reversal operation $\hat{T}\equiv\hat{K}$ is complex-conjugation in the
eigenbasis of the steady state. Explicitly:

$\displaystyle|\psi_{K}\rangle=\sum_{n}\sqrt{p_{n}}|n\rangle|n\rangle.$ (133)

Furthermore, $\hat{O}\to\hat{O}^{T}$ denotes matrix transposition in the
eigenbasis of the steady state.

An arbitrary time-reversal operation can be decomposed as
$\hat{T}\equiv\hat{V}\hat{K}$, where $\hat{V}$ commutes with $\hat{\rho}_{\rm
ss}$. Explicitly, an arbitrary thermofield doubled state corresponding to a
hidden TRS must always have the expression

$\displaystyle|\psi_{T}\rangle$ $\displaystyle=\sum_{n}\sqrt{p_{n}}|n\rangle
e^{i\theta_{n}}|n\rangle,$ (134)
$\displaystyle\equiv\sum_{n}\sqrt{p_{n}}|n\rangle\hat{V}\hat{K}|n\rangle.$
(135)

Under the above rules, the thermofield doubled state with an arbitrary time-
reversal operation for a unitary $\hat{V}$ satisfies the following identities:

$\displaystyle\hat{O}_{A}|\psi_{T}\rangle$
$\displaystyle=\hat{O}_{A}\hat{V}_{B}|\psi_{K}\rangle\to\hat{O}\hat{\rho}_{\rm
ss}^{1/2}\hat{V}^{T}$ (136) $\displaystyle\hat{O}_{B}|\psi_{T}\rangle$
$\displaystyle=(\hat{O}\hat{V})_{B}|\psi_{K}\rangle\to\hat{\rho}_{\rm
ss}^{1/2}(\hat{O}\hat{V})^{T}$ (137)

We now verify the formula $\mathcal{J}[\hat{O}]=\hat{\rho}_{\rm
ss}^{1/2}\tilde{O}^{\dagger}\hat{\rho}_{\rm ss}^{-1/2}$ used in the main text.
First of all, any hidden TRS must leave the steady state invariant, and we can
rewrite this formula as

$\displaystyle\mathcal{J}[\hat{O}]\equiv\hat{\rho}_{\rm
ss}^{1/2}\hat{O}^{T}\hat{\rho}_{\rm ss}^{-1/2}$ (138)

We can then proceed to verify Eq. (130) directly:

$\displaystyle(\hat{\rho}_{\rm ss}^{1/2}\hat{O}\hat{\rho}_{\rm
ss}^{-1/2})_{B}|\psi_{T}\rangle$
$\displaystyle\to\hat{\rho}^{1/2}_{ss}\hat{V}^{T}(\hat{\rho}_{\rm
ss}^{1/2}\hat{O}^{T}\hat{\rho}_{\rm ss}^{-1/2})^{T}.$ (139)

Taking advantage of the fact that $\hat{V}$ commutes with $\hat{\rho}_{\rm
ss}$, we have

$\displaystyle\hat{\rho}^{1/2}_{ss}\hat{V}^{T}(\hat{\rho}_{\rm
ss}^{1/2}\hat{O}^{T}\hat{\rho}_{\rm
ss}^{-1/2})^{T}=\hat{O}\hat{V}^{T}\hat{\rho}_{\rm
ss}^{1/2}=\hat{O}\hat{\rho}_{\rm ss}^{1/2}\hat{V}^{T},$ (140)

which is just the operator representation of $\hat{O}_{A}|\psi_{T}\rangle$
according to the rules Eqs. (136-137).

##  Appendix E From CQDB to hidden TRS

In this appendix, we will show that, for systems with modular symmetry, CQDB
and hidden TRS are equivalent. Since CQDB already implies modular symmetry by
itself Kossakowski  _et al._ (1977); Bratteli and Haagerup (1978), this will
demonstrate that CQDB necessarily implies hidden TRS. Therefore, CQDB is a
strict subphenomenon of hidden TRS.

Indeed, consider an arbitrary driven-dissipative system described by a
Lindblad master equation. The definition of the exchange superoperator tells
us that

$\displaystyle C_{X,\mathcal{J}[Y]}(t)$
$\displaystyle=C_{X,Y}^{\text{TFD}}(t),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \forall t\geq
0.$ (141)

The above correlation function, for negative times, is harder to rephrase as a
two-site quantity:

$\displaystyle C_{X,\mathcal{J}[Y]}(-t)$
$\displaystyle=\langle(\hat{\rho}^{-1/2}_{ss}\hat{Y}\hat{\rho}^{1/2}_{ss})(t)\tilde{X}^{\dagger}\rangle.$
(142)

However, the above expression simplifies considerably in systems with modular
symmetry. Indeed, let us now assume that the driven-dissipative system in
question is symmetric with respect to the modular Hamiltonian
$\hat{H}_{\rho}$, as is always the case with systems that have CQDB
Kossakowski  _et al._ (1977). Then, we can write
$(\hat{\rho}^{-1/2}_{ss}\hat{Y}\hat{\rho}^{1/2}_{ss})(t)=\hat{\rho}^{-1/2}_{ss}\hat{Y}(t)\hat{\rho}^{1/2}_{ss}$.
Substituting this identity into Eq. (142), we get

$\displaystyle C_{X,\mathcal{J}[Y]}(-t)$
$\displaystyle=C_{X,Y}^{\text{TFD}}(-t),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \forall t\geq
0.$ (143)

Therefore, for any driven-dissipative system with modular symmetry,

$\displaystyle C_{X,\mathcal{J}[Y]}(t)$
$\displaystyle=C_{X,Y}^{\text{TFD}}(t)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \forall
t\in\mathbb{R}.$ (144)

Therefore, since $\mathcal{J}$ is a bijection of the observable algebra, CQDB
and hidden TRS are equivalent for this class of systems, as time-symmetry of
one set of correlation functions implies time-symmetry of the other.

##  Appendix F Complex-$P$ and hidden TRS correspondence theorem

We now consider a general many-body bosonic Lindblad master equation, with,
say, $n$ bosonic modes. Recall that classical detailed balance for this class
of master equations can be formulated in the complex-$P$ representation as the
time-symmetry of the correlation function

$\displaystyle
C^{\mathcal{P}}_{X,Y}(t)\equiv\begin{cases}\overline{X(t)Y(0)}&t\geq 0\\\
\overline{\tilde{Y}(-t)\tilde{X}(0)}&t<0\end{cases}$ (145)

where, here, $X(\vec{\alpha},\vec{\beta}),Y(\vec{\alpha},\vec{\beta})$ are
arbitrary multivariate holomorphic functions, and averages are understood to
be taken with respect to the stationary complex-$P$ distribution, e.g.

$\displaystyle\overline{X}\equiv\int_{\Sigma}d^{n}\alpha
d^{n}\beta\,P_{ss}(\vec{\alpha},\vec{\beta})X(\vec{\alpha},\vec{\beta}),$
(146)

where $\Sigma$ is a $2n$-dimensional closed integration surface (the many-body
analogue of the pair of integration contours
$\mathcal{C},\mathcal{C}^{\prime}$ mentioned in the main text). Furthermore,
$X(\vec{\alpha},\vec{\beta})\to\tilde{X}(\vec{\alpha},\vec{\beta})$ is a well-
defined classical time-reversal operation, that is, induced by a bijection of
the phase-space squaring to one (c.f. Eq. (91) in the main text).

Recall that hidden TRS for a quantum system is defined as the time-symmetry of
the thermofield doubled correlation function, which, e.g. for positive times,
is given by a symmetric bilinear form:

$\displaystyle
C_{\hat{X},\hat{Y}}^{\text{TFD}}(t)\equiv\langle\\!\langle\hat{X}(t),\hat{Y}\rangle\\!\rangle_{\mathcal{T}},$
(147)

and for negative times (with a time-reversal invariant steady state), is
expressible using the same bilinear form:

$\displaystyle
C_{\hat{X},\hat{Y}}^{\text{TFD}}(-t)\equiv\langle\\!\langle\hat{X},\hat{Y}(t)\rangle\\!\rangle_{\mathcal{T}},$
(148)

where the relevant bilinear form (which depends on a particular choice of
time-reversal operator $\hat{T}$) is defined in the main text as follows:

$\displaystyle\boxed{\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{T}}\equiv
C^{\text{TFD}}_{\hat{X},\hat{Y}}(t\equiv 0)}$ (149)

Clearly, the condition of hidden TRS is equivalent to identifying the left-
hand side of Eq. (147) with the left-hand side of Eq. (148), which, upon
examining the right-hand sides of said equations, is equivalent to the
symmetry of the Heisenberg-picture time-evolution superoperator
$\mathcal{E}_{t}\equiv e^{-t\bar{\mathcal{L}}}$ with respect to the bilinear
form $\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathcal{T}}$.

One can form an analogous construction to express classical correlations in
the complex-$P$ distribution in terms of a bilinear form. Indeed, for any pair
of normally-ordered quantum observables $\hat{X},\hat{Y}$ and a classical
time-reversal operation $X\to\tilde{X}$, one can write

$\displaystyle
C_{X,\tilde{Y}}^{\mathcal{P}}(t)=\langle\\!\langle\hat{X}(t),\hat{Y}\rangle\\!\rangle_{\mathcal{P}},$
(150)

and for negative times (with a stationary complex-$P$ distribution which is
time-reversal invariant), we can also write:

$\displaystyle
C_{X,\tilde{Y}}^{\mathcal{P}}(-t)=\langle\\!\langle\hat{X},\hat{Y}(t)\rangle\\!\rangle_{\mathcal{P}},$
(151)

Where we have introduced a new symmetric bilinear form

$\displaystyle\boxed{\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{P}}\equiv
C_{X,\tilde{Y}}^{\mathcal{P}}(t\equiv 0)}$ (152)

where $X,Y$ are the normally-ordered symbols of $\hat{X},\hat{Y}$, i.e. the
classical representatives of the observables $\hat{X},\hat{Y}$ in the
complex-$P$ representation. Explicitly, if
$\hat{X}=\sum_{IJ}c_{IJ}(\hat{a}_{1}^{\dagger})^{i_{1}}\cdots(\hat{a}_{n}^{\dagger})^{i_{n}}\hat{a}_{1}^{j_{1}}\cdots\hat{a}_{n}^{j_{n}}$,
then the normally-ordered symbol $X$ is

$\displaystyle
X=\sum_{IJ}c_{IJ}\beta^{i_{1}}\cdots\beta^{i_{n}}\alpha_{1}^{j_{1}}\cdots\alpha_{n}^{j_{n}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \text{etc..}$
(153)

where $I\equiv\\{i_{1},\cdots,i_{n}\\}$ and $J\equiv\\{j_{1},\cdots,j_{n}\\}$
are multi-indices. The condition of classical detailed balance in the
complex-$P$ representation, i.e. the time-symmetry of Eq. (145), is equivalent
to identifying the left-hand side of Eq. (150) with the left-hand side of Eq.
(151), which is equivalent to the symmetry of the Heisenberg-picture time-
evolution superoperator $\mathcal{E}_{t}\equiv e^{-t\bar{\mathcal{L}}}$ with
respect to the bilinear form
$\langle\\!\langle\cdot,\cdot\rangle\\!\rangle_{\mathcal{P}}$.

With these definitions in hand, we return to the general problem of interest:
a Markovian multi-mode bosonic system where each mode is subject to loss. For
simplicity, we start by assuming each mode has the same loss rate, implying a
master equation of the form

$\displaystyle\frac{d}{dt}\hat{\rho}=-i[\hat{H},\hat{\rho}]+\sum_{j=1}^{N}\kappa\mathcal{D}[\hat{a}_{j}].$
(154)

If this system has hidden TRS with respect to a particular quantum time-
reversal operation $\hat{T}$, then as discussed in the main text (c.f. Eq.
(43)) there must exist a $N\times N$ unitary matrix $U$ such that:

$\displaystyle\mathcal{J}[\hat{a}_{j}]=\sum_{kj}U_{kj}\hat{a}_{k},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ U^{2}\equiv 1.$
(155)

Here (as always) $\mathcal{J}$ is the exchange superoperator.

What we will show in this section is a remarkable coincidence between the two
families of bilinear forms, under the above jump operator constraint. That is,
Eq. (155) implies that we can identify both bilinear forms, i.e.

$\displaystyle\boxed{\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{P}}\equiv\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{T}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\forall\hat{X},\hat{Y}}$ (156)

where the classical time-reversal operation on the complex-$P$ side is none
other than the change-of-Kraus representation given in Eq. (155):
$\widetilde{(\vec{\alpha},\vec{\beta})}\equiv(U\vec{\alpha},U\vec{\beta})$. In
what follows, we first prove this result in the single mode case, then extend
to the multi-mode case where each mode has an identical damping rate $\kappa$.
Finally, we extend the result to the more general case where each mode has a
different damping rate $\kappa_{j}$.

####  F.0.1 Single-mode case

Consider Eq. (154) in the single mode limit $N=1$. If this system has hidden
TRS with respect to a particular quantum time-reversal operation $\hat{T}$,
then the jump operator constraint reduces to $\mathcal{J}[\hat{a}]=u\hat{a}$,
with $u=\pm 1$ a scalar quantity. We show in what follows that this in turn
implies the following identity:

$\displaystyle\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{P}}\equiv\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{T}},$
(157)

where the bilinear form on the right-hand side is defined using the classical
time-reversal operation $\widetilde{(\alpha,\beta)}=(u\alpha,u\beta)$. To see
this, note that both the left- and right-hand sides are bilinear with respect
to $\hat{X},\hat{Y}$, and so it suffices to verify the above identity in a
basis of normally-ordered monomials. That is, without loss of generality, we
may assume that

$\displaystyle\hat{X}$
$\displaystyle=(\hat{a}^{\dagger})^{k}\hat{a}^{l},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \hat{Y}=(\hat{a}^{\dagger})^{p}\hat{a}^{q}.$ (158)

We begin by computing, e.g.

$\displaystyle\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{T}}$
$\displaystyle=\langle\psi_{T}|(\hat{a}^{\dagger})^{k}\hat{a}^{l}(\hat{b}^{\dagger})^{p}\hat{b}^{q}|\psi_{T}\rangle.$
(159)

Substituting-in the definition of the exchange superoperator, we get

$\displaystyle\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{T}}$
$\displaystyle=\langle(\hat{a}^{\dagger})^{k}\hat{a}^{l}\mathcal{J}[(\hat{a}^{\dagger})^{p}\hat{a}^{q}]\rangle,$
(160)

where, here, $\langle\hat{O}\rangle\equiv\text{Tr}[\hat{\rho}_{\rm
ss}\hat{O}]$ denotes a steady-state expectation value. Now, we utilize the
fact that
$\mathcal{J}[\hat{O}\hat{O}^{\prime}]=\mathcal{J}[\hat{O}^{\prime}]\mathcal{J}[\hat{O}]$
for generic $\hat{O},\hat{O}^{\prime}$:

$\displaystyle\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{T}}$
$\displaystyle=\langle(\hat{a}^{\dagger})^{k}\hat{a}^{l}(u\hat{a})^{q}\mathcal{J}[(\hat{a}^{\dagger})^{p}]\rangle.$
(161)

Finally, by direct computation, we also have that
$\mathcal{J}[\hat{O}^{\dagger}]=\hat{\rho}_{\rm
ss}\mathcal{J}[\hat{O}]^{\dagger}\hat{\rho}_{\rm ss}^{-1}$ for generic
$\hat{O}$, and so (via the cyclic nature of the trace)

$\displaystyle\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{T}}$
$\displaystyle=\langle(u\hat{a}^{\dagger})^{p}(\hat{a}^{\dagger})^{k}\hat{a}^{l}(u\hat{a})^{q}\rangle$
$\displaystyle=\int_{\mathcal{C}}d\alpha\int_{\mathcal{C}^{\prime}}d\beta\,P_{ss}(\alpha,\beta)\tilde{\beta}^{p}\beta^{k}\alpha^{l}\tilde{\alpha}^{q}$
(162)

Immediately, we recognize here the normally-ordered symbols of
$\hat{X},\hat{Y}$, which are, explicitly:

$\displaystyle\tilde{Y}(\alpha,\beta)=\tilde{\beta}^{p}\tilde{\alpha}^{q},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
X(\alpha,\beta)=\beta^{k}\alpha^{l}.$ (163)

With the above observation, we have thus proved Eq. (157), which establishes
the equivalence of generalized quantum detailed balance, that is, hidden TRS,
with classical detailed balance in the complex-$P$ representation. Note that
the potential conditions, as well as the original CQA method, as originally
formulated in Stannigel  _et al._ (2012), both correspond in this context to
the special case of a trivial TRS, i.e. $U=1$.

####  F.0.2 Many-body case

Now, consider the multi-mode master equation Eq. (154). If this system has
hidden TRS with respect to a particular quantum time-reversal operation
$\hat{T}$, then we have the constraint Eq. (155) on the jump operators, which
we write as

$\displaystyle\mathcal{J}[\hat{a}_{j}]=\hat{\alpha}_{j},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\hat{\alpha}_{j}\equiv\sum_{jk}U_{jk}\hat{a}_{k}$ (164)

What we will demonstrate in this section is that this jump operator constraint
implies the following identity:

$\displaystyle\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{P}}\equiv\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{T}},$
(165)

where the bilinear form on the right-hand side is defined using the classical
time-reversal operation
$\widetilde{(\vec{\alpha},\vec{\beta})}=(U\vec{\alpha},U\vec{\beta})$. To see
this, note that both the left- and right-hand sides are bilinear with respect
to $\hat{X},\hat{Y}$, and so it suffices to verify the above identity in a
basis of normally-ordered monomials. That is, without loss of generality, we
may assume that

$\displaystyle\hat{X}=(\hat{a}_{1}^{\dagger})^{k_{1}}\cdots(\hat{a}_{n}^{\dagger})^{k_{n}}\hat{a}_{1}^{l_{1}}\cdots\hat{a}_{n}^{l_{n}},$
(166)
$\displaystyle\hat{Y}=(\hat{a}_{1}^{\dagger})^{p_{1}}\cdots(\hat{a}_{n}^{\dagger})^{p_{n}}\hat{a}_{1}^{q_{1}}\cdots\hat{a}_{n}^{q_{n}}.$
(167)

We begin by computing, e.g.

$\displaystyle\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{T}}$
$\displaystyle=\langle\psi_{T}|(\hat{a}_{1}^{\dagger})^{k_{1}}\cdots(\hat{a}_{n}^{\dagger})^{k_{n}}\hat{a}_{1}^{l_{1}}\cdots\hat{a}_{n}^{l_{n}}$
(168) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\cdot(\hat{b}_{1}^{\dagger})^{p_{1}}\cdots(\hat{b}_{n}^{\dagger})^{p_{n}}\hat{b}_{1}^{q_{1}}\cdots\hat{b}_{n}^{q_{n}}|\psi_{T}\rangle.$
(169)

Substituting-in the definition of the exchange superoperator, we get

$\displaystyle\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{T}}$
$\displaystyle=\langle(\hat{a}_{1}^{\dagger})^{k_{1}}\cdots(\hat{a}_{n}^{\dagger})^{k_{n}}$
(170) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\cdot\mathcal{J}[(\hat{a}_{1}^{\dagger})^{p_{1}}\cdots(\hat{a}_{n}^{\dagger})^{p_{n}}\hat{a}_{1}^{q_{1}}\cdots\hat{a}_{n}^{q_{n}}]\rangle,$
(171)

where, here, $\langle\hat{O}\rangle\equiv\text{Tr}[\hat{\rho}_{\rm
ss}\hat{O}]$ denotes a steady-state expectation value. Again, we now utilize
the fact that
$\mathcal{J}[\hat{O}\hat{O}^{\prime}]=\mathcal{J}[\hat{O}^{\prime}]\mathcal{J}[\hat{O}]$
for generic $\hat{O},\hat{O}^{\prime}$:

$\displaystyle\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{T}}$
$\displaystyle=\langle(\hat{a}_{1}^{\dagger})^{k_{1}}\cdots(\hat{a}_{n}^{\dagger})^{k_{n}}$
(172) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\cdot\hat{\alpha}_{1}^{q_{1}}\cdots\hat{\alpha}_{n}^{q_{n}}\mathcal{J}[(\hat{a}_{1}^{\dagger})^{p_{1}}\cdots(\hat{a}_{n}^{\dagger})^{p_{n}}]\rangle.$
(173)

Finally, by direct computation, we also have that
$\mathcal{J}[\hat{O}^{\dagger}]=\hat{\rho}_{\rm
ss}\mathcal{J}[\hat{O}]^{\dagger}\hat{\rho}_{\rm ss}^{-1}$ for generic
$\hat{O}$, and so

$\displaystyle\langle\\!\langle\hat{X},\hat{Y}\rangle\\!\rangle_{\mathcal{T}}$
$\displaystyle=\langle(\hat{\alpha}_{1}^{\dagger})^{p_{1}}\cdots(\hat{\alpha}_{n}^{\dagger})^{p_{n}}$
(174) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
\cdot(\hat{a}_{1}^{\dagger})^{k_{1}}\cdots(\hat{a}_{n}^{\dagger})^{k_{n}}\hat{a}_{1}^{l_{1}}\cdots\hat{a}_{n}^{l_{n}}\hat{\alpha}_{1}^{q_{1}}\cdots\hat{\alpha}_{n}^{q_{n}}\rangle$
$\displaystyle=\int_{\Sigma}\,d^{n}\alpha
d^{n}\beta\,P_{ss}(\vec{\alpha},\vec{\beta})\tilde{\beta}_{1}^{p_{1}}\cdots\tilde{\beta}_{n}^{p_{n}}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\cdot\beta_{1}^{k_{1}}\cdots\beta_{n}^{k_{n}}\alpha_{1}^{l_{1}}\cdots\alpha_{n}^{l_{n}}\tilde{\alpha}_{1}^{q_{1}}\cdots\tilde{\alpha}_{n}^{q_{n}}$
(175)

Immediately, we recognize here the normally-ordered symbols of
$\hat{X},\hat{Y}$, which are, explicitly:

$\displaystyle\widetilde{Y}(\vec{\alpha},\vec{\beta})$
$\displaystyle=\tilde{\beta}_{1}^{p_{1}}\cdots\tilde{\beta}_{n}^{p_{n}}\tilde{\alpha}_{1}^{q_{1}}\cdots\tilde{\alpha}_{n}^{q_{n}}$
(176) $\displaystyle X(\alpha,\beta)$
$\displaystyle=\beta_{1}^{k_{1}}\cdots\beta_{n}^{k_{n}}\alpha_{1}^{l_{1}}\cdots\alpha_{n}^{l_{n}}.$
(177)

With the above observation, we have thus proved Eq. (165), which establishes
the equivalence of generalized quantum detailed balance, that is, hidden TRS,
with classical detailed balance in the complex-$P$ representation.

Finally, consider the general case where each mode has a different damping
rate $\kappa_{j}$. In this case, the corresponding classical TRS is modified
to be

$\displaystyle U_{ij}\to\sqrt{\frac{\kappa_{i}}{\kappa_{j}}}U_{ij},$ (178)

which can be proven by trivially repeating the steps above.

##  Appendix G Doubled system correlation function symmetry for TRS $\psi=\pi$

In the main text we showed that the TFD correlation function
$C_{yz}^{\mathrm{TFD}}(t)$ is time symmetric for the TRS $\psi=\pi$. We must
still explicitly verify that the other two correlation functions, $C_{xy}^{\rm
TFD}(t)$ and $C_{xz}^{\rm TFD}(t)$, are time symmetric for the same TRS.

The expansion of the TFD correlation functions as in Eq. (19) is practically
useful for computing correlation functions if the time-dependent operators are
easily found because it reduces the problem to computing matrix elements. For
the qubit system, the time-dependent operators are readily written in terms of
the Liouvillian eigenmodes as

$\hat{\sigma}_{k}(t)\equiv\sum_{n}e^{\lambda_{n}t}{\rm
Tr}(\hat{\sigma}_{k}\hat{r}_{n})\hat{l}_{n}^{\dagger}.$ (179)

This definition ensures that the operators reproduce the correct time averages
and single system correlation functions under the hypotheses of the quantum
regression theorem. We can then immediately write the time evolved operators
as

$\displaystyle\hat{\sigma}_{x}(t)$ $\displaystyle=e^{-\kappa
t/2}\hat{\sigma}_{x}$ (180) $\displaystyle\hat{\sigma}_{y}(t)$
$\displaystyle=\langle\hat{\sigma}_{y}\rangle+\sum_{n=2}^{3}e^{\lambda_{n}t}{\rm
Tr}(\hat{\sigma}_{y}\hat{r}_{n})\hat{l}_{n}^{\dagger}$ (181)
$\displaystyle\hat{\sigma}_{z}(t)$
$\displaystyle=\langle\hat{\sigma}_{z}\rangle+\sum_{n=2}^{3}e^{\lambda_{n}t}{\rm
Tr}(\hat{\sigma}_{z}\hat{r}_{n})\hat{l}_{n}^{\dagger}$ (182)

With these in hand we proceed to compute the matrix elements.

We label the pointer states of the steady state by $|\pm\rangle$ satisfying
$\hat{\rho}_{\rm ss}|\pm\rangle=p_{\pm}|\pm\rangle$ for eigenvalues
$p_{\pm}=\frac{1}{2}(1\pm\sqrt{4b^{2}+1}/\sqrt{2b^{2}+1})$. The eigenstates
lie in the $YZ$ plane of the Bloch sphere, namely

$|+\rangle=\cos\theta|g\rangle-i\sin\theta|e\rangle$ (183)

for some angle $\theta$ whose precise value as a function of $b$ does not
concern us (except that $0<\theta<\pi/2$ so that these do not reduce to pure
$\hat{\sigma}_{y}$ or $\hat{\sigma}_{z}$ eigenstates). The matrix elements of
$\hat{\sigma}_{x}=\hat{\sigma}_{x}(0)$ and $\hat{\sigma}_{x}(t)$ follow
immediately as

$\displaystyle\langle\pm|\hat{\sigma}_{x}(t)|\pm\rangle$ $\displaystyle=0$
(184) $\displaystyle\langle-|\hat{\sigma}_{x}(t)|+\rangle$
$\displaystyle=\langle+|\hat{\sigma}_{x}(t)|-\rangle^{*}=ie^{-\kappa t/2}$
(185)

and the matrix elements in the time-reversed pointer state basis are easily
found for the TRS $\psi=\pi$ (c.f. Eq. (31) in the main text):

$\displaystyle\langle\tilde{\pm}|\hat{\sigma}_{x}|\tilde{\pm}\rangle$
$\displaystyle=0$ (186)
$\displaystyle\langle\tilde{-}|\hat{\sigma}_{x}|\tilde{+}\rangle$
$\displaystyle=\langle\tilde{+}|\hat{\sigma}_{x}|\tilde{-}\rangle^{*}=-i.$
(187)

We thus arrive at an interesting result already: the doubled system
classically correlated state has identically zero correlation functions
$C_{xy}^{\rm cl}(t)=0=C_{xz}^{\rm cl}(t)$. We therefore need to compute only
the off-diagonal elements of $\hat{\sigma}_{y}(t)$ and $\hat{\sigma}_{z}(t)$.
Given the TRS $\psi=\pi$ and the form of the pointer states, it is
straightforward to compute these in the time-reversed pointer state basis. The
relevant result is that they are real and thus

$\displaystyle\langle\tilde{-}|\hat{\sigma}_{y}|\tilde{+}\rangle$
$\displaystyle=\langle\tilde{+}|\hat{\sigma}_{y}|\tilde{-}\rangle,$ (188)
$\displaystyle\langle\tilde{-}|\hat{\sigma}_{z}|\tilde{+}\rangle$
$\displaystyle=\langle\tilde{+}|\hat{\sigma}_{z}|\tilde{-}\rangle,$ (189)

as required by their hermiticity. To see why this must be true without
explicitly computing the matrix elements, note that in the time-reversed
pointer state basis, $\hat{\sigma}_{x}(t)$ is off-diagonal and imaginary and
thus plays the role of an effective “$\hat{\sigma}_{y}$” in this basis.
Therefore $\hat{\sigma}_{y}$ and $\hat{\sigma}_{z}$ must be linear
combinations of the effective “$\hat{\sigma}_{x}$” and “$\hat{\sigma}_{z}$”
and hence their off-diagonal matrix elements must be real and equal.

Furthermore, we can draw the same conclusion about the matrix elements of the
time-dependent $\hat{\sigma}_{y}(t)$ and $\hat{\sigma}_{z}(t)$:

$\displaystyle\langle-|\hat{\sigma}_{y}(t)|+\rangle$
$\displaystyle=\langle+|\hat{\sigma}_{y}(t)|-\rangle,$ (190)
$\displaystyle\langle-|\hat{\sigma}_{z}(t)|+\rangle$
$\displaystyle=\langle+|\hat{\sigma}_{z}(t)|-\rangle.$ (191)

Again $\hat{\sigma}_{x}$ is off-diagonal and imaginary and because
$\hat{l}_{2}$ and $\hat{l}_{3}$ have only $\hat{\sigma}_{y}$ and
$\hat{\sigma}_{y}$ components, the time-dependent operators must remain as
linear combinations of the effective “$\hat{\sigma}_{x}$” and
“$\hat{\sigma}_{z}$” which have real and equal off-diagonal matrix elements.

Finally putting everything together, the TFD correlation functions are

$\displaystyle C_{xy}^{\rm TFD}(t)$ $\displaystyle=ie^{-\kappa
t/2}\begin{cases}\left[\langle\tilde{-}|\hat{\sigma}_{y}|\tilde{+}\rangle-\langle\tilde{+}|\hat{\sigma}_{y}|\tilde{-}\rangle\right]&t\geq
0\\\
\left[\langle+|\hat{\sigma}_{y}(t)|-\rangle-\langle-|\hat{\sigma}_{y}(t)|+\rangle\right]&t<0\end{cases}$
$\displaystyle=0$ (192)

and

$\displaystyle C_{xz}^{\rm TFD}(t)$ $\displaystyle=ie^{-\kappa
t/2}\begin{cases}\left[\langle\tilde{-}|\hat{\sigma}_{z}|\tilde{+}\rangle-\langle\tilde{+}|\hat{\sigma}_{z}|\tilde{-}\rangle\right]&t\geq
0\\\
\left[\langle+|\hat{\sigma}_{z}(t)|-\rangle-\langle-|\hat{\sigma}_{z}(t)|+\rangle\right]&t<0\end{cases}$
$\displaystyle=0$ (193)

which are obviously time symmetric. Therefore all TFD correlation functions of
the Rabi-driven qubit are indeed time symmetric for the TRS $\psi=\pi$, Eq.
(31).

##  Appendix H Mapping to a cascaded quantum system

In this section, we show that the dual Liouvillian
$\overline{\mathcal{L}}^{*}$, as defined and discussed in the main text, can
be reinterpreted as a "perfect absorber" of the output radiation of
$\overline{\mathcal{L}}$ Stannigel  _et al._ (2012). We start with the
following identity in the main text (which is just a rewriting of the
definition of $\mathcal{J}$):

$\displaystyle\hat{H}_{{\rm eff},A}\left|\psi_{T}\right>$
$\displaystyle=\mathcal{J}[\hat{H}_{{\rm eff}}]_{B}\left|\psi_{T}\right>$
(194) $\displaystyle\hat{c}_{l,A}\left|\psi_{T}\right>$
$\displaystyle=\mathcal{J}[\hat{c}_{l}]_{B}\left|\psi_{T}\right>$ (195)

where $\hat{H}_{\rm eff}$ is the effective Hamiltonian in our master equation,
and $\hat{c}_{l}$ are the jump operators. We denote the Hermitian (anti-
Hermitian) parts of an operator $\hat{A}$ as $\textrm{Re}\left[\hat{A}\right]$
($i\textrm{Im}\left[\hat{A}\right]$). One can then tautologically rewrite Eq.
(194) as

$\displaystyle\bigg{[}\hat{H}_{A}$
$\displaystyle-\text{Re}[\mathcal{J}[\hat{H}_{\text{eff}}]]_{B}$ (196)
$\displaystyle-\bigg{(}\frac{i}{2}\sum_{l=1}^{M}\hat{c}_{l,A}^{\dagger}\hat{c}_{l,A}-i\text{Im}[\mathcal{J}[\hat{H}_{{\rm
eff}}]]_{B}\bigg{)}\bigg{]}\left|\psi_{T}\right>=0.$

A short but nontrivial calculation in App. I then shows that

$\displaystyle i\textrm{Im}\left[\mathcal{J}[H_{\text{eff}}]\right]$
$\displaystyle=-\frac{i}{2}\sum_{l=1}^{M}\mathcal{J}[c_{l}]^{\dagger}\mathcal{J}[c_{l}],$
(197)

so that Eq. (196) simplifies to:

$\displaystyle\bigg{[}H_{A}-$
$\displaystyle\text{Re}[\mathcal{J}[H_{\text{eff}}]]_{B}$ (198)
$\displaystyle-\frac{i}{2}\sum_{l=1}^{M}(c_{l,A}^{\dagger}c_{l,A}-\mathcal{J}[c_{l}]_{B}^{\dagger}\mathcal{J}[c_{l}]_{B})\bigg{]}|\psi_{T}\rangle=0.$

Now, separately note the following algebraic identity (which is independent of
Eqs. (194-195):

$\displaystyle(c_{l,A}^{\dagger}c_{l,A}-\mathcal{J}[c_{l}]_{B}^{\dagger}\mathcal{J}[c_{l}]_{B})$
(199)
$\displaystyle=\left(\hat{c}^{\phantom{\prime}}_{l,A}+\mathcal{J}[\hat{c}_{l}]_{B}\right)^{\dagger}\,(\hat{c}^{\phantom{\prime}}_{l,A}-\mathcal{J}[\hat{c}_{l}]_{B})$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+(\hat{c}_{l,A}^{\dagger}\mathcal{J}[\hat{c}_{l}]_{B}-h.c.).$

Using the above equality along with Eq. (195) to simplify Eq. (198), one
obtains:

$\displaystyle\bigg{[}H_{A}-\text{Re}[\mathcal{J}[H_{\text{eff}}]]_{B}-\frac{i}{2}\sum_{l=1}^{M}(\hat{c}_{l,A}^{\dagger}\mathcal{J}[\hat{c}_{l}]_{B}-h.c.)\bigg{]}|\psi_{T}\rangle$
$\displaystyle=0,$

as desired.

##  Appendix I Anti-Hermitian part of the effective Hamiltonian for an
electromagnetic absorber

In this appendix, we compute the anti-hermitian part of the effective
Hamiltonian for the absorber system discussed in Appendix H:

$\displaystyle i\text{Im}\,\mathcal{J}[\hat{H}_{\text{eff}}]$
$\displaystyle=\frac{1}{2}(\mathcal{J}[\hat{H}_{\text{eff}}]-\mathcal{J}[\hat{H}_{\text{eff}}]^{\dagger})$
$\displaystyle=\frac{1}{2}(\mathcal{J}[\hat{H}_{\text{eff}}]-\hat{\rho}_{\rm
ss}^{-1}\mathcal{J}[\hat{H}_{\text{eff}}^{\dagger}]\hat{\rho}_{\rm ss})$ (200)

We now apply $\mathcal{J}^{2}$ to both sides:

$\displaystyle i\text{Im}\,\mathcal{J}[\hat{H}_{\text{eff}}]$
$\displaystyle=\frac{1}{2}\mathcal{J}[\hat{H}_{\text{eff}}-\hat{\rho}_{\rm
ss}\hat{H}_{\text{eff}}^{\dagger}\hat{\rho}_{\rm ss}^{-1}]$
$\displaystyle=\frac{i}{2}\mathcal{J}[-i(\hat{H}_{\text{eff}}\hat{\rho}_{\rm
ss}-\hat{\rho}_{\rm ss}\hat{H}_{\text{eff}}^{\dagger})\hat{\rho}_{\rm
ss}^{-1}]$ (201)

Now, since $\hat{\rho}_{\rm ss}$ is a steady state of the Lindbladian
$\mathcal{L}$, we have

$\displaystyle-i(\hat{H}_{\text{eff}}\rho_{ss}-\hat{\rho}_{\rm
ss}H_{\text{eff}}^{\dagger})+\sum_{l}\hat{c}_{l}\hat{\rho}_{\rm
ss}\hat{c}_{l}^{\dagger}\equiv 0.$ (202)

Therefore, plugging the above identity into Eq. (201), we get

$\displaystyle i\text{Im}\,\mathcal{J}[\hat{H}_{\text{eff}}]$
$\displaystyle=\frac{i}{2}\mathcal{J}[-\sum_{l}\hat{c}_{l}\rho_{ss}\hat{c}_{l}^{\dagger}\hat{\rho}_{\rm
ss}^{-1}].$ $\displaystyle=-\frac{i}{2}\sum_{l}\hat{\rho}_{\rm
ss}^{-1}\mathcal{J}[\hat{c}_{l}^{\dagger}]\hat{\rho}_{\rm
ss}\mathcal{J}[\hat{c}_{l}]$
$\displaystyle=-\frac{i}{2}\sum_{l}\mathcal{J}[\hat{c}_{l}]^{\dagger}\mathcal{J}[\hat{c}_{l}],$
(203)

which is the expression utilized in Appendix H.

##  Appendix J The potential conditions: manifestation of trivial TRS in the
complex-$P$ representation

The main goal of this appendix is to reproduce a well-known result in
classical probability theory which is known for Fokker-Planck equations.
Specifically, these results apply to Fokker-Planck equations of the form

$\displaystyle\partial_{t}P(\vec{x},t)$
$\displaystyle\equiv\partial_{\mu}[C^{\mu}(\vec{x})P(\vec{x},t)]+\partial_{\mu}\partial_{\nu}[D^{\mu\nu}(\vec{x})P(\vec{x},t)],$
(204)

where $P(\vec{x},t)$ is the probability distribution of a real-valued random
process on $\mathbb{R}^{d}$, with stationary expectation values of observables
$X(\vec{x})$ described in the standard way, e.g.

$\displaystyle\overline{X}\equiv\int d^{d}xP_{ss}(\vec{x})X(\vec{x}).$ (205)

The statement about these Fokker-Planck equations that we wish to generalize
to the complex-$P$ representation is as follows Pavliotis (2014): provided
that $D^{\mu\nu}(\vec{x})$ is positive definite for all $\vec{x}$, then the
potential conditions are equivalent to detailed balance with respect to a
trivial time-reversal operation $\tilde{x}\equiv\vec{x}$. By the potential
conditions, we mean the statement that the stationary probability current
vanishes.

In this appendix, we generalize the above result for the pseudo-Fokker-Planck
equations encountered in the complex-$P$ representation:

$\displaystyle\partial_{t}P(\vec{\alpha},\vec{\beta},t)$
$\displaystyle\equiv\partial_{\mu}[C^{\mu}(\vec{\alpha},\vec{\beta})P(\vec{\alpha},\vec{\beta})]$
$\displaystyle+\partial_{\mu}\partial_{\nu}[D^{\mu\nu}(\vec{\alpha},\vec{\beta})P(\vec{\alpha},\vec{\beta},t)].$
(206)

A complex pseudo-probability current may then be defined for the complex-$P$
distribution in the way described in Sec. VIII:

$\displaystyle\partial_{t}P(\vec{\alpha},\vec{\beta},t)\equiv\partial_{\mu}J^{\mu}(\vec{\alpha},\vec{\beta},t)$
(207)

However, there are significant departures from standard probability theory:
the complex-$P$ distribution is complex-valued, and thus violates the axioms
of classical probability. In particular, stationary moments are given by

$\displaystyle\overline{X}\equiv\int_{\Sigma}d^{n}\alpha d^{n}\beta
P_{ss}(\vec{\alpha},\vec{\beta},t)X(\vec{\alpha},\vec{\beta}).$ (208)

Here, $\Sigma$ is a 2$n$-dimensional closed integration surface (the many-body
analogue of the pair of integration contours
$\mathcal{C},\mathcal{C}^{\prime}$ mentioned in the main text). Furthermore,
as the diffusion tensor $D^{\mu\nu}(\vec{\alpha},\vec{\beta})$ is generically
complex, it may fail to be positive-definite.

Despite these differences, in this appendix we will nonetheless prove that the
standard classical result in Pavliotis (2014) still applies (however, here
there will be no assumption on the diffusion tensor
$D^{\mu\nu}(\vec{\alpha},\vec{\beta})$): the potential conditions are
equivalent to detailed balance with respect to a trivial time-reversal
operation:

$\displaystyle\widetilde{(\vec{\alpha},\vec{\beta})}\equiv(\vec{\alpha},\vec{\beta}).$
(209)

This result is critical for bosonic many-body systems with local onsite loss,
as, with a nontrivial $U$ parameter, hidden TRS corresponds to a nontrivial
classical TRS in the complex-$P$ representation, and thus potentially
corresponds to a richer symmetry than that encapsulated by the potential
conditions.

We first establish the forward implication: if the stationary
pseudoprobability current Eq. (207) vanishes, then a straightforward
calculation shows that

$\displaystyle\mathcal{L}(XP_{\text{ss}})=P_{\text{ss}}\mathcal{L}^{*}(X),$
(210)

where $\mathcal{L}$ is the Liouvillian for the pseudo Fokker-Planck equation
Eq. (206), and $\mathcal{L}^{*}$ is the adjoint Liouvillian, obtained via
integration by parts Drummond and Gardiner (1980). The proof almost exactly
follows Pavliotis (2014), except while carrying out the calculation we notice
that positive-definiteness of the diffusion tensor
$D^{\mu\nu}(\vec{\alpha},\vec{\beta})$ is not needed in the proof, and thus
the assumption of positive-definiteness may be relaxed.

From Eq. (210), detailed balance is immediate, as, by integrating by parts on
the surface $\Sigma$ (which is valid, as $X,Y$ are holomorphic), the dual
Liouvillian $\mathcal{L}^{*}$ can then be shown to be symmetric with respect
to the bilinear form

$\displaystyle\langle X,Y\rangle\equiv\int_{\Sigma}d^{n}\alpha
d^{n}\beta\,P_{ss}XY.$ (211)

From this and the expressions

$\displaystyle\overline{X(t)Y(0)}$ $\displaystyle=\langle
e^{-t\mathcal{L}^{*}}(X),Y\rangle,$ (212) $\displaystyle\overline{Y(t)X(0)}$
$\displaystyle=\langle X,e^{-t\mathcal{L}^{*}}(Y)\rangle,$ (213)

we have detailed balance. For the converse direction, the proof follows
analogously to the classical case as well: one starts by proving the following
(assuming $P_{ss}$ is nonvanishing on $\Sigma$):

$\displaystyle\langle-\mathcal{L}^{*}(X),Y\rangle=\langle
D^{\mu\nu}\partial_{\mu}X,\partial_{\nu}Y\rangle+\langle
X,P_{ss}^{-1}\partial_{\mu}YJ^{\mu}_{ss}\rangle$ (214)

The above is a generalization of the lemma in Section 4.6 of Pavliotis (2014),
to pseudo Fokker-Planck equations, and may be proven by invoking the
holomorphicity of $X,Y$. With the above, one can compute the asymmetry of the
Liouvillian:

$\displaystyle\langle\mathcal{L}^{*}(X),Y\rangle-\langle
X,\mathcal{L}^{*}(Y)\rangle$ (215) $\displaystyle=2\int_{\Sigma}d^{n}\alpha
d^{n}\beta\,Y(\vec{\alpha},\vec{\beta})\partial_{\mu}X(\vec{\alpha},\vec{\beta})J^{\mu}_{ss}(\vec{\alpha},\vec{\beta})$

Now, if detailed balance holds with respect to a trivial TRS, then the
Liouvillian $\mathcal{L}^{*}$ is symmetric, and thus the right-hand side must
vanish for all pairs of holomorphic functions $X,Y$. The only way that this
can happen is if the stationary current vanishes everywhere on the integration
surface, i.e.

$\displaystyle J^{\mu}_{ss}(\vec{\alpha},\vec{\beta})\equiv
0\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \forall(\vec{\alpha},\vec{\beta})\in\Sigma.$ (216)

However, the above is precisely the statement of the potential conditions.

## References

  * Gardiner (2009) C. Gardiner,  _Stochastic Methods: A Handbook for the Natural and Social Sciences_, 4th ed., Springer Series in Synergetics (Springer-Verlag, Berlin Heidelberg, 2009).
  * Agarwal (1973) G. S. Agarwal, Z. Phys. 258, 409 (1973).
  * Carmichael and Walls (1976) H. J. Carmichael and D. F. Walls, Z. Phys. B 23, 299 (1976).
  * Alicki (1976) R. Alicki, Rep. Math. Phys. 10, 249 (1976).
  * Kossakowski  _et al._ (1977) A. Kossakowski, A. Frigerio, V. Gorini, and M. Verri, Commun. Math. Phys. 57, 97 (1977).
  * Majewski (1984) W. A. Majewski, J. Math. Phys. 25, 614 (1984), https://doi.org/10.1063/1.526164 .
  * Majewski and Streater (1999) W. A. Majewski and R. F. Streater, J. Phys. A. Math Gen 31, 7981 (1999).
  * Denisov  _et al._ (2002) A. Denisov, H. M. Castro-Beltran, and H. J. Carmichael, Phys. Rev. Lett. 88, 243601 (2002).
  * Fagnola and Umanita (2007) F. Fagnola and V. Umanita, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10, 335 (2007), arXiv: 0707.2147.
  * Fagnola and Umanità (2010) F. Fagnola and V. Umanità, Commun. Math. Phys. 298, 523 (2010).
  * Duvenhage and Snyman (2018) R. Duvenhage and M. Snyman, in  _Ann. Henri Poincaré_ , Vol. 19 (Springer, 2018) pp. 1747–1786.
  * Carlen and Maas (2017) E. A. Carlen and J. Maas, J. Func. Anal. 273, 1810 (2017).
  * Ramezani  _et al._ (2018) M. Ramezani, F. Benatti, R. Floreanini, S. Marcantoni, M. Golshani, and A. T. Rezakhani, Phys. Rev. E 98, 052104 (2018).
  * Note (1) The only attempt at such connections in the past were limited to systems that were easily solvable by other means (e.g. linear bosonic systems, or systems that could be reduced to a classical master equation) Agarwal (1973).
  * Goldstein and Lindsay (1995) S. Goldstein and J. M. Lindsay, Math. Z 219, 591 (1995).
  * Takahashi and Umezawa (1996) Y. Takahashi and H. Umezawa, Int. J. Mod. Phys. B 10, 1755 (1996).
  * Stannigel  _et al._ (2012) K. Stannigel, P. Rabl, and P. Zoller, New J. Phys. 14, 063014 (2012).
  * Roberts and Clerk (2020) D. Roberts and A. A. Clerk, Phys. Rev. X 10, 021022 (2020).
  * Drummond and Gardiner (1980) P. D. Drummond and C. W. Gardiner, J. Phys. A 13, 2353 (1980).
  * Drummond and Walls (1980) P. D. Drummond and D. F. Walls, J. Phys. A 13, 725 (1980).
  * Bartolo  _et al._ (2016) N. Bartolo, F. Minganti, W. Casteels, and C. Ciuti, Phys. Rev. A 94, 033841 (2016).
  * Elliott and Ginossar (2016) M. Elliott and E. Ginossar, Phys. Rev. A 94, 043840 (2016).
  * Zhu and Monroe (2020) D. Zhu and C. Monroe, Proc. Nat. Acad. Sci. (2020).
  * Dykman and Krivoglaz (1979) M. I. Dykman and M. A. Krivoglaz, Zh. Eksp. Teor. Fiz. 77, 60 (1979), [Sov. Phys. - JETP 50, 30 (1979).
  * Dykman and Smelyanskii (1988) M. I. Dykman and V. N. Smelyanskii, Zh. Eksp. Teor. Fiz. 94, 61 (1988), [Sov. Phys. - JETP 67, 1769 (1988)].
  * Marthaler and Dykman (2006) M. Marthaler and M. Dykman, Phys. Rev. A 73, 042108 (2006).
  * Dykman (2012) M. Dykman, in  _Fluctuating Nonlinear Oscillators_ , edited by M. Dykman (Oxford, 2012).
  * Guo  _et al._ (2013) L. Guo, V. Peano, M. Marthaler, and M. I. Dykman, Phys. Rev. A 87, 062117 (2013).
  * Guo (2013) L. Guo,  _Quantum Effects in Driven Nonlinear Systems_ , Ph.D. thesis, Karlsruher Institut fur Technologie (2013).
  * Zhang and Dykman (2019) Y. Zhang and M. I. Dykman, Phys. Rev. E 100, 052148 (2019).
  * Breuer and Petruccione (2002) H.-P. Breuer and F. Petruccione,  _The Theory of Open Quantum Systems_ (Oxford University Press, 2002).
  * Gardiner and Zoller (2000) C. W. Gardiner and P. Zoller,  _Quantum Noise_ (Springer, Berlin, 2000).
  * Cipriani (1997) F. Cipriani, J. Funct. Anal. 147, 259 (1997).
  * Note (2) The Hamiltonian in a Lindblad master equation is in general not unique. Our condition holds for the case where jump operators are chosen to be traceless.
  * Bratteli and Haagerup (1978) O. Bratteli and U. Haagerup, Commun. Math. Phys. 59, 79 (1978).
  * Ekert  _et al._ (2002) A. K. Ekert, C. M. Alves, D. K. L. Oi, M. Horodecki, P. Horodecki, and L. C. Kwek, Phys. Rev. Lett. 88, 217901 (2002).
  * Islam  _et al._ (2015) R. Islam, R. Ma, P. M. Preiss, M. Eric Tai, A. Lukin, M. Rispoli, and M. Greiner, Nature 528, 77 (2015).
  * Kaufman  _et al._ (2016) A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Science 353, 794 (2016).
  * Note (3) Note that we are implicitly using the fact that our bilinear form is non-degenerate (as $\hat{\rho}_{\rm ss}$ is full rank), something which guarantees the uniqueness of the dual.
  * Parthasarathy (1992) K. R. Parthasarathy,  _An Introduction to Quantum Stochastic Calculus_, Modern Birkhäuser Classics (Birkhäuser Basel, 1992).
  * Note (4) Note that the condition in Eq. (43) on the jump operators implies the exchange superoperator acts as a unitary on the subspace spanned by the Lindblad operators $c_{l}$. Since $\mathcal{J}$ squares to one, the unitary operation in Eq. (43), which is the restriction of $\mathcal{J}$ to the subspace spanned by the Lindblad operators $c_{l}$, also squares to one. It follows that its eigenvalues can only be $1$ or $-1$.
  * Bratteli and Robinson (1997) O. Bratteli and D. W. Robinson,  _Operator Algebras and Quantum Statistical Mechanics_ (Springer Berlin Heidelberg, 1997).
  * Gardiner (1993) C. W. Gardiner, Phys. Rev. Lett. 70, 2269 (1993).
  * Carmichael (1993) H. J. Carmichael, Phys. Rev. Lett. 70, 2273 (1993).
  * Mirrahimi  _et al._ (2014) M. Mirrahimi, Z. Leghtas, V. V. Albert, S. Touzard, R. J. Schoelkopf, L. Jiang, and M. H. Devoret, New J. Phys. 16, 045014 (2014).
  * Leghtas  _et al._ (2015) Z. Leghtas, S. Touzard, I. M. Pop, A. Kou, B. Vlastakis, A. Petrenko, K. M. Sliwa, A. Narla, S. Shankar, M. J. Hatridge, M. Reagor, L. Frunzio, R. J. Schoelkopf, M. Mirrahimi, and M. H. Devoret, Science 347, 853 (2015).
  * Wang  _et al._ (2016) C. Wang, Y. Y. Gao, P. Reinhold, R. W. Heeres, N. Ofek, K. Chou, C. Axline, M. Reagor, J. Blumoff, K. M. Sliwa, L. Frunzio, S. M. Girvin, L. Jiang, M. Mirrahimi, M. Devoret, and R. J. Schoelkopf, Science 352, 1087 (2016).
  * Puri  _et al._ (2017) S. Puri, S. Boutin, and A. Blais, npj Quantum Information 3, 18 (2017).
  * Grimm  _et al._ (2020) A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada, S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, and M. Devoret, Nature 584, 1 (2020).
  * Wolinsky and Carmichael (1988) M. Wolinsky and H. J. Carmichael, Phys. Rev. Lett. 60, 1836 (1988).
  * Walls and Milburn (2008) D. F. Walls and G. J. Milburn,  _Quantum Optics_ (Springer-Verlag, Berlin, 2008).
  * Risken and Frank (1996) H. Risken and T. Frank,  _The Fokker-Planck Equation: Methods of Solution and Applications_, 2nd ed., Springer Series in Synergetics (Springer-Verlag, Berlin Heidelberg, 1996).
  * Kamenev (2011) A. Kamenev,  _Field Theory of Non-Equilibrium Systems_ (Cambridge University Press, 2011).
  * Sieberer  _et al._ (2015) L. M. Sieberer, A. Chiocchetta, A. Gambassi, U. C. Täuber, and S. Diehl, Phys. Rev. B 92, 1515 (2015).
  * Sieberer  _et al._ (2016) L. M. Sieberer, M. Buchhold, and S. Diehl, Rep. Prog. Phys. 79, 096001 (2016).
  * Lindblad (1976) G. Lindblad, Comm. Math. Phys. 48, 119 (1976).
  * Pavliotis (2014) G. A. Pavliotis,  _Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations_, Texts in Applied Mathematics (Springer-Verlag, New York, 2014).