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Maximal-entropy-production-rate nonlinear quantum dynamics compatible with second law, reciprocity, fluctuation-dissipation, and time-energy uncertainty relations

In view of the recent quest for well-behaved nonlinear extensions of the traditional Schroedinger-von Neumann unitary dynamics that could provide fundamental explanations of recent experimental evidence of loss of quantum coherence at the microscopic level, in this paper, together with a review of the general features of the nonlinear quantum (thermo)dynamics I proposed in a series of papers [see references in G.P. Beretta, Found.Phys. 17, 365 (1987)], I show its exact equivalence with the maximal-entropy-production variational-principle formulation recently derived in S. Gheorghiu-Svirschevski, Phys.Rev. A 63, 022105 (2001). In addition, based on the formalism of general interest I developed for the analysis of composite systems, I show how the variational derivation can be extended to the case of a composite system to obtain the general form of my equation of motion, that turns out to be consistent with the demanding requirements of strong separability. Moreover, I propose a new intriguing fundamental ansatz: that the time evolution along the direction of steepest entropy ascent unfolds at the fastest rate compatible with the time-energy Heisenberg uncertainty relation. This ansatz provides a possible well-behaved general closure of the nonlinear dynamics, compatible with the nontrivial requirements of strong separability, and with no need of new physical constants. In any case, the time-energy uncertainty relation provides lower bounds to the internal-relaxation-time functionals and, therefore, upper bounds to the rate of entropy production.


I. INTRODUCTION
In my 1981 doctoral thesis [1] and a subsequent series of papers [2][3][4][5][6][7][8] (see also Refs. [9,10]), I addressed the problem of deriving a well-behaved extension of the Schrödinger-von Neumann unitary dynamics meeting the very demanding set of strict requirements that appear to be necessary if one is willing to accept [11]: (1) the "broader quantum kinematics" ansatz that ρa unit-trace, nonnegative-definite, hermitian operator on the Hilbert space H associated with a system according to standard quantum mechanics -represents a "true" quantum state (in the sense analogous to that of the wave function of standard QM) not only when ρ 2 = ρ but also when ρ 2 = ρ, even if the system is strictly isolated and uncorrelated; and (2) that −k B Tr(ρ ln ρ) represents the "physical" entropy (as opposed to a statistical or information-theoretic entropy related to incoherent stochastic mixtures of true states).
The equation of motion I designed and proposed for the state operator ρ of a general system consisting of M distinguishable indivisible subsystems has the form where the notation is defined in Section X. For a single indivisible system, it reduces to the form ]} where I is the identity operator on the Hilbert space H of the system, H the usual Hamiltonian operator associated with the system in standard QM, the G i 's are additional (not always necessary) hermitian operators commuting with H that we call the non-Hamiltonian generators of the motion and that depend on the structure of the system, such as the i-th-type particle-number operators N i for a Bose-Einstein or Fermi-Dirac field (in which case H is a Fock space), or the total momentum component operators P i of a free particle (in which case Gheorghiu-Svirschevski [12] proved Galilei invariance).
As I realized in Ref. [6], the internal-relaxation times τ J (ρ) need not be constants (as initially assumed in Ref. [2]) but can be any positive functionals of ρ. However, here I note that the condition of separability imposes precise restrictions on these functionals. In Section VII, I derive a lower bound for τ J (ρ) which corresponds to the highest entropy production rate compatible with the time-energy Heisenberg uncertainty principle. Taking the internal-relaxation times τ J (ρ) exactly equal to these nontrivial lower bounds turns out to be compatible with the separability condition, and completes the nonlinear dynamics in a general way with no need to assume the existence of new physical constants.
Equation (1) satisfies all the necessary requirements (listed in Appendix A) for a self-consistent and welldefined extension of the Schrödinger-von Neumann equation of motion compatible with thermodynamics requirements, and turns out to have very intriguing mathematical properties and consistent physical consequences.
In this paper, in addition to providing a self-contained review of results already proved in previous work, I give a more explicit account of the steepest-entropy-ascent geometric construction on my nonlinear equation of motion for quantum thermodynamics than available in the original papers. As a result, I show that the equivalent variational formulation in Ref. [12] can be extended to the composite system case. I also show that, in order for the general form of my equation to satisfy strong separability conditions conceptually equivalent to those recently discussed by Czachor [15], the relaxation-time functionals must satisfy nontrivial conditions. I emphasize that the nonlinearity of the dynamics is such that one should not expect that the equation valid for a single elementary (indivisible) constituent of matter be valid also for a system with an internal structure, because the equation of motion must reflect such structure not only through the Hamiltonian operator. However, of course, the superoperatorD M must reduce toD 1 [see Eq. (94) below] when the system has only one indivisible subsystem. For a composite system, my original general formulation To make the paper self-contained, I briefly review the fundamental unifying extensions to all nonequilibrium (dissipative) states of the Onsager reciprocity relations [13] and Callen's fluctuation-dissipation theorem [14] that derive from Eq. (2) [6] as well as, in general, from Eq. (1) [8].
In Appendixes A and B, I discuss a set of criteria and the definition of stability of equilibrium that a fundamen-tal dynamics must satisfy in order to be compatible with the second law. In the other Appendixes, I discuss some extensions of the mathematics of Eqs. (1) and (2).

II. THE AUGMENTED STATE DOMAIN ANSATZ
The fundamental ansatz that the postulates of quantum mechanics can be successfully supplemented by the first and second principles of thermodynamics by assuming a broader state domain that includes not only ρ 2 = ρ but also ρ 2 = ρ state operators, provided that the functional −k B Tr(ρ ln ρ) is taken for the physical entropy, was first proposed (without a dynamical law) by Hatsopoulos and Gyftopoulos [11].
Twenty five years ago, the hypothesis of a state domain augmented with respect to that of traditional QM was perceived as countercurrent to the prevailing approaches to dissipative quantum dynamics within the frameworks of statistical, stochastic, phenomenological, informationtheoretic, chaotic-behavior and bifurcation theories. For this reason, the broader quantum kinematics ansatz [11] and, five years later, my equation of motion have been considered initially as unphysical and substantially ignored, mainly because their motivation appeared to be derived from theoretical reasoning only.
In search for direct experimental evidence, I computed the effects of the irreversible atomic relaxation implied by the nonlinear equation of motion onto some basic quantum-electrodynamics results on absorption, stimulated emission, and resonance fluorescence from a single two-level atom [4,5]. The results were obtained in the near-equilibrium linear limit and, of course, in terms of the yet undetermined internal-relaxation-time functional τ (ρ) that is part of the equation of motion. To my knowledge no one has yet attempted to verify these results experimentally and estimate τ (ρ).
The recent new experimental evidence of loss of quantum coherence [16,17] and the impressive effort devoted to study nonlinear modifications of the standard Schrödinger equation in the last ten years [18], finally seem to make more acceptable, if not require, the ρ 2 = ρ augmented state domain Hatsopoulos-Gyftopoulos ansatz.
Once the ρ 2 = ρ ansatz is accepted, the nonlinear equation of motion I proposed completes the dynamics and holds the promise to provide a microscopic-level explanation of the recent experimental evidence of loss of quantum coherence. It is with this motivation that Gheorghiu-Svirschevski [12] has "rediscovered" the simplest form of my equation together with many of its known features. Ref. [12] contributes to confirm the validity of my equation, including existence and uniqueness of solutions, and derives useful results in the nearequilibrium linear limit. However, the question of defining the form of the relaxation-time functional is left unresolved.

III. THE STEEPEST-ENTROPY-ASCENT ANSATZ FOR AN INDIVISIBLE SYSTEM
In Refs. [3,6,7], I emphasized that Eqs. (1) and (2) have an important geometric interpretation. The Hamiltonian term, −i[H, ρ]/h, and the dissipative term,D(ρ, H, G i ), compete with each other in the sense that the first drives the state operator toward a unitary motion tangent to the local constant entropy surface, whereas the second drives it toward the local direction of steepest entropy ascent along the surface with constant mean values of the generators of the motion. In this section, I show more explicitly than in the original papers how Eq. (2) can be constructed directly from the steepest-entropyascent ansatz.
Let H (dimH ≤ ∞) be the Hilbert space and H the Hamiltonian operator that are associated with the given (indivisible) elementary constituent system in standard QM. For simplicity, we first consider a system composed of a single elementary constituent. The generalization to M constituents is given in Section X.
We assume that the (true) quantum states are oneto-one with the linear hermitian operators ρ on H with Tr(ρ) = 1 and ρ ≥ 0. As done in Ref. [2], we introduce the square-root state operator √ ρ obtained from ρ by substituting its eigenvalues with their positive square roots [19].
As a first step to force positivity and hermiticity of the state operator ρ we assume that the equation of motion may be written as where the operator E, in general non-hermitian, is defined later. We consider the space L(H) of linear operators on H equipped with the real scalar product so that for any (time-independent) hermitian R in L(H) the corresponding mean-value state functional, its local gradient operator with respect to √ ρ, and its rate of change are Moreover, the entropy state functional, its gradient operator with respect to √ ρ, and its rate of change are Now it is easy to see from Eq. (7)  √ ρR 1 = √ ρH (to conserve energy), plus the additional non-Hamiltonian generators of the motion as already discussed.
It is noteworthy that with c(ρ) any real functional of ρ, yields [through Eq.
(3)] the Hamiltonian part of the equation of motion, and is orthogonal to L{ √ ρR i } and to the entropy gradient ∂s(ρ)/∂ √ ρ as well. Introducing the notation [6] ∆F = F − Tr(ρF )I , for F and G hermitian in L(H) and {·, ·} the usual anticommutator, and defining the shortest characteristic time associated with the Hamiltonian part of the equation of motion by the relation (see Appendix C) we find, from Eq. (11) and ( √ ρ| √ ρ) = 1, so that only the choice c = −e(ρ) = −Tr(ρH) is compatible with the Heisenberg time-energy uncertainty relation τ H (ρ) 2 ∆H∆H ≥h 2 /4 which is then satisfied with strict equality. Therefore, and We may note that the foregoing discussion on the choice of c(ρ) is somewhat artificial because any c(ρ) cancels out in Eq. (3). However, the line of thought becomes important when we apply it later to the dissipative part of the dynamics.
For the general rate of change of √ ρ, we let and assume E D orthogonal to L{ √ ρR i } but in the direction of the entropy gradient operator. We cannot take E D directly proportional to ∂s(ρ)/∂ √ ρ as such, because the entropy gradient in general has a component along L{ √ ρR i } which would not preserve the mean values of the generators of the motion. We must take E D proportional to the component of the entropy gradient orthogonal to L{ √ ρR i }, namely, for the indivisible system, where, for convenience, we defined the entropy operator with B obtained from ρ by substituting its nonzero eigenvalues with unity, i.e., more formally, B = B 2 = P Ran ρ = I − P Ker ρ = P ⊥Ker ρ , so that The Gram determinant at the denominator, ρG i by a Gram-Schmidt orthogonalization procedure followed by normalization. For example, Then, It is noteworthy that operators A i are nonlinear functions of ρ and, therefore, in general, vary with ρ as it evolves with time.
We finally obtain the following equivalent expressions for the rate of entropy change where the Gram determinant is also strictly positive except when ρ satisfies Eq. (27).
or, equivalently, where in which case we say that the state is nondissipative, and the solution of the equation of motion is which includes the usual ρ 2 = ρ Schrödinger dynamics when C is the null operator [β = ν i = 0 in Eq. (27)] and, therefore, Tr(B) = 1.
Indeed, Equation (2) preserves the rank and nullity of ρ [2] or, as said in Ref. [12], preserves the cardinality of the set of nonzero eigenvalues of ρ. Because the equation of motion attracts the state operator towards the highest entropy state or limit cycle compatible with the initial mean values of the generators of the motion and the number of zero eigenvalues, a minor perturbation of the state that changes an initially zero eigenvalue to an arbitrarily small nonzero value would cause an irreversible departure of the state towards a different (higher entropy) equilibrium state or limit cycle. Hence, as long as there are zero eigenvalues of ρ, i.e., unless B = I, all equilibrium states and limit cycles are unstable. Without this conclusion, we could not claim that the equation of motion entails the second law of thermodynamics (see Appendixes A and B, and the original papers, for further discussion of this important point).
The Hamiltonian operator may be set to vary with time in order to model (adiabatic) energy exchange with the surroundings. Then the unitary part of the equation of motion, −i[H(t), ρ]/h, may cause energy exchange, but not entanglement nor entropy exchange with the surroundings. The dissipative term, instead, describes internal relaxation only and does not contribute to energy change in any case.

V. HIGHEST-ENTROPY EQUILIBRIUM STATES
The only globally stable equilibrium states of the dynamics generated by Equation 2 are The definition of global stability, which is stronger than Lyapunov or local stability and is required by the second law [20,21], is given and discussed in Appendix B. The proof was discussed in Refs. [2,4] and relied on a technical conjecture discussed in Ref. [20] and later found to be proved by Hiai, Ohya, and Tsukada [22]. For a system with generators of the motion I, H, N i , Eq. (29) represents in general the grand canonical thermodynamic equilibrium states. It reduces to the canonical equilibrium states if N i = n i I for all i's (to the microcanonical if also H = eI).
As is well known, states given by Eq. (29) are solutions of the constrained maximization problem max s(ρ) (30a) subject to r i (ρ) = r i and ρ ≥ 0 , where s(ρ) = −k B Tr(ρ ln ρ), r 1 (ρ) = Tr(ρI), r 2 (ρ) = Tr(ρH), r i (ρ) = Tr(ρG i ), r 1 = 1, r 2 and r i 's given. The inequality constraint ρ ≥ 0 can be eliminated recasting the problem in terms of √ ρ, . The method of Lagrange multipliers then gives the condition (necessary but not sufficient) which, using Eqs. (6) and (9), becomes It is noteworthy that Eq. (33) is satisfied with obvious identification of multipliers by Eq. (27c), where Therefore, each nondissipative state satisfies the necessary condition (33) although it is not a solution of the maximization problem (31) unless B = I.

VI. ONSAGER'S RECIPROCITY AND CALLEN'S FLUCTUATION-DISSIPATION NONEQUILIBRIUM RELATIONS
First, I introduce a general representation of state operators particularly useful for representing nonequilibrium states. Any ρ can be written as [6] where the hermitian operators in the set {I, X j } span the real space L h (H) of linear hermitian operators on H, and B is the usual idempotent operator defined in Eq. (21b). Indeed, the operator B ln ρ is always well-defined and belongs to L h (H), so that where the second equality follows from where, for simplicity, we use the expression (24b) for D in terms of the orthonomal set { √ ρA i } [an equivalent expression in terms of Gram determinants is readily obtained using Eq. (36) in Eq. (20c)].
Notice from Eq. (38) that the partial derivative may be interpreted as the generalized affinity or force representing the entropy change that corresponds to an independent change in the mean value of the linear observable X j . Next, we define the dissipative rate of change of the linear mean-value functional associated with operator X i , and Eq. (39) yields the linear interrelations between dissipative rates and generalized affinities, where the coefficients may be interpreted as generalized (Onsager) dissipative conductivities and are nonlinear functions of ρ, This equations are at the same time a proof and a generalization of Onsager's reciprocity relations [13] extended to all nonequilibrium (dissipative) states. Moreover, they give explicit expressions for the nonlinear dependence of the dissipative conductivities L ij (ρ) on the state operator ρ, the generators of the motion and the internalrelaxation-time τ (ρ). Using Eqs. (25) and (36), the rate of entropy change may be rewritten as a quadratic form of the generalized affinities, The symmetric matrix [{L ij (ρ)}] is a Gram matrix [see Eq. (43c)] and, as such, it is nonegative definite, i.e., its determinant with strict positivity only if all operators are linearly independent, in which case Eq. (42) may be solved to yield and the rate of entropy change can be written as a quadratic form of the dissipative rates Equations (43) are also a proof and generalization of Callen's fluctuation-dissipation theorem [14] extended to all nonequilibrium (dissipative) states. Indeed, we interpret ∆X i ∆X j as the codispersion (covariance) of measurement results of observables X i and X j when the system is in state ρ and ∆X i ∆X i as the dispersion (or fluctuations) of measurement results of observable X i . As in Callen's fluctuation-dissipation theorem, the expressions in Eq. (43) relate codispersions with generalized conductivities.
A judicious choice of the set { √ ρX j } may greatly simplify these relations. In particular, if { √ ρX j } is any orthogonal extension of the orthonormal subset { √ ρA i }, in the sense that ( √ ρX j | √ ρA i ) = 0 for all A i 's and j > a, then for all i and j > a, which relates directly [through the internal-relaxation time τ (ρ)] covariances and fluctuations in the observables X i with their associated dissipative conductivities. Onsager's [13] and Callen's [14] theorems are keystones of our understanding of irreversibility. Indeed, the proof I proposed emerges not from the analysis of the Hamiltonian term of the equation of motion supplemented with the so-called assumption of microscopic reversibility, but from the dissipative term of the equation of motion, i.e., the only term responsible for irreversibility, with no additional assumptions.
Onsager's result [13] was obtained from empirical observations on nonequilibrium phenomena very close to stable thermodynamic equilibrium, so that the list of X i 's was indeed very short, and the result valid only for a limited class of states. Our result generalizes the validity of Onsager's reciprocity relations to all nonequilibrium states, close and far from stable thermodynamic equilibrium. Of course, the price we have to pay to describe nonequilibrium states far from stable equilibrium is that we must use a much larger, possibly infinite list of X i 's in Eq. (34).

VII. INTERNAL RELAXATION TIME VS TIME-ENERGY HEISENBERG UNCERTAINTY: MAXIMAL-ENTROPY-PRODUCTION-RATE ANSATZ
By analogy with what done for the Hamiltonian characteristic time τ H (ρ), we define the shortest characteristic time of the dissipative rate of change of √ ρ by the From Eqs. (19) and (20) and From this relation we may extract a possible ansatz on the functional form or at least a lower bound for the internal-relaxation-time functional τ (ρ) in Eq. (2) by assuming that also τ D (ρ), like τ H (ρ), should satisfy the Heisenberg uncertainty relation This implies or, using Eqs. (25) and (44), Relation (53) implies an upper bound to the rate of entropy production, By analogy with the Hamiltonian time, a possible ansatz that I propose should be carefully considered is that also τ D (ρ) and, therefore, τ (ρ) be equal exactly to the lower bound [i.e., strict equality in Eqs. (53), (54) and (55)], corresponding to the maximal entropy production rate compatible with the time-energy uncertainty relation.
This corresponds to a truly maximal entropy production dynamics. The square-root state operator √ ρ not only moves in the direction of steepest entropy ascent, but it does so at the highest rate compatible with the uncertainty principle.
This ansatz is very intriguing, conceptually appealing and fraught with far-reaching physical consequences. But, like any other hypotheses on the functional form of τ (ρ), it should be verified against known behavior in worked-out specific cases, such as the examples considered in Refs. [5,9,12] and the variety of physical problems for which nonlinear modifications of the Schrödinger equation have been deemed necessary but have not yet been resolved [18].
If proved valid, this ansatz would complete the dynamics without need of additional physical constants and would provide a fundamental microscopic foundation of the macroscopic observation that Nature always evolves at the fastest possible rate along the most direct path towards maximum entropy compatible with the system's structure and external constraints. Interpreted in this way, my nonlinear dynamics would provide a unifying fundamental microscopic foundation of all phenomenological theories of irreversible processes advanced in the last seventy years after the pioneering work of Onsager.

VIII. VARIATIONAL PRINCIPLE FORMULATION
Following Gheorghiu-Svirschevski [12], the direction of steepest-entropy-ascent can also be found by considering the constrained maximization problem whereṙ i and τ 2 D are given real functionals of ρ. The necessary solving condition in terms of Lagrange multipliers is Using Eqs. (7) and (10), it becomes where the multipliers must be determined by substitution in the system of constraint equations. It easy to verify that our expression of E D = −D/2τ (ρ) in Eqs. (19) and (20) is an explicit solution of Eq. (58) in the caseṙ i = 0 and τ D given by the explicit expression in Eq. (51). See Appendix E forṙ i = 0.

IX. BLOCH NEAR-EQUILIBRIUM LINEAR LIMIT
In Ref. [2] we have shown that in the neighbourhood of each equilibrium state ρ e {given by Eq. (27) with the additional condition [B, H] = 0}, if we assume τ (ρ) constant (e.g., τ e = τ (ρ e ) = 0), the state operators in the subset with B = B e and Tr(ρR i ) = Tr(ρ e R i ) (i.e., with the same nullity, rank and mean values of the generators as the equilibrium state) obey a linearized form of the equation of motion that has the form of a Bloch relaxation equation, so that the solution of the equation of motion is Gheorghiu-Svirschevski [12], derived a linearized form for the case with no non-Hamiltonian generators valid also when [ρ, H] = 0 (but assuming τ → τ e = 0), and showed that it yields a generalized Fokker-Planck equation for a free particle, and a Langevin equation for a harmonic oscillator.
It is however noteworthy that such linearized limit behavior should be revised if we take τ (ρ) equal to the lower bound imposed by the time-energy uncertainty relation [Eq. (53) with strict equality], because then, if ∆H∆H = 0, τ (ρ) → 0 as ρ → ρ e .

X. DIVISIBLE SYSTEMS DYNAMICS. DEFINITIONS AND NOTATION
The composition of the system is embedded in the structure of the Hilbert space as a direct product of the subspaces associated with the individual elementary constituent subsystems, as well as in the form of the Hamiltonian operator.
In this section, we consider a system composed of M distinguishable and indivisible elementary constituent subsystems. For example, each subsystem may be a different elementary particle or a Fermi-Dirac or Bose-Einstein field (in which case the corresponding H J is a Fock space). The Hilbert space is and the Hamiltonian operator where H J is the Hamiltonian operator on H J associated with the J-th subsystem when isolated and V (on H) the interaction Hamiltonian among the M subsystems. For convenience, we denote by H J the direct product of the Hilbert spaces of all subsystems except the J-th one, so that the Hilbert space of the overall system and the identity operator I = I J ⊗I J . The subdivision into elementary constituents, considered as indivisible, and reflected by the structure of the Hilbert space H as a direct product of subspaces, is particularly important because it defines the level of description of the system and specifies its elementary structure. This determines also the structure of the nonlinear dynamical law I proposed. In a sense, this is a price we have to pay in order to free ourselves from the assumption of linearity of the law of motion.
In other words, the form of the superoperatorsD M and D 1 are different depending on whether the system is or is not subdivisible into indivisible subsystems. For example, consistently with the conditions listed in Appendix A, we request that the superoperatorD M satisfies the strong separability conditions [9,15] where subsystems A and B are obtained by partitioning the set of constituents 1, 2, . . . , M into two disjoint subsets of M A and M B constituents, respec- It is noteworthy that trying to apply the same conditions to superoperatorD 1 would be physically meaningless, because if the system were subdivisible thenD 1 would not be its dissipative evolution superoperator.
Conditions (64b) and (64c) prevent locality problems because they guarantee that changes of the Hamiltonian (or the other generators of the motion) in one of two noninteracting subsystems cannot affect the mean values of local observables of the other subsystem. For example, assume that subsystems A and B are correlated but not interacting. We may switch on a measurement apparatus within B and therefore alter the Hamiltonian H B . By virtue of Eqs. (64b) and (64c), the rate of change of the reduced state operator ρ A = Tr B (ρ) does not depend on H B and, therefore, all functionals of ρ A (local observables) remain unaffected by the change in B, i.e., no faster-than-light communication can occur between B and A (of course, if we exclude the projection postulate).
In addition, we must consider the following additional non-trivial conditions of separate energy conservation of noninteracting subsystems for any ρ (see Appendix A, Condition 6) and of separate entropy nondecrease for uncorrelated (possibly interacting) subsystems (see Appendix A, Condition 7) where For each type of particle in the system, we may write without loss of generality the number-of-particles-of-i-thtype operator associated with the system as [23] where N iJ denotes the number-of-particles-of-i-th-type operator associated with the J-th subsystem.
In general, we assume that the set of linear hermitian operators I, H, G i on H, always including I and H, are the generators of the motion of the composite system. For example, the list of non-Hamiltonian generators G i may coincide with that of the number operators defined in Eq. (67).
For each subsystem J we denote by L(H J ) the space of linear operators on H J equipped with the real scalar product For a given state operator ρ on H, given linear hermitian F and G on H, and each subsystem J, in addition to that already defined in Eqs. (4), (12) and (13), we define the following convenient notation [2,6] where Tr J denotes the partial trace over H J , Tr J over H J , ρ J is the reduced state operator of elementary subsystem J and ρ J that of the composite of all other subsystems.
In view of the special role they play in the equation of motion, we interpret the operators (H) J and (S) J , respectively, as "internal perception" operators representing the way the overall-system Hamiltonian and entropy operators are "felt" locally within the J-th constituent subsystem. For a given ρ J , and given linear hermitian F J and G J on H J , we further define the notation where the operators √ ρ J , B J and S J are obtained by substituting each nonzero eigenvalue p i of ρ J with its positive square root, unity and −k B ln p i . In general, S J = (S) J . The entropy is defined for a subsystem J only if it is not correlated with the other subsystems, i.e., if ρ can be written as then the subsystem entropy is given by the nonlinear state functional of the reduced state operator, and we also have (∆S) J = ∆S J . When subsystem J is correlated, instead, its entropy is not defined; however, the functional may be interpreted as the subsystem's local "internal perception" of the overall-system entropy. Only when J is uncorrelated, The energy is defined for a subsystem J only if it is not interacting with the other subsystems, i.e., if H can be written as then it is given by the mean-value state functional and we also have (∆H) J = ∆H J . When subsystem J is interacting, instead, its energy is not defined, however the functional may be interpreted as the subsystem's local "internal perception" of the overall-system energy. Only when J is non-interacting, e J (ρ) = e(ρ) = e J (ρ J ) + e J (ρ J ). The number-of-particles-of-i-th-type of the overall system in state ρ and of each subsystem J are given by the mean-value state functionals n i (ρ) = Tr(ρN i ) and , respectively, and from Eq. (67) we clearly have n i (ρ) = M J=1 n iJ (ρ J ). It is noteworthy that, for any ρ, For generality, we assume that the generators of the motion on H are I, H and G i , with [G i , H] = 0 but not necessarily with the separated structure of the number operators. Moreover, for each state ρ and each constituent J, we denote by {R iJ , i = 0, 1, 2, . . . , z J } a set of hermitian operators in L(H) such that the operators in the set are linearly independent and span the linear manifold generated by the operators If the latter operators are linearly independent, for subsystem J, then the set {R iJ } may be chosen to coincide with the generators of the motion. If they are not independent, then it could be a smaller subset (in which it is convenient, though not necessary, to maintain R 0J = I and, if possible, R 1J = H). In any case, we call the operators in the set {R iJ } the (instantaneous) "generators of the motion of subsystem J". The structure of the dissipative termD M is invariant under transformation from one set {R iJ } to any other {R ′ iJ } with the same defining properties.
For each instantaneous generator R iJ of subsystem J, we define the local "internal perception" mean functional We denote by the linear span of the operators { √ ρ J (R iJ ) J } or, that is the same, the linear span of the operators is always strictly positive.
The set {R iJ } can be conveniently chosen so that

XI. THE STEEPEST-ENTROPY-ASCENT ANSATZ FOR A COMPOSITE SYSTEM
Maintaining the validity of Eqs. (3) and (18), we now assume as a first step to guarantee the strong separability condition (64) [see Appendix G for a discussion related to the form of Eq. (94)]. The second step is to make sure that E DJ , which in general may be a function of the operators ρ, H and G i on H, reduces to a function of the operators ρ J , H J and G iJ on H J only, whenever the constituent is, at the same time, uncorrelated (ρ = ρ J ⊗ρ J ), non-interacting (H J ⊗I J + I J ⊗H J ) and not coupled through the non-Hamiltonian generators (G iJ ⊗I J +I J ⊗G iJ ).
To preserve the formal analogy with the notation for the indivisible system, we define the local "partial gradient" operators Now, it is easy to show that the structure of dρ/dt assumed with Eqs. (3), (18) and (94) yields [24] dr iJ (ρ) dt = Tr( dρ dt Finally, we assume that each E DJ is: (A) orthogonal to L{ √ ρ J (R iJ ) J }, so that all the rates of change in Eq. (97) and the first term in the rate of entropy change in Eq. (98) are zero; and (B) in the direction of the local partial gradient of the "internally perceived" overall-system entropy functional, s J (ρ), i.e., we take where, similarly to what done for the indivisible system, All the results found for the single constituent extend to the composite system in a straightforward way.
The rate of entropy change The dynamics reduces to the Schrödinger-von Neumann unitary Hamiltonian dynamics when The conjecture about the time-energy Heisenberg uncertainty relation in Section VII can be extended as well by assuming that each E DJ gives rise to a characteristic time τ DJ (ρ) defined by (see Appendix C) which should independently satisfy the uncertainty relation Therefore, using Eqs. (99) and (100), we obtain the lower bounds to the internal relaxation times, or, using, Eqs. (101) and (44), and the upper bound to the entropy production rate Taking strict equality in each Eq. (107) yields the maximum entropy production rate compatible with the uncertainty relations.
The variational formulation can be found, after assuming the structure of Eq. (94), by solving the constrained maximization problem whereṙ iJ and τ 2 DJ are given real functionals of ρ. By virtue of Eqs. (97) and (98), the necessary solving conditions in terms of Lagrange multipliers for each E DJ become clearly verified by the E DJ 's given in Eqs. (99) and (100). If two subsystems A and B are non-interacting but in correlated states, the reduced state operators obey the equations The strong separability conditions (64) are satisfied provided the internal-relaxation-time functionals are either all constants or satisfy the following set of nontrivial conditions.
It is noteworthy that Conditions 113 are satisfied by the maximal-entropy-production-rate ansatz, i.e., if we assume that each τ J (ρ) is given by Eq. (107)  It is interesting, however, that dρ A /dt in general may depend not only on the "local" (reduced) state operator ρ A but also on the overall state ρ through the operators (B ln ρ) J (with J ∈ A), thus determining a collectivebehavior effect on the local dynamics originating from existing correlations. In fact, operators D J are in terms of (∆S) J which may differ, when the subsystems are correlated, from operators (∆S A ) J . In other words, the lack of interactions between two subsystems does guarantee that the energy of each subsystem is conserved [Eqs. (65)] and that the reduced dynamics of each subsystem is local, in the sense of independent of the Hamiltonian operator (and the other generators) of the other subsystem, but does not necessarily imply that each subsystem evolves independently of existing correlations with the other subsystem.
Regarding the time evolution of correlations, in Ref. [2] we defined the correlation functional between two subsystems A and B, which is nonnnegative definite in general, and zero only if ρ = ρ A ⊗ρ B . The rate of change of the correlation can be written as Based on our understanding of the equation of motion, we conjectured thatσ AB | D should always be nonnegative, if it is true that the dissipative term can only destroy correlations between subsystems, but cannot create them. However, this conjecture remains to be proved. The corresponding entropy inequality to my knowledge has not yet been studied.

XII. CONCLUSIONS
I reviewed most previous results in Refs. [1][2][3][4][5][6] on the well-behaved nonlinear equation of motion I proposed for quantum thermodynamics that entails the second law and nonequilibrium steepest-entropy-ascent dynamics with Onsager's reciprocity and Callen's fluctuationdissipation relations. Together with the variational principle formulation derived in Ref. [12] and the observation in Refs. [11,26] that only the functional −k B Tr(ρ ln ρ) can represent the physical entropy, we may conclude that all the results so far confirm that my equation of motion has all the necessary features to provide a self-consistent and conceptually-sound resolution of the century-old dilemma on the nature of entropy and irreversibility, alternative to Boltzmann's statistical approach and valid also for systems with few degrees of freedom.
The nonlinear dynamics encompasses within a unified framework all the successful results of quantum mechanics, equilibrium and nonequilibrium thermodynamics. It also holds the promise to provide a fundamental framework within which to address the currently unexplained evidence on loss of quantum coherence, to design new fundamental experiments, to examine new applications on the lines of those developed in Refs. [5,9,12], and to further investigate the dependence of the internal-relaxation-time functional on the state operator and physical constants, as well as possibly verify the ansatz proposed in this paper by which each indivisible subsystem follows the direction of steepest perceived entropy ascent at the highest rate compatible with the time-energy uncertainty principle.
The equation of motion satisfies the set of conditions discussed in Appendix A and, therefore, preserves most of the traditional conceptual keystones of physical thought, including a strongest form of the non-relativistic principle of causality, by which future states of a strictly isolated system should unfold deterministically from initial states along smooth unique trajectories in state domain defined for all times (future as well as past [25]). Interestingly, while the maximum entropy states are attractors in forward time, the unitary-solutions boundary limit cycles of standard quantum mechanics are attractors in backward time.
As pointed out by Onsager and Machlup [14], the fluctuation-dissipation relations cannot be derived in any rigorous way from the traditional Hamiltonian dynamical principles, unless these are complemented by some additional postulate closely related to the additional principles, assumptions, or approximations needed to derive the Onsager reciprocity relations. This is sometimes referred to as the irreversibility paradox. In other words, in order to infer any feature of irreversibility (including its very existence) from the irreducibly reversible dynamical principle of standard Hamiltonian mechanics, we must complement it with some additional postulate that seems to contradict it.
Within our nonlinear quantum (thermo)dynamics based on Eqs. (1) and (2) the paradox is resolved. The augmented state domain ansatz broadens the set of conceivable states but includes the standard pure states, and the nonlinear equation of motion describes irreversibile time evolutions and entails reciprocity and fluctuationdissipation relations, but maintains the standard unitary dynamics of pure states.
Finally, from the Heisenberg time-energy uncertainty principle I derived a lower bound for the internalrelaxation-time functionals from which follows an upper bound for the rate of entropy production. Consequently, I proposed a physically intriguing maximalentropy-production-rate ansatz, by which each indivisible subsystem follows the direction of steepest perceived entropy ascent at the highest rate compatible with the time-energy uncertainty principle. If this ansatz is experimentally verified, the nonlinear dynamics is complete and self-consistent, with no need of new physical constants.

ACKNOWLEDGEMENTS
Work supported in part by grants from the italian Ministero dell'Istruzione, dell'Università e della Ricerca (MIUR) and the italian Istituto Nazionale di Fisica della Materia (INFM).

APPENDIX A: CRITERIA FOR A GENERAL (NONLINEAR) QUANTUM DYNAMICS COMPATIBLE WITH THERMODYNAMICS
Within a quantum theory that accepts the augmented set of true quantum states described by state operators ρ without the restriction ρ 2 = ρ, and a nonlinear dynamical law for a strictly isolated system, the following demanding set of conditions should be satisfied in order for the theory to be compatible or, better, imply the second law of thermodynamics without contradicting the fundamental results of standard quantum mechanics (QM). Obviuosly these are the criteria I followed in designing Eqs. (1) and (2), and discussed at length in Refs. [3,8].
Conditions 6, 7 and 8 are closely related to the condition recently referred to as strong separability [15]. I also added a condition on correlation and entanglement to reflect the need to avoid, and possibly resolve, physical inconsistencies related to nonlocality issues, as well as a strong causality condition that is nontrivial and quite demanding both from the conceptual and the technical mathematical points of view.
Certainly, when viewed from different perspectivese.g., different physical interpretations of the augmented state domain ρ 2 = ρ ansatz, of the role of the nonlinear extension of the Schrödinger equation of motion, of the Shannon-von Neumann entropy functional −k B Tr(ρ ln ρ) versus other nonextensive functionals, of the role of the system's environment and the measuring apparati, and so on -some authors might view this set of conditions as too strong in many respects. Nevertheless, my equation of motion demonstrates that at least a satisfactory dynamics exists which satisfies all such conditions and, in my view, features a number of intriguing, unifying and far-reaching implications.

Causality. Forward and backward in time
Considering the set P of all linear, hermitian, nonnegative-definite, unit-trace operators ρ on H, every solution of the equation of motion, i.e., every trajectory u(t, ρ) which at time t = 0 passes through state ρ in P, should lie entirely in P for all times t, −∞ < t < +∞.

Standard QM unitary evolution of ρ 2 = ρ states
The unitary time evolution of the states of QM according to the Schrödinger equation of motion must be compatible with the more general dynamical law. These trajectories, passing through any state ρ such that ρ 2 = ρ and entirely contained in the state domain of Quantum Mechanics, must be solutions also of the more general dynamical law. In view of the fact that the states of QM are the extreme points of the augmented state domain, the trajectories of QM must be boundary solutions (limit cycles) of the dynamical law.
If the complete dynamics preserves the feature of uniqueness of solutions throughout the augmented state domain, then pure states can only evolve according to the Schrödinger equation of motion. In this case, no trajectory can enter or leave the state domain of QM and by continuity, there must be trajectories that approach indefinitely these boundary solutions (of course, this can only happen backward in time, as t → −∞).

Conservation of energy and number of particles
If the system is isolated, the value of the energy functional e(ρ) = Tr(ρH), where H is the standard QM Hamiltonian operator, must remain invariant along every trajectory. If the isolated system consists of a variable amount of a single type of particle with a number operator N that commutes with H, then also the value of the number-of-particle functional n(ρ) = Tr(ρN ) must remain invariant along every trajectory. If the isolated system consists of r types of particles each with variable amount and each with a number operator N i that commutes with the Hamiltonian H, then also the value of each number-of-particle functional n i (ρ) = Tr(ρN i ) must remain invariant along every trajectory. Depending on the type of system, there may be other time-invariant functionals.

Stability of the thermodynamic equilibrium states. Second law
A state operator ρ represents an equilibrium state if dρ/dt = 0 when the system is isolated [e.g., H = H(t)]. For each given set of feasible values of the energy functional e(ρ) and the number-of-particle functionals n i (ρ) (i.e., the functionals that must remain invariant according to Condition 2 above), among all the equilibrium states that the dynamical law may admit there must be one and only one which is globally stable (definition and discussion in Appendix B). This stable equilibrium state must represent the corresponding state of equilibrium thermodynamics and, therefore, must be of the form given by Eq. (29). All the other equilibrium states that the dynamical law may admit must not be globally stable.

Entropy nondecrease. Irreversibility
The principle of nondecrease of entropy must be satisfied, i.e., the rate of change of the entropy functional −k B Tr(ρ ln ρ) must be nonnegative along every trajectory. Two distinguishable subsystems A and B are in independent states if the state operator ρ = ρ A ⊗ρ B , so that the entropy operator For permanently noninteracting subsystems, every trajectory passing through a state in which the subsystems are in independent states must maintain the subsystems in independent states along the entire trajectory. When two uncorrelated systems do not interact with each other, each must evolve in time independently of the other.
In addition, if at some instant of time two subsystems A and B, not necessarily non-interacting, are in independent states, then the instantaneous rates of change of the subsystem's entropy functionals −k B Tr(ρ A ln ρ A ) and −k B Tr(ρ B ln ρ B ) must both be nondecreasing in time.

Correlations, entanglement and locality. Strong separability
Two non-interacting subsystems A and B initially in correlated states (possibly due to a previous interaction that has then been turned off) should each proceed in time towards less correlated states or, at least, maintain the same level of quantum entanglement. The generation of quantum entanglement between interacting subsystems should emerge only through the Schrödinger-von Neumann term −i[H, ρ]/h of the equation of motion, whereas the other terms, that might entail loss of correlations between subsystems, must not be able to create them. This condition is perhaps too strong and its validity for my equation is still only conjectural. In any case, the dynamics should not generate locality problems, i.e., faster-than-light communication between noninteracting subsystems, even if in entangled or correlated states. In other words, when subsystem A is not interacting with subsystem B, it should never be possible to influence the local observables of A by acting only on the interactions within B, such as switching on and off parameters or measurement devices within B. This does not mean that existing correlations between A and B established by past interactions should have no influence whatsoever on the time evolution of the local observables of either A or B. In particular, I see no physical reason to request that two different states ρ and ρ ′ such that ρ ′ A = ρ A should evolve in such a way that dρ ′ A /dt = dρ A /dt whenever A is isolated (but not uncorrelated) from the rest of the overall system. For example, state ρ ′ could be the maximum entropy stable equilibrium state (and, therefore, ρ ′ = ρ ′ A ⊗ρ ′ B , dρ ′ /dt = 0) whereas in state ρ subsystems A and B could be correlated and evolving in time towards the stable equilibrium state or ρ could even be a pure entangled state evolving along a unitary trajectory according to Condition 2 above, and therefore it would never reach stable equilibrium.

APPENDIX B: LYAPUNOV STABILITY AND THERMODYNAMIC STABILITY
The condition concerning the stability of the thermodynamic equilibrium states is extremely restrictive and requires further discussion.
In order to implement Condition 5 in Appendix A, we need to establish the relation between the notion of stability implied by the second law of thermodynamics [20,21] and the mathematical concept of stability. An equilibrium state is stable, in the sense required by the second law, if it can be altered to a different state only by interactions that leave net effects in the state of the enviromment. We call this notion of stability global stability. The notion of stability according to Lyapunov is called local stability.
We denote the trajectories generated by the dynamical law on our state domain by u(t, ρ), i.e., u(t, ρ) denotes the state at time t along the trajectory that at time t = 0 passes through state ρ. A state ρ e is an equilibrium state if and only if u(t, ρ e ) = ρ e for all times t. An equilibrium state ρ e is locally stable (according to Lyapunov) if and only if for every ǫ > 0 there is a δ(ǫ) > 0 such that d(ρ, ρ e ) < δ(ǫ) implies d(u(t, ρ), ρ e ) < ǫ for all t > 0 and every ρ, i.e., such that every trajectory that passes within the distance δ(ǫ) from state ρ e proceeds in time without ever exceeding the distance ǫ from ρ e . Conversely, an equilibrium state ρ e is unstable if and only if it is not locally stable, i.e., there is an ǫ > 0 such that for every δ > 0 there is a trajectory passing within distance δ from ρ e and reaching at some later time farther than the distance ǫ from ρ e .
The Lyapunov concept of instability of equilibrium is clearly equivalent to that of instability stated in thermodynamics according to which an equilibrium state is unstable if, upon experiencing a minute and short lived influence by some system in the environment (i.e., just enough to take it from state ρ e to a neighbouring state at infinitesimal distance δ), proceeds from then on spontaneously to a sequence of entirely different states (i.e., farther than some finite distance ǫ).
It follows that the concept of stability in thermodynamics implies that of Lyapunov local stability. However, it is stronger because it also excludes the concept of metastability. Namely, the states of equilibrium thermodynamics are global stable equilibrium states in the sense that not only they are locally stable but they cannot be altered to entirely different states even by means of interactions which leave temporary but finite effects in the environment. Mathematically, the concept of metastability can be defined as follows. An equilibrium state ρ e is metastable if and only if it is locally stable but there is an η > 0 and an ǫ > 0 such that for every δ > 0 there is a trajectory u(t, ρ) passing at t = 0 between distance η and η + δ from ρ e , η < d(u(0, ρ), ρ e ) < η + δ, and reaching at some later time t > 0 a distance farther than η + ǫ, d(u(t, ρ), ρ e ) ≥ η + ǫ. Thus, the concept of global stability implied by the second law is as follows. An equilibrium state ρ e is globally stable if for every η > 0 and every ǫ > 0 there is a δ(ǫ, η) > 0 such that every trajectory u(t, ρ) with η < d(u(0, ρ), ρ e ) < η + δ(ǫ, η), i.e., passing at time t = 0 between distance η and η + δ from ρ e , remains within d(u(t, ρ), ρ e ) < η + ǫ for every t > 0, i.e., proceeds in time without ever exceeding the distance η + ǫ.
The second law requires that for each set of values of the invariants Tr(ρH) and Tr(ρG i ) (as many as required by the structure of the system), and of the parameters embedded in the Hilbert space H and the Hamiltonian H describing the external forces (such as the size of a container), there is one and only one globally stable equilibrium state. Thus, the dynamical law may admit many equilibrium states that all share the same values of the invariants and the parameters, but among all these only one is globally stable, i.e., all the other equilibrium states are either unstable or metastable.
Interestingly, we may use this condition to show that a unitary (Hamiltonian) dynamical law would be inconsistent with the second-law stability requirement. A unitary dynamical law in the augmented kinematics would be expressed by an equation of motion ihρ = [H, ρ] with trajectories u(t, ρ) = U (t)ρU −1 (t) with U (t) = exp(−itH/h). Such a dynamical law would admit as equilibrium states all the states ρ e such that ρ e H = Hρ e . Of these states there are more than just one for each set of values of the invariants. With respect to the metric d(ρ 1 , ρ 2 ) = Tr|ρ 1 − ρ 2 |, it is easy to show that every trajectory u(t, ρ) would be equidistant from any given equilibrium state ρ e , i.e., d(u(t, ρ), ρ e ) = d(u(0, ρ), ρ e ) for all t and all ρ. Therefore, all the equilibrium states would be globally stable and there would be more than just one for each set of values of the invariants, thus violating the second-law requirement.
The entropy functional −k B Tr(ρ ln ρ) plays a useful role in proving the stability of the states of equilibrium thermodynamics [Eq. (29)] provided that the dynamical law guarantees that −k B Tr[u(t, ρ) ln u(t, ρ)] ≥ −k B Tr(ρ ln ρ) for every trajectory, i.e., provided Condition 6 above is satisfied. The proof of this is nontrivial and is given in Ref. [20] where, however, we also show that the entropy functional is not a Lyapunov function, even if, in a strict sense that depends on the continuity and the conditional stability of the states of equilibrium thermodynamics, it does provide a criterion for the stability of these states. Anyway, even if the entropy were a Lyapunov function, this would suffice only to guarantee the local stability of the states of equilibrium thermodynamics but not to guarantee, as required by the second law, the instability or metastability of all the other equilibrium states.
APPENDIX C: CHARACTERISTIC TIMES Using Eq. (3), the rate of change of the mean functional f (ρ) = Tr(ρF ) may be written as For the Schrödinger-von Neumann evolution, the characteristic time of change of f (ρ) may be defined as [ For this reason, in Eq. (14) we take τ H equal to the lower bound of the τ F H 's. By analogy, but considering the understanding of the contribution of each subsystem to the overall-system dynamics embedded in our nonlinear dynamics, we define the characteristic time of the dissipative change of the mean functional f (ρ) = Tr(ρF ) due to the J-th constituent subsystem as , it follows that the characteristic times τ F DJ are bounded by the value attained for an operator F such that √ ρ J (∆F ) J is in the same direction as E DJ , i.e., such that Therefore, (C7) For this reason, in Eqs. (50) and (105) we take τ D and τ DJ equal to the respective lower bounds of the τ F DJ 's. where the dependences of f j , X j and τ on ρ are evidenced in order to emphasize the nonlinearity.

APPENDIX E: EXTENSION TO TIME-VARYING RATES OF THE GENERATORS OF THE MOTION
For the purposes of quantum thermodynamics, in my view the dissipative part of the equation of motion should not account for rates of change of the mean values of the generators of the motion other than through the Hamiltonian term, consistently with all the results of standard QM.
Nevertheless, the mathematical extension of Eqs. (1) and (2) to (artificially) imposed ratesṙ i = 0 (arbitrarily specified as functions of time) is straightforward and may be useful in applications or other frameworks [7].
It does emerge naturally from the maximization problem (56); it does so, however, implicitly, through substitution of Eq. (58) back into the constraint Equations (56b). Instead, the explicit form in terms of projections and, therefore, the equivalent expressions by means of Gram determinants amount to assuming, for the operator D in the equation of motion, instead of Eq. (2c) or the equivalent Eqs. (20), where the explicit expression of the j-th term in the summation is given by Eq. (20b) where we interchange everywhere √ ρ ln ρ with √ ρR j and Geometrically, the additional terms are in the steepestr j -ascent direction compatible with maintaining constant the mean values of the other generators and the entropy.
Analogous obvious extension to the composite system case can be obtained by assuming for the operator D J , instead of Eq. (1), with the α J j 's such thaṫ

APPENDIX F: EXTENSION TO OTHER ENTROPY OR MEAN VALUE FUNCTIONALS
For the purposes of quantum thermodynamics, in my view the necessary entropy functional is −k B Tr(ρ ln ρ), for the reasons in Refs. [2,3,6,8,11,26].
However, in view of the recent literature on nonextensive quantum theories, as suggested also in Ref. [12], it may be useful to note that the entire formalism of my equation of motion can be readily reformulated in the case of any other well-behaved entropy functional s(ρ) and set of nonlinear generator functionals r i (ρ).
For the single constituent system it suffices to substitute throughout the operator (−2k B √ ρI+) √ ρS = (−2k B √ ρ) − 2k B √ ρ ln ρ (notice that in the Gram determinants the addenda in parantheses cancel out) with the new entropy gradient operator, ∂s(ρ)/∂ √ ρ, and the operators 2 √ ρR i with [operators which when symmetrized ({A, A † }/2) are equal to] the gradient operators of the new generator functionals, ∂r i (ρ)/∂ √ ρ. However, for a composite system, consistently with the nonextensivity of these theories, it may be difficult or not at all possible to identify the operators corresponding to (S) J and (R i ) J representing the subsystems' local perceptions of the entropy and the generators.

APPENDIX G: A NOTEWORTHY EQUATION FOR A COMPOSITE SYSTEM THAT FAILS TO MEET A SEPARABILITY CONDITION
It is interesting to note that the role of the √ ρ J operators in Eq. (1) [and that of √ ρ in Eq. (2)] is formally useful but only auxiliary, because wherever there is a √ ρ J , another √ ρ J comes in front or behind it. In fact, in my doctoral thesis [1] the equation of motion was in terms of ρ J only. I realized the usefulness of the √ ρ J formalism only later, in connection with the proof of the steepest-entropy-ascent geometric property [3].
By allowing a more relevant role of the √ ρ operator, an alternative to the construction in Eq. (94) may appear formally better, and suggest the apparently alternative equation of motion based on the following definitions In fact, it can be readily verified that for a given However, the resulting dynamics would fail to satisfy at least the important property expressed by Eq. (65), because it can be verified that if H = H J ⊗I J + I J ⊗H J but ρ = ρ J ⊗ρ J then separate conservation of the noninteracting subsystem's energy would not be guaranteed.