . a = − .
5, the curves of F φ are also coincident for a = 1and a = −
1. The inset of Fig. 6(a) shows that theinitial value of F φ will be larger when the system is faraway from the equilibrium environment. In Fig. 6(b), F φ is plotted as a function of the dimensionless quan-tity λt for the different nonequilibrium parameter a inthe non-Markovian dynamic region ( ν = 4 λ ). It is seenthat the initial value of F φ has the different value of thenonequilibrium parameter a . F φ appears oscillating re-covery phenomenon with time t due to the feedback ef-fects of the non-Makovian environment. Similar to thecase of the inset of Fig. 6(a), the initial value of F φ will ( a ) λ t ℱ ϕ ( b ) λ t ℱ ϕ ( c ) λ t ℓ ( t ) =- = =- = = ( d ) λ t ℓ ( t ) FIG. 6: (Color online) QFI F φ and coherence C l as afunction of dimensionless quantity λt for the initialstate | ψ (cid:105) . (a), (c) ν = 0 . λ ; (b), (d) ν = 4 λ . The memorydecay rate κ = 8 λ. The insets show the evolution of ashort time. The other parameter are θ = π and φ = 0.be larger as | a | increases, as shown the inset of Fig. 6(b).In Fig. 6(c), C l is plotted as a function of the dimen-sionless quantity λt for different the a in the Markovianregion ( ν = 0 . λ ). In Fig. 7 (d), we plot the time evo-lution of the C l for θ = π , φ = 0 and ν = 4 λ . It canbe seen that the C l oscillates damply to zero. This fig- ( a ) λ t ℱ ϕ ( b ) λ t ℱ ϕ FIG. 7: (Color online) Left: QFI F φ as a function of λt when a = 0, κ = 0, θ = π and under different values of ν : ν = 0 . λ (red line), ν = 0 . λ (blue line), ν = 0 . λ (black line), ν = 0 . λ (green line). Right: QFI F φ as a function of λt when a = 0, κ = 0, ν = 0 . λ andunder different values of θ : θ = π (red line), θ = π (blueline), θ = π (black line), θ = π (green line).ure shows that the C l in the nonequilibrium environmentis greater than the equilibrium environment in the shorttime. But the C l will be close to zero in a long evolutiontime. Obviously, the nonequilibrium parameter a playsthe important role in the dynamic revolutions of the F φ and C l . When | a | is greater, the initial value of coherenceis larger, as shown the inset of Fig. 6(d).As a comparison, when the environment is out of equi-librium, the C l and F φ will be delay in both Markovianand non-Markovian regions, which indicates that the big-ger ν , the slower down the evolution of C l and F φ . Forexample, comparing Fig. 6(a) and Fig. 6(c), it can beseen that the tendency of C l and F φ is very similar in ν = 0 . λ . The tendency of Fig. 6(b) and Fig. 6(d) is likein ν = 4 λ .In the above Figures 1-6, we have discussed in detailthe influences of different noise parameters on the dynam-ical behaviors of QFI and QC of the qubit in a nonequi-librium environments. In order to better understand thedifference between nonequilibrium and equilibrium envi-ronments, we also draw the dynamical curves of the QFIof the qubit in the equilibrium environment, as shown inthe Figures 7(a) and (b). Figures 7(a) shows that thesecurves share the same beginning for different ν values and Figures 7(b) indicates that these curves also sharethe same beginning for different initial states when thenonequilibrium environment parameter( a = 0) and thememory decay rate( κ = 0). In the equilibrium environ-ment, the dynamical curves of C l ( t ) are similar to that ofthe F φ so we omit it in order to reduce the space.Comparing Figure 7 and Figures 1-6, we can knowthat, in the equilibrium environment, the different noiseparameters only alter their time evolution but not theirinitial values. However, in the nonequilibrium environ-ment, the different noise parameters alter not only theirtime evolution but also their initial values. V. CONCLUSION
In summary, we have investigated the non-Markovianity, QFI and quantum coherence of asingle qubit coupled to a nonequilibrium environmentand have obtained the expressions of QFI and quantumcoherence as well as the relationship between QFI andquantum coherence. Their relationship ( Eq. (12))indicates that the quantum coherence can enlarge theQFI and can effectively enhance the quantum metrology,which gives us a novel understanding of the relationshipamong quantum resources. We have also discussed indetail the influences of the different noise parameters onthe non-Markovianity, QFI and quantum coherence. Theresults show that, the suitable environment parameters,including the nonequilibrium parameter, the memorydecay rate and the jumping rate, are beneficial toenhance the non-Markovianity and slower the decayingof QFI and quantum coherence. And we utilized thenon-Markovianity to explain the dynamical behaviors ofthe QFI and the quantum coherence. Besides, in theequilibrium environment, the different noise parametersonly alter their time evolution but not their initialvalues. However, in the nonequilibrium environment,the different noise parameters alter not only their timeevolution but also their initial values.These results give us an active way to suppress deco-herence, which is quite significant in quantum informa-tion processing and quantum metrology. These would bea very interesting issue that needs further attention. Wehope that our work is valuable to understand the com-mon feature of quantum resources and the relationshipbetween QFI and quantum coherence. In future work,we will study the essential relations between quantumresources and try to provide a unified framework of them.
ACKNOWLEDGMENTS
Project supported by the Scientific Research Project ofHunan Provincial Education Department, China (GrantNo 16C0949) and the National Science Foundation ofChina (Grant No 11374096). [1] R. A. Fisher, 1925 Theory of statistical estimation.
Proc.Camb. Philos. Soc. [2] S. L. Braunstein and Carlton M. Caves 1994 Phys. Rev.
Lett Chin. Phys. B Chin. Phys. B Phys. Scripta Phys. Rev. A Scientific Reports New Journalof Physics Quantum Inf.Process Phys. Rev.
Lett
Phys. Rev.
Lett
Phys. Rev. A Phys. Rev. A Chin. Phys. B Phys. Rev. A Phys. Rev. A Phys. Rev. E Phys. Rev. A Communications of the Acm Journal of Statistical Physics
Probabilistic and Statistical As-pects of Quantum Theory (Amsterdam: North-Holland/American Elsevier)[22] M. G. A. Pairs 2009
Int. J. Quant. Inf Rev. mod.phys Phys. Rev.
Lett113