Adiabatic theorem for closed quantum systems initialized at finite temperature
AAdiabatic theorem for closed quantum systems initialized at finite temperature
Nikolai Il‘in , Anastasia Aristova , , and Oleg Lychkovskiy , , Skolkovo Institute of Science and Technology, Bolshoy Boulevard 30, bld. 1, Moscow 121205, Russia Department of Mathematical Methods for Quantum Technologies,Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow 119991, Russia and Laboratory for the Physics of Complex Quantum Systems, Moscow Institute of Physics and Technology,Institutsky per. 9, Dolgoprudny, Moscow region, 141700, Russia (Dated: November 30, 2020)The evolution of a driven quantum system is said to be adiabatic whenever the state of the systemstays close to an instantaneous eigenstate of its time-dependent Hamiltonian. The celebrated quantumadiabatic theorem ensures that such pure state adiabaticity can be maintained with arbitrary accuracy,provided one chooses a small enough driving rate. Here, we extend the notion of quantum adiabaticity toclosed quantum systems initially prepared at finite temperature. In this case adiabaticity implies that the(mixed) state of the system stays close to a quasi-Gibbs state diagonal in the basis of the instantaneouseigenstates of the Hamiltonian. We prove a sufficient condition for the finite temperature adiabaticity.Remarkably, it implies that the finite temperature adiabaticity can be more robust than the pure stateadiabaticity, particularly in many-body systems. We present an example of a many-body system where, inthe thermodynamic limit, the finite temperature adiabaticity is maintained, while the pure state adiabaticity breaks down.
Introduction.
A concept of quantum adiabatic evolutionwas introduced by Born and Fock in the early days of quan-tum mechanics [1, 2]. The concept pertains to a drivenclosed quantum system described by a time-dependentHamiltonian. The evolution of the system is called adia-batic as long as the state of the system stays close to thetime-dependent instantaneous eigenstate of the Hamilto-nian. The celebrated adiabatic theorem [2, 3] states thatadiabaticity can be maintained with any prescribed accu-racy, provided the driving rate (i.e. the rate of change ofthe Hamiltonian) is chosen small enough. The adiabatictheorem enjoys a glorious history and a wide range of the-oretical and practical applications, including dynamics ofchemical reactions [4], population transfer between molec-ular vibrational levels [5, 6], theory of quantum topologicalorder [7], quantized charge transport [8], quantum mem-ory [9] and quantum adiabatic computation [10–12].Nowadays there is a wealth of experimental techniquesavailable to manipulate large quantum systems consistingof cold atoms in optical lattices, ions in ion traps, ar-rays of superconducting qubits and quantum dots etc [13].However, these systems are rarely prepared in pure states.Rather, they are typically initialized at some finite tempera-ture determined by the preparation protocol. Therefore theconventional concept of adiabaticity [1–3], which we referto as pure state adiabaticity (PSA) in what follows, callsfor extension to the case of finite temperature.Here we define the finite temperature adiabaticity as theproperty by which the state of a system initially preparedat finite temperature stays close to the quasi-Gibbs statein the course of the unitary quantum evolution. The time-dependent quasi-Gibbs state, defined by eq. (12) below,is diagonal in the instantaneous eigenbasis of the Hamilto-nian and has the same spectrum as the initial thermal state.Clearly, if the driving rate is so low that the conditions for PSA for any eigenstate are met, then the finite tem-perature adiabaticity is also present, irrespectively of thetemperature. It turns out that, in fact, the finite temperatureadiabaticity can be present at much higher driving rates.This follows from the finite temperature adiabatic condi-tion proven in the present paper. Remarkably, the energygaps do not enter this conditions directly, in contrast to thecase of PSA. Instead, the role of the energy gaps is playedby the temperature. This can be of particular importancefor many-body systems, where energy gaps vanish in thethermodynamic limit, and the pure state adiabaticity typi-cally breaks down whenever the driving rate is kept finitebut the system size is increased [14, 15]. We provide aparticular example of a many-body system where the finitetemperature adiabaticity survives the thermodynamic limit,despite the pure state adiabaticity being broken.The rest paper is organised as follows. We start fromintroducing required definitions and notions (most impor-tantly, the notion of the quasi-Gibbs state). Then we statethe adiabatic theorem for closed quantum systems preparedin thermal states and discuss its scope and implications.After that we illustrate the theorem by applying it to a par-ticular many-body system. We conclude the paper by thesummary and outlook. Technical details are relegated tothe Supplementary material [16].
Preliminaries
We describe an isolated driven quantumsystem by means of a time-dependent Hamiltonian. To in-troduce time dependence in a way convenient for our pur-poses, we consider a Hamiltonian H s dependent on a pa-rameter s and assume that s varies in time. Without lossof generality, we can assume that s is a linear function oftime, s = ωt, (1)where ω is the driving rate. The adiabatic limit is defined a r X i v : . [ qu a n t - ph ] N ov as ω → , t → ∞ , ωt = const > . (2)Let E ns and Φ ns be respectively eigenenergies and eigen-vectors of H s , H s Φ ns = E ns Φ ns , n = 1 , , ..., d, (3)where d is the dimension of the Hilbert space. We assumethat E ns and Φ ns are continuously differentiable in s .Importantly, H s can be represented as H s = U s (cid:101) H s U † s , (4)where U s is a continuously differentiable unitary operator, U = 1 , and (cid:101) H s is an auxiliary operator with the sameeigenvalues as H s and the same eigenvectors as H , (cid:101) H s = (cid:88) n E ns | n (cid:105)(cid:104) n | , (5)where | n (cid:105) ≡ Φ n . Note that time dependence enters (cid:101) H s only through E ns . An important object in our study is theoperator V s ≡ − iU † s ∂ s U s . (6)To characterize the spectrum, we define µ s = max n (cid:12)(cid:12)(cid:12)(cid:12) E n +10 − E n E n +1 s − E ns (cid:12)(cid:12)(cid:12)(cid:12) (7)and ν s = max n | ∂ s ln( E n +1 s − E ns ) | . (8)Often the spectrum of the driven Hamiltonian does notchange with time, which we refer to as isospectral driving .In this case (cid:101) H s = H , µ s do not actually depend on s , and ν s is identically zero. A particular simple instance of theisospectral driving is the uniform isospectral driving with H s = e isV H e − isV . (9)Here V coincides with V s defined by eq. (6).The state of the system ρ t satisfies the von Neumannequation i∂ t ρ t = [ H ωt , ρ t ] . (10)We assume that at t = 0 the system is initialized in a ther-mal state, ρ = e − βH /Z , Z ≡ tr e − βH , (11) β being the inverse temperature.If the system were prepared in an eigenstate (in partic-ular, in the ground state, i.e. “at zero temperature”), theadiabatic theorem [2, 3, 11] would imply that for any given s one can choose sufficiently small ω so that the state of the Note that U s is not an evolution operator. system at a (large) time t = s/ω is close (within a given er-ror margin) to the corresponding instantaneous eigenstate.This is what we refer to as pure state adiabaticity (PSA).When we turn to the case of finite temperatures, the firstquestion we have to address is what state one should com-pare the dynamical state ρ t with. If the conditions for PSAare met for any eigenstate, then ρ t stays close to the quasi-Gibbs state given by (see also a recent ref. [17]) θ βt ≡ Z − (cid:88) n e − βE n | Φ nωt (cid:105)(cid:104) Φ nωt | . (12)We will prove that, in fact, this is also the case under dif-ferent (and, generally, less stringent) conditions that thosefor PSA.It should be emphasized that the quasi-Gibbs state (12)is diagonal in the time-dependent instantaneous eigenba-sis of the Hamiltonian, but its spectrum does not changewith time and coincides with the spectrum of the initialGibbs state. The latter feature emerges because the spec-trum of the density matrix ρ t cannot be changed by the uni-tary evolution (10). For this reason the quasi-Gibbs state(12) is, in general, different from the instantaneous Gibbsstate ρ βt ≡ e − βH ωt / tr e − βH ωt , whose spectrum varies withtime.In what follows we will need to quantify the differencebetween two mixed quantum states. To this end, we employthe trace distance D tr ( ρ , ρ ) ≡ (1 /
2) tr | ρ − ρ | , (13)which is known to have a straightforward operationalmeaning [18–21]. Adiabatic theorem for finite temperatures.
Now we are ina position to state the following
Theorem:
The trace distance between the dynamicalstate of the system ρ t (initialized in the Gibbs state (11)and evolving according to the von Neumann equation(10)) and the quasi-Gibbs state θ βt (defined by eq. (12))is bounded from above by D tr (cid:16) ρ t , θ βt (cid:17) ≤ (cid:113) √ ωβ (cid:32) µ ωt (cid:107) V ωt (cid:107) + (cid:90) ωt µ s (cid:48) (cid:107) ∂ s (cid:48) V s (cid:48) (cid:107) ds (cid:48) + (cid:90) ωt ν s (cid:48) µ s (cid:48) (cid:107) V s (cid:48) (cid:107) ds (cid:48) + √ (cid:90) ωt µ s (cid:48) (cid:107) V s (cid:48) (cid:107) ds (cid:48) (cid:33) / . (14)Here V s , µ s and ν s are defined according to eqs. (6), (7)and (8), respectively, and (cid:107) . . . (cid:107) refers to the operatornorm. For our purposes, the operator norm (cid:107) . . . (cid:107) can be defined as the maximumamong absolute values of eigenvalues of the corresponding operator.
This theorem implies that ρ t converges to θ βt in the adi-abatic limit (2), provided the term in brackets remains fi-nite. The proof of the theorem can be found in the Supple-ment [16].Observe that the r.h.s. of the bound (14) vanishes in thelimit of infinite temperature, β = 0 . This is consistent withthe simple fact that at the infinite temperature ρ t = θ β =0 t = /d , and the evolution is adiabatic at any driving rate.The theorem admits a particularly simple form in thecase of the uniform isospectral driving (9): Corollary:
For the isospectrally and uniformly drivenHamiltonian (9) the bound (14) reads D tr (cid:16) ρ t , θ βt (cid:17) ≤ (cid:114) √ ωβ (cid:107) V (cid:107) (cid:16) √ ωt (cid:107) V (cid:107) (cid:17) . (15)The corollary immediately implies that ρ t converges to θ βt in the adiabatic limit (2) whenever (cid:107) V (cid:107) is finite.Remarkably, energy gaps do not directly enter thebounds (14) and (15), in contrast to typical sufficient con-ditions for PSA [11] (see, however, [22, 23]). This is cru-cial for the robustness of adiabaticity in the thermodynamiclimit, since the energy gaps vanish with increasing the sys-tem size. The system size may also enter the bounds (14)and (15) through (cid:107) V s (cid:107) , (cid:107) ∂ s V s (cid:107) and (for the bound (14))through µ s , ν s . When the above quantities are finite inthe thermodynamic limit, the finite temperature adiabatic-ity survives in this limit even if the PSA fails. Below weconsider a many-body system exhibiting such behavior. Example.
Consider a thin straight wire with N electronsand a quantum sensor which can be moved around the wire,see Fig. 1. We consider a toy model of the sensor consist-ing of a single quantum spin S with a magnetic moment µ magn (not to be confused with µ s defined in eq. (7)). Theinteraction between the spin and the electrons is mediatedby the magnetic field produced by the electron motion. We consider the case of zero net current of electrons. Still,the interaction persists even in this case due to fluctuationsof the current, both classical and quantum. The Hamilto-nian of the system reads H α = H e + H Seα , (16)where H e is the Hamiltonian of electrons (we do not needits explicit form here), and H Seα = − µ magn πr J ( − sin α S x + cos α S y ) (17)is the Hamiltonian of the magnetic field-mediated interac-tion between electrons and the spin. Here ( S x , S y , S z ) arethe components of the spin operator, J is the operator ofthe electron current, r is the distance from the sensor to the We disregard the magnetic fields of the magnetic moments of electrons.
FIG. 1. (Color online) A quantum sensor with a single spin pos-sessing a magnetic moment is moved around a wire along a circu- lar trajectory. The net current through the wire is zero, howeverthe electrons in the wire are still magnetically coupled to the spindue to fluctuations of the current, see eqs. (16), (17). The many-body adiabaticity of the electron-spin system at finite temperatureis robust with respect to increasing the system size (i.e. the lengthof the wire). In contract, the pure state adiabaticity breaks downin the thermodynamic limit at any finite driving rate. wire and α is the polar angle determining the position ofthe sensor, see Fig. 1.We further assume that the sensor is moved along a cir-cular trajectory around the wire with r = const and α = ωt . Then the Hamiltonian (16) can be cast in the form (9), H α = e − iαS z H α =0 e iαS z , therefore the bound (15) with V = − S z applies. This bound implies that it suffices tochoose ω ≤ ε √ β S (1 + √ α S ) (18)to move the sensor up to the angle α along the circulartrajectory while maintaining adiabaticity with precision ε , D tr (cid:16) ρ t , θ βt (cid:17)(cid:12)(cid:12)(cid:12) t = α/ω ≤ ε .Remarkably, the sufficient adiabatic condition (18) doesnot depend on the number of electrons. Thus the finite tem-perature adiabaticity is robust in the thermodynamic limit N → ∞ , L → ∞ , A = const , ρ ≡ N/ ( LA ) = const ,where L and A are, respectively, the length and the crosssection of the wire, and ρ is the number density of electronsin the wire.In contract, the pure state adiabaticity breaks down inthe thermodynamic limit. This can be easily seen if peri-odic boundary conditions along the z direction are imposedon the electron wave functions. In this case the Hamilto-nian (16) commutes with the current operator, the latter be-ing related to the total momentum of electrons, P e , J = eρAm e N P e , (19)where e and m e are the charge and the mass of the elec-tron. As a result, the dynamics of the spin is governed bythe effective Hamiltonian (17), where J now refers to theeigenvalue of the current operator (19) in the eigenstate thesystem is initialized in. Since in the typical eigenstate fromthe Gibbs ensemble this eigenvalue is O (1 / √ N ) , the driv-ing rate necessary to maintain the pure state adiabaticityalso scales as / √ N and vanishes in the thermodynamiclimit (see a detailed analysis in the Supplement [16]). Summary and outlook
To summarize, we have introduced the notion of fi-nite temperature adiabaticity of an isolated quantum sys-tem and proved the finite temperature adiabatic theorem(14). The sufficient adiabatic condition which follows fromthis theorem does not contain energy gaps, in contrast tomost of the adiabatic conditions for pure state adiabaticity.This indicates that the finite temperature adiabaticity canbe more robust in the thermodynamic limit then the purestate adiabaticity. We confirm this expectation for the spe-cific model (16). It should be noted that this robustnessis consistent with earlier numerical observations that mi-crocanonical mixed states are more robust to adiabaticitybreaking than pure states [24].It should be emphasized that our notion of adiabaticityrefers to the many-body state of the system and is differentfrom the notion of local adiabaticity [25–31]. The latternotion applies to the reduced density matrix of a subsystemcoupled to a reservoir. The many-body adiabaticity impliesthe local adiabaticity, but not vice versa.A considerable limitation of the bounds (14), (15) isthat they contain the operator norms. For continuous sys-tems operator norms of certain physically relevant oper-ators (e.g. momentum) are infinite, which renders thebounds void. In fact, the operator norm can be replaced bythe better behaved thermal averages in some of the termsin eqs. (14), (15), as we discuss in the Supplement [16].However, at the moment we are not able to avoid the op-erator norms altogether, and leave the improvement of thebounds (14), (15) in this direction for further work.
Acknowledgements.
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Properties of µ s and ν s Here we prove a Lemma about µ s and ν s required for the proof of the finite temperature adiabatic theorem. We introducea shorthand notation ∆ mn ( s ) ≡ E ms − E ns . (S1)We assume that at a given s the spectrum is ordered: ∆ mn ( s ) ≥ m > n. (S2)Let us show that µ s and ν s defined respectively by eqs. (7) and (8) of the main text, satisfy the following Lemma: µ ( s ) = max ≤ n Consider an arbitrary set of real numbers A n , n = 1 , , . . . , d and introduce q = max m,n (cid:12)(cid:12)(cid:12)(cid:12) A m − A n E ms − E ns (cid:12)(cid:12)(cid:12)(cid:12) . (S5)We are going to prove that in fact q = max k (cid:12)(cid:12)(cid:12)(cid:12) A k +1 − A k E k +1 s − E ks (cid:12)(cid:12)(cid:12)(cid:12) . (S6)This equality entails eq. (S3) for A n = E n and eq. (S4) for A n = ∂ s E ns .To prove eq. (S6), we start from an obvious observation that q ≥ max k (cid:12)(cid:12)(cid:12)(cid:12) A k +1 − A k E k +1 s − E ks (cid:12)(cid:12)(cid:12)(cid:12) . (S7)Let us show that, in fact, the strict inequality is impossible. To this end we assume the opposite, i.e. that (cid:12)(cid:12)(cid:12)(cid:12) A k +1 − A k E k +1 s − E ks (cid:12)(cid:12)(cid:12)(cid:12) < q ∀ k. (S8)Then for any m > n we obtain | A m − A n | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m − (cid:88) k = n ( A k +1 − A k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ m − (cid:88) k = n | A k +1 − A k | < q m − (cid:88) k = n ( E k +1 s − E ks ) = q ( E ms − E ns ) , (S9)where the ordering of energies, (S2), is used to get rid of the modulus. Eq. (S9) is inconsistent with eq. (S8). Thus theequality (S6) is true, q.e.d. Proof of the finite temperature adiabatic theorem Here we prove the bound (14) from the main text.We introduce σ t = U † ωt ρ t U ωt , (S10)which evolves according to ∂ t σ t = − i [ (cid:101) H ωt + ωV ωt , σ t ] , σ = e − βH /Z . (S11)We denote by dot the derivative of a function with respect to its argument s = ωt . For example, ˙ U s = ∂ s U s , ˙ U ωt = ∂ s U s | s = ωt but ∂ t U ωt = ω ˙ U ωt .We first estimate the quantity D ≡ − tr( (cid:113) θ βt √ ρ t ) = 1 − tr( √ ρ √ σ t ) . (S12)Note that ∂ t √ σ t = − i [ (cid:101) H ωt + ωV ωt , √ σ t ] . (S13)Therefore ∂ t D = − tr( √ ρ ∂ t √ σ t ) = i √ Z tr( e − βH / [ (cid:101) H ωt + ωV ωt , √ σ t ]) = iω √ Z tr([ e − βH / , V ωt ] √ σ t )= iω √ Z (cid:88) n, k ( e − β E n − e − β E k ) (cid:104) n | V ωt | k (cid:105) (cid:104) k |√ σ t | n (cid:105) = iβω √ Z (cid:88) n, k f nk ( ωt )∆ nk ( ωt ) (cid:104) n | V ωt | k (cid:105) (cid:104) k |√ σ t | n (cid:105) , (S14)where | n (cid:105) = | Φ n (cid:105) , | k (cid:105) = | Φ k (cid:105) are eigenstates of H and, consequently, of (cid:101) H ωt for arbitrary t , E n , E k are eigenenergiesof H , and f nk ( ωt ) ≡ e − β E n − e − β E k β ∆ nk ( ωt ) / n (cid:54) = k, f nn = 0 . (S15)Note that we will occasionally drop an argument of the function f nk when this does not lead to ambiguities.We notice that ( E nωt − E kωt ) (cid:104) k |√ σ t | n (cid:105) = −(cid:104) k | [ (cid:101) H ωt , √ σ t ] | n (cid:105) = − i (cid:104) k | ∂ t √ σ t | n (cid:105) + ω (cid:104) k | [ V ωt , √ σ t ] | n (cid:105) . (S16)Substituting this expression to eq. (S14) and integrating it over time one obtains D = βω √ Z (cid:88) n, k (cid:90) t f nk ( ωt (cid:48) ) (cid:104) n | V ωt (cid:48) | k (cid:105) ( (cid:104) k | ∂ t (cid:48) √ σ t (cid:48) | n (cid:105) + iω (cid:104) k | [ V ωt (cid:48) , √ σ t (cid:48) ] | n (cid:105) ) dt (cid:48) . (S17)Integrating (S17) by parts one gets D = βω √ Z (cid:88) n, k (cid:32) f nk ( ωt ) (cid:104) n | V ωt | k (cid:105)(cid:104) k |√ σ t | n (cid:105) − ω (cid:90) t f nk ( ωt (cid:48) ) (cid:104) n | ˙ V ωt (cid:48) | k (cid:105)(cid:104) k |√ σ t (cid:48) | n (cid:105) dt (cid:48) + ω (cid:90) t f nk ( ωt (cid:48) ) (cid:104) n | V ωt (cid:48) | k (cid:105) ˙∆ nk ( ωt (cid:48) )∆ nk ( ωt (cid:48) ) (cid:104) k |√ σ t (cid:48) | n (cid:105) dt (cid:48) + iω (cid:90) t f nk ( ωt (cid:48) ) (cid:104) n | V ωt (cid:48) | k (cid:105)(cid:104) k | [ V ωt (cid:48) , √ σ t (cid:48) ] | n (cid:105) dt (cid:48) (cid:33) = βω √ Z ( K + K + K + K ) , (S18)where we us ˙ f nk ( ωt ) = − f nk ( ωt ) ˙∆ nk ( ωt ) / ∆ nk ( ωt ) and K = (cid:88) n, k f nk ( ωt ) (cid:104) n | V ωt | k (cid:105)(cid:104) k |√ σ t | n (cid:105) (S19) K = − ω (cid:88) n, k (cid:90) t f nk ( ωt (cid:48) ) (cid:104) n | ˙ V ωt (cid:48) | k (cid:105)(cid:104) k |√ σ t (cid:48) | n (cid:105) dt (cid:48) (S20) K = ω (cid:88) n, k (cid:90) t f nk ( ωt (cid:48) ) (cid:104) n | V ωt (cid:48) | k (cid:105) ˙∆ nk ( ωt (cid:48) )∆ nk ( ωt (cid:48) ) (cid:104) k |√ σ t (cid:48) | n (cid:105) dt (cid:48) (S21) K = iω (cid:88) n, k (cid:90) t f nk ( ωt (cid:48) ) (cid:104) n | V ωt (cid:48) | k (cid:105)(cid:104) k | [ V ωt (cid:48) , √ σ t (cid:48) ] | n (cid:105) dt (cid:48) (S22)Obviously, | D | ≤ βω √ Z ( | K | + | K | + | K | + | K | ) . (S23)Let us estimate | K | : | K | ≤ (cid:32)(cid:88) n, k f nk (cid:104) n | V ωt | k (cid:105) (cid:104) k | V ωt | n (cid:105) (cid:33) / (cid:32)(cid:88) n, k (cid:104) k |√ σ t | n (cid:105)(cid:104) n |√ σ t | k (cid:105) (cid:33) / . (S24)The term in the second bracket reads (cid:88) n, k (cid:104) k |√ σ t | n (cid:105)(cid:104) n |√ σ t | k (cid:105) = tr σ t = tr ρ = 1 . (S25)To estimate the first term we need to estimate f nk : f nk ( ωt ) = (cid:32) e − β E n − e − β E k β ( E n − E k ) / nk (0)∆ nk ( ωt (cid:48) ) (cid:33) ≤ (cid:32) e − β E n − e − β E k β ( E n − E k ) / (cid:33) µ ( ωt ) , (S26)where eq. (S3) is used to establish the inequality. Further, by the Lagrange’s Mean Value Theorem there exists a ∈ (0 , such that (cid:32) e − β E n − e − β E k β ( E n − E k ) / (cid:33) = e − β ( aE n +(1 − a ) E k ) ≤ e − β min { E n ,E k } ≤ e − βE n + e − βE k , n (cid:54) = k. (S27)Combining inequalities (S26) and (S27) and extending them to the trivial case n = k (where f nn = 0 by definition (S15))we get f nk ( ωt ) ≤ µ ( ωt ) (cid:16) e − βE n + e − βE k (cid:17) . (S28)We use this bound to proceed further: (cid:88) n, k f nk (cid:104) n | V ωt | k (cid:105) (cid:104) k | V ωt | n (cid:105)≤ µ ( ωt ) (cid:88) n, k (cid:16) e − βE n (cid:104) n | V ωt | k (cid:105) (cid:104) k | V ωt | n (cid:105) + e − βE k (cid:104) n | V ωt | k (cid:105) (cid:104) k | V ωt | n (cid:105) (cid:17) = 1 µ ( ωt ) (cid:88) n, k (cid:16) (cid:104) n | V ωt | k (cid:105) (cid:104) k | V ωt e − βH | n (cid:105) + (cid:104) n | V ωt | k (cid:105) (cid:104) k | e − βH V ωt | n (cid:105) (cid:17) = 2 1 µ ( ωt ) tr( V ωt e − βH ) (S29)Next use the inequality | tr AB | (cid:54) (cid:107) A (cid:107) tr B, (S30)valid for any B > and diagonalisable A (be reminded that (cid:107) ... (cid:107) stands for the operator norm), to obtain tr V ωt e − βH ≤ (cid:107) V ωt (cid:107) tr e − βH . (S31)Finally | K | ≤ (cid:112) Z µ ωt (cid:107) V ωt (cid:107) . (S32) K can be bounded in an analogous way: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) n, k f nk ( ωt (cid:48) ) (cid:104) n | ˙ V ωt (cid:48) | k (cid:105)(cid:104) k |√ σ t (cid:48) | n (cid:105) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:112) Z µ ωt (cid:48) (cid:107) ˙ V ωt (cid:48) (cid:107) (S33)and | K | ≤ ω (cid:112) Z (cid:90) t µ ωt (cid:48) (cid:107) ˙ V ωt (cid:48) (cid:107) dt (cid:48) = (cid:112) Z (cid:90) ωt µ s (cid:48) (cid:107) ˙ V s (cid:48) (cid:107) ds (cid:48) . (S34)Let us estimate | K | : | K | ≤ (cid:90) t (cid:88) n, k f nk (cid:32) ˙∆ nk ( ωt (cid:48) )∆ nk ( ωt (cid:48) ) (cid:33) (cid:104) n | V ωt (cid:48) | k (cid:105) (cid:104) k | V ωt | n (cid:105) / (cid:32)(cid:88) n, k (cid:104) k |√ σ t (cid:48) | n (cid:105)(cid:104) n |√ σ t (cid:48) | k (cid:105) (cid:33) / dt (cid:48) , (S35) f nk ( ωt (cid:48) ) (cid:32) ˙∆ nk ( ωt (cid:48) )∆ nk ( ωt (cid:48) ) (cid:33) = (cid:32) e − β E n − e − β E k β ( E n − E k ) / nk (0) ˙∆ nk ( ωt (cid:48) )∆ nk ( ωt (cid:48) ) (cid:33) ≤ ν ( ωt (cid:48) ) µ ( ωt (cid:48) ) ( e − βE n + e − βE k ) , (S36)where eqs. (S3), (S4) are used to establish the inequality. Thus we obtain | K | ≤ (cid:112) Z (cid:90) ωt ν s (cid:48) µ s (cid:48) (cid:107) V s (cid:48) (cid:107) ds (cid:48) (S37)Finally, let us estimate K : | K | ≤ ω (cid:90) t dt (cid:48) (cid:32)(cid:88) n, k f nk (cid:104) n | V ωt (cid:48) | k (cid:105) (cid:104) k | V ωt (cid:48) | n (cid:105) (cid:33) / (cid:32)(cid:88) n, k (cid:104) k | [ V ωt (cid:48) , √ σ t (cid:48) ] | n (cid:105)(cid:104) n | [ √ σ t (cid:48) , V ωt (cid:48) ] | k (cid:105) (cid:33) / . (S38)The term in the first bracket has been already bounded, see eq. (S29). The term in the second bracket reads tr[ V ωt (cid:48) , √ σ t (cid:48) ] [ √ σ t (cid:48) , V ωt (cid:48) ] = 2 tr V ωt (cid:48) σ t (cid:48) − (cid:16) σ / t (cid:48) V ωt (cid:48) σ / t (cid:48) (cid:17) ≤ (cid:107) V ωt (cid:48) (cid:107) tr σ t = 2 (cid:107) V ωt (cid:48) (cid:107) , (S39)and we get | K | ≤ (cid:112) Z ω (cid:90) t µ ωt (cid:48) (cid:107) V ωt (cid:48) (cid:107) dt (cid:48) = 2 (cid:112) Z (cid:90) ωt µ s (cid:48) (cid:107) V s (cid:48) (cid:107) ds (cid:48) . (S40)Finally we collect all pieces (S32)–(S40) together and bound D according to eq. (S23): D ≤ ωβ √ (cid:18) µ ωt (cid:107) V ωt (cid:107) + (cid:90) ωt µ ( t (cid:48) ) (cid:107) ∂ t (cid:48) V t (cid:48) (cid:107) dt (cid:48) + (cid:90) ωt ν ( t (cid:48) ) µ ( t (cid:48) ) (cid:107) V t (cid:48) (cid:107) dt (cid:48) + √ (cid:90) ωt µ ( t (cid:48) ) (cid:107) V t (cid:48) (cid:107) dt (cid:48) (cid:19) . (S41)The last thing we need to do is to connect D with the trace distance D tr (cid:16) ρ t , θ βt (cid:17) . This can be done thanks to theinequality proven in [19] which reads D tr ( ρ , ρ ) ≤ (cid:113) − (cid:0) tr √ ρ √ ρ (cid:1) ≤ √ D. (S42)Eq. (14) of the main text follows from eqs. (S41) and (S42), q.e.d.As was mentioned in the main text, the presence of operator norms in the final result makes the bound inapplicable inthe cases where V or ˙ V are unbounded operators. In fact, the operator norms are superficial for estimating K , K and K above and can be substituted by thermal averages with respect to the initial Gibbs state. For K this can be seen from theeq. (S29), and analogously for K and K . However, we were not able to avoid the operator norm when estimating K .0 Spin moved around a wire: pure state adiabaticity Here we derive a condition for pure state adiabaticity in the electron-spin system (see (16),(17) of the main text) underthe assumption of periodic boundary conditions for electron wave functions in the z -direction. To be specific, we choose S = 12 . (S43)Using eq. (19) of the main text, we rewrite the Hamiltonian as H α = H e + γ e − iαS z S y e iαS z , (S44)where γ = − µ magn πr eρAm e N P e . (S45)The total momentum of electrons, P e , commutes with H α , therefore we treat it as a c -number. We initialize the system inan eigenstate of H α , Ψ = | electrons (cid:105) ⊗ ψ , (S46)where | electrons (cid:105) is an eigenstate of H e and P e , while ψ is an eigenstate of S y , S y ψ = 12 ψ . (S47)The time-dependent many-body wave function Ψ t satisfies the Schr¨odinger equation i∂ t Ψ t = H ωt Ψ t . (S48)It is easy to see that Ψ t factors as follows: Ψ t = (cid:0) e − iH e t | electrons (cid:105) (cid:1) ⊗ ψ t , (S49)where ψ t satisfies the Schr¨odinger equation with the effective spin Hamiltonian H Seωt , i∂ t ψ t = H Seωt ψ t , H Seωt = γ e − iωtS z S y e iωtS z . (S50)The figure of merit of the pure state adiabaticity is the adiabatic fidelity between the dynamical wave function Ψ t andthe instantaneous eigenfunction Φ α of the Hamiltonian (S44): F t ≡ |(cid:104) Φ α = ωt | Ψ t (cid:105)| . (S51)From (S49) one immediately obtains that F t is given by F t = |(cid:104) ϕ α = ωt | ψ t (cid:105)| , (S52)where ϕ α is the eigenstate of H Seα satisfying ϕ = ψ .The dynamics of ψ t can be easily inferred from eq. (S50) by transformation to the rotating frame. As a result oneobtains − F t = ω ω + γ sin (cid:18) α (cid:113) γ /ω (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) α = ωt . (S53)We say that the adiabaticity is maintained up to some target α with the accuracy ε if for t ≤ α/ω − F t ≤ ε. (S54)Let us assume that the target α is greater than π . Then the sine squared in eq. (S53) will become equal to unity somewhereon the way to the target α . Taking this into account, we conclude from eq. (S53) that the maximal driving rate ω ε thatallows to maintain adiabaticity with the accuracy ε is given by ω ε = γ (cid:114) ε − ε . (S55)Since P e ∼ p F √ N in the majority of states in the Gibbs ensemble, it follows from eqs. (S45) and (S55) that for thesestates ω ε ∼ / √ N (S56)in the thermodynamic limit N → ∞ , ρ = const= const