Entropy and Canonical Ensemble of Hybrid Quantum Classical Systems

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J. and PA), systems III ial,i Section in Finally, ewl eiethe derive will we ,2 5 2, 1, 28 chosen – 33 ]). 2 start by recalling the basic definitions in the purely- classical subsystem are just ξ-functions times the iden- classical or purely-quantum cases. tity, i.e. Aˆ(ξ)= A(ξ)Iˆ; those observables defined on the In classical systems, a basis of MEEs is sim- quantum subsystem only are operators that lack the ξ- ply the phase space MC , the set of all positions dependence. and momenta of the classical particles: MC = We are going to consider two different approaches to {(Q, P ) | Q ∈ Rn, P ∈ Rn}, where n is the number of the definition of the entropy, one based on the usual ap- classical degrees of freedom. Any point in this phase proach to classical systems, and another one inspired by space defines an exclusive event from any other event. the quantum case. Observables are real functions on this MC. Statistical mechanics for classical systems can then be described by using ensembles on this phase space, i.e. (generalized)1 A. A Gibbs-entropy for hybrid systems? probability distribution functions (PDFs) FC In quantum systems, the states are rays of a Hilbert The formal similarities of one of the best known hybrid space H, i.e. the analogous to the classical phase space dynamical models, Ehrenfest dynamics, with the classical is the projective space, MQ = PH. We will represent one (see [18, 27] for details) may lead to consider hybrid its points as the projectors on 1-dimensional subspaces of systems as formally closer to classical than to quantum |ψihψ| ~ the Hilbert spaceρ ˆψ = hψ|ψi , with |ψi∈H\{0}. Even dynamics. Indeed, Ehrenfest dynamics can be given a though all of the states in MQ are physically legitimate, Hamiltonian structure (see [27, 35]) in terms of they are not mutually exclusive. Indeed, if the system • a Hamiltonian function constructed as has been measured to be, with probability one, in a state hψ|Hˆ (ξ)ψi ρˆψ1 , the probability of measuring it to be in other state f (ξ, ρˆ ) = Tr(Hˆ (ξ)ˆρ )= , (2) H ψ ψ hψ|ψi ρˆψ2 is not zero, unless they are orthogonal:ρ ˆψ1 , ρˆψ2 are MEE only if hψ | ψ i = 0. As a consequence, consid- 1 2 • ering generalized probability density functions FQ over and a Poisson bracket obtained as the combination the Hilbert space (or over the projective space of rays) to of the Poisson bracket of Classical Mechanics and define ensembles, following the classical analogy, results the canonical Poisson bracket of quantum systems in over-counting the same outcome for a hypothetical ex- (see [36, 37]). periment in a non-trivial way. One way to see this clearly This fact makes Ehrenfest dynamical description of F is that many different Q can correspond to exactly the hybrid system formally analogous to a classical Hamil- same ensemble (i.e. they are physically indistinguish- tonian dynamical system. When considering the defini- able). The correct way to get a sample space of MEEs is tion of hybrid statistical systems, we can then consider therefore considering a basis of orthogonal events. From a hybrid (generalized) PDF F defined over the hybrid this idea, von Neumann [34] derived the density matrix H phase space MH = MC ×MQ, in an analogous man- formalism, which contains all the physically relevant sta- ner to the definition of classical statistical systems. The tistically non-redundant information in a compact way. Hamiltonian nature of the dynamics allows to define a A density matrix can be obtained from a PDF FQ in the Liouville equation for FH in a straightforward manner quantum state space as: (see [18, 27]). Within that framework, it is also tempting to borrow ρˆ[F ]= dµ (ˆρ )F (ˆρ )ˆρ , (1) Q Z Q ψ Q ψ ψ the notion of entropy from Classical Statistical Mechan- ics and define a Gibbs-like function associated with the where we represent by dµQ the volume element on MQ. density function FH in the form: Analogously, in the following, we will represent by dµC the volume element on MC . S [F ]= −k dµ (ξ,ρ )F (ξ,ρ ) log (F (ξ,ρ )) , G H B Z H ψ H ψ H ψ We move on now to QC theories. Despite the various MH proposals referenced above, one can perhaps establish a (3) common denominator. The classical part is described by where kB represents the Boltzman costant and dµH rep- a set of position Q ∈ Rn and momenta P ∈ Rn variables, resents the volume element on MH which can be written that we will hereafter collectively group as ξ = (Q, P ). in terms of the classical and quantum volume elements The quantum part is described by a complex Hilbert as dµH = dµC ∧ dµQ. space H. Observables are Hermitian operators on H, Notice that this entropy function is well defined for and they may depend parametrically on the classical vari- classical systems, where the points of phase-space cor- ables, Aˆ(ξ): H → H. Those observables defined on the respond to mutually exclusive events. Therefore, when considering SG we are adding all points of the phase space MH as if they were mutually exclusive. Thus we treat them as classical statistical systems, where being at a 1 We introduce the adjective generalized to refer to the set of gen- given point in phase space excludes the possibility of be- eralized functions (or distributions) and include, for instance, ing at a different point. Hence, we are not weighting cor- Dirac delta functions. rectly the quantum subsystems from the physical point 3 of view, ruining the function ability to measure physical For these quantum conditional probabilities p(a|ξ), all information for the hybrid system. the requirements of Gleason’s theorem [42] apply, and one Despite this fact, this entropy function has been im- may therefore define, at each ξ-point, a density matrixρ ˆξ. plicitly assumed several times when considering hybrid It provides the probabilities of measuring an eigenvalue or even purely-quantum statistical systems (see [27, 38– a of observable Aˆ(ξ), given ξ, through the usual Born ξ 40]), when defining the so called Schrödinger-Gibbs (SG) rule: p(a|ξ) = Tr[ˆρ πâ(ξ)], whereπ â(ξ) is the projector ensemble or the corresponding Schrödinger microcanoni- onto the eigen-subspace associated to a. From this, we cal ensemble. Thus, SG represents a canonical ensemble can define the hybrid density matrix as the ξ–dependent where the probability density is written by assigning to matrix: each state the Gibbs weight associated with the expec- ξ tation value of the Hamiltonian, instead of the operator ρˆ(ξ)= FC(ξ)ˆρ , (5) itself. But the bad physical properties of SG lead to very strange and un-physical properties for the correspond- such that p(a, ξ) = Tr(ˆρ(ξ)ˆπ(a)). Notice that, strictly ing thermodynamic functions. In particular, this was the speaking, Gleason theorem ensures the existence and case when the Schrödinger-Gibbs ensemble was analyzed uniqueness of the density matrixρ ˆξ only for Hilbert in [41]. Nonetheless, notice that SG is a mathematically spaces of dimension at least 3. However, the recent de- consistent entropy function, despite the unphysical prop- velopments based on positive-operator-valued measures erties of the Statistical Mechanics it defines. (POVM) (see for instance [43, 44]) allow to prove a more general formulation of Gleason theorem for quantum states which is valid in dimension 2, but in that B. Gibbs-von Neumann entropy case the construction is not based on orthogonality of the rank-one projectors but on a more global set of effects. From our analysis above, it is clear that the straightfor- In conclusion, the probability distribution on the set ward extension of Gibbs classical entropy function to hy- of MEEs of hybrid states can be written as a family of brid systems leads to inconsistencies because the points quantum density operators parameterized by the classical of hybrid phase space do not define mutually exclusive degrees of freedom,ρ ˆ(ξ). For each ξ,ρ ˆ(ξ) is a self-adjoint events as the classical phase space points do. In order and non-negative operator, which is normalized on the to do statistical mechanics in a consistent way with the full hybrid sample space: nature of its quantum subsystem, one must reconsider the notion of mutually exclusive events, and combine the dµ (ξ)Tr(ˆρ(ξ))=1 . (6) Z C classical and the quantum notions of MEE. The combined MC hybrid phase space is now M = M ×M . But, we H C Q This is an immediate consequence of the normalization must consider that two hybrid states (ξ1, ρˆψ1 ), (ξ2, ρˆψ2 ) ∈ ξ M represent MEEs if and only if ξ 6= ξ or hψ |ψ i = of FC(ξ) = Trˆρ(ξ) dµC (ξ)FC(ξ)=1 and ofρ ˆ H 1 2 1 2 MC 0. (Trˆρξ = 1). Given aR hybrid state determined by the The next step is to define a probability distribution on classical point ξ (which has probability Trˆρ(ξ)), and a the set of MEEs of MH . Following von Neumann idea quantum state represented by the projectorπ ˆ, the prob- and the mathematical construction of Gleason theorem ability of measuring the system to be in that state is [42], we can build a hybrid density matrix to represent given by Tr(ˆρ(ξ)ˆπ). These ξ−dependent density matrices the hybrid probability in a consistent way. As the phys- have already been used before, for example by Aleksan- ical properties of the hybrid system, in general, combine drov [25], or obtained by taking the partial classical limit the states of MC and MQ (for instance, the total energy in the Wigner transformation of the full quantum den- of the system), we cannot expect both sets to be inde- sity matrix, in the quantum-classical Liouville equation pendent from the probabilistic point of view. Nonethe- method [26]. less, we can assume that we can simultaneously mea- Let us consider now how to define the entropy of these sure any classical observable and any hybrid observable hybrid states. For any bivariate distribution p(x, y) of of the form Aˆ(ξ). This fact permits to define the condi- two sets of random variables (X, Y ), the entropy S(p) tional probabilities p(a|ξ): the probability of measuring decomposes as an eigenvalue a of operator Aˆ(ξ), given that the classical subsystem is at state ξ ∈M . The probabilities associ- C S(p)= S(pX )+ pX (x)S(pY |x) , (7) ated to the hybrid measurement can then be decomposed Xx into the marginal probability associated to the classical phase space, FC (ξ), and the conditional probabilities as- where pX (x)= y p(x, y) is the marginal distribution of sociated to the measurement of Aˆ(ξ), given ξ: X, and pY |x is theP conditional probability of Y given x. This general result must be applicable to the decomposi- p(a, ξ)= FC(ξ)p(a|ξ). (4) tions (4) and (5) . Therefore, the entropy of the hybrid system must be equal to the sum of the (classical) entropy (SC ) of the marginal classical distribution FC(ξ) 4 and the average, over FC(ξ), of the (von Neumann) en- determined by the choice of E, that is used to define the tropy associated to the conditional probability ρξ, i.e.: (inverse of the) temperature. Note that this ensemble had been perhaps implicitly assumed before, but seldom 2 SC (FC) explicitly written and, to our knowledge, never derived. Notice that the orthogonal projectors of its spectral de- S[ˆρ(ξ)] = −k dµ (ξ)F (ξ) log(F (ξ)) + z B Z C }| C C { composition coincide with those of the adiabatic basis. MC The problem can be addressed as a constrained op- dµ (ξ)F (ξ) −k Tr ρˆξ logρ ˆξ (8) timization problem: find the density matrix that maxi- Z C C B MC mizes S in Eq. (9), subject to the constraints: SvN(ˆρξ) | {z } C [ˆρ(ξ)]:= dµ (ξ)Tr(ˆρ(ξ)) − 1=0 , (12) It is immediate then to rewrite this as: N Z C MC S[ˆρ(ξ)] = −k dµ (ξ) Tr(ˆρ(ξ) logρ ˆ(ξ)) , (9) ˆ B Z C CE [ˆρ(ξ)]:= dµC (ξ)Tr(ˆρ(ξ)H(ξ)) − E =0 . (13) MC Z MC which is our proposal for the hybrid QC entropy. To These can be incorporated via Lagrange multipliers, the best of our knowledge, this is the first rigorous defining the full optimization functional to be: proposal of an entropy function for a hybrid quantum- classical system. If the classical subsystem is pure, (i.e. S := S − λN CN − λECE . (14) FC(ξ)= δ(ξ − ξ0)) the classical entropy vanishes and the entropy above reduces to von Neumann entropy. Analo- Without loss of generality, let us work in the (ξ- gously, when the quantum state is pure and independent dependent) basis of eigenstates of the Hamiltonian (the ξ of the classical state, the von Neumann entropy of ρ adiabatic basis). First, we will consider the optimization vanishes, and the expression above reduces to the clas- over a reduced set of density matrices: those which are sical entropy function. Therefore, the entropy function diagonal in this adiabatic basis. The terms in Eq. (14) (9) combines the classical and quantum information in a then read: consistent way, and has the correct classical and quantum limits. S[{ρ }]= −k dµ (ξ) ρ (ξ) log(ρ (ξ) , ii B Z C ii ii MC Xi (15)

III. THE MAXENT PRINCIPLE FOR HYBRID C [{ρ }]= dµ (ξ) ρ (ξ) − 1 (16) N ii Z C ii QC SYSTEMS. MC Xi C [{ρ }]= dµ (ξ) H (ξ)ρ (ξ) − E (17) A. MaxEnt principle for the hybrid entropy E ii Z C i ii function MC Xi Taking derivatives and setting them to zero leads imme- The maximum entropy principle is one of the standard diately to procedures to derive the canonical ensemble at both the −1 −βHi(ξ) classical or the quantum level. Firstly, one must assume ρii(ξ)= ZHCE(β) e , (18) that the system is in equilibrium. Then, one can find the canonical ensemble as the solution of the MaxEnt where β = λE . kB problem: given a certain thermodynamic system and an We consider now a general density matrix ρ˜ˆ(ξ), whose S entropy function , find the equilibrium ensemble which non-diagonal elements may be non-zero, fulfilling the two S maximizes among those with a fixed value of the aver- constraints (12) and (13). Since it is Hermitian with non- E Hˆ ξ age energy = h ( )i. negative eigenvalues, it satisfies Klein’s lemma [46]: In the following, we will prove that the canonical ensemble that results of this maximization, for the hybrid − Tr(ρ˜ˆ(ξ) log(ρ˜ˆ(ξ)) ≤− ρ˜ (ξ) log(˜ρ (ξ)) (19) case, is given by: ii ii Xi e−βHˆ (ξ) ρˆHCE(ξ)= (10) ZHCE(β) 2 ˆ For example, it was given in Ref. [47], where it was claimed to Z (β)= dµ (ξ) Tr(e−βH(ξ)) (11) HCE Z C be the partial classical limit of the fully quantum canonical en- MC semble. It was also presented as the zero-th order term in a ˆ classical-limit expansion of the partial Wigner transformation of where H(ξ) is the Hamiltonian (typically decomposed the quantum canonical ensemble in Ref. [26]. Finally, in foot- c ˆ ˆ into a classical and a quantum part, as fH (ξ)I + HQ(ξ)), note 30 of Ref. [48], some of the current authors already hinted, without proof, the result demonstrated here. ZHCE(β) is the partition function, and β is a constant, 5 whereρ ˜ii(ξ) are its diagonal elements (the equality only • If the QC coupling is turned off (the quantum holds if it is actually diagonal). As the constraints (12) Hamiltonian HˆQ is independent of the classical and (13) in the adiabatic basis only depend on the di- variables and vice versa), the HCE becomes the agonal elements ofρ ˆ(ξ), we may conclude that for any product of the classical and quantum canonical en- non-diagonal density matrix that fulfills the constraints sembles, which maximize the sum of their respec- there exists a diagonal one (defined to be the one whose tive entropies independently. diagonal entries are the same) that also fulfills the constraints and has a larger entropy. The global maximum, therefore, has to be found among the diagonal ones, and is the one given in Eq. (18). This concludes the proof. C. Dynamics.

B. Properties of the HCE Another extra condition that an equilibrium ensemble must obviously verify is missing in the previous list: sta- Let us now check that the ensemble thus defined fulfills tionarity under the dynamics of the micro-states. How- some very natural requirements: ever, up to now we have disregarded the dynamics, and derived the canonical ensemble from very broad assump- • Additivity. If two systems are in the canonical en- tions, freed from dynamical arguments. The dynamics is semble equilibrium at the same temperature, they neither relevant for the definition of the entropy function must also be at equilibrium when we consider them nor affects directly the solution of the MaxEnt condition. to form a single systems with two (independent) For instance, MaxSG defines the MaxEnt solution for the subsystems. Extensive variables as the energy and entropy function SG (Eq. (3)), independently of the dy- entropy must be additive. namics of the microstates we consider. The existence of This can be proven for the HCE in the following dynamics having it as an equilibrium point would be an way. If Hˆ1(ξ1) and Hˆ2(ξ2) are the Hamiltonians of extra requirement for the definition of a thermodynami- both systems, the combined one is: cal ensemble. On the other hand, we also proved above that the Max- Hˆ (ξ)= Hˆ1(ξ1) ⊗ Î2 + Î1 ⊗ Hˆ2(ξ2), (20) Ent solution of the true hybrid entropy function (9) is the HCE. This implies that the only possible ensemble which where ξ = (ξ1, ξ2). can be considered to represent the canonical ensemble of As the two terms of (20) trivially commute, a hybrid system is the HCE. Is there a dynamics that makes it also stationary? Trivially, the commutator with

ˆ ˆ 1 1 ˆ1 2 Hˆ (ξ) (i.e. a generalized von Neumann equation) does, e−βH(ξ) = e−βH (ξ ) ⊗ e−βH (ξ ) , (21) but many others may also be possible. We will analyze and because of this, this issue in a forthcoming publication.

ˆ dµ (ξ , ξ )Tr e−βH(ξ) = IV. CONCLUSIONS Z C 1 2 MC1 ×MC2 ˆ ˆ dµ (ξ )Tr e−βH(ξ1) dµ (ξ )Tr e−βH(ξ2) It has been the purpose of this paper to shed some Z C 1 Z C 2 MC1 MC2 light into the issue of the entropy and the canonical equi- (22) librium expression for hybrid systems. We have ﬁrst dis- cussed the deﬁnition for the entropy of an ensemble of Thus we can just write hybrid systems. We have done it by making very general assumptions on the hybrid theory, but without any con-

ρˆ(ξ)=ˆρ1(ξ1) ⊗ ρˆ2(ξ2) (23) sideration for the particular dynamics. We have considered two different alternatives, one based on probability densities on the hybrid phase space and another based This factorization ofρ ˆ(ξ) immediately implies the on projectors and the notion of hybrid mutually exclu- additivity of the internal energy (13) and of the sive events. The first case leads to a Gibbs-like function entropy (9). which treats the hybrid system as a direct analogue of • The classical canonical ensemble, which maximizes a classical system. We have shown how that entropy Gibbs entropy, is recovered when only one quantum function assigns the wrong weight to hybrid events and energy state exists. because of this fails to produce a physically meaning- ful Thermodynamics. The second proposal departs from • The quantum canonical ensemble, which maximizes the information-theory definition of entropy, and care- von Neumann entropy, is recovered when only one fully considers the principle of mutually exclusive events. classical point is allowed. The resulting hybrid entropy function weights correctly 6 the hybrid exclusive events and defines a physically con- ACKNOWLEDGMENTS sistent thermodynamical entropy. Then, we have derived the HCE as the one that fulfills the MaxEnt principle with respect to the hybrid entropy function, using it for the first time for hybrid quantum- The authors would like to thank Profs. Floria and classical systems. Furthermore, we verified that the HCE Zueco for their very useful suggestions. Partial financial reproduces the classical and quantum cases when the support by MINECO Grant FIS2017-82426-P is acknowl- suitable limits are considered. Hence, we can claim edged. C. B. acknowledges financial support by Gob- that the MaxEnt principle is applicable and consistent ierno de Aragón through the grant defined in ORDEN for hybrid quantum-classical systems. IIU/1408/2018.

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