Distinguishing decoherence from alternative quantum theories by dynamical decoupling
Christian Arenz, Robin Hillier, Martin Fraas, Daniel Burgarth
DDistinguishing decoherence from alternative quantum theories by dynamicaldecoupling
Christian Arenz, Robin Hillier, Martin Fraas, and Daniel Burgarth Institute of Mathematics, Physics, and Computer Science,Aberystwyth University, Aberystwyth SY23 2BZ, UK Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, UK Mathematisches Institut der Universit¨at M¨unchen,Theresianstrasse 39, D-80333 M¨unchen, Germany (Dated: October 18, 2018)A long standing challenge in the foundations of quantum mechanics is the verification of alternativecollapse theories despite their mathematical similarity to decoherence. To this end, we suggest anovel method based on dynamical decoupling. Experimental observation of non-zero saturation ofthe decoupling error in the limit of fast decoupling operations can provide evidence for alternativequantum theories. The low decay rates predicted by collapse models are challenging, but highfidelity measurements as well as recent advances in decoupling schemes for qubits let us explorea similar parameter regime to experiments based on macroscopic superpositions. As part of theanalysis we prove that unbounded Hamiltonians can be perfectly decoupled. We demonstrate thison a novel dilation of a Lindbladian to a fully Hamiltonian model that induces exponential decay.
I. INTRODUCTION
Despite of its puzzling nature and persistent founda-tional problems, such as the infamous measurement prob-lem, quantum mechanics remains one of the most preciseand successful physical theories to date. This makes ithard to develop alternative theories (for an overview werefer to [1–3]), which are either bound to agree with quan-tum mechanics on all measurable aspects – and thereforebeing indistinguishable from it – or must disagree withit only at the most subtle level, which means that suchtheories are hard to falsify experimentally. While in ourdaily life quantum effects do not appear to play a role,this does not imply that it is an incomplete theory, as theonset of classicality can – at least up to a certain degree[4, 5] – be explained from within quantum theory, usingthe concept of decoherence.Decoherence arises from the coupling of a quantumobject with other degrees of freedom, which washes outquantum mechanical features. Besides being a major ob-stacle to quantum computing, decoherence is also an ob-stacle to the tests of theories alternative to quantum me-chanics, since it tends to obscure the – already minimal– deviations they predict from the usual Schr¨odinger dy-namics. Even worse, since most alternative theories aimto explain the onset of classicality, they predict featuresidentical in their mathematical nature to decoherence [6].The main aim of this article is to demonstrate that whilethese models might be mathematically identical, they are physically distinguishable, irrespectively of decoherence.At first, this seems impossible. Especially in quantum in-formation theory, the Church of the Larger Hilbert Space– the idea that any noisy dynamics or state might equallywell be represented by a noiseless one on a dilated space –is so deeply rooted that such a distinction seems heretic.A method to distinguish decoherence from alternativequantum theories (AQT) which is obvious but impracti- cal is to derive ab initio predictions of decoherence andcompare these with experiments. Unfortunately, the pre-dictive power of decoherence models till date is low, asthey contain many free parameters to fit. We thereforeaim to develop methods which are independent of the de-tails of the decoherence involved, as well as of the specificAQT considered.Our work is based on a very simple idea, namely thatdynamical decoupling [7] – a popular method to suppressquantum noise – only works for systems which are truly coupled to environments [8], but not for systems whichhave intrinsic noise terms, as arriving from axiomaticmodifications of Schr¨odinger’s equation [6, 9–11].This seems to leave us with an amazingly simple strat-egy to distinguish decoherence from AQT: apply decou-pling, and if it works, then the noise was due to stan-dard quantum theory; if it does not work, it can provideevidence for AQT. Is this therefore the most successful“failed” experiment ever? Of course not: we need tobe convinced that the experiment did not work despitegood effort , in other words, we need to know quantita-tively how much the experiment can fail while still beingin the realms of standard decoherence; and how much itcan succeed despite being in the realms of AQT. Thisposes an additional problem. It is a common view thatdynamical decoupling only works for environments in-ducing non-exponential decay (sometimes referred to as‘non-Markovian’, although this term is used ambiguouslyin the literature). This means that if the observed quan-tum dynamics shows exponential behaviour, we wouldnot be able to distinguish it from AQT. On the otherhand, most AQTs predict exponential decay [6].The reason for this common view is that exponentialdecay can only be obtained from an unbounded interac-tion with the environment [12], for which standard erroranalysis of dynamical decoupling fails [13]. Perhaps sur-prisingly, we will prove in section IV that in general evenunbounded Hamiltonians can be decoupled and hence a r X i v : . [ qu a n t - ph ] A ug distinguished from intrinsic decoherence. This generalproof is illustrated by an analytically solvable exampleIV A. We can conclude that non-exponential dynamicsis in general not the underlying mechanism of dynami-cal decoupling. This result extends the applicability ofdecoupling to a vast class of system-environment inter-actions and has applications in quantum engineering be-yond the scope of this paper.Finally, dynamical decoupling arises in the limit of in-finitely fast quantum gates, so in practice it is never per-fect. How fast should these operations be so that deco-herence and AQT can be distinguished? Below, we pro-vide numerical simulations of two common models andasymptotical bounds (referring to [8] for a detailed math-ematical analysis) regarding these questions. As we willsee below, the convergence speed can depend strongly onthe initial bath state, which implies that model indepen-dent bounds , e.g., depending only on the observed decayrates of the system, cannot be provided. Nevertheless,experimental evidence can be provided if a saturationof fidelity is observed under increasingly fast operations.For the parameter range explorable by our scheme, wecan do the following rough estimate. The strongest in-trinsinc decay rates for qubits predicted by AQT are ofthe order of 10 − s − corresponding to a half-life timeof several years [11]. Precision measurements of qubitson the other hand are very well developed meaning thatcoherence decay of the order of percent can be detected.This means that if one aims to keep a qubit from de-tectable decay for several days, the first AQT modelscould be detected or excluded. At present qubit coher-ence times can be prolonged by dynamical decoupling upto six hours [14]. This is still a few orders of magni-tude off the theoretical predictions, which is comparableto the usual AQT tests in the macroscopic superpositionregime.Our results pave the way to test AQT in low-dimensional systems, including qubits, where AQT pre-dicts very weak effects [1], but where dynamical decou-pling is very efficient, and where accurate tomographycan be performed [15]. This is a different parameterregime compared to tests using macroscopic superposi-tions [16–19], where AQT predict stronger effects butdynamical decoupling is challenging (see, however, [20]). II. DYNAMICAL DECOUPLING FORBOUNDED HAMILTONIANS
Dynamical decoupling is a highly successful strategyto protect quantum systems from decoherence [7]. Itsparticular strength is that it is applicable even if the de-tails of the system-environment coupling are unknown.In the context of quantum information the theoreticalframework was developed in [13, 21] and the efficiency ofdifferent decoupling schemes was studied and improvedfor several environmental models in [22–27]. Many ex-periments, such as [28–30], demonstrate the applicability of dynamical decoupling in an impressive way by pro-longing coherence times a few orders of magnitude. Ad-ditionally, dynamical decoupling can be combined withthe implementation of quantum gates which makes it aviable option to error correction [31, 32]. The idea ofdynamical decoupling is to rapidly rotate the quantumsystem by means of classical fields to average the system-environment coupling to zero.More precisely consider the unitary decoupling oper-ations v taken from the set V of | V | unitary d × d ma-trices satisfying | V | (cid:80) v ∈ V vxv † = d tr( x ) x . An example of such a set for a single qubit are thePauli matrices V = { , σ x , σ y , σ z } . While usually dy-namical decoupling is discussed in the realm of a unitarytime evolution, we already allow a noisy dynamics gen-erated by a Lindbladian L because we later want to seewhat happens for AQT. This dynamics is now modifiedby decoupling operations v i ∈ V with i = 1 , ..., n appliedinstantaneously in time steps ∆ t . After time t = n ∆ t the system has evolved according toΛ t,n ( · ) = n (cid:89) i =1 Ad( v i ) exp(∆ t L )Ad( v † i )( · ) , (1)where Ad( v i )( · ) = v i ( · ) v † i and the product is time-ordered. The generalization to time-dependent gener-ators is straight forward and will be used later in theexamples. Throughout this paper we consider perfectdecoupling operations, while bounds for the non-perfectcase can be found for example in [33–35]. The decouplingoperations are chosen uniformly random from V , whichhas some advantage over deterministic schemes [13, 24].Notice that our definition of random dynamical decou-pling differs slightly from [13]. The time evolution (1)becomes a stochastic process with expected dynamics de-termined by ¯ L := 1 | V | (cid:88) v ∈ V Ad( v ) L Ad( v † ) . (2)This leads to the decoupling condition ¯ L = 0, which onerequires in order to successfully suppress decoherence.Note that this condition is independent of whether weuse a deterministic or random decoupling scheme [21].The idea behind this condition is that it ensures the can-cellation of L in first order in ∆ t ||L|| . For ∆ t →
0, keep-ing the total time t fixed, the time evolution (1) becomestherefore effectively the identity.Hamiltonian dynamics L ( · ) = i [ H, · ] can always be su-pressed through dynamical decoupling. In the section IVwe prove that this is even true for unbounded Hamilto-nians. But what happens for AQT? Note first of all thatfor AQT models that modify the Schr¨odinger equationin a nonlinear way, it was argued in [6] that under theassumption of the no-signalling principle the resultingdynamics is described by a time independent Lindbladoperator L ( · ) = d − (cid:88) j =1 γ j (2 L j ( · ) L † j − ( L † j L j ( · ) + ( · ) L † j L j )) , (3)yielding the averaged Lindbladian¯ L ( · ) = d − (cid:88) j =0 γ j (cid:32) | V | (cid:88) v ∈ V vL j v † ( · ) vL † j v † − d tr( L † j L j )( · ) (cid:33) . (4)We will henceforth refer such AQT dynamics as intrinsicdecoherence . In order to avoid confusion, we will write extrinsic decoherence for decoherence arising in standardquantum theory. Surprisingly if the dynamics includesintrinsic decoherence, the decoupling condition can never be fulfilled. Intuitively the irreversible nature of the non-unitary dynamics, i.e. the increase of entropy, makesit impossible to counteract the loss of coherence withunitary decoupling pulses. For a detailed mathematicalproof we refer to [8]. This is a remarkable result since itenables us to distinguish two different seemingly equaldecoherence mechanisms. We remark that the gener-alization to time-dependent Lindbladians is straightfor-ward allowing our technique also to discriminate non-exponential collapse models from extrinsic decoherence.In the limit of arbitrarily fast decoupling operations(∆ t →
0) dynamical decoupling works perfectly for ex-trinsic decoherence. However, in practice even dynamicaldecoupling of extrinsic decoherence can never be perfectmeaning that higher orders in ∆ t ||L|| enter the result-ing dynamics. To detect the presence of intrinsic de-coherence we therefore need to develop an extrapola-tion for ∆ t →
0. Furthermore to distinguish extrinsicand intrinsic decoherence we need bounds. Using a cen-tral limit theorem, such bounds are developed in [8] forthe expectation of the decoupling error ¯ (cid:15) , while here wewill focus on specific examples. The decoupling error (cid:15) = tr { ( − Λ t,n ) † ( − Λ t,n ) } /d compares the free evolu-tion under random dynamical decoupling with the iden-tity operation. In the limit ∆ t →
0, keeping the totaltime t fixed, the decoupling error becomes [8], (cid:15) = 1 d tr (cid:0) ( − exp( ¯ L t )) † ( − exp( ¯ L t )) (cid:1) , (5)where for extrinsic decoherence the time evolution of thetotal system is followed by the partial trace over the en-vironment yielding (cid:15) = 0 for ∆ t →
0. Note that the de-coupling error can be estimated in an experiment by per-forming process tomography [36]. Simpler fingerprints todistinguish AQT which do not require process tomogra-phy can easily be derived for specific systems. In thefollowing we emphasize the physics calculating boundsfor two common models.
III. MODELS AND BOUNDS
To demonstrate our method we consider two differenttypes of decoherence of a single qubit, namely amplitudedamping and pure dephasing.
A. Two qubit model
To begin with suppose that one observes a dynamicsdescribed by an amplitude damping (AD) channel, givenby the Lindblad operator L AD ( · ) = − γ ( σ + σ − ( · ) + ( · ) σ + σ − − σ − ( · ) σ + ) , (6)with σ ± the raising and lowering Pauli operators. Withinthe extrinsic decoherence model such amplitude dampingdynamics can be obtained by a time dependent interac-tion with an ancilla qubit ( A ) initialized in its groundstate. The total Hamiltonian reads H ( t ) = g ( t )( σ + ⊗ σ ( A ) − + σ − ⊗ σ ( A )+ ) , (7)with the time dependent coupling constant g ( t ) = γ/ (cid:112) exp(2 γt ) −
1. The Hamiltonian H ( t ) commuteswith itself at all times such that the time evolution of thecomposite system can easily be integrated. After trac-ing over the ancilla qubit one obtains precisely the twoKraus operators which describe the amplitude dampingchannel generated by (6). Note that at t = 0 the interac-tion strength g ( t ) diverges while the time evolution op-erator remains well defined. Clearly there are other pos-sible choices of the system-bath Hamiltonian that leadto the same dynamics. For example within the Born-Markov approximation the same Lindblad operator (6) isobtained by a time independent interaction of the qubitwith a bath of harmonic oscillators at zero temperature.However as a toy model, (7) has the advantage of beingsimpler. Such time-dependent dilations may also find ap-plications in other context.Now we turn to the question how well dynamical de-coupling can distinguish between extrinsic decoherence,given by the Hamiltonian (7), and pure intrinsic decoher-ence given by the Lindbladian (6). Using (4) one findsfor the intrinsic decoherence case the averaged Lindbladoperator ¯ L AD ( · ) = − γ ( · ) − σ − ( · ) σ + − σ + ( · ) σ − ) whichdetermines the dynamics in the limit of infinitely fast de-coupling operations. The first observation is that ¯ L AD does not vanish. With (5) we can furthermore derive thefollowing asymptotic behaviour for the decoupling errorin the intrinsic decoherence case (cid:15) intAD → (cid:0) − e − γt (cid:0) − e − γt (cid:1)(cid:1) , ∆ t → , (8)and for γt (cid:29) /
4. In Fig. 1 weevaluated the averaged decoupling error for intrinsic andextrinsic decoherence as a function of ∆ t for a fixed totaltime t = γ − . We see that for the Hamiltonian model D ec oup li ng e rr o r Δ t ( γ -1 )1 0.1 0.01Intrinsic decoherence Extrinsic decoherence FIG. 1. (Colour online) Averaged decoupling error underrandom dynamical decoupling as a function of ∆ t on an in-verse logarithmic scale for the total time t = γ − . The circlescorrespond to pure intrinsic decoherence described by (6), thetriangles to extrinsic decoherence given by (7) and the dashedline shows the asymptotic behavior (8) for the intrinsic deco-herence case for ∆ t →
0. The average was taken over 100trajectories. (7) the decoupling error tends to zero. The asymptoticbehaviour of the averaged trajectories allows us to dis-tinguish intrinsic from extrinsic decoherence: for purelyintrinsic decoherence we have (8), while for purely extrin-sic it is 0, and everything in-between must correspond toa mixture of the two. The actual speed of convergenceto the limit in the extrinsic case depends on the chosendilation [13], so that we cannot say how small ∆ t has tobe chosen in order to distinguish with certainty. B. Spin-boson model
Next, we consider a more realistic and experimentallyrelevant model describing pure dephasing (PD) in the σ z basis of the qubit. The Lindbladian reads L PD ( t )( · ) = − γ ( t )4 [ σ z , [ σ z , · ]] , (9)where the time dependent damping rate γ ( t ) will be spec-ified later. As extrinsic decoherence such PD would arisefrom an interaction with a bosonic heat bath given by H = (cid:88) k ω k a † k a k + σ z (cid:88) k ( g k a † k + g ∗ k a k ) , (10)where a † k , a k are the bosonic creation and annihilationoperators of the k th field mode and g k are couplingconstants quantifying the interaction strength to eachharmonic oscillator. After tracing over the bath de-grees of freedom [37–39] one finds for the time dependentdamping rate γ ( t ) = 4 (cid:82) t ds (cid:82) ∞ dωI ( ω ) coth (cid:0) ω T (cid:1) cos( ωs )where the continuum limit was performed and the spec-tral density I ( ω ), which contains the statistical proper-ties of the bath, and the temperature T of the bath wereintroduced. For an intrinsic dephasing mechanism given by(9) the decoupling operations V do not affectthe dynamics vσ z v † = ± σ z for all v ∈ V suchthat L PD = ¯ L PD . Therefore the decoupling er-ror in the intrinsic decoherence case is governed bythe dynamics generated by L PD and with (cid:15) PD = tr (cid:16) ( − exp( (cid:82) t dt (cid:48) L PD ( t (cid:48) ))) † ( − exp( (cid:82) t dt (cid:48) L PD ( t (cid:48) ))) (cid:17) one finds independently of ∆ t , (cid:15) intPD = 12 (cid:20) − exp (cid:18) − (cid:90) t γ ( t (cid:48) ) dt (cid:48) (cid:19)(cid:21) , (11)showing that the asymptotic decoupling error is given by1 /
2. Based on the spin-boson Hamiltonian (10) it wasshown in [24] that under random dynamical decouplingthe spectral density gets renormalized by a factor thatensures for ∆ t → D ec oup li ng e rr o r Δ t ( ω c-1 )10 1 0.1Intrinsic decoherence Extrinsic decoherence FIG. 2. (Colour online) Averaged decoupling error underrandom dynamical decoupling as a function of ∆ t on an in-verse logarithmic scale evaluated for t = 50 ω − c . The tri-angles correspond to extrinsic decoherence given by the spinboson model (10) where the dashed line corresponds to in-trinsic decoherence (9) which is independent of ∆ t here (11).The average was taken over 100 trajectories. Because the decoupling operations V give the samespectral density as in [24] we can easily evaluate the av-eraged decoupling error for extrinsic and intrinsic deco-herence ( Fig. 2 ). We chose an ohmic spectral densitywith a sharp cut off I ( ω ) = 1 / κωθ ( ω − ω c ) with κ = 0 .
25a measure of the coupling strength to the environmentand ω c = 100 the cut off frequency. We calculated theaveraged decoupling error in the low temperature limit ω c /T = 10 .Note that for ∆ t (cid:38) . ω − c decoherence gets acceler-ated as reported in [24] in the extrinsic case since thedecoupling error is higher than the decoupling error thatis obtained for the dynamics generated by L P D . IV. DYNAMICAL DECOUPLING OFUNBOUNDED HAMILTONIANS
Many physical environments are modelled as infinitedimensional system, often with unbounded interactions.In order to discuss dynamical decoupling of such systems,we find it enlightening to start with a specific, analyt-ically solvable model, before providing a general proofthat generally even unbounded time-independent Hamil-tonians can be decoupled.
A. Shallow pocket model
We now provide an analytically solvable model of anunbounded, time-independent Hamiltonian which, with-out approximations, leads to a time-independent dephas-ing Lindbladian, but can be decoupled arbitrarily well.It is an example of an exact time-independent dilationdescribing a small system coupled to a fictitious particleon a line. After tracing over the decrees of freedom of theparticle we obtain a time independent Lindblad genera-tor for the reduced dynamics of the system. The particlecannot store energy internally – hence the name – andthe dynamics is governed by an interaction Hamiltonian H = g σ z ⊗ x = g (cid:18) x − x (cid:19) , (12)where x is the position operator and the small systemis a qubit for simplicity and g a coupling constant. TheHamiltonian is diagonal and the evolution of a joint den-sity matrix is ρ ( t, x ) = (cid:18) ρ (0 , x ) ρ (0 , x ) e igxt c.c. ρ (0 , x ) (cid:19) . (13)A reduced dynamic displaying exponential decay isachieved by choosing an initial state ρ ⊗ | ψ (cid:105)(cid:104) ψ | where (cid:104) x | ψ (cid:105) = (cid:114) γπ x + iγ . (14)After integrating out the particle degree of freedom weobtain, through the Fourier transform of a Lorentzian, apurely exponential decay of the off diagonal terms, ρ ( t ) = (cid:18) ρ (0) ρ (0) e − gγt c.c. ρ (0) (cid:19) , (15)which corresponds to a time-independent dephasingLindbladian L ( · ) = − g γ σ z , [ σ z , · ]] . (16)The model can be perfectly decoupled using Z controls v = , v = σ x . In fact v Hv † = − H and hence v exp( i ∆ tH ) v † v exp( i ∆ tH ) v † = . (17) F i d e lit y t FIG. 3. (Colour online) Schematic representation of the fi-delity for exponential dephasing (dotted green line) to stayin a coherent superposition of ground and excited state. Thesolid blue line shows the dynamics of the qubit under dynam-ical decoupling.
This model displays similar effects as the above ones,which means that the explicit time-dependence of theHamiltonian/Lindbladian of the first two examples is notrelevant to the discussion. In Fig. 3 we show the fidelity F ( t ) = ( e − gγt + 1) (dotted green line) of being in acoherent superposition of ground and excited state ob-tained from the dynamics generated by the Lindbladian(16). The solid blue line shows the reduced dynamicsof the shallow pocket model under dynamical decoupling(17).The shallow pocket model is a counterexample to dy-namical decoupling working for non-exponential decayonly. For a fixed decoupling time τ the fidelity neverdrops below F ( τ ). The model also highlights some of theunpleasant mathematical properties required for mod-elling strict exponential decay: the initial state of thesystem is not in the domain of the interaction [40], whichin turn is unbounded below and above [12]. Such prop-erties indicate that the general proof below requires acertain degree of mathematical precision. B. General proof
It is a fact of nature and an ubiquitous challenge inthe mathematical treatment of quantum mechanics thatunbounded Hamiltonians cannot be defined everywhere[41, Chapter VIII]. A definition domain D ( H ) has tobe specified in order to make a clear sense of an un-bounded Hamiltonian H . For example the notion of self-adjointness, properties of a sum H + H , etc has to takethe definition domain into account. Starting with a pi-oneering work of von Neumann a machinery has beendeveloped with a purpose to circumvent these problemswhen dealing with a derived quantum mechanical phe-nomena. This is precisely our case, we show that when-ever a Hamiltonian which couples a finite-dimensionalsystem of size d to an infinite-dimensional bath can bereasonably defined then it can be decoupled perfectly.All Hamiltonians under our consideration have a sum-like structure consisting of the system \ bath free Hamil-tonians and the interactions. A core of an operator [41]is then a natural notion to make sense of this sum inthe most general setting. We postpone this technicaldiscussion by few paragraphs and start with a natural– albeit less general – setting where this notion is notneeded. It includes for example the case when the inter-action Hamiltonian is relatively bounded with respect tothe free Hamiltonian.We assume that a Hamiltonian describing the systemis a densely defined self-adjoint operator of the form H = H S ⊗ ⊗ H B + (cid:80) α S α ⊗ R α on the tensor productHilbert space H SB = H S ⊗H B , with H B itself self-adjointon a dense domain D ( H B ) and D ( H ) = C d ⊗ D ( H B ).For simplicity we only consider deterministic decouplingschemes here, while the random case can be provedusing [42, Th.2.2] (c.f. forthcoming work for details).The announced perfect decoupling of such a Hamilot-nian might be surprising given that the usual derivationof dynamical decoupling hinges on a perturbative expan-sion exp( i ∆ tA ) ∼ i ∆ tA + O (∆ t ) and a limit formula (cid:18) An + O ( n − ) (cid:19) n → exp( A ) . (18)In particular all standard error bounds [13] become infi- nite for unbounded Hamiltonians. These apparent prob-lems can be circumvented by means of a deep general-ization of the above limit formula due to Chernoff [43],c.f. also [44, Chapter 8.]: Let F ( t ) , || F ( t ) || ≤ be a fam-ily of operators on a Hilbert space H with F (0) = ( F ( t ) − )( ψ ) /t → Aψ as t → , for every ψ ∈ H in a core of A . Then we have lim n →∞ F (cid:18) tn (cid:19) n ( ψ ) = exp( tA ) ψ, ψ ∈ H . (19)We apply Chernoffs theorem with F ( t ) =Π v ∈ V v exp( iHt/ | V | ) v † and H as above. Then for ψ ∈ D ( H ),( F ( t ) − ψ ) t → i (cid:32) | V | (cid:88) v ∈ V vHv † (cid:33) ψ = i ( ⊗ H B ) ψ, (20)due to the decoupling property of V , as t → ψ in the domain of all vHv † ’s. Note that the con-vergence in (20) is not obvious since the use of the Taylorseries is not well defined for unbounded operators. Alongthe lines of [45] it can be proven instead on the grouplevel, by rearranging the exponentials in such a way thatStone’s theorem can be used. Consider for example as asystem a qubit with V the Pauli group. We can evaluatethe limit (20) using( F ( t ) − ψ ) t = 1 t (cid:0) e − iσ z Hσ z t − (cid:1) ψ + 1 t e − iσ z Hσ z t (cid:0) e − iσ y Hσ y t − (cid:1) ψ + 1 t e − iσ z Hσ z t e − iσ y Hσ y t (cid:0) e − iσ x Hσ x t − (cid:1) ψ, + 1 t e − iσ z Hσ z t e − iσ y Hσ y t e − iσ x Hσ x t (cid:0) e − iHt − (cid:1) ψ, (21)with ψ ∈ C ⊗ D ( H B ). By assumption all vHv † are self-adjoint on this domain, so we can apply Stone’s theoremfor each summand of (21) yielding the desired result (20)as t goes to zero. We conclude that perfect dynamicaldecouplinglim n →∞ tr B (Λ t,n ρ )) = tr B (cid:0) e it ⊗ H B ρe − it ⊗ H B (cid:1) = tr B ( ρ ) , (22)is possible where ρ is the density operator of the systemand the bath.Notice that many examples including the shallowpocket model verify the above assumptions of self-adjointness. Nevertheless, we aim for even bigger gen-erality and to achieve this we introduce the notion of acore into our discussion. A core of an operator is a sub-space of its domain such that restriction of the operatorto the core and subsequent closure gives back the originaloperator. Clearly the domain itself is a core, but it mightbe too big in certain applications like the present one. We may assume that H is formally given as abovewith some unknown dense domain D ( H ), with H B andeach R α selfadjoint on certain dense domains D ( H B ) and D ( R α ), which might be different, but with all H B and R α having a common core C . This is the minimal as-sumption to make in order to have the sum definition of H well-defined at all. Under this assumption the sum (cid:80) v ∈ V vHv † is then also well-defined on C d ⊗ C and itsclosure is exactly (an extension of) ⊗ H B . For any ψ ∈ C d ⊗ C the conditions of Chernoff’s theorem, andin particular ( F ( t ) − ψ ) /t → ( ⊗ H B ) ψ , are thensatisfied, so (20) follows again.Clearly if H is self-adjoint with domain C d ⊗ D ( H B )then all vHv † are also self-adjoint on that domain, butthere are cases of H with different domains, and that iswhen the above criterion with cores is needed.We now discuss the question of how small ∆ t needsto be to efficiently decouple. For bounded operators, themotion induced by the decoupling field needs to be faster than the fastest time-scale characterizing the unwantedinteractions [21]. In the unbounded case, such a simpletime-scale defined only by the interaction cannot be pro-vided, as the convergence speed also crucially depends onthe state, given by the speed of convergence of ChernoffsTheorem (19). Clearly there exist a τ ( ψ, (cid:15) ) = tn largerthan zero for which F ( τ ) n ψ is up to an error (cid:15) given byexp( tA ) ψ . Assuming that system and bath are initiallyuncorrelated, we may (through purification) without lossof generality assume that the initial bath state ψ B ispure. We can then define τ ( (cid:15) ) = inf ψ S τ ( ψ S ⊗ ψ B ) > V. CONCLUSION
So far we have considered the two extreme cases inwhich either extrinsic or intrinsic decoherence is presentassuming the two mechanisms take place with the samedecay rate. Clearly in an experimental situation both, amixture L = L int + L ext of extrinsic and intrinsic deco-herence could be present. In this case, the asymptoticbehavior of the gate error would be between those twoextremal cases. It seems difficult to determine a gen- eral precise value, but estimates for the amount of in-trinsic decoherence can be obtained based on the bounds (cid:107)L int (cid:107) ≤ (cid:107)L int (cid:107) . The effective Lindbladian L int can bedetermined using process tomography. For intrinsic de-coherence decay rates predicted by collapse models weare at present a few orders of magnitude away from theregime in which this becomes feasible. But with currentadvances in qubit design and a world-wide effort to in-crease the number of clean qubits this could come withinreach soon.Our results pave the way towards the experimental ver-ification of alternative quantum theories (AQT) – despitethe presence of (extrinsic) decoherence. Even if the quan-tum noise is due to some unbounded coupling to an infi-nite dimensional environment we proved that the systemevolution can be decoupled and hence distinguished fromAQT. Furthermore, this decoupling of unbounded Hamil-tonians has applications in quantum engineering beyondthe scope of this paper. It is fascinating to contemplatethat in the vast experimental evidence for dynamical de-coupling such AQTs have already been discovered. Theanalysis of such experiments requires a detailed mathe-matical analysis, parts of which we have provided in [8]and parts of it remain to be done in future. Acknowledgements. – We thank Gavin Morley, TerryRudolph, Y. 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