Thermal equilibrium of a macroscopic quantum system in a pure state
Sheldon Goldstein, David A. Huse, Joel L. Lebowitz, Roderich Tumulka
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J un Thermal Equilibrium of a Macroscopic QuantumSystem in a Pure State
Sheldon Goldstein ∗ , David A. Huse † ,Joel L. Lebowitz ‡ , and Roderich Tumulka § June 23, 2015
Abstract
We consider the notion of thermal equilibrium for an individual closed macro-scopic quantum system in a pure state, i.e., described by a wave function. Themacroscopic properties in thermal equilibrium of such a system, determined by itswave function, must be the same as those obtained from thermodynamics, e.g.,spatial uniformity of temperature and chemical potential. When this is true wesay that the system is in macroscopic thermal equilibrium (MATE). Such a sys-tem may however not be in microscopic thermal equilibrium (MITE). The latterrequires that the reduced density matrices of small subsystems be close to thoseobtained from the microcanonical, equivalently the canonical, ensemble for the whole system. The distinction between MITE and MATE is particularly relevantfor systems with many-body localization (MBL) for which the energy eigenfuc-tions fail to be in MITE while necessarily most of them, but not all, are in MATE.We note however that for generic macroscopic systems, including those with MBL,most wave functions in an energy shell are in both MATE and MITE. For a classi-cal macroscopic system, MATE holds for most phase points on the energy surface,but MITE fails to hold for any phase point.Key words: many-body localization, quantum statistical mechanics, canonicaltypicality, thermal equilibrium subspace, macro-observables, thermalization. ∗ Department of Mathematics, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway,NJ 08854-8019, USA. E-mail: [email protected] † Department of Physics, Princeton University, Jadwin Hall, Washington Road, Princeton, NJ 08544-0708, USA. E-mail: [email protected] ‡ Departments of Mathematics and Physics, Rutgers University, Hill Center, 110 Frelinghuysen Road,Piscataway, NJ 08854-8019, USA. E-mail: [email protected] § Department of Mathematics, Rutgers University, Hill Center, 110 Frelinghuysen Road, Piscataway,NJ 08854-8019, USA. E-mail: [email protected] Introduction
Thermal behavior of closed macroscopic systems in pure states has been widely studied inrecent years; see e.g., [7, 30, 16, 43, 31, 32, 35, 23, 15, 33, 36, 34, 8], after some pioneeringwork even earlier [38, 46, 39, 6, 40, 44]. In particular, the importance of the eigenstatethermalization hypothesis (ETH) [6, 40] has become widely appreciated, see e.g., [35, 32,23, 15, 17, 25, 34, 8]. It asserts, in one version, that all energy eigenstates (in a suitableenergy shell) are thermal, i.e., assigning probability distributions to observables that arecharacteristic of thermal equilibrium. The ETH holds for many (but not all) macroscopicquantum systems. When it holds then the system, starting out of equilibrium, willthermalize (at least in the time average; see Section 4 below). Thus if a system doesnot thermalize it must have energy eigenfunctions that fail to be thermal.An important case of this is that of many-body localization (MBL) [1, 3, 26], forwhich the Hamiltonian has (at least some) eigenfunctions that are in some way localized,so that, for any wave function, the component from these eigenfunctions will not spreadbut stay localized forever. For systems with MBL it has been argued that most (if notall) energy eigenfunctions (in suitable energy intervals) fail to be thermal, and there arein fact models for which this can be analytically [19], numerically [28], or perturbatively[2, 3, 37] seen to be the case. At the same time it has been argued [15] that most energyeigenstates must rather generally be thermal, in particular even for systems with MBL.This situation is thus rather puzzling: How can the fact that most energy eigen-functions for (some) MBL systems are not thermal be reconciled with the results in [15]showing that in wide generality most of them must be. The answer to this question,we point out, lies in the fact that there are basically two notions of thermal equilibriumfor pure states: a macroscopic notion of thermal equilibrium that we call MATE, anda more refined microscopic one that we call MITE. While most, but definitely not all,energy eigenstates of a system with MBL are in MATE (see Sec. 4), none, or nearlynone, are in MITE [2, 3, 28, 19]. Nonetheless, most pure states in the energy shell arein both MATE and MITE, even for systems with MBL (see Sec. 2 and 3). It is withthese two notions of thermal equilibrium and their consequences that we are concernedhere.To be more precise, consider a finite, macroscopic, closed quantum system withHilbert space H . Let H mc be a micro-canonical energy shell, i.e., the subspace of H spanned by the energy eigenstates with eigenvalue in an energy interval that is smallon the macroscopic scale but contains many eigenvalues. The micro-canonical densitymatrix ˆ ρ mc is defined by ˆ ρ mc = (dim H mc ) − ˆ P mc with ˆ P mc the projection to H mc . Asusual, pure states in H mc are superpositions of energy eigenstates in H mc . Both MITEand MATE can be expressed as subsets of the unit sphere in H mc , S ( H mc ) = { ψ ∈ H mc : k ψ k = 1 } . (1)We will often simply say “micro” for “microscopic” and “macro” for “macroscopic”;“most” means “all but a few” or “all except a set of small measure” (i.e., “an over-whelming majority of”); measures are taken to be normalized; the small measure, in2act, tends to 0 in the thermodynamic limit. We define MATE and MITE in Sections 2and 3, respectively, and discuss the properties and differences of the two notions in Sec-tion 4. We also note there that these notions can also be applied to mixed states. Weprovide a deeper and more detailed discussion elsewhere [13]. The definition of MATE is based on macro observables ˆ M , . . . , ˆ M K . To be specific,macro observables can reasonably be based on a partition of the system’s availablevolume Λ ⊂ R into cells Λ i that are small on the macro scale but still large enoughto each contain a large number of degrees of freedom. Examples of natural choices ofˆ M ’s are, for each cell, the number of particles of each type, the total energy, the totalmomentum, and/or the total magnetization. These are the variables usually consideredin the thermodynamic or hydrodynamic description of a fluid or magnet.Following von Neumann [46], we take the ˆ M j to commute with each other and to besuch that the gaps between the eigenvalues are of the order of the macroscopic resolu-tion (so that the eigenvalues are highly degenerate). This can be achieved by suitably“rounding off” and coarse-graining the operators representing the macro observables[46, 17, 27]. Taking H mc to be an eigenspace of a “macro energy” operator, and thusto commute with the other macro observables, all ˆ M j can be regarded as operators on H mc . Their joint spectral decomposition defines an orthogonal decomposition H mc = M ν H ν , (2)and the subspaces H ν (“macro spaces”), the joint eigenspaces of the macro observables,correspond to the different macro states and have very high dimension. The decom-position is in some ways analogous to a partition, in classical mechanics, of an energyshell Γ mc in phase space into disjoint subsets Γ ν corresponding to different macro states[4, 9, 14, 22]. It is generally the case [4, 21, 14], both in classical and quantum mechan-ics, that one of the macro states, corresponding to thermal equilibrium, is dominant,i.e., that one of the Γ ν ’s, denoted Γ eq , has most of the phase space volume of Γ mc , andthat one of the H ν , denoted H eq , has most of the dimensions of H mc , i.e.,dim H eq dim H mc = 1 − ε (3)with ε ≪ Realistic values of ε , say for a liter of air under atmospheric pressure,are smaller than 10 − [13]; more generally, ε is exponentially small in the number of An exception to the existence of a dominant macro space is provided by first-order phase transitions,such as in the ferromagnetic Ising model in a vanishing external magnetic field, where H ν has theappropriate majority of spins up and H ν ′ has the appropriate majority of spins down, each havingnearly 50% of the dimension of H mc for a suitable energy interval. ε .The system is said to be in MATE whenever its wave function ψ lies within a δ -neighborhood of H eq with suitably small δ >
0, i.e., in the setMATE = n ψ ∈ S ( H mc ) : h ψ | ˆ P eq | ψ i > − δ o (4)with ˆ P eq the projection to H eq . Thus for a state ψ that is in MATE, the probabilityis close to one that all macro observables take on their thermal equilibrium values. Aconcept of thermal equilibrium along these lines was used before in, e.g., [18, 35, 15, 10,11, 12]. It is known [15] that, if ε ≪ δ , then MATE has most of the surface area of S ( H mc ), so most pure states are in MATE. This follows from the fact that the averageof h ψ | ˆ P eq | ψ i over S ( H mc ) is equal to (3); since h ψ | ˆ P eq | ψ i cannot exceed 1, it must beclose to 1 for most ψ (see Section 4 for more detail). It can similarly be shown, see (21)below, that most energy eigenstates are in MATE. To be sure, there are states in theenergy shell which are not in MATE; for example, one could take a tensor product ofstates of two regions having (what look macroscopically like) different temperatures.An alternative definition due to Tasaki [45] (in the same direction as [5, 43]), notstrictly but approximately equivalent and denoted TMATE here, avoids the step ofrounding off to make the macro observables commute, which may pose substantial dif-ficulty to carry out in practice. Instead, take ˆ M , . . . , ˆ M K to be the macro observablesbefore rounding off and coarse graining (mathematically, any self-adjoint operators), let V j = tr( ˆ ρ mc ˆ M j ) be the thermal equilibrium value of ˆ M j , and let ∆ M j be the macroresolution of the observable represented by ˆ M j . Using 1 A to denote the characteristicfunction of the set A , we defineˆ P j = 1 [ V j − ∆ M j ,V j +∆ M j ] ( ˆ M j ) (5)to be the projection associated with the eigenvalues of ˆ M j that lie within the macroresolution of the thermal equilibrium value. Then letTMATE = K \ j =1 n ψ ∈ S ( H mc ) : h ψ | ˆ P j | ψ i > − δ o . (6)Note that h ψ | ˆ P j | ψ i is the probability of finding, in a quantum measurement of ˆ M j on asystem in state ψ , a value that is ∆ M j -close to V j . If this probability is > − δ for atleast the fraction 1 − η of S ( H mc ) for each j , then TMATE has at least size 1 − Kη (interms of normalized surface area), which is close to 1 if η ≪ K − . We note further thatMATE as in (4) can essentially also be written as the right-hand side of (6) if the ˆ M j are taken again as commuting and coarse-grained on the scale ∆ M j , so that ˆ P j becomesthe projection to the eigenspace of ˆ M j with the dominant (most degenerate) eigenvalue,and H eq is the intersection of these eigenspaces. Indeed, the right-hand side of (6) isthen contained in that of (4) with δ replaced by Kδ , and that of (4) is then containedin that of (6). 4 MITE
While MATE implies thermal behavior only for macro observables, MITE involves also“micro” observables, more precisely, those observables concerning only a region smallerthan a certain length scale ℓ . The definition of MITE is inspired by canonical typicality ,the observation [7, 29, 30, 16] that for any not-too-large subsystem S and most wavefunctions ψ in the energy shell H mc , the reduced density matrix of S is close to thethermal equilibrium density matrix of S ,ˆ ρ ψS ≈ ˆ ρ mc S , (7)where ˆ ρ ψS = tr S c | ψ ih ψ | (8)is the reduced density matrix of S obtained by tracing out the complement S c of S , andˆ ρ mc S = tr S c ˆ ρ mc . (9)If S is small enough then ˆ ρ mc S ≈ ˆ ρ ( β ) S (10)for suitable β >
0, where the right-hand side is the partial trace,ˆ ρ ( β ) S = tr S c ˆ ρ ( β ) , (11)of the canonical density matrixˆ ρ ( β ) = 1 Z e − β ˆ H with Z = tr e − β ˆ H . (12)As a consequence, for small S , it does not matter whether one starts from ˆ ρ mc or ˆ ρ ( β ) (this fact is a version of equivalence of ensembles), and we also haveˆ ρ ψS ≈ ˆ ρ ( β ) S . (13)We will call (11) the canonical or thermal density matrix for S . We note that if ˆ ρ ψS ≈ ˆ ρ mc S for some subsystem S then the same is true for every smaller subsystem S ′ contained in S (“subsubsystem property” of (7)), just by taking another partial trace on both sidesof the approximate equation ˆ ρ ψS ≈ ˆ ρ mc S .The system is said to be in MITE ℓ (MITE on the length scale ℓ ) whenever ψ ∈ S ( H mc ) satisfies (7) for every subsystem S corresponding to a spatial region of diameterdiam( S ) ≤ ℓ , i.e., MITE ℓ = \ S : diam( S ) ≤ ℓ n ψ ∈ S ( H mc ) : ˆ ρ ψS ≈ ˆ ρ mc S o (14) The density matrix Z − S exp( − β ˆ H S ) with ˆ H S the Hamiltonian of S is sometimes called the canonicalor thermal density matrix for S ; it agrees with (11) if the interaction between S and its complementcan be neglected. If the interaction cannot be neglected, then (11) is the correct density matrix to use. ≈ (such as the trace norm of the difference being smallerthan a given value). The subsubsystem property implies that every ψ in MITE ℓ lies alsoin MITE ℓ ′ for any smaller scale 0 < ℓ ′ < ℓ . MITE is then defined to mean MITE ℓ with ℓ the largest ℓ small enough to ensurethat (10) holds for every subsystem S with diam( S ) ≤ ℓ . As a practical value, forexample, we may take ℓ = 10 − diam(Λ) , (15)where Λ ⊂ R is the volume of the whole system. Thus a state ψ is in MITE if for everysubsystem of diameter ℓ or smaller, the reduced density matrix is close to the thermalequilibrium reduced density matrix.Most ψ ∈ S ( H mc ) lie in MITE. Indeed, canonical typicality (in the sense of ˆ ρ ψS ≈ ˆ ρ mc S for most ψ ) holds for subsystems of up to nearly half the size of Λ (that is, half thedegrees of freedom, or square root of the Hilbert space dimension, in practice usuallyhalf of the volume; see [29, 30, 13] for a discussion). We can choose a moderate number r (e.g., r = 8 for cube-shaped Λ) of overlapping regions S i ⊂ Λ (e.g., also cubes) of nearlyhalf the volume so that most ψ satisfy ˆ ρ ψS i ≈ ˆ ρ mc S i for all 1 ≤ i ≤ r simultaneously, andso that every region S with diam( S ) ≤ diam(Λ) is contained in one of the S i . By thesubsubsystem property, also ˆ ρ ψS ≈ ˆ ρ mc S for such regions S , so most ψ lie in MITE diam(Λ) / and a fortiori in MITE = MITE ℓ with ℓ as in (15).A concept along the lines of MITE was used before in, e.g., [32, 23, 36, 25]. We organize our discussion under a number of subheadings.
MITE implies MATE. —To discuss this point, it is helpful to first introduce a commonframework for formulating MITE and MATE. For any observable ˆ A , let µ ψ ˆ A denote theprobability distribution defined by ψ ∈ S ( H mc ) over the spectrum of ˆ A , µ ψ ˆ A ( B ) = h ψ | B ( ˆ A ) | ψ i (16)for all sets B ⊆ R (with 1 B ( ˆ A ) the projection to the subspace spanned by the eigenvec-tors of ˆ A with eigenvalue in B ). Likewise, let µ mcˆ A denote the probability distributiondefined by the micro-canonical ensemble, µ mcˆ A ( B ) = tr (cid:0) B ( ˆ A ) ˆ ρ mc (cid:1) (17)for all B ⊆ R ; µ mcˆ A is the average of µ ψ ˆ A with ψ taken to be uniformly distributed in S ( H mc ).Both MITE and MATE are ultimately of the following form (see also [43]): For acertain family A of observables, consider the set of ψ ∈ S ( H mc ) for which µ ψ ˆ A ≈ µ mcˆ A forall ˆ A ∈ A . MATE is obtained by taking A = A MATE = { ˆ M , . . . , ˆ M K } , and MITE ℓ by6aking A = A MITE ℓ = ∪ S A S with the union taken over all regions in Λ of diameter ≤ ℓ and A S the set of all self-adjoint operators on H S , more precisely, A S = (cid:8) ˆ A ⊗ ˆ I S c : ˆ A self-adjoint on H S (cid:9) , (18)where ˆ I denotes the identity operator and S c again the complement of S . Indeed, forˆ A = ˆ A ⊗ ˆ I S c and ψ ∈ MITE ℓ , µ mcˆ A ( B ) = tr (cid:0) B ( ˆ A ) ˆ ρ mc (cid:1) = tr (cid:0) B ( ˆ A ) ˆ ρ mc S (cid:1) ≈ tr (cid:0) B ( ˆ A ) ˆ ρ ψS (cid:1) = µ ψ ˆ A ( B ) (19)for all B ⊆ R by (14).From this perspective it is obvious that MITE implies MATE when based on rea-sonable choices: Suppose that L ≤ ℓ , (20)where L is the length scale of the macro observables—the diameter of the cells Λ i onwhich the macro observables ˆ M j were defined at the beginning of Section 2; that is,suppose that (7) holds at least up to the length scale of the macro observables. This iscommonly the case; e.g., for a cubic meter of gas at room conditions, we can realisticallytake L ≈ − m and ℓ ≈ − m. As a consequence of (20), A MATE ⊂ A MITE , so if ψ ∈ MITE then ψ ∈ MATE.
ETH. —To come back to the eigenstate thermalization hypothesis (ETH), it comes intwo variants: MATE-ETH and a more refined version MITE-ETH, according to whetherthe energy eigenstates are required to be in MATE or MITE. It is MITE-ETH that failsdramatically in some MBL systems, according to the findings of [2, 28, 19]; there itis shown for certain MBL systems that a substantial fraction of the energy eigenstates(in a micro-canonical energy interval), or even all of them, lie outside of MITE. Atthe same time, it is easy to see that for every macroscopic quantum system (MBL ornot), MATE-ETH must be almost satisfied, in the sense that most energy eigenstates | n i ∈ S ( H mc ) are in MATE: Assuming that (3) holds with ε ≪ δ , we obtain, writing D = dim H mc , that 1 D D X n =1 h n | ˆ P eq | n i = 1 D tr( ˆ P eq ) = 1 − ε , (21)and since h n | ˆ P eq | n i cannot exceed 1, most of these terms must be close to 1.If MATE-ETH holds strictly, i.e., if all energy eigenstates in H mc are in MATE, thenevery state ψ ∈ S ( H mc ) will sooner or later reach MATE and spend most of the timein MATE in the long run. That is because [15], writing f ( t ) = lim T →∞ T R T f ( t ) dt for7ime averages, | n i for the energy eigenstate with eigenvalue E n , and ψ t = e − i ˆ Ht ψ , h ψ t | ˆ P eq | ψ t i = X n,n ′ h ψ | n i e iE n t h n | ˆ P eq | n ′ i e − iE n ′ t h n ′ | ψ i (22)= X n (cid:12)(cid:12) h ψ | n i (cid:12)(cid:12) h n | ˆ P eq | n i (23) ≥ X n (cid:12)(cid:12) h ψ | n i (cid:12)(cid:12) (1 − δ ) (24)= 1 − δ , (25)provided ˆ H is non-degenerate, i.e., E n = E n ′ for n = n ′ (using e iEt = 1 if E = 0 and= 0 otherwise). Since its time average is close to 1, h ψ t | ˆ P eq | ψ t i must be close to 1 formost t in the long run.It follows from this that systems with MBL for which the transport coefficients van-ish, so that an initial state ψ with a non-uniform temperature will remain so indefinitely,cannot have all of its energy eigenfunctions in MATE. Since most energy eigenstates arein MATE, such ψ must be a superposition of predominantly those rare eigenstates thatare not in MATE.This leads to the question whether there are macroscopic systems for which all energy eigenstates are in MATE—i.e., whether MATE-ETH ever strictly holds. It isknown that this is so for a random Hamiltonian whose eigenbasis is uniformly chosenamong all orthonormal bases [15]; see also [46, 17]. Some numerical evidence [20] pointsto the existence of systems with realistic interactions for which all energy eigenstatesare in MITE and thus also in MATE.For MITE-ETH, there are several results [35, 23, 33], all of which assume that theHamiltonian is non-degenerate and has non-degenerate energy gaps, i.e., E m − E n = E m ′ − E n ′ unless ( either m = m ′ and n = n ′ or m = n and m ′ = n ′ , (26)a condition that is generically fulfilled. We note here two results, the first of which [23]asserts that if all energy eigenstates in H mc are in MITE, then most ψ ∈ S ( H mc ) willsooner or later reach MITE and spend most of the time in MITE in the long run. Moreprecisely, those ψ will behave this way for which the effective number of significantlyparticipating energy eigenstates is much larger than dim H S for any small S . The secondresult [35] shows that all (rather than most) ψ will ultimately reach MITE and staythere most of the time, under two assumptions, first again that all energy eigenstates arein MITE, and second Srednicki’s [41, 42] extension of the ETH to off-diagonal elements,i.e., that for ˆ A ∈ A (here, A = ∪ S A S as in (18)), h m | ˆ A | n i ≈ m = n (27) In fact, the assumption of non-degeneracy can be dropped: If we number the eigenvalues as E n with E n = E n ′ for n = n ′ and let | n i denote the normalized projection of ψ to the eigenspace of E n ,then the calculation (22)–(25) still applies. (cid:18) h ψ t | ˆ A | ψ t i − h ψ t | ˆ A | ψ t i (cid:19) = X m = n (cid:12)(cid:12) h ψ | m i (cid:12)(cid:12) (cid:12)(cid:12) h m | ˆ A | n i (cid:12)(cid:12) (cid:12)(cid:12) h n | ψ i (cid:12)(cid:12) , (28)and if (cid:12)(cid:12) h m | ˆ A | n i (cid:12)(cid:12) < ε ≪ m = n , then the time variance (28) is smaller than ε .If all | n i are in MITE, a calculation similar to (22)–(25) shows that h ψ t | ˆ A | ψ t i ≈ tr( ˆ ρ mc ˆ A ) . (29)It follows that, for most t in the long run, h ψ t | ˆ A | ψ t i ≈ tr( ˆ ρ mc ˆ A ) for all ˆ A ∈ ∪ S A S (inparticular for projections), so ψ t ∈ MITE for most t in the long run. Mixed states. —Once we have the notions of MITE and MATE for pure states ψ ,they are easily generalized to mixed states ˆ ρ : MATE occurs if tr( ˆ P eq ˆ ρ ) > − δ , andMITE if ˆ ρ S ≈ ˆ ρ mc S for all subsystems S defined by spatial regions of diameter ≤ ℓ . Notethat neither MATE nor MITE requires that ˆ ρ be close to ˆ ρ mc or ˆ ρ ( β ) . Thermal equilibrium in classical mechanics. —Only one of the two notions MITE andMATE can be satisfied for pure states in classical mechanics, namely MATE. That isbecause a “pure state” corresponds in classical mechanics to a point X in phase space,while a “mixed state” corresponds to a probability distribution over phase space. Since X specifies the positions and momenta of all particles, it also provides a pure state forany subsystem. In contrast, in quantum mechanics ˆ ρ ψS can be mixed, and in fact is mixedexcept for product states. So in classical mechanics it is never true for a system in a purestate that a subsystem S could have a state close to a thermodynamic ensemble suchas the marginal (obtained by integrating out the variables not belonging to S ) of themicro-canonical distribution (i.e., uniform over the energy shell) or the canonical one forthe whole system. In contrast, MATE is analogous to Boltzmann’s [4, 9, 14, 22] notionof thermal equilibrium for a closed classical system, based on a partition of phase spaceinto macro states Γ ν . (Note that there is no difference between MATE and TMATEclassically, as all observables commute.) Abstract MITE. —A natural mathematical generalization that is often interesting toconsider is based on dropping the idea that S corresponds to a region in 3-space andregarding S as an abstract subsystem defined by any splitting of Hilbert space into atensor product, H mc ⊆ H S ⊗ H S c , (30)where S c can be thought of as just an index for another Hilbert space. For example, S may comprise the spin degrees of freedom and S c the position degrees of freedom, or S may comprise the oxygen atoms and S c all other atoms in the system. Given a list ofsubsystems S , . . . , S r , one can defineMITE S ,...,S r = r \ i =1 n ψ ∈ S ( H mc ) : ˆ ρ ψS i ≈ ˆ ρ mc S i o . (31)9anonical typicality implies that, if r is not too large and each S i is not too large (a suf-ficient condition is dim H S i ≪ √ dim H mc ), then most ψ ∈ S ( H mc ) are in MITE S ,...,S r ;see Theorem 1 in [29, 30] and Proposition 1 in [13] for a precise and quantitative state-ment of this fact.One can also consider the set MITE most comprising those ψ ∈ S ( H mc ) for whichˆ ρ ψS ≈ ˆ ρ mc S holds for most abstract subsystems S of dimension ≤ d . If d ≪ √ dim H mc ,then also MITE most has most of the surface area of S ( H mc ) [13]. On the other hand,given any pure state ψ ∈ S ( H mc ), ˆ ρ ψS ≈ ˆ ρ mc S cannot hold for all abstract subsystems S of dimension ≤ d simultaneously [24, 13]. Perhaps the most surprising aspect of the situation is that the various criteria for thermalequilibrium of pure states proposed in the literature fall into two groups that differsubstantially in how much they demand.Arguably, the essence of thermal equilibrium is what characterizes it in thermody-namics: that a system appears stationary on the macro level, and that temperatureand all chemical potentials are spatially uniform. This corresponds to MATE, whichmay therefore be regarded as the direct expression of thermal equilibrium. On the otherhand, since MITE is the stronger statement, and since it is usually true that macroscopicquantum systems approach MITE (MBL systems being an exception), it is natural toconsider MITE, and it would seem artificial to not regard it as a new kind of thermalequilibrium property emerging from quantum entanglement.
Acknowledgments.
J. L. Lebowitz was supported in part by the National Science Foun-dation [grant DMR1104500].
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