aa r X i v : . [ qu a n t - ph ] S e p Quantum Fluctuation Relations for Ensembles ofWave Functions
Michele Campisi
Institute of Physics, University of Augsburg, Universit¨atsstr. 1, D-86135 Augsburg,Germany
Abstract.
New quantum fluctuation relations are presented. In contrast with thethe standard approach, where the initial state of the driven system is describedby the (micro)canonical density matrix, here we assume that it is described by a(micro)canonical distribution of wave functions, as originally proposed by Schr¨odinger.While the standard fluctuation relations are based on von Neumann measurementpostulate, these new fluctuation relations do not involve any quantum collapse, butinvolve instead a notion of work as the change in expectation of the Hamiltonian. uantum Fluctuation Relations for Ensembles of Wave Functions
1. Introduction
In the last two decades the field of non-equilibrium thermodynamics has undergonea tremendous advancement due to the discovery of exact non-equilibrium relations(named fluctuation relations) which characterize non-equilibrium processes well beyondthe regime of linear response, and provide a deep insight into statistical nature andmicroscopic origin of the second law of thermodynamics. The most prominent exampleof such exact relations is the Jarzynski equality [1] which allows for obtaining thefree energy landscape of small systems, like a single DNA molecule, from very manymeasurements of work done on the system as it is driven out of equilibrium, e.g., bystretching the molecule [2]. A related result, known as Crooks work fluctuation theorem[3], relates the free energy to the probability of performing work W during the processand the probability of performing work − W during the time-reversed process. Theseresults, which have been first obtained within the framework of classical mechanics werelater derived also within the quantum mechanical framework [4, 5, 6, 7].The crucial ingredient needed for obtaining the fluctuation relations in the quantumcase is the so called two measurements scheme [8, 9]. In this scheme the system energyis measured at the beginning and end of the driving protocol and the work is defined asthe difference of the outcomes of these measurements: W = E τm − E n (1)where E tk denotes an eigenvalue of the (time-dependent) Hamilton operator ˆ H ( λ t ) attime t . As usual, here it is assumed the Hamilton operator changes in time due tothe time dependence of an external parameter λ t . This scheme relies on the vonNeumann measurement postulate according to which the measurement process inducesthe collapse of the wave function on one of the eigenstates of the measured observable,i.e, ˆ H ( λ ) and ˆ H ( λ τ ) in the present case. Notably, experimental verification andapplication of the quantum fluctuation relations based on the two-measurement schemehave not been accomplished yet, while alternative strategies aimed at avoiding the twoprojective measurements have been proposed. Two prominent examples propose toreplace them with many weak measurements during the driving protocol [10], or withstate tomography of one or two qubit ancillae appropriately coupled to the driven system[11, 12, 13, 14]With this work we establish new quantum fluctuation relations, which look exactlylike the standard quantum fluctuation relations but substantially differ from them due toa different underlying definition of quantum work, and a different ensemble specifyingthe initial condition. As in the standard case [8, 9] we assume an initial statisticalensemble, but at variance with the ordinary quantum statistical mechanics, we assumethat the statistical ensemble is described by a distribution of wave functions as originallysuggested by Schr¨odinger [15], later pursued by Khinchin [16] and recently advocatedby an increasing number of authors [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29].We will establish fluctuation relations for the microcanonical [19, 23, 26, 25, 27, 28]and canonical [17, 30] wave function ensembles. Most remarkably these new fluctuation uantum Fluctuation Relations for Ensembles of Wave Functions expectation of theHamiltonian operator w = h ψ ( τ ) | ˆ H ( λ τ ) | ψ ( τ ) i − h ψ (0) | ˆ H ( λ ) | ψ (0) i . (2)Accordingly they do not involve von Neumann measurement postulate. In Eq. (2) | ψ (0) i is a wave function randomly chosen from the distribution, and | ψ ( τ ) i is its timeevolution. Just as with the classical fluctuation theorems, the stochastic nature of w comes from the fact that the initial state | ψ (0) i is randomly drawn from a distribution,while its evolution is deterministic.So, the interpretation framework that is adopted here is that experimentallyobserved quantities correspond to their quantum mechanical expectation , an approachthat is at least as common in the scientific literature and effective as that involvingwave function collapses. To give one example, Kubo’s linear response theory [31], isa theory of quantum expectations which mentions no collapses. This same philosophyhas been advocated by G. Jona-Lasinio and C. Presilla [30], who pointed out that thewave function ensembles could be good candidates for the study of mesoscopic systems,where robust coherence phenomena are involved.
2. Wave function ensembles
We consider a quantum system with a finite dimensional Hilbert space of dimension N .Each wave function | ψ i can be represented by an N dimensional complex vector c , andthe system Hamilton operator ˆ H ( λ ) can be represented by a N × N Hermitean matrix H ( λ ). Following Ref. [32] we introduce the suggestive notation c = x + i p (3) h ( x , p ; λ t ) = ( x − i p ) T H ( λ t )( x + i p ) (4)where x and p are the real and imaginary parts of c , h ( x , p , λ t ) denotes the expectationof the Hamilton operator on the state x + i p , z T denotes transpose of z , and matrixmultiplication is implied. We stress that x and p should not be confused with positionsand momenta.Below we shall consider statistical ensembles ρ ( x , p ) defined on the wave function“phase space” ( x , p ). Given an observable ˆ B , with matrix representation B , its ensembleaverage is its wave function expectation b ( x , p ) = ( x − i p ) T B ( x + i p ), averaged overthe wave function distribution, namely h ˆ B i = Z d x d p ρ ( x , p ) b ( x , p ) . (5) In the microcanonical wave function ensemble [19, 23, 26, 25, 27, 28] all wave functions( x , p ) with a given expectation of energy E = h ( x , p ; λ ) have same probability, whereas uantum Fluctuation Relations for Ensembles of Wave Functions λ it reads: ρ µ ( x , p ; E, λ ) = δ ( E − h ( x , p ; λ )) δ (1 − | x + i p | )Ω( E, λ ) (6)where δ denotes Dirac’s delta function, andΩ( E, λ ) = Z d x d p δ ( E − h ( x , p ; λ )) δ (1 − | x + i p | ) (7)is the density of states. Note the formal similarity with the classical microcanonicalensemble. The main the difference is the presence of the extra factor δ (1 − | x + i p | )which restricts the integration to the “physical Hilbert space”, namely the subspaceof normalized wave functions, also known as the projective Hilbert space. Note thatat variance with the textbook quantum microcanonical ensemble [33, 34], in whichonly those eigenstates of the Hamiltonian in a narrow interval around the energy E contribute, here all eigenstates participate to the ensemble. ‡ For this reason variousauthors claim that the ensemble in Eq. (7) provides a more realistic description of thethermodynamics of isolated systems [25, 26, 27, 28]. Another pleasing property of thisensemble is that, at variance with the standard microcanonical ensemble, it does notrequire a dense energy spectrum, and can therefore be well applied to small quantumsystems with well separated energy levels. Indeed, the ensemble depends continuously onthe real parameters
E, λ , which makes the derivation of the associated thermodynamicsstraightforward also in case of small systems [24]. Ref. [28] shows that this ensemble welldescribes the statistics of a small thermally isolated system after repeated non-adiabaticperturbations.
In the canonical wave function ensemble [17, 30], wave functions are weighted with theGibbs factor e − βh ( x , p ; λ ) : ρ c ( x , p ; β, λ ) = e − βh ( x , p ; λ ) δ (1 − | x + i p | ) Z ( β, λ ) (8)where Z ( β, λ ) = Z d x d p e − βh ( x , p ; λ ) δ (1 − | x + i p | ) (9)Note again the formal similarity with the classical canonical ensemble. In Ref. [30]this ensemble is called the Schr¨odinger-Gibbs ensemble. According to [18] this ensemblecan give realistic predictions in case of mesoscopic systems where robust coherencephenomena are involved. ‡ To see this, consider for example a spin-1 particle in a (possibly large) magnetic field, ˆ H ( λ ) = λ ˆ J z ,and consider the microcanonical ensemble of states with expectation E = 0. Besides the statewith null angular momentum (the only state contributing to the standard microcanonical ensemble),superposition containing both the up and down states now contribute to the ensemble as well. uantum Fluctuation Relations for Ensembles of Wave Functions
3. Rationale for wave function ensembles
A criterion for establishing the goodness of a statistical ensembles as a candidate modelof equilibrium thermodynamics is whether the ensemble is invariant under the timeevolution. As will become clearer in the next section this is indeed the case for thecanonical and microcanonical wave function ensemble.Another criterion, which traces back to Boltzmann [35], is whether the ensembleendows the parameter space with a “thermodynamic structure”. To be more explicit,given a statistical ensemble ρ (Γ , X i ), (defined on a phase space Γ and on a parameterspace X i ), one checks whether there exist an integrating factor γ ( X i ), such that γ δQ = exact differential (10)where δQ is the heat differential as calculated in the ensemble. This equation is known asthe heat theorem , and is the most fundamental equation of thermodynamics. Prominentexamples of textbooks that take this viewpoint in establishing the foundations ofquantum statistics are those of Schr¨odingier [15], and Khinchin [16]. § To calculate δQ use the standard formula δQ = dE + F dλ (11)where E = h ˆ H i (12) F = − * ∂ ˆ H∂λ + (13)denote the ensemble averages of energy and of the generalized force conjugated to theexternal parameter. Note that in case of a single parameter λ , mathematics ensures thatan integrating factor always exists. A differential form in two dimensions (i.e., E and λ , in Eq. (11)), always admits an integrating factor. However, the system Hamiltonianmay depend on many external parameters λ i , hence δQ = dE + P i F i dλ i , which makesthe question of the existence of an integrating factor non-trivial. In the canonical case we have E = E ( β, λ ) = Z d x d p ρ c ( x , p ; β, λ ) h ( x , p ; λ ) (14) F = F ( β, λ ) = − Z d x d p ρ c ( x , p ; β, λ ) ∂h ( x , p ; λ ) ∂λ (15)In this case β is an integrating factor for δQ and S c ( β, λ ) = βE ( β, λ ) + ln Z ( β, λ ) isthe associated generating function. The argument follows step by step the classical § Interestingly both books also advocate the use of wave function ensembles. uantum Fluctuation Relations for Ensembles of Wave Functions S c ( β, λ ) are: ∂S c ∂β = E + β ∂E∂β + 1 Z ∂Z∂β = β ∂E∂β (16) ∂S c ∂λ = β ∂E∂λ + 1 Z ∂Z∂λ = β ∂E∂λ + βF (17)therefore dS c = β (cid:18) ∂E∂β dβ + ∂E∂λ dλ + F dλ (cid:19) = β ( dE + F dλ ) = βδQ (18)The derivation can be straightforwardly repeated in the case of many parameters. Weremark that there are however infinitely many integrating factors for δQ . So havingfound one does not ensure by itself that it can be interpreted as inverse temperature,and that the associated generator of the exact differential can be interpreted asentropy. Take for example g ( β, λ ) = f ( S c ( β, λ )) with any monotonic function f . Then dg = f ′ ( S c ( β, λ )) dS C = f ′ ( S c ( β, λ )) βδQ , where f ′ is the derivative of f . This says that f ′ ( S c ( β, λ )) β , is also an integrating factor for δQ . In order to pick the “thermodynamic”integrating factor, we need an extra ingredient. We thus further require that the entropybe additive. Namely, if two non interacting and non-entangled systems have separatelythe entropies S and S , the entropy of the total system should be S + S . Therequirement of non-entanglement is very crucial here. It restricts the Hilbert space ofthe compound system, from a tensor product of dimension N N to the direct productof dimension N + N . In this “classical” phase space the canonical wave functiondistribution of the compound system reduces to the product of the canonical wavefunction distributions for each subsystem, so does the partition function Z . Noting thatthe energy is additive, it follows that S c ( β, λ ) is additive as well, which singles it out asa good candidate for thermodynamic entropy. Accordingly β is the inverse temperature. In the microcanonical case E = Z d x d p ρ µ ( x , p ; E, λ ) h ( x , p ; λ ) (19) F = F ( E, λ ) = − Z d x d p ρ µ ( x , p ; E, λ ) ∂h ( x , p ; λ ) ∂λ (20)An integrating factor for δQ is in this case the function Ω( E, λ ) / Φ( E, λ ), where, inanalogy with classical mechanicsΦ(
E, λ ) = Z d x d p θ ( E − h ( x , p ; λ )) δ (1 − | x + i p | ) (21)denotes the volume of physical Hilbert space with energy expectation below E . As inclassical mechanics, we have Φ( E, λ ) = R EE Ω( E ′ , λ ) dE ′ , where E is the ground stateenergy. The symbol θ denotes the Heaviside step function. The proof follows, mutatismutandis , the classical argument (the generalized Helmholtz theorem) [37], which can uantum Fluctuation Relations for Ensembles of Wave Functions / Φ is S µ ( E, λ ) = ln Φ(
E, λ ). In this case the requirement ofadditivity does not seem to single S µ ( E, λ ) so straightforwardly as in the canonical case.The reason is that, unlike the exponential, the theta function does not factorize in theproduct of two theta functions. Classically this problem can be easily circumvented uponnoticing that the integrating factor Ω / Φ equals the average kinetic energy per degreeof freedom (equipartition theorem [38]), which singles it out as the thermodynamictemperature. In quantum mechanics however there is no equipartition theorem to helpus. We leave the resolution of this question to future studies.It should be remarked that our present analysis contrasts with Ref. [24], wherethermodynamics was derived from the logarithm of the density of states, namelyln Ω(
E, λ ). We remark that this choice does not comply with the heat theorem, Eq.(24), namely, there does not exist, in general a function γ ( E, λ ), such that γ ( E, λ ) δQ would equal the differential of ln Ω( E, λ ). This very same question appears also at theclassical level, where it has been long ignored due to the fact that in most cases of interestthe “surface entropy” (logarithm of the density of states) and the “volume entropy”(logarithm of the integrated density of states), give practically undistinguishable resultsfor sufficiently large systems [37, 39].
4. Fluctuation relations
Fluctuation relations for the wave function ensembles follow straightforwardly uponnoticing that in the ( x , p ) representation, the Schr¨odinger equation i ~ ˙ c = H ( λ t ) c (22)assumes the form of classical Hamilton’s equation˙ x = ∂∂ p h ( x , p ; λ t ) (23)˙ p = − ∂∂ x h ( x , p ; λ t ) (24)with the function h ( x , p ; λ t ) being the generator of the dynamics [32]. In analogy withthe classical case, we introduce the following notion of quantum work w = h ( x τ , p τ ; λ τ ) − h ( x , p ; λ ) (25)where ( x τ , p τ ) denotes the evolved of ( x , p ), according to Hamilton’s equations (24).Physically, w is the change in the expectation of the Hamilton operator ˆ H , due to theevolution of the wave function | ψ i , see Eq. (2). Note that w can be expressed as anintegrated power: w = Z τ dt ˙ λ t ∂h ( x t , p t , λ t ) ∂λ t (26)In equilibrium, namely for a constant λ , energy conservation and Liouville theoremensure that surfaces of constant energy expectation in the physical Hilbert space will uantum Fluctuation Relations for Ensembles of Wave Functions w be performed on a system prepared in a wavefunction ensemble ρ ( x , p ) can be written as p ( w ) = Z d x d p ρ ( x , p ) δ ( w − h ( x τ , p τ , λ τ ) + h ( x , p , λ )) (27)Noticing that the evolution (24) conserves the normalization, | x τ + i p τ | = | x + i p | = 1(unitarity of quantum evolution) and is volume preserving, d x τ d p τ = d x d p (classical Liouville theorem), one can repeat step by step the derivations of classicalmicrocanonical [40] and canonical [2] fluctuation relations, upon requiring that theHamilton operator is time reversal invariant. k In the microcanonical case one obtains: p E ( w ) e p E + w ( − w ) = Ω( E + w, λ τ )Ω( E, λ ) (28)where p E ( w ) is the probability of doing work w when the initial state is randomlydrawn from the distribution ρ µ ( x , p ; E, λ ) under the driving protocol λ t t ∈ [0 , τ ] , and e p E + w ( − w ) is the probability of doing work − w when the initial state is randomly drawnfrom ρ µ ( x , p ; E + w, λ τ ) under the protocol λ τ − t , t ∈ [0 , τ ].In the canonical case one obtains: p ( w ) e p ( − w ) = Z ( β, λ τ ) Z ( β, λ ) e βw = e − β (∆ F − w ) (29)where p ( w ) is the probability of doing work w when the initial state is randomly drawnfrom the distribution ρ c ( x , p ; β, λ ) under the driving protocol λ t t ∈ [0 , τ ] , and e p ( − w )is the probability of doing work − w when the initial state is randomly drawn from ρ c ( x , p ; β, λ τ ) under the protocol λ τ − t , t ∈ [0 , τ ]. In analogy with the classical case wehave introduced the notation ∆ F = F ( β, λ τ ) − F ( β, λ ), with F ( β, λ ) = − β − ln Z ( β, λ ).We stress that this free energy F ( β, λ ) may considerably differ from the usual free energy F st ( β, λ ) = − β − ln Tr e − β ˆ H ( λ ) , see Fig. 1.a.
5. Illustrative example
To better clarify the differences and similarities between the standard quantumfluctuation relations and the quantum fluctuation relations for wave function ensembles,we consider the Landau-Zener(-St¨uckelberg-Majorana) [42, 43, 44, 45] problemˆ H ( λ t ) = λ t σ z + ∆ σ x , λ t = vt/ . (30)It governs the dynamics of a two-level quantum system whose energy separation, vt , varies linearly in time, and whose states are coupled via the interaction energy∆. For example, a spin-1 / µ in a magnetic field ~B t = − (∆ /µ, , vt/ µ ). Here, σ x and σ z denote Pauli matrices. k Formally that means that at each time t , the Hamilton operator ˆ H ( λ t ) commutes with time-reversaloperator Θ, which changes the sign of momenta and leaves spatial coordinates unchanged [41] uantum Fluctuation Relations for Ensembles of Wave Functions c = ( a, b ) T , with a, b ∈ C , denote a point in theHilbert space (a wave function). The energy expectation h ( a, b, λ ) over the state c reads h ( a, b ; λ ) = λ ( | a | − | b | ) + ∆( a ∗ b + ab ∗ ), where ∗ denotes complex conjugation.Accordingly, the partition function reads: Z ( β, λ ) = Z da db e − β [ λ ( | a | −| b | )+∆( a ∗ b + ab ∗ )] δ (1 − | a | − | b | ) (31)As is well known, the projective Hilbert space of a two-level system can be mappedonto a sphere of unit radius in R , the Bloch sphere. Accordingly, the partitionfunction Z ( β, λ ) can be expressed as an integral over the Bloch sphere. This isaccomplished by the following change of variables, a = e iφ r cos γ/ , b = e iφ re iδ sin γ/ r ∈ [0 , ∞ ) , φ ∈ [0 , π ] , γ ∈ [0 , π ] , δ ∈ [0 , π ], leading to: Z ( β, λ ) = 18 Z dr dφ dδ dγ sin γ r e − βr [ λ cos γ +∆ sin δ sin γ ] δ (1 − r )= π Z dδ dγ sin γ e − β [ λ cos γ +∆ sin δ sin γ ] (32)where γ, δ are the Bloch angles. To perform the integration we first consider the case∆ = 0. Physically this corresponds to a spin-1 / z direction with intensity λ/µ . By the change of variable y = cos γ weobtain, for ∆ = 0, Z = π sinh( βλ ) / ( βλ ). When ∆ = 0, this corresponds to a magneticfield oriented along some direction ˆ n and an intensity √ λ + ∆ /µ . Because of spatialisotropy, the partition function can only depend on the intensity of the field and not onits orientation, hence we obtain, Z ( β, λ ) = π sinh( β √ λ + ∆ ) β √ λ + ∆ . (33)This expression should be contrasted with the standard expression Z st ( β, λ ) =Tr e − β ˆ H ( λ ) = 2 cosh( β √ λ + ∆ ). Figure 1.a shows a comparison of the resulting freeenergies, F = − β − ln Z , F st = − β − ln Z st . As already highlighted in Ref. [17] theygive rise to distinct thermodynamics.It is worth stressing that, just like the standard ensemble, the wave functionensemble is a mixed state which can, accordingly, be represented by a density matrix[30]: ˆ ρ ( β, λ ) = R d x d p ρ ( x , p , β, λ )( x − i p )( x + i p ) T . In the present case it reads, in the σ z basisˆ ρ ( β, λ ) = π Z dδ dγ sin γ e − β [ λ cos γ +∆ sin δ sin γ ] Z ( β, λ ) cos ( γ/
2) sin γe − iδ / γe iδ / ( γ/ ! (34)In the case ∆ = 0 we get [17]ˆ ρ ( β, λ ) = 12 / ( βλ ) − coth( βλ ) 00 1 − / ( βλ ) + coth( βλ )) ! , ∆ = 0 . (35)This density matrix should be contrasted with the standard canonical density matrixˆ ρ st ( β, λ ) = diag( e − βλ , e βλ ) /Z st ( β, λ ). By replacing λ with √ λ + ∆ , one gets thedensity matrix for the case ∆ = 0, in the corresponding energy eigenbasis. uantum Fluctuation Relations for Ensembles of Wave Functions . . . − − − p r o b a b ili t y w, W (units of ∆) p ( w ) p st ( W ) / − − − − −
10 0 0 . . F , F s t ( un i t s o f ∆ ) k B T/ √ λ + ∆ FF st − T b ) a ) Figure 1.
Panel a: Free energy of a two-level-system described by the Hamiltonian(30) with a fixed λ , as computed in the canonical (Schr¨odinger-Gibbs) wave functionensemble, Eq. (8), and in the standard Gibbs canonical ensemble. Panel b: Probabilityhistograms of work w , Eq. (25), and standard work W , Eq. (1), as computed in thecanonical (Schr¨odinger-Gibbs) wave function ensemble, Eq. (8), and in the standardGibbs canonical ensemble, respectively. The standard work probability p st ( W ) isrescaled by a factor 10 for better visualization. Inset: sketch of the driving protocol, i.e.a half Landau-Zener sweep. The parameters used are: β = ∆ − , v = ∆ / ~ , T = 5 ~ / ∆. In Fig. 1.b we report results concerning the work statistics. We considered herea “half” Landau-Zener sweep, i.e., Eq. (30) from time t = − T , to time t = 0, see theinset of Fig. 1.b. The unitary quantum evolution operator can be expressed in terms ofspecial functions [46, 47]. The figure shows both the statistics p ( w ) originating from theexpression of work in Eq. (25) in the canonical wave function ensemble, Eq. (8), andthe standard work statistics p st ( W ) originating from the two-measurement expressionof work in Eq. (1) in the standard canonical ensemble e − β ˆ H ( λ ) /Z st ( β, λ ). ¶ Note theprominent difference that the wave function work pdf p ( w ) is a smooth function whereasthe standard work pdf p st ( W ) is a discrete sum of 4 Dirac deltas [9] (the two most leftpeaks of p st ( W ) are barely visible in Fig. 1.b). Note also that the support of p ( w ) issmaller than the support of p st ( W ). Stronger driving (i.e. larger v ’s) result in broaderdistributions p ( w ). The support of p ( w ) cannot however become wider that that of p st ( W ), which, independent of v , is given by [ − p ( vT / + ∆ − ∆ , p ( vT / + ∆ +∆].Notwithstanding their differences both distributions satisfy formally equivalentfluctuation relations. To better clarify this, let us focus on the average exponentiatedwork. As predicted by the theory and confirmed by our numerical calculation, we have: h e − βw i = Z dwp ( w ) e − βw = e − β ∆ F (36) h e − βW i st = Z dW p st ( W ) e − βW = e − β ∆ F st (37) ¶ To be more precise, Fig. 1.b shows the quantities R w + d/ w − d/ p ( w ′ ) dw ′ , and R W + d/ W − d/ p st ( W ′ ) dW ′ , (with d the width of the bars), i.e., discrete versions of p ( w ) and p st ( W ). p st ( W ) is rescaled by a factor 10in Fig. 1.b, for a better visualization. uantum Fluctuation Relations for Ensembles of Wave Functions
6. Concluding remarks
We have obtained fluctuation relations for microcanonical and canonical wave functionensembles. They look exactly as the standard relations, but substantially differ fromthem because they involve a notion of work as the change in the expectation of theenergy, rather than the difference of two eigenvalues emerging from quantum collapses.These ensembles in fact have been proposed in a framework where one is interested inthe the expectation of quantum observables [30]. As highlighted with the illustrativeexample, this notion of work gives rise to smooth work probability densities, in starkcontrast with the discrete standard probability densities. Also it gives information aboutthe equilibrium “free energy” (“entropy”) as calculated in the canonical (microcanonical)wave function ensemble. These substantially differ from their standard counterpart, seeFig. 1.Other authors are currently developing alternative formulations of quantumfluctuation relations which do not rely on quantum collapses. Among them is thework of Ref. [48] which presents a study of entropy production based on the Wignerrepresentation of quantum states.We have expressed some considerations regarding the rational foundations of thewave function ensembles. Further investigation is certainly necessary in order to reach amore satisfactory understanding of the physical basis for these ensembles. One questionto be pursued regards the lack of ergodicity of the Hamiltonian flow on the surfaceof constant energy expectation in the physical Hilbert space, which marks a starkdistinction with the classical case. Another important question that deserves furtherstudy is whether these ensembles converge to the usual statistical ensembles in somelimit, e.g. classical, and/or thermodynamic limit. Experiments will have the final wordin regard to their scope of applicability. Certainly they have proved very important inrecent advancements in the foundations of quantum statistics [21, 22].
Ackowledgements
This work was supported by the German Excellence Initiative “Nanosystems InitiativeMunich (NIM)”.
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