Entropy and canonical ensemble of hybrid quantum classical systems J. L. Alonso,1, 2, 3 C. Bouthelier,1, 2 A. Castro,1, 2, 4 J. Clemente-Gallardo,1, 2, 3 and J. A. Jover-Galtier1, 2, 5 1Departamento de F´ısica Te´orica, Universidad de Zaragoza, Campus San Francisco, 50009 Zaragoza (Spain) 2Instituto de Biocomputaci´on y F´ısica de Sistemas Complejos (BIFI), Universidad de Zaragoza, Edificio I+D, Mariano Esquillor s/n, 50018 Zaragoza (Spain) 3Centro de Astropart´ıculas y F´ısica de Altas Energ´ıas (CAPA), Departamento de F´ısica Te´orica, Universidad de Zaragoza, Zaragoza 50009, (Spain) 4Fundaci´on ARAID, Av. de Ranillas 1-D, planta 2ª, oficina B, 50018 Zaragoza (Spain) 5Centro Universitario de la Defensa de Zaragoza, Academia General Militar, 50090 Zaragoza (Spain) (Dated: October 16, 2020) In this work we generalize and combine Gibbs and von Neumann approaches to build, for the first time, a rigorous definition of entropy for hybrid quantum-classical systems. The resulting function coincides with the two cases above when the suitable limits are considered. Then, we apply the MaxEnt principle for this hybrid entropy function and obtain the natural candidate for the Hybrid Canonical Ensemble (HCE). We prove that the suitable classical and quantum limits of the HCE coincide with the usual classical and quantum canonical ensembles since the whole scheme admits both limits, thus showing that the MaxEnt principle is applicable and consistent for hybrid systems. I. INTRODUCTION hybrid theory. The focus of this work is on the statistical mechan- Hybrid quantum-classical (QC) systems are the natu- ics of hybrid systems, regardless of the dynamics chosen ral approximation to those quantum systems containing for their description. In particular, we consider two open some degrees of freedom that can be well approximated as questions: first, what is the correct definition of the en- classical variables. This possibility arises when there are tropy of a hybrid system? And then, given this definition, two different energy or mass scales, as it happens, for in- can we use the MaxEnt formalism and obtain the canon- stance, in molecular and condensed matter systems where ical ensemble of a hybrid system, as we do for classical the nuclei are heavy and slow, while the electrons are light or quantum ones? Apparently, these are purely funda- and fast. Hybrid models have also been proposed to ex- mental questions, but their answers are crucial for many plain the measurement process [1, 2]: the measurement applications, in particular for the ab initio modeling of device is modeled as a classical system coupled to the molecules and materials and their numerical simulation quantum system to be measured. In field theory, hybrid methods at finite temperature (for example, see [28–33]). quantum-classical systems have also been considered as We determine, in a simple way, the equilibrium ensem- candidates to describe quantum matter fields interacting ble that the numerical methods must reproduce and the with a (classical) gravitational field, as a semiclassical entropy function they must consider. approximation or even a fundamental theory (see [3, 4]). The structure of the paper is as follows. In Section The correct mathematical formalism for the dynamics II we will first discuss the proper definition of the hybrid and statistics of these hybrid models is not obvious. Two entropy function. Then, in Section III we will derive the different points of view can be taken. On the one hand, hybrid canonical ensemble (HCE) as the one that max- a practical one: the construction of a hybrid theory that imizes this entropy, subject to the constraint of a given approximates, as closely as possible, the full quantum dy- expectation value for the energy (MaxEnt principle). The namics of the problem. Such methods can be applied to resulting ensemble had been perhaps implicitly assumed a very large array of problems in condensed matter and before, but few times explicitly spelled, and never, to our molecular physics and chemistry, as non-adiabatic pro- knowledge, derived from the general principle of entropy maximization. We will also briefly discuss some relevant arXiv:2004.01986v2 [physics.chem-ph] 15 Oct 2020 cesses play a fundamental role [5–9]. On the other hand, a fundamental, theoretical point of view: the construc- properties of the resulting ensemble. Finally, in Section tion of a mathematically and physically consistent the- IV we will summarize our main conclusions. ory for hybrid systems, according to different demands of consistency [1, 10–27], independently of how well it may approximate the full quantum dynamics. This second II. THE ENTROPY OF A HYBRID QC approach is compulsory when the full quantum dynam- SYSTEM. ics is not known, as in the case of a system of quantum matter fields interacting with gravity. In any case, it is A correct statistical mechanical definition of any sys- not clear what is the best possible dynamics from any tem departs from the definition of a sample space: a set of those two points of view. Here, we assume the sec- of statistically independent states, i.e. a basis of mutu- ond one, and add to the discussion on the construction ally exclusive events (MEE), which can be unequivocally of a mathematically consistent and physically motivated characterized by the results of an experiment. Let us 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 ).