Suppression of Decoherence and Disentanglement by the Exchange Interaction
SSuppression of Decoherence and Disentanglement by the Exchange Interaction
Amrit De, Alex Lang, Dong Zhou, and Robert Joynt
Department of Physics, University of Wisconsin - Madison, WI 53706 (Dated: November 2, 2018)Entangled qubit pairs can serve as a quantum memory or as a resource for quantum communi-cation. The utility of such pairs is measured by how long they take to disentangle or decohere.To answer the question of whether qubit-qubit interactions can prolong entanglement, we calculatethe dissipative dynamics of a pair of qubits coupled via the exchange interaction in the presenceof random telegraph noise and 1 /f noise. We show that for maximally entangled (Bell) states, theexchange interaction generally suppresses decoherence and disentanglement. This suppression ismore apparent for random telegraph noise if the noise is non-Markovian, whereas for 1 /f noise theexchange interaction should be comparable in magnitude to strongest noise source. The entangledsinglet-triplet superposition state of 2 qubits ( ψ ± Bell state) can be protected by the interaction,while for the triplet-triplet state ( φ ± Bell state), it is less effective. Thus the former is more suitablefor encoding quantum information.
PACS numbers: 03.67.-Pp, 03.65.-Yz, 05.40.-a
I. INTRODUCTION
Much theoretical and experimental effort has been di-rected towards studying the viability of quantum infor-mation processing (QIP) in recent years due to a se-ries of remarkable QIP algorithms [1–3]. Initial concernsabout quantum coherence being too fragile to be use-ful have been partially dispelled with the discovery ofquantum error-correcting codes [4–7], quantum thresh-old theorems[7–10], decoherence-free subspaces [11–13]and dynamical decoupling by using optimized pulsed se-quences [14–17].Qubit decoherence can be attributed to various sourcesand has been investigated using models such as spin-bath models [18–20], hyperfine interaction models [21–23] and phonon induced decoherence [24–26]. Anothercommon source of decoherence in solid state devices aretwo level systems (TLS) that generate random telegraphnoise (RTN), which with a wide distribution of switchingrates can give rise to 1 /f noise [27, 28]. In a numberof recent semiconductor quantum dot(QD) experiments,RTN is observed when the potential in the dot lines upwith the electrochemical potential in the reservoir, caus-ing electrons to randomly tunnel back and forth betweenthe dot and the reservoir [29–31]. This puts limitationson the performance of a QD qubit, as such a randomtelegraphic current can modulate the QD’s orbital wave-function which can create magnetic noise via spin-orbitcoupling mechanisms. This type of noise is also known tobe important in superconducting qubits [32], and couldaffect the performance of other types of qubits as well[33–36].This suggests that ways be invented to prolong quan-tum coherence and entanglement in qubit pairs in thepresence of such noise sources. In this paper we presentan extremely encouraging set of results – that the ex-change interaction between the qubits can be used tosuppress decoherence as well as disentanglement due toRTN. This is a natural choice as the Heisenberg exchange interaction between the qubits is often used in any caseto implement various gate operations such as controlled-NOT and SWAP gates [37–40]. Thus new circuit ele-ments are not required. Typical proposals to suppressdecoherence from TLS that rely on spin echo techniques[27, 28, 41, 42] require additional resources and systemmonitoring. Our proposal can be used either as an alter-native or even in addition to pulsing.Our aim in this paper is to show that for the maxi-mally entangled Bell states, the interaction between thequbits can be used to suppress decoherence and disen-tanglement. Without any loss of generality, we first ana-lytically and numerically show this effect using a modelwhere a single RTN source is coupled to only one of thetwo qubits. We then consider a model with two uncorre-lated RTN sources, with each of them coupled to a qubitand show that the interaction suppresses quantum dis-sipation in this case as well. Our analytical results sug-gests that a straightforward generalization can be madeto the case of multiple uncorrelated RTN sources. Wesubsequently show that the qubit interaction suppressesdecoherence even if the qubits are coupled to a large num-ber of uncorrelated fluctuators with a 1 /f noise powerspectrum. These results are extremely important forQIP as much of its vaunted capabilities are due to thefact that unlike the classical bit, multiple qubits exhibitquantum entanglement which allows multiple states tobe addressed simultaneously[43]. Alternatively, an effec-tive single qubit can be created from the exchange cou-pled singlet-triplet states, which will be less susceptibleto RTN.The exchange interaction in our work is taken to bean externally controllable parameter. Testolin et al. [44]have proposed a model to show how the two level fluctua-tors itself affect the exchange interaction as a function oftime, while Das [45] has shown that the interactions canlead to periodic disentanglement and entanglement be-tween the qubits in contact with different environments.In general, non-Hermitian Hamiltonians are often used a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b to describe decay processes in open quantum systems[46–50]. Our calculations are done using a recently de-veloped quasi-Hamiltonian formalism which is suitablefor describing the non-unitary temporal evolution of aquantum system acted on by a classical stochastic pro-cess [36, 51, 52]. Similar approaches have been used inRefs.[53, 54]. In many instances, Quantum and classicalnoise models can yield the same solution for decoherenceif the noise power spectrums have the same line-shape.For example, in [55, 56] the analytical expression ob-tained for a qubit’s decoherence in the presence of a ther-mal bosonic reservior in a lossy cavity with Lorentzianbroadening is exactly the same as what we obtain for clas-sical pure dephasing noise. More recently, Saira et al [53]have shown that within the Born approximation, manyfully quantum mechanical noise models can be exactlymapped onto classical stochastic noise models, includingthose for RTN.This paper is organized as follows. First we describeour single fluctuator model and the quasi-Hamiltonianmethod in sec.II. Then in sec.III, we give results and theexact Bloch vector solutions for some important cases.Since for QIP applications the entanglement dynamics atimmediate times is often the most important, in sec.IVwe obtain analytical results for the short time behav-ior for the Bell states for any arbitrary set of noise pa-rameters. Our results are then extended to the case oftwo uncorrelated fluctuators in sec.V. This treatment oftwo fluctuators is then subsequently extended to treat anarbitrary number of uncorrelated fluctuators in sec.VI,where we calculate the temporal dynamics of two inter-acting qubits in the presence of eight RTSs with a 1 /f distribution. Finally we present our conclusions and im-plications for qubit design in sec.VII. II. MODEL AND METHOD
It is well known that in various materials and systemssudden step like transitions occur at random intervalsof time between two or more discrete voltage levels[57].In the case of semiconductors, these random telegraphicsignals (RTS) are often attributed to trapping and releaseof charge carriers by defect sites. If the fuctuators arestatistically independent, then the RTS can be expressedas a sum of the contributions from individual fluctuations g (cid:48) ( t ) = (cid:88) i g i s i ( t ) (1)where, s i ( t ) is a RTN sequence that switches between ± g i is the noise vector for the i th fluctuator and | g | = g is the noise strength. Their au-tocorrelation function is (cid:104) s i ( t ) s j ( t ) (cid:105) ∝ exp( − γ i | t − t | ) δ ij and γ i is the switching rate of the i th fluctua-tor. Each individual fluctuator has a Lorentzian powerspectrum and a broad distribution of γ results in a 1 /f noise power spectrum[57]. Analytical solutions are how-ever more tractable for few RTN sources, hence in this paper we first primarily focus on the dissipative effectsof a single fluctuator. We then discuss the two fluctua-tor case, the results of which are then extended to treata larger number of uncorrelated fluctuators with a 1 /f noise power spectrum.We use the following Hamiltonian to describe the twoqubit quantum system, H = H o + H noise + H int (2)= B · [ S + S ] + s ( t ) g · S + J S · S (3)where, S = I ⊗ σ and S = σ ⊗ I respectively repre-sents qubits one and two, σ is the triad of Pauli matri-ces, B is the steady magnetic field chosen to be in the z direction for all our calculations, and J is the Heisen-berg interaction strength. Note that in this model theRTN is only coupled to one qubit. Earlier calculationson non-interacting qubits suggest that this is sufficientto describe all the qualitative effects [58]. The angle θ ,between the noise vector g and magnetic field B is calledthe working point of the qubit. The Hamiltonian H iswritten for a given realization s ( t ) of the noise. Physicalquantities are calculated by averaging over all sequences.Since we are primarily interested in disentanglementof the qubits, it is convenient to rewrite the two qubitHamiltonian using the maximally entangled Bell states ψ ± = ( | (cid:105) ± | (cid:105) ) / √ φ ± = ( | (cid:105) ± | (cid:105) ) / √ φ − , φ + , ψ − , ψ + ] basis is H (cid:48) = (cid:20) H φ H † xy H xy H ψ (cid:21) , (4)where H φ = J I − [ B z + g z s ( t )] σ x , H xy = g x s ( t ) I − ig y s ( t ) σ x and H ψ = − [ J I + 2 Jσ z + g z s ( t ) σ x ].In general, density matrices are suitable for treatingopen quantum systems where one has a statistical mix-ture of pure states. For two qubits, the time dependentdensity matrix can be expressed as ρ ( t ) = 14 I + (cid:88) ( i,j ) (cid:54) =(0 , n ij ( t ) σ i ⊗ σ j (5)the coherent time evolution of which is governed by thevon Neumann equation. Here I is the 4 × n ( t ) is the fifteen-component generalized Blochvector. We have used a notation, where two indices i and j (where i, j = 0 , x, y, z and ( i, j ) (cid:54) = (0 , σ i ⊗ σ j = λ k are the generators of SU (4) and σ = I is the 2 × | n | can be thought of as ameasure of purity. For instance, | n | = 0 is the completelymixed state.In the recently-developed quasi-Hamiltonian formal-ism, the dissipative temporal dynamics of an open quan-tum system under the influence of classical noise is calcu-lated by transforming a random time-dependent Hamil-tonian into a time-independent non-Hermitian Hamilto-nian [36, 51, 52]. This is done as follows. For a given noiserealization the density matrix is unitarily time evolved as ρ (∆ t ) = U · ρ (0) · U † , where U = exp( − iH ( t )∆ t/ (cid:126) ). Sub-stituting these in Eq.5 and using the identity T r ( λ i λ j ) =4 δ ij [58], one obtains the following temporal transfer ma-trix equation n ( t ) = N (cid:89) m =1 T m · n (0) . (6)where, N = t/ ∆ t and the transfer matrix elements aregiven by T ij = T r (cid:2)
U λ i U † λ j (cid:3) /
2. As the RTN is modeledas a classical stochastic process, its dynamics is governedby the master equation, ˙ W ( t ) = VW ( t ) [59], where V is a matrix of transition rates (such that the sum of eachof its columns is zero) and W is the flipping probabilitymatrix for the TLS. If the average occupation of the twostates is the same (for unbiased fluctuators), then W ( t ) = 12 (cid:20) e − γt − e − γt − e − γt e − γt . (cid:21) (7)Here, γ is the switching rate for the TLF. The combinedtemporal dynamics of the quantum and classical TLS,averaged over all noise sequences, can be described by n ( t ) = (cid:104) f | Γ N | i (cid:105) n (0) = (cid:104) f | exp( − iH q t ) | i (cid:105) n (0) (8)where, | i (cid:105) and | f (cid:105) are the initial and final state vectorsfor the TLS that satisfy W | i ( f ) (cid:105) = | i ( f ) (cid:105) . For an un-biased TLS ( i.e. with equal occupation probabilities), | i (cid:105) = | f (cid:105) = [1 , / √
2. Here Γ N = Γ N Γ N − ... Γ and inthe small time approximation, any of the matrices Γ j = W (cid:12) T = (cid:20) (1 − γ ∆ t ) T s =+1 γ ∆ t T s =+1 γ ∆ t T s = − (1 − γ ∆ t ) T s = − (cid:21) (9)Where (cid:12) denotes a Hadamard product and T is a squarematrix, each of whose columns consists of the transfermatrices [ T s =+1 , T s = − ]. Γ is now a 30 ×
30 matri-ces in the combined generalized Bloch vector and TLSspaces. H q = lim ∆ t → i ( Γ − I ) / ∆ t is a time-independentnon-Hermitian quasi-Hamiltonian[52]. As H q is not her-mitian, the time-evolution operator exp( − iH q t ) is notunitary, which makes it suitable for treating dissipativeprocesses in open quantum systems.There are thus two relevant time scales that determinethe noise characteristics of the system. The correlationtime of the environment (which is proportional to γ − )and the time period for the noise induced Rabi-like os-cillations of the qubit ( which is proportional to g − ).If the noise correlation time of the environment is muchless than the time period of the noise induced Rabi-likeoscillation ( g << γ, ), we say that the noise is Marko-vian. Whereas if g >> γ we say that the noise is non-Markovian. In the former case Redfield theory can be applied to describe dissipation, while in the latter casethe methods such as the quasi-Hamiltonian method isrequired. III. RESULTS AND DISCUSSION
At the pure dephasing point ( i.e., θ = 0), H xy = 0in Eq.4 and the two qubit dynamics can be effectivelydecoupled into two single qubit problems, whose dynam-ics is governed by H ψ and H φ . The effective one particlequasi-Hamiltonian for ψ ± can be extracted using the pro-cedure outlined in the previous section. H ψq = H ψq + H ψqJ (10)= i [ γ ( σ x − σ ) ⊗ L + g z σ z ⊗ L x ] − i [2 Jσ ⊗ L z ]where, L x,y,z ∈ SO (3) and L is the three dimensionalidentity matrix. In the absence of J , the exact solutionfor the non-zero component of the Bloch vector usingEq.8 is n z ( t ) = (cid:104) cos(Ω t ) + γ Ω sin(Ω t ) (cid:105) e − γt (11)where Ω = (cid:112) g z − γ . Note that as one crosses overfrom the non-Markovian to the Markovian noise regime( γ > g z ), the trigonometric functions in n z ( t ) becomehyperbolic functions and the Bloch vector’s oscillationsbetween ψ + and ψ − would then not be seen. When g z << γ this reduces to the usual Redfield form n z ( t ) ≈ exp( − t/T ) where, T = γ/g .For the effect of J on the Bloch vector, we approximatethe matrix exponential in Eq.8 using the Zassenhaus ex-pansion [60] as follows e − iH ψq t ≈ e − iH ψq t e − iH ψqJ t e [ H ψq ,H ψqJ ] t / (12)This gives the following expression for the Bloch vectorcomponent, valid for either small J, g z or at short times φ - φ + ψ + ψ - (a) (b) J FIG. 1: (color online) Effective Bloch spheres and theensemble averaged Bloch vector trajectory at the puredephasing point shown for (a) ψ ± and for (b) φ ± with B z = 1. The free precession of Jσ z about ψ + is also shownin (a) . n (cid:48) z ( t ) ≈ (cid:104) ζ cos( Jg z t ) + g z Ω sin(Ω t ) sin(2 Jt ) sin( Jg z t ) (cid:105) e − γt . (13)where ζ = cos(Ω t ) + γ Ω sin(Ω t ). Now, in case of the twoqubit Bloch vector for ψ ± , the non zero components are n xx = n yy = n z and n zz = −
1. If in addition the ini-tial state lies entirely in the ψ ± subspace, then the entirehistory of the qubit pair can be visualized in the cor-responding effective Bloch sphere, with ψ ± at the poles(see Fig.1).Let us first consider the dynamics on the ψ ± subspace,taking ψ + as the initial state. In the absence of J inthe strong coupling limit, the effective Bloch vector os-cillates between ψ + and ψ − with the quantum coherenceand entanglement dissipating in time (see Figs.1-a and2-a). This oscillation between ψ + and ψ − is due to the g z s ( t ) σ x component of H ψ . In a given noise realization, s ( t ) causes rotations about the x -axis which switch ran-domly between the two orientations. However the en-semble averaged Bloch vector always travels in a straightline from pole to pole for ψ ± (as shown in Fig.1-a) andeventually diminishes to the center, since the averagingrestores the chiral symmetry. In Fig. 2 we show | n | , themagnitude of the Bloch vector and the concurrence as afunction of time and J for the ψ + Bell state. For bipar-tite systems, the concurrence provides a measure of theentanglement between two qubits and is defined as [61] C = max (cid:2) √ κ − √ κ − √ κ − √ κ , (cid:3) (14)where, κ , κ , κ and κ are the eigenvalues of ρ ( σ y ⊗ σ y ) ρ ∗ ( σ y ⊗ σ y ) in decreasing order. | n | , as stated above,provides a measure of the purity. The two quantities tendto track each other but are not in one-to-one correspon-dence. The dissipative dynamics of the entangled qubitsis shown in three different noise coupling regimes –in thestrong coupling limit (non-Markovian noise, g > γ ), in J=0
Time (1/B) C on c u r en c e ( C ) & B l o c h V e c . ( n ) |g|=0.1 , = 0.01 (a) J=0.05 (a)(a) J=0.1 0 50 100 1500123
Time (1/B) |g|=0.1 , = 0.1 (b) J=0.05 (b) J=0.1 (b) J=0 0 50 100 1500123 J=0
Time (1/B) |g|=0.1, = 1 (c) J=0.1 (c) J=1 (c) |n(t)|C(t)
FIG. 2: (color online) Bloch vector magnitude andconcurrence as a function of time and J s for ψ + shown for(a) non-Markovian, (b) intermediate and (c) Markoviannoise coupling regimes at θ = 0, and B z = 1 . Note that theBloch vector amplitudes are offset by +1 for clarity. γ , g and J are in units of B . Time (1/B) C on c u r en c e ( C ) & B l o c h V e c . ( n ) |g|=0.1, = 0.01 (a) J=0.05 (a) J=0.1 (a) J=0 0 50 100 1500123 J=0
Time (1/B) |g|=0.1, = 0.1 (b) J=0.05 (b) J=0.1 (b) J=0
Time (1/B) |g|=0.1, = 1 (c) J=0.05 (c) J=0.1 (c) |n(t)|C(t)
FIG. 3: (color online) Same as Fig.2 but at θ = π/ the intermediate regime ( g ≈ γ ) and in the weak couplinglimit (Markovian noise g < γ ). Note that the purity ofthe state as measured by | (cid:126)n | is zero only at the center ofthe Bloch sphere whereas the concurrence is zero every-where on the straight line that passes through the centerand connects | (cid:105) to | (cid:105) (i.e. the x -axis).As the exchange interaction is turned on, the Jσ z com-ponent of H ψ causes the Bloch vector to precess about ψ + (see Fig.1-a). This effect competes with the effect ofthe noise, which is to drive the system toward the ori-gin of the effective Bloch sphere. Therefore the effectiveensemble averaged Bloch vector tends to remain closerto ψ + with increasing J ; this delays decoherence andthe disentanglement. This behavior holds true even inthe intermediate noise and weak noise coupling regimesas shown in Figs.2-b and c. However in the Markovianlimit this suppression of decoherence occurs only at larger J values, namely when J ∼ γ. This is shown in Fig.2-c.The overall dissipation is also much slower for Markoviannoise.We next consider an equal mixture of dephasing andrelaxational noise, a working point of θ = π/ . . The re-sults are shown in Fig.3. Here no analytic solution is pos-sible even for J = 0 and we must diagonalize the quasi-Hamiltonian H q numerically. Qualitatively, the temporalbehavior of dissipative process is the same as that of thepure dephasing case for all three noise coupling regimes,but with overall longer decoherence times.FIG. 4: (color online) Time period for the (a) envelopefunction decay and (b) oscillations as a function of J and θ . J is in units of B . Clearly, the presence of the interaction J suppressesdisentanglement and decoherence for the ψ + initial state.In the non-Markovian case, however, there are severaltime scales, one for the envelope function decay, whichfits well to a form ∼ exp ( − t/T env ) and one for the oscil-lation period T osc . In Fig.4, we show T env and T osc as afunction of θ and J .In Fig.4-(a) it is seen that T env increases monotonicallyas a function of θ . On the other hand, T env initially de-creases as a function of J , reaches a minimum and thenincreases monotonically. This non-monotonic behavior iseasily understood physically. Slow dephasing noise (small θ ) tends to mix ψ − in with the initial state ψ + , movingthe Bloch vector’s trajectory ”south” on through the ef-fective Bloch sphere. Since the noise is non-Markovian,this can actually cause oscillations between ψ + and ψ − .On the other hand, J tends to move the trajectory ”east”or ”west” for a fixed realization of the noise, causing theoscillation to just miss the south pole. This results inthe speeding up of the oscillations on averaging. For suf-ficiently large J, however, the trajectory tends to stayentirely in the northern hemisphere, which slows downthe oscillations. Here we have the basic mechanism forexchange interaction induced suppression of decoherence.If the initial state is an eigenstate of the interaction andthe noise connects this state to a state belonging to adifferent eigenstate with a different eigenvalue, then theinteraction term causes the trajectory to undergo tightoscillations near the initial state.To show this, we look at how the the action of J de-pends on the initial state. Take φ + as the initial state.Then the question is how this mixes with φ − . The cru-cial difference is that φ + and φ − both belong to thetriplet manifold, i.e., the exchange interaction has thesame eigenvalue for the two states.When θ = 0, (pure dephasing), the qubit’s dissipativedynamics is independent of J , as seen in H φ (Eq.4). The J=0
Time (1/B) C on c u r en c e ( C ) & B l o c h V e c . ( n ) |g|=0.1, = 0.01 (a) J=0.1 (a) J=0 0 50 100 1500123 J=0 Time (1/B) |g|=0.1, = 0.1 (b) J=0.1 (b) J=0
Time (1/B) |g|=0.1, = 1 (c) J=0.1 (c) |n(t)|C(t)
FIG. 5: (color online) Bloch vector magnitude andconcurence for φ + as a function of time and J s shown for(a)non-Markovian, (b)intermediate and (c)Markovian noisecoupling regimes at θ = π/ B z = 10 − . The Blochvector amplitudes are offset by +1 for clarity. γ , g and J arein units of B . J=0
Time (1/B) C on c u r en c e ( C ) & B l o c h V e c . ( n ) |g|=0.1, = 0.01 (a) J=0.1 (a) J=0
Time (1/B) |g|=0.1, = 0.1 (b) J=0.1 (b) J=0
Time (1/B) |g|=0.1, = 1 (c) J=0.1 (c) |n(t)|C(t)
FIG. 6: (color online) Same as Fig.5 but at θ = π/ effective one particle quasi-Hamiltonian is therefore H φq = i [ γ ( σ x − σ ) ⊗ L + (2 B z σ + gσ z ) ⊗ L x ] . (15)The exact solution of which gives n y ( t ) = (cid:104) cos(Ω t ) + γ Ω sin(Ω t ) (cid:105) cos(2 B z t ) e − γt (16) n z ( t ) = − (cid:104) cos(Ω t ) + γ Ω sin(Ω t ) (cid:105) sin(2 B z t ) e − γt (17)The ensemble averaged temporal trajectory of the Blochvector at a large magnetic field is shown in Fig.1-b. Inthe absence of a magnetic field, the Bloch vector onlytravels in a straight line from pole to pole. However, asin the case of ψ ± , the overall magnitude of the Blochvector (hence the decoherence) for φ ± is independent ofthe magnetic field at θ = 0 as seen from Eq.17. The cor-responding non-zero two qubit Bloch vector componentsare, n xy = n yx = n y and n xx = − n yy = n z .If θ (cid:54) = 0, then B z suppresses the decoherence. How-ever the effects of J are not visible if B z >> J , hence weset B z = 10 − for the next set of calculations. In Figs.5and 6, the decoherence and disentanglement dynamicsis shown for φ + as a function of time and J , at variousnoise coupling regimes for θ = π/ θ = π/ J , the decoherence has the samequalitative behavior as that of Eq. 17. With the onsetof J the decoherence is suppressed initially, however atlonger times it exhibits a crossover behavior in the non-Markovian and intermediate noise coupling regimes. Thiscrossover occurs later in time with increasing θ . Thus thepicture is more complicated than in the case of ψ ± andthe beneficial effect of the exchange interaction is weaker.The results presented in this paper using the quasi-Hamiltonian method have also been confirmed throughnumerical simulations. A single numerical run consistsof generating a sequence of random flips such that thenumber of flips within a given time interval t follows aPoisson distribution, P n ( t ) = ( γt ) n e − γt /n !. The timedependent density matrix is then exactly solved by nu-merically integrating the von Neumann equation in smallsteps of ∆ t using the time evolution operators U and U † .The final numerical result is then obtained after produc-ing thousands of runs (5000 per dissipative curve in thispaper) each with a different RTN sequence, and then av-eraging density matrix over all sequences. This allowsus to numerically simulate the quasi-Hamiltonian resultswhich are inherently averaged over all RTN sequences.The results from the numerical simulation and quasi-Hamiltonian method are in exact agreement to withinround-off error. IV. ANALYTICAL RESULTS FOR SHORTTIME EXPANSION AT ANY WORKING POINT
Often tolerance levels for decoherence levels are quitestringent: a fidelity loss of more than one part in 10 candestroy the result of a computation [62, 63]. Hence e-folding times such as T and also the above T env are notnecessarily the most relevant:it is important to look atshort time behavior. We now analytically show that atshort times, J always suppress decoherence for the Bellstates ( φ ± , ψ ± ) for an arbitrary working point θ , and forany set of noise parameters.The von Neumann equation can be expanded as follows ρ ( t ) = ρ (0) − i (cid:90) t [ H ( t (cid:48) ) , ρ ( t (cid:48) )] dt (cid:48) (18)= ρ (0) − i (cid:90) t [ H ( t ) , ρ (0)] dt + (cid:90) t (cid:90) t [ H ( t ) [ H ( t ) , ρ (0)]] dt dt + ... The ensemble average of ρ ( t ) has to be then calculatedover all noise sequences (cid:104) ρ ( t ) (cid:105) = (cid:88) N ( − i ) N (cid:90) t (cid:90) t ... (cid:90) t N (cid:104) C N (cid:105) dt dt ...dt N (19)where C N is the N t h order nested commutator, C N =[ H ( t N ) , ... [ H ( t ) , [ H ( t ) , ρ (0)]]]. The ensemble average of (cid:104) C N (cid:105) has to be then calculated over all noise sequences.This is done using the noise autocorrelation functionswhich are given by (cid:104) s ( t ) s ( t ) ...s ( t N ) (cid:105) = (cid:104) f | N (cid:89) j =1 s ( t j ) ⊗ W j | i (cid:105) (20)= 1 + ( − N N/ (cid:89) k =1 e − γ | t k − t k − | where s ( t j ) = 1 for the first row of W and is − W j is as in Eq.7 with t → t j +1 − t j . Includ-ing a factor of ( g/ N in the above expression accountsfor the noise amplitude. A. Short time decoherence, for ψ ± at any θ The ensemble averaged non-zero Bloch vector com-ponents for the ψ ± Bell states, obtained using ρ (0) = | ψ ± (cid:105)(cid:104) ψ ± | in the short time expansion of Eq. 19, arelisted in the Appendix along with further details. Asseen in these equations, the Bloch vector components n ox , n xo , n oy and n yo depend on J and do not contributeto the suppression of decoherence at pure relaxation( θ = π/
2) or pure dephasing ( θ = 0) points. Whereas n xx and n yy depend on J . They are equal for puredephasing and contribute to the suppression of decoher-ence at all working points except at pure relaxation. TheBloch vector components n zx , n xz , n zy and n yz disappearat either pure relaxation or dephasing points. These crossterms along with n xy and n yx do not contribute to thesuppression of decoherence.The following approximate expression (for short times)can be obtained for the magnitude of the ψ ± Bloch vec-tor by taking the sum of the squares of the above Blochvector components n ψ ≈ − g z (cid:16) − γ t (cid:17) t (21) − (cid:2) g x + 20 g z g x + g z (8 g z + γ − J ) (cid:3) t where, n ψ = (cid:80) j n j was truncated after after the t term.Note that the lowest order contribution from J is ob-tained only when Eq. 19 is expanded upto fourth orderin time. From the functional form of Eq.53, it is clearlyseen that the decoherence will be suppressed with theonset of the exchange interaction. B. Short time decoherence, for φ ± at any θ Similarly, in case of the φ ± Bell state, ρ (0) = | φ ± (cid:105)(cid:104) φ ± | is used in the short time expansion of Eq. 19. As inthe previous case, the lowest order contribution from J is only obtained when Eq.19 is expanded upto fourth or-der in time. The non-zero Bloch vector components arelisted in the Appendix. As seen in these equations, theBloch vector components n ox , n xo , n oz and n zo dependon J , however at zero magnetic field none of these com-ponents will not contribute to the suppression of deco-herence in the short time expansion. n oz and n zo respec-tively contribute only at pure dephasing and relaxationpoints. Whereas n ox and n xo contribute only at interme-diate working points. Note that even though all of thesecomponents depend on the sign of J , the overall magni-tude of the Bloch vector (and hence the decoherence) isindependent of the sign of J (just as in for n ψ ( t )).In case of the φ ± Bell state, the following expressionfor the square magnitude of the Bloch vector in the shorttime expansion of Eq. 19 n φ ≈ − g x (cid:16) − γ t (cid:17) t (22) − (cid:2) g z + 20 g x g z + g x (8 g x + γ − B z ) (cid:3) t where, n φ = (cid:80) j n j was also truncated after t term. Itis seen in Eq.23 that unlike the case of n ψ ( t ), J does notcontribute in suppressing the decoherence at immediatetimes. Instead B z aids in maintaining the quantum co-herence in its place. Even though it appears that the J dependent terms will not suppress the initial decoher-ence due to their higher order time dependencies, ourcalculations show that the initial t -decoherence can becompensated for with a sufficiently strong J at slightlylonger times. V. QUASI HAMILTONIAN RESULTS FOR TWOFLUCTUATORS
In this section we will demonstrate that even in thepresence of two uncorrelated fluctuators coupled to thequbits, the exchange interaction still suppresses the de-coherence.The Hamiltonian for two qubits coupled to two uncor-related RTSs is, H = [ B + s ( t ) g ] · S + [ B + s ( t ) g ] · S + J S · S (23) where, g k and s k ( t ) is the respective noise strength andRTN sequence for the k th fluctuator. θ k , is angle between g k and B and is the working point of the k th qubit. Themagnetic field is taken to be in the z -direction.Here we will only obtain analytical results for the caseof pure dephasing for ψ ± . The two qubit dynamics is thenonce again reduced to an effective single qubit problemgoverned by H ψ = 2 Jσ z + [ g z s ( t ) − g z s ( t )] σ x . Theflipping probability matrix for two uncorrelated and un-biased fluctuators is W = W ⊗ W , where W hasthe same functional form as Eq.7 but with γ replaced by γ . Analogous to Eq.9, the combined transfer matrixfor the two fluctuators and the effective qubit at the j th instance of time is Γ j = W (cid:12) T (24)where T is a square matrix whose each column consists ofthe transfer matrices [ T ++ , T + − , T − + , T −− ]. Here, it isimplied that T ± , ± = T s = ± ,s = ± = L +[ JL z ± g z L x ± g z L x ]∆ t . Following the procedure outlined at the endof sec.II we obtain the following quasi Hamiltonian H (cid:48) ψq = H ψqa + H ψqb + H ψqi (25)where, H ψqa = iγ ( σ x − σ ) ⊗ σ ⊗ L + ig z σ z ⊗ σ ⊗ L x (26) H ψqb = iγ σ ⊗ ( σ x − σ ) ⊗ L + ig z σ ⊗ σ z ⊗ L x (27) H ψqi = i Jσ ⊗ σ ⊗ L z (28) H ψqa , H ψqb and H ψqd commute with each other. Hence inthe absence of J , the time dependent Bloch vector can besolved for exactly. Analogous to Eq.8, for two unbiasedand uncorrelated fluctuators, the time dependent Blochvector is n ( t ) = ( (cid:104) f | ⊗ (cid:104) f | ) exp( − iH q t )( | i (cid:105) ⊗ | i (cid:105) ) n (0) (29)where, | i (cid:105) = | f (cid:105) = [1 , / √ n z ( t ) = (cid:89) k =1 (cid:20) cos(Ω k t ) + γ k Ω k sin(Ω k t ) (cid:21) e − γ k t (30)where Ω k = (cid:112) g zk − γ k .Next, for understanding the the effect of J on the Blochvector, the matrix exponential in Eq.29 is calculated withthe inclusion of H ψqi using the Zassenhaus expansion (seeEq.12). We derive the following approximate expressionfor the dephasing of the non-zero component of the effec-tive Bloch vector, valid only at short times or for small J or for small g zk n (cid:48) z ( t ) ≈ (cid:89) k =1 (cid:20) ζ k cos( Jg zk t ) + g z,k Ω k sin(Ω t ) sin(2 Jt ) sin( Jg zk t ) (cid:21) e − γ k t . (31) where ζ k = cos(Ω k t ) + γ k Ω k sin(Ω k t ). As clearly seen fromEqs.30 and 31, the Bloch vectors temporal dynamics inthe presence of two uncorrelated fluctuators is simply aproduct of the single fluctuators states shown in Eqs.11and 13. This suggests that the temporal dynamics of in-teracting qubits in the presence of multiple uncorrelatedfluctuators (which can result in 1 /f noise) can be just aswell understood in the single fluctuator picture.So far, in this section, we have analytically shown how J suppresses pure dephasing for ψ ± Bell state. The mostgeneral quasi Hamiltonian for two interacting qubits inthe presence of two uncorrelated fluctuators with for anyarbitrary set of working point, noise parameters, fieldstrength and for any initial state is H q = H qy + H qg + H qb + H qJ (32)where H qγ = i [ γ ( σ x − σ ) ⊗ σ + γ σ ⊗ ( σ x − σ )] ⊗ L (cid:48) ⊗ L (cid:48) (33) H qg = i [ (cid:126)g · (cid:126)L (cid:48) ] ⊗ L (cid:48) ⊗ σ z ⊗ σ + iL (cid:48) ⊗ [ (cid:126)g · (cid:126)L (cid:48) ] ⊗ σ ⊗ σ z (34) H qB = iB z L (cid:48) ⊗ [ L (cid:48) z ⊗ L (cid:48) + L (cid:48) ⊗ L (cid:48) z ] (35) H qJ = iJL (cid:48) ⊗ [ L (cid:48) x ⊗ Λ zx + Λ zx ⊗ L (cid:48) x + ... (36) L (cid:48) y ⊗ Λ xz + Λ xz ⊗ L (cid:48) y + L (cid:48) z ⊗ Λ xy + Λ xy ⊗ L (cid:48) z ] .Here, Λ zx = ( σ z + σ ) ⊗ σ x , Λ xz = σ x ⊗ ( σ z + σ )and Λ xy = σ x ⊗ σ x − σ y ⊗ σ y . L (cid:48) i = x,y,z is the 4 × SO (3)-generators L i , whose first row and firstcolumn is padded with zeros. L (cid:48) is 4 × (cid:126)g = [ g x , g y , g z ] and (cid:126)L (cid:48) = [ L (cid:48) x , L (cid:48) y , L (cid:48) z ]. It is importantto note that the quasi Hamiltonian, H q in Eq.32 is a64 ×
64 matrix which when projected down using Eq.30,acts on the 16 component Bloch vector instead of the 15component generalized
Bloch vector.The decoherence and the disentanglement dynamicsfor any initial Bell state can be calculated for any givenset of noise parameters, by exponentiating the abovequasi-Hamiltonian (Eq.32) numerically. As an exam-ple, in Fig.7 we have shown the dissipative dynamics ofthe ψ + entangled qubits in the Markovian, intermedi-ate and non-Markovian noise regimes for an equal mix-ture of dephasing and relaxational noise for both qubits i.e. θ = θ = π/
4. Note that if the qubit working pointsare held equal, then the qualitative temporal behavior ofdissipative process is the same as that of pure dephasingin all three noise regimes, but with overall longer decoher-ence times. As seen in the figure, the decoherence and thedisentanglement is significantly delayed with increasing J in the non-Markovian and intermediate noise regimes.Similar to the single fluctuator case, in the Markovianlimit the suppression of decoherence occurs when J ∼ γ .For the sake of simplicity, all noise parameters are heldequal for both fluctuators in Fig.7. However, J will sup-press the decoherence for any arbitrary choice of θ s, g sand γ s for both ψ ± and φ ± initial Bell states. This canbe easily verified numerically by using Eqs.32 and Eq.29. J=0
Time (1/B) C on c u r en c e ( C ) & B l o c h V e c . ( n ) |g |=0.1, = 0.01 (a) J=0.1 (a)(a) J=0
Time (1/B) |g |=0.1, = 0.1 (b) J=0.1 (b)(b) Time (1/B) |g |=0.1, = 1 (c) J=0.1 (c) J=1 (c) |n(t)|C(t)
FIG. 7: (color online) Bloch vector magnitude andconcurrence as a function of time and J s for ψ + in thepresence of two uncorrelated RTSs shown for (a)non-Markovian, (b) intermediate and (c) Markovian noisecoupling regimes at θ = π/
4, and B z = 1 . For allcalculations shown in this figure, g = g , γ = γ and θ = θ . The the Bloch vector amplitudes are offset by+1 for clarity. γ , g and J are in units of B . VI. INTERACTING QUBITS IN THEPRESENCE OF /f NOISE
We now generalize our results for the interactingqubits, for a full spectrum of uncorrelated fluctuators.The general quasi Hamiltonian for two interacting qubitsin the presence of n uncorrelated fluctuators, for any ar-bitrary set of parameters, with m ( where m < n ) fluctu-ators coupled to one qubit and n − m fluctuators coupledto the second qubit isˆ H q = ˆ H qy + ˆ H qg + ˆ H qb + ˆ H qj (37)where ˆ H qγ = i (cid:88) j γ i ( τ ( j ) x − I n ) ⊗ L (cid:48) ⊗ L (cid:48) (38)ˆ H qg = i m (cid:88) j =1 ( (cid:126)g j · (cid:126)L (cid:48) ) ⊗ L (cid:48) ⊗ τ ( j ) z + ... (39) i n (cid:88) j = m +1 L (cid:48) ⊗ ( (cid:126)g j · (cid:126)L (cid:48) ) ⊗ τ ( j ) z ˆ H qB = iB z I n ⊗ ( L (cid:48) z ⊗ L (cid:48) + L (cid:48) ⊗ L (cid:48) z ) (40)ˆ H qJ = iJI n ⊗ ( L (cid:48) x ⊗ Λ zx + Λ zx ⊗ L (cid:48) x + ... (41) L (cid:48) y ⊗ Λ xz + Λ xz ⊗ L (cid:48) y + L (cid:48) z ⊗ Λ xy + Λ xy ⊗ L (cid:48) z ) . I n is an identity matrix of dimension 2 n and it is impliedthat τ ( j ) x ( z ) = σ (1)0 ⊗ ...σ ( j − ⊗ σ ( j ) x ( z ) ⊗ σ ( j +1)0 ⊗ ...σ ( n )0 . (42)The time dependent 16 component Bloch vector is nowobtained from the projected temporal dynamics of thequantum and n classical TLSs as follows n ( t ) = (cid:104) f n |⊗ ... (cid:104) f |⊗(cid:104) f | exp( − iH q t ) | i (cid:105)⊗| i (cid:105) ... ⊗| i n (cid:105) n (0) . (43)As the dimension of the quasi Hamiltonian scales as2 n , calculating the coupled qubit dynamics in the pres-ence of a very large number of fluctuators can quicklybecome computationally very intensive. We thereforecalculate the magnitude of the Bloch vector and the con-currence, using the quasi Hamiltonian in Eq.37, for aset of eight fluctuators with 4 of them coupled to eachqubit. A random 1 /f distribution of γ s is taken (rang-ing from 0 .
95 to 0 . /f noise power spectrum (where the power spectrumis S ( ω ) = (cid:80) γ i / ( γ i + ω )) as shown in Figs. 8-(a) and(b). For the sake simplicity and to limit the parameterspace we have held the noise strengths, g j s and their re-spective working point the same for all the fluctuators.The coupled qubits dissipative dynamics is shown in Fig.8 for the ψ + state at θ i = π/
4, for g j = 0 . , . ,
1. Inall three cases it is seen that the exchange interaction, J , suppresses decoherence and disentanglement. As ex-pected the rate of decoherence rate itself increases with (B) S ( ) (a) -4 -2 0024 l og [ S ( ) ] log( ) (b) Time (1/B) C on c u r en c e ( C ) & B l o c h V e c . ( n ) B z = 1, |g j | = 0.01 (c) J=0.1 (c) J=0 0 50 1000123
Time (1/B) C on c u r en c e ( C ) & B l o c h V e c . ( n ) B z = 1, |g j | = 0.1 (d) J=0.1 (d) J=0 0 5 100123 J=0
Time (1/B) C on c u r en c e ( C ) & B l o c h V e c . ( n ) B z = 1, |g j | = 1 (e) J=1 (e) |n(t)|C(t)
FIG. 8: (color online) (a)Power spectrum, S ( ω ), for a 1 /f distribution of 8 uncorrelated fluctuators. (b) Inset showingthe noise power spectrum on a logarithmic scale. Magnitudeof the Bloch vector and concurrence for the ψ + state shownas a function of time and J in the presence of the samefluctuators with all g i s held constant at (c) g j = 0 .
01 (d) g j = 0 . g j = 1, at B z = 1. θ = π/ γ , g , J are in units of B . increasing noise strength, g j . For a full spectrum of fluc-tuators, if g j is smaller than the smallest γ then the noisedue to the RTSs falls purely in the Markovian noise cou-pling regime, as shown in Fig. 8-(c). If g j falls somewherein between the selected range of γ s, then one has a mix-ture of Markovian and non-Markovian noise sources asshown in Fig. 8-(d). However in this case, the Marko-vian noise sources tend to dominate and the oscillatorybehavior, typically seen for non-Markovian noise, tendsto get washed out. Whereas for the case of g j = 1 in Fig.8-(e), where one has a mixture of Markovian and inter-mediate noise sources, small oscillations superposed ontop of a smoothly decaying function can be seen. This issimilar to the dissipative behavior seen for a single qubitin the presence of broad spectrum noise [36].As in the previous case of one and two fluctuators, oncethe exchange interaction is turned on, the decoherenceand disentanglement dynamics is suppressed for multi-ple fluctuators as well. This suppression is proportionalto the strength of J . However, for the 1 /f noise powerspectrum, it is seen that the effect of J in suppressing thedecoherence is apparent only when it is at least the sameorder of magnitude as g j . Finally, if all the uncorrelatedfluctuators are at the pure dephasing point, then the de-coherence and disentanglement dynamics of the coupled qubits are simply products of the dissipative dynamicsdue to each individual fluctuator. The short time behav-ior can be calculated for any number of fluctuators withany arbitrary distribution of γ s and g s by taking the k sum in Eq 31 from 1 to n . VII. SUMMARY
In summary we have suggested a new way to prolongquantum entanglement and coherence via qubit-qubit in-teraction. We have analytically and numerically shownthat the exchange interaction suppresses decoherence anddisentanglement for the entangled Bell states. This isshown to be true for a single RTS, for two RTSs and for1 /f noise as long as the fluctuators are uncorrelated. Ourcalculations are carried out using the quasi-Hamiltonianmethod, which is seen to be a particularly powerfulmethod while dealing the many degrees of freedom asso-ciated with 1 /f noise. For the single fluctuator case, thesuppression of decoherence is most apparent when J ∼ γ and hence is more effective for non-Markovian noise. For1 /f noise, we conclude that J should be about the sameorder of magnitude as the strongest noise source, g , inorder to alleviate decoherence. As the suppression of de-coherence at immediate times is key to performing highfidelity gate operations, we have analytically shown thatthe exchange interaction induced suppression of decoher-ence at short times is more effective for the ψ ± Bell statethen it is for the φ ± state. This is true for any arbitraryqubit working point, fluctuator switching rate and noisestrength. If however a large magnetic field is used, thenthe φ ± Bell state is more suitable for encoding quan-tum information. These results are vital for quantuminformation processing as they can be used to developalternative methods (in addition to existing proposals)to enhance qubit lifetimes.
VIII. ACKNOWLEDGEMENTS
This work is supported by the DARPA/MTO QuESTprogram through a grant from AFOSR.
IX. APPENDIX
In sec.IV, for the ψ ± Bell states, the ensemble aver-aged (cid:104) ρ ( t ) (cid:105) is obtained by using ρ (0) = | ψ ± (cid:105)(cid:104) ψ ± | in theshort time expansion (upto N = 4) of Eq. 19. For any θ ,the lowest order contribution from J is obtained onlywhen Eq. 19 is expanded upto fourth order in time.Note that the first order term (cid:104) [ H ( t ) , ρ (0)] (cid:105) = 0. Us-ing n ij ( t ) = T r [ σ i ⊗ σ j × (cid:104) ρ ( t ) (cid:105) ], the following ensembleaveraged non-zero Bloch vector components are obtainedfor ψ ± n ox = − n xo = − B z Jg x g z t (44) n oy = 3 γt − − γt n yo = 43 Jg x g z (2 − γt ) t (45) n oz = − n zo = − B z Jg x t (46) n xy = 3 γt − γt − n yx = 23 B z g x ( γt − t (47) n yz = 3 γt − − γt n zy = 23 B z g x g z (3 γt − t (48) n xz = − g x g z [3 t − γt − (3 B z + 6 J + Ω ) t ] (49) n zx = 23 g x g z [3 t − γt + ( B z + 10 J − Ω ) t ] (50) n xx = 1 − g z (cid:18) t − γt (cid:19) (51) − (cid:20) B z g x − g z (2 J + Ω ) (cid:21) t n yy = 1 − g z + g x ) (cid:18) t − γt (cid:19) (52)+ 43 (cid:2) g x (Ω − Bz ) + g z (Ω + 2 J ) (cid:3) t n zz = − g x [3 t − γt − ( B z + Ω ) t ] (53) where, Ω = (cid:112) g x + g z − γ and g x = g z tan( θ ).Similarly for the φ ± Bell state, ρ (0) = | φ ± (cid:105)(cid:104) φ ± | is usedin the short time expansion of Eq. 19. 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