# Hybrid completely positive Markovian quantum-classical dynamics

Lajos Diósi  Wigner Research Center for Physics, H-1525 Budapest 114 , P.O.Box
49, Hungary  Eötvös Loránd University, H-1117 Budapest, Pázmány Péter stny.
1/A

###### Abstract

A concise and self-contained derivation of hybrid quantum-classical dynamics
is given in terms of Markovian master equations. Many previously known results
are re-derived, revised, some of them completed or corrected. Using as simple
method as possible, our goal is a brief introduction to state-of-the-art of
hybrid dynamics, with a limited discussion of the implications for
foundations, and without discussion of further relevance in measurement
problem, quantum-gravity, or chemistry, numeric methods, etc. Hybrid dynamics
is defined as special case of composite quantum dynamics where the observables
of one of the two subsystems are restricted for the commuting set of diagonal
operators in a fixed basis. With this restriction, the derivation of hybrid
dynamical equations is clear conceptually and simple technically. Jump and
diffusive dynamics follow in the form of hybrid master equations. Their
stochastic interpretation (called unravellings) is derived. We discuss gauge-
type ambiguities, problems of uniqueness, and covariance of the diffusive
master equation. Also conditions of minimum noise and of monitoring the
quantum trajectory are derived. We conclude that hybrid formalism is
equivalent with standard Markovian theory of time-continuous quantum
measurement (monitoring) on one hand, and is a motivating alternative
formalism on the other hand.

##  I Introduction

In real word, quantum and classical phenomena coexist and evolve according to
their own rules. They do interact, of course, and we know very well the
mathematical models of some particular interactions. The action of a classical
system on the quantum one is modeled if we make the Hamiltonian $\hat{H}$
depend on the classical system’s variables $x$. The _backaction_ , i.e. the
quantum system’s impact on a classical one is also known well from the quantum
measurement: the quantum system rules the pointer position $x$ of a classical
meter. This backaction is extremely specific. More general backaction is the
central problem to understand and to describe mathematically when quantum and
classical systems are interacting.

The central object of composite quantum-classical systems is the hybrid state,
represented by the hybrid density Aleksandrov (1981):

$\hat{\rho}(x)=\hat{\rho}_{x}\rho(x),$ (1)

where $\rho(x)$ is the normalized probability density of the classical
variables $x$ and $\hat{\rho}_{x}$ is the density operator of the quantum
system conditioned on the value $x$ of the classical variable.

The efforts and results concern the possible evolution equations for
$d\hat{\rho}(x)/dt$. The bottleneck is the backaction although its elementary
pattern is known from all quantum theory textbooks. The von Neumann
measurement of the complete orthogonal set $\\{\hat{P}_{x}\\}$ of Hermitian
projectors is imposes the following change of the hybrid state (1):

$\Bigl{\\{}\hat{\rho}\rightarrow\frac{\hat{P}_{x}\hat{\rho}\hat{P}_{x}}{\mathsf{tr}(\hat{P}_{x}\hat{\rho})}\mbox{
with prob
}\rho(x)=\mathsf{tr}(\hat{P}_{x}\hat{\rho})\Bigr{\\}}\Longleftrightarrow\Bigl{\\{}\hat{\rho}(x)\longrightarrow\hat{P}_{x}\hat{\rho}_{x}\hat{P}_{x}\Bigr{\\}},$
(2)

As we see, the textbook stochastic jumps are _equivalent_ with a single
deterministic map of the hybrid state. If we construct a stochastic dynamics
underlying the process on the lhs, we have an equivalent determinsitic hybrid
dynamics for $\hat{\rho}(x)$. Von Neumann’s statistical interpretation of
quantum states (also called the Born rule) follows from the statistical
interpretation of the hybrid state.

For longtime, the dynamics of von Neumann measurement, l.h.s. of (2), was
missing from the textbooks, as irrelevant. Lately, by very different
motivations, it was constructed in the continuous limit of discrete von
Neumann measurements. The wining formalism has been Markovian stochastic
equations, not the hybrid formalism. Refs. Gisin (1984); Diósi (1988);
Belavkin (1988) were milestones, reviews are in Wiseman and Diósi (2001); Brun
(2002); Breuer et al. (2002); Jacobs and Steck (2006); Wiseman and Milburn
(2009).

Whether hybrid dynamics, developped on their own, yield more than the time-
continuous measurements in hybrid formalism, i.e.: time-continuous extension
of the r.h.s. of (2)? Before the answer, we introduce ourselves into the
calculus of hybrid dynamics.

A large body of investigations of quite different concepts, motivations,
methods, level of mathematical rigor, etc., emerged in forty years, from the
earliest attempts to couple quantum and canonical classical systems
Aleksandrov (1981); Gerasimenko (1982); Boucher and Traschen (1988) through
the present author’s Diósi (1995, 1996); Diósi and Halliwell (1998); Diosi
(1999); Diósi et al. (2000); Diósi (2011, 2012, 2014) and other’s results
Blanchard and Jadczyk (1993, 1995); Alicki and Kryszewski (2003); Oppenheim
(2018); Oppenheim and Weller-Davies (2022); Oppenheim et al. (2022a, b);
Layton et al. (2022); Oppenheim et al. (2023) directly related to the present
work, and many other contributions, e.g., in refs. Anderson (1995); Prezhdo
and Kisil (1997); Peres and Terno (2001); Salcedo (2012); Tilloy (2018);
Kapral (2006); Barceló et al. (2012); Elze (2012, 2013); Burić et al. (2014);
Hall et al. (2016); Bondar et al. (2019); Donadi and Bassi (2022).

Here we use a unique approach, elementary derivations, economic presentation,
in order to give a complete but concise account of Markovian hybrid quantum-
classical dynamics modeled by hybrid master equations (HMEs). Sec. II derives
the canonical jump HME and its stochastic interpretation, called _unraveling_.
From this HME, sec. III derives the limit of continuous, diffusive HME which,
in sec. IV is put into the general covariant form. Condition of minimum
irreversibility, covariant and non-covariant unravelings, and monitoring are
contained in the subsections. Comparison with and comments on previous results
will be concentrated in the sec. V. At last, sec. VI contains conclusion of
the author. Appendix A shows an elemantary example, B contains application to
canonical classical subsystems, with a simplest example.

##  II Hybrid master equation

Quantum theory is universal in our approach and classical systems are special
case of quantum ones: operators are restricted for a commutative subset, i.e.,
they are diagonal in a fixed basis. Accordingly, consider the Hilbert space
$\mathcal{H}_{QC}=\mathcal{H}_{Q}\otimes\mathcal{H}_{C}$ where
$\mathcal{H}_{C}$ will host our classical system in a fixed basis
$\\{|x\rangle\\}$. The composite quantum state $\widehat{\rho}$, and composite
Hamiltonian $\widehat{H}$ (as well as the composite observables) are diagonal
in the fixed basis:

$\displaystyle\widehat{\rho}$ $\displaystyle=$
$\displaystyle\sum_{x}\hat{\rho}(x)\otimes|x\rangle\langle x|,$ (3)
$\displaystyle\widehat{H}$ $\displaystyle=$
$\displaystyle\sum_{x}\hat{H}(x)\otimes|x\rangle\langle x|.$ (4)

The block-diagonality of the objects $\hat{\rho}(x),\hat{H}(x)$ ensures
classicality of the subsystem in $\mathcal{H}_{C}$, i.e., that its state is
and remains diagonal in the fixed basis. The block-diagonal objects
$\hat{\rho}(x),\hat{H}(x)$ will be called respectively the hybrid state and
hybrid Hamiltonian. Note the notational difference: operators on
$\mathcal{H}_{QC}$ wear wider hats than operators on $\mathcal{H}_{Q}$. We are
looking for a Markovian evolution equation for $\widehat{\rho}$, which is
completely positive (CP) and preserves the block-diagonal form of
$\widehat{\rho}$.

Since CP maps represent the most general quantum dynamics, we start from the
CP map $\Lambda$ of the composite state $\widehat{\rho}$ and request that it
preserve block-diagonality, i.e., classicality of the subsystem in
$\mathcal{H}_{C}$. Conveniently, we can write the general form of the map for
the hybrid representation $\hat{\rho}(x)$ of $\widehat{\rho}$. With Einstein
convention of summation, it reads

$\hat{\rho}_{\Lambda}(x)=\sum_{y}D_{\beta\alpha}(x,y)\hat{L}_{\alpha}\hat{\rho}(y)\hat{L}^{\dagger}_{\beta},$
(5)

where $\\{\hat{L}_{\alpha}\\}$ is an operator basis in $\mathcal{H}_{Q}$. The
Hermitian matrix $D$ satisfies

$\sum_{x}D_{\beta\alpha}(x,y)\hat{L}^{\dagger}_{\beta}\hat{L}_{\alpha}=\hat{I}$
(6)

for all $y$. If we diagonalize $D$ at each point $(x,y)$, the form of the map
becomes

$\hat{\rho}_{\Lambda}(x)=\sum_{y,\alpha}\lambda_{\alpha}(x,y)\hat{L}_{\alpha}(x,y)\hat{\rho}(y)\hat{L}_{\alpha}^{\dagger}(x,y),$
(7)

where $\\{\hat{L}_{\alpha}(x,y)\\}$ is an operator basis depending on $(x,y)$.
Hence all $\lambda_{\alpha}(x,y)\geq 0$ for all $\alpha$ and $(x,y)$ otherwise
the map $\Lambda$ cannot be CP. Now we can absorb each factor
$\lambda_{\alpha}$ into $\hat{L}_{\alpha}$ and return to the full operator
formalism. We can write the CP map, preserving the block-diagonal form of
$\widehat{\rho}$ into the standard quantum mechanical form

$\widehat{\rho}_{\Lambda}=\widehat{L}_{\alpha}\widehat{\rho}\widehat{L}_{\alpha}^{\dagger},$
(8)

with the linearly independent Kraus-operators defined by

$\widehat{L}_{\alpha}=\sum_{y}\hat{L}_{\alpha}(x,y)\otimes|x\rangle\langle
y|,$ (9)

where $\hat{L}_{1}(x,y),\hat{L}_{2}(x,y),\dots$ are linearly independent. If
$\Lambda$ is a semigroup and generates time evolution of $\widehat{\rho}$
then, according to the GKLS theorem Lindblad (1976); Gorini et al. (1976), the
evolution is governed by the quantum master equation of the following from:

$\frac{d\widehat{\rho}}{dt}=-i[\widehat{H},\widehat{\rho}]+\widehat{L}_{\alpha}\widehat{\rho}\widehat{L}_{\alpha}^{\dagger}-\mathbb{H}\widehat{L}_{\alpha}^{\dagger}\widehat{L}_{\alpha}\widehat{\rho},$
(10)

where the $\widehat{L}_{\alpha}$’s are called the Lindblad generators. They
can (and will) be chosen such that
$\widehat{I},\widehat{L}_{1},\widehat{L}_{2},\dots$ are linearly independent.
As expected, this quantum master equation preserves block-diagonality by
construction hence it obtains a closed equation in the hybrid formalism
Blanchard and Jadczyk (1993, 1995); Alicki and Kryszewski (2003); Diósi
(2014); Oppenheim et al. (2022a); Layton et al. (2022); Oppenheim et al.
(2023):

$\frac{d\hat{\rho}(x)}{dt}=-i[\hat{H}(x),\hat{\rho}(x)]+\sum_{y}\left(\hat{L}_{\alpha}(x,y)\hat{\rho}(y)\hat{L}^{\dagger}_{\alpha}(x,y)-\mathbb{H}\hat{L}^{\dagger}_{\alpha}(y,x)\hat{L}_{\alpha}(y,x)\hat{\rho}(x)\right),$
(11)

where $\mathbb{H}$ means the Hermitian part of the subsequent expression. This
is the canonical form of the CP Markovian HME where the classical system is
discrete. The hybrid generators are only restricted by their linear
independence and their linear independence of $\hat{I}\delta(x,y)$ as well.
Their dependence on $(x,y)$ can otherwise be arbitrary. Through our work
$\delta(x,y)$ will denote the usual discrete delta-function. The backaction is
encapsulated by the generators $\hat{L}_{\alpha}(x,y)$. Note the basic lesson:
when the quantum-classical interaction is mutual influence between the
subsystems then the hybrid system can never be reversible, the evolution is
governed by hybrid master (kinetic) equations. Quantitative lower bounds on
their irreversibility (noise) will be derived in sec. III.

The choice of the hybrid Hamiltonian and the hybrid generators is unique upto
arbitrary complex functions $\ell_{\alpha}(x)$ since the HME (11) is invariant
for the following shifts:

$\displaystyle\hat{L}_{\alpha}(x,y)$ $\displaystyle\rightarrow$
$\displaystyle\hat{L}_{\alpha}(x,y)+\ell_{\alpha}(x)\delta(x,y),$
$\displaystyle\hat{H}(x)$ $\displaystyle\rightarrow$
$\displaystyle\hat{H}(x)-\frac{i}{2}\bigl{(}\ell_{\alpha}^{\star}(x)\hat{L}_{\alpha}(x,x)-\mathsf{h.c.}\bigr{)},$
(12)

Special cases may allow much larger groups of such ‘gauge’ freedom when the
dependence of the generators on $(x,y)$ is degenerate, like in secs. III, IV
of continuous HMEs.

###  II.1 Stochastic unravelling

Master equations are deterministic. Like any quantum or classical master
equation, also our HME (11) possesses statistical interpretation. The solution
$\rho(x)$ of a classical master (kinetic) equation can be decomposed into a
unique stochastic process $x_{t}$ of random trajectories. Quantum master
equations can be decomposed into stochastic processes $\psi_{t}$ of random
quantum trajectories, the decomposition is called unraveling. Quantum
unravellings are not unique. Here we define the hybrid of the classical and
quantum stochastic decompositions (unravellings).

Let us formulate the mathematical condition of hybrid unraveling. If
$\hat{\rho}(z,t)$ is the solution of the HME (11) then it is the stochastic
mean $\mathbb{M}$ over the contribution of hybrid trajectories
$(x_{t},\psi_{t})$:

$\hat{\rho}(z,t)=\mathbb{M}\psi_{t}\psi_{t}^{\dagger}\delta(z,x_{t}),$ (13)

where the $x_{t}$ and $\psi_{t}$ are correlated jump stochastic processes, one
in the space of classical cooordinates and the other in the Hilbert space of
state vectors. Consider first the unique unraveling of the distribution
$\rho(x)=\mathsf{tr}\hat{\rho}(x)$ of the classical subsystem. The trace of
the HME (11) yields the following classical master (kinetic) equation:

$\frac{d\rho(x)}{dt}=\sum_{\alpha}\sum_{y}\Bigl{(}T_{\alpha}(x,y)\rho(y)-T_{\alpha}(y,x)\rho(x)\Bigr{)},$
(14)

with the $\psi$-dependent transition (jump) rate from $x$ to $y$ for each
$\alpha$:

$T_{\alpha}(y,x)=\mathsf{tr}\left(\hat{L}_{\alpha}(y,x)\hat{\rho}_{x}\hat{L}^{\dagger}_{\alpha}(y,x)\right)=\bigl{\langle}\hat{L}^{\dagger}_{\alpha}(y,x)\hat{L}_{\alpha}(y,x)\bigr{\rangle}_{x}.$
(15)

We introduce the total transition rate from $x$:

$T(x)=\sum_{\alpha}\sum_{y}T_{\alpha}(y,x).$ (16)

To unravel the quantum subsystem, we need the anti-Hermitian (frictional)
hybrid Hamiltonian defined by

$-i\hat{H}_{\mathrm{fr}}(x)=-\frac{1}{2}\sum_{y}\hat{L}^{\dagger}_{\alpha}(y,x)\hat{L}_{\alpha}(y,x).$
(17)

The corresponding frictional Schrödinger equation is

$\frac{d\psi}{dt}=\left(-i\hat{H}(x)-i\hat{H}_{\mathrm{fr}}(x)+\frac{1}{2}T(x)\right)\psi,$
(18)

where $\tfrac{1}{2}T(x)$ restores the norm $\psi^{\dagger}\psi$ since
$\tfrac{1}{2}T(x)=-\mathbb{I}\langle\hat{H}_{\mathrm{fr}}\rangle$, cf. eqs.
(15-17); symbol $\mathbb{I}$ means the imaginary part.

The hybrid unraveling consists of the following two correlated jump stochastic
(piecewise deterministic) processes, one for $x_{t}$, another for $\psi_{t}$
(cf. Oppenheim et al. (2023)):

$\displaystyle x$ $\displaystyle=$ $\displaystyle\mbox{ const.},$ (19)
$\displaystyle\frac{d\psi}{dt}$ $\displaystyle=$
$\displaystyle\left(-i\hat{H}(x)-i\hat{H}_{\mathrm{fr}}(x)\psi+\tfrac{1}{2}T(x)\right)\psi,$
(20)
$\hskip-32.0pt\mbox{jumps}\left\\{\begin{array}[]{ccc}x&\rightarrow&x^{\prime}~{},\\\
\psi&\rightarrow&\frac{\hat{L}_{\alpha}(x,x^{\prime})\psi}{\sqrt{T_{\alpha}(x^{\prime},x)}}\end{array}\right.\mbox{at
rate }T_{\alpha}(x^{\prime},x).$ (21)

The proof that the unraveling satisfies the condition (13) is the following.
With the notation $\hat{P}=\psi\psi^{\dagger}$, the condition reads

$d\hat{\rho}(z,t)=d\mathbb{M}\hat{P}_{t}\delta(z,x_{t}),$ (22)

where $d\hat{\rho}(z,t)$ is determined by the HME (11). The unraveling yields
two terms for the change of $\hat{P}\delta(z,x)$ in time $dt$:

$\displaystyle\hat{P}\delta(z,x)\rightarrow$
$\displaystyle\bigl{(}1-T(x)dt\bigr{)}\times\left(\hat{P}-i[\hat{H}(x),\hat{P}]dt-i[\hat{H}_{\mathrm{fr}}(x),\hat{P}]_{+}dt+T(x)\hat{P}dt\right)\delta(z,x)$
(23)
$\displaystyle+\sum_{\alpha}T_{\alpha}(x^{\prime},x)dt\times\frac{\hat{L}_{\alpha}(x,x^{\prime})\hat{P}\hat{L}_{\alpha}^{\dagger}(x,x^{\prime})}{T_{\alpha}(x^{\prime},x)}\delta(z,x^{\prime}).$

The first term comes from the deterministic eqs. (19,20), the second term
represents the average of jumps (21). The $\psi$-dependent transition rates
$T_{\alpha}(x^{\prime},x)$ and $T(x)$ cancel, we are are left with an
expression linear in $\hat{P}\delta(z,x)$. Taking its mean obtains this:

$\frac{d}{dt}\mathbb{M}\hat{P}\delta(z,x)=\mathbb{M}\left(-i[\hat{H}(x),\hat{P}]-i[\hat{H}_{\mathrm{fr}}(x),\hat{P}]_{+}\right)\delta(z,x)+\mathbb{M}\hat{L}_{\alpha}(x,x^{\prime})\hat{P}\hat{L}_{\alpha}^{\dagger}(x,x^{\prime})\delta(z,x^{\prime})$
(24)

If we recall the expression (17) of $\hat{H}_{\mathrm{fr}}$ then we can
recognize that $\mathbb{M}\hat{P}\delta(z,x)$ satisfies the HME (11).

The unraveling (19-21) is ambiguous. The shifts (II) leave the HME (11)
invariant but we get different unravelings. Nevertheless, the unraveling
becomes invariant and unique with the replacement (cf. Diósi (1986)):

$\hat{L}_{\alpha}(x,y)\rightarrow\hat{L}_{\alpha}(x,y)-\langle\hat{L}_{\alpha}(x,y)\rangle\delta(x,y).$
(25)

One would think that in eq. (21) the state $\psi_{t}$ and $x_{t}$ are jumping
together. This is not true if $\hat{L}_{\alpha}(x,x)\neq 0$ since
$\hat{L}_{\alpha}(x,x)$ generates nontrivial jump of $\psi_{t}$ and no jump
for $x_{t}$. So, to synchronize the jumps of $\psi$ and $x$ either we consider
the invariant modification (25) of the unraveling or we just request that
$\hat{L}_{\alpha}(x,x)$ vanishes for all $\alpha$ and $x$.

###  II.2 Monitoring the quantum trajectory

The quantum trajectory $\psi_{t}$ is not observable in general since its
detection inevitably perturbs it. That is the problem of _monitoring_ the
quantum system. Can we, just by monitoring the classical subsystem’s $x_{t}$,
monitor the evolution of $\psi_{t}$? Generally we cannot. If the number of
generators $\hat{L}_{\alpha}(x,y)$ is more than one, detecting a jump of
$x_{t}$ does not tell us which generator made $\psi_{t}$ jump. The jump of
$x_{t}$ leaves the jump of $\psi_{t}$ in eq. (21) undetermined. This ambiguity
can be fixed in a natural class of special hybrid systems.

Let us add a vectorial structure $\\{x^{\alpha}\\}$ to the classical discrete
space and assume the following specific form of the hybrid generators
$\hat{L}_{\alpha}(x,y)$:

$\hat{L}_{\alpha}(x^{\alpha},y^{\alpha})\prod_{\beta\neq\alpha}\delta(x^{\beta},y^{\beta}),$
(26)

also with $\hat{L}_{\alpha}(x^{\alpha},x^{\alpha})=0$. The jump rates
$T_{\alpha}(x^{\alpha},y^{\beta})$ vanish if $\alpha\neq\beta$, cf. eq. (15).
Now, if we observe a jump of $x_{t}^{\alpha}$ we can uniquely determine the
jump of $\psi_{t}$. With the vectorized classical variables $x^{\alpha}$, the
unraveling (19-21) of the HME (11) becomes unique and the quantum trajectory
can be monitored.

##  III From discrete to diffusive hybrid master equation

Although discrete classical systems are important, most classical systems of
interest, like the Hamiltonian ones, are continuous. Therefore we are going to
construct the continuous limit of the obtained discrete HME (11). As is known,
the only continuous classical Markovian process is diffusion with additional
deterministic drift. We start with the generic HME (11) on the discrete subset
$\\{x^{n}\\}$ of the multidimensional continuum and generate a diffusion
process in the continuous limit $\epsilon\rightarrow 0$.

Note that some hybrid generators can be of the form
$\hat{L}_{\alpha}(y,x)=\hat{I}L_{\alpha}(x,y)\equiv L_{\alpha}(x,y)$, where
$\hat{I}$ is the unit operator (and we used to spare them in notations). For
some of such classical generators we introduce new label $n$ and the classical
generators $L_{n}(x,y)$ So we have hybrid generators $\hat{L}_{\alpha}(x,y)$
and the classical ones $L_{n}(x,y)$. The classical terms are intended to
generate diffusion terms for the classical subsystem in the continuous limit.
The two classes of generators will be chosen as follows:

$\displaystyle\hat{L}_{\alpha}(x,y)$ $\displaystyle=$
$\displaystyle\hat{L}_{\alpha}(y,y)\prod_{n}\frac{\delta(x^{n}-y^{n},\epsilon)+\delta(y^{n}-x^{n},\epsilon)}{\sqrt{2}},$
(27) $\displaystyle L_{n}(x,y)$ $\displaystyle=$
$\displaystyle\frac{\delta(x^{n}-y^{n},\epsilon)-\delta(y^{n}-x^{n},\epsilon)}{\sqrt{2}\epsilon}\prod_{m\neq
n}\frac{\delta(x^{m}-y^{m},\epsilon)+\delta(y^{m}-x^{m},\epsilon)}{\sqrt{2}}.$
(28)

We introduce the positive semi-definite complex decoherence matrix
$D_{\mathrm{Q}}^{\alpha\beta}$, the positive semi-definite real diffusion
matrix $D_{\mathrm{C}}^{nm}$ and the arbitrary complex matrix
$G_{\mathrm{CQ}}^{n\alpha}$ of backaction. Consider the following Hermitian
block-matrix and let it be positive semi-definite:

$\mathcal{D}=\begin{bmatrix}D_{\mathrm{Q}}&G_{\mathrm{CQ}}^{\dagger}\\\
G_{\mathrm{CQ}}&D_{\mathrm{C}}\end{bmatrix}\geq 0.$ (29)

We can see that nonzero backaction will require both nonzero decoherence and
diffusion. In addition to the constraints $D_{\mathrm{Q}}\geq
0,D_{\mathrm{C}}=D_{\mathrm{C}}^{\dagger}\geq 0$ that we always take for
granted, there are further constraints on matrices
$D_{\mathrm{Q}},D_{\mathrm{C}},G_{\mathrm{CQ}}$, equivalent to (29), to be
shown in sec. IV.1.

The following HME generates CP maps (since it reduces to the HME (11) if we
diagonalize $\mathcal{D}$):

$\displaystyle\frac{d\hat{\rho}(x)}{dt}=-i[\hat{H}(x),\hat{\rho}(x)]$
$\displaystyle+D_{\mathrm{Q}}^{\beta\alpha}\sum_{y}\Bigl{(}\hat{L}_{\alpha}(x,y)\hat{\rho}(y)\hat{L}_{\beta}^{\dagger}(x,y)-\mathbb{H}\hat{L}_{\beta}^{\dagger}(x,y)\hat{L}_{\alpha}(y,x)\hat{\rho}(x)\Bigr{)}$
(30)
$\displaystyle+D_{\mathrm{C}}^{nm}\sum_{y}\Bigl{(}L_{n}(x,y)L_{m}(x,y)\hat{\rho}(y)-L_{n}(y,x)L_{m}(y,x)\hat{\rho}(x)\Bigr{)}$
$\displaystyle+\overline{G}_{\mathrm{CQ}}^{n\alpha}\sum_{y}\left(L_{n}(x,y)\hat{L}_{\alpha}(x,y)\hat{\rho}(y)-L_{n}(y,x)\mathbb{H}\hat{L}_{\alpha}(x,y)\hat{\rho}(x)\right).$

In the continuous limit $\epsilon\rightarrow 0$, the terms with
$\hat{L}_{\alpha},\hat{L}_{\beta}$ contribute to standard GKLS structures:

$\hat{L}_{\alpha}(x)\hat{\rho}(x)\hat{L}_{\beta}^{\dagger}(x)-\mathbb{H}\hat{L}_{\beta}^{\dagger}(x)\hat{L}_{\alpha}(x)\hat{\rho}(x),$
(31)

with notation $\hat{L}_{\alpha}(x)\equiv\hat{L}_{\alpha}(x,x)$. The terms with
$L_{n},L_{m}$ yield diffusion of the classical variable $x$. With the notation
$\epsilon_{n}$ meaning a vector whose only nonzero component is the
$n^{\prime}th$ one, and it is $\epsilon$, the yield at $n\neq m$ reads:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\frac{\hat{\rho}(x\\!+\\!\epsilon_{n}\\!\\!+\\!\epsilon_{m})\\!+\\!\hat{\rho}(x\\!-\\!\epsilon_{n}\\!\\!\\!-\\!\epsilon_{m})\\!-\\!\hat{\rho}(x\\!-\\!\epsilon_{n}\\!\\!+\\!\epsilon_{m})\\!-\\!\hat{\rho}(x\\!+\\!\epsilon_{n}\\!\\!\\!-\\!\epsilon_{m})}{2\epsilon^{2}}$
$\displaystyle\underset{\epsilon\rightarrow
0}{\longrightarrow}\frac{1}{2}\partial_{n}\partial_{m}\hat{\rho}(x),$ (32)

where $\partial_{n}\equiv\partial/\partial x^{n}$. We get the similar yield
for $n=m$:

$\frac{1}{2\epsilon^{2}}\Bigl{(}\hat{\rho}(x+\epsilon_{n})+\hat{\rho}(x-\epsilon_{n})-2\hat{\rho}(x)\Bigr{)}\underset{\epsilon\rightarrow
0}{\longrightarrow}\frac{1}{2}\partial_{n}\partial_{n}\hat{\rho}(x).$ (33)

The cross-terms with $\hat{L}_{\alpha},L_{n}$ generate the non-trivial
backaction of the quantum system on the classical part:

$\displaystyle\frac{1}{2\epsilon}\Bigl{(}\hat{L}_{\alpha}(x+\epsilon_{n})\hat{\rho}(x+\epsilon_{n})-\hat{L}_{\alpha}(x-\epsilon_{n})\hat{\rho}(x-\epsilon_{n})\Bigr{)}+\mathsf{h.c.}$
$\displaystyle\underset{\epsilon\rightarrow
0}{\longrightarrow}\partial_{n}\left(\hat{L}_{\alpha}(x)\hat{\rho}(x)\right)+\mathsf{h.c.}$
(34)

Using these limits in eq. (30) and adding a deliberate classical drift of
velocity $V(x)$, we obtain the continuous limit of the discrete HME (11):

$\displaystyle\frac{d\hat{\rho}(x)}{dt}=$
$\displaystyle-i[\hat{H}(x),\hat{\rho}(x)]+D_{\mathrm{Q}}^{\beta\alpha}\Bigl{(}\hat{L}_{\alpha}(x)\hat{\rho}(x)\hat{L}_{\beta}^{\dagger}(x)-\mathbb{H}\hat{L}_{\beta}^{\dagger}(x)\hat{L}_{\alpha}(x)\hat{\rho}(x)\Bigr{)}$
(35)
$\displaystyle+\tfrac{1}{2}D_{\mathrm{C}}^{nm}\partial_{n}\partial_{m}\hat{\rho}(x)+\left(\overline{G}_{\mathrm{CQ}}^{n\alpha}\partial_{n}\left(\hat{L}_{\alpha}(x)\hat{\rho}(x)\right)+\mathsf{h.c.}\right)-\partial_{n}\left(V^{n}(x)\hat{\rho}(x)\right).$

The three _constant_ parameter matrices
$D_{\mathrm{Q}},D_{\mathrm{C}},G_{\mathrm{CQ}}$ are constrained by the semi-
definiteness $\mathcal{D}\geq 0$ of the block-matrix (29) formed by them.

The above HME is not yet the general diffusive one. Obviously we can add an
extra $x$-dependent decoherence $\Delta D_{\mathrm{Q}}(x)$ as well as an extra
diffusion $\Delta D_{\mathrm{C}}(x)$ as long as $\mathcal{D}\geq$ holds true
after the replacements $D_{\mathrm{Q}}\rightarrow D_{\mathrm{Q}}+\Delta
D_{\mathrm{Q}}(x)$ and $D_{\mathrm{C}}\rightarrow D_{\mathrm{C}}+\Delta
D_{\mathrm{C}}(x)$. That is, the validity of the diffusive HME (35) extends
for $x$-dependent parameters $D_{\mathrm{Q}}(x)$ and $D_{\mathrm{C}}(x)$
provided $\mathcal{D}(x)\geq 0$. In the next section we show that also the
backaction matrix $G_{\mathrm{CQ}}$ can depend on $x$.

##  IV Covariant hybrid master equation

The form (35) of the HME is explicitly covariant under the global linear
transformations of the operator basis $\hat{L}_{\alpha}(x)$ and the classical
variables $x$. We are interested in explicit covariant form under local, i.e.:
$x$-dependent complex linear transformation of the operator basis and under
general coordinate transformations of $x$. The form of such HME (35) should be
this (cf. Oppenheim et al. (2022a)):

$\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}]+D_{\mathrm{Q}}^{\beta\alpha}\Bigl{(}\hat{L}_{\alpha}\hat{\rho}\hat{L}_{\beta}^{\dagger}-\mathbb{H}\hat{L}_{\beta}^{\dagger}\hat{L}_{\alpha}\hat{\rho}\Bigr{)}+\tfrac{1}{2}\partial_{n}\partial_{m}(D_{\mathrm{C}}^{nm}\hat{\rho})+\partial_{n}(\overline{G}_{\mathrm{CQ}}^{n\alpha}\hat{L}_{\alpha}\hat{\rho}+\mathsf{h.c.})-\partial_{n}\left(V^{n}\hat{\rho}\right)$
(36)

where every object is function of $x$, the fact our above notations hide for
the sake of compactness.

_Correction._ The covariance of the Eq. (36) and its unravellings in Sec. IV.B
does not extend for the general transformations of the classical variables but
for the linear ones. To achieve general covariant forms, the scalar hybrid
density and the co(ntra)variant drift vector should be used via two respective
replacements:

$\displaystyle\hat{\rho}$ $\displaystyle\rightarrow$
$\displaystyle\frac{1}{\sqrt{g}}\hat{\rho},$ $\displaystyle V^{n}$
$\displaystyle\rightarrow$ $\displaystyle
V^{n}+\frac{1}{2}\partial_{m}\left(\sqrt{g}D_{\mathrm{C}}^{nm}\right),$

where $g=1/\mathrm{det}D_{\mathrm{C}}$. These replacements allow us to recast
the non-covariant Fokker–Planck structure in Eq. (36) into the equivalent
covariant one:

$\displaystyle\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}]+D_{\mathrm{Q}}^{\beta\alpha}\Bigl{(}\hat{L}_{\alpha}\hat{\rho}\hat{L}_{\beta}^{\dagger}-\mathbb{H}\hat{L}_{\beta}^{\dagger}\hat{L}_{\alpha}\hat{\rho}\Bigr{)}$
$\displaystyle+\frac{1}{2\sqrt{g}}\partial_{n}\\!\\!\left(\sqrt{g}D_{\mathrm{C}}^{nm}\partial_{m}\hat{\rho}\right)\\!+\\!\frac{1}{\sqrt{g}}\partial_{n}\\!\\!\left(\sqrt{g}(\overline{G}_{\mathrm{CQ}}^{n\alpha}\hat{L}_{\alpha}\hat{\rho}\\!+\\!\mathsf{h.c.}\\!\\!-\\!\\!V^{n})\hat{\rho}\right),$

where we recognize the first and second order covariant derivations in a
Riemann space of contravariant metric tensor $D_{\mathrm{C}}^{nm}$. For
general covariance of the stochastic unraveling (49,50), the two Ito type SDEs
should be changed for the equivalent Stratonovich ones and the additional
constraint $\partial_{n}(dW^{n}/\sqrt{g})=0$ will have to be imposed Diósi
(2023). The general covariant formalism can then be implemented in the rest of
Sec. IV.B, the scope requires a separate work though.

The coefficients $D_{\mathrm{Q}}(x),D_{\mathrm{C}}(x),G_{\mathrm{CQ}}(x)$
satisfy the same constraint (29) $\mathcal{D}(x)\geq 0$ as before, now
understood for all $x$:

$\mathcal{D}(x)=\left[\\!\begin{array}[]{c|c}\\\
D_{\mathrm{Q}}^{\alpha\beta}(x)&~{}~{}~{}\overline{G}_{\mathrm{CQ}}^{m\beta}(x){}{}{}{}\\\
\\\ \hline\cr\\\ \\\
\\!\\!G_{\mathrm{CQ}}^{n\alpha}(x)\\!\\!&D_{\mathrm{C}}^{nm}(x)\\\ \\\ \\\
\end{array}\right]\geq 0.$ (37)

We prove the equivalence of the covariant HME (36) with (35). By suitable
choice of the operator basis $\hat{L}_{\alpha}(x)$ and the classical
coordinates $x^{n}$, one can always transform $G_{\mathrm{CQ}}(x)$ into
constant matrix. This results in the HME (35) which, as we argued there, is
valid for $x$-dependent $D_{\mathrm{Q}},D_{\mathrm{C}}$.

The covariant diffusive HME (36) is the most general continuous HME to
generate CP dynamics. Every object in it is $x$-dependent. The condition (37)
is necessary and sufficient when the generators $\hat{L}_{\alpha}(x)$ are
linearly independent and also linearly independent of the unit operator
$\hat{I}$. The hybrid Hamiltonian $\hat{H}$ and the classical drift $V$ are
arbitrary. The Hermitian matrix $D_{\mathrm{Q}}\geq 0$ of decoherence, the
real matrix $D_{\mathrm{C}}\geq 0$ of diffusion must form the positive semi-
definite block matrix (37) in corners with the matrix $G_{\mathrm{CQ}}$ (and
$G_{\mathrm{CQ}}^{\dagger}$) of backaction. We list three useful alternatives
that can always be achieved by trasformation of the reference frames: a fixed
operator basis $\\{\hat{L}_{\alpha}\\}$, or simultaneously diagonal
$D_{\mathrm{Q}}$ and $D_{\mathrm{C}}$ with zeros and ones, or
$G_{\mathrm{CQ}}$ with zeros and ones in the main diagonal and zeros
elsewhere. (These coordinate transformations may request the embeddig of $x$
in higher dimensions then they are of.)

In addition to the explicit covariance, there is a further ‘gauge’ freedom,
the descendant of the shifts (II) in the discrete HME:

$\displaystyle\hat{L}_{\alpha}$ $\displaystyle\rightarrow$
$\displaystyle\hat{L}_{\alpha}+\ell_{\alpha},$ $\displaystyle\hat{H}$
$\displaystyle\rightarrow$
$\displaystyle\hat{H}-i\frac{i}{2}\bigl{(}\ell_{\alpha}^{\star}\hat{L}_{\alpha}-\mathsf{h.c.}\bigr{)},$
$\displaystyle V^{n}$ $\displaystyle\rightarrow$ $\displaystyle
V^{\alpha}-(\overline{G}_{\mathrm{CQ}}^{n\alpha}\ell_{\alpha}+\mathsf{h.c.}),$
(38)

where $\ell_{\alpha}(x)$ is an arbitrary complex function.

###  IV.1 Minimum-noise threshold

We are going to breakdown the condition $\mathcal{D}\geq 0$ (37) into
constraints between $\mathcal{D}$’s building blocks. Necessary conditions are
the following:

$\displaystyle\mathsf{range}D_{\mathrm{Q}}$ $\displaystyle\geq$
$\displaystyle\mathsf{range}G_{\mathrm{CQ}}^{\dagger}G_{\mathrm{CQ}},$ (39)
$\displaystyle\mathsf{range}D_{\mathrm{C}}$ $\displaystyle\geq$
$\displaystyle\mathsf{range}G_{\mathrm{CQ}}G_{\mathrm{CQ}}^{\dagger}.$ (40)

They express that the range of decoherence $D_{\mathrm{Q}}$ cannot be narrower
than the range of $\hat{L}_{\alpha}$’s that are coupled to the classical
$x^{n}$’s by backaction $G_{\mathrm{CQ}}$. And similarly, the range of
classical diffusion $D_{\mathrm{C}}$ cannot be smaller than the range of
$x^{n}$’s that are coupled to the $\hat{L}_{\alpha}$’s. Nonzero backaction
means mandatory noise: both decoherence and diffusion.

Suppose that we have a given nonzero matrix $G_{\mathrm{CQ}}$ of backaction
and we are interested in a certain minimum of the total irreversibility, i.e.,
a certain minimum of the block-matrix $\mathcal{D}$, implying a certain
minimum of the decoherence $D_{\mathrm{Q}}$ and diffusion $D_{\mathrm{C}}$ as
we see below. Obviously, the strict positivity $\mathcal{D}>0$ means more
noise than the minimum. We are interested in the maximum degenerate
$\mathcal{D}$ meaning the lowest $\mathsf{rank}\mathcal{D}$ which is limited
by the rank $r_{\mathrm{CQ}}=\mathsf{rank}G_{\mathrm{CQ}}$, otherwise
$\mathcal{D}\geq 0$ cannot be true. Therefore we define the minimum-noise
threshold by this:

$\mathsf{rank}\mathcal{D}=\mathsf{rank}G_{\mathrm{CQ}}.$ (41)

Then the inequalities (39) saturate,
$\mathsf{range}D_{\mathrm{Q}}=\mathsf{range}G_{\mathrm{CQ}}^{\dagger}G_{\mathrm{CQ}}$
and
$\mathsf{range}D_{\mathrm{C}}=\mathsf{range}G_{\mathrm{CQ}}G_{\mathrm{CQ}}^{\dagger}$,
also meaning that
$\mathsf{rank}D_{\mathrm{Q}}=\mathsf{rank}D_{\mathrm{C}}=r_{\mathrm{CQ}}$.
Both the number of coupled independent generators $\hat{L}_{\alpha}$ and
coordinates $x^{n}$ coincide with $r_{\mathrm{CQ}}$.

For a given $x$, transform $G_{\mathrm{CQ}}$ into the frame where its elements
are zero expect for an $r_{\mathrm{CQ}}\times r_{\mathrm{CQ}}$ unit matrix in
the upper-left corner. Then, because of (39), both $D_{\mathrm{Q}}$ and
$D_{\mathrm{C}}$ respectively must have an $r_{\mathrm{CQ}}\times
r_{\mathrm{CQ}}$ strictly positive matrix in their upper-left corners and
zeros elsewhere. Recall the general condition $\mathcal{D}\geq 0$. If we drop
rows and columns of zeros then the said non-zero $r_{\mathrm{CQ}}\times
r_{\mathrm{CQ}}$ sub-matrices, denoted invariably, must be each other’s
inverses Diósi (1995):

$D_{\mathrm{C}}(x)D_{\mathrm{Q}}(x)=I.$ (42)

This is the condition of the minimum-noise threshold (41) in the special frame
fitted to the backaction matrix.

We can now identify quantitatively and directly the lower bound of
irreversibility that hybrid systems must undergo even if the quantum and
classical subsystems were reversible in themselves. At a given backaction
strength $G_{\mathrm{CQ}}$ (scaled now to be the unity) and at the minimum
noise threshold, the quantum decoherence strength $D_{\mathrm{Q}}$ and
classical diffusion strength $D_{\mathrm{C}}$ are inverses of each other,
lower decoherence requests higher diffusion and vice versa.

It is possible to decompose the constraint $\mathcal{D}\geq 0$ as well as the
minimum noise condition (41) into covariant relationships (i.e.: valid in any
reference frame) for the three parameter matrices. Two equivalent forms of
$\mathcal{D}\geq 0$ are the following Oppenheim et al. (2022a):

$\displaystyle
G_{\mathrm{CQ}}\frac{1}{D_{\mathrm{Q}}}G_{\mathrm{CQ}}^{\dagger}$
$\displaystyle\leq$ $\displaystyle D_{\mathrm{C}},$ (43) $\displaystyle
G_{\mathrm{CQ}}\frac{1}{D_{\mathrm{Q}}}D_{\mathrm{Q}}G_{\mathrm{CQ}}^{\dagger}$
$\displaystyle=$ $\displaystyle G_{\mathrm{CQ}}G_{\mathrm{CQ}}^{\dagger};$
(44)

and

$\displaystyle
G_{\mathrm{CQ}}^{\dagger}\frac{1}{D_{\mathrm{C}}}G_{\mathrm{CQ}}$
$\displaystyle\leq$ $\displaystyle D_{\mathrm{Q}},$ (45) $\displaystyle
G_{\mathrm{CQ}}^{\dagger}\frac{1}{D_{\mathrm{C}}}D_{\mathrm{C}}G_{\mathrm{CQ}}$
$\displaystyle=$ $\displaystyle G_{\mathrm{CQ}}^{\dagger}G_{\mathrm{CQ}},$
(46)

where $1/D_{\mathrm{Q}}$ and $1/D_{\mathrm{C}}$ are generalized inverses. The
threshold condition $\mathsf{rank}\mathcal{D}=\mathsf{rank}G_{\mathrm{CQ}}$ of
minimum noise corresponds to the saturation of inequalities into equalities.
Note that $(1/D_{\mathrm{Q}})D_{\mathrm{Q}}=\mathsf{range}D_{\mathrm{Q}}$ and
$(1/D_{\mathrm{C}})D_{\mathrm{C}}=\mathsf{range}D_{\mathrm{C}}$; hence the
second lines in eqs. (43,45) correspond to the mentioned identities
$\mathsf{range}D_{\mathrm{Q}}=\mathsf{range}G_{\mathrm{CQ}}^{\dagger}G_{\mathrm{CQ}}$,
and
$\mathsf{range}D_{\mathrm{C}}=\mathsf{range}G_{\mathrm{CQ}}G_{\mathrm{CQ}}^{\dagger}$,
respectively. Under the above covariant parametric constraints the covariant
HME (36) and its unravellings (49,50) include all possible dynamics that
contain noise on and above the threshold of consistency. (The appendix A shows
the sharpness of the above conditions on a simplest HME.)

###  IV.2 Stochastic unravellings

When we construct unravellings of the HME (36), we follow the steps of sec.
II.1 and construct the two correlated stochastic processes for $x_{t}$ and
$\psi_{t}$ satisfying the condition of unraveling (13). The two processes will
diffusive ones this time. First, take the trace of the diffusive HME (36) and
obtain the classical Fokker–Planck equation of the classical subsystem:

$\displaystyle\frac{d\rho(x)}{dt}=\\!\\!\\!\\!\\!\\!$
$\displaystyle\frac{1}{2}\partial_{n}\partial_{m}\left(D_{\mathrm{C}}^{nm}(x)\rho(x)\right)-$
(47)
$\displaystyle\\!\\!\\!\\!\\!-\partial_{n}\\!\left(V^{n}(x)\rho(x)-2\mathbb{Re}\overline{G}_{\mathrm{CQ}}^{n\alpha}(x)\langle\hat{L}_{\alpha}(x)\rangle\rho(x)\right),$

where $\mathbb{Re}$ denotes the real part of the subsequent expression. The
unraveling of this equation will be a unique Brownian motion with a unique
$\psi$-dependent drift. To unravel the quantum subsystem, we need the non-
Hermitian (frictional) hybrid Hamiltonian defined by

$-i\hat{H}_{\mathrm{fr}}(x)=-\frac{1}{2}D_{\mathrm{Q}}^{\beta\alpha}\Bigl{(}\left(\hat{L}_{\beta}^{\dagger}(x)-\langle\hat{L}_{\beta}^{\dagger}(x)\rangle\right)\left(\hat{L}_{\alpha}(x)-\langle\hat{L}_{\alpha}(x)\rangle\right)+\left(\langle\hat{L}_{\beta}^{\dagger}(x)\rangle\hat{L}_{\alpha}(x)-\mathsf{h.c.}\right)\Bigr{)}.$
(48)

The hybrid unraveling consists of two correlated diffusive stochastic
processes, one for $x_{t}$, another for $\psi_{t}$:

$dx^{n}=V^{n}(x)dt-2\mathbb{Re}\overline{G}_{\mathrm{CQ}}^{n\alpha}(x)\langle\hat{L}_{\alpha}(x)\rangle
dt-dW^{n}(x),$ (49)
$d\psi\\!=\\!-i(\\!\hat{H}(\\!x\\!)\\!+\\!\hat{H}_{\mathrm{fr}}(\\!x\\!))\psi
dt\\!+\\!(\hat{L}_{\alpha}(\\!x\\!)\\!-\\!\langle\hat{L}_{\alpha}(\\!x\\!)\rangle)\psi
d\overline{\xi}^{\alpha}(\\!x\\!),$ (50)

where $dW^{n}(x)$ is real, and $d\xi^{\alpha}(x)$ is complex zero-mean Ito-
differential of auxiliary stochastic processes, correlated as follows:

$\displaystyle dW^{n}dW^{m}$ $\displaystyle=$ $\displaystyle
D_{\mathrm{C}}^{nm}dt,$ $\displaystyle d\xi^{\alpha}d\overline{\xi}^{\beta}$
$\displaystyle=$ $\displaystyle D_{\mathrm{Q}}^{\alpha\beta}dt,$ (51)
$\displaystyle dW^{n}d\xi^{\alpha}$ $\displaystyle=$ $\displaystyle
G_{\mathrm{CQ}}^{n\alpha}dt.$

Let us use the vector symbols $dW,d\xi$, then we get the equivalent compact
form of correlations:

$\begin{bmatrix}d\xi d\xi^{\dagger}&d\xi dW^{T}\\\
dWd\xi^{\dagger}&dWdW^{T}\\\ \end{bmatrix}=\mathcal{D}dt.$ (52)

We prove that the unraveling (49,50) satisfies the condition (13). Like in
sec. II.1, we use the notation $\hat{P}=\psi\psi^{\dagger}$ and the same form
(22) of the condition to be proved:

$d\hat{\rho}(z,t)=d\mathbb{M}\hat{P}_{t}\delta(z-x_{t}).$ (53)

The Ito-differential on the rhs contains three terms and we are going to
express them by the equations of $d\psi$ and $dx$ of the unraveling (see also
Layton et al. (2022)). First, to calculate $d\hat{P}=d\psi\psi^{\dagger}+\psi
d\psi^{\dagger}+d\psi d\psi^{\dagger}$ we use the stochastic equation (50) of
$d\psi$:

$\displaystyle d\hat{P}=$ $\displaystyle-i[\hat{H}(z),\hat{P}]dt$ (54)
$\displaystyle+D_{Q}^{\beta\alpha}\left(\hat{L}_{\alpha}(x)\hat{P}\hat{L}_{\beta}^{\dagger}(x)-\mathbb{H}\hat{L}_{\beta}^{\dagger}(x)\hat{L}_{\alpha}(x)\hat{P}\right)dt$
$\displaystyle+\Bigl{(}\bigl{(}\hat{L}_{\alpha}(x)-\langle\hat{L}_{\alpha}(x)\rangle\bigr{)}\hat{P}d\overline{\xi}^{\alpha}(x)+\mathsf{h.c.}\Bigr{)}dt.$

Second, we calculate $d\delta(z-x)$ using stochastic equation (49) of $dx$:

$d\delta(z-x)=\tfrac{1}{2}D_{\mathrm{C}}^{nm}\partial_{n}\partial_{m}\delta(z-x)dt+\left(V^{n}(x)-2\mathbb{Re}\overline{G}_{\mathrm{CQ}}^{n\alpha}\langle\hat{L}_{\alpha}(x)\rangle\right)\partial_{n}\delta(z-x)dt+\left(\partial_{n}\delta(z-x)\right)dW^{n}(x),$
(55)

where the partial derivations refer to $x$ obviously. From here, we get the
three terms of $d\mathbb{M}\hat{P}\delta(z-x)$ after using
$\partial\delta(z-x)/\partial x^{n}=-\partial\delta(z-x)/\partial z^{n}$, then
taking the stochastic mean over $dW,d\xi$ and $x$:

$\displaystyle\mathbb{M}d\hat{P}\delta(z-x)$ $\displaystyle=$
$\displaystyle-i[\hat{H}(z),\hat{\rho}(z)]dt+D_{\mathrm{Q}}^{\beta\alpha}(z)\left(\hat{L}_{\alpha}(z)\hat{\rho}(z)\hat{L}_{\beta}^{\dagger}(z)-\mathbb{H}\hat{L}_{\beta}^{\dagger}(z)\hat{L}_{\alpha}(z)\hat{\rho}(z)\right)dt,$
$\displaystyle\mathbb{M}\hat{P}d\delta(z-x)$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}\partial_{n}\partial_{m}(D_{\mathrm{C}}^{nm}(z)\hat{\rho}(z))-\partial_{n}\bigl{(}V^{n}(z)-2\mathbb{Re}\overline{G}_{\mathrm{CQ}}^{n\alpha}(z)\langle\hat{L}_{\alpha}(z)\rangle\hat{\rho}(z)\Bigr{)}dt,$
(56) $\displaystyle\mathbb{M}d\hat{P}d\delta(z-x)$ $\displaystyle=$
$\displaystyle\partial_{n}\Bigl{(}\overline{G}_{\mathrm{CQ}}^{n\alpha}(z)\left(\hat{L}_{\alpha}(z)-\langle\hat{L}_{\alpha}(z)\rangle\right)\hat{\rho}(z)\Bigr{)}dt+\mathsf{h.c.}$

By taking the sum of these three equations we recognize that
$\mathbb{M}\hat{P}\delta(z-x)$ satisfies the HME (36).

The hybrid unravelling corresponds to the time-continuous measurement of the
observables
$\left(\overline{G}_{\mathrm{CQ}}^{n\alpha}\hat{L}_{\alpha}(x)+\mathsf{h.c.}\right)$.
Hybrid unravelling contains an autonomous drift of the classical variables and
general feedbacks: the Hamiltonian, the decoherence matrix, the measured
observable, and the measurement noise can depend on the measured signal $x$.
These dependences could be part of time-continuous measurement but usually
they are not. (Except for typical feedback Hamiltonians, linear in $dx/dt$.)

As we can easily inspect, the unraveling (49,50) is invariant for the shifts
(IV). But in general, it is not covariant under the linear transformations of
the operator basis $\\{\hat{L}_{\alpha}(x)\\}$ of the HME! Therefore the
unraveling of the HME is not unique, inherits the ambiguity of unravellings
Wiseman and Diósi (2001) of the quantum mechanical GKLS dynamics. Observe that
we left the non-Hermitian correlation $d\xi^{\alpha}(x)d\xi^{\beta}(x)$
unspecified whereas the stochastic processes $(x_{t},\psi_{t})$ depend on it.
At deliberate choices of $d\xi^{\alpha}d\xi^{\beta}$ we get different
unravellings of the same HME (36).

Covariance of the unravelling can be achieved if the complex noise
$d\xi^{\alpha}$ is covariant. The simplest way is if we set

$d\xi^{\alpha}d\xi^{\beta}=0,$ (57)

see Gisin and Percival (1992); Percival (1998), also Diósi (1986). Another
option of covariant (and real) $d\xi^{\alpha}$ will be shown in sec. IV.3.

Just like unravellings of standard GKLS master equations, the hybrid
unravellings are not necessarily be in terms of pure states $\psi_{t}$. We
shall discuss generic mixed state unravellings later in this section. Before
that, a specific case is considered which is a closest generalization of the
pure state unravelings (49,50). Extension from pure states $\psi_{t}$ for
mixed states $\hat{\sigma}_{t}$ is straightforward since the formalism is very
similar. The pure state density operator $\hat{P}$ gives way to $\hat{\sigma}$
and the equation of $d\psi$ is replaced by an equation of $d\hat{\sigma}$.
Accordingly, the definition of unraveling reads

$\hat{\rho}(z,t)=\mathbb{M}\hat{\sigma}_{t}\delta(z-x_{t}),$ (58)

The eq. (49) of $dx^{n}$ is the same as before, the eq. (50) give way to the
equation of $d\hat{\sigma}$:

$d\hat{\sigma}=-i[\hat{H}(x),\hat{\sigma}]dt+D_{\mathrm{Q}}^{\beta\alpha}\left(\hat{L}_{\alpha}(x)\hat{\sigma}\hat{L}_{\beta}^{\dagger}(x)-\mathbb{H}\hat{L}_{\beta}^{\dagger}(x)\hat{L}_{\alpha}(x)\hat{\sigma}\right)dt+\left((\hat{L}_{\alpha}(x)-\langle\hat{L}_{\alpha}(x)\rangle)\hat{\sigma}d\overline{\xi}^{\alpha}(x)+\mathsf{h.c.}\right)dt.$
(59)

With the replacement $\hat{P}\rightarrow\hat{\sigma}$, the equations of
(47-IV.2) of the previous proof apply exactly in the same form and conclude
that the process $(x_{t},\hat{\sigma}_{t})$ unravels the HME (36). Notice the
purification feature of (59) known from standard unravellings of GKLS master
equations. If the state is not pure, i.e.: $\mathsf{tr}\hat{\sigma}^{2}<1$
then

$\displaystyle\frac{d}{dt}\mathbb{M}\mathsf{tr}\hat{\sigma}^{2}$
$\displaystyle=$
$\displaystyle\frac{2}{dt}\mathbb{M}\mathsf{tr}(\hat{\sigma}d\hat{\sigma})+\frac{1}{dt}\mathsf{tr}(d\hat{\sigma})^{2}=$
(60) $\displaystyle=$ $\displaystyle
2D_{\mathrm{Q}}^{\beta\alpha}\mathsf{tr}\left(\hat{\sigma}\hat{L}_{\alpha}\hat{\sigma}\hat{L}_{\beta}^{\dagger}-\mathbb{H}\hat{\sigma}\hat{L}_{\beta}^{\dagger}\hat{L}_{\alpha}\hat{\sigma}\right)+\mathsf{tr}\left((\hat{L}_{\alpha}-\langle\hat{L}_{\alpha}\rangle)\hat{\sigma}^{2}(\hat{L}_{\beta}^{\dagger}-\langle\hat{L}_{\beta}^{\dagger}\rangle)\right)$
$\displaystyle=$ $\displaystyle
2D_{\mathrm{Q}}^{\beta\alpha}\mathsf{tr}\left(\left(\sqrt{\hat{\sigma}}(\hat{L}_{\beta}-\langle\hat{L}_{\beta}\rangle)\sqrt{\hat{\sigma}}\right)^{\dagger}\left(\sqrt{\hat{\sigma}}(\hat{L}_{\alpha}-\langle\hat{L}_{\alpha}\rangle)\sqrt{\hat{\sigma}}\right)\right)>0.$

An arbitrary initial mixed state $\hat{\sigma}_{t}$ will be purified
asymptotically until $\mathsf{tr}\hat{\sigma}^{2}=1$ and then the mixed state
eq. (59) becomes equivalent with the pure state eq. (20).

Given a covariant HME (36), the family of unravellings is larger than the
above family of perfect purifying ones. There are partially purifying
unravellings if the HME is above the threshold of minimum noise, cf. (45):

$D_{\mathrm{Q}}=G_{\mathrm{CQ}}^{\dagger}\frac{1}{D_{\mathrm{C}}}G_{\mathrm{CQ}}+\Delta
D_{\mathrm{Q}}\equiv D_{\mathrm{Qmin}}+\Delta D_{\mathrm{Q}}$ (61)

where $\Delta D_{\mathrm{Q}}\geq 0$. Then the eq. (59) of mixed state
unravelling remains the same but the correlation $d\xi
d\xi^{\dagger}=D_{\mathrm{Q}}dt$ will be reduced to the minimum noise
threshold

$d\xi
d\xi^{\dagger}=D_{\mathrm{Qmin}}dt=G_{\mathrm{CQ}}^{\dagger}\frac{1}{D_{\mathrm{C}}}G_{\mathrm{CQ}}dt.$
(62)

The other correlations $dWd\xi=G_{\mathrm{CQ}}dt$ and
$dWdW^{T}=D_{\mathrm{C}}dt$ are unchanged. The price we pay for the reduced
noise is illustrated if we group the terms of (59) as follows:

$\displaystyle d\hat{\sigma}=-i[\hat{H}(x),\hat{\sigma}]dt$ $\displaystyle+$
$\displaystyle\tfrac{1}{2}D_{\mathrm{Qmin}}^{\beta\alpha}\left(\hat{L}_{\alpha}(x)\hat{\sigma}\hat{L}_{\beta}^{\dagger}(x)-\mathbb{H}\hat{L}_{\beta}^{\dagger}(x)\hat{L}_{\alpha}(x)\hat{\sigma}\right)dt+\left((\hat{L}_{\alpha}(x)-\langle\hat{L}_{\alpha}(x)\rangle)\hat{\sigma}d\overline{\xi}^{\alpha}(x)+\mathsf{h.c.}\right)dt$
(63) $\displaystyle+$ $\displaystyle\tfrac{1}{2}\Delta
D_{\mathrm{Q}}^{\beta\alpha}\left(\hat{L}_{\alpha}(x)\hat{\sigma}\hat{L}_{\beta}^{\dagger}(x)-\mathbb{H}\hat{L}_{\beta}^{\dagger}(x)\hat{L}_{\alpha}(x)\hat{\sigma}\right)dt.$

The first line corresponds to the perfect purifying unravelling, at the
minimum of noise (62) whereas the decoherence term in the second line counters
the purification. A highly mixed state becomes purer, a low level of mixture
becomes higher. Mixedness may have a stationary value for certain HMEs and
certain unravellings. We notice that the family of mixed state unravellings is
even larger since we can always set

$d\xi d\xi^{\dagger}=\eta D_{\mathrm{Qmin}},~{}~{}~{}~{}(0<\eta\leq 1).$ (64)

###  IV.3 Monitoring the diffusive quantum trajectory

We are going to show that, similarly to the jump trajectories in sec. II.2,
also diffusive quantum trajectories $\psi_{t}$ can be monitored if the
classical trajectories $x_{t}$ are observed. As we shall see, this option of
monitoring constrains the parameters of the HME (36) and singles out a unique
unraveling among the infinite many.

Monitoring the quantum trajectory $\psi_{t}$ via monitoring the classical
$x_{t}$ is possible if and only if $d\psi$ (50) uniquely depends on $dx$ (49).
Hence, the vector $d\xi$ must be a linear function of the vector $dW$:

$d\xi=F_{\mathrm{QC}}dW.$ (65)

This deterministic relationship removes the ambiguity of the unravelling
because it also specifies $d\xi d\xi$ that was free correlation, see eqs.
(52), in general unravelings. The above relationship should be consistent with
the correlations (52). They imply two equations:
$D_{\mathrm{Q}}=F_{\mathrm{QC}}D_{\mathrm{C}}F_{\mathrm{QC}}^{\dagger}$ and
$G_{\mathrm{CQ}}^{\dagger}=F_{\mathrm{QC}}D_{\mathrm{C}}$. These equations
possesses the solution:

$F_{\mathrm{QC}}=G_{\mathrm{CQ}}^{\dagger}\frac{1}{D_{\mathrm{C}}},$ (66)

and a constraint on the HME’s parameters:

$D_{\mathrm{Q}}=G_{\mathrm{CQ}}^{\dagger}\frac{1}{D_{\mathrm{C}}}G_{\mathrm{CQ}}.$
(67)

This constraint is the condition that monitoring $\psi_{t}$ be possible.
Namely, we insert the solution (66) into (65) to obtain the desired map of
$dW$ into $d\xi$:

$d\xi=G_{\mathrm{CQ}}^{\dagger}\frac{1}{D_{\mathrm{C}}}dW.$ (68)

It is remarkable that this equation defines covariant $d\xi$, removes the
ambiguity of $d\xi d\xi$.

The condition (67) of monitoring coincides with the saturated condition (45)
but, importantly, the other condition (46) of minimum noise is not necessary
for monitoring. Accordingly, the option of monitoring is still guaranteed
above the noise threshold: $\mathsf{range}D_{\mathrm{C}}$ can be larger than
$\mathsf{range}G_{\mathrm{CQ}}G_{\mathrm{CQ}}^{\dagger}$. At some $x$’s, there
can be some componenents of $x$ that are not coupled to the quantum subsystem,
represent above threshold noise, redundant for and do not prevent the
monitoring of $\psi_{t}$.

Using (68), we can eliminate $d\xi$ from the equations of unraveling. Then
both $dx$ and $d\psi$ (or $d\hat{\sigma}$) are driven by the same real noise
$dW$. Accordingly, the eq. (50) becomes the following Layton et al. (2022):

$d\psi=-i\left(\hat{H}(x)+\hat{H}_{\mathrm{fr}}(x)\right)\psi
dt+dW^{n}(x)[D_{\mathrm{C}}^{-1}(x)]_{nm}G_{\mathrm{CQ}}^{m\alpha}(x)\left(\hat{L}_{\alpha}(x)-\langle\hat{L}_{\alpha}(x)\rangle\right)\psi.$
(69)

In the special reference frame where
$G_{\mathrm{CQ}}^{n\alpha}=\delta^{n\alpha}$ and we consider the minimum noise
threshold where the non-degenerate $D_{\mathrm{Q}},D_{\mathrm{C}}$ are real
and inverses of each other, we have
$d\xi=D_{\mathrm{C}}^{-1}dW=D_{\mathrm{Q}}dW$, the eqs. (49,50) of unraveling
reduce to the following ones :

$\displaystyle dx^{\alpha}$ $\displaystyle=$ $\displaystyle
V^{\alpha}(x)dt-2\mathbb{Re}\langle\hat{L}_{\alpha}(x)\rangle
dt+dW_{\alpha}(x),$ (70) $\displaystyle d\psi$ $\displaystyle=$
$\displaystyle-i(\hat{H}(x)-i\hat{H}_{\mathrm{fr}}(x))\psi dt$
$\displaystyle+\bigl{(}\hat{L}_{\alpha}(x)-\langle\hat{L}_{\alpha}(x)\rangle\bigr{)}\psi
D_{\mathrm{Q}}^{\alpha\beta}dW_{\beta}(x).$

Notice the new notation $dW_{\alpha}\equiv dW^{n}|_{n=\alpha}$. We recognize
the equations of correlated time-continuous measurements (monitoring) of the
observables $\hat{L}_{\alpha}+\mathsf{h.c.}$ where $x_{t}$ is the measured
signal. Here the model is a bit more general because the signal can have an
autonomous drift, the Hamiltonian and the monitored observables can depend on
the measured signal.

##  V Discussion

In this work we revisited, clarified and completed earlier results on the
Markovian master and stochastic equations of hybrid quantum-classical
dynamics, paying attention to simplicity and brievity.

The starting concept and the derivation of the HME in sec. II is most similar
to the rigorous formulations of Blanchard and Jadczyk Blanchard and Jadczyk
(1993, 1995). The shift-invariance (II) of the HME and the related
nonuniqueness of the unraveling was not recognized before. Using vectorized
classical variables is useful alternative to the sophisticated conditions of
monitoring the jump quantum trajectory in Oppenheim et al. (2023).

Derivation in sec. III of the diffusive HME (35) from the discrete (11) is
completely new, in attempt to replace the complicated though perhaps more
rigorous procedure of Oppenheim et al. Oppenheim et al. (2022a). Our
derivation is based on the discrete forerunners of $\delta(x-y)$ and
$\partial/\partial x^{n}$. These are nontrivial while allowing for an
elementary derivation. Importantly, the naive choice
$\hat{L}_{\alpha}(x,y)=\hat{L}_{\alpha}(x)\delta(x-y)$ and
$L_{n}(x,y)=|x\rangle\partial_{n}\langle x|$ instead of (27,28) turn out to be
incorrect because off-block-diagonal terms play a role 111J. Oppenheim private
communication.. The naive choice gave correct structure of the HME but below
the correct threshold of minimum noise by a factor $1/2$.

Sec. IV re-derives the HME (36) which was derived already by Oppenheim et al.
Oppenheim et al. (2022a) and Layton et al. Layton et al. (2022). These works
did not mention the covariance of their result, neither put it in the usual
form of co- and contravariant indices. Our work emphasizes and exploits that
the HME is explicit covariant for local linear (i.e.: not necessarily unitary)
transformations of the Lindblad generators and for general transformations of
the classical coordinates.

Sec. IV.1 presents a pretty compact condition (41) of the threshold for
minimum noise. The phenomenon and equation (42) of trade-off between
decoherence and diffusion was recognized in Diósi (1995), and has been
extended recently for for the general diffusive HME in Oppenheim et al.
(2022a), see eqs. (43-46) using general matrix inverses to cover degenerate
matrices of decoherence, diffusion, and backaction; their degeneracies are not
exceptions but typical.

Our important new contribution in sec. IV.2 is that the pure state diffusive
unravellings of a diffusive HME are always possible and are exactly as
ambiguous as the standard unravellings of GKLS master equations. Since the
ambiguity coincides with that of the unravelings of pure quantum GKLS master
equations, we can fix them in the same way. The choice $d\xi d\xi=0$ is well-
known in theory of quantum state diffusion, used for covariance in Diósi
(1986) and developed by Gisin and Percival Gisin and Percival (1992); Percival
(1998) in the GKLS and Ito fomalisms. The full multitude of unravellings,
applicable to the hybrid dynamics as well, is discussed by Wiseman and the
author in Wiseman and Diósi (2001).

Sec. IV.3 postulates the covariant condition (65) of quantum trajectory
monitoring, not used before. The resulting eqs. of monitoring coincide with
those in Layton et al. (2022). Their claim that these eqs. are in one-to-one
correspondence with the HME is confirmed by covariance when the HME is at the
threshold of minimum noise. The option of monitoring is not restricted for the
HME of minimum noise, diffusion (not decoherence) can be higher than the
threshold.

##  VI Conclusion

Rarely stated explicitly, but the interaction between quantum and classical
systems has no other consistent mathematical model than time-continuous
quantum measurement and feedback, where measurement outcomes form the
variables of the classical system. This echoes von Neumann’s visionary
postulate. To obtain a classical variable correlated with an unknown quantum
state the only consistent mathematical model is the von Neumann quantum
measurement. Not too surprizing, the equations of hybrid stochastic
unravelings, both discrete and continuous, coincide with the respective
equations of time-continuous measurement, provided the classical system is
identified with the measurement outcomes, as in the elementary case (2).

The unravellings (statistical interpretation) of hybrid master equations are
mathematical equivalents of time-continuous quantum measurements as mentioned
e.g. in Diósi (2014). The advantage of hybrid master equations and
unravellings over time-continuous quantum measurement is not yet conceptual.
The hybrid formalism may be fairly convenient in many applications, let them
be e.g. foundations or improved semi-classical gravity. No doubt, it may
develop its own metaphysics as well.

###### Acknowledgements.

The author is obliged for the extended valuable discussions with Jonathan
Oppenheim and Isaac Layton. This research was funded by the Foundational
Questions Institute and Fetzer Franklin Fund, a donor-advised fund of the
Silicon Valley Community Foundation (Grant No’s. FQXi-RFPCPW-2008, FQXi-
MGA-2103), the National Research, Development and Innovation Office (Hungary)
“Frontline” Research Excellence Program (Grant No. KKP133827), and the John
Templeton Foundation (Grant 62099).

##  Appendix A Diffusive HME of two-level quantum system

Consider a two-level quantum system coupled to a classical system of a single
variable $x$. In Pauli’s formalism, we can write the hybrid density (1) in the
general form:

$\hat{\rho}(x)=\tfrac{1}{2}(1+\hat{s}(x))\rho(x),$ (71)

where
$\hat{s}(x)=s_{1}(x)\hat{\sigma}_{1}+s_{2}\hat{\sigma}_{2}+s_{3}\hat{\sigma}_{3}$
and the length $s=|\mathbf{s}|$ of the Bloch-vector
$\mathbf{s}=(s_{1},s_{2},s_{3})$ must satify $s\leq 1$. Let the diffusive HME
be the following simple one:

$\frac{d\hat{\rho}(x)}{dt}=\hat{\sigma}_{3}\hat{\rho}(x)\hat{\sigma}_{3}-\hat{\rho}(x)+G[\hat{\sigma}_{3},\hat{\rho}^{\prime}(x)]_{+}+\tfrac{1}{2}\hat{\rho}^{\prime\prime}(x).$
(72)

This corresponds to $D_{\mathrm{Q}}=D_{\mathrm{C}}=1$, and $G$ is real. We
prove that $|G|$ cannot be larger than $1$.

Substitute eq. (71) and multiply both sides by $2$, yielding:

$\frac{d}{dt}\left((1+\hat{s})\rho\right)=[\hat{\sigma}_{3}(1+\hat{s})\hat{\sigma}_{3}-(1+\hat{s})]\rho+G[\hat{\sigma}_{3},[(1+\hat{s})\rho]^{\prime}]_{+}+\tfrac{1}{2}[(1+\hat{s})\rho]^{{}^{\prime\prime}}.$
(73)

An equivalent form reads:

$\frac{d\hat{s}}{dt}\rho+\hat{s}\frac{d\rho}{dt}=-2\hat{s}_{\perp}\rho+2G\hat{\sigma}_{3}\rho^{\prime}+\tfrac{1}{2}\hat{s}\rho^{{}^{\prime\prime}}+\hat{s}^{\prime}\rho^{\prime}+\tfrac{1}{2}\hat{s}^{{}^{\prime\prime}}\rho.$
(74)

We take trace of both sides, yielding the equation
$d\rho/dt=+2G(s_{3}\rho)^{\prime}+\tfrac{1}{2}\rho^{{}^{\prime\prime}}$. If we
substitute it back, we get

$\frac{d\hat{s}}{dt}\rho=-2\hat{s}_{\perp}\rho+2G\hat{\sigma}_{3}\rho^{\prime}+\hat{s}^{\prime}\rho^{\prime}+\tfrac{1}{2}\hat{s}^{{}^{\prime\prime}}\rho-2G\hat{s}(s_{3}\rho)^{\prime},$
(75)

where $\hat{s}_{\perp}=s_{1}\hat{\sigma}_{1}+s_{2}\hat{\sigma}_{2}$. We
multiply both sides by $\hat{s}$ and take $\tfrac{1}{2}$ times their trace
again:

$\frac{1}{2}\frac{ds^{2}}{dt}\rho=-2s_{\perp}^{2}\rho+2Gs_{3}(1-s^{2})\rho^{\prime}+(s^{2})^{\prime}\rho^{\prime}+\tfrac{1}{2}\mathbf{s}\mathbf{s}^{{}^{\prime\prime}}\rho-2Gs^{2}s_{3}^{\prime}\rho.$
(76)

Suppose $s^{2}=1$, then $ds^{2}/dt$ cannot be positive and $(s^{2})^{\prime}$
must vanish. Thus we have the following inequality:

$0\geq-2s_{\perp}^{2}+\tfrac{1}{2}\mathbf{s}\mathbf{s}^{{}^{\prime\prime}}-2Gs^{2}s_{3}^{\prime}.$
(77)

Now we insert the anzatz $\mathbf{s}(x)=(\cos(x),0,\sin(x))$ into the
inequality, leading to

$0\geq\frac{1}{2}\frac{ds^{2}}{dt}=-2\cos^{2}(x)-2G\cos(x)-\tfrac{1}{2},$ (78)

which must be satisfied for all $x$. This is equivalent with the upper bound
on the backaction coupling:

$G^{2}\leq 1.$ (79)

##  Appendix B Hamiltonian HME

Let $\hat{H}(x)$ be a our hybrid Hamiltonian where the classical subsystem is
canonical. The first $N$ canonical variables $\\{x^{n};n=1,\dots,N\\}$ are the
coordinates and the second $N$ ones $\\{x^{n};n=N+1,\dots,2N\\}$ are the
momenta. The HME takes this form:

$\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}]+\mathbb{H}\\{\hat{H},\hat{\rho}\\}+\dots~{},$
(80)

where the ellipsis stands for mandatory decoherence and diffusion terms. The
bracket $\\{\hat{H},\hat{\rho}\\}$ is the Poisson bracket of classical
canonical theory:

$\\{\hat{H},\hat{\rho}\\}=\varepsilon^{nm}(\partial_{n}\hat{H})(\partial_{m}\hat{\rho}),$
(81)

where $\varepsilon^{nm}$ is the $2N\times 2N$ symplectic matrix. To determine
the decoherence and diffusion terms, we note the hybrid Hamiltonian can always
have the following form:

$\hat{H}(x)=\hat{H}_{\mathrm{Q}}+H_{\mathrm{C}}(x)+\hat{H}_{\mathrm{CQ}}(x)$
(82)

with

$\hat{H}_{\mathrm{CQ}}(x)=h^{\alpha}(x)\hat{L}_{\alpha},$ (83)

where $h^{\alpha}(x)$’s are real functions and the
$\hat{L}_{\alpha}^{\prime}s$ are linearly independent Hermitian operators,
also linearly independent of $\hat{I}$. Accordingly, the HME contains a purely
Hamiltonian term $-i[\hat{H}_{\mathrm{Q}}+\hat{H}_{\mathrm{CQ}},\hat{\rho}]$,
a purely classical term $\\{H_{\mathrm{C}},\hat{\rho}\\}$, a backaction term
$\mathbb{H}\\{\hat{H}_{\mathrm{CQ}},\hat{\rho}\\}$, and the mandatory
decoherence and diffusive terms. It is easy to see that
$\\{H_{\mathrm{C}},\hat{\rho}\\}$ corresponds to classical drift velocity
$V^{n}=\varepsilon^{nm}\partial_{m}H_{\mathrm{C}}$. By identifying the
backaction term

$\mathbb{H}\\{\hat{H}_{\mathrm{CQ}},\hat{\rho}\\}=\mathbb{H}\\{h^{\alpha}\hat{L}_{\alpha},\hat{\rho}\\}=\varepsilon^{nm}\mathbb{H}(\partial_{n}h^{\alpha})\hat{L}_{\alpha}(\partial_{m}\hat{\rho})$
(84)

in eq. (80) with the covariant HME’s backaction term
$2\mathbb{H}\partial_{n}(\overline{G}_{\mathrm{CQ}}^{n\alpha}\hat{L}_{\alpha}\hat{\rho})$,
we read out the matrix of backaction:

$G_{\mathrm{CQ}}^{n\alpha}=\overline{G}_{\mathrm{CQ}}^{n\alpha}=-\tfrac{1}{2}\varepsilon^{nm}\partial_{m}h^{\alpha}.$
(85)

Hence, the Hamiltonian HME takes this general form:

$\frac{d\hat{\rho}}{dt}=-i[\hat{H}_{\mathrm{Q}}+\hat{H}_{\mathrm{CQ}},\hat{\rho}]+\\{H_{\mathrm{C}},\hat{\rho}\\}+\mathbb{H}\\{\hat{H}_{\mathrm{CQ}},\hat{\rho}\\}+D_{\mathrm{Q}}^{\beta\alpha}\Bigl{(}\hat{L}_{\alpha}\hat{\rho}\hat{L}_{\beta}^{\dagger}-\mathbb{H}\hat{L}_{\beta}^{\dagger}\hat{L}_{\alpha}\hat{\rho}\Bigr{)}+\tfrac{1}{2}\partial_{n}\partial_{m}(D_{\mathrm{C}}^{nm}\hat{\rho}),$
(86)

and the matrices $D_{\mathrm{Q}},D_{\mathrm{C}},G_{\mathrm{CQ}}$ should
satisfy the non-negativity constraints discussed in secs. III,IV. Note that
the decoherence and diffusion terms do not retain the covariance for canonical
transformations of the classical coordinates $\\{x^{n}\\}$.

Let us apply this Hamiltonian HME to the simple hybrid system of two identical
harmonic oscillators, one is quantized the other is classical. The classical
canonical variables are $(x^{1},x^{2})\equiv(q,p)$, the quantized ones are
$(\hat{Q},\hat{P})$. Let the hybrid Hamiltonian (82) consist of

$\displaystyle\hat{H}_{\mathrm{Q}}$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}(\hat{Q}^{2}+\hat{P}^{2}),$ (87) $\displaystyle
H_{\mathrm{C}}(q,p)$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}(q^{2}+p^{2}),$ (88)
$\displaystyle\hat{H}_{\mathrm{CQ}}(q,p)$ $\displaystyle=$ $\displaystyle
g(q\hat{Q}+p\hat{P}).$ (89)

In expansion (83) of $\hat{H}_{\mathrm{CQ}}$ we choose
$\hat{L}_{1}=\hat{Q},~{}\hat{L}_{2}=\hat{P}$ then
$h^{1}=gq=gx^{1},~{}h^{2}=gp=gx^{2}$ hence $G^{12}=-G^{21}=g/2$ while
$G^{11}=G^{22}=0$. The minimum noise condition reads, e.g.,
$GD_{\mathrm{Q}}^{-1}G^{T}=D_{\mathrm{C}}$ which means that both
$D_{\mathrm{Q}}$ and $D_{\mathrm{C}}$ are $2\times 2$ diagonal matrices and
satisfy

$D_{\mathrm{Q}}^{11}D_{\mathrm{C}}^{22}=D_{\mathrm{Q}}^{22}D_{\mathrm{C}}^{11}=g^{2}/4.$
(90)

The resulting HME of $\hat{\rho}(q,p)$ takes this form:

$\displaystyle\frac{d\hat{\rho}}{dt}$ $\displaystyle=$
$\displaystyle-i[\tfrac{1}{2}(\hat{Q}^{2}+\hat{P}^{2})+g(q\hat{Q}+p\hat{P}),\hat{\rho}]+(q\partial_{p}\hat{\rho}-p\partial_{q}\hat{\rho}+g\mathbb{H}(\hat{Q}\partial_{p}\hat{\rho}-\hat{P}\partial_{q}\hat{\rho})$
(91)
$\displaystyle+\tfrac{1}{2}D_{\mathrm{Q}}^{11}[\hat{Q},[\hat{Q},\hat{\rho}]]+\tfrac{1}{2}D_{\mathrm{Q}}^{22}[\hat{P},[\hat{P},\hat{\rho}]]+\tfrac{1}{2}D_{\mathrm{C}}^{11}\partial_{q}^{2}\hat{\rho}+\tfrac{1}{2}D_{\mathrm{C}}^{22}\partial_{q}^{2}\hat{\rho}.$

We see that both $\hat{Q}$ and $\hat{P}$ contribute to non-vanishing
decoherence terms and both $q$ and $p$ must diffuse. As we noticed, the
mandatory decoherence and diffusion terms violate the canonical invariance of
the classical subsystem. And, of course, they remain non-unique even after
taking the constraints (90) into the account .

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