# Unified collision model of coherent and measurement-based quantum feedback

Alfred Harwood  Department of Physics & Astronomy, University College London,
Gower Street, WC1E 6BT, London, United Kingdom  Matteo Brunelli  Department of
Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland
Alessio Serafini  Department of Physics & Astronomy, University College
London, Gower Street, WC1E 6BT, London, United Kingdom

###### Abstract

We introduce a general framework, based on collision models and discrete CP-
maps, to describe on an equal footing coherent and measurement-based feedback
control of quantum mechanical systems. We apply our framework to prominent
tasks in quantum control, ranging from cooling to Hamiltonian control. Unlike
other proposed comparisons, where coherent feedback always proves superior, we
find that either measurements or coherent manipulations of the controller can
be advantageous depending on the task at hand. Measurement-based feedback is
typically superior in cooling, whilst coherent feedback is better at assisting
quantum operations. Furthermore, we show that both coherent and measurement-
based feedback loops allow one to simulate arbitrary Hamiltonian evolutions,
and discuss their respective effectiveness in this regard.

Introduction.—The designation ‘coherent feedback’ (CF) is used to describe a
broad class of quantum control strategies where the controller is itself a
quantum system which processes quantum information. It can be contrasted with
‘measurement-based feedback’ (MF), in which the controller processes classical
information resulting from measurement outcomes. The theory of MF goes back to
the work of Belavkin, who first extended ideas from classical control theory
to quantum systems Belavkin (1983). CF has instead its origins in the ‘all-
optical feedback’ and ‘feedback without measurements’ Wiseman and Milburn
(1994); Wiseman (1994), based on the input-output formalism Gardiner and
Collett (1985). There, the output from a cavity is used to modulate the
cavity’s dynamics (without measurements being performed) and the process has a
clear, directional ‘feedback loop’ structure 111Later, the term ‘coherent
quantum feedback’ was used to describe more general interactions between two
quantum systems aimed at a certain control task Lloyd (2000); Jacobs  _et al._
(2014). In our work, we focus on the definition involving explicit loops, as
made clear in the text.. MF has long been one of the key techniques for
achieving quantum control Smith  _et al._ (2002); Sayrin  _et al._ (2011);
Wiseman and Milburn (2009); Serafini (2012) and CF has also emerged as a
powerful alternative over the past few years, with applications in quantum
optics Hamerly and Mabuchi (2012); Iida  _et al._ (2012); Shi and Nurdin
(2015), optomechanics Bennett  _et al._ (2014); Wang and Safavi-Naeini (2017);
Harwood  _et al._ (2021); Woolley and Clerk (2014); Li  _et al._ (2017);
Naumann  _et al._ (2014), nanomechanics Schmid  _et al._ (2022), NV centres
Hirose and Cappellaro (2016) and circuit QED setups Zhang  _et al._ (2012); Ma
_et al._ (2021); Crowder  _et al._ (2021).

Both quantum feedback strategies have been widely studied theoretically,
especially in the context of quantum optics Wiseman and Milburn (2009); Nurdin
_et al._ (2009); Yamamoto (2014); Hamerly and Mabuchi (2012); Wiseman and
Milburn (1994); Wiseman (1994); Harwood  _et al._ (2021); Xiang  _et al._
(2020); Liu  _et al._ (2016); Whalen  _et al._ (2017). It is often claimed
that CF is inherently superior to MF, and for certain definitions, regimes and
problems, this is true Yamamoto (2014); Nurdin  _et al._ (2009); Hamerly and
Mabuchi (2012); Jacobs  _et al._ (2014). On the other hand, other results seem
to suggest a less clearcut scenario and point at the usefulness of quantum
measurements for certain tasks Liu  _et al._ (2016), which can even outperform
CF, e.g. for stabilising the squeezing of a cavity mode Harwood and Serafini
(2020). In particular, the alleged superiority of CF relies on a comparison
with measurement-based unconditional, averaged dynamics without considering
the corresponding conditional dynamics, whose stochastic jumps cannot be
reproduced by unitary means (a way to express the notorious measurement
problem), or miss the potential of non-destructive quantum measurements, upon
which system and controller evolve according to the von Neumann postulate and
then interact with each other once again afterwards. A broader approach is
then required to fully take into account the possibilities offered by quantum
measurements.

To this aim, we shall present a unified framework for describing coherent and
measurement-based feedback loops as cascaded collision models. Collision
models (CMs) are a class of schemes for modelling open quantum dynamics where
the system repeatedly interacts with an environment, which is refreshed after
each interaction Rau (1963); Bruneau  _et al._ (2014); Scarani  _et al._
(2002); Ciccarello  _et al._ (2021); Campbell and Vacchini (2021). They have
found widespread application, ranging from quantum thermodynamics Strasberg
_et al._ (2017); Lorenzo  _et al._ (2015a, b); Seah  _et al._ (2019), to
quantum optics Ciccarello (2017) and non-Markovian quantum systems Giovannetti
and Palma (2012); Rybár  _et al._ (2012); Ciccarello  _et al._ (2013);
Kretschmer  _et al._ (2016). We should mention that a discretised model for
coherent feedback, encompassing an exhaustive treatment of delays, was also
proposed in Whalen  _et al._ (2017).

In this work we treat the environment of the collision model as a ‘controller’
which undergoes _two_ sequential interactions with the system before being
refreshed. In between these two interactions, the controller is processed,
either through measurements (in the case of MF), or coherently (in the case of
CF), a key restriction being that the system cannot be manipulated directly,
but only indirectly through the controller. This sequence gives the model the
directional structure of a feedback loop. By assuming that the controller is
always refreshed to the same state after each iteration and by keeping the
system-environment interaction the same in both cases, we are ensuring that
the only extra resource that MF has access to is the ability to perform
measurements. We therefore argue that this framework provides a fair
comparison between MF and CF. If the system is subject to additional noise, we
assume that the controller does not have access to its source (as would be the
case in quantum feed _forward_ —a distinction made in Jacobs  _et al._
(2014)).

We use our framework to investigate the efficacy of MF and CF for relevant
examples of control tasks and uncover a highly non-trivial relationship
between the two feedback strategies, depending on the task at hand. Control
tasks can broadly be grouped into two classes: _state control_ —where the goal
is to prepare the system in a specific state—and _operator control_ —where the
goal is to simulate the effect of a particular CP-map (often a unitary)
without knowledge of the input state Jurdjevic and Sussmann (1972); Schirmer
_et al._ (2003); Albertini and D’Alessandro (2003); d’Alessandro (2021); Dong
and Petersen (2010); Schirmer and Solomon (2002). Through analytic
optimisation of some of these tasks, we show that MF is often better at
achieving state stabilisation and control, especially when the system is
subject to noise, whereas CF tends to outperform MF in operator control, where
the preservation of quantum coherence in all bases is key. As examples of
state control, we consider the steady-state cooling of a $d$-level quantum
system (qudit) subject to noise and the stabilisation of a qubit in an excited
state, counteracting decay. For operator control, we investigate the
archetypal task of implementing a bit-flip on an unknown pure state qubit
input. We also investigate the task of complete unitary controllability (being
able to generate _any_ unitary evolution) in the limit of weak system-
controller interaction.

A Collision Model of Quantum Feedback.—We start by outlining a common
framework encompassing both MF and CF, as sketched in Fig. 1. CF is described
by iterated cycles each comprising the following sequence of events: i) The
system is subject to noise given by a CP-map $\mathcal{E}$, representing any
internal dynamics and inaccessible noise; ii) The system and controller
interact through a unitary $U_{1}$; iii) A unitary operation, chosen from a
set $\\{V_{j}\\}$ (constrained, in practice, by experimental limitations), is
performed on the controller. Here, we expand the definition of the controller
to include the subsystems which interact directly with the main system, as
well as any other additional subsystems; iv) The system and controller
interact again, through another unitary $U_{2}$; v) The state of the
controller is refreshed to its initial state $\eta$, so that it retains no
memory of the previous interaction.

MF is implemented in the same way, with step iii) being replaced with the
following step: iii′) A non-destructive measurement, described by a POVM, is
performed on the controller (the available measurements being determined by
experimental limitations). After the measurement, depending on the measurement
result $\mu$, a unitary $V_{\mu}$, from the set $\\{V_{j}\\}$ is performed on
the controller and available ancillas.

Figure 1: A schematic diagram for a single “feedback collision”. A system is
subject to noise channel $\mathcal{E}$ and then interacts twice with a
controller. In between these collisions, either measurement-plus-unitary or
just a unitary are implemented on the controller, thus realising measurement-
based feedback (MF) or coherent feedback (CF), respectively. The feedback
dynamics is obtained by repeated iterations of this circuit. We include both
clean (“cold”) and noisy (“hot”) controllers, initialised respectively in pure
or mixed states.

This model allows for a fair comparison between MF and CF since the system
noise, controller state and system-controller couplings are kept the same in
both cases, and so is the set of unitaries that can be applied to the
controller. The only difference between the two cases is the presence or
absence of an in-loop measurement. Thus, comparing these two classes allows us
to ask ‘does performing a measurement on the controller act as a resource for
a particular task, or is it a hindrance?’. Indeed, in a sense that is made
clear in the Supplemental Material (SM), this model allows CF to be framed as
MF in the limit of infinitely weak measurements, putting both strategies on
different ends of the same spectrum.

Typically, in MF schemes, the output system state will depend on the
measurement record. If the measurement record is not kept, then the system
will evolve deterministically to an “unconditional” state resulting from
averaging over the measurement outcomes. If the measurement record is kept,
then the “conditional” system state will typically evolve stochastically
(although there exist cases where the subsequent operations can undo the
measurements’ stochasticity).

Feedback for cooling I: Noisy Controller.—As a first case study, we apply our
framework to the pivotal task of feedback cooling. We take the system and
controller to have the same Hilbert space dimension $d$ and both couplings
$U_{1}$ and $U_{2}$ to be partial swaps, as commonly used in CMs Ciccarello
_et al._ (2021); McCloskey and Paternostro (2014); Campbell  _et al._ (2019),
i.e., we set $U_{s}=\sqrt{\tau}\mathbbm{1}-i\sqrt{1-\tau}\hat{S}$, where
$\hat{S}$ is the unitary swap operator and $0\leq\tau\leq 1$ is the
transmissivity; this choice of system-controller interaction will be used for
the remainder of this work. We assume that the noise $\mathcal{E}$, takes the
form of a depolarising channel acting on the system as
$\mathcal{E}(\rho)=\lambda\rho+(1-\lambda)\frac{1}{d}\mathbbm{1}$ where
$\lambda$ determines the strength of the noise. In this section, we consider
the case where the controller is in the maximally mixed—or infinite
temperature—state $\eta=\frac{1}{d}\mathbbm{1}$, which we refer to as “noisy
controller”. The relevant figure of merit is the entropy of the steady state
of the system (i.e., a state which is unchanged by the application of a single
iteration of the MF or CF protocol). If the steady state exists and is unique
(as will be the case in our examples), any input state will tend to the steady
state upon repeated applications of the protocol.

For CF, the set of allowed in-loop operations on the controller is the set of
single qudit unitaries. It is shown in the SM, through a general argument
hinging on the subadditivity of entropy that, regardless of the in-loop
unitary used, CF is incapable of counteracting the noise on the system and
leads to a steady state which is the maximally mixed state
$\rho_{S}=\frac{1}{d}\mathbbm{1}$.

For MF, we consider a protocol with an in-loop projective measurement in an
arbitrary basis $\\{\ket{j},j=0,\ldots,d-1\\}$. After measurement, a unitary
is applied to the controller which maps all post-measurement states to the
same state, which we will label $\ket{0}$. Considering the averaged dynamics
of this MF protocol, the system reaches the steady state

$\rho_{S}=\frac{1}{d}\frac{d(1-\tau)+\tau-\lambda\tau^{2}}{1-\lambda\tau^{2}}\ket{0}\bra{0}+\sum_{j=1}^{d-1}\frac{1}{d}\frac{\tau-\lambda\tau^{2}}{1-\lambda\tau^{2}}\ket{j}\bra{j}\,,$
(1)

which is clearly lower entropy than the maximally mixed state for all values
of the parameters except for the trivial case where $\tau=1$ and the system
and controller do not interact. Thus, we have shown that that introducing
projective measurements into the feedback loop leads to lower system
entropies. If a cold environment (in the form of a pure controller) is not
available as a resource, MF outperforms all possible CF protocols. In this
case measurements act as a powerful tool for preparing the controller’s state,
which cannot be reproduced through unitary means.

We now consider the conditional dynamics of this MF protocol. In general, the
stochasticity of measurement outcomes, unless averaged upon, does not yield a
steady state and instead results in oscillating entropies as shown in Fig. 2.
However, in the case of an ideal full swap ($\tau=0$) the post-measurement
unitary can completely undo the stochasticity of the measurement, leading to
the steady state $\ket{0}$, i.e., perfect cooling. Examining Fig. 2 one can
see that the asymptotic cooling performance, in terms of steady state entropy,
decreases with $\tau$ (as lower $\tau$ imply the possibility of injecting low
entropy states from the controller) and, as one should expect, decreases with
the noise strength $\lambda$. One may also see that the entropy ‘jumps’ become
larger for higher system dimensions. Besides, this study also allows us to
estimate the cooling rate of the model, i.e., the typical number of iterations
needed to approach the unconditional entropy, as $1/(1-\tau)$ (in that the
interaction with the controller dominates the cooling process).

By applying a quantum entropy power inequality Audenaert  _et al._ (2016);
Carlen  _et al._ (2016), one may show that, for an input diagonal in the
measurement basis, the entropy of the conditional output state is minimised
when, after measuring the controller, all measurement outcomes are mapped to
the system’s dominant eigenvector (corresponding to the system’s largest
eigenvalue). Furthermore, when the input is diagonal in the measurement basis,
the output is diagonal in the same basis, with the same dominant eigenvector.
Thus, for repeated application, starting with maximally mixed input state or
any input state with dominant eigenvector $\ket{0}$, our MF protocol is
optimal when considering conditional dynamics.

Figure 2: Normalised (with base-$d$ logarithms) von Neumann entropy of
conditional (continuous lines) and steady-state unconditional (horizontal
dotted lines) measurement-based feedback cooling for different values of
$\tau$, $\lambda$ and $d$, for a maximally mixed initial state. The $x$-axis
reports the number of iterations.

Feedback for cooling II: Clean Controller.—While MF outperforms CF when
cooling with a maximally mixed controller, this is not always the case when
the controller is in a pure state. Here we consider the controller to be in
the pure state $\eta=\ket{0}\bra{0}$, which we refer to as a “clean
controller”. In the case of CF, no in-loop unitary is needed, because the
system is swapped with the pure controller, thereby getting cooled. In the SM,
we prove analytically that this CF loop is optimal for qubits assuming
$\ket{0}$ is an eigenvector of the steady state. Here, we present the linear
entropies for qubit systems in view of their more compact expressions (more
general expressions for arbitrary dimension can be found in the SM). The
steady-state linear entropy for the MF protocol is
$S_{MF}=\frac{1}{2}-\frac{\left(\tau^{2}-1\right)^{2}}{2\left(\tau^{2}\lambda-1\right)^{2}}$,
while for CF protocol we have
$S_{CF}=\frac{1}{2}-\frac{8\tau^{2}\left(\tau-1\right)^{2}}{\left(\left(1-2\tau\right)^{2}\lambda-1\right)^{2}}$.
Comparing the two expressions, we find that $S_{MF}<S_{CF}$ for $\tau<1/3$ and
$S_{MF}>S_{CF}$ for $\tau>1/3$. In other words, there are two regimes: one
where MF outperforms CF and one where CF outperforms MF. Note that when
$\tau=1/2$, $S_{CF}=0$ and CF is able to stabilise a pure state. On the other
hand, no MF protocol with projective measurements can stabilise a pure state
for $\tau=1/2$ (see SM). Conversely, for $\tau=0$, MF is always capable of
stabilising a pure state using the protocol described earlier, whilst CF
cannot stabilise any state other than the maximally mixed state, as the action
of a unitary alone cannot undo the effect of the depolarising map. A
comparison of MF and CF for cooling at intermediate controller noises (where
the controller is neither pure nor maximally mixed) can be found in the SM.

Protecting an excited state from decaying.—After a task based on entropy, we
now consider one focused on the system’s energy, namely protecting an excited
state from environment-induced decay. For simplicity, we consider a qubit
initially prepared in the excited state $\ket{1}$ (with $\ket{0}$ being the
ground state) subject to decay, modelled by $\mathcal{E}$ with Kraus operators
$E_{0}=\sqrt{\gamma}\ket{0}\bra{1},\;E_{1}=\sqrt{1-\gamma}\ket{1}\bra{1}+\ket{0}\bra{0}$.
A feedback loop, either CF or MF, is employed to counter the effect of decay.
We take the controller to be noisy. Our figure of merit is then the steady-
state occupation of the $\ket{1}$ state. For CF, we restrict the in-loop
unitaries to rotations $U=\cos\chi\mathbbm{1}+i\sin\chi\sigma_{y}$. and assume
that the controller is initialised in the maximally mixed state. Investigating
numerically, we find that the optimal CF protocol depends on the partial swap
strength; for $\tau>\frac{1}{2}$ $(\tau<\frac{1}{2})$, the optimal protocol
involves setting $\chi=0$ $(\chi=\frac{\pi}{2})$.

For $\tau>\frac{1}{2}$, increasing the noise strength decreases the steady-
state occupation of the excited state, as expected. However, for strong
system-controller interactions $\tau<\frac{1}{2}$, the optimal performance
(when $\chi=\frac{\pi}{2}$) is actually _improved_ by stronger damping. This
is because the action of the in-loop $\frac{\pi}{2}$ rotation is more
effective at populating the $\ket{1}$ state when more of the state is
initially prepared in the $\ket{0}$ state. In this sense, CF allows for the
purifying effect of the amplitude damping map to be harnessed for the purpose
of increasing the excited state population. When $\chi=0$ and the in-loop
unitary is the identity, the steady-state occupation of the excited state is

$\rho_{11}^{\chi=0}=\frac{2\tau\left(1-\tau\right)}{4\left(\tau-1\right)(\gamma-1)\tau+\gamma},$
(2)

which is a decreasing function of $\gamma$ for $\tau>\frac{1}{2}$. Conversely,
when $\chi=\frac{\pi}{2}$, the steady-state occupation is

$\rho_{11}^{\chi=\frac{\pi}{2}}=\frac{1-\tau}{2(\gamma-1)\tau-\gamma+2}\,,$
(3)

which is an increasing function of $\gamma$ for $\tau<\frac{1}{2}$.

We compare this to an intuitive MF protocol which measures the controller in
the $\\{\ket{1},\ket{0}\\}$ basis, does nothing if the result is $\ket{1}$,
and applies an in-loop rotation with $\chi=\frac{\pi}{2}$ if the result is
$\ket{0}$. This protocol results in an excited steady-state occupation

$\rho_{11}^{MF}=\frac{2-\tau^{2}-\tau}{2(\gamma-1)\tau^{2}+2}\,.$ (4)

This expression is greater than equation (3) for all values of $\tau$ and
$\gamma$, so that MF outperforms CF in the regime of strong system-controller
coupling $\tau<\frac{1}{2}$. However, for $\tau>\frac{1}{2}$, equation (4) is
slightly lower than (2). In particular, (2) is greater than (4) when
$\tau>\frac{1}{2}\frac{-7\gamma+\sqrt{\gamma(17\gamma-24)+16}+4}{2-2\gamma}$.
In this regime, CF will outperform MF, though numerical investigations suggest
that the advantage is small. Conversely, in the regime where MF outperforms
CF, the advantage tends to be larger. Figure (3) shows $\rho_{11}$ the
occupation of the excited state, against the number of iterations of the
feedback loop. Both MF and CF protocols described above are presented. The
unconditional (averaged) MF trajectory is shown, as well as fifty conditional
trajectories for each setup.

Figure 3: Occupation of the excited state against number of iterations of MF
or CF. The MF trajectories shown in red and green are the unconditional
trajectories. For each setup fifty conditional, filtered trajectories are
shown in light grey.

Operator control: Performing a bit flip operation.—As a first paradigmatic
task in operator control, we shall consider the implementation of a bit flip
operation on a qubit system ($d=2$); we stress that the bit flip must be
induced on the system by only acting on the controller. We allow in-loop
operations to be any single qubit unitary, we assume that the system is not
subject to any noise (i.e., the map $\mathcal{E}$ in Fig. 1 is the identity)
and that the controller is initialised to a generic mixed state
$\eta=\eta_{0}\ket{0}\bra{0}+(1-\eta_{0})\ket{1}\bra{1}$. The system input
state is of the general form
$\ket{\psi}=\cos{\frac{\chi}{2}}\ket{0}+e^{i\phi}\sin{\frac{\chi}{2}}\ket{1}$
and the figure of merit will be the Haar-averaged fidelity of the output state
to the desired state, i.e.,
$A=(4\pi)^{-1}\int\mathrm{d}\phi\mathrm{d}\chi\sin\chi\bra{\psi_{X}}\rho_{S}^{out}\ket{\psi_{X}}$,
where $\ket{\psi_{X}}=\sigma_{x}\ket{\psi}$. Crucially, the feedback protocol
must be implemented without knowledge of the exact input state (i.e., without
knowledge of $\chi$ and $\phi$).

First, we consider an intuitive CF protocol, where the in-loop operation
performed is a $\sigma_{x}$ unitary. When the state $\ket{\psi}$ is put
through this protocol, we obtain the average fidelity for coherent feedback
(see SM)

$A_{CF}=1-\frac{2}{3}\tau\,.$ (5)

We will now show that no MF protocol involving projective measurements can
outperform this CF protocol. Our general MF protocol will be as follows. After
interacting through the partial swap, the controller will be measured in the
$\\{\ket{0},\ket{1}\\}$ basis (notice that all choices of basis are equivalent
due to the unitary covariance of bit-flips and of the Haar measure). If the
measurement result is $\ket{0}$, a general unitary $U$ will be applied to the
controller and if the result is $\ket{1}$, a different unitary denoted by $V$
will be applied instead. After the system and controller interact again
through the partial swap, the Haar measure averaged fidelity to the desired
output state for this general MF protocol is a function of $U$ and $V$. It is
shown in the SM that this fidelity is maximised when $U=V=\sigma_{x}$, which
yields the following output fidelity for the optimal MF protocol

$A_{MF}=\frac{2}{3}-\frac{1}{3}\tau.$ (6)

It is simple to see that this fidelity is always smaller than that achieved
through CF (except for the trivial case $\tau=1$). Furthermore, we find that
the performance of MF improves with (non-projective) weaker measurements, but
cannot outperform CF for _any_ in-loop POVM (see SM). Therefore, CF is
superior to MF in assisting bit-flipping.

Operator control in the limit of weak interactions.—Having considered the task
of implementing a specific unitary, we now investigate the general task of
implementing an arbitrary unitary evolution on the system. To this aim, we
consider the setup discussed above in the limit of infinitesimally weak
system-controller interactions, i.e., we let
$\theta\xrightarrow[]{}\text{d}\theta$ in the transmissivity of the partial
swap $\tau=\cos^{2}\theta$. Expanding the interaction and keeping only first-
order terms, we find that one iteration of MF or CF results in the following
transformation on the system density operator:

$\rho_{S}\xrightarrow[]{}\rho_{S}-i[(\Lambda_{j}(\eta)+\eta),\rho_{S}]\text{d}\theta,$
(7)

where $\Lambda_{j}$ represents the particular CP-map applied in-loop (unitary
for CF and measurement followed by unitary feedback for MF) from the set of
possible in-loop CP-maps $\\{\Lambda_{j}\\}$. Therefore, in this weak limit,
the system evolves unitarily for both MF and CF, under a unitary given by
$V_{j}=e^{-iH_{j}\text{d}t}$ where $H_{j}=(\Lambda_{j}(\eta)+\eta)$. By
iteratively applying this process, both MF and CF can simulate any Hamiltonian
on the system, provided that it falls within the algebra generated by
commutation of elements of the set $\\{\Lambda_{j}(\eta)+\eta\\}$ (assuming
that the $\Lambda_{j}$ can be changed after each iteration as necessary). If
the set of in-loop CP-maps generated by MF or CF is non-trivial, this algebra
is typically the entire space of Hermitian matrices, since all pairs of
Hermitian matrices, except a set of measure zero can generate the entire space
of Hermitian matrices by commutation Lloyd (1995). Thus, in the limit of weak
interactions, as long as the set $\\{\Lambda_{j}(\eta)+\eta\\}$ is non-
trivial, both MF and CF can be used to simulate the effect of any Hamiltonian
on the system and for this purpose are equivalent. However, the dramatic
ability of measurements of changing the controller’s spectrum can be leveraged
here to achieve faster evolutions to target states than in CF schemes, as
$\Lambda_{j}(\eta)$ can in principle be any quantum state in MF. In this
regard, let us also notice that in the limiting case where $\eta$ is maximally
mixed, the set $\\{\Lambda_{j}(\eta)+\eta\\}$ for CF will contain only one
element which will be proportional to the identity (since the action of any
unitary leaves the identity unchanged), and so CF will not be able to exert
any Hamiltonian control.

Conclusion.—We have introduced a common framework for coherent and
measurement-based quantum feedback based on a collision model, featuring
repeated interactions with an environment which acts as a controller. This
allows MF and CF to be compared on equal footing, under the assumption of non-
destructive measurements. Our framework is broadly applicable to practical
cases, ranging from quantum optics and optomechanics in the continuous
variable regime to linear optics Bernardes  _et al._ (2015); Marletto  _et
al._ (2020), extending to all conceptual situations where CMs find application
Campbell and Vacchini (2021). We note that, since the input-output formalism
can be framed as a CM Ciccarello (2017), if the continuous-time limit of this
model is taken it fully captures the quantum optical notion of coherent
feedback as described in, e.g. Harwood and Serafini (2020); Harwood  _et al._
(2021) or in the equivalent SLH formulation Nurdin  _et al._ (2009); Nurdin
and Gough (2015).

The case studies addressed provide one with strong indications of the general
advantages and disadvantages of each type of feedback.

Acknowledgements.— M.B. acknowledges funding from the Swiss National Science
Foundation (PCEFP2$\\_$194268).

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Supplemental Material

##  I Coherent Feedback Cooling with noisy controller

We prove here that for ‘noisy’ controllers initialised in the maximally mixed
state, coherent feedback cannot provide any advantage. All of our examples use
the model introduced in the paper. In this model, the system and controller
are both $d$-dimensional qudits which interact twice through a partial swap
unitary. Inside the coherent feedback loop, we have no ancillary systems, and
the only allowed in-loop operations are single qudit unitaries. Throughout
these notes, the subscript $T$ will indicate the total two-qudit, system-
controller joint state and the subscript $S$ will refer to the system alone.
Our setup for cooling with coherent feedback at high temperature unfolds as
follows:

  1. 1.

The controller is initialised to the maximally mixed state
$\eta^{in}=\frac{1}{d}\mathbbm{1}$. The system input $\rho_{S}^{in}$ is the
result of the previous iteration of the process.

  2. 2.

The system is subject to noise in the form of a depolarising channel.

  3. 3.

The system and controller interact through a partial swap unitary.

  4. 4.

A unitary $U$ is applied to the controller.

  5. 5.

The system and controller interact again through a partial swap unitary.

The von Neumann entropy of the total input state $\rho_{S}^{in}\otimes\eta$ is
given by:

$S_{T}^{in}=S(\rho_{S}^{in}\otimes\eta)=S(\rho_{S}^{in})+S(\eta^{in})\,.$ (8)

After the application of the noise, the system is still in a separable state
and the total entropy is:

$S_{T}=S(\mathcal{E}(\rho_{S}^{in}))+S(\eta^{in})\,.$ (9)

Since the rest of the protocol can be described by unitary operations, the
global entropy is left unchanged so

$S_{T}^{out}=S_{T}=S(\mathcal{E}(\rho_{S}^{in}))+S(\eta^{in})\,.$ (10)

Using the subadditivy of entropy, we can obtain the following bound on the
entropy of the system output state:

$S(\rho_{S}^{out})\geq
S_{T}^{out}-S(\eta^{out})=S(\mathcal{E}(\rho_{S}^{in}))-\Delta S(\eta)\,.$
(11)

where $\Delta S(\eta)=S(\eta^{out})-S(\eta^{in})$ is the change in entropy of
the controller. Since $\eta^{in}$ is maximally mixed, the maximum value of
$\Delta S(\eta^{in})$ is $0$. This gives us the bound on the entropy of the
output:

$S(\rho_{S}^{out})\geq S(\mathcal{E}(\rho_{S}^{in}))\geq S(\rho_{S}^{in})\,.$
(12)

The entropy of the system output state can never be lower than the entropy of
the input. Furthermore, the input and output entropies are only equal when
$S(\mathcal{E}(\rho_{S}^{in}))=S(\rho_{S}^{in})$ which is only true when
$\rho_{S}^{in}$ is maximally mixed, meaning that the only steady state is the
maximally mixed state.

##  II  Measurement-Based Feedback for Cooling with a noisy controller

In this section we consider measurement-based feedback cooling for a two-qudit
setup with an initially maximally mixed environment/controller
$\eta=\frac{1}{d}\mathbbm{1}$. The system undergoes a depolarising map, before
system and controller interact through a first partial swap. The controller is
then measured in the basis $\\{\ket{j}\\}$ and a unitary $V_{j}$ is applied to
the controller, depending on the measurement outcome. The system and
controller then interact again through a partial swap, and the controller is
traced over. Throughout the supplemental material, we use the convention that
the partial swap is written $U_{s}=c\mathbbm{1}-is\hat{S}$ where
$c=\cos\theta$ and $s=\sin\theta$, in contrast to the main paper, where it is
written in terms of the transmissivity $\tau=c^{2}$.

###  II.1 Unconditional MF

We will now derive the steady state for the case of measurement-based feedback
averaged over all trajectories. This means that the in-loop measurement and
feedback can be represented as a CP-map with elements
$\\{V_{j}\ket{j}\bra{j}\\}$. In particular, we will find the steady state for
the case where, through the action of $V_{j}$ , all measurement outcomes are
mapped to the same state, which we will label $\ket{0}$. This corresponds to a
CP-map with elements $\\{\ket{0}\bra{j}\\}$. In the next section we will show
that this is optimal when one filters and conditions the state on the
measurement outcome. We will first derive expressions for the most general
projective MF protocol, and then restrict to the optimal case. Throughout, we
will denote the action of the initial depolarising map as
$\rho_{N}=\mathcal{E}(\rho_{in})=\lambda\rho_{in}+\frac{1-\lambda}{d}\mathbbm{1}$.
Notice that $\rho_{N}$ has the same eigenvectors as $\rho_{in}$ as well as the
same ordering (ie. the eigenvector corresponding to the largest eigenvalue of
$\rho_{in}$ also corresponds to the largest eigenvalue of $\rho_{N}$).

After the action of the depolarising map and the first partial swap, the
system and controller are correlated in the following state:

$\rho_{T}=c^{2}\rho_{N}\otimes\eta+s^{2}\eta\otimes\rho_{N}-isc[S,\rho_{N}\otimes\eta]\,,$
(13)

where $S$ is the swap unitary. After the measurement, and application of the
unitaries $\\{V_{j}\\}$, the global state is

$\rho_{T}=\sum_{j}p_{j}\rho_{j}\otimes\ket{\psi_{j}}\bra{\psi_{j}}\,,$ (14)

where $\ket{\psi_{j}}=V_{j}\ket{j}$ and $p_{j}$ are the probabilities of each
outcome. The (un-normalised) system state for each measurement outcome is
given by

$p_{j}\rho_{j}=\frac{1}{d}c^{2}\rho_{N}+s^{2}\eta(\rho_{N})_{jj}-isc[\ket{j}\bra{j},\rho_{N}]\,,$
(15)

where $(\rho_{N})_{jj}=\bra{j}\rho_{N}\ket{j}$. We have also used the fact
that $\text{Tr}_{C}[S,A\otimes B]=[B,A]$. After a final interaction through a
partial swap, the controller is traced out, leaving the output state:

$\rho_{out}=\sum_{i}p_{i}\bigg{(}c^{2}\rho_{i}+s^{2}\ket{\psi_{i}}\bra{\psi_{i}}-isc[\ket{\psi_{i}}\bra{\psi_{i}},\rho_{i}]\bigg{)}\,.$
(16)

The optimal protocol, in which all measurement results are mapped to the
$\ket{0}$ state through the unitaries $V_{j}$, amounts to setting
$\ket{\psi_{j}}=V_{j}\ket{j}=\ket{0}$ for all $j$. Using
$\sum_{j}p_{j}\rho_{j}=c^{2}\rho_{N}+s^{2}\eta$, we find the output state:

$\rho_{out}=c^{4}\rho_{N}+s^{2}c^{2}\eta+s^{2}\ket{0}\bra{0}-isc^{3}[\ket{0}\bra{0},\rho_{N}]\,.$
(17)

The fact that $\eta=\frac{1}{d}\mathbbm{1}$ allows us to write
$\rho_{N}=\lambda\rho_{in}+(1-\lambda)\eta$. We now assume a steady state,
which involves setting $\rho_{in}=\rho_{out}=\rho_{ss}$. Re-arranging for
$\rho_{ss}$ gives:

$\rho_{ss}=\frac{1}{1-c^{4}\lambda}\bigg{(}(c^{4}(1-\lambda)+s^{2}c^{2})\eta+s^{2}\ket{0}\bra{0}-\lambda
isc^{3}[\ket{0}\bra{0},\rho_{ss}]\bigg{)}\,.$ (18)

The only solution to this equation is one where $\rho_{ss}$ is diagonal in a
basis $\\{\ket{j}\\}$ which contains $\ket{0}$. To see this, we can act on the
above equation with $\bra{j}$ and $\ket{k}$ from the left and right (for
$j\neq k$) and obtain:

$\bra{j}\rho_{ss}\ket{k}=\frac{1}{1-c^{4}\lambda}(-\lambda
isc^{3}(\delta_{j0}\bra{0}\rho_{ss}\ket{k}-\bra{j}\rho_{ss}\ket{0}\delta_{0k})\,.$
(19)

Note that the right hand side of this equation is only nonzero when either $j$
or $k$ is equal to zero. For $j=0$, $k\neq 0$, we obtain:

$\bra{0}\rho_{ss}\ket{k}=\frac{1}{1-c^{4}\lambda}(-\lambda
isc^{3}(\bra{0}\rho_{ss}\ket{k})\,.$ (20)

The only solution to this equation is when $\bra{0}\rho_{ss}\ket{k}=0$.
Therefore $\rho_{ss}$ is diagonal in the basis $\\{\ket{j}\\}$. This means
that the eigevalues of $\rho_{ss}$ are given by $\bra{j}\rho_{ss}\ket{j}$. The
eigenvalue associated with $\ket{0}$ is:

$\alpha_{0}=\frac{1}{1-c^{4}\lambda}\bigg{(}\frac{c^{4}(1-\lambda)+s^{2}c^{2}}{d}+s^{2}\bigg{)}\,.$
(21)

The remaining $d-1$ eigenvalues are degenerate, each with value:

$\alpha_{j}=\frac{1}{1-c^{4}\lambda}\bigg{(}\frac{c^{4}(1-\lambda)+s^{2}c^{2}}{d}\bigg{)}\,.$
(22)

Thus, the steady state under this protocol takes the form:

$\rho_{S}=\frac{1}{d}\frac{d(1-c^{2})+c^{2}-\lambda c^{4}}{1-\lambda
c^{4}}\ket{0}\bra{0}+\sum_{j=1}^{d-1}\frac{1}{d}\frac{c^{2}-\lambda
c^{4}}{1-\lambda c^{4}}\ket{j}\bra{j}\,.$ (23)

The linear entropy for this steady state is:

$S_{L}=1-\sum_{j}\alpha_{j}^{2}=(1-\frac{1}{d})-\frac{\left(c^{2}-1\right)^{2}(d-1)}{d\left(c^{4}\lambda-1\right)^{2}}.$
(24)

We note that, for states like these with one large eigenvalues and $d-1$
degenerate eigenvalues, both Von Neumann and linear entropies are solely
functions of the largest eigenvalue and are thus equivalent for the purposes
of comparison. This is always the case for qubits, where the linear entropy
and Von Neumann entropy are equivalent. Because of this, and in view of their
more compact expressions, we will use the linear entropy for the remainder of
this investigation (though we will give expressions for the eigenvalues of
each state, from which the Von Neumann entropy can easily be calculated).

###  II.2 Conditional MF

In this section, we will consider filtered measurement feedback, meaning that
the measurement result is recorded and the system’s conditional state evolves
stochastically. After the measurement is performed along with the in-loop
unitary operation, the system and controller are in the joint state
$\rho_{j}\otimes\ket{\psi_{j}}\bra{\psi_{j}}$ for a measurement result
labelled by $j$ (where $\rho_{j}$ and $\ket{\psi_{j}}$ are defined as in the
previous section). This is as opposed to averaged MF, where the system and
controller are in the mixture of states given by (14). After the second
partial swap and tracing out of the controller, the system output will be:

$\rho_{out,j}=\text{Tr}_{C}[U_{s}\rho_{j}\otimes\ket{\psi_{j}}\bra{\psi_{j}}U_{s}^{\dagger}]\,.$
(25)

We will now show that, for a $\rho_{in}$ which is initially diagonal in the
measurement basis (as would be the case for the maximally mixed state, which
represents the uncontrolled steady state of the system, as well as for the
unconditional steady states determined in the previous section), the entropy
of this output is minimised when $\ket{\psi_{j}}$ is set to be the dominant
eigenvector of $\rho_{in}$ (i.e., the eigenvector associated to its largest
eigenvalue), regardless of the measurement result. Notice also that the choice
of basis for the ‘atomic’ projective measurement (i.e., a measurement that
resolves each individual basis state) is irrelevant, given the unitary
invariance of the problem (here, we are disregarding more general POVMs or
adaptive measurements, which may become the subject of further inquiry in the
future). If $\rho_{in}$ is diagonal in the measurement basis, the commutator
in equation (15) is equal to zero and we can write

$\rho_{j}=\frac{1}{p_{j}}\big{(}\frac{1}{d}c^{2}\rho_{N}+s^{2}\eta(\rho_{N})_{jj}\big{)}\,.$
(26)

Note that each $\rho_{j}$ has the same set of eigenvectors in the same
ordering as $\rho_{in}$, so the eigenvector corresponding to the largest
eigenvalue is the same for both $\rho_{in}$ and $\rho_{j}$.

We now make use of the entropy power inequality for the output of a partial
swap gate derived in Audenaert  _et al._ (2016), in the form of the following
majorisation relation:

$\lambda(\rho_{out,j})\prec
c^{2}\lambda(\rho_{j})+(1-c^{2})\lambda(\ket{\psi_{j}}\bra{\psi_{j}})\,,$ (27)

where we have used $\lambda(\rho)$ to indicate the spectrum of $\rho$, ordered
from the largest eigenvalue to the smallest. The right and left hand sides of
this relation are equal when $\ket{\psi_{j}}$ is pointing along the direction
of the dominant eigenvector of $\rho_{j}$, which in turn is the dominant
eigenvector of $\rho_{in}$. Thus, to minimise the conditional output entropy
for an input diagonal in the measurement basis, we must set all
$\ket{\psi_{j}}$ equal to the eigenvector corresponding to the largest
eigenvalue of $\rho_{in}$.

Let us now also briefly consider some quantitative examples of the conditional
stochastic evolution occurring under this filtered, measurement-based
feedback. This provides insight into the dependence of the optimal cooling
process on the various system parameters. Let us remind that the latter are
the depolarising strength $\lambda$ (from complete depolarisation for
$\lambda=0$ to the identity channel for $\lambda=1$), the partial swap angles
parametrised by the ‘connectivity’ $c=\cos\theta$ and the system dimension
$d$. Fig. 4 shows the evolution of the von Neumann entropy (with base $d$
logs, such that $1$ is always maximum entropy for all dimensions) for various
choices of parameters, always starting from the maximally mixed state (which
corresponds to the uncontrolled steady state). The unconditional
(‘unfiltered’) steady state entropies are also reported for comparison and, as
may be seen, always fall within the range of two possible, typical
unconditional values, deviating relatively little from them, such that the
stabilised, unfiltered strategy proves to be very effective in these
instances. Indeed, in all of these case studies, regardless of the Hilbert
space dimension, only one of the system eigenvalues is different from the
other, accruing probability at the expense of the other eigenvalues, which
remain equal (therefore, any POVM capable of resolving the dominant
eigenvector would also be optimal). When the dominant eigenvector is detected
on the controller branch of the feedback loop, its proportion in the system
branch decreases, and therefore the measurement is disadvantageous in terms of
cooling the system. At each run of the feedback loop, this occurs with
probability

$p_{0}=\frac{c^{2}}{d}+s^{2}\alpha_{\lambda}\;,$ (28)

where $s=\sin\theta$
$\alpha_{\lambda}=\lambda\,\alpha_{in}+\frac{(1-\lambda)}{d}$ and
$\alpha_{in}$ is the dominant eigenvalue of the input state. For completeness,
let us also report the values of the largest output eigenvector upon
measurement of the dominant eigenvector in the feedback loop ($\alpha_{0,0}$)
upon measurement of another outcome ($\alpha_{0,1}$), to be contrasted with
the average, unconditional $\alpha_{0}$ of Eq. (21)

$\displaystyle\alpha_{0,0}$
$\displaystyle=\frac{c^{2}\alpha_{\lambda}}{p_{0}d}+s^{2}\;,$ (29)
$\displaystyle\alpha_{0,1}$
$\displaystyle=\frac{c^{2}}{(1-p_{0})d}\left(c^{2}d\alpha_{\lambda}-\alpha_{\lambda}+s^{2}\
\right)+s^{2}\;.$ (30)

All of these three functions are monotonically increasing in $\lambda$ and
decreasing in $c^{2}$. Therefore, as one should expect, the asymtpotic cooling
performance will be more effective for higher $\lambda$, corresponding to less
noise, and lower $c^{2}$, corresponding to a larger connectivity between
system and controller, which allows one to swap a substantial part of the
final state with a low entropy one. Hence, as shown in Fig. 4, higher
connectivities can offset larger noise parameters. Observe also that higher
dimensions, typically, make for larger spread in the normalised entropy around
the unconditional value. Further, it is worthwhile noticing that the model
also allows one to observe that cooling will be achieved in a number of steps
of the order of $1/s^{2}$: the transmittivity between controller and system is
the parameter that determines the model’s cooling rate.

Figure 4: Normalised (with base-$d$ logarithms) von Neumann entropy of
conditional (continuous lines) and steady-state unconditional (horizontal
dotted lines) measurement-based feedback cooling for different values of $c$,
$\lambda$ and $d$, for a maximally mixed initial state. The $x$-axis reports
the number of loop iterations performed.

##  III Coherent Feedback for Cooling with a clean controller

We will consider a setup identical to the previous section, except that the
environment is initialised to a pure state $\ket{0}\bra{0}$. We will restrict
to qubits and find the lowest entropy steady state achievable through CF which
is diagonal in the $\\{\ket{0},\ket{1}\\}$ basis.

We express the in-loop unitary as using the general decomposition:

$U=\begin{pmatrix}e^{i\varphi_{1}}\cos\chi&e^{i\varphi_{2}}\sin\chi\\\
-e^{-i\varphi_{2}}\sin\chi&e^{-i\varphi_{1}}\cos\chi\end{pmatrix}\,.$ (31)

The steady state which satisfies the condition that it is diagonal in the
basis containing $\ket{0}$ takes the form:

$\rho_{ss}=\text{diag}(e_{1},1-e_{1})$ (32)

with

$e_{1}=\frac{-2c^{4}(\lambda+1)p^{2}(q+1)+2c^{2}\left(p^{2}(\lambda(q+2)+q+1)-\lambda\right)+\lambda-2\lambda
p^{2}+1}{\lambda\left(4c^{2}\left(p^{2}\left(\left(c^{2}-1\right)q+c^{2}-2\right)+1\right)+4p^{2}-2\right)-2}$
(33)

where $p=\cos{\chi}$ and $q=\cos{2\varphi_{1}}$. The linear entropy for this
state is minimised when $\chi$ and $\varphi_{1}$ are both integer multiples of
$\pi$, which is satisfied when the in-loop unitary $U$ is equal to the
identity. In this case the steady state has linear entropy:

$S_{L}=\frac{1}{2}-\frac{8c^{4}\left(c^{2}-1\right)^{2}}{\left(\left(1-2c^{2}\right)^{2}\lambda-1\right)^{2}}\,.$
(34)

If this CF protocol (‘do nothing’ in the loop) is applied to a setup with a
system and environment of dimension $d$, the steady state will have its
largest eigenvalue (with eigenvector $\ket{0}$) equal to:

$\beta_{0}=\frac{4\left(c^{2}-1\right)c^{2}(d+\lambda-1)+\lambda-1}{d\left(\left(1-2c^{2}\right)^{2}\lambda-1\right)}\,.$
(35)

The other $(d-1)$ eigenvalues are:

$\beta_{j}=\frac{\left(1-2c^{2}\right)^{2}(\lambda-1)}{d\left(\left(1-2c^{2}\right)^{2}\lambda-1\right)}\,.$
(36)

For dimension $d$, this steady state will have linear entropy:

$S_{L}=(1-\frac{1}{d})-\frac{16c^{4}\left(c^{2}-1\right)^{2}(d-1)}{d\left(\left(1-2c^{2}\right)^{2}\lambda-1\right)^{2}}\,.$
(37)

which is always less than the linear entropy of the maximally mixed state.
Note that when $c=\frac{1}{\sqrt{2}}$, $\beta_{0}=1$ and $\beta_{j\neq 0}=0$
and the steady state is pure. This is because, when $c=\frac{1}{\sqrt{2}}$,
the partial swap is the square root of the full swap, so applying it twice
enacts a full swap, replacing the system mode with the pure environmental
mode.

##  IV  Measurement-based Feedback for cooling with a clean controller

We will now look at the effect of our averaged MF protocol when the
environment is not maximally mixed. We will consider a qubit system with a
generic environmental input state $\eta$ which will have an arbitrary
temperature and apply the same MF protocol as in the high temperature case.
After one iteration of our MF protocol, the output state is:

$\rho_{out}=c^{4}\rho_{N}+s^{2}c^{2}\eta+s^{2}\ket{0}\bra{0}-isc^{3}[\ket{0}\bra{0},\rho_{N}]\,.$
(38)

Previously, we assumed that $\eta$ was maximally mixed, but it need not be. We
will restrict to qubits and choose $\ket{0}$ to point along the direction of
the dominant eigenvector of $\eta$, so we can write
$\eta=\eta_{0}\ket{0}\bra{0}+(1-\eta_{0})\ket{1}\bra{1}$. Solving for steady
state gives:

$\displaystyle\rho_{ss}$
$\displaystyle=\frac{1}{1-c^{4}\lambda}(c^{4}(1-\lambda)\frac{1}{d}\mathbbm{1}+(s^{2}c^{2}\eta_{0}+s^{2})\ket{0}\bra{0}+s^{2}c^{2}(1-\eta_{0})\ket{1}\bra{1}$
(39) $\displaystyle-i\lambda sc^{3}[\ket{0}\bra{0},\rho_{ss}])$ (40)

As before, this state is diagonal in the basis $\\{\ket{j}\\}$. It has
eigenvalues

$\alpha_{0}=\frac{1}{1-c^{4}\lambda}\left(c^{4}(1-\lambda)\frac{1}{2}+s^{2}c^{2}\eta_{0}+s^{2}\right)\,,$
(41)
$\alpha_{1}=\frac{1}{1-c^{4}\lambda}\left(c^{4}(1-\lambda)\frac{1}{2}+s^{2}c^{2}(1-\eta_{0})\right)\,.$
(42)

The linear entropy of this state is:

$S_{MF}=\frac{1}{2}-\frac{\left(c^{2}-1\right)^{2}\left(c^{2}(2\eta_{0}-1)+1\right)^{2}}{2\left(c^{4}\lambda-1\right)^{2}}\,.$
(43)

A maximally mixed environment corresponds to $\eta_{0}=1/2$, which recovers
the expression from earlier. For a pure environment, we set $\eta_{0}=1$ and
we obtain the following expression from the paper:

$S_{MF}=\frac{1}{2}-\frac{(c^{4}-1)^{2}}{2(c^{4}\lambda-1)^{2}}\,.$ (44)

We will now prove that, even with a pure environment, no MF protocol can
prepare a pure steady state when $c=\frac{1}{\sqrt{2}}$. Recall that, for
$c=\frac{1}{\sqrt{2}}$ and a pure environment CF produces a pure steady state,
so this is an interesting point of comparison. Again, we will restrict our
investigation to qubits. As we have seen before, after projective measurement
and the action of a unitary, the system and controller are in a joint state:

$\rho_{T}=\sum_{j}p_{j}\rho_{j}\otimes\ket{\psi_{j}}\bra{\psi_{j}}\,,$ (45)

where

$\rho_{j}=\frac{1}{p_{j}}\big{(}\frac{1}{2}c^{2}\rho_{N}+s^{2}\eta(\rho_{N})_{jj}-isc[\ket{j}\bra{j},\rho_{N}]\big{)}\,.$
(46)

First, we will show that $\rho_{j}$ cannot be pure. We do this by writing
$\rho_{j}$ in the basis containing $\ket{j}$ and restrict to the case of
interest, when $c=s=\frac{1}{\sqrt{2}}$. This gives us:

$\rho_{S}^{(1)}=\frac{1}{2}\begin{pmatrix}1+\rho_{00}&\rho_{01}(1+i)\\\
\rho_{10}(1-i)&1-\rho_{00}\end{pmatrix}$ (47)

where $\rho_{ij}$ are the matrix elements of $\rho_{N}$. This matrix has
eigenvalues:

$\lambda_{+/-}=\frac{1}{2}\big{(}1\pm\sqrt{\rho_{00}^{2}+2\rho_{01}\rho_{10}}\big{)}\geq
0\,.$ (48)

If this state is pure, then one of its eigenvalues will be equal to $0$, and
the other equal to $1$. This requires
$\sqrt{\rho_{00}^{2}+2\rho_{01}\rho_{10}}=1$. Since both eigenvalues must be
greater than or equal zero, $\rho_{00}^{2}+2\rho_{01}\rho_{10}\leq 1$. The
equality is reached only in the case where $\rho_{00}=1$, which could only be
the case if $\rho_{N}$ was pure. However, $\rho_{N}$ cannot be pure since it
has been subject to the depolarising map. Even if $\rho_{in}$ was pure,
$\rho_{N}$ would not be pure for any non-neglible value of the noise parameter
$\lambda$. Thus, we can conclude that the $\rho_{j}$’s are not pure.

After the second system-controller interaction, the controller is traced out
and the output state is given by:

$\rho_{out}=\sum_{i}p_{i}\bigg{(}c^{2}\rho_{i}+s^{2}\ket{\psi_{i}}\bra{\psi_{i}}-isc[\ket{\psi_{i}}\bra{\psi_{i}},\rho_{i}]\bigg{)}=\sum_{i}p_{i}\sigma_{i}\,.$
(49)

By applying the same argument that we used to prove that $\rho_{j}$ could not
be pure, we can prove that $\sigma_{j}$ also cannot be pure. Thus, since
$\rho_{out}$ is a mixture of mixed states, it cannot be pure. This means that
no averaged MF protocol can achieve a pure steady state for a partial swap
interaction when the coupling is characterised by $c=\frac{1}{\sqrt{2}}$.

##  V Cooling Comparison with intermediate noise

We will now apply our CF protocol from earlier, where the in-loop unitary is
the identity, to a setup with a non-zero temperature environment with state
$\eta=\eta_{0}\ket{0}\bra{0}+(1-\eta_{0})\ket{1}\bra{1}$. This leads to a
steady state where which is diagonal in the $\\{\ket{0},\ket{1}\\}$ basis with
the following eigenvalue corresponding to the $\ket{0}$ eigenvector:

$\beta_{0}=\frac{4\left(c^{2}-1\right)c^{2}(2\eta_{0}+\lambda-1)+\lambda-1}{2\left(1-2c^{2}\right)^{2}\lambda-2}\,.$
(50)

This steady state has a linear entropy

$S_{CF}=\frac{1}{2}-\frac{8c^{4}\left(c^{2}-1\right)^{2}(1-2\eta_{0})^{2}}{\left(\left(1-2c^{2}\right)^{2}\lambda-1\right)^{2}}\,.$
(51)

We will now compare the performance MF and CF for intermediate environmental
temperatures by comparing equations (43) and (51). Equation (43) gives the
steady state system entropy when the MF protocol involves measuring the
ancilla and, regardless of the outcome, preparing it in the same state before
the system and environment interact for the second time. This equation assumed
that, after measurement, the controller is prepared in the state
$\ket{0}\bra{0}$. Note that the protocol allows for some flexibility as the
controller could be prepared in any state after measurement. It is optimal to
use MF to prepare the controller in the state corresponding to the largest
eigenvalue of the environmental input. For $\eta_{0}<1/2$, this corresponds to
$\ket{0}\bra{0}$ and for $\eta>1/2$, this corresponds to $\ket{1}\bra{1}$. For
our comparison, we assume that the optimal MF protocol is used. The steady
state entropy for this protocols, along with steady state CF entropy given by
equation (51) is plotted in Fig. 5 for different environmental states, as
parameterised by $\eta_{0}$. We find that for the setup with $c=0.5$, MF
outperforms CF for all environmental temperatures. For a weaker system-
environment interaction, characterised by $c=0.9$ and noise parameter
$\lambda=0.5$, we find that MF outperforms CF for high temperature
environments with $0.357<\eta_{0}<0.764$, but CF outperforms MF at low
temperatures characterised by $\eta_{0}<0.357$ and $\eta_{0}>0.764$. Broadly,
we can make the following observation: for low temperature environments and
weak couplings, the act of measurement disturbs the coherent process which
allows for low entropy environmental states to be transferred to the system,
meaning that MF is inferior to CF. However, with strong couplings and noisy
environments, the purification from the act of measurement compensates for
this and leads MF to be superior to CF.

Figure 5: (a) Shows the steady state entropy against the largest eigenvalue of
the environmental state for MF and CF setups, as given by equations (43) and
(51) where $c=0.5$ and $\lambda=0.25$. (b) Shows the same expressions for
setups where $c=0.9$ and $\lambda=0.5$.

##  VI Coherent Feedback for Protecting an Excited State from Amplitude
Damping

We will now investigate the ability of CF to protect a qubit from amplitude
damping. The qubit will be subject to amplitude damping channel, after which
the CF loop will be applied. We will assume that the highest energy state is
the $\ket{1}$ state, so that amplitude damping channel is given by Kraus
operators:

$E_{0}=\sqrt{\gamma}\ket{0}\bra{1}\,,\quad
E_{1}=\sqrt{1-\gamma}\ket{1}\bra{1}+\ket{0}\bra{0}\,.$ (52)

This channel will be applied to the system before the CF loop is applied to
attempt to counter it. Using our toy model again, the two system-controller
interactions will be partial swaps. We will restrict the in-loop unitaries to
rotations of the form:

$U=\begin{pmatrix}\cos\chi&\sin\chi\\\ -\sin\chi&\cos\chi\\\ \end{pmatrix}\,.$
(53)

We will assume that the controller is ‘noisy’ and initialised in the maximally
mixed state. Our figure of merit will be the steady-state occupation of the
$\ket{1}$ state. Investigating numerically, we find that the optimal CF
protocol depends on the partial swap strength, characterised by $c$. For
$c^{2}<\frac{1}{2}$, the optimal protocol involves setting
$\chi=\frac{\pi}{2}$ and for $c^{2}>\frac{1}{2}$, the optimal protocol
involves setting $\chi=0$. Plots of the steady-state occupation of the
$\ket{1}$ state, against $\chi$, for different values of the $c^{2}$ are
plotted in Figure 6.

Plot 6 shows the performance of the setup with weak damping, characterised by
$\gamma=0.2$ and plot 6 shows the performance of the same setup with stronger
damping, characterised by $\gamma=0.8$. Notice that, for weak system-
controller interactions (where $c^{2}>\frac{1}{2}$), increasing the damping
noise strength decreases the steady-state occupation of the excited state, as
expected. However, for strong system-controller interactions,(where
$c^{2}<\frac{1}{2}$), the optimal performance (when $\chi=\frac{\pi}{2}$) is
actually _improved_ by stronger damping. This is because the action of the in-
loop $\frac{\pi}{2}$ rotation is more effective at populating the $\ket{1}$
state when more of the state is initially prepared in the $\ket{0}$ state. In
this sense, CF allows for the purifying effect of the amplitude damping map to
be harnessed for the purpose of increasing the excited state population.

Figure 6: The steady-state occupation of the $\ket{1}$ state for CF setups
with different coupling strengths, characterised by $c^{2}$. Figure 6 is for a
setup with amplitude damping characterised by $\gamma=0.2$ and Figure 6 is for
a setup characterised by $\gamma=0.8$. Note that, for strong couplings
($c^{2}<\frac{1}{2}$), the setup with stronger amplitude damping actually has
a _higher_ excited state occupation.

When $\chi=0$ and the in-loop unitary is the identity, the steady-state
occupation of the $\ket{1}$ state is:

$\rho_{11}^{\chi=0}=\frac{2c^{2}\left(1-c^{2}\right)}{4\left(c^{2}-1\right)(\gamma-1)c^{2}+\gamma},$
(54)

which is a decreasing function of $\gamma$ for $c^{2}>\frac{1}{2}$.
Conversely, when $\chi=\frac{\pi}{2}$, the steady-state occupation of the
$\ket{1}$ state is:

$\rho_{11}^{\chi=\frac{\pi}{2}}=\frac{1-c^{2}}{2(\gamma-1)c^{2}-\gamma+2}\,,$
(55)

which is an increasing function of $\gamma$ for $c^{2}<\frac{1}{2}$.

##  VII Measurement-based Feedback for Protecting an Excited State from
Amplitude Damping

We will compare this to an intuitive MF protocol which measures the controller
in the $\\{\ket{1},\ket{0}\\}$ basis, does nothing if the result is $\ket{1}$,
and applies an in-loop rotation with $\chi=\frac{\pi}{2}$ if the result is
$\ket{0}$, resulting in a controller which is always prepared in the $\ket{1}$
state post-measurement. This protocol results in a steady-state occupation of
the $\ket{1}$ state:

$\rho_{11}^{MF}=\frac{2-c^{4}-c^{2}}{2(\gamma-1)c^{4}+2}\,.$ (56)

This expression is greater than equation (55) for all values of $c$ and
$\gamma$, so we can say that MF outperforms CF in the regime of
$c^{2}<\frac{1}{2}$. However, for some setups with $c>\frac{1}{2}$, equation
(56) is slightly lower than (54). In particular, (54) is greater than (56)
when:

$c>\frac{1}{2}\sqrt{\frac{-7\gamma+\sqrt{\gamma(17\gamma-24)+16}+4}{2-2\gamma}}\,.$
(57)

In this regime, CF will outperform MF, though numerical investigations suggest
that the advantage is small. Conversely, in the regime where MF outperforms
CF, the advantage tends to be larger. Figure (3) shows $\rho_{11}$ the
occupation of the $\ket{1}$ state, against $N$, the number of iterations of
the feedback loop. Both MF and CF protocols described above are presented. In
one setup, MF outperforms yields a higher steady-state occupation of the the
excited state, and in the other, CF achieves a higher steady-state occupation.
The unconditional (averaged) MF trajectory is shown, as well as fifty
conditional trajectories for each setup.

##  VIII Coherent Feedback for performing a bit-flip

The input for the system will be a pure state given by

$\ket{\psi}=\cos{\frac{\chi}{2}}\ket{0}+e^{i\phi}\sin{\frac{\chi}{2}}\ket{1}\,.$
(58)

The environment is initialised to a state of the form
$\eta=\eta_{0}\ket{0}\bra{0}+(1-\eta_{0})\ket{1}\bra{1}$. The system and
environment interact though a partial swap unitary given by
$U_{s}=c\mathbbm{1}-is\hat{S}$. After this interaction, a $\sigma_{x}$ unitary
is performed on the controller, before the system and controller interact
again through $U_{s}$ and the controller is traced out. For such a setup, the
final output state is $\rho_{S}^{out}$, which does not depend on the value of
$\eta_{0}$ and takes the form:

$\rho_{out}=\frac{1}{2}\left(\begin{array}[]{cc}(c^{4}-s^{4})\cos(\chi)+1&e^{-i\phi}\sin(\chi)\left(c^{2}+s^{2}e^{2i\phi}\right)\\\
e^{-i\phi}\sin(\chi)\left(s^{2}+c^{2}e^{2i\phi}\right)&\left(s^{4}-c^{4}\right)\cos(\chi)+1\\\
\end{array}\right)$ (59)

The desired final output state is $\rho_{X}=\ket{\psi_{X}}\bra{\psi_{X}}$
where

$\ket{\psi_{X}}=\sigma_{x}\ket{\psi}=\cos{\frac{\chi}{2}}\ket{1}+e^{i\phi}\sin{\frac{\chi}{2}}\ket{0}\,.$
(60)

The fidelity of the output state to the desired state is a function of $\chi$
and $\phi$ and is given by

$F_{CF}(\chi,\phi)=\bra{\psi_{X}}\rho_{S}^{out}\ket{\psi_{X}}=\frac{1}{4}(c^{2}\left(\cos(2\phi)-2\cos(2\chi)\cos^{2}(\phi)\right)-3c^{2}+4)\,.$
(61)

As our figure of merit, we take this ouput fidelity, averaged over the Haar
measure for the input states, given by:

$A_{CF}=\frac{1}{4\pi}\int F_{CF}(\chi,\phi)\sin{(\chi)}d\chi
d\phi=1-\frac{2}{3}c^{2}\,.$ (62)

where $\chi$ is integrated from $0$ to $\pi$ and $\phi$ is integrated from $0$
to $2\pi$.

##  IX Measurement-based feedback for performing a bit-flip

###  IX.1 Projective Measurements

Our general projective MF protocol is as follows. The system is initialised in
the pure state $\ket{\psi}$ given above, and the controller initialised in the
state $\eta=\eta_{0}\ket{0}\bra{0}+(1-\eta_{0})\ket{1}\bra{1}$. They interact
through $U_{s}$. Then, a measurement is made on the controller. If the result
is $\ket{0}$, the unitary $U$ is applied to the controller and if the result
is $\ket{1}$, a unitary $V$ is applied instead. This process is equivalent to
the POVM with elements $\\{U\ket{0}\bra{0},V\ket{1}\bra{1}\\}$ being applied
to the controller. After this, the system and controller interact again
through $U_{s}$. The output system state $\rho_{S,MF}^{out}$ is too lengthy to
print here, as is the fidelity
$F_{MF}=\bra{\psi_{X}}\rho_{S,MF}^{out}\ket{\psi_{X}}$. However, after the
Haar measure average is taken, we obtain the more reasonable expression:

$\displaystyle A_{MF}$ $\displaystyle=\frac{1}{4\pi}\int
F_{MF}(\chi,\phi)\sin{(\chi)}d\chi d\phi$ (63)
$\displaystyle=\frac{1}{12}(6-2c^{4}-s^{4}\cos{2\theta_{u}}-s^{4}\cos{2\theta_{v}})\,,$

where we have used the decomposition of $2\times 2$ unitary matrices to write
magnitudes of the matrix elements $u_{jk}$ and $v_{jk}$ as

$|u_{00}|=\cos{\theta_{u}}\quad|u_{10}|=\sin{\theta_{u}}\quad|v_{11}|=\cos{\theta_{v}}\quad|v_{01}|=\sin{\theta_{v}}\,.$
(64)

We find that the Haar measure averaged fidelity is maximised when
$\theta_{u}=\theta_{v}=\frac{\pi}{2}$, meaning that both $U$ and $V$ are equal
to $\sigma_{x}$ (up to a phase which doesn’t affect the outcome). Plugging
these optimal unitaries gives the maximum average output fidelity:

$A_{MF}=\frac{1}{3}(1+s^{2})=\frac{2}{3}-\frac{1}{3}c^{2}$ (65)

It is straightforward to see that this expression is lower than $A_{CF}$ for
all values of $\theta$, except when $c=1$ and there is no feedback present.

###  IX.2 General POVMs

In the previous section, we considered MF using projective measurements in-
loop, but we can also consider more general POVMs. Here, we will consider the
action of general POVMs in between the two system-environment partial swap
interactions. The polar decomposition can be used to write the Kraus operators
of any POVM, represented with $K_{j}$ as $K_{j}=U_{j}P_{j}$, where $U_{j}$ is
a unitary matrix and $P_{j}$ is a positive semidefinite matrix . Note that, to
define a POVM, we must have $\sum_{j}K_{j}^{\dagger}K_{j}=\mathbbm{1}$ which
implies $\sum_{k}P_{j}^{\dagger}P_{j}=\mathbbm{1}$. Thus, we can view any
general POVM as the action of measurement characterised by $\\{P_{j}\\}$,
followed by the action of a unitary $U_{j}$ which depends on the measurement
outcome (this observation was made in Jacobs  _et al._ (2014)). Since, in MF,
we are already allowing for the action of a unitary depending on the
measurement outcome, we can absorb $U_{j}$ into these feedback unitaries and
consider the measurement process as entirely characterised by $\\{P_{j}\\}$.

Furthermore, since our figure of merit is averaged over the Haar measure,
which is unitarily invariant, we can assume that $P_{j}$ are diagonal in the
$\\{\ket{0},\ket{1}\\}$ basis. Therefore, the most general qubit POVM can be
described using the Kraus operators:

$P_{0}=a\ket{0}\bra{0}+b\ket{1}\bra{1}\quad\,,\quad
P_{1}=\sqrt{1-a^{2}}\ket{0}\bra{0}+\sqrt{1-b^{2}}\ket{1}\bra{1}\quad\,.$ (66)

For $0\leq a\leq 1$ and $0\leq b\leq 1$. Note that when $a=b$, both elements
of the POVM are proportional to the identity and correspond to no measurements
being performed. When $a=1$ and $b=0$ the POVM corresponds to a projective
measurement. Thus, the difference between $a$ and $b$ acts as a measure of the
strength of the measurement.

We will absorb the action of $U_{j}$ into $U$ and $V$, so that the entire
process is captured by a POVM with elements $\\{UP_{0},VP_{1}\\}$. Again, by
using the decomposition of $2\times 2$ unitary matrices, we come to the
optimal protocol, which is when $U=V=\sigma_{x}$. Applying this optimal
protocol yields the following expression for the Haar-measure averaged
fidelity:

$A_{MF}=\frac{1}{3}\big{(}\sqrt{1-a^{2}}\sqrt{1-b^{2}}s^{2}+abs^{2}+2-c^{2}\big{)}\,.$
(67)

This expression yields the expression previously obtained for projective
measurement feeback when $a=1$ and $b=0$. When $a=b=1/2$, we have
$P_{0}=P_{1}=\frac{1}{2}\mathbbm{1}$, which corresponds to no measurement
being performed (or ‘infinitely weak’ measurement) and only the action of the
unitaries. In this case, (67) yields the expression previously obtained for
coherent feedback. It is in this sense CF can be viewed as MF in the limit of
infinitely weak measurements.

It is straightforward to show that the output fidelity (67) is maximised when
$a=b$, thus proving that no measurement-based feedback process can outperform
the coherent feedback protocol in this task. However for performing a bit-
flip, POVMs corresponding to weaker measurements (where the value of $a$ is
closer to $b$) can achieve better performance than stronger measurements, as
they disturb the input state less.

##  X Operator Control in the Limit of weak interactions

We have looked at comparing MF and CF with a partial swap coupling

$U_{s}^{\theta}=\cos{\theta}\mathbbm{1}-i\sin{\theta}\hat{S}$ (68)

where $\hat{S}$ is the full swap. When $\theta=0$, both MF and CF are
trivially the same, as the system and controller do not interact at all. We
will now investigate the effect of MF and CF in the limit of an infinitesimal
interaction angle $\theta\xrightarrow[]{}d\theta$.

Expanding $U_{s}$ to lowest order in $\theta$:

$\lim_{\theta\xrightarrow[]{}d\theta}U_{s}^{\theta}=\mathbbm{1}-id\theta\hat{S}+o((d\theta)^{2})\,.$
(69)

Then, to first order we have:

$U_{s}U_{T}U_{s}=U_{T}-id\theta\\{S,U_{T}\\}\,.$ (70)

for any operator $U_{T}$. The effect of one iteration of CF is

$\rho\xrightarrow{}\text{Tr}_{C}[U_{s}U_{T}U_{s}\rho\otimes\eta
U_{s}^{\dagger}U_{T}^{\dagger}U_{s}^{\dagger}]\,,$ (71)

where $U_{T}=\mathbbm{1}\otimes U$ is a unitary which acts only on the
controller. Expanding $U_{s}$ to first order in $\theta$ and discarding higher
order terms gives:

$\text{Tr}_{C}\big{[}U_{T}\rho\otimes\eta
U_{T}^{\dagger}-id\theta\\{S,U_{T}\\}\rho\otimes\eta U_{T}^{\dagger}\
+id\theta U_{T}\rho\otimes\eta\\{S^{\dagger},U_{T}^{\dagger}\\}\big{]}\,.$
(72)

Using the fact that $\text{Tr}_{C}\big{(}SA\otimes B=BA\big{)}$ and the fact
that the partial trace is cyclically invariant over the subspace which is
being traced over, we obtain:

$\text{Tr}_{C}[\\{S,U_{T}\\}\rho\otimes\eta U_{T}^{\dagger}]=U\eta
U^{\dagger}\rho+\eta\rho\,,$ (73)
$\text{Tr}_{C}[U_{T}\rho\otimes\eta\\{S^{\dagger},U_{T}^{\dagger}\\}]=\rho
U\eta U^{\dagger}+\rho\eta\,.$ (74)

Combining these, we obtain the transformation:

$\rho_{S}\xrightarrow[]{}\rho_{S}+i[\rho_{S},U\eta
U^{\dagger}]d\theta+i[\rho_{S},\eta]d\theta\;.$ (75)

To represent more general in-loop operations, we can replace $U$ with a Kraus
operator $K_{j}$ acting on the controller. Due to the linearity of the trace,
we can sum over these Kraus operators and in this way we obtain the expression

$\rho_{S}\xrightarrow[]{}\rho_{S}+i[\rho_{S},\Phi(\eta)]d\theta+i[\rho_{S},\eta]d\theta\;,$
(76)

where $\Phi(\eta)=\sum_{j}K_{j}\eta K_{j}^{\dagger}$. This expression can be
used to describe measurement-based feedback when $K_{j}$ are used to describe
the POVM operators, followed by the action of a unitary.