The classical-quantum limit

The standard notion of a classical limit, represented schematically by $\hbar\rightarrow 0$, provides a method for approximating a quantum system by a classical one. In this work we explain why the standard classical limit fails when applied to subsystems, and show how one may resolve this by explicitly modelling the decoherence of a subsystem by its environment. Denoting the decoherence time $\tau$, we demonstrate that a double scaling limit in which $\hbar \rightarrow 0$ and $\tau \rightarrow 0$ such that the ratio $E_f =\hbar /\tau$ remains fixed leads to an irreversible open-system evolution with well-defined classical and quantum subsystems. The main technical result is showing that, for arbitrary Hamiltonians, the generators of partial versions of the Wigner, Husimi and Glauber-Sudarshan quasiprobability distributions may all be mapped in the above double scaling limit to the same completely-positive classical-quantum generator. This provides a regime in which one can study effective and consistent classical-quantum dynamics.


Introduction
The classical limit describes the emergence of classical physics from quantum theory.Typically justified in a variety of ways, the most famous of these is to consider action large compared to the reduced Planck's constant, .This leads to the ubiquitous statement that the classical limit is taking → 0. As well as explaining the success of classical mechanics for the description of macroscopic systems, the classical limit provides an important theoretical tool for simplifying the analysis of quantum systems that are too complex to study directly.
The classical limit allows one to replace a quantum system with an entirely classical description.However, many systems of interest operate in a regime where both classical and quantum features are important, and this leads to the following question: Can you take a limit of a quantum system such that one subsystem behaves classically, while the rest remains quantum?A limit of this kind would have a wide variety of applications, from providing first principle derivations of quantum control and measurement set-ups, to formalising approaches in quantum chemistry, where the nuclear degrees of freedom are treated as classical, and the electronic degrees of freedom are treated quantum mechanically [1][2][3].Beyond this, it would be interesting if recently proposed models of classicalquantum theories of gravity [4][5][6][7] could arise as effective descriptions of quantum gravity.
In this work, we tackle the problem of taking such a classical limit.Since this involves mapping two quantum subsystems to a quantum subsystem and an effective classical subsystem, we call this a "quantumquantum to classical-quantum" limit, or classical-quantum limit for short.Such a limit could also be referred to as a semi-classical limit, since the resulting effective theory contains both classical and quantum degrees of freedom, or a classical limit for subsystems.
Two important requirements to make on any classical-quantum limit of quantum theory is that it be physically motivated and consistent.Although the standard → 0 classical limit is often well-motivated physically, we shall see that as a classical-quantum limit it fails to be consistent, in that it fails to describe an effective classical subsystem.In this case, the resulting dynamics is known as the quantum-classical Liouville equation [3,8,9], and does not lead to well-defined classical evolution on the subsystem in question [10].The first attempt at a consistent limit procedure was made in the pioneering work by Diósi [11] who considered two particles, each having a different Planck's constant.An example of consistent dynamics was then derived for a constant back-reaction.The limiting procedure allowed him to derive one of the first examples of consistent classical-quantum dynamics, but the master equation would not arise from physical considerations.
In the present work we demonstrate that a physically motivated and consistent limit procedure exists, starting from standard unitary quantum mechanics in a closed system.The key observation we make is the following: such closed system dynamics will always generate entanglement between subsystems, and thus always lead to a breakdown of classicality, independently of any parameter such as that is usually used to quantify classicality.This means that the standard notions of a classical limit, such as → 0, must be supplemented by an additional mechanism that removes the entanglement generated between subsystems.
In our framework, the classicality of a subsystem is guaranteed by decoherence due to its external environment.Already well understood to play an important role in the quantum-to-classical transition [12,13], the coupling to the environment in our framework leads to an associated decoherence timescale τ of the subsystem in question.By taking τ → 0, one can ensure that this subsystem is classical at all times.The key conceptual takeaway from this work is that a double scaling limit, in which → 0 while τ → 0 such that the ratio E f = /τ is fixed, provides a version of a classical limit that may be consistently applied to subsystems i.e. a classical-quantum limit.
The main technical result of this work is computing the explicit form of the dynamics in this double scaling limit, for arbitrary bipartite Hamiltonians for which a classical limit is possible, which is given in Equation 18.In order to prove the consistency of this dynamics, we show that the dynamics is a special case of the recently characterised completely-positive form [4,14], which guarantees that the effective classical subsystem is well-defined.Equation 18 thus provides a regime in which the general form of continuous dynamics introduced in [14] could arise as an effective theory.The resulting dynamics is generically an irreversible open-system dynamics, with decoherence and diffusion controlled by the parameter E f .The complete-positivity ensures the classical-quantum dynamics may be directly unravelled in terms of continuous classical trajectories in phase space and quantum trajectories in Hilbert space [15], which are given in Equations ( 29) and (30).
Alongside the main results, we find a number of related results: • A partial version of the Glauber-Sudarshan quasiprobability distribution is introduced, and identified as the correct representation to require positivity of for effective classical-quantum dynamics in the Hilbert space.
• The dynamics of partial versions of the Glauber-Sudarshan and Husimi quasiprobability distribution are explicitly computed to O( 0 ).
• The classical-quantum limit is shown to lead to dynamics independent on the choice of partial Q, P or W representation.
• The double scaling limit applied to a single system is shown to give a stochastic classical limit, of the kind described by [16].
• A simplified form of dynamics is found in Equation 33 for classical-quantum Hamiltonians that selfcommute, which takes the form of O( 0 ) partial Glauber-Sudarshan dynamics with the minimal additional decoherence and diffusion for complete-positivity.
• The explicit form of dynamics for the classical-quantum limit of two quantum harmonic oscillators is computed.
• The double scaling limit on a single system is shown to recover the standard → 0 limit in the low diffusion limit E f → 0.
• Two distinct behaviours of the effective classical-quantum dynamics are found in the E f → 0 limit, namely a quantum Zeno type behaviour, and a coherent quantum control limit.
The results in this paper establish that the classical limit of a subsystem has a far richer structure than the classical limit of a single system.We also provide technical tools for the study of effective classical-quantum theories, which may be useful for categorising the large body of existing proposals for constructing hybrid theories [17][18][19][20][21][22][23][24].
The structure of the paper is as follows: in Section 2 we introduce the Wigner and partial Wigner representations, and demonstrate using the latter how the → 0 limit is insufficient in providing a classical limit of a subsystem.In Section 3 we introduce a discrete time, decoherence channel model of an environment, and show how this leads to well-defined stochastic evolution in a double scaling limit.In Section 4 we present our main result, which is the derivation of the general form of classical-quantum dynamics under a bipartite Hamiltonian, under this double scaling limit.In Section 5 we introduce two other partial quasiprobability representations, the partial Glauber-Sudarshan and partial Husimi representations.These are used to illustrate two technical notions useful for characterising effective classical-quantum dynamics, which we use to determine that the positivity of the partial Glauber-Sudarshan distribution is a sufficient and necessary measure of the effective classicality of a subsystem.In Section 6 we study the main form of dynamics in the three different quasiprobability distributions introduced, and show the equivalence between them.In Section 7 the main results are unravelled in terms of stochastic trajectories in phase space and Hilbert space.Finally, in Sections 8 and 9, special cases of the general form of dynamics are given, in particular the self-commuting classical-quantum Hamiltonian case, and the low diffusion E f → 0 limit.

2
The standard → 0 limit To motivate the need for an alternate notion of a classical limit, we first begin by looking at where the standard classical limit succeeds and fails as a technique for deriving classical equations of motion.To do so, we will look at the simplest model of a quantum system with a classical limit, i.e. a single quantum system characterised by a canonical commutation relation with parameter .However, the results that follow may be reinterpreted in the standard way for general order parameters controlling the degree of classicality, such as coupling strength g or number of systems N [25].
Consider a single quantum system, that we denote C, with Hilbert space H C and trace-one, positive semi-definite density operators ρ that form a set of states S(H C ).We will take this quantum system to be characterised by the canonical commutation relation [q, p] = i , and interpret the operators q and p as the position and momentum of the system.This system will have an associated Hamiltonian Ĥ which generates the free, unitary evolution of the C system in the absence of any interactions with other systems.
A typical method of studying the classical limit of such a system is via the Wigner representation, which provides an alternate description of quantum mechanics in terms of functions of phase space [26][27][28].Defining the operators Âq,p = 1 π dy e −ipy/ |q − 1 2 y q + 1 2 y|, where |q denotes the eigenstates of the position operator q, one may map operators acting on H C to functions of phase space M by taking the trace with respect to Âq,p i.e.Ô → tr[ Âq,p Ô].The most important example is the Wigner function W (q, p), the phase space representation of the quantum state By the properties of Âq,p and ρ, it follows that W (q, p) is real-valued and is normalised when integrated over phase space i.e.W (q, p) dqdp = 1.Unlike a probability distribution, it is not guaranteed to be non-negative for all q, p in phase space, and hence is termed a quasiprobability distribution.To study how unitary dynamics are represented in the Wigner representation, one must also consider the Wigner representation of the Hamiltonian Ĥ, given by the real-valued H W (q, p) = tr[ Âq,p Ĥ].The free unitary dynamics in the Wigner representation is then given by the Moyal bracket where here the star product of two phase space functions f = f (q, p) and g = g(q, p) is given to be interpreted in terms of the series expansion of the exponential, with the arrows denoting whether each derivative acts on the function on the left or the right.
The Wigner representation is an entirely equivalent description of quantum mechanics, and does not a priori have anything to do with classical dynamics.However, by considering the dynamics to lowest order in , one arrives at an equation familiar from classical mechanics.Specifically, to lowest order in , the dynamics (2) takes the form where { • , • } denotes the Poisson bracket, and H denotes the classical Hamiltonian i.e. the O( 0 ) part of H W [25].This equation is the Liouville equation, and describes how classical probability distributions evolve under Hamiltonian flow.This leads to the statement that → 0 gives the classical limit of a quantum system.Of course, it is not actually meaningful to send a dimensionful quantity like to zero, although it is often a convenient shortcut in practice.The statement that classical equations of motion are recovered in the → 0 limit is more precisely understood as the statement that for a given W (q, p), the higher order derivatives terms containing in the expansion are negligible compared to the Liouville equation terms given above.
Even if one did not already know the form of the Liouville equation, one is still led to the → 0 limit by considering when the dynamics preserves the classicality of initial states.For a quantum state ρ of the C system to be viewed as effectively classical, it is necessary for the corresponding Wigner function to be positive i.e.W (q, p) ≥ 0 ∀q, p ∈ M Correspondingly, for the dynamics of the C system to be effectively classical, it must also be positive i.e. preserve the positivity of all normalised functions of phase space.As should be expected, the general quantum dynamics of equation ( 2) is not positive, except in the cases that the Hamiltonian is at most quadratic in q and p.To see this, one may appeal to the Pawula theorem, which characterises the general form of linear, trace-preserving and positive dynamics for real-valued functions of phase space [29].The Pawula theorem states that unless the dynamics contains an infinite number of higher order derivatives with respect to q and p, any positive dynamics must be of the Fokker-Planck form, with at most second order derivatives in q and p (see Appendix for more details).Since the series expansion of (2) typically truncates at a finite number of terms (i.e. for Hamiltonians polynomial in position and momentum), the dynamics in such cases cannot be positive.
Considered in this way, the → 0 limit may be understood as a method of enforcing positivity preservation on the quantum dynamics of a single system when represented in phase space.In particular, since the higher order derivative terms in equation ( 2) responsible for violating positivity also are higher order in , by truncating the expansion to lowest order in the dynamics reduces to a dynamics that maps initial probability distributions to final probability distributions, and hence preserves the classicality of initial states.
However, let us now consider the same approach when the C system is just a subsystem of a larger quantum system.Denoting the other subsystem Q, we again denote states of the joint system as ρ ∈ S(H Q ⊗ H C ).Here again we take the closed system unitary evolution to be governed by an arbitrary Hamiltonian Ĥ, which may include both self-Hamiltonians and an interaction Hamiltonian between the C and Q subsystems.We now wish to study whether the above procedure results in a well-defined classical-quantum limit -a limit in which the C subsystem may be treated classically, while the generic Q system is still described using standard quantum mechanics.For notational convenience in what follows, we will reserve the use of hats for operators with support on the C system Hilbert space H C ; operators acting on H Q alone will be left without.
To adapt the standard classical limit procedure to the case where C is a subsystem, one may use a partial Wigner representation [3,10].This provides an equivalent representation of quantum mechanics in which one part of the system is described in terms of a phase space, while the other part remains described by operators in Hilbert space.Specifically, we map operators that act on H Q ⊗ H C to phasespace dependent operators on H Q by taking the partial trace with respect to Âq,p i.e.Ô → tr C [ Âq,p Ô].The only difference from the Wigner representation is that the trace is performed over the C subsystem alone, leaving an operator valued function of phase space.In this representation, the bipartite quantum state ρ is represented by the partial Wigner distribution ̺ W (q, p), which is an operator-valued function of the phase space associated to the C system, given by By the properties of Âq,p and ρ, it follows that ̺ W (q, p) is a Hermitian-valued operator and is normalised when integrated over phase space and traced over Hilbert space i.e. tr̺ W (q, p) dqdp = 1.Analogously to how the real-valued Wigner function is not guaranteed to be positive, the Hermitian operator-valued function ̺ W (q, p) is not guaranteed to be positive semi-definite for all points in phase space.To study the unitary closed system dynamics of the bipartite quantum system in this representation, one may consider the partial Wigner representation of the Hamiltonian Ĥ, given by the Hermitian operator-valued function of phase space H W (q, p) = tr C [ Âq,p Ĥ].The closed system unitary dynamics then takes the form which is analogous to (2) except for the fact that here the quantities are operators that act on H Q .This dynamics will appear frequently in what follows, and it is thus convenient to define the associated generator L W i.e. the generator of closed system evolution under the Hamiltonian Ĥ in the partial Wigner representation where here we use • to denote the input to the generator.
If the argument was to follow as before, then taking → 0 of the closed system unitary dynamics in the partial Wigner representation would lead to one system becoming effectively classical, while the other remaining quantum.Considering equation (8) to O(1) in , the resulting equation is known as the quantum-classical Liouville equation [3,8,9].Here H is the O( 0 ) part of H W and we will refer to this as the classical-quantum Hamiltonian.The first term takes the form of a Liouville von-Neumann term, that describes unitary evolution of density operators.The second term, sometimes referred to as the Alexandrov-Gerasimenko bracket, is a version of the Poisson bracket that is symmetric in the ordering of the phase space dependent operators H and ̺ W .As before, this form of dynamics will appear repeatedly, and it will be useful to define the corresponding generator which is simply the generator L W of (8) truncated to O(1) in .
However, there is a key difference between the Liouville and quantum-classical Liouville dynamics: while the Liouville equation preserves the classicality of the C system, the quantum-classical Liouville equation does not.To see this, note that for a quantum state ρ on the bipartite Hilbert space H C ⊗ H Q to be effectively classical on the C subsystem, it is necessary for the corresponding operator-valued partial Wigner distibution to be positive semi-definite at all points in phase space i.e Taking (11) as a necessary condition for effective classicality of the C subsystem guarantees that the state may be written as a positive probability distribution over phase space multiplied by a corresponding quantum state on H Q at each point, and is the natural generalisation of (5) to operator-valued functions.For a dynamics to preserve the classicality of the C subsystem, it therefore must be the completely-positive on all initial operator-valued functions of phase space.However, while the Liouville equation preserves the positivity of real-valued functions, the quantum-classical Liouville equation does not preserve the positivity of operator-valued functions of phase space [10].This may be seen by appealing to the recently proved analogue of the Pawula theorem for operator-valued functions [14] (see also [30] for a later discussion of this result).Known as the CQ Pawula theorem, it showed that every trace-preserving, normalised and completely-positive Markovian dynamics on operator-valued functions of phase space is also separated into two classes, with one class truncating at second order in derivatives in phase space, and the other containing an infinite number of higher derivative terms (see Appendix A).
Since the full dynamics of ( 8) typically truncates at a finite number of derivative terms, the → 0 limit helps to bring the resulting form of equations closer to a completely-positive form, by removing all derivative terms second order and higher.However, as we show in Appendix A, even with these higher order derivatives removed, the resulting dynamics are still not of the required form for complete-positivity.
The problem ultimately lies in the fact that while → 0 suppresses non-classicality arising from the higher order derivatives in q and p, it has no effect on suppressing the entanglement that is generated between the C and Q quantum subsystems.Since entanglement may be generated for even linear coupling between subsystems, the quantum-classical Liouville equation ( 9) must also generically describe entanglement build up between the C and Q subsystems, and thus the generation of states that are not effectively classical on the C subsystem.
Before moving on to see how one may resolve this, we should first address a technical detail regarding the kinds of Hamiltonian that we consider.Up to this point, we have implicitly assumed that H W , referring to either the Wigner or partial Wigner representation of the Hamiltonian Ĥ, may be written as . This assumption holds when Ĥ is a function of q and p, and in such typical cases, the classical or classical-quantum Hamiltonian H coincides with the function of phase space obtained by making the substitutions q → q, p → p.In general however, the Hamiltonian Ĥ may also depend explicitly on , and in these cases there is no guarantee that H W = H + O( 2 ).However, if H W contains any terms of O( −1 ) or higher inverse powers of , there is no well-defined classical limit as → 0 [25], and one may check that the dynamics truncated to O( 0 ) in such cases are not positive.The only remaining case is thus where H W contains O( ) terms.We consider this in Appendix C, and find that it amounts to only a minor modification of the dynamics when . For conceptual clarity we therefore present the following analysis under the assumption that H W = H + O( 2 ) -but this, up to a known modification, describes all possible Hamiltonians which permit a classical limit.
3 Decoherence timescale τ and a double scaling limit The preceding section introduced the formalism of the Wigner and partial Wigner representations, and showed how the standard → 0 limit is insufficient to describe a classical limit of a subsystem due to the presence of entanglement.In this section, we introduce a simple model of the effect of an environment on the C subsystem, and show how this leads one to a double scaling limit involving the decoherence timescale τ of this subsystem.
We begin by noting that it is well-understood that → 0 is not sufficient to ensure classicality, even in single systems.The key observation is that when is small, but finite, the evolution generated by the Liouville equation will generally map an initial state W (q, p, t i ) in which the higher order terms are negligible to a state at later times W (q, p, t f ) in which they are not [12].The resolution to this problem was to acknowledge that in practice, all quantum systems are open systems, and thus interact with their environments.In this case, the interaction with an environment leads to dispersion in the system, preventing any later states of the Wigner quasiprobability distribution W (q, p, t f ) from becoming overly peaked in phase space and thus preventing the higher order terms contributing, an analysis that has been put on more rigorous footing in recent work [31,32].More generally, acknowledging the role of the environment, which generically acts to decohere the system, has turned out to be an extremely successful way of explaining a number of features in the quantum-to-classical transition [13].
In what follows, we shall follow the above philosophy by modelling the effect of the environment on the subsystem that will be classicalised.The basic idea is that the interactions with an environment will lead to decoherence on the C subsystem that can break the entanglement with the Q subsystem.In other words, the decoherence induced by an environment will act to replace the quantum correlations between the C and Q subsystems with purely classical correlations, which will ensure that the resulting ̺ W is positive.
In order to include the effect of an environment, without overly increasing the complexity of the analysis, we will assume that the effective action of the environment is to collapse the C subsystem into a classically definite state.The classically definite states will be taken to be coherent states, which are the states with minimum uncertainty in q and p. Allowing them to have some squeezing, such that the ratio of the variances in position and momentum is given by s 2 = ∆q ∆p , we will denote the coherent state with expectation values q = q and p = p as |α s (q, p) [33,34].The environment is then modelled as performing a coherent state POVM with measurement operators M s q,p = (2π ) −1/2 |α s (q, p) α s (q, p)|.Assuming for now that the observer has no access to the environmental degrees of freedom, the effect of the environment is a decoherence channel ρ → dqdp M s q,p ρ M s q,p .In the partial Wigner representation this amounts to a convolution of ̺ W with a normalised Gaussian with variance s 2 in q and s −2 in p.Such a convolution is known as a Weierstrass transform, and has the following representation as a differential operator This differential operator D provides a representation of the decoherence action of an environment, and will prove extremely useful.
Although we have specified the action of the environment as collapsing the states of the system to coherent states, we have not specified over which timescale.To do so, we will specify explicitly that the environment collapses the state of the system over a time τ .This timescale τ is to be understood to be the decoherence timescale of the C subsystem i.e. the time over which the interaction with the environment has decohered the C subsystem to being in a classically definite state.
The joint specification of the map D and associated timescale τ leads to something akin to a trotterised picture of dynamics, in which the effect of the environment is modelled by a decoherence channel that acts at discrete time intervals τ .For now leaving aside the unitary dynamics generated by Ĥ, this explicitly means that the total dynamics in the partial Wigner representation is given by the application of the differential operator D at times 0, τ, 2τ, . . .and so on, with no evolution in between.Although different from the standard continuous time dynamics of typical open systems treatments [35], the advantage of this discrete time approach is that after each action of the decoherence channel, the state is in a guaranteed classical state.
In order to arrive at a well defined continuum limit of this discrete time model of decoherence, one would wish to take the decoherence timescale to zero i.e. τ → 0. Since the environment acts to select classical states on the C subsystem, taking this limit ensures that the subsystem is in a classical state at all times.However, taking this limit while remains finite would lead to the state of the system becoming infinitely spread in phase space in an infinitesimally short time.To prevent this from occurring, we observe that simultaneously taking the limit that → 0 allows, in principle, for an infinite series of convolutions to still give a finite effect on the resulting distribution.To see which rates it may be sensible to take and τ to zero, one may consider t/τ environmental decoherence steps, which gives the overall effect of the environment G t after finite time t as When the ratio /τ depends on either or τ , the overall effect of the environment at finite times will either diverge or vanish.Remarkably however, one can see that when the ratio /τ is fixed, the differential operator G t corresponds exactly to the semi-group corresponding to a classical diffusion process, with diffusion in q given by s 2 /τ and diffusion in p given by s −2 /τ .Despite starting from a strong measurement at each step in time, the resulting equations of motion describe continuous evolution in time and phase space.Moreover, one can see that no other dependence of /τ will give a well-defined, non-trivial limit.This motivates the following double-scaling limit as a candidate method of taking the classical-quantum limit: where we have chosen E f to denote the constant with dimensions of energy that describes the fixed ratio of the two.As before, the taking of dimensionful quantities to zero should be more carefully interpreted as statements about the relevant scales in the system.Here we may interpret the above double-scaling limit as the statement that the action associated to any observables of interest on the C subsystem are large compared to the scale of , and change over much longer timescales than the decoherence time τ of the C subsystem.The ratio of the reduced Planck's constant and the decoherence time give a measure of the size of the fluctuations in the system due to the environment, which is captured by the constant E f .
In the context of single systems, the first discussion of a double scaling limit relating to classical limits and decoherence appears to be in the conclusion of [16], which describes a double scaling limit of and a measurement rate of a continuous measurement procedure that leads to diffusive classical evolution, leading to a notion of a stochastic classical limit.While conceptually similar, it is important to emphasise a technical difference that is important when moving to the full classical-quantum limit: the double scaling limit of ( 14) describes a continuous time limit of a series of strong measurements, rather than weak measurements as is usually considered in continuous measurement set-ups [36].A similar theoretical set-up in the context of the quantum Zeno effect was explored in [37].More recently, a related model, albeit with a different double scaling limit, was considered in the context of holography [38].

Main results
In this section, we use the discrete time model of decoherence and associated double scaling limit of the previous section to arrive at the general form of dynamics when one takes the classical-quantum limit we have introduced.This main result is the given in equation (18).Since some of the technical steps are rather long, we reproduce here only the key conceptual points, and refer the reader to Appendix B for more details.
In section 3, the effect of the environment was considered in isolation.However, the key question of interest is to consider how the environment and the free evolution of a generic quantum system interplay in the double-scaling limit we have arrived at.To study this, we must consider the total evolution after a time τ , which should include both the environmental decoherence effects given by D and the free evolution generated in the partial Wigner representation by L W .The obvious question then arises of which ordering to choose of the two processes.This point will become clearer in section 6 where alternative partial quasiprobability representations are considered, but it sufficient at this time to simply consider a symmetrised total evolution, in which the action of the environment is divided equally between one part before the free evolution generated by the Hamiltonian, and one part afterwards The total evolution operator E τ describes the action of both the environment and the free evolution on the partial Wigner representation of the bipartite quantum system CQ, and we use the superscript and subscript to indicate the functional dependence on both τ and .
The evolution map E τ describes the total change in the partial Wigner representation over a finite decoherence time τ with a finite value of .In order to take the classical-quantum limit described in (14) one must consider the infinitesimal evolution in τ generated when τ and are taken to zero such that = E f τ .To do so, we first set = E f τ in E τ , and consider the generator of the evolution map in the τ → 0 limit, which takes the form The first term is the standard form of the generator often used to formally construct time-local dynamics [39,40].We shall see that this part of the generator captures a large proportion of the dynamics, and importantly the back-reaction of the quantum system on the classical one.However, by construction, this part of the generator only captures the τ -dependent part of the dynamics.In fact, one can check that there is an additional τ -independent component E 0 , generated by As discussed in Appendix B, this term may be accounted for by reintroducing = E f τ , and computing the generator of this component, corresponding to the second term in (16).Since this term only effects the quantum system, the reappearance of is to be expected: while → 0 should be interpreted as the assumption that the relevant classical observables are much larger than , no assumption is made on the scale of relevant quantum observables.From this point on, any appearance of should be interpreted as characterising the quantum features of the Q subsystem.
Computing the above generator explicitly, we arrive at where here the ad denotes the adjoint operation with respect to the generators of the classical-quantum dynamics i.e. (ad A B)̺ = (AB − BA)̺.The complex structure of this generator owes itself to the fact that the generators of the exponential maps that make up E τ do not commute with themselves for all τ .This means that when the derivative in ( 16) is computed one must take care in using the correct definition of their derivatives with respect to τ , just as one must do when computing derivatives of exponentials of matrices, as is commonly considered in the derivation of the Baker-Campbell-Hausdorff formula [41].
To reduce the complexity of the generator above, one may explicitly compute the adjoint action in the various series above, which allows one to map the expressions involving the adjoint action of classicalquantum generators (i.e.ad −i ), to expressions involving the adjoints of quantum operators (i.e.ad −i E f H ). Upon doing so, one arrives at the following form of dynamics where and Here C nm denote numerical coefficients given by which we show in Appendix B are related to the binomial coefficients.
To give some intuition about the dynamics, we sketch the role of each line as follows.The top line describes purely unitary evolution of the quantum system, governed by both the classical-quantum Hamiltonian H and an effective Hamiltonian H ef f that depends on s and E f .This additional Hamiltonian term arises due to the fluctuations induced by the environment [42], and is analogous to the Lamb and Stark shifts that renormalise the bare system Hamiltonian in standard open systems treatments [35].The second line describes both the free classical evolution and the back-reaction of the quantum system upon it, and we shall see that in Section 8 that this reduces to the symmetrised Poisson bracket appearing in (8) for a special class of classical-quantum Hamiltonians.The third line describes how random fluctuations in the classical degrees of freedom are correlated with random fluctuations in the unitary dynamics of the quantum system i.e. noisy Hamiltonian quantum dynamics.The fourth and fifth lines describe the Lindblad portion of the dynamics, which acts to decohere the quantum system into a basis determined by the Lindblad operators L H q and L H p .Finally, the final line describes the previously described diffusion in the classical degrees of freedom, with overall strength proportional to E f and relative strengths in position and momentum determined by the parameter s.
To understand whether the evolution laws given by the above generator are consistent, it is important to check that the dynamics are linear, trace-preserving, and completely-positive on a suitable set of operatorvalued functions of phase space.While this seems likely a priori, given that the generator above was derived from free evolution and environmental decoherence in a full quantum theory, it is often the case in the study of open quantum systems that approximations lead to violations of one or more of these conditions [35].In order to check this, we note that the simplified form of the dynamics given in ( 18) is of the canonical classical-quantum form of dynamics, first written in general form by [43] (see also [30] for a later discussion of this).Any dynamics of this form is linear and trace-preserving, and these properties are straightforward to directly check by hand.In order to check the positivity of a dynamics of this form, one must check a series of positivity conditions given by the classical-quantum Pawula theorem [43].The first step is to pick a basis of operators, and phase space degrees of freedom, in which to read off certain decoherence, back-reaction and diffusion matrices.Picking the basis (q, p), (L H q , L H p ) for convenience, one may refer to the general form given in Appendix B to see that that the decoherence D 0 , back-reaction D 1 and diffusion D 2 are given by The most basic requirements for positivity in the classical-quantum Pawula theorem are the same as those of the Lindblad and Fokker-Planck equations.For the quantum degrees of freedom, these are the requirements that for all points in phase space the total Hamiltonian H + H ef f is Hermitian and the decoherence matrix D 0 is positive semi-definite.For the classical degrees of freedom, it is that the real matrix D 2 is positive semi-definite for all points in phase space.One may check that these properties do indeed hold, with The key result of the CQ Pawula theorem, is that the remaining conditions sufficient for complete-positivity of a classical-quantum dynamics are that , where I denotes the identity matrix of the dimension of the classical degrees of freedom, and D −1 2 denotes the pseudoinverse of D 2 .The first condition ensures that any classical degree of freedom that experiences quantum back-reaction has noise in it, and this holds here since D 2 is full-rank.The second condition, known as the decoherence-diffusion trade-off [44], ensures that decoherence in the quantum system is sufficiently large to to be compatible with the rate of information gain about it by the classical system.One may explicitly check this condition with the above matrices and see that the decoherence-diffusion trade-off is satisfied, and in fact is saturated as Thus, analogously to the standard classical limit of the Wigner distribution, the classical-quantum limit presented here arrives at a dynamics that is positive on all initial operator-valued functions of phase space.

Effective classical-quantum states and subset positivity
The analysis of both the insufficiency of the → 0 limit and the apparent success of the double scaling limit have thus far has been presented using the partial Wigner quasiprobability distribution ̺ W .However, the positive semi-definiteness of ̺ W is only a necessary condition for the classicality of a subsystem.In this section, we introduce a general notion of partial quasiprobability representations, and show that the positivity of a partial Glauber-Sudarshan quasiprobability distribution ̺ P provides both necessary and sufficient conditions for the C subsystem to be effectively classical.The considerations in this section and the next do not change the main result of equation (18), and those interested in understanding this general form of classical-quantum dynamics better may instead choose to go straight to section 7 or 8.
We start by defining in general terms the notion of a partial quasiprobability distribution.Recall that a more general treatment of measurements is that of POVMs { Êi }, where Êi denote the POVM elements.A partial quasiprobability representation R is the assignment to every state ρ and every set of POVM elements Êi the operator-valued functions ̺ R and E R i acting on H Q , in such a way that Here the trace on the left-hand side is over the C and Q subsystem Hilbert spaces, while the trace on the right-hand side is just over H Q .By definition, every measurement may be represented in this way, and thus the partial quasiprobability representation provides an entirely equivalent description of bipartite quantum mechanics.The partial Wigner representation described in Section 2 provides an example of this.Note that here the same map is applied to both states ρ and POVM elements Êi to generate the representation, but in general the states and observables are treated differently.
To identify when a given set of bipartite quantum states { ρλ } and measurements {{ Êi }, { Fi }, . ..} may be described using an effectively classical subsystem, it is necessary to study the positivity of their representations.This was first demonstrated in [45], where the criterion for whether a given set of quantum states and measurements could be modelled classically was identified as when the representations of both the states and POVM elements were non-negative real-valued functions of phase space.To generalise to the case of an effectively classical subsystem, we will say that a set of states and POVMs admit an effective classical-quantum description whenever there exists a representation R in which ̺ R and E R i are positive semi-definite for all z in phase space, by direct analogue with the purely classical case.As in the case of defining an effective classical description of a quantum system [45,46], only a restricted set of all measurements and states in quantum theory permit an effective classical-quantum description.
For a restricted set of measurements, many quantum states may admit an effective classical-quantum description of the combined set of measurements and states.However, a special class of states are those which may be modelled using a classical-quantum description for all possible bipartite measurements on the system.Translated to the technical language above, we will call a bipartite density operator ρ an effective classical-quantum state whenever there exists a representation where the corresponding partial quasiprobability distribution is positive semi-definite ̺ R 0 and the representation of all POVMs are positive semi-definite.This provides an operationally relevant definition of states with an effective classical subsystem, since it means that regardless of the form of measurement performed on the joint bipartite quantum system, the statistics are reproducible via an underlying classical-quantum (or partially non-contextual) model [45].
A second notion that we will introduce is that of subset-positivity.In Section 2, the notion of positivity of dynamics was introduced, and used to argue for the validity of the Liouville equation as classical equation of motion, and against the quantum-classical Liouville equation as having describing a genuinely classical subsystem.The key property is that the positivity of the dynamics was considered on the set of all positive semi-definite operator-valued functions, that we will denote S.However, there also exist dynamics which although they do not preserve the positivity of all initial real or operator-valued functions of phase-space states, do preserve positivity of on a subset of initial conditions.For a given subset of all positive semidefinite functions Λ ⊂ S, we state that a dynamics is Λ-positive if it is positive for all initial states belonging to that subset.Since subset-positive dynamics need not positive on all initial states, it also need not be characterised by the Pawula [29] or CQ Pawula [14] theorems.
To illustrate these two notions, we may consider two important examples of partial quasiprobability representations, derived from the well-known Q and P representations from quantum optics [47][48][49].In particular, we may define the partial Husimi distribution ̺ Q explicitly via and the partial Glauber-Sudarshan distribution ̺ P implicitly by ρ = ̺ P (q, p) ⊗ |α s (q, p) α s (q, p)|dqdp.
Like ̺ W , both ̺ Q and ̺ P are normalised to 1 when traced over Hilbert space and intergrated over phase space, and are useful for illustrating different properties of a given bipartite quantum state ρ.The partial Husimi ̺ Q is an operationally relevant quantity, giving the actual probabilities and corresponding quantum states on Q of a coherent state POVM with measurement operators Mq,p on the C subsystem, and is consequently positive semi-definite for all q, p.A consequence of the non-orthogonality of the coherent states is that the set of all operator-valued functions ̺ Q form a strict subset H ⊂ S of positive operator-valued functions, in particular not including those with uncertainty in position and momentum less than the Heisenberg bound [19,27,33].By contrast, the partial Glauber-Sudarshan ̺ P is not necessarily positive semi-definite at all points in phase space [34], but when it is, one may see from its definition that the bipartite quantum state is separable between the classical and quantum subsystems i.e. contains no entanglement [50].
Aside from guaranteeing that there is no entanglement between the C and Q subsystems, the positive semi-definiteness of the partial Glauber-Sudarshan representation turns out to provide sufficient and necessary conditions for the underlying bipartite quantum state to be an effective classical-quantum state.To see this, we may substitute the definition of ̺ P given in equation ( 25) into equation (23) to see that the representation of POVM elements in the partial P representation are in fact given by the partial Q representation, and thus are always positive semi-definite.By the definition given above, if the partial Glauber-Sudarshan representation ̺ P for a bipartite state ρ is positive, this state must therefore be an effective classical-quantum state.The positivity of ̺ P therefore provides sufficient and necessary conditions for the underlying bipartite quantum system to have an effectively classical C subsystem.Since the partial Wigner ̺ W is related to ̺ P by a convolution (see Appendix D), any positive semidefinite ̺ P necessarily implies that ̺ W is also positive, justifying the original claim that ̺ W 0 is a neccessary condition for an effective classical subsystem.
For the other notion introduced, we note that unitary dynamics in the partial Husimi representation provides an example of subset-positive dynamics.Since the partial Husimi representation is always positive, the unitary dynamics in Hilbert space induces a positive dynamics on partial Husimi distributions [19].However, this map is not positive on all initial states, but instead is H-positive.While this subset-positive dynamics has many interesting features, the positivity should not be conflated with the interpretation as having an effectively classical subsystem, for the reason that all dynamics, even those that generate large amounts of entanglement, may be represented in this way.

Equivalence between partial quasiprobability representations
In this section, we shed some light onto why the dynamics of equation ( 18) is completely-positive on all operator valued functions of phase space, and on the original choice of operator ordering in the definition of E τ , by studying the dynamics of the partial Husimi ̺ Q and partial Glauber-Sudarshan ̺ P distributions introduced in the previous section.In doing so, we demonstrate that the classical-quantum limit we have arrived at preserves the effective classicality of the C subsystem.
To study the total dynamics of the partial Glauber-Sudarshan and partial Husimi distributions in the classical-quantum limit, we first note that the decoherence channel used to model the environment in these representations is identical to that of the partial Wigner distribution, and so may be modelled as before as D. To study the unitary dynamics generated by the Hamiltonian in each representation, we show in Appendix D how one may find the generators of the partial Husimi L Q and the partial Glauber-Sudarshan L P by mapping first to the Wigner distribution by the differential operator D ∓ 1 2 , using the free Wigner evolution, and then mapping back using the inverse D ± 1 2 , for ̺ Q and ̺ P respectively.Considering the corresponding generators to O(1) in , we find the following generator of partial Husimi evolution which was first written down in [19], though without the final term, and the following generator of partial Glauber-Sudarshan evolution ) Using these, one may then construct the generator of evolution E τ as in (15) and take the double-scaling limit as described previously to find the generator of the dynamics.However, in order to derive the same evolution map, and thus the same generator, one can check that one must choose different operator orderings depending on the representation!In particular, one can see from the above argument using D ± 1 2 to map between representations, that three distinct operator orderings of the free evolution and the environmental decoherence steps lead to the same evolution map: The key observation to understand the difference in operator ordering in each case is to note that the environment plays a different role in each partial quasiprobability representation in order to maintain classicality.As discussed in section 5, the unitary dynamics in the partial Husimi representation are only positive on initial states with sufficient spread in phase space.Consequently, in this representation the decohering action of the environment must be taken before the unitary evolution, such that any arbitrarily peaked states in phase space are first convoluted before they are evolved.Conversely, in the partial Glauber-Sudarshan representation, the state ̺ P is only positive when all entanglement has been removed; in this case the environment acts after the unitary evolution to ensure any entanglement built up by the unitary evolution is destroyed at the end of each step.Since the partial Wigner representation ̺ W lies exactly half-way between ̺ Q and ̺ P by Wierstrass transform (see Appendix D for more details), the original symmetrised dynamics postulated in ( 15) turns out to be exactly that which performs both steps in half-measure.In all of these cases, the map that is defined is completely-positive on all positive semi-definite operator-valued functions.
The above analysis also guarantees that the dynamics of equation ( 18) preserves the effective classicality of the C subsystem.As discussed in Section 5, the positivity of the partial Glauber-Sudarshan probability distribution provides sufficient and necessary conditions for the quantum state of the bipartite system to be an effective classical-quantum state.Thus, by here explicitly showing that the dynamics of ̺ P are also positive, we guarantee that the C subsystem may be treated as effectively classical in the double scaling limit.This may be equivalently argued using the fact that the map between the different representations becomes the identity in the limit that → 0, and thus that ̺ W coincides with ̺ P in the classical limit.
For the same reason, ensuring that the dynamics in the three representations all agree, as it does above, provides an important consistency check on the validity of any classical-quantum dynamics arising from a classical limit.

Classical-quantum trajectories
We assumed up to this point that the observer has no access to the environmental degrees of freedom that store the information about the C subsystem.However, one could assume that the observer has sufficient information about the environment to reconstruct the outcome of the effective coherent state POVM that it induces at each time step [28,51,52].In this case, the observer has access to the classical system's trajectory in phase space, and their best estimate of the quantum state deduced from the motion of the classical system leads to a quantum trajectory in Hilbert space.
The general form of classical-quantum trajectories, corresponding to the unravellings of continuous classical-quantum master equations, was first given in [15] (see also [30] for a later discussion of these points).A key result of this work was that when the trade-off is saturated in the form D 0 = D † 1 D −1 2 D 1 , any initially pure state of the quantum system remains pure conditioned on the classical trajectory.Since this is the case here, one may use the general form of unravellings to write down the coupled evolution of the effective classical and quantum system in this classical-quantum limit.Defining a column vector Z t = (q t , p t ) T for the classical degrees of freedom, and the operator-valued column vector L = (L H q , L H p ) T , the stochastic dynamics takes the form Here σ denotes any 2 × 2 matrix such that σσ T = D 2 , and dW t = (dW 1 t , dW 1 t ) T denotes a column vector of two uncorrelated Wiener processes.The form of equations make clear that the semi-classical limit we present here does not lead to any loss of quantum information, provided an observer has access to the full classical trajectory [15].Since this originates from a full quantum theory, we see that in principle the irreversibility introduced by tracing out an environment may be partially recovered in the classical limit.One may also use these equations as a starting point for simulating the semi-classical theory we present here, and we refer the reader to [15] for some basic examples of the simulation of classical-quantum trajectories.

Two special cases of dynamics
The general form of generator, given in equation 18 and their corresponding unravellings in ( 29) and ( 30) are the main results from this work, describing the general form of dynamics for a bipartite Hamiltonian Ĥ in the double-scaling classical limit on one subsystem.To gain some more insight into what this dynamics predicts, we will consider now two special cases.
The first case we will consider is the effect of the double-scaled classical limit on a single system.To study this, one can take a bipartite quantum Hamiltonian of the form Ĥ = ( p2 2m + V (q)) ⊗ I i.e. a Hamiltonian with trivial action on the Q subsystem.The corresponding classical-quantum Hamiltonian may be computed to be H = p 2 2m + V (q), and defines the operators Using these definitions in the general dynamics (18) one finds that the unitary, Lindbladian, and mixed derivative-commutator terms all vanish, and the mixed derivative-anticommutator terms combine to give the Poisson bracket.This gives the following stochastic dynamics on the classical system This example shows that the idea that that the limit is specific to subsystems is not neccesary -rather the double scaling limit we find provides a general notion of a "stochastic classical limit", that happens to also give consistent evolution when it is applied to subsystems alone.Although the existence of stochastic classical limits are somewhat of a folk wisdom in physics, the earliest concrete proposal we have found in the literature is a discussion in [16].
A second limiting case of the above dynamics is to consider the dynamics under the approximation for z = (q, p).This is true exactly when H(q, p) is self-commuting i.e. when [H(z), H(z ′ )] = 0 for all z, z ′ in phase space, and has an error of O( n ) when the classical-quantum Hamiltonian takes the form H = p 2 2mC + p2 2mQ + V (q − q).Making this approximation, we find that L H q = ∂H ∂q , L H p = ∂H ∂p and The dynamics in (18) then reduces in form to the following where we have defined the Lindblad operator L = sL H q + is −1 L H p such that the decoherence part of the dynamics is diagonalised.The first line gives the unitary evolution and Alexandrov bracket from the quantum-classical Liouville equation (9).However, the second line, formed from H ef f and the mixed derivative-commutator terms, contain exactly the additional terms associated to the dynamics of the partial Glauber-Sudarshan representation i.e. the first two lines give L P | O( 0 ) , previously found in (27).We thus see that the total dynamics is exactly the dynamics of the partial Glauber-Sudarshan representation to lowest order in , with additional terms corresponding to noise in the classical and quantum systems.Since the approximation made above occurs at the level of the operators, the complete positivity of the dynamics is unchanged, and thus may still be unravelled by using the simplified forms of the operators L H q , L H p and H ef f in equations ( 29) and (30).The majority of work in the literature on completely-positive classical-quantum dynamics, including earlier work by the present authors, concluded that the natural form of dynamics would take the form of the quantum-classical Liouville equation with minimal additional noise terms to ensure positivity [4,11,15].However, as the above example shows, when derived in a physical manner from a full quantum theory, a more natural model is instead the O(1) partial Glauber-Sudarshan dynamics of ( 27) supplemented with the minimal terms necessary for positivity.This result seems particularly reasonable when one considers that it is the positivity in the partial Glauber-Sudarshan distribution, and not the partial Wigner distribution, that guarantees the classicality of the C subsystem, as discussed in Section 6.
9 The E f → 0 limit The double scaling limit we have presented leads generically to irreversible dynamics, with the parameter characterising the diffusion in the classical system given by E f .A question we now turn to is whether one may recover a deterministic evolution, as in the standard classical limit, by tuning this free parameter.
The first example to look at is the result of the double scaling classical limit on a single system.The dynamics in this case was computed earlier in equation (31), taking the form of Hamiltonian dynamics with additional diffusion in both position and momentum proportional to E f .In the limit E f → 0, one thus recovers the Liouville equation (4), i.e. deterministic evolution under the classical Hamiltonian.This additional E f → 0 limit may be physically interpreted as saying that if one considers large macroscopic scales, any noise due to the environment is negligible, and thus reversible Hamiltonian dynamics is recovered.Since the Liouville equation ( 4) was previously obtained directly from the standard → 0 limit, we see that when applied to single systems, the stochastic notion of a classical limit that we have presented reduces to the standard notion in the E f → 0 limit.
Given that the E f → 0 limit recovers a deterministic classical limit on a single system, it is interesting to consider whether the same may be true when one considers the classical limit of a subsystem.To explore this question, we will first consider the limiting case described in equation (33) for self-commuting classical-quantum Hamiltonians.In this case, the parameter appears in two places: proportional to the the strength of classical diffusion, and inversely proportional to the strength of the decoherence on the quantum system.One thus sees that while taking E f to be small reduces the amount of classical diffusion, it leads to very large decoherence on the quantum degrees of freedom in a basis determined by the Lindblad operator L. In the limit E f → 0, decoherence acts to instantaneously select an eigenstate of the operator L, and then freeze the quantum system in this eigenstate, via the quantum zeno effect [37,53,54].In doing so, the quantum system is essentially classicalised, with any superpositions being supressed by the strong decoherence.Since the backreaction on the classical system is determined by the eigenvalues of the operator L, the classical system then undergoes deterministic evolution with drift given by the eigenstate that the quantum system is frozen in.Such a dynamics may be understood to be reversible on a subset of initial quantum states that are eigenstates of the Lindblad operator L, but in general is highly non-deterministic, with a generic initial quantum state being rapidly decohered by the interaction.
The above example illustrates that in the E f → 0 limit, dynamics arising from classical-quantum Hamiltonians that are self-commuting exhibit rapid decoherence in the quantum system.It turns out however that this is not a generic feature of dynamics in the E f → 0 limit, which we may illustrate with the following example.
A classical-quantum limit of two quantum harmonic oscillators.Consider a system of two interacting quantum particles in one dimension, with the Q subsystem characterised by the canonical commutation relation [Q, P ] = i and the C subsystem as usual by [q, p] = i .The system will be taken to have free evolution given by the bipartite quantum Hamiltonian Ĥ = p2 /2m C + P 2 /2m Q + λ(q − Q) 2 .Taking the semi-classical limit of the C subsystem gives a classical-quantum Hamiltonian For this model, one may compute the Lindblad and effective Hamiltonian operators of equations ( 19) and ( 20) exactly, exploiting the fact that the adjoint action closes under the set of linear operators in I, Q, P to obtain These explicit forms of Lindblad and effective Hamiltonian operators may be used in the master equation (18) or the unravelling equations ( 29) and (30) to study the dynamics for arbitrary E f .Remarkably however, we see that in the limit E f → 0, the non-trivial Lindblad operator L H q and effective Hamiltonian H ef f both vanish.Moreover, the product of the Lindblad operators vanish faster than rate at which the decoherence strength increases.In other words, in the E f → 0 limit, we find that the harmonic oscillator dynamics reduces to unitary dynamics on the quantum system under the classical-quantum Hamiltonian H, and the classical system experiences no back-reaction: In this limit, the strong monitoring by an environment on the C subsystem thus acts to effectively remove the backreaction of the quantum system on the classical one, leaving simply coherent control of the quantum system by the classical one, despite the strength of interaction remaining fixed.This effect is reminiscent of dynamical decoupling, where the application of unitary pulses on a quantum system may reduce the interaction with an external environment [55].Incidentally, one can check that the → 0 limit of the operators defined in equations ( 34) and ( 35) are still well-defined, and reduces to the form of dynamics given in (33); the apparent difference in limiting behaviour as E f becomes small is due to the non-commutativity of the two limits E f → 0 and → 0. Note also that the small mass limit m Q → 0 seems to reproduce the results of an earlier work on classical-quantum limits in closed systems [22].
The two examples above show that in the low diffusion limit, E f → 0, the classical-quantum dynamics we find can exhibit two very different behaviours; one in which the quantum system rapidly decoheres, and affects the classical system, the other in which it evolves with unitary evolution, and has no backreaction on the classical system.In the regime that a classical system appears to evolve without diffusion, it thus appears to be the case that any degrees of freedom that are affecting the evolution of the system must be rapidly decohered and effectively classical, or do not influence the dynamics of the classical system, and undergo unitary evolution depending on the classical state of the system.It would be interesting to study how generic the later case is, and indeed whether there exist other behaviours aside from these two.

Discussion
The main results, given in master equation form in (18) or stochastic unravelling form in ( 29) and ( 30), provide a physical derivation of consistent effective classical-quantum dynamics from a full quantum theory.A special limiting case of this, given in equation ( 33), provides a form of dynamics close to the quantum-classical Liouville equation that may be directly unravelled in classical trajectories in phase space and quantum trajectories in Hilbert space.Beyond the coupled quantum harmonic oscillators example given, understanding the kinds of dynamics obtained via this classical-quantum limit in further models, from optics to condensed matter theory, would be of great interest.With the form of Lindblad and effective Hamiltonian operators computed, the average and statistical properties of such systems may be numerically simulated using the stochastic unravellings of ( 29) and (30).
Another important research direction is to understand in greater detail the conditions under which the above dynamics are a good approximation to a full quantum dynamics.While the work in this paper demonstrated that a classical-quantum limit gives a rich dynamical structure, the analysis was the analogue of the steps leading to the Liouville equation.Understanding whether the various approaches characterising the conditions under which classical dynamics arise [12,31,32,56] can be generalised to the more complex case of a classical-quantum limit is an important open question.
The methods presented here rely on the assumption that the environment may be modelled in a particularly simple way, as a series of discrete time decoherence channels on the subsystem that is classicalised.It would be interesting to understand whether the results we obtain here may also be derived directly from continuous time models of an environment.Moreover, the effect of the environment in this proposed classical-limit procedure is particularly simplistic, characterised only by the total strength of phase space diffusion E f and a parameter s quantifying the relative strength of diffusion in position and momentum.
In real systems, the environment may induce a large number of additional effects on the dynamics such as friction, and in such cases we expect the corresponding classical-quantum dynamics to be modified to reflect this.For this reason, the classical-quantum limit presented here is likely to be one of many, and understanding the landscape of effective classical-quantum dynamics is of interest.
In this regard, it would be useful to understand the effect of relaxing our demand that the state always be effectively classical-quantum state and the dynamics be Markovian.There are many physical situations where a system may have an effective classical-quantum description for almost all times, but for short time-scales, it may not.In this regard, the notion of almost always classical-quantum, or approximately classical-quantum, are likely to be important concepts.This is partly motivated by attempts to understand the regimes in which the consistent classical-quantum dynamics of [4,14] provides an effective theory in which to describe evolution laws in the classical-quantum limit.

A Pawula and CQ Pawula theorems
For convenience, we reproduce the two theorems relevant for characterising positivity of dynamics in classical limits, the Pawula theorem [29] and the CQ Pawula theorem [43], as well as explaining how the Liouville equation ( 4), quantum-classical Liouville equation ( 9), and classical-quantum generator (18) satisfy (or not) the various forms.Pawula (1957) The general form of Markovian, linear, trace-preserving and positive dynamics is either of Fokker-Planck form or it contains an infinite number of higher order derivative terms in phase space.The i, j, . . .indices run from 1 to n, the number of phase space degrees of freedom z i , and there is summation of repeated indices.Here, D 1,i are the elements of a real vector of length n, D 1 , and D 2,ij are the elements of a real positive semi-definite n × n matrix D 2 .All of the D coefficients are allowed to have dependence on phase space.
CQ Pawula (2023) The general form of Markovian, linear, trace-preserving and completely-positive dynamics is either of the form where or it contains an infinite number of higher order derivative terms in phase space.Here, the i, j, . . .indices run from 1 to n, the number of phase space degrees of freedom z i , while the α, β, . . .When the Lindblad operators are not chosen traceless and orthogonal, the above conditions on the dynamics can be shown to still be sufficient for complete-positivity, even when dependent on phase space.In this case, the role of classical drift vector D C 1 is essentially played by the component of the L α proportional to the identity.

A.1 Liouville equation
The Liouville equation ( 4) satisfies the Fokker-Planck form given by (A.1) for where H is the classical Hamiltonian.

A.2 Quantum-classical Liouville equation
The quantum-classical Liouville equation, when written in the form of (A.2) with phase space dependent Lindblad operators, has where H is the classical-quantum Hamiltonian.Since D 2 and D 0 are zero everywhere, but D 1 is not, the positivity conditions (A.3) are not satisfied, and thus the dynamics is not completely-positive.

A.3 Classical-quantum dynamics of L
By the same reasoning as above, one may read from (18) the three matrices D 0 , D 1 and D 2 given in (22) by taking the Lindblad operators L H q and L H p .The remaining degrees of freedom are H = (H + H ef f )/ and D C 1 = (0, 0) T .

B Derivation of the generator L
To compute L given by equation ( 16) we first write out the evolution map explicitly as The most important part of this to notice immediately is that the first term in the middle exponential has no τ dependence -this part is ultimately responsible for most of the subsequent structure of this generator.To see why this leads to the second term in L, additional to the usual one, note that After N = t/τ evolution steps, the total contribution of this part of the dynamics is and thus is generated by the unitary term − i [H, • ] if we restore = E f τ .Although in principle the unitary steps occur in between steps generated by the τ -dependent part of the generator, any changes to the generator from these are of O(τ ), and thus vanish in the τ → 0 limit, meaning that the resulting dynamics is captured by the generator To compute the main part of the generator L, we take the derivative of E τ to give where here D = ∂p 2 .This gives the first component of the generator L as where the O(τ ) terms disappear in the τ → 0 limit and we have used the fact that lim τ →0 e ± E f 2 Dτ is the identity operator.Noting that we may use the following equality between the exponential of the adjoint and the adjoint of the exponential e ad B A = e B Ae −B (B.6) we find the quoted form of the generator in equation (17).
To compute the form of the generator given in ( 18) is a little more work.Denoting the following term T 1 where we introduce an aribtrary CQ state ̺ to make the action of this generator explicit, one may use the equality between the exponential of the adjoint and the adjoint of the exponential as in equation (B.6) to rewrite this as 1 2 e and then use it again, noting that H )e One may then compute this expression explicitly, taking care to note that whenever a derivative is made of the exponential of a z = q, p dependent operator, that and Using these formulae, one may show that 1 2 e which gives the overall generator T 1 as The other component of L that remains to be computed we will denote T 2 and is given Since the fraction e ad −1 ad is to be interpreted as describing a power series, and using the symmetry of second derivatives of H to rewrite the Alexandrov-bracket as the derivatives of anticommutators, we may rewrite this generator more explicitly as To compute this infinite series, we will first need to find the commutation relations of the algebra generated by − i E f [H, •], as one would do for the case for a Lie algebra of a Lie group -for some related work in the purely quantum case, see [57].To simplify this subsequent analysis, we will use a shorthand L(A, • , B) to denote a generic component of a Lindblad decoherence generator i.e.
One may then compute the commutation relations between − i E f [H, •] and the various terms that appear: (a) with the derivative of an anticommuator and (c) with a unitary generator Since the above generators are closed under the repeated action of ad −i E f [H, • ] , these relations are sufficient to compute the above series.
To actually compute the series, we will consider each kind of generator (a)-(c) separately.Starting with (a), the derivative of an anticommutator, we note that the adjoint action of the commutator with H is equivalent to the adjoint action with H on the operator in question.This gives the first part of T 2 as where we have again used the series expansion of e ad −1 ad .
To compute T (b) 2 , the part corresponding to the Lindblad terms, we first write down the form of the O(E −n f ) order term, which is given

21)
We will now show by induction that this is true for all n ≥ 1.When n = 1, all the Lindblad terms come from the application of (B.17) on the Alexandrov-bracket, which one can check agrees with the above expression (being careful to include the factor of 1/(1 + 1)! coming from (B.15)).For an arbitrary term of order n + 1, it follows from (B.17)-(B.19) that all terms must either come from the application of 1 n+2 ad −i E f [H, • ] to the expression for the previous nth order term, or applied to the nth order term of (B.20).This gives the (n + 1)th order term in total as Considering first the Lindblad terms with one entry ad n and the other ad 0 , we see that the numerical prefactors of these terms are given  26) which noting that L is linear each of its arguments can be simplified to The final component of T 2 to compute is the unitary part, which we will keep track of by defining an associated Hamiltonian H qp+pq via T  gives the form quoted in (18).To put this in canonical form, we note that ad [A, • ] [B, • ] = [ad A B, • ], which follows from the Jacobi identity, and thus using the series expansion of e x −1 x and resumming we find Considering an H W with O( ) terms thus simply leads to an additional unitary term, and does not affect the resulting complete-positivity of the dynamics.

D Relating states and dynamics in the partial quasiprobability representations
A well known property of the three common quasiprobability distributions is that they may be related via convolution.Specifically, the Wigner distribution W may be obtained from the Glauber-Sudarshan P distribution by a convolution with a Gaussian with variance 1 2 s 2 in q and 1 2 s −2 in p, and in turn the Husimi Q representation may be obtained from the Wigner representation by the same convolution [28,33,34].These relations are unchanged by when one considers instead the partial quasiprobability representations ̺ W , ̺ P , ̺ Q , and so using the differential operator representation of the convolution we may write them as ̺ W (q, p) = D 1 2 ̺ P (q, p) ̺ Q (q, p) = D For the different representations to be all equivalent, the mapping between the quasiprobability distributions must be bijective, and thus the convolutions must be invertible.While this is not possible for general functions on phase space, in this case it is possible on the restricted domain formed by the sets of all possible partial Husimi and Wigner distributions [27].In terms of the differential operator D, these inverse maps may be written in terms of the differential operator which gives ̺ P (q, p) = D − 1 2 ̺ W (q, p) ̺ W (q, p) = D − 1 2 ̺ Q (q, p) (D.4) Having specified the maps between states in the three representations, one may construct the dynamics in any representation from another by mapping the state, evolving in that representation, and then mapping back to the original representation.In particular, using the form of the generator in the partial Wigner representation L W , given in (8), one may construct generators for L Q and L P , which take the form (D.8) as given in (26).Similarly, one may compute the same for the partial Glauber-Sudarshan dynamics, which differs only by a minus sign, giving (D.9) as in (27).
indices run from 1 to p, the number of traceless and orthogonal Lindblad operators L α in Hilbert space.We assume summation over repeated indices of either kind.The various D coefficients are organised as follows: D αβ 0 are the elements of an p × p complex positive semi-definite matrix D 0 , D α 1,i are the elements of a complex n × p matrix D 1 , which has conjugate transpose D † 1 , while D α 1,i * denotes the complex conjugate of D α 1,i .Additionally, D C 1,i the elements of a real vector of length n, D C 1 , and D 2,ij are the elements of a real positive semi-definite n × n matrix D 2 , which has the generalised inverse D −1 2 .Finally, H is Hermitian operator.All the D coefficients and H may have arbitrary dependence on z.
1)! , (B.23) with the term on the left hand side coming from k = n − 1 or 0 terms, and the right hand side coming from the bottom two lines.Analogously, for a generic Lindblad term with one entry ad m and the other ad n−m for 0 < m < n we have two terms coming from the sum over k, which indeed is the expression (B.21) with n → n + 1.Since this expression is only the nth order term, we may write T b as the sum over all these terms T (b) 2

(c) 2 =
−i[H qp+pq , • ].From (B.17)-(B.19) it is apparent that any contributions to H qp+pq are generated by the action of ad −i E f [H, • ] on derivatives of anticommutator terms, given by (B.20), and then the subsequent action of ad −i E f [H, • ] on the unitary terms generated by these.The numerical factor coming from the repeated action in (B.15) may be kept track of by simply noting that the O(E −n f ) terms have a factor 1/(n + 1)!.This lets us write down the Hamiltonian H qp+pq as

C 2 (
Including O( ) contributions in the classical-quantum HamiltonianIf instead of assumingH W = H + O( 2 ) we had assumed H W = H + H 1 + O( 2 ), we would find that the equations of motion for the Liouville equation are unchanged, but that there is a change in the quantum-classical Liouville equation.Specifically, the O( 0 ) part of the partial Wigner generator now takes the formL W = − i [H, • ] − i[H 1 , • ] + 1 {H, • } − { • , H}) + O( ).(C.1)Following the same steps as before in computing the generator, the only change is found at the level of the T 2 component given in (B.14), which now has the additional termT •] − i[H 1 , • ] .(C.2)

2 ∂p 2
L W , (D.6) where we have used the relation e adB A = e B Ae −B .To compute the generators to O(1) in , one can use the definition of the exponential of the adjoint, and the generator of L W , and expand in orders of .Taking first the generator of partial Husimi dynamics, we find Computing the adjoint action explicitly gives 19th the sum over m indicating the initial creation of a unitary term via (B.17), and the sum over n giving the subsequent action via (B.19).Using the fact that ad n may be written out pictorially to show that they are generated by a version of the Pascal triangle, here with the same addition rules but with the boundary elements given by the integers Z i.e.Finally, combining the components − i [H, • ], T 1 and T 2 , using the definition