# Deterministic preparation of optical qubits with coherent feedback control

Amy Rouillard  School of Chemistry and Physics, University of KwaZulu-Natal,
Durban, South Africa  Tanita Permaul  School of Chemistry and Physics,
University of KwaZulu-Natal, Durban, South Africa  Sandeep K. Goyal
<EMAIL_ADDRESS>Department of Physical Sciences, Indian Institute of
Science Education & Research (IISER) Mohali, Sector 81 SAS Nagar, Manauli PO
140306 Punjab India.  Thomas Konrad<EMAIL_ADDRESS>School of Chemistry
and Physics, University of KwaZulu-Natal, Durban, South Africa  National
Institute of Theoretical and Computational Sciences (NITheCS), KwaZulu-Natal,
South Africa

###### Abstract

We propose a class of preparation schemes for orbital angular momentum and
polarisation qubits carried by single photons or classical states of light
based on coherent feedback control by an ancillary degree of freedom of light.
The preparation methods use linear optics and include the transcription of an
arbitrary polarisation state onto a two-level OAM system (swap) for arbitrary
OAM values $\pm\ell$ within a light beam, i.e. without spatial interferometer.
The preparations can be carried out with unit efficiency independent from the
potentially unknown initial state of the system. Moreover, we show how to
translate measurement-based qubit control channels into coherent feedback
schemes for optical implementation.

##  I Introduction

The control of quantum systems is an important prerequisite for all quantum
technologies, i.e. quantum computation and communication [1] as well as
quantum metrology [2, 3]. Closed-loop quantum control techniques, probing the
quantum system, can be classified in two groups, measurement-based techniques
[4] and coherent-feedback techniques [5, 6, 7, 8]. For quantum communication,
which involves non-classical states of light, measurement based-control is
technically challenging since non-invasive measurements that do not absorb
photons require coupling to matter or other photons, which is in general weak
and difficult to realise [2] or comes with low efficiency [9]. Therefore
coherent control feedback techniques have the potential to improve quantum
communication and all quantum technologies using light as a medium.

Here, we give a recipe to translate any measurement-based control channel for
two-level systems into a coherent feedback control scheme that can be
implemented with optics given that the control channel is described by two
Kraus operators. Our scheme can be generalised to systems with Hilbert spaces
of finite dimension and arbitrary number of Kraus operators.

Moreover, we present a class of control methods based on coherent feedback
where a particular degree of freedom of light can be controlled by a second
degree of freedom. Such methods enable, e.g., the deterministic preparation of
any pure polarisation state in a Mach-Zehnder interferometer for a photon with
initially mixed polarisation. Similarly, any coherent superposition
$\alpha\ket{-\ell}+\beta\ket{\ell}$ can be prepared from an unknown possibly
incoherent mixture of light modes with orbital angular momentum (OAM) values
$\pm\ell$ by means of a spatial interferometer (path-degree of freedom) or,
using polarisation in a single light beam. Unlike other methods of preparation
for spatial modes, e.g. by means of spatial light modulation, the present one
suggests low noise and works ideally without photon loss. For the special case
of equally weighted superpositions, $\alpha=\beta=1/\sqrt{2}$, we show how to
produce such a target state for all pairs of OAM values $\pm\ell$,
simultaneously.

All preparation schemes described below work equally well for single-photon
and classical states of light including thermal states with finite
temperature. The underlying coherent feedback control method is robust and can
be used to protect a target state against noise [8].

The conditions required for a control channel to successfully drive an
arbitrary initial state into a desired target state are reviewed in Section
II. The optical implementation scheme, for various degrees of freedom of
light, is informed by a decomposition technique [10] described in Section III.
In Section IV this technique is applied to unitary couplings that realise
coherent feedback control, and the translation from measurement-based to
coherent feedback control is presented. Section IV also details the optical
realisation of the coherent feedback control of polarisation and OAM qubits.
Methods which allow for the repeated application of coherent feedback control
are discussed in Section V. Such repetitions also protect target states
against noise [8] and can steer the system into target dynamics [11]. In
Section VI we give three examples of coherent feedback control and their
respective optical implementations. A discussion of the results in Section VII
concludes this article.

##  II Coherent feedback control

In order to prepare light in a given target state $\ket{T}\in\mathcal{H}_{s}$,
where $\mathcal{H}_{s}$ is the Hilbert space of the system, the following
theorem can be applied [8]. Consider a trace-preserving quantum channel $
described by an $n$-element set of Kraus operators $K_{i}$ satisfying the fix-
point condition

$\displaystyle K_{i}\ket{T}=z_{i}\ket{T}$ (1)

for all $i=0,\dots,n-1$ with $z_{i}\in\mathbb{C}$. In addition, let the Kraus
operators obey a second condition,

$\displaystyle\operatorname{span}\\{K_{i}^{\dagger}\ket{T}\\}_{i=0,\dots,n-1}=\mathcal{H}_{s}\,.$
(2)

Then an arbitrary initial state of the system $\rho$ converges to target state
$\ket{T}\in\mathcal{H}_{s}$ under repeated application of the channel $\$$,

$\displaystyle\rho\rightarrow\$(\rho)=\sum_{i=0}^{n-1}K_{i}\rho
K_{i}^{\dagger}\,.$ (3)

The second condition (2) ensures that the system is driven towards the target
state [8], while fix point condition (1) arrests it there. Various
construction methods of control channels are discussed in [8, 11].

Any such channel $\$$ can be implemented by a unitary time evolution $U$ which
couples the system to a suitable ancilla system (quantum controller) in
initial state $\ket{0}$, such that

$\displaystyle\$(\rho)=\operatorname{Tr}[U(\ket{0}\bra{0}\otimes\rho)U^{\dagger}]\,.$
(4)

The quantum controller probes the system’s state and accentuates coherent
feedback accordingly. If we compare Eqs. (3) and (4) the Kraus operators are
revealed as function of the time evolution,

$\displaystyle K_{i}=\braket{i}{U}{0}\,,$ (5)

where $(\ket{i})_{i=1,\ldots n}$ is an orthonormal basis of the quantum
controller’s Hilbert space. Obviously, there is no way to map all initial
states of system and controller onto a single state by a unitary (i.e.
reversible) evolution. Consequently, the information about the arbitrary, and
possibly unknown, initial state of the system must be transferred to the
controller system during their coupling [12]. This implies that after one
application of the channel $\$$, the controller will be in an unknown state
and must be reset to $\ket{0}$ if $\$$ is to be repeated using the same
controller.

The resetting of the controller presents the main challenge in the repeated
implementation of the control scheme and will be discussed in Section V
(abbreviated to Sec. V). It is possible to design a control channel that
reaches the target within a single application, an example of such a channel
is given in Sec. VI.1. However, it is still useful to investigate methods
which allow for multiple iteration of the channel as this will protect the
state against noise.

##  III Cosine-Sine decomposition

The time evolution $U$ can be constructed from elementary single- and two-
partite unitary gates. For this purpose, we employ the Cosine-Sine (CS)
decomposition of an arbitrary $U$ into a product of conditional unitaries and
Hadamard gates. When applied to light, these conditional unitaries correspond
to operations on a single degree of freedom of light (system) conditioned on
the state of another ancillary degree of freedom (controller).

An arbitrary $(m+n)\times(m+n)$ unitary matrix $U_{m+n}$ ($n\geq m$) can be
decomposed into $n\times n$ and $m\times m$ unitaries, $L_{n},R_{n}$ and
$L_{m},R_{m}$ respectively, as well as a cosine-sine (CS) matrix, according to
the CS decomposition [10], given by

$U_{m+n}=\begin{pmatrix}L_{m}&0\\\
0&L^{\prime}_{n}\end{pmatrix}(\mathcal{S}_{2m}\oplus\mathds{1}_{n-m})\begin{pmatrix}R^{\dagger}_{m}&0\\\
0&R^{\prime\dagger}_{n}\end{pmatrix}\,,$ (6)

where $\mathds{1}_{n-m}$ is the identity matrix in $(n-m)$-dimensions and the
so-called CS matrix $\mathcal{S}_{2m}$ reads

$\mathcal{S}_{2m}=\begin{pmatrix}C_{m}&S_{m}\\\ -S_{m}&C_{m}\end{pmatrix}$ (7)

with $C_{m}\equiv\sum_{i=1}^{m}\cos\theta_{i}\ket{i}\bra{i}$ and
$S_{m}\equiv\sum_{i=1}^{m}\sin\theta_{i}\ket{i}\bra{i}$. The direct sum
$\mathcal{S}_{2m}\oplus\mathds{1}_{n-m}$ can also be written in the form of a
block diagonal matrix

$\displaystyle\begin{pmatrix}\mathcal{S}_{2m}&0\\\
0&\mathds{1}_{n-m}\end{pmatrix}\,.$ (8)

In this work we will only consider cases where $m=n$, so that

$U=\begin{pmatrix}L&0\\\ 0&L^{\prime}\end{pmatrix}\begin{pmatrix}C&S\\\
-S&C\end{pmatrix}\begin{pmatrix}R^{\dagger}&0\\\
0&R^{\prime\dagger}\end{pmatrix}\,,$ (9)

where, for convenience, we drop the indices which indicate dimension. The
matrix $\mathcal{S}$ can be further decomposed, see Appendix (App.) A, as

$\mathcal{S}=\begin{pmatrix}C&S\\\
-S&C\end{pmatrix}=\Big{(}P_{\frac{\pi}{4}}^{\dagger}H\otimes\mathds{1}\Big{)}\begin{pmatrix}\Theta&0\\\
0&\Theta^{\dagger}\end{pmatrix}\Big{(}HP_{\frac{\pi}{4}}\otimes\mathds{1}\Big{)},$
(10)

where $P_{\phi}=\exp(i\phi)\ket{0}\bra{0}+\exp(-i\phi)\ket{1}\bra{1}$ may
represent a phase shift and
$H=\ket{0}(\bra{0}+\bra{1})/\sqrt{2}+\ket{1}(\bra{0}-\bra{1})/\sqrt{2}$ is the
Hadamard transformation – both acting on a two-level system. In addition,
$\Theta=C+iS$ can be implemented as a state-dependent phase shift of an
$n$-level system.

Any unitary operator acting on a $2n$-dimensional Hilbert space can therefore
be written

$U=\begin{pmatrix}L&0\\\
0&iL^{\prime}\end{pmatrix}H\otimes\mathds{1}\begin{pmatrix}\Theta&0\\\
0&\Theta^{\dagger}\end{pmatrix}H\otimes\mathds{1}\begin{pmatrix}R^{\dagger}&0\\\
0&-iR^{\prime\dagger}\end{pmatrix}\,.$ (11)

The CS decomposition thus points to the physical realisation of a ($2n\times
2n$) unitary operator in terms of the evolution of a closed system formed by
two subsystems, i.e., a two-level system coupling to an $n$-level system. This
agrees with a sequence of evolutions of a system of $n$ spatial modes of light
depending on two paths (as e.g. realised in a Mach-Zehnder) or depending on
its polarisation. Hence, the CS decomposition has an operational meaning for a
composite system, which is represented in Fig. 1.

$R^{\prime\dagger}$$R^{\dagger}$$\Theta^{\dagger}$$\Theta$$L^{\prime}$$L$
Figure 1: Operational meaning of the CS decomposition. The solid (dashed) line
represents the basis state $\ket{0}$ ($\ket{1}$) of the controller. Each box
on the solid (dashed) line depicts an operation acting on the system, given
that the controller is in the state $\ket{0}$ ($\ket{1}$). Initially the
system is subjected to unitary evolution $R^{\dagger}$ ($R^{\prime\dagger}$)
if the controller is in state $\ket{0}$ ($\ket{1}$). Thereafter the controller
undergoes a Hadamard transformation (thick horizontal line) followed by a
sequence of conditional unitaries acting on the system with a second Hadmard
transformation of the controller in between.

##  IV Optical implementation of coherent feedback

Based on a realisation method for optical qubit channels [13], we provide a
recipe for the design of the optical implementation of coherent feedback given
the corresponding Kraus operators from Sec. II. The construction of the latter
is discussed in [8, 11]. We note that the corresponding control channel can
also be realised by measurement, constituting measurement-based feedback
control [11]. Therefore what follows is also a recipe to translate
measurement-based feedback control into coherent feedback control.

The discussion is limited to the control of two-level systems (qubits) by
qubit controllers, corresponding to control channels with a pair of Kraus
operators. However, an analogous scheme can be readily constructed for
$d$-level systems controlled by $n$-level ancilla systems, i.e., control
channels involving $n$ Kraus operators [13].

A simple deterministic preparation channel for qubits that reaches target
state $\ket{T}$ in a single shot, is given by $K_{0}=\ket{T}\bra{T}$ and
$K_{1}=\ket{T}\bra{T^{\perp}}$ with $\langle T^{\perp}\ket{T}=0$. In general,
let $K_{0}$ and $K_{1}$ be Kraus operators which meet conditions (1) and (2),
from which we can construct a unitary of the form

$U=\begin{pmatrix}K_{0}&A\\\ K_{1}&B\end{pmatrix},$ (12)

where the matrices $A\equiv\braket{0}{U}{1}$ and $B\equiv\braket{1}{U}{1}$ are
appropriately chosen such that $U$ is unitary. In addition, Eq. (5) prescribes
$K_{0}\equiv\braket{0}{U}{0}$ and $K_{1}\equiv\braket{1}{U}{0}$. This is in
agreement with the form of $U$ in Eq. (12), which acts on the Hilbert space
$\mathcal{H}_{c}\otimes\mathcal{H}_{s}$, where $c$ and $s$ stands for
controller and system, respectively.

The CS decomposition of $U$ is given by Eq. (9), by comparing this to Eq. (12)
we can write the Kraus operators as

$\displaystyle K_{0}$ $\displaystyle=LCR^{\dagger}$ (13) $\displaystyle K_{1}$
$\displaystyle=iL^{\prime\prime}SR^{\dagger}\,,$

where $L^{\prime\prime}=iL^{\prime}$. This is simply a singular value
decomposition which exists for all Kraus operators $K_{i}$. Unitary
$R^{\prime}$ can be freely chosen and this can be exploited in order to
simplify the optical implementation of $U$. For this purpose, we choose
$R^{\prime}=-iR$. This converts the conditional unitary $R^{\dagger}\oplus-
iR^{\prime\dagger}$ to the local unitary $\mathds{1}\otimes R^{\dagger}$.

Our aim is now to design an optical set-up to implement the unitary operator

$U=\begin{pmatrix}L&0\\\
0&L^{\prime\prime}\end{pmatrix}H\otimes\mathds{1}\begin{pmatrix}\Theta&0\\\
0&\Theta^{\dagger}\end{pmatrix}H\otimes R^{\dagger}$ (14)

where $L$, $L^{\prime\prime}$, $R^{\dagger}$ and $\Theta=C+iS$ are determined
by the singular value decomposition of the Kraus operators $K_{0}$ and
$K_{1}$, given in Eq. (13).

Thus far the system and controller degree of freedom have not been specified.
Below we discuss two specific examples of controller and system, namely path
and polarisation, followed by the polarisation and orbital angular momentum.

###  IV.1 Path and polarisation

If we consider the paths of a photon as the controller and its polarisation as
the system degree of freedom, then the CS decomposition of $U$, Eq. (14), can
be implemented using balanced beam-splitters and local unitary operations on
the polarisation. The implementation of the latter require just one half-wave
plate and two quarter-wave plates mounted coaxially [14, 15]. The
decomposition of $U$ represents a generalized Mach-Zehnder interferometer and
can be made robust against noise associated with the vibration of optical
elements by employing a Sagnac interferometer, shown in Fig. 2.

Figure 2: Optical realisation of coherent feedback control for path and
polarisation. Light entering along a single path (purple) passes through a
sequence of quarter-wave plates (green) and half-wave plates (blue) that
implements $R^{\dagger}$ on the polarisation degree of freedom. A beams-
splitter ($BS$) separates the beam into two paths (blue and red). Mirrors are
labelled $M$. The light beam exits the interferometer along two paths
(purple).

In order to determine the configuration of half- and quarter-wave plates which
realises $L$, for example, we first expressed $L$ in terms of Euler angles
$(\xi,\eta,\zeta)$,

$\displaystyle L(\xi,\eta,\zeta)$
$\displaystyle=e^{-i\frac{1}{2}\xi\sigma_{y}}e^{i\frac{1}{2}\eta\sigma_{z}}e^{-i\frac{1}{2}\zeta\sigma_{y}}\,.$
(15)

The relationship between the Euler angles and the rotation angles of a half-
wave plate ($H_{\phi}$) and two quarter-wave plates ($Q_{\phi}$) is given by
[15]:

$\displaystyle L(\xi,\eta,\zeta)$
$\displaystyle=Q_{\frac{\pi}{4}+\frac{\xi}{2}}H_{-\frac{\pi}{4}+\frac{\xi+\eta-\zeta}{4}}Q_{\frac{\pi-\zeta}{4}}\,.$
(16)

It is possible to reorder the optical elements by exploiting the fact that

$\displaystyle
Q_{\phi}H_{\phi^{\prime}}=H_{\phi^{\prime}}Q_{2\phi^{\prime}-\phi}\,.$ (17)

The same decomposition procedure can be performed for $L^{\prime\prime}$,
$\Theta$, $R$ and $R^{\prime}$.

###  IV.2 Polarisation and orbital angular momentum

In this section we present a linear optical scheme to implement the coherent
feedback control of a two-dimensional subspace of the OAM degree of freedom of
light. Here the polarisation degree of freedom is considered as controller,
hence, the scheme can be made non-interferometric. In order to determine the
optical implementation of $U$ for polarisation and OAM we first express $U$ by
means of local unitaries and controlled unitaries of the form
$C_{A}:=\mathds{1}\oplus A$. Eq. (14) can be written in terms of controlled
unitary operations $C_{L^{\prime\prime}L^{\dagger}}$ and
$C_{(\Theta_{2}^{\dagger})^{2}}$ as

$\displaystyle U$ $\displaystyle=\begin{pmatrix}\mathds{1}&0\\\
0&L^{\prime\prime}L^{\dagger}\end{pmatrix}H\otimes
L\Theta\begin{pmatrix}\mathds{1}&0\\\
0&(\Theta^{\dagger})^{2}\end{pmatrix}H\otimes R^{\dagger}$
$\displaystyle=C_{L^{\prime\prime}L^{\dagger}}\left(H\otimes
L\Theta\right)C_{(\Theta^{\dagger})^{2}}\left(H\otimes R^{\dagger}\right)\,.$
(18)

In order to determine the optical implementation of the controlled unitary
$C_{L^{\prime\prime}L^{\dagger}}$, it is first diagonalised:

$\displaystyle C_{L^{\prime\prime}L^{\dagger}}=(\mathds{1}\otimes
W)(\mathds{1}\oplus\Phi)(\mathds{1}\otimes W^{\dagger})\,,$ (19)

where $W\Phi W^{\dagger}=L^{\prime\prime}L^{\dagger}$ and
$\Phi=\sum_{j}\exp(i\phi_{j})\ket{j}\bra{j}$, so that

$U=\left(\mathds{1}\otimes W\right)C_{\Phi}\left(H\otimes
W^{\dagger}L\Theta\right)C_{(\Theta^{\dagger})^{2}}\left(H\otimes
R^{\dagger}\right)\,.$ (20)

The controlled unitary operations $C_{\Phi}$ and $C_{(\Theta^{\dagger})^{2}}$
have polarisation as control bit, where $\ket{0}\equiv\ket{V}$ (vertically
polarised light) and $\ket{1}\equiv\ket{H}$ (horizontally polarised light),
and as target bit the OAM subspace spanned by
$\\{\ket{-\ell}\equiv\ket{0},\ket{+\ell}\equiv\ket{1}\\}$. Such controlled
unitaries can be implemented using a linear optical device named polarisation
selective Dove prism (PSDP) [16].

(a) Modified Dove prism. Three uniaxial crystals are sandwiched together to
form a cuboid. All three components are constructed from the same uniaxial
crystal, however their optic axes are aligned in different directions [16].
Vertically polarised light (  $\leftrightarrow$ ) passes through the device,
while horizontally polarised light ( $\bullet$ ) is reflected inside the
trapezoid (grey).

(b) Modified Dove prism between two quarter-wave plates forms the PSDP device.
The rotated PSDP which implements the controlled operation in Eq. (21)
consists of a modified Dove prism rotated by an angle $\alpha$, between two
quarter-wave plates that are each rotated by $\alpha/2$.

Figure 3: Diagram showing the components of the polarisation selective Dove
prism (PSDP).

A PSDP consists of a modified Dove Prism (DP) mounted between two half-wave
plates, cp. Fig. 3. A rotated PSDP realises the following controlled operation

$\displaystyle\mathcal{C}^{PSDP}_{l,\alpha}:=\begin{pmatrix}\mathds{1}&0\\\
0&\sigma_{x}P_{2\ell\alpha}^{\dagger}\end{pmatrix}=\begin{pmatrix}\mathds{1}&0\\\
0&P_{2\ell\alpha}\sigma_{x}\end{pmatrix}\,,$ (21)

where $\alpha$ is the angle of rotation and $P_{2\ell\alpha}^{\dagger}$ is the
previously defined phase shift operator. The rotated PSDP device is configured
by rotating the modified DP by an angle $\alpha$ and the half-wave plates by
$\alpha/2$.

It is therefore possible to implement the controlled unitaries $C_{\Phi}$ and
$C_{(\Theta^{\dagger})^{2}}$ using a local phase shift on polarisation and two
PSDPs. This is further outlined in App. B. If $L^{\prime}L^{\dagger}$ can be
written in the form $\exp(i2\phi)\sigma_{x}P^{\dagger}_{\theta}$, then a phase
shifter implementing $P^{\dagger}_{\phi}$ on polarisation and a single rotated
PSDP can be used to implement $C_{L^{\prime}L^{\dagger}}$ in Eq. (18), without
the need to diagonalise it.

The Hadamard operation on polarisation can be performed using a half-wave
plate rotated by an angle $\pi/8$. The final requirement for the optical
realisation of $U$ is local unitary operations on the OAM degree of freedom.
It is not currently known how to implement arbitrary unitary operations on
OAM. However, cylindrical-lens mode converters [17] can be used to perform
unitary operations on the $\\{\ket{\ell=-1},\ket{\ell=+1}\\}$ subspace of OAM.
The so called $\pi$\- and $\pi/2$-converters are analogous to the half-wave
and quarter-wave plates, respectively [18] and can therefore be used to
construct arbitrary unitary operations on this subspace of OAM. In addition, a
rotated Dove prism can be used to implement the $P_{2\ell\alpha}\sigma_{x}$
operation on the two-level OAM subspace associated with $\ell$, where $\alpha$
is the angle of rotation of the Dove prism.

##  V Iterative application of control channels

In Sec. IV we showed how to construct coherent feedback control schemes from
control channels which satisfy the conditions for convergence to a target
state given in Eqs. (1) and (2). Such schemes may require repetition for
various reasons. Some control schemes work weakly, meaning convergence to the
target state occurs over multiple iterations. Examples of weak schemes are
given in Sec. VI.2 and Sec. VI.3. In addition, repeated coherent feedback
control protects the system against noise [8]. Here we discuss two methods to
achieve repeated application of coherent feedback control.

If the system starts in the pure state $\ket{\psi_{0}}$, the initial state of
system and controller is given by $\ket{\Psi_{0}}:=\ket{0}\ket{\psi_{0}}$. The
output state $\ket{\Psi_{1}}:=U\ket{\Psi_{0}}$ of a coherent feedback scheme
then reads

$\displaystyle\ket{\Psi_{1}}$
$\displaystyle=\sum_{i}\ket{i}\bra{i}U\ket{0}\ket{\psi_{0}}$ (22)
$\displaystyle=\ket{0}K_{0}\ket{\psi_{0}}+\ket{1}K_{1}\ket{\psi_{0}}\,,$

where the Kraus operators $K_{0}$ and $K_{1}$ obey the conditions (1) and (2)
to drive the system into a target state. In order to iterate the control
channel, the controller must be reset to its initial state $\ket{0}$.

This could be accomplished by submitting each of the two components in Eq.
(22) into an additional control device adjusting the state of the controller
where necessary. However, this would require $2^{N}-1$ devices for $N$
iterations. We discuss in the following two ways to circumvent this
exponential growth of machinery. The first method involves the resetting of
the controller by filtering. This results in losses that we propose to
compensate by parametric amplification. The second method involves the
transfer of the controller state to an additional degree of freedom.

###  V.1 Resetting of controller by filtering

One way to reset the controller is to project it onto the state
$\ket{+}\equiv\tfrac{1}{\sqrt{2}}(\ket{0}+\ket{1})$ to create a superposition
of the Kraus operators:

$\ket{\Psi_{1}}\xrightarrow{\ket{+}\bra{+}}\ket{\Psi_{1}^{\prime}}=\frac{1}{\sqrt{2}}\ket{+}(K_{0}\ket{\psi}+K_{1}\ket{\psi}).$
(23)

A subsequent rotation restores the initial controller state $\ket{0}$,

$\ket{\Psi_{1}^{\prime}}\xrightarrow{}\ket{\Psi_{1}^{\prime\prime}}=\frac{1}{\sqrt{2}}\ket{0}(K_{0}\ket{\psi}+K_{1}\ket{\psi}).$
(24)

After the projection (23) the information about the controller states is
deleted, and the state change no longer corresponds to a trace-preserving
channel that satisfies the conditions for convergence given in Eqs. (1) and
(2). However, we show that for the specific examples of coherent feedback
schemes discussed in Sec. VI, the system will still converge to the target
state (see App. D). The state of the composite system after the second round
of coherent control is given by
$\ket{\Psi_{2}}:=U\ket{\Psi_{1}^{\prime\prime}}$. The projection (23) will, in
general, lead to photon losses. In order to compensate for these we consider
parametric amplification.

The preparation of light modes or polarisation by coherent feedback control
can be applied to coherent (classical) states of light. In this case,
amplification is an adequate means to compensate for losses. Parametric
amplification is available for both spatial light modes and polarisation as
system degree of freedom. This technique uses a non-linear crystal and a pump
beam to amplify an input signal beam. The intensity gain factor of the
parametric amplifier can be appropriately adjusted in order to restore the
original input beam intensity.

The parametric intensity gain is given by [19],

$G(L)=1+\left(\frac{\Gamma}{g}\sinh(gL)\right)^{2}.$ (25)

Here $L$ is the length of the crystal (interaction length), and $g$ and
$\Gamma$ are generalised wave numbers that depend on the parameters of the
non-linear process, i.e., the pump beam intensity, the phase matching, the
angular frequencies, the wave numbers in the medium as well as the non-linear
susceptibility (see App. E).

Resetting by means of filtering is available for different controller degrees
of freedom. For the path degree of freedom, the light from the two output
ports of the interferometer shown in Fig. 2 can be combined according to Eq.
(23) using a balanced beam-splitter. However, one path will still have to be
discarded and this intensity loss could be compensated by amplification. For
polarisation, the intensity losses are due to the use of a linear polariser
(filter) which performs the projection onto $(\ket{V}+\ket{H})/\sqrt{2}$ (Eq.
(23)). A polarisation rotator can then be employed to reset the initial
polarisation $\ket{V}$.

We are not limited to the use of classical light, as amplification still works
using low numbers of photons, but with finite success probabilities. In this
regime, the output photons would only reach a limited fidelity, in agreement
with the no-cloning theorem. It has been shown in [20] that stimulated
emission, and thus parametric amplification, is capable of producing quantum
clones with near optimal fidelity. For an optimal universal $N\rightarrow M$
cloner, the optimal fidelity is given by [21],

$F=\frac{NM+N+M}{M(N+2)}.$ (26)

For the case of $N,M\rightarrow\infty$ in Eq. (26), the classical fidelity of
$1$ is recovered.

Possibly, the noise present in the amplification process on the single photon
level is reduced with each repetition of the control channel. This may lead to
a high fidelity asymptotically, but with finite success probability. However,
we do not know whether the coherent feedback method introduced here works for
non-classical states of light with more than one photon. This is subject of
further investigation.

###  V.2 Storage in an additional degree of freedom

Here we make use of an additional degree of freedom in order to reset the
controller. Taking into account an additional ancilla “$a$” in initial state
$\ket{0}_{a}$, the composite state
$\ket{\Psi_{1}}:=(U\ket{0}_{c}\ket{\psi_{0}}_{s})\ket{0}_{a}$ after the system
interacted with the controller reads

$\ket{\Psi_{1}}=(\ket{0}K_{0}\ket{\psi_{0}}+\ket{1}K_{1}\ket{\psi_{0}})\ket{0}_{a}\,$
(27)

where $\ket{\psi_{0}}$ is the initial state of the system. In a first step, we
mark the controller basis states using basis states of the ancilla $a$, e.g.,
by a C-NOT operation acting on “$a$” conditioned on the controller,

$\ket{\Psi_{1}}\rightarrow\ket{\Psi_{1}^{\prime}}=\ket{0}K_{0}\ket{\psi}\ket{0}_{a}+\ket{1}K_{1}\ket{\psi}\ket{1}_{a}.$
(28)

In this case, the initial controller state can be restored by a C-NOT
conditioned on the state of subsystem “$a$”:

$\ket{\Psi_{1}^{\prime}}\rightarrow\ket{\Psi_{1}^{\prime\prime}}=\ket{0}\left(K_{0}\ket{\psi}\ket{0}_{a}+K_{1}\ket{\psi}\ket{1}_{a}\right).$
(29)

In order to restore the initial controller state $\ket{0}$ unitarily, the
information about the unknown state of the controller after the application of
the control channel must be stored in the ancillary degree of freedom. When
this resetting process is extended to $N$ iterations of the control channel,
the dimension of the ancilla is given by $2^{N}$. Below we discuss this
mechanism at the example of the time-bins and OAM as ancillas.

####  V.2.1 Time-bins as additional degree of freedom

We consider coherent feedback control with polarisation and OAM of a light
pulse as the controller and the system, respectively. The ancilla (time-bins)
is in initial state $\ket{t=0}$, where $t$ represents the time delay between
light pulses. Distinct time-bins can be created by varying the path lengths
that the two different polarisation components experience by means of a delay
loop. This marks each polarisation state with a time-bin state, analogous to
Eq. (28). The initial polarisation can then be restored in each pulse before
it re-enters the coherent feedback scheme for the next iteration (Eq. (29)).
Given that the output state of the pulse after the first application of the
coherent feedback is
$\ket{\Psi_{1}}:=(U\ket{V}_{c}\ket{\psi_{0}}_{s})\ket{0}_{a}$, then the total
transformation described can be written as

$\displaystyle\ket{\Psi_{1}}=\ket{V}K_{0}\ket{\psi_{0}}\ket{0}_{a}+\ket{H}K_{1}\ket{\psi_{0}}\ket{0}_{a}$
(30)
$\displaystyle\longrightarrow\ket{\Psi_{1}^{\prime\prime}}=\ket{V}(K_{0}\ket{\psi_{0}}\ket{0}_{a}+K_{1}\ket{\psi_{0}}\ket{\tau}_{a})\,,$

where $\tau$ is a delay period. A proposed method to iteratively implement Eq.
(30) is shown in Fig. 4.

EOM1
---
EOM2
---
PBS
---
coherent
---
feedback
delay loop
---
Figure 4: Setup to create disjoint time-bins depending on the order of Kraus
operators in $N$ iterations of the channel which is realised by coupling
polarisation (controller) to OAM (system). The setup uses electro-optic
modulators (EOM) that change the polarisation of a pulse when switched on, and
a polarising beam-splitter (PBS) that reflects (transmits) vertically
(horizontally) polarised light [22]. EOM1 keeps the horizontally polarised
component in the delay loop for the required number of rounds in each
iteration. EOM2 ensures that the polarisation of each time-bin is reset to the
initial state before re-entering the channel.

The delay time $\tau$ must change in each iteration, in such a way that the
pulses do not overlap. This allows for the selective change of the
polarisation in order to restore the initial controller state for each time-
bin. One way to create distinct pulses is to double the number of round trips
in the delay loop from one iteration to the next. This is demonstrated in App.
F.

During the second iteration the two distinct pulses in
$\ket{\Psi_{1}^{\prime\prime}}$ are taken (at different times) as inputs in
the coherent feedback scheme so that the state
$\ket{\Psi_{2}}:=U\ket{\Psi_{1}^{\prime\prime}}$ is given by

$\displaystyle\ket{\Psi_{2}}=$
$\displaystyle\left(\ket{V}K_{0}^{2}\ket{\psi}+\ket{H}K_{1}K_{0}\ket{\psi}\right)\ket{0}_{a}$
(31)
$\displaystyle+\left(\ket{V}K_{0}K_{1}\ket{\psi}+\ket{H}K_{1}^{2}\ket{\psi}\right)\ket{\tau}_{a}.$

The polarisation states are marked by the conditional operation on the
controller and ancilla “$a$”,

$\displaystyle\ket{V}\bra{V}\otimes\mathds{1}+\ket{H}\bra{H}\otimes\sum_{n=0}^{2^{N-1}-1}\ket{n\tau+2^{N-1}\tau}\bra{n\tau}\,,$
(32)

where $N$ is the number of applications of the coherent feedback control which
have already occurred, and therefore corresponds to the index of the composite
state. Here $N=2$ so that

$\displaystyle\ket{\Psi_{2}^{\prime}}=$
$\displaystyle\ket{V}K_{0}^{2}\ket{\psi}\ket{0}_{a}+\ket{H}K_{1}K_{0}\ket{\psi}\ket{2\tau}_{a}$
(33)
$\displaystyle+\ket{V}K_{0}K_{1}\ket{\psi}\ket{\tau}_{a}+\ket{H}K_{1}^{2}\ket{\psi}\ket{3\tau}_{a}.$

It is clear that the leading pulses in the first half of the pulse train are
in the desired polarisation state. However, the second half of all pulses in
the pulse train require resetting. This can be achieved by the conditional
operation on the controller and ancilla “$a$”,

$\displaystyle\mathds{1}\otimes\sum_{n=0}^{2^{N-1}-1}\ket{n\tau}\bra{n\tau}+\sigma_{x}\otimes\sum_{n=2^{N-1}}^{2^{N}-1}\ket{n\tau}\bra{n\tau}\,,$
(34)

so that

$\displaystyle\ket{\Psi_{2}^{\prime\prime}}=\ket{V}$
$\displaystyle\left(K_{0}^{2}\ket{\psi}\ket{0}_{a}+K_{1}K_{0}\ket{\psi}\ket{\tau}_{a}\right.$
(35)
$\displaystyle\quad\left.+K_{0}K_{1}\ket{\psi}\ket{2\tau}_{a}+K_{1}^{2}\ket{\psi}\ket{3\tau}_{a}\right).$

After a sufficiently high number $N$ of applications of the control channel,
the system converges to the target state $\ket{T}$ [8]. Therefore, after $N$
iterations of coherent control and controller reset the composite system is in
a product state of the form

$\ket{\Psi_{N}^{\prime\prime}}=\ket{V}\ket{T}\left(\sum_{m=0}^{2^{N}-1}\alpha_{m}\ket{m\tau}_{a}\right).$
(36)

The amplitudes of the $2^{N}$ pulses are given by
$\alpha_{m}=\bra{T}\mathcal{K}_{m}\ket{\psi}$ as follows from writing the
final state in terms of the initial state,
$\ket{\Psi_{N}^{\prime\prime}}=\ket{V}\left(\sum_{m}\mathcal{K}_{m}\ket{\psi_{0}}\ket{m\tau}_{a}\right)$.
Here $\mathcal{K}_{m}$ refers to the $m$th permutation of the product
$\Pi_{j=1}^{N}K_{i}^{(j)}$ of the two Kraus operators, with $i=0,1$.

Although the system degree of freedom of each pulse is in the target state,
the pulses have low amplitudes. In measurements on the system, collecting the
accumulated signal over a large time period compensates the low amplitudes.

####  V.2.2 OAM as additional degree of freedom

We consider coherent feedback control with path as the controller,
polarisation as the system, and employ OAM as an additional ancillary degree
of freedom. The input state has an even OAM mode so that the composite system
is $\ket{0}_{c}\ket{\psi_{0}}_{s}\ket{\ell=0}_{a}$. Here $\ket{0}_{c}$
corresponds to the initial state of the path degree of freedom, which is the
upper path, and $\ket{\psi_{0}}_{s}$ is the initial state of the system.

After the coherent feedback, one unit of OAM is added to the light in the
lower path (with $\ket{1}_{c}$) by use of a spiral phase plate. This is done
in order to mark the controller, analogous to Eq. (28). The controller can
then be reset using an inverse even-odd OAM mode sorter, so that the total
transformation is given by

$\displaystyle\ket{\Psi_{1}}=(\ket{0}K_{0}\ket{\psi_{0}}+\ket{1}K_{1}\ket{\psi_{0}})\ket{\ell=0}_{a}$
(37)
$\displaystyle\longrightarrow\ket{\Psi_{1}^{\prime\prime}}=\ket{0}(K_{0}\ket{\psi_{0}}\ket{\ell=0}_{a}+K_{1}\ket{\psi_{0}}\ket{\ell=1}_{a}).$

For the next iteration of the coherent feedback control, the OAM values must
be doubled as to only obtain even OAM values. To accomplish this we must have
a method of multiplying the OAM values of an input beam. In [23],
multiplication is achieved by using superpositions of circular-sector
transformations of the input beam. This method works best for low OAM values,
as the conversion efficiencies decrease for increasing OAM values e.g., when
doubling the values $\ell=+1$, $\ell=+2$ and $\ell=+3$, conversion
efficiencies are $0,97$, $0,93$ and $0,86$, respectively [23].

coherent
---
feedback
S
---
M
---
D
---
Figure 5: Setup to reset controller degree of freedom for coherent feedback
with polarisation (system) and path (controller) using OAM as additional
controller. The marking of the controller states is achieved by S, a spiral
phase plate, on the lower path. The initial controller state is restored
through the use of M, the inverse OAM mode sorter for even and odd OAM values.
Before the next application of the coherent feedback, the OAM values of the
light must be doubled with the OAM doubler D, so only even OAM modes are
present.

The setup to iteratively implement the resetting transformation (Eq. (37)) is
shown in Fig. 5. A description of the $N$th iteration of the transformation is
now briefly considered. The conditional operation on the controller and
ancilla “$a$” that marks the path states with different OAM values reads

$\sum_{\ell=0}^{2^{N}-2}\left(\ket{0}\bra{0}\otimes\ket{\ell}\bra{\ell}+\ket{1}\bra{1}\otimes\ket{\ell+1}\bra{\ell}\right).$
(38)

Since the ingoing OAM modes are all even, the upper path will contain only
even OAM modes, while the lower path contains only odd OAM modes. The initial
path state is restored by

$\sum_{l=0}^{2^{N-1}-1}\left(\mathds{1}\otimes\ket{2\ell}\bra{2\ell}+\sigma_{x}\otimes\ket{2\ell+1}\bra{2\ell+1}\right),$
(39)

which can be implemented by a mode sorter for even and odd OAM values run in
reverse. Before entering the coherent feedback device the OAM value of the
light is doubled.

The system converges to the target state $\ket{T}$ for a sufficiently high
number $N$ of iterations as described in the previous subsection. The
composite system, after $N$ iterations of coherent feedback and controller
reset but before the final doubling of OAM values, is given by

$\ket{\Psi_{N}^{\prime\prime}}=\ket{0}\ket{T}\left(\sum_{m=0}^{2^{N}-1}\alpha_{m}\ket{\ell=m}_{a}\right).$
(40)

Each OAM mode has an amplitude $\alpha_{m}=\bra{T}\mathcal{K}_{m}\ket{\psi}$.
After convergence to the target state, the output yields a scalar beam with
the desired polarisation. In this case, it is easy to discard the OAM degree
of freedom, as we are only interested in the polarisation.

##  VI Examples of coherent feedback schemes

In this section we discuss three coherent feedback schemes. The first is a
basic control mechanism which allows a target state to be reached in a single
iteration. The second is based on the swap operation and allows any target
state to be reached provided it is encoded in the controller degree of
freedom. The third scheme allows for the preparation of an equal superposition
of $+\ell$ and $-\ell$ OAM states.

###  VI.1 Basic control scheme

Let us consider the simple control channel mentioned in Sec. IV, with the
Kraus operators $K_{0}=\ket{T}\bra{T}$ and $K_{1}=\ket{T}\bra{T^{\perp}}$.
This scheme allows the target state to be reached in a single iteration. The
singular value decomposition of the Kraus operators reads,
$K_{0}=V\ket{0}\bra{0}V^{\dagger}$ and
$K_{1}=iV\sigma_{y}\ket{1}\bra{1}V^{\dagger}$, where
$V=\ket{T}\bra{0}+\ket{T^{\perp}}\bra{1}$. Hence, according to Eq. (14), the
CS-decomposition of the unitary that implements the control channel determined
by $K_{0}$ and $K_{1}$ is given by

$\displaystyle U$ $\displaystyle=\mathds{1}\otimes
V\begin{pmatrix}\mathds{1}&0\\\
0&\sigma_{y}\end{pmatrix}H\otimes\mathds{1}\begin{pmatrix}e^{i\frac{\pi}{4}}P_{\frac{\pi}{4}}^{\dagger}&0\\\
0&e^{-i\frac{\pi}{4}}P_{\frac{\pi}{4}}\end{pmatrix}H\otimes V^{\dagger}\,.$
(41)

If polarisation is the system degree of freedom and the path is the
controller, then the above unitary can be implemented using balanced beam-
splitters as well as half- and quarter-wave plates as described in Sec. IV.1.
If the system degree of freedom is OAM and the controller is polarisation,
then the scheme could be realised using local operations in polarisation and
PSPDs as described in Sec. IV.2. In the simplest case the target state is
given by $\ket{0}\equiv\ket{-\ell}$, so that $V=\mathds{1}$ and (up to a
global phase factor)

$\displaystyle U$ $\displaystyle=\begin{pmatrix}\mathds{1}&0\\\
0&\sigma_{x}P^{\dagger}_{\frac{\pi}{2}}\end{pmatrix}P^{\dagger}_{\frac{\pi}{4}}HP_{\frac{\pi}{4}}\otimes\mathds{1}\begin{pmatrix}\mathds{1}&0\\\
0&P_{\frac{\pi}{2}}\end{pmatrix}H\otimes P_{\frac{\pi}{4}}^{\dagger}\,.$ (42)

The local operations $P^{\dagger}_{\frac{\pi}{4}}HP_{\frac{\pi}{4}}$ and $H$
on polarisation can be decomposed into half- and quarter-wave plates as
$Q_{\frac{\pi}{2}}H_{\frac{\pi}{8}}Q_{0}$ and $H_{\frac{\pi}{8}}$,
respectively. The operations which act on OAM are $\ell$-value dependent. The
first controlled unitary in Eq. (42) is realised by a PSDP rotated by
$\frac{\pi}{4\ell}$, according to Eq. (21). The second controlled unitary
requires two PSDPs mounted coaxially. Finally, the local phase operation
$P_{\frac{\pi}{4}}^{\dagger}$ on OAM can be realised by a two coaxially
mounted Dove prisms, one rotated by $\frac{\pi}{8\ell}$.

This example is in essence a mechanism to prepare the system in the state
$\ket{0}$. The local unitary $V^{\dagger}$ can be absorbed into the initial
(possibly unknown) system state, while $V$ simply rotates $\ket{0}$ to the
target state. This local operation may not be readily available for OAM, as
discussed earlier, which limits the applications of this control scheme. In
the following examples we explore optical set-ups which allow for more general
target states to be reached without the use of inaccessible local OAM
operations.

(a) Implementation of the weak swap, where
$\gamma=\frac{2\alpha+\lambda}{4}-\frac{\pi}{8}$.

(b) Implementation of the target state dependent control scheme. This optical
set-up allows the state $(\ket{-\ell}+\ket{+\ell})/\sqrt{2}$ to be prepared if
the polarisation state (controller) is in an equal superposition of vertical
and horizontal polarisation.

Figure 6: Optical implementation of two coherent feedback control schemes on
polarisation (controller) and the subspace of OAM spanned by
$\\{\ket{-\ell},\ket{+\ell}\\}$ (system). Half-wave plates are shown in green,
quarter-wave plates in blue and modified Dove prisms are represented by a
cuboid. The angle $\alpha=\frac{\lambda}{2\ell}$ depends on the parameter
$\lambda$ and on the OAM subspace considered.

###  VI.2 Optical implementation of the weak swap

In [8] it is shown that the so called weak swap unitary

$\displaystyle U=\exp\left(-i\lambda S\right)\,,$ (43)

where $\lambda\in\mathbb{R}$ and $S$ is the swap operator leads to convergence
to any target state $\ket{T}$ encoded in the controller system since the Kraus
operators $\\{K_{i}=\braket{i}{U}{T}_{c}\\}_{i}$ satisfy condition (1) and
(2). By adding a unitary transformation of the controller state,
$U_{c}\ket{0}_{c}=\ket{T}_{c}$, to the weak swap $U\rightarrow
UU_{c}\otimes\mathds{1}$, this case can be reduced to coherent control with
initial controller state $\ket{0}_{c}$. It might be worth noting, that a
scheme where the state of the controller determines the target, allows the
preparation of an unknown target state that might be the result a quantum
computation. The CS decomposition of the weak swap on
$\mathcal{H}_{c}\otimes\mathcal{H}_{s}=\mathcal{H}_{2}\otimes\mathcal{H}_{2}$
(up to a phase factor) is given by

$\displaystyle U$ $\displaystyle=\begin{pmatrix}\mathds{1}&0\\\
0&\sigma_{x}\end{pmatrix}(H\otimes\mathds{1})\begin{pmatrix}\mathds{1}&0\\\
0&P_{\lambda}^{\dagger}\end{pmatrix}(P^{\dagger}_{\frac{\lambda}{2}}H\otimes\mathds{1})\begin{pmatrix}\mathds{1}&0\\\
0&\sigma_{x}\end{pmatrix}\,,$ (44)

where $L=P^{\dagger}_{\frac{\lambda}{2}}$,
$L^{\prime\prime}=\sigma_{x}P^{\dagger}_{\frac{\lambda}{2}}$,
$\Theta=e^{-i\frac{\lambda}{2}}P_{\frac{\lambda}{2}}$, $R=\mathds{1}$ and
$R^{\prime}=\sigma_{x}$. The implementation of the weak swap for path
(controller) and polarisation (system) can be achieved using beam-splitters,
half- and quarter-wave plates, as discussed in Sec. IV.1.

The optical implementation of the weak swap for polarisation (controller) and
a two-dimensional subspace of OAM spanned by $\\{\ket{-\ell},\ket{+\ell}\\}$
(system) is depicted in Fig. 6(a). An unrotated PSDP can be used to implement
a C-NOT, $\mathds{1}\oplus\sigma_{x}$, independent of the choice of $|\ell|$
which determines the OAM subspace. It is clear from Eq. (21) that the
controlled unitary $\mathds{1}\oplus P_{\lambda}^{\dagger}$ can be implemented
using two PSDP’s. One of these PSDP’s should be rotated by
$\alpha=\tfrac{\lambda}{2l}$ and mounted coaxially with the second unrotated
PSDP. In order to reduce the number of half- and quarter-wave plates required,
we combine the action of one of the rotated half-wave plates from the rotated
PSDP with the action of $P^{\dagger}_{\frac{\lambda}{2}}H$ (App. C).

By selecting an appropriate angle of rotation $\alpha$, namely
$\alpha=\frac{\pi}{4\ell}$, the optical scheme shown in Fig. 6(a) allows us to
perform swap operation $S$ between polarisation and a particular OAM subspace,
in one iteration. We note that the swap will also be performed for the two-
level OAM subspace associated with $\ell^{\prime}=(4n+1)\ell$,
$n\in\mathbb{Z}$, since $2\alpha l^{\prime}=\frac{\pi}{2}+2\pi n$.

###  VI.3 Target state dependent control mechanism

In [8] it is shown that the unitary

$\displaystyle
U(\lambda)=\exp\left(-i\frac{\lambda}{2}(\sigma_{y}\otimes\sigma_{y}+\sigma_{z}\otimes\sigma_{z})\right)\,,$
(45)

with coupling parameter $\lambda\in\mathbb{R}$ leads to convergence to the
target state $\ket{T}=(\ket{0}+\ket{1})/\sqrt{2}$ encoded in the controller
system, since the Kraus operators $\\{K_{i}=\braket{i}{U}{T}\\}_{i}$ satisfy
condition (1) and (2). A different target state can be reached by choosing
different combinations of Pauli operators as generators.

The following decomposition represents the optical implementation of this
unitary (45):

$\displaystyle U(\lambda)$ $\displaystyle=\begin{pmatrix}\mathds{1}&0\\\
0&\sigma_{x}\end{pmatrix}(H\otimes\mathds{1})\begin{pmatrix}\mathds{1}&0\\\
0&P_{\lambda}^{\dagger}\end{pmatrix}(H\otimes\mathds{1})\begin{pmatrix}\mathds{1}&0\\\
0&\sigma_{x}\end{pmatrix}\,.$ (46)

The optical implementation for polarisation (controller) and a two-dimensional
subspace of OAM spanned by $\\{\ket{-\ell},\ket{\ell}\\}$ (system) is shown in
Fig. 6(b).

The unitary in Eq. (46) is almost identical to the weak swap discussed in the
previous example, Eq. (44). However, as a consequence of the absence of phase
shift $P_{\frac{\lambda}{2}}^{\dagger}$ acting on polarisation, a fixed
optical implementation achieves $U(\lambda)$ of varying $\lambda$ values
depending on the OAM subspace considered. If we fix the angle $\alpha$ then
the apparatus implements the unitary $U(2\alpha\ell)$ on the subspace spanned
by $\\{\ket{-\ell},\ket{\ell}\\}$.

Consequently, the target state can be reached in all subspaces with close to
unit fidelity, if a sufficient number of iterations are performed. This fails
if $\alpha\ell$ is equal to an integer multiple of $\pi$.

For a chosen optical set-up, i.e. for fixed $\alpha$, the fidelity,
$F^{\ell}_{n}$, for a particular $\ell$ subspace after $n$ iterations is given
by the overlap between the final state of the system and the target state. The
fidelity increases in an exponential fashion [8]

$\displaystyle F^{\ell}_{n}=1-(1-F^{\ell}_{0})(1-\sin^{2}(\alpha\ell))^{n}\,,$
(47)

where $F^{\ell}_{0}$ is the initial overlap between the system and target
state. From this we can also determine the number of iterations necessary to
reach a certain fidelity $F$

$\displaystyle
n=\frac{1}{1-\sin^{2}(\alpha\ell)}\ln\left(\frac{1-F}{1-F_{0}^{\ell}}\right)\,.$
(48)

##  VII Discussion

In conclusion, we have presented a class of schemes to prepare the OAM and
polarisation qubits using coherent feedback control. Our results are valid for
single photons as well as classical beams of light and require mostly linear
optical setups. The biggest obstacle in realizing the coherent feedback
control in optical systems is to perform the non-local unitaries jointly on
the system and controller. This was accomplished by using the CS-
decomposition, which reduces the joint unitaries into simple unitaries acting
on individual degrees of freedom of light. Our coherent feedback control
methods allow to prepare arbitrary superpositions of two OAM modes, even
without spatial interferometers, and in principle without photon losses. This
is an important step forward compared to other preparation methods, such as
using spatial light modulators. A generalisation to the preparation of
arbitrary structured light modes seems possible.

While for massive systems, measurement-based feedback can be used to prepare
target states without losses, non-destructive and efficient measurements are
not easily available for photons. Our results show how to translate
measurement based feedback into coherent feedback coupling various degrees of
freedom of light. This recipe might also be employed for composite massive
systems.

Most of the schemes discussed here enable the preparation of a desired state
in a single shot. However, coherent feedback control requires in general
iterative interaction of the system and controller. This is for example the
case to protect a system against noise or for steering the system into target
dynamics. For this purpose, the controller needs to be reset to its initial
state, or we need a fresh controller after each iteration. While readily
available for systems with strong coupling, for optical systems the situation
is more severe.

In optical systems, resetting the controller or using fresh controller leads
to exponential increase in the resources. To overcome this problem we have
suggested two methods, one involving coherent amplification of light and the
other using an additional degree of freedom. Both the methods have their own
limitations which result in low fidelities and inefficient implementation of
coherent feedback control. However, using these techniques the resources
required are linear with the number of interactions.

The coherent feedback control methods discussed here can in principle
compensate weak control by repeated application of the control channel, for
example the weak swap or the state dependent control scheme. This compensation
mechanism is important for photons that weakly couple to their environment and
might lead to further applications in quantum communication tasks.

##  VIII Acknowledgements

SKG acknowledges the financial support from SERB-DST (File No.
ECR/2017/002404). AR and TP acknowledges the financial support of the National
Research Foundation of South Africa. We thank G.P. Teja, Akshay Menon and
Benjamin Perez-Garcia for useful discussions.

##  Appendix A Decomposition of the Sine-Cosine matrix

In the section we provide details of the derivation of Eq. (10).

$\displaystyle\mathcal{S}$ $\displaystyle=\begin{pmatrix}C&S\\\
-S&C\end{pmatrix}$ (49) $\displaystyle=\mathds{1}\otimes C+i\sigma_{y}\otimes
S$ (50)
$\displaystyle=\mathds{1}\otimes\left(\frac{\Theta+\Theta^{\dagger}}{2}\right)+\sigma_{y}\otimes\left(\frac{\Theta-\Theta^{\dagger}}{2}\right)$
(51)
$\displaystyle=\left(\frac{\mathds{1}+\sigma_{y}}{2}\right)\otimes\Theta+\left(\frac{\mathds{1}-\sigma_{y}}{2}\right)\otimes\Theta^{\dagger}$
(52)

It is straightforward to show that

$\displaystyle\frac{\mathds{1}+\sigma_{y}}{2}$
$\displaystyle=P_{\tfrac{\pi}{4}}^{\dagger}H\ket{0}\bra{0}HP_{\tfrac{\pi}{4}}$
(53)

and

$\displaystyle\frac{\mathds{1}-\sigma_{y}}{2}$
$\displaystyle=P_{\tfrac{\pi}{4}}^{\dagger}H\ket{1}\bra{1}HP_{\tfrac{\pi}{4}}\,,$
(54)

so that

$\displaystyle\mathcal{S}$
$\displaystyle=\left(P_{\tfrac{\pi}{4}}^{\dagger}H\otimes\mathds{1}\right)\begin{pmatrix}\Theta&0\\\
0&\Theta^{\dagger}\end{pmatrix}\left(HP_{\tfrac{\pi}{4}}\otimes\mathds{1}\right)\,.$
(55)

##  Appendix B Decomposition of controlled unitary operations into PSDPs

We wish to implement an arbitrary controlled unitary of the form

$\displaystyle C_{\Theta}=\mathds{1}\oplus\begin{pmatrix}e^{i\theta_{1}}&0\\\
0&e^{i\theta_{2}}\\\ \end{pmatrix}\,.$ (56)

An appropriate phase shift on the first subsystem symmetrise the controlled
operation as follows

$\displaystyle C_{\Theta}$
$\displaystyle=e^{i\theta^{\prime}_{1}}\left(\mathds{1}\oplus
P_{\theta^{\prime}_{2}}\right)\left(P_{\theta^{\prime}_{1}}^{\dagger}\otimes\mathds{1}\right)\,,$
(57)

where $\theta^{\prime}_{1}=\frac{\theta_{1}+\theta_{2}}{4}$ and
$\theta^{\prime}_{2}=\frac{\theta_{1}-\theta_{2}}{2}$. Discarding the global
phase factor we write the controlled unitary $C_{\Theta}$ as the product

$\displaystyle C_{\Theta}$
$\displaystyle=\left(\mathds{1}\oplus\sigma_{x}\right)\left(\mathds{1}\oplus\sigma_{x}P_{\theta^{\prime}_{2}}\right)\left(P_{\theta^{\prime}_{1}}^{\dagger}\otimes\mathds{1}\right)\,.$
(58)

The C-NOT operation $\mathds{1}\oplus\sigma_{x}$ can be implemented using a
single PSDP (not rotated) while the the second controlled operation
$\mathds{1}\oplus\sigma_{x}P_{\theta^{\prime}_{2}}$ can be implemented using a
PSDP rotated by an angle $\theta^{\prime}_{2}$ (Eq. (21)).

##  Appendix C Wave plate configuration

In this section we give the explicit calculation which provides the
decomposition of $H_{\frac{\alpha}{2}}P_{\frac{\lambda}{2}}H$ into quarter-
and half-wave plates.

$\displaystyle P_{\frac{\lambda}{2}}$
$\displaystyle=e^{i\frac{\lambda}{2}\sigma_{z}}$ (59) $\displaystyle H$
$\displaystyle=e^{-i\frac{\pi}{4}\sigma_{y}}\sigma_{z}=\sigma_{z}e^{i\frac{\pi}{4}\sigma_{y}}$
(60) $\displaystyle H_{\frac{\alpha}{2}}$
$\displaystyle=e^{-i\alpha\sigma_{y}}\sigma_{z}$ (61)
$\displaystyle\Rightarrow H_{\frac{\alpha}{2}}P_{\frac{\lambda}{2}}H$
$\displaystyle=e^{-i\alpha\sigma_{y}}e^{i\frac{\lambda}{2}\sigma_{z}}e^{i\frac{\pi}{4}\sigma_{y}}$
(62)

From Eq. (16) we have:

$\displaystyle H_{\frac{\alpha}{2}}P_{\frac{\lambda}{2}}H$
$\displaystyle=Q_{\alpha+\frac{\pi}{4}}H_{\frac{2\alpha+\lambda}{4}-\frac{\pi}{8}}Q_{\frac{3\pi}{8}}\,.$
(63)

##  Appendix D Examples of iterative coherent feedback using filtering

In this section we provided the target fidelity $F_{n}$ after $n$ iterations
of control and filtering for the three examples discussed in Sec. VI.

$\displaystyle F_{n}$
$\displaystyle:=\frac{\left|\braket{T}{\widetilde{K}^{n}}{\psi_{0}}\right|^{2}}{\braket{\psi_{0}}{(\widetilde{K}^{\dagger})^{n}\widetilde{K}^{n}}{\psi_{0}}}\,,$
(64)

where $\widetilde{K}:=({K_{0}+K_{1}})/{\sqrt{2}}$, $\ket{\psi_{0}}$ is the
initial state of the system and $\braket{T}{T_{\perp}}=0$.

###  D.1 Basic control scheme

In this example the Kraus operators are given by

$\displaystyle K_{0}$ $\displaystyle=\ket{T}\bra{T}$ (65) $\displaystyle
K_{1}$ $\displaystyle=\ket{T}\bra{T_{\perp}}$ (66)

so that

$\displaystyle\widetilde{K}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{T}\bra{T}+\ket{T}\bra{T_{\perp}}\right)\,.$
(67)

Taken to the $n$th power

$\displaystyle\widetilde{K}^{n}=\frac{1}{2^{\frac{n}{2}}}\left(\ket{T}\bra{T}+\ket{T}\bra{T_{\perp}}\right)\,.$
(68)

Provide that $\braket{T}{\psi_{0}}+\braket{T_{\perp}}{\psi_{0}}\neq 0$, the
target fidelity after each iteration can be shown to be unity, which is to be
expected since this scheme works in a single round.

###  D.2 Weak swap

The Kraus operators associated with the weak swap are given by [8]

$\displaystyle K_{0}$
$\displaystyle=e^{-i\lambda}\ket{T}\bra{T}+\cos\lambda\ket{T_{\perp}}\bra{T_{\perp}}$
(69) $\displaystyle K_{1}$ $\displaystyle=\sin\lambda\ket{T}\bra{T_{\perp}}$
(70)

so that

$\displaystyle\widetilde{K}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(e^{-i\lambda}\ket{T}\bra{T}+\cos\lambda\ket{T_{\perp}}\bra{T_{\perp}}+\sin\lambda\ket{T}\bra{T_{\perp}}\right)\,.$
(71)

Taken to the $n$th power

$\displaystyle\widetilde{K}^{n}=\left(\frac{e^{-i\lambda}}{\sqrt{2}}\right)^{n}\Bigg{(}\ket{T}$
$\displaystyle\bra{T}+e^{in\lambda}\cos^{n}\lambda\ket{T_{\perp}}\bra{T_{\perp}}$
$\displaystyle+\sin\lambda\sum_{k=0}^{n-1}e^{i(k+1)\lambda}\cos^{k}\lambda\ket{T}\bra{T_{\perp}}\Bigg{)}$
(72)

so that

$\displaystyle F_{n}$ $\displaystyle=\frac{1}{1+\Lambda(\cos\lambda)^{2n}}\,,$
(73)

where

$\displaystyle\Lambda=\frac{|\braket{T_{\perp}}{\psi_{0}}|^{2}}{|\braket{T}{\psi_{0}}+e^{i\lambda}\sin\lambda\sum_{k=0}^{n-1}(e^{i\lambda}\cos\lambda)^{k}\braket{T_{\perp}}{\psi_{0}}|^{2}}\,.$
(74)

The target fidelity $F_{n}$ converges to one for large $n$ provided that
$\lambda$ is not an integer multiple of $\pi$ and $\Lambda<\infty$.

###  D.3 Target state dependent control mechanism

The Kraus operators in this example are given by [8]

$\displaystyle K_{0}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{T}\bra{T}+\sin\lambda\ket{T}\bra{T_{\perp}}+\cos\lambda\ket{T_{\perp}}\bra{T_{\perp}}\right)$
(75) $\displaystyle K_{1}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{T}\bra{T}-\sin\lambda\ket{T}\bra{T_{\perp}}+\cos\lambda\ket{T_{\perp}}\bra{T_{\perp}}\right)$
(76)

so that

$\displaystyle\widetilde{K}$
$\displaystyle=\ket{T}\bra{T}+\cos\lambda\ket{T_{\perp}}\bra{T_{\perp}}\,.$
(77)

Taken to the $n$th power

$\displaystyle\widetilde{K}^{n}=\ket{T}\bra{T}+\cos^{n}\lambda\ket{T_{\perp}}\bra{T_{\perp}}\,.$
(78)

so that

$\displaystyle F_{n}$
$\displaystyle=\frac{1}{1+(\cos\lambda)^{2n}\frac{|\braket{T_{\perp}}{\psi_{0}}|^{2}}{|\braket{T}{\psi_{0}}|^{2}}}\,.$
(79)

The target fidelity $F_{n}$ converges to one for large $n$ provided that
$\lambda$ is not an integer multiple of $\pi$ and $|\braket{T}{\psi_{0}}|>0$.

##  Appendix E Parametric Amplification

The process of parametric amplification involves the interaction of three
fields, the signal $E_{1}(z,t)$, the idler $E_{2}(z,t)$ and the pump field
$E_{3}(z,t)$. We follow the method presented in [19]. For the case of
monochromatic plane waves, where the pump beam is undepleted during the
nonlinear interaction and there is no initial idler field, the signal field
evolution along the crystal is given by

$\frac{\partial^{2}A_{1}}{\partial z^{2}}=-i\Delta k\ \frac{\partial
A_{1}}{\partial z}+\Gamma^{2}A_{1}.$ (80)

Here $A_{1}$ refers to the complex amplitude of the signal field, $\Delta
k=k_{3}-k_{2}-k_{1}$ is the wave vector mismatch and $\Gamma$ is the coupling
constant from the nonlinear wave equations defined as

$\Gamma^{2}=\frac{4d_{eff}^{2}\omega_{1}^{2}\omega_{2}^{2}}{k_{1}k_{2}c^{4}}|A_{3}|^{2}.$
(81)

The constant $d_{eff}$ relates to the nonlinear susceptibility of the crystal,
$\omega_{i}$ and $k_{i}$ refer to the angular frequencies and the wave numbers
of the fields in the medium, respectively, here $c$ is the speed of light.
Since the beam intensity can be given by
$I_{i}=\tfrac{1}{2}n_{i}\varepsilon_{0}c|A_{i}|^{2}$, the signal and idler
intensities after the interaction length of the nonlinear crystal are

$\displaystyle I_{1}(L)$
$\displaystyle=I_{1}(0)\left(1+\left(\tfrac{\Gamma}{g}\text{sinh}(gL)\right)^{2}\right)$
(82) $\displaystyle I_{2}(L)$
$\displaystyle=I_{1}(0)\tfrac{\omega_{2}}{\omega_{1}}\left(\tfrac{\Gamma}{g}\text{sinh}(gL)\right)^{2},$

where $I_{1}(0)$ is the initial signal field intensity, $\varepsilon_{0}$ is
the electric permittivity and $g$ is a generalised wave number given by

$g=\sqrt{\Gamma^{2}-\frac{\Delta k^{2}}{4}}.$ (83)

The parametric gain $G$ is therefore defined as the ratio of the signal
intensity before and after the nonlinear process,
$G=\frac{I_{1}(L)}{I_{1}(0)}$.

##  Appendix F Creating distinct time-bins

If the number of round trips in the delay loop (Fig. 4) is doubled in each
iteration, the delay time $T_{N}$ a specific pulse spends in the delay loop in
$N$ iterations reads

$T_{N}(\mathbf{a})=\sum_{n=1}^{N}a_{n}\ 2^{n}\ \tau\,.$ (84)

Here the vector $\mathbf{a}=(a_{1},a_{2}\ldots a_{N})$ contains information
about which path the pulse took in each iteration. The component $a_{i}$ is
$0$ or $1$, depending on whether the pulse entered the delay loop in the $i$th
interation. The time period $\tau$ is the duration of a single round-trip in
the delay loop. Since each vector $\mathbf{a}$ is the binary representation of
a specific number $T_{N}/\tau$, the corresponding delay $T_{N}$ is unique and
hence the time bins do not overlap.

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