# Noiseless Loss Suppression for Entanglement Distribution

Cory M. Nunn  National Institute of Standards and Technology, Gaithersburg,
Maryland 20899, USA  University of Maryland Baltimore County, Baltimore, MD
21250, USA  Daniel E. Jones  DEVCOM Army Research Laboratory, Adelphi, MD
20783, USA  Todd B. Pittman  University of Maryland Baltimore County,
Baltimore, MD 21250, USA  Brian T. Kirby  DEVCOM Army Research Laboratory,
Adelphi, MD 20783, USA  Tulane University, New Orleans, LA 70118, USA

###### Abstract

Recent work by Mičuda et al. Mičuda  _et al._ (2012) suggests that pairing
noiseless amplification with noiseless attenuation can conditionally suppress
loss terms in the direct transmission of quantum states. Here we extend this
work to entangled states: first, we explore bipartite states, specifically the
two-mode squeezed vacuum (TMSV) and NOON states; and second, we examine
$M$-partite states, concentrating on W and Greenberger–Horne–Zeilinger (GHZ)
states. In analogy with the original proposal, our results demonstrate that in
each case under consideration, a correct combination of attenuation and
amplification techniques before and after transmission through a pure loss
channel can restore the initial quantum state. However, we find that for both
W and NOON states, the noiseless attenuation is redundant and not required to
achieve loss term suppression. This work clarifies the role of noiseless
attenuation when paired with noiseless amplification for entanglement
distribution and provides an operational example of how GHZ and W state
entanglement differs.

##  I Introduction

Quantum networking offers the potential for various advanced applications such
as distributed quantum computing Cirac  _et al._ (1999), long-baseline
interferometry Gottesman  _et al._ (2012), and precise time synchronization
Kómár  _et al._ (2016). Photon loss in a quantum network’s channels, links,
and components limits the ability to realize these applications in practice.
Traditional methods to counter these issues, such as quantum error correction
and quantum repeaters, can have demanding technical requirements including the
need for quantum memories, multiple copies of a quantum state, or quantum
information processing at the nodes. Unlike classical communication, the
accumulated photon losses in a quantum network cannot be compensated for with
phase-insensitive amplification due to the inherent noise added and the no-
cloning principle Pandey  _et al._ (2013). However, these limitations can be
partially mitigated using heralded noiseless amplification of light Ralph and
Lund (2009), which conditionally boosts the amplitude of a coherent state
without adding noise, presenting a promising avenue for overcoming these
challenges.

Noiseless linear amplification (NLA) has been used in a variety of schemes to
reduce the effects of loss on quantum communication for both continuous-
variable Xiang  _et al._ (2010) and discrete Osorio  _et al._ (2012)
encodings. For instance, it has been used to probabilistically amplify
coherent states Xiang  _et al._ (2010); Ferreyrol  _et al._ (2010); Usuga  _et
al._ (2010); Zavatta  _et al._ (2011); Donaldson  _et al._ (2015); Haw  _et
al._ (2016) and two-mode squeezed vacuum (TMSV) Ulanov  _et al._ (2015); Dias
_et al._ (2020); Karsa  _et al._ (2022), as well as polarization Kocsis  _et
al._ (2013), path Monteiro  _et al._ (2017), and time-bin states Bruno  _et
al._ (2016). In addition, various schemes utilizing NLA have been used to
perform entanglement distillation Mauron and Ralph (2022); Ulanov  _et al._
(2015); Monteiro  _et al._ (2017); Dias  _et al._ (2020); Liu  _et al._
(2022); Slussarenko  _et al._ (2022). While noiseless amplification alone can
improve the performance of quantum state distribution systems, some schemes
also utilize quantum scissors Seshadreesan  _et al._ (2019); Dias  _et al._
(2020); Slussarenko  _et al._ (2022) or photon subtraction techniques Usuga
_et al._ (2010); Zavatta  _et al._ (2011); Donaldson  _et al._ (2015).
Unfortunately, NLA alone is ultimately limited in its ability to overcome loss
in the direct transmission of superpositions of Fock states due to the
inevitable mixing incurred by these channels. More precisely, the NLA process
is not the inverse map of the loss process, and hence it cannot eliminate the
noise added by it Mičuda  _et al._ (2012).

An alternative approach devised by Mičuda et al. supplements the use of NLA
with a companion process called noiseless attenuation Mičuda  _et al._ (2012).
In the proposed noiseless loss suppression (NLS) protocol, quantum states are
noiselessly attenuated prior to their distribution across a lossy channel, and
then noiseless amplification restores their amplitudes at the end of the
channel. Crucially, this type of heralded noiseless attenuation can be
faithfully inverted by the noiseless amplifier, as these operations are
inverse maps of one another. When the pre-attenuation is sufficiently strong
such that the effects of photon loss are negligible in comparison, the full
NLS protocol can transmit quantum states with arbitrarily high fidelity at the
cost of reduced probability of success. Mičuda et al. showed that, in
principle, any quantum state from an arbitrarily large Hilbert space could be
transmitted across a single lossy channel in this way. Here, we investigate
the potential use of NLS for entanglement distribution of multi-partite
entanglement through independent lossy channels.

We begin in Section II by reviewing the original NLS protocol of Mičuda et al.
for a single optical mode with states encoded in a superposition of vacuum and
single-photon states. Then, to provide insight into our main results, in
Section III we consider the application of the NLS protocol to bipartite
entangled states by introducing a possible experimental setup involving a
bipartite entangled state consisting of only a single photon with physical
noiseless amplifiers based on quantum scissors. We then abstract away the
experimental configuration and generalize these bipartite states to $M$ modes
sharing a single photon (i.e., W states) in Section IV and to $n$-photon
Greenberger–Horne–Zeilinger (GHZ) states in Section V. Next, in Section VI we
consider the application of NLS to TMSV states where higher-order photon terms
complicate the analysis. Then in Section VII we again consider multiple
photons per mode by studying the NLS protocol on NOON states. In Section VIII
we provide general arguments describing which states require noiseless
attenuation in addition to noiseless amplification in order to recover their
initial form, followed by a discussion and conclusion in Section IX.

##  II Noiseless Loss Suppression

The original NLS protocol devised by Mičuda et al. Mičuda  _et al._ (2012) is
shown in Fig. 1(a). Here, an input state $\hat{\rho}_{in}$ is sent across a
single lossy channel $\mathcal{L}_{\tau}$ with amplitude transmittance $\tau$.
Unaided, photon loss over this channel will generate noise and reduce the
coherence of the transmitted output $\hat{\rho}_{out}$. The NLS scheme
mitigates these effects with the introduction of two probabilistic devices:
the noiseless attenuator and the noiseless amplifier. The noiseless
attenuation is described by the quantum filter $\nu^{\hat{n}}$ where
$\hat{n}=\hat{a}^{\dagger}\hat{a}$ is the number operator and $\nu<1$, which
reduces Fock state amplitudes by $\ket{n}\rightarrow\nu^{n}\ket{n}$. This
precedes the lossy channel, essentially suppressing the higher-photon-number
terms which are most susceptible to photon loss. A noiseless amplifier, which
similarly transforms Fock state amplitudes as $\ket{n}\rightarrow
g^{n}\ket{n}$ with $g>1$, is then placed at the end of the channel with gain
$g=1/(\nu\tau)$ chosen to restore the original amplitudes. In the limit
$\nu\rightarrow 0$, noise terms generated by photon loss are eliminated, and
the input state is faithfully (but conditionally) transmitted, i.e.
$\hat{\rho}_{out}=\hat{\rho}_{in}$.

Figure 1: Basic illustration of the noiseless loss suppression (NLS) protocol
Mičuda  _et al._ (2012). Panel (a) shows the case of a single-mode input state
$\hat{\rho}_{in}$ transmitted over one quantum channel, while (b) shows an
extension of NLS to multimode states sent over $M$ channels. Each channel is
preceded by a noiseless attenuator with an attenuation factor $\nu_{k}<1$, and
followed with noiseless amplification by gain $g_{k}>1$. With appropriate
choices of attenuation and gain, any added noise generated by the lossy
channels $\mathcal{L}(\tau_{k})$ can be mitigated.

Following Ref. Mičuda  _et al._ (2012), we will illustrate the mechanics of
NLS in more detail with a simple example: the single-rail ($M=1$) qubit
$\ket{\psi}=c_{0}\ket{0}+c_{1}\ket{1}$. Sending this state through the lossy
channel $\mathcal{L}_{\tau}$, we obtain a mixed output with added noise:

$\mathcal{L}_{\tau}\left(\ket{\psi}\bra{\psi}\right)=\ket{\tilde{\psi}}\bra{\tilde{\psi}}+(1-\tau^{2})|c_{1}|^{2}\ket{0}\bra{0}$
(1)

where $\ket{\tilde{\psi}}=c_{0}\ket{0}+\tau c_{1}\ket{1}$. To restore the
single-photon amplitude, we can noiselessly amplify this state with gain
$g=1/\tau$. This transforms the Fock state amplitudes as
$\ket{0}\rightarrow\ket{0}$ and $\ket{1}\rightarrow g\ket{1}$, which yields
the following output:

$\hat{\rho}_{amp}\propto\ket{\psi}\bra{\psi}+(1-\tau^{2})|c_{1}|^{2}\ket{0}\bra{0}.$
(2)

The original signal has been restored, but the extra vacuum noise term
remains. Further amplification would reduce the size of the noise term
relative to the signal but would over-amplify the single-photon amplitude and
alter the $\ket{\psi}\bra{\psi}$ term in the mixture.

However, this can be corrected if the lossy channel $\mathcal{L}_{\tau}$ is
preceded by a noiseless attenuator. Prior to $\mathcal{L}_{\tau}$, we now have
a pure attenuated state $\ket{\psi^{\prime}}=c_{0}\ket{0}+\nu c_{1}\ket{1}$.
Then after amplifying by $g=1/(\nu\tau)$ at the end of the channel, we obtain:

$\hat{\rho}_{out}\propto\ket{\psi}\bra{\psi}+(1-\tau^{2})\nu^{2}|c_{1}|^{2}\ket{0}\bra{0}.$
(3)

Now the vacuum noise term is suppressed by the attenuation factor $\nu^{2}$.
In the limit $\nu\rightarrow 0$, which requires more extreme attenuation and
amplification, arbitrarily high fidelity can be achieved. Note, however, that
both noiseless attenuation and heralded amplification have a success rate that
scales with the attenuation and gain introduced into the channel. Hence, the
NLS protocol introduces a trade-off between channel fidelity and success rate,
with an exact single-mode NLS success probability expression provided in
Mičuda  _et al._ (2012).

The benefits of NLS still hold true for any superposition of Fock states in a
single mode, so long as noiseless amplification can be faithfully implemented
at the end of the channel. Because the ideal noiseless amplification operator
$g^{\hat{n}}$ is unbounded, this requires that we restrict our input states to
a finite subspace spanned by Fock states $\ket{n}$ for $n\leq N$ where $N$ is
the maximum photon number in the channel Pandey  _et al._ (2013); Gagatsos
_et al._ (2014). Then for the full sequence of noiseless attenuation, loss,
and then noiseless amplification shown in Figure 1(a), this complete channel
$\mathcal{M}$ can be described as follows:

$\mathcal{M}(\hat{\rho}_{in})=\sum_{j=0}^{N}G_{N}(g)\hat{A}_{j}\nu^{\hat{n}}\hat{\rho}_{in}\nu^{\hat{n}}\hat{A}^{\dagger}_{j}G_{N}(g)$
(4)

where the Kraus operators $\hat{A}_{j}$ are given by:

$\hat{A}_{j}=\sum_{m=0}^{N-j}\sqrt{\binom{m+j}{j}}(1-\tau^{2})^{j/2}\tau^{m}\ket{m}\bra{m+j}$
(5)

and each correspond to the loss of $j$ photons from the input state. Noiseless
amplification is implemented by the filter
${G_{N}(g)=g^{-N}\sum_{n=0}^{N}g^{n}\ket{n}\bra{n}}$, which replicates the
action of the operator $g^{\hat{n}}$ within the smaller Hilbert space of
states with a maximum photon number $N$. The success probability for this
filter $G_{N}$ is bounded below by $g^{-2N}$.

Following Ref. Mičuda  _et al._ (2012), it is straightforward to show that
with a choice of gain $g=1/(\nu\tau)$, the resulting output state takes the
following form:

$\hat{\rho}_{out}\propto\hat{\rho}_{in}+\sum_{j=1}^{N}{\nu^{2j}\hat{B}_{j}\hat{\rho}_{in}\hat{B}^{\dagger}_{j}}$
(6)

with newly defined composite operators
$\hat{B}_{j}=\hat{A}_{0}^{-1}\hat{A}_{j}$. The inverse Kraus operator
$\hat{A}_{0}^{-1}=\sum_{m=0}^{N}\tau^{-m}\ket{m}\bra{m}$ is equivalent to
noiseless amplification by $g=1/\tau$, much like $\hat{A}_{0}$ is equivalent
to noiseless attenuation by $\nu=\tau$. As these are exact inverses of one
another, the lossless $j=0$ term leaves the input state unchanged. Meanwhile,
the remaining noise terms corresponding to $j>0$ photon losses are suppressed
by a factor $\nu^{2j}$. In the single-photon example, the loss of $j=1$
photons produced a single vacuum noise term, which acquired an additional
factor of $\nu^{2}$. More generally, this channel tends to the identity
$\mathcal{M}\rightarrow\mathcal{I}$ in the limit $\nu\rightarrow 0$, and input
states can be transmitted with arbitrarily high fidelity.

Here we propose a natural extension of NLS to entangled multimode states,
shown in Figure 1(b). The input $\hat{\rho}_{in}$ propagates along $M$ lossy
paths, each with a distinct transmittance $\tau_{k}$ for $k=1,...,M$. As
pictured, the original protocol is repeated for every mode: each path $k$ is
preceded by a noiseless attenuator with $\nu_{k}<1$ and followed by an
amplifier with $g_{k}>1$. The operators for mode $k$ commute with those of all
other modes, so we can apply the NLS protocol $M$ times successively, once per
each mode. All together, this transforms the state as:

$\mathcal{M}_{tot}(\hat{\rho}_{in})=\sum_{j_{1}=0}^{N}\cdots\sum_{j_{M}=0}^{N}\left(\prod_{k=0}^{M}g_{k}^{-2N}\nu_{k}^{2j_{k}}\right)\hat{B}^{(M)}_{j_{M}}\cdots\hat{B}^{(1)}_{j_{1}}\hat{\rho}_{in}\hat{B}^{(1)\dagger}_{j_{1}}\cdots\hat{B}^{(M)\dagger}_{j_{M}}$
(7)

where the operators $\hat{B}^{(k)}_{j_{k}}$ are similarly defined for each
mode $k$:

$\hat{B}^{(k)}_{j_{k}}=\sum_{m=j}^{N}\sqrt{\binom{m}{j}}(1-\tau_{k}^{2})^{j_{k}/2}(\nu_{k}\tau_{k}g_{k})^{m-j_{k}}\ket{m-j_{k}}\bra{m}$
(8)

Note, that unlike in the definition of $\hat{B}_{j}$ in the single mode case
(which follows the treatment in Mičuda  _et al._ (2012)), this definition of a
multimode $\hat{B}_{jk}^{(k)}$ does not enforce any particular relationship
between $g_{k}$ and $\nu_{k}$ (i.e., they are free variables). With the
greater number of degrees of freedom in this multimode system, more
investigation is required to find the optimal values of attenuation and
amplification for NLS.

One possible strategy is to mimic the original NLS protocol and choose
${g_{k}=1/(\nu_{k}\tau_{k})}$ for each mode. As in the single-mode case, each
$\hat{B}^{(k)}_{0}$ is equal to the identity when $j_{k}=0$ photons are lost.
Thus when $j_{k}=0$ for all $k$ modes, the entire input state is left
unchanged. For all other cases, we define

$\tilde{\rho}_{j_{1}\cdots
j_{m}}\equiv\hat{B}^{(M)}_{j_{M}}\cdots\hat{B}^{(1)}_{j_{1}}\hat{\rho}_{in}\hat{B}^{(1)\dagger}_{j_{1}}\cdots\hat{B}^{(M)\dagger}_{j_{M}}$
(9)

and replace each of the separate sums over $j_{k}$ in Eq. 7 with a sum over
all possible sequences $\bm{j}=\\{j_{1},...,j_{M}\\}$. The resulting output
state from the composite channel becomes

$\hat{\rho}_{out}\propto\hat{\rho}_{in}+\sum_{\bm{j}}\left(\prod_{k=1}^{M}{\nu_{k}^{2j_{k}}}\right)\tilde{\rho}_{j_{1}\cdots
j_{M}}$ (10)

Each sequence of photon losses generates a unique noise term with a density
operator $\tilde{\rho}_{j_{1}\cdots j_{M}}$. We again stress that this
particular result assumes $g_{k}=1/(\nu_{k}\tau_{k})$ and hence the relative
weight of the terms are set by one free variable per mode (either $\nu_{k}$
or, equivalently in this case, $g_{k}$).

Though one might suspect otherwise, this naive approach works perfectly well
despite any path entanglement that may exist in the system. Following Eq. 6,
each instance of $\mathcal{M}$ returns the original state $\hat{\rho}_{in}$
with added noise. The total channel $\mathcal{M}_{tot}$ in Eq. 10 thus has a
similar form. Now every noise term corresponding to $j_{k}$-photon losses in
mode $k$ will be suppressed by a factor of $\nu_{k}^{2j_{k}}$, with the
original state corresponding to the unique case of $j_{k}=0$ photons lost in
all $M$ modes. Therefore, even if the $M$ modes are entangled, we can still
compensate for photon loss locally with $M$ pairs of noiseless attenuators and
amplifiers.

However, while the naive approach to NLS will work for any state, it is not
the best solution in every case. In the remainder of this paper, we will
examine the performance of NLS for several common forms of entangled input
states and consider the appropriate values of attenuation and gain (e.g., look
for solutions where $g_{k}\neq 1/(\nu_{k}\tau_{k})$). Interestingly, there are
some cases in which NLS can be accomplished while omitting noiseless
attenuation, allowing for compensation with only noiseless amplifiers at the
end of each channel.

##  III Bipartite single-photon entanglement

Here we consider the most straightforward extension of the original NLS
protocol to entangled states: its application to the superposition of a single
photon in two spatial modes. Practically, single-photon entangled states of
this type result from transmitting a single photon through a beamsplitter. In
analogy with the single-mode states considered in the original paper, the
vacuum modes of single-photon entangled states encode quantum information.
Single-photon entanglement of this type has applications throughout quantum
information science and has motivated fundamental questions about whether its
nonlocal properties are detectable in experiments Tan  _et al._ (1991);
Dunningham and Vedral (2007); Das  _et al._ (2021). Note that this state has
been considered in a related context in both Monteiro  _et al._ (2017);
Slussarenko  _et al._ (2022), but differs from the states used in Ulanov  _et
al._ (2015), which are of the form $|00\rangle-\gamma|11\rangle$. In section
IV, we will explore the physical significance of each of these entangled
states in the context of NLS.

To develop physical intuition for our more generalized results to follow, we
will now examine a specific proposed optical experiment and examine the
properties of the output state. The proposed experiment is pictured in Figure
2 and consists of an $M=2$ mode experiment. Here a single photon is prepared
in an equal superposition of the two optical modes in the state
$|\psi\rangle=(1/\sqrt{2})(|01\rangle+|10\rangle)$ (i.e, a single photon is
passed through a 50/50 beamsplitter, up to a phase). The original NLS protocol
Mičuda  _et al._ (2012) is then applied independently in each mode in
parallel. At the beginning of each channel, noiseless attenuation is applied
by using beamsplitters with amplitude transmittance $\nu_{1}$ and $\nu_{2}$
and conditioning on the detection of zero photons in the two detectors Nunn
_et al._ (2021), resulting in:

$\ket{\psi^{\prime}}\propto\nu_{1}\ket{10}+\nu_{2}\ket{01}.$ (11)

Channel loss is then applied via another set of beamsplitters, each with
transmittance given by $\tau_{1,2}$. In contrast to noiseless attenuation, the
output modes are not detected and will be traced over, generating a mixed
state with the noise terms that NLS aims to suppress. Before tracing over the
loss modes, the state is given by:

$\displaystyle\ket{\psi^{\prime\prime}}$
$\displaystyle\propto\nu_{1}\left(i\sqrt{1-\tau_{1}^{2}}\ket{00}\ket{1}_{R}\ket{0}_{R^{\prime}}+\tau_{1}\ket{10}\ket{0}_{R}\ket{0}_{R^{\prime}}\right)+\nu_{2}\left(i\sqrt{1-\tau_{2}^{2}}\ket{00}\ket{0}_{R}\ket{1}_{R^{\prime}}+\tau_{2}\ket{01}\ket{0}_{R}\ket{0}_{R^{\prime}}\right)$
(12)
$\displaystyle=\left(\nu_{1}\tau_{1}\ket{10}+\nu_{2}\tau_{2}\ket{01}\right)\ket{0}_{R}\ket{0}_{R^{\prime}}+i\ket{00}\left(\nu_{1}\sqrt{1-\tau_{1}^{2}}\ket{1}_{R}\ket{0}_{R^{\prime}}+\nu_{2}\sqrt{1-\tau_{2}^{2}}\ket{0}_{R}\ket{1}_{R^{\prime}}\right)$

where $R$ and $R^{\prime}$ indicate the reflected modes of the channel loss
beampslitters in modes $1$ and $2$ respectively. Finally, NLA is implemented
using the “quantum scissors”-based protocol considered in Xiang  _et al._
(2010); Winnel  _et al._ (2020) as pictured in the “Noiseless Amplification”
box of Fig. 2. Within the amplification step an ancilla photon is introduced
into one input of a tunable beamsplitter, one output of which is mixed with
the input signal at a separate 50/50 beamsplitter, where a Bell measurement is
performed. Successful amplification is heralded by the detection of one and
only one photon at either of the detectors (an anti-coincidence event). The
gain of each noiseless amplifier is determined by the amplitude transmittance
$0\leq t_{j}\leq 1$ of the tunable beamsplitter, with
${g_{j}=t_{j}/\sqrt{1-t_{j}^{2}}}$ Winnel  _et al._ (2020).

The resulting density matrix after post-selection on the outcomes pictured in
Fig. 2 and tracing over modes $R$ and $R^{\prime}$ is given by:

$\displaystyle\hat{\rho}\propto$
$\displaystyle\left[\nu_{1}^{2}(1-\tau_{1}^{2})+\nu_{2}^{2}(1-\tau_{2}^{2})\right](1-t_{1}^{2})(1-t_{2}^{2})|00\rangle\langle
00|+$
$\displaystyle\left(\nu_{1}\tau_{1}t_{1}\sqrt{1-t_{2}^{2}}|01\rangle+\nu_{2}\tau_{2}\sqrt{1-t_{1}^{2}}t_{2}|10\rangle\right)\left(\nu_{1}\tau_{1}t_{1}\sqrt{1-t_{2}^{2}}\langle
01|+\nu_{2}\tau_{2}\sqrt{1-t_{1}^{2}}t_{2}\langle 10|\right)$ (13)

assuming all $t_{j}$, $\nu_{j}$, and $\tau_{j}$ are real and a $\pi/2$ phase
shift is added upon reflection from each beamsplitter.

Figure 2: Proposed Optical Realization: This diagram illustrates the setup for
the proposed quantum optics experiment. A single photon is initially generated
in a balanced superposition of two distinct spatial modes. This superposition
is achieved using a 50/50 beam splitter (BS) followed by a $\phi=\pi/2$ phase
delay, ensuring the modes’ desired phase relationship. The experiment proceeds
through several stages as per the Noiseless Loss Suppression (NLS) protocol,
which includes noiseless attenuation (NA), simulation of channel loss (L), and
noiseless amplification (NLA). Each stage is represented as a distinct box
within the figure. The amplitude transmittance of each beam splitter is
annotated directly on the respective elements. It is important to note that
all beam splitters in this setup are designed to impart no phase shift upon
transmission while introducing a phase shift of $i$ ($\pi$/2 radians) on
reflection. Detectors are strategically placed to monitor conditional outcomes
and are labeled accordingly.

Then, in order to recover a result analogous to the single-mode treatment in
Mičuda  _et al._ (2012) where the resultant state can be expressed as a
mixture between the initial state and vacuum terms, we require the following
condition

$\displaystyle\sqrt{\mu}=$
$\displaystyle\sqrt{2}\nu_{1}\tau_{1}t_{1}\sqrt{1-t_{2}^{2}}=\sqrt{2}\nu_{2}\tau_{2}t_{2}\sqrt{1-t_{1}^{2}}$
(14) $\displaystyle=$
$\displaystyle\sqrt{2}\nu_{1}\tau_{1}g_{1}\sqrt{\left(1-t_{1}^{2}\right)\left(1-t_{2}^{2}\right)}=\sqrt{2}\nu_{2}\tau_{2}g_{2}\sqrt{\left(1-t_{1}^{2}\right)\left(1-t_{2}^{2}\right)}$

Note that $0\leq\sqrt{\mu}\leq\sqrt{2}$ due to the restrictions on $\nu_{j}$,
$\tau_{j}$, and $\phi_{j}$. For convenience, we define

$\sigma=\left[\nu_{1}^{2}(1-\tau_{1}^{2})+\nu_{2}^{2}(1-\tau_{2}^{2})\right](1-t_{1}^{2})(1-t_{2}^{2})$
(15)

so that the state becomes

$\hat{\rho}\propto\mu|\psi\rangle\langle\psi|+\sigma|00\rangle\langle 00|.$
(16)

Our definition of $\mu$ can be equivalently expressed as the following
restriction on attenuation and gain parameters in each channel

$\frac{\nu_{1}\tau_{1}g_{1}}{\nu_{2}\tau_{2}g_{2}}=1$ (17)

which is analogous to the gain constraint $g=(\nu\tau)^{-1}$ in the single-
mode case (see Ref. Mičuda  _et al._ (2012) and Sec. II) which recovers the
form of the initial state with added vacuum terms. Note that applying the same
constraint to each channel individually satisfies the two-mode relationship
found here with $\nu_{1}\tau_{1}g_{1}=\nu_{2}\tau_{2}g_{2}=1$. More generally,
however, the number of free parameters has doubled, so other configurations of
gain and attenuation that recover the equivalent output state are possible.
It’s only important that the overall amplitudes in modes 1 and 2 are balanced.
This “balancing condition” in Eq. 17 ensures the transmitted state has equal
$\ket{01}$ and $\ket{10}$ amplitudes like the input state and is reminiscent
of local filtering operations used in Procrustean filtering Kwiat  _et al._
(2001); Bennett  _et al._ (1996) and nonlocal compensation of polarization-
dependent loss Kirby  _et al._ (2019); Jones  _et al._ (2018).

To complete the NLS protocol, we must ensure that the vacuum noise terms can
be made arbitrarily small while maintaining the form of Eq. 16, e.g., while
enforcing the balancing condition given by Eq. 17. To this end, we consider
the vacuum-signal ratio $\sigma/\mu$ which goes to zero as $\sigma\rightarrow
0$ or $\mu\rightarrow\infty$. The ratio is given by:

$\frac{\sigma}{\mu}=\frac{\sigma}{\sqrt{\mu}\sqrt{\mu}}=\frac{\nu_{1}^{2}(1-\tau_{1}^{2})+\nu_{2}^{2}(1-\tau_{2}^{2})}{2\nu_{1}\tau_{1}g_{1}\nu_{2}\tau_{2}g_{2}},$
(18)

where we have used both equivalent definitions of $\sqrt{\mu}$ in the
denominator ($\nu_{1}\tau_{1}g_{1}=\nu_{2}\tau_{2}g_{2}$ in the resulting
expression) as it makes the general behavior of the system more transparent
but does not impact the results. As expected, whenever the channels are
lossless ($\tau_{1}=\tau_{2}=1$), the ratio goes to zero.

The first case of interest is when the single-mode NLS protocol is applied to
each channel independently, with ${g_{j}=(\nu_{j}\tau_{j})^{-1}}$. Inserting
this into Eq. 18, we obtain:

$\frac{\sigma}{\mu}\bigg{|}_{g_{j}=(\nu_{j}\tau_{j})^{-1}}=\frac{\nu_{1}^{2}(1-\tau_{1}^{2})+\nu_{2}^{2}(1-\tau_{2}^{2})}{2},$
(19)

which, for a fixed channel loss (fixed $\tau$), goes to zero as both
$\nu_{j}\rightarrow 0$. Hence, the original proposal from Mičuda  _et al._
(2012) works in the two-mode setting when applied to each channel
independently.

Figure 3: Plot of the vacuum-signal ratio $\sigma/\mu$ as a function of
channel transmittance $\tau=\tau_{1}=\tau_{2}$ and NLA gain $g=g_{1}=g_{2}$.
Increasing the gain can make the vacuum amplitude arbitrarily small, even with
significant channel loss (small $\tau$).

However, the two-mode setting allows for an additional approach to make the
ratio in Eq. 18 arbitrarily small: fixing the $\nu_{j}$ terms and increasing
the $g_{j}$. This can be understood intuitively by setting $\nu_{1}=\nu_{2}$,
resulting in

$\frac{\sigma}{\mu}\bigg{|}_{\nu_{1}=\nu_{2}}=\frac{(1-\tau_{1}^{2})+(1-\tau_{2}^{2})}{2\tau_{1}g_{1}\tau_{2}g_{2}}.$
(20)

In this case, the balancing condition only requires that
$g_{1}\tau_{1}=g_{2}\tau_{2}$, allowing arbitrarily high gains in the
denominator that drive the ratio $\sigma/\mu$ toward zero. Furthermore, the
$\nu$ terms cancel out entirely, for which $\nu_{j}=1$ is a special case with
no noiseless attenuation. In other words, the extra degrees of freedom
afforded by the multi-mode case allow for a version of NLS which only uses
noiseless amplification at the end of each channel. Figure 3 shows the vacuum-
signal ratio $\sigma/\mu$ for a specific example of this case (Eq. 20) with
$\tau=\tau_{1}=\tau_{2}$ and $g=g_{1}=g_{2}$. With higher gains, the vacuum
amplitude $\sigma$ can be made arbitrarily small relative to the signal.
Intuitively this occurs because in each term in the state $\ket{01}+\ket{10}$,
there is a single photon component that can be amplified relative to the other
term, and hence it’s possible to re-weight these terms through amplification
alone. Contrast this with the single-mode case in section II that uses the
state $\ket{0}+\ket{1}$; where the inability to change the coefficient of the
vacuum term arbitrarily using amplification necessitates the use of noiseless
attenuation.

In short, the NLS protocol can be applied as normal in each channel to
faithfully transmit the state with low noise. However, the effect of noiseless
attenuation is made completely redundant by the amplifiers for this kind of
entangled input state. Noiseless attenuation provides no advantage as it only
reduces the probability of success, forcing us to increase the gain at the end
of each channel to compensate for the reduced amplitude. This can be seen by
considering the probability of success of the NLS protocol, which is taken
directly from the form of Eq. 16 to be
$P_{s}=\text{Tr}\\{\mathcal{M}(\hat{\rho}_{in})\\}=\mu+\sigma$. Since both
$\mu$ and $\sigma$ decrease as $\nu_{i}$ decreases, the probability of success
of the NLS scheme is reduced as more noiseless attenuation is applied. As we
will see in later sections, the redundancy of the noiseless attenuation also
holds more generally for M-mode W-states but not for M-mode GHZ states or TMSV
states. In section VIII, we will provide an abstract argument for when this is
possible with a general input state.

##  IV W state

Here we generalize the results of the previous section for one photon in a
superposition between two modes to one photon in a superposition of $M$ modes.
States of this form are known as $W$ states, and have the property that any
two modes remain entangled even when all other modes are discarded Dür  _et
al._ (2000). The nonlocality of single-photon $W$ states has been studied
closely Heaney  _et al._ (2011), with applications in long-baseline telescopy
Gottesman  _et al._ (2012) and sensing Guha and Shapiro (2013).

An $M$-mode $W$ state with a single-photon can be generated analogously to the
two-mode state by replacing the beamsplitter with an $M$-mode multiport
splitter, which generates

$\displaystyle|W\rangle$
$\displaystyle\propto|100...0\rangle+|010...0\rangle+...+|000...1\rangle$ (21)
$\displaystyle=\sum_{j=1}^{M}|j\rangle,$

where we define $|j\rangle$ as the state with the single photon in mode $j$
and all other modes unoccupied, and $|0\rangle$ will indicate the vacuum state
for all modes. Normalization is omitted and will be included at the end of the
calculation. If each channel now undergoes noiseless attenuation (NA) of
magnitude $\nu_{j}$, the state becomes

$\displaystyle|W_{\text{NA}}\rangle$
$\displaystyle\propto\sum_{j=1}^{M}\nu_{j}|j\rangle.$ (22)

We now include channel loss by introducing a beamsplitter in each mode with
transmittance $\tau_{j}$ where a phase of $i$ is added upon reflection and no
phase is added in transmission. We further include the loss modes (the modes
where a reflected photon enters) as $|j\rangle_{R}$ which indicates a single
photon in the reflected port of the $j$th attenuation beam splitter and
$|0\rangle_{R}$ which means all reflected modes are in the vacuum states.
After splitting all $M$ modes, the $W$ state then becomes

$\displaystyle|W_{\text{L}}\rangle$
$\displaystyle\propto\sum_{j=1}^{M}\tau_{j}\nu_{j}|j\rangle\otimes|0\rangle_{R}+\sum_{k=1}^{M}i\sqrt{1-\tau_{k}^{2}}\nu_{k}|0\rangle\otimes|k\rangle_{R},$
(23)

where the first sum of terms corresponds to instances where the photons are
transmitted, and the second sum corresponds to when the photons are lost
(reflected). We have assumed an $i$ phase term for reflection. We can now
include NLA in each channel with gain $g_{j}$, which acting in the $j$th mode
takes $|j\rangle\rightarrow g_{j}|j\rangle$ and
$|0\rangle\rightarrow|0\rangle$. The resulting $W$ state is then

$\displaystyle|W_{\text{NLA}}\rangle$
$\displaystyle\propto\sum_{j=1}^{M}g_{j}\tau_{j}\nu_{j}|j\rangle\otimes|0\rangle_{R}+\sum_{k=1}^{M}i\sqrt{1-\tau_{k}^{2}}\nu_{k}|0\rangle\otimes|k\rangle_{R}$
(24)

If we now trace out the loss modes, we have the density matrix given by:

$\hat{\rho}\propto\sum_{j,k=1}^{M}g_{j}g_{k}\tau_{j}\tau_{k}\nu_{j}\nu_{k}|j\rangle\langle
k|+\left(\sum_{l=1}^{M}\left(1-\tau_{l}^{2}\right)\nu_{l}^{2}\right)|0\rangle\langle
0|.$ (25)

As an illustrative case, we consider using the same gain setting as in the
single-mode NLS protocol, meaning each ${g_{j}=(\nu_{j}\tau_{j})^{-1}}$. We
obtain the final output state

$\displaystyle\hat{\rho}_{out}$
$\displaystyle\propto\sum_{j,k=1}^{M}|j\rangle\langle
k|+\left(\sum_{l=1}^{M}\left(1-\tau_{l}^{2}\right)\nu_{l}^{2}\right)|0\rangle\langle
0|$ (26) $\displaystyle=|W\rangle\langle
W|+\frac{1}{M}\left(\sum_{l=1}^{M}\left(1-\tau_{l}^{2}\right)\nu_{l}^{2}\right)|0\rangle\langle
0|$

where we recover the original result that driving $\nu_{i}\rightarrow 0$
removes the vacuum terms.

More generally, we can recover the original $W$ state form whenever the
following condition holds

$g_{k}\tau_{k}\nu_{k}=C_{W},\quad k\in\\{1,2,\ldots,M\\}$ (27)

where $C_{W}$ is some arbitrary constant. This is equivalent to the balancing
condition in Eq. 17, in which the product ${g_{k}\tau_{k}\nu_{k}}$ must be
equal for two modes, now generalized to $M$ modes.

With this balancing condition enforced, the resulting density matrix is given
by

$\hat{\rho}\propto C_{W}^{2}|W\rangle\langle
W|+\frac{1}{M}\left(\sum_{l=1}^{M}\left(1-\tau_{l}^{2}\right)\nu_{l}^{2}\right)|0\rangle\langle
0|.$ (28)

For fixed $\nu_{j}$ and $\tau_{j}$, the vacuum terms can still be made
arbitrarily small relative to the $C_{W}^{2}$ coefficient of the $W$ state
term by increasing the gain subject to the condition of Eq. 27. The original
NLS protocol can be applied to each mode independently, but it is advantageous
to do away with the noiseless attenuators ($\nu=1$) and use only noiseless
amplifiers. Hence, the same behavior for bipartite entanglement in Sec. III
appears for the more general W state, wherein noiseless amplification alone
can faithfully preserve the form of the input state.

Figure 4: W state fidelity $F$ and probability of success $P_{s}$ after NLS
with $(\nu=\frac{1}{g\tau})$ and without $(\nu=1)$ noiseless attenuation, as a
function of gain $g$ with all channels equal $\tau_{k}=\tau_{l}$ and
$g_{k}=g_{l}$). In each plot $M=3$ and channel transmittance $\tau^{2}=0.25$
for all modes. Note that the regime $g<1/\tau=2$ is unphysical for the case
that includes noiseless attenuation, hence the switch from red to blue at the
vertical dashed line in (a) (where blue and red are both overlapping for
$g\geq 2$).

This can be formally seen by considering the fidelity of the output state
after NLS

$F=\frac{C_{W}^{2}}{C_{W}^{2}+\frac{1}{M}\sum_{l=1}^{M}\left(1-\tau_{l}^{2}\right)\nu_{l}^{2}},$
(29)

as well as the probability of success $P_{s}$ given by

$P_{s}=\left(\prod_{k=1}^{M}g_{k}^{-2}\right)\left(C_{W}^{2}+\frac{1}{M}\sum_{l=1}^{M}\left(1-\tau_{l}^{2}\right)\nu_{l}^{2}\right).$
(30)

Here we have assumed that noiseless amplification is performed with the filter
${G_{N}(g)=g^{-N}\sum_{n=0}^{N}g^{n}\ket{n}\bra{n}}$ described in Sec. II,
with $N=1$ 111All probability of success calculations beyond Sec. III assume
NLA is implemented with the filter $G_{N}$, with $N$ set to the maximum number
of photons in each channel. For the W and GHZ states $(N=1)$, we could also
have used the physical amplifiers from in Sec. III, which are identical to the
$G_{N}$ filter apart from an additional prefactor. However, the simple $G_{N}$
filter is much easier to generalize to states with $N>1$ photons, allowing for
better comparison with NOON and TMSV states. Crucially, the success
probability of the $G_{N}$ filter exhibits a very similar exponential
dependence on amplifier gain that we would also expect from the physical
amplifiers in Ref. Ralph and Lund (2009). . These $F$ and $P_{s}$ expressions
encapsulate two special cases of interest: the original NLS protocol with
$g_{j}=(\nu_{j}\tau_{j})^{-1}$ (hence, $C_{W}=1$), and the absense of
noiseless attenuation (all $\nu_{j}=1$ and $g_{j}=C_{W}/\tau_{j}$). In the
former, noise terms are suppressed by a factor $\nu_{l}$ and fidelity
approaches one in the limit $\nu_{l}\rightarrow 0$. In the latter, $C_{W}$ can
be increased without bound by appropriately adjusting each $g_{k}$, and we see
fidelity can still be pushed arbitrarily close to one as
$C_{W}\rightarrow\infty$, even without noiseless attenuation.

In fact, as can be seen in Appendix A, when Eq. 29 is expressed such that all
channels are equal (all $\tau_{k}=\tau$ and $g_{k}=g$), the two special cases
with and without noiseless attenuation are exactly equivalent. This is seen in
Fig. 4a, where both equivalent expressions for fidelity approach unity as gain
increases. However, the probability of success differs when noiseless
attenuation is omitted.

In Fig. 4b, $P_{s}$ is plotted as a function of gain for $M=3$ lossy channels
with and without noiseless attenuation with all channels equal (all
$\tau_{k}=\tau_{l}$ and $g_{k}=g_{l}$). As seen in the two-mode case in Sec.
III, noiseless attenuation of $W$ states drastically reduces the probability
of success while providing no advantage over noiseless amplification alone.

##  V GHZ State

To this point we have only considered states where a single-photon is present
in a superposition over some number of modes. Here we will generalize further
to states of the GHZ form which are a superposition of $M$ photons, one in
each of $M$ modes, with the $M$ mode vacuum. The GHZ states represent a type
of entanglement at odds with W-state entanglement. In particular, the GHZ
states can be reduced to a completely separable state through measurement on a
single qubit in the computational basis and cannot be transformed into a W
state (or the reverse, W to GHZ) through local operations assisted with
classical communication (LOCC) Dür  _et al._ (2000). In an intuitive sense,
the $N$-mode GHZ state is actually more similar to the single-photon state
used in the original NLS protocol Mičuda  _et al._ (2012), as all information
is encoded in a superposition of presence or complete absence of a photon in
all modes, reminiscent of the $|0\rangle+|1\rangle$ state in a single mode. As
we will see, the GHZ states, in contrast to the W-states of the prior section,
do in fact require the noiseless attenuation step in order to perform NLS.

We begin by defining the $M$-mode GHZ state:

$\displaystyle|\text{GHZ}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\left(|000...0\rangle+|111...1\rangle\right).$
(31)

The first step of the NLS protocol is to apply noiseless attenuation of
magnitude $\nu_{j}$ to each mode independently, resulting in

$\displaystyle|\text{GHZ}_{\text{NA}}\rangle$
$\displaystyle\propto|000...0\rangle+\nu_{1}\nu_{2}\nu_{3}...\nu_{M}|111...1\rangle$
(32) $\displaystyle=|000...0\rangle+\prod_{k=1}^{M}\nu_{k}|111...1\rangle.$

We now include channel loss by including a beamsplitter in each mode with
transmittance $\tau_{j}$ where $j$ labels the mode and a phase of $i$ is added
on reflection.

$\displaystyle|\text{GHZ}_{\text{L}}\rangle$
$\displaystyle\propto|000...0\rangle\otimes|000...0\rangle_{R}+\left(\sum_{\mathbf{j}\in
B^{M}}\left[\prod_{k=1}^{M}\left(\tau_{k}\right)^{j_{k}}\left(i\sqrt{1-\tau_{k}^{2}}\right)^{\neg
j_{k}}\nu_{k}\right]|j_{1}j_{2}j_{3}...j_{M}\rangle\otimes|\neg\left(j_{1}j_{2}j_{3}...j_{M}\right)\rangle_{R}\right).$
(33)

Here, $\mathbf{j}$ denotes an $M$-tuple representing a binary string of length
$M$. The set of all such $M$-tuples, corresponding to all possible binary
strings of length $M$, is represented as $B^{M}$. Each M-tuple can be written
as $(j_{1},j_{2},j_{3},...,j_{M})$, where the subscript indicates the position
of a bit in the binary string. We use the logical $\neg$ symbol to invert a
bit string (e.g., $\neg 010=101$), have included the additional beamsplitter
loss (reflection) modes as $|\cdot\rangle_{R}$, and include a phase shift of
$i$ upon reflection.

Finally, we can now include noiseless attenuation in each channel with gain
$g_{j}$, which acting in the $j$th mode takes $|j\rangle\rightarrow
g_{j}|j\rangle$ and $|0\rangle\rightarrow|0\rangle$. The resulting GHZ state
is then

$\displaystyle|\text{GHZ}_{\text{NLA}}\rangle$
$\displaystyle\propto|000...0\rangle\otimes|000...0\rangle_{R}$ (34)
$\displaystyle+\left(\sum_{\mathbf{j}\in
B^{M}}\left[\prod_{k=1}^{M}\left(g_{k}\right)^{j_{k}}\left(\tau_{k}\right)^{j_{k}}\left(i\sqrt{1-\tau_{k}^{2}}\right)^{\neg
j_{k}}\nu_{k}\right]|j_{1}j_{2}j_{3}...j_{M}\rangle\otimes|\neg\left(j_{1}j_{2}j_{3}...j_{M}\right)\rangle_{R}\right).$

For ease of calculation we rewrite this as

$\displaystyle|\text{GHZ}_{\text{NLA}}\rangle$
$\displaystyle\propto\left(|000...0\rangle+\left[\prod_{k=1}^{M}g_{k}\tau_{k}\nu_{k}\right]|111...1\rangle\right)\otimes|000...0\rangle_{R}$
(35) $\displaystyle+\left(\sum_{\mathbf{j}\in B^{M}\setminus
1^{M}}\left[\prod_{k=1}^{M}\left(g_{k}\right)^{j_{k}}\left(\tau_{k}\right)^{j_{k}}\left(i\sqrt{1-\tau_{k}^{2}}\right)^{\neg
j_{k}}\nu_{k}\right]|j_{1}j_{2}j_{3}...j_{M}\rangle\otimes|\neg\left(j_{1}j_{2}j_{3}...j_{M}\right)\rangle_{R}\right).$

where $B^{M}\setminus{1^{M}}$ is the set of all possible binary strings of
length $M$ excluding the one where all bits are 1. We can now trace out the
reflected modes to obtain:

$\displaystyle\hat{\rho}$
$\displaystyle\propto\left(\ket{000...0}+\left[\prod_{k=1}^{M}g_{k}\tau_{k}\nu_{k}\right]\ket{111...1}\right)\left(\bra{000...0}+\left[\prod_{k=1}^{M}g_{k}\tau_{k}\nu_{k}\right]\bra{111...1}\right)$
(36) $\displaystyle+\left(\sum_{\mathbf{j}\in
B^{M}\setminus{1^{M}}}\left[\prod_{k=1}^{M}\left(g_{k}^{2}\right)^{j_{k}}\left(\tau_{k}^{2}\right)^{j_{k}}\left(1-\tau_{k}^{2}\right)^{\neg
j_{k}}\nu_{k}^{2}\right]|j_{1}j_{2}j_{3}...j_{M}\rangle\langle
j_{1}j_{2}j_{3}...j_{M}|\right)$

We see that whenever the condition

$\prod_{k=1}^{M}g_{k}\tau_{k}\nu_{k}=1$ (37)

is met, we recover

$\displaystyle\hat{\rho}\propto|\text{GHZ}\rangle\langle\text{GHZ}|+\frac{1}{2}\left(\sum_{\mathbf{j}\in
B^{M}\setminus{1^{M}}}\left[\prod_{k=1}^{M}\left(g_{k}^{2}\right)^{j_{k}}\left(\tau_{k}^{2}\right)^{j_{k}}\left(1-\tau_{k}^{2}\right)^{\neg
j_{k}}\nu_{k}^{2}\right]|j_{1}j_{2}j_{3}...j_{M}\rangle\langle
j_{1}j_{2}j_{3}...j_{M}|\right).$ (38)

This condition is akin to that of the single-mode case, which this reduces to
for $k=1$, becoming $g\tau\nu=1$, the same as found in Mičuda  _et al._ (2012)
and in section II.

If we perform single-mode NLS in each channel with
$g_{k}=(\nu_{k}\tau_{k})^{-1}$, the fidelity of the output state is given by

$\displaystyle F=$
$\displaystyle\frac{1+\frac{1}{4}\prod_{k=1}^{M}(1-\tau_{k}^{2})\nu_{k}^{2}}{1+\frac{1}{2}\left(\sum_{\mathbf{j}\in
B^{M}\setminus{1^{M}}}\left[\prod_{k=1}^{M}\left((1-\tau_{k}^{2})\nu_{k}^{2}\right)^{\neg
j_{k}}\right]\right)}$ (39)

with a probability of success:

$\displaystyle P_{s}=$
$\displaystyle\left(\prod_{k=1}^{M}g_{k}^{-2}\right)\left[1+\frac{1}{2}\left(\sum_{\mathbf{j}\in
B^{M}\setminus{1^{M}}}\left[\prod_{k=1}^{M}\left((1-\tau_{k}^{2})\nu_{k}^{2}\right)^{\neg
j_{k}}\right]\right)\right]$ (40)

Following our analysis of the $W$ state, we can explore the effect of
noiseless amplification alone on GHZ states by setting all $\nu_{j}=1$ and
introducing an arbitrary constant $g_{j}\tau_{j}=C_{GHZ}$. In this case, the
output state takes a similar form:

$\displaystyle\hat{\rho}$
$\displaystyle\propto\frac{1}{2}\left(\ket{000...0}+C_{GHZ}^{M}\ket{111...1}\right)\left(\bra{000...0}+C_{GHZ}^{M}\bra{111...1}\right)$
(41) $\displaystyle+\frac{1}{2}\left(\sum_{\mathbf{j}\in
B^{M}\setminus{1^{M}}}\left[\prod_{k=1}^{M}\left(C_{GHZ}^{2}\right)^{j_{k}}\left(1-\tau_{k}^{2}\right)^{\neg
j_{k}}\right]|j_{1}j_{2}j_{3}...j_{M}\rangle\langle
j_{1}j_{2}j_{3}...j_{M}|\right)$

As seen before, driving the gain higher in the limit
$C_{GHZ}\rightarrow\infty$ will suppress the noise terms generated by photon
loss. However, the balancing condition in Eq. 37 is only met for $C_{GHZ}=1$,
so the input state is no longer preserved at higher gain values. General
expressions for fidelity and probability of success are given in Appendix A.

Figure 5: GHZ state fidelity and probability of success after NLS with
$(\nu=\frac{1}{g\tau})$ and without $(\nu=1)$ noiseless attenuation, as a
function of gain. Channel transmittance $\tau^{2}=0.25$ for all $M=3$ modes.
Note that the regime $g<1/\tau=2$ is unphysical for the case that includes
noiseless attenuation. The original output fidelity for pure loss without
noiseless attenuation or amplification $(\nu=g=1)$ is indicated by the solid
gray line.

Each of these cases, with and without noiseless attenuation, are compared in
Fig. 5 for $M=3$ modes with identical loss. Ordinary NLS with
$g_{k}=(\nu_{k}\tau_{k})^{-1}$ allows for fidelities arbitrarily close to one
as gain is increased, at the cost of vanishing probability of success. In
contrast, direct amplification without noiseless attenuation reduces fidelity
for low values of gain $g\lesssim 6$ and only ever reaches $F=\frac{1}{2}$ in
the limit of infinite gain. In that limit, the amplified output state is
simply the product state $\ket{111}$, so all entanglement has been lost.
Notably, the probability of success tends to a nonzero value at infinite gain
$P_{s}\rightarrow\tau^{2M}/2$. This is also what we would expect using
physical single-photon amplifiers like that in Section III. When the gain is
infinite, the incoming signal photons are effectively just replaced with
unentangled ancilla photons, and probability of success is determined entirely
by the transmission of the signal photons across all lossy channels
$\tau^{2M}$, which heralds successful amplification once they are detected.

To this point, we have only considered states with at most one photon per
mode. In the next two sections, we relax this condition and study states with
multiple photons per mode and assess the impact of NLS.

##  VI Two-Mode Squeezed Vacuum

Entanglement in quantum optics experiments is commonly generated via a
nonlinear process that results in a two-mode squeezed vacuum state (TMSV).
These states are given by:

$\ket{\text{TMSV}}=\sqrt{1-\gamma}\sum_{n=0}^{\infty}{\gamma^{n}}\ket{n}_{1}\ket{n}_{2}.$
(42)

States of this form are often considered in the context of entanglement
distillation with noiseless amplifiers, which is central to many continuous
variable quantum communication protocols Ralph and Lund (2009); Ulanov  _et
al._ (2015); Seshadreesan  _et al._ (2019); Liu  _et al._ (2022). Here we also
consider noiseless amplification of TMSV states as a benchmark against a full
NLS protocol (noiseless attenuation included), comparing their utility for
high-fidelity transmission across a lossy channel.

In order to apply the NLS protocol to this state, we first attenuate modes $1$
and $2$ by $\nu_{1}$ and $\nu_{2}$, resulting in another TMSV with a smaller
amplitude

$\ket{\text{TMSV}_{\text{NA}}}\propto\sum_{n=0}^{\infty}{\gamma^{n}}(\nu_{1}\nu_{2})^{n}\ket{n}_{1}\ket{n}_{2}.$
(43)

Photon loss is then applied using beamsplitters with transmission coefficients
$\tau_{1}$ and $\tau_{2}$ and then traced over resulting in a mixed state of
the form:

$\hat{\rho}\propto\sum_{j,k=0}^{\infty}\ket{\psi_{jk}}\bra{\psi_{jk}}$ (44)

where the loss of $j$ photons from mode $1$ and $k$ photons from mode $2$
corresponds to the following (unnormalized) state:

$\ket{\psi_{jk}}=\sum_{n\geq j,k}c_{njk}\ket{n-j}_{1}\ket{n-k}_{2}$ (45)

with the coefficients:

$c_{njk}\equiv\sqrt{\binom{n}{j}\binom{n}{k}}(\gamma\nu_{1}\nu_{2})^{n}(\tau_{1})^{n-j}(\tau_{2})^{n-k}(i\sqrt{1-\tau_{1}^{2}})^{j}(i\sqrt{1-\tau_{2}^{2}})^{k}.$
(46)

After noiseless amplification with the filters $G_{N}(g_{l})$ in each mode
$l=1,2$, we modify these coefficients with two extra gain factors and
introduce a maximum number of photons $N$ in each mode:

$c^{\prime}_{njk}\equiv\begin{cases}c_{njk}g_{1}^{n-j}g_{2}^{n-k}&\text{if
}n-j\leq{N}\text{ and }n-k\leq N\\\ 0&\text{otherwise}\end{cases}$ (47)

The first term of the output state $\hat{\rho}$ corresponds to the case of no
photon loss $(j=k=0)$. The Fock state coefficients of this lossless term are:

$c^{\prime}_{n00}=(\gamma\nu_{1}\nu_{2}\tau_{1}\tau_{2}g_{1}g_{2})^{n}$ (48)

which corresponds to a pure TMSV state. To ensure these coefficients are the
same as for the initial state, we arrive at the following condition:

$(\nu_{1}\tau_{1}g_{1})(\nu_{2}\tau_{2}g_{2})=1,$ (49)

which is equivalent to the GHZ state balancing condition in the previous
section. Satisfying this condition, the output once again consists of the
original state plus added noise. As expected, if we implement NLS by setting
$g_{1}=1/(\nu_{1}\tau_{1})$ and $g_{2}=1/(\nu_{2}\tau_{2})$ and consider the
limits $\nu_{1,2}\rightarrow 0$, the coefficients scale as

$c^{(NLS)}_{njk}\propto\nu_{1}^{j}\nu_{2}^{k},$ (50)

and all terms with $j>0$ ($k>0$) will disappear as $\nu_{1}\rightarrow 0$
($\nu_{2}\rightarrow 0$). Hence, we conclude that the TMSV behaves similarly
to the GHZ states in that the application of NLS requires noiseless
attenuation and the mixing due to photon loss can be made arbitrarily small.

The fidelity for a general output state can be calculated as follows:

$\displaystyle F$
$\displaystyle=\frac{\sum_{j,k}|\langle\psi|\psi^{\prime}_{jk}\rangle|^{2}}{\sum_{j,k}|\psi^{\prime}_{jk}|^{2}}$
(51)
$\displaystyle=\frac{(1-\gamma^{2})\sum_{j=0}^{\infty}\left(\sum_{n=j}^{N+j}|c^{\prime}_{njj}|\gamma^{(n-j)}\right)^{2}}{\sum_{j,k=0}^{\infty}\sum_{n\geq
j,k}|c^{\prime}_{njk}|^{2}}$ (52)

where $\ket{\psi^{\prime}_{jk}}$ are defined as in Eq. 45 with the modified
coefficients $c^{\prime}_{njk}$ given in Eq. 47. The probability of success is
given by:

$\displaystyle P_{s}$
$\displaystyle=(g_{1}g_{2})^{-2N}(1-\gamma^{2})\sum_{j,k}|\psi^{\prime}_{jk}|^{2}.$
(53)

Following Eq. 50, all but the $j=k=0$ terms in the above expressions will
vanish in the limit of $\nu_{1,2}\rightarrow 0$, leaving only sums over the
coefficients $c^{\prime}_{n00}=\gamma^{n}$ with
${g_{l}=(\nu_{l}\tau_{l})^{-1}}$. Evaluating Eq. 52 in this limit confirms
that fidelity approaches unity for sufficiently large $N$.

Similar to the GHZ state, we compare the fidelity and probability of success
for two special cases. The first is the typical NLS protocol with each
${g_{l}=(\nu_{l}\tau_{l})^{-1}}$, which satisfies the balancing condition in
Eq. 49. The second case uses no noiseless attenuation $(\nu_{l}=1)$ and
defines the arbitrary constant $C_{TMSV}=g_{1}\tau_{1}=g_{2}\tau_{2}$, which
does increase state purity as $C_{TMSV}\rightarrow\infty$ but does not satisfy
the same balancing condition.

Figure 6: TMSV state fidelity (a) and probability of success (b) after NLS
with $(\nu=\frac{1}{g\tau})$ and without $(\nu=1)$ noiseless attenuation, as a
function of gain. TMSV amplitude is $\gamma=0.2$, truncated at $N=1$. Channel
transmittance $\tau^{2}=0.25$ for all modes. Note that the regime $g<1/\tau=2$
is unphysical for the case that includes noiseless attenuation.

The main results are shown in Fig. 6. The output fidelity and probability of
success are plotted for a TMSV input with amplitude $\gamma=0.2$, equal
channel losses of $\tau_{1}=\tau_{2}=0.5$, and a maximum photon number $N=1$
for noiseless amplification. Because of this truncation, the output state will
lack higher-order terms and thus fidelity can never reach unity in the limit
of high gain. This can be overcome to an extent by increasing $N$, but at the
expense of probability of success since $P_{s}\propto g^{-2N}$. Even so, Fig.
6a shows very high fidelity can be achieved when retaining only single-photon
terms, with $F\rightarrow 1-\gamma^{4}$ in the limit of infinite gain. This is
in stark contrast to the case where noiseless attenuation is omitted, in which
fidelity tends to a small finite value with marginally improved probability of
success. Noiseless amplification on its own simply weights the highest-number
Fock states above all others, so $\rho\rightarrow\ket{N}\bra{N}$ in the limit
of high gain. Purity increases but all entanglement is lost, and fidelity with
the original TMSV is minimal. In fact, fidelity only worsens with increased
gain, as the initial fidelity for a TMSV state experiencing channel losses
$\tau_{k}$ is already quite high without applying any noiseless attenuation or
amplification $(\nu=g=1)$. This is owed to the large vacuum term in the
initial TMSV, ensuring large overlap with a state sent through a pure loss
channel, despite the degraded entanglement. So while the full NLS protocol
only appears to provide a modest increase in fidelity, it is the only one of
the above methods which preserves the form and desired properties of the TMSV.

##  VII NOON states

The NOON states are a type of bipartite entangled state of $N$ photons where
all are either in one mode or the other. These states are of particular
interest in quantum metrology due to the scaling of interference effects, and
hence phase sensitivity, with photon number Dowling (2008); Lang and Caves
(2014). Beyond practical applications, NOON states are also of fundamental
interest since, in the limit of large $N$, the entanglement is between
macroscopic systems.

The NOON state is given by:

$\ket{\psi}=\frac{1}{\sqrt{2}}\left(\ket{n}_{1}\ket{0}_{2}+\ket{0}_{1}\ket{n}_{2}\right)$
(54)

To apply NLS, we first separately apply noiseless attenuation to each mode
with attenuation coefficients $\nu_{1}$ and $\nu_{2}$, resulting in

$\ket{\psi_{NA}}\propto\nu_{1}^{n}\ket{n}_{1}\ket{0}_{2}+\nu_{2}^{n}\ket{0}_{1}\ket{n}_{2}.$
(55)

Photon loss in each mode is now included through the use of beamsplitters with
amplitude transmittances $\tau_{1,2}$, resulting in

$\displaystyle\ket{\psi_{L}}\propto$
$\displaystyle\nu_{1}^{n}\sum_{j=0}^{n}\sqrt{\binom{n}{j}}\tau_{1}^{n-j}(i\sqrt{1-\tau_{1}^{2}})^{j}\ket{n-j}_{1}\ket{j}_{R}\ket{0}_{2}\ket{0}_{R^{\prime}}$
(56)
$\displaystyle+\nu_{2}^{n}\sum_{j=0}^{n}\sqrt{\binom{n}{j}}\tau_{2}^{n-j}(i\sqrt{1-\tau_{2}^{2}})^{j}\ket{0}_{1}\ket{0}_{R}\ket{n-j}_{2}\ket{j}_{R^{\prime}}.$
(57)

where $R$ and $R^{\prime}$ are the output ports of the photon-loss
beamsplitters in modes $1$ and $2$, respectively. We now complete the NLS
protocol by tracing out the $a$ and $b$ modes and applying noiseless
amplification, resulting in

$\displaystyle\hat{\rho}_{out}\propto$
$\displaystyle\left(\nu_{1}^{n}\tau_{1}^{n}g_{1}^{n}\ket{n}_{1}\ket{0}_{2}+\nu_{2}^{n}\tau_{2}^{n}g_{2}^{n}\ket{0}_{1}\ket{n}_{2}\right)\left(c.c.\right)$
(58)
$\displaystyle+\nu_{1}^{2n}\sum_{j=1}^{n}\binom{n}{j}(\tau_{1}g_{1})^{2(n-j)}(1-\tau_{1}^{2})^{j}\ket{n-j}_{1}\ket{0}_{2}\bra{n-j}_{1}\bra{0}_{2}$
(59)
$\displaystyle+\nu_{2}^{2n}\sum_{j=1}^{n}\binom{n}{j}(\tau_{2}g_{2})^{2(n-j)}(1-\tau_{2}^{2})^{j}\ket{0}_{1}\ket{n-j}_{2}\bra{0}_{1}\bra{n-j}_{2}.$
(60)

The first term, corresponding to zero photon loss, will be proportional to the
original NOON state if we satisfy the familiar balancing condition
$\nu_{1}\tau_{1}g_{1}=\nu_{2}\tau_{2}g_{2}=C_{N}$, analogous to the $W$ state
case (see Eq. 27).

The fidelity of the output state is given by:

$F=\frac{C_{N}^{2n}}{C_{N}^{2n}+\frac{1}{2}\sum_{l=1}^{2}\sum_{j=1}^{n}\binom{n}{j}C_{N}^{2(n-j)}\left[(1-\tau_{l}^{2})\nu_{j}^{2}\right]^{j}}$
(61)

with the corresponding probability of success:

$P_{s}=\left(\frac{1}{g_{1}g_{2}}\right)^{2n}\left(C_{N}^{2n}+\frac{1}{2}\sum_{l=1}^{2}\sum_{j=1}^{n}\binom{n}{j}C_{N}^{2(n-j)}\left[(1-\tau_{l}^{2})\nu_{j}^{2}\right]^{j}\right).$
(62)

Once again, we can suppress losses in multiple ways. We can set
$g_{k}=1/(\nu_{k}\tau_{k})$ for each mode as in the original NLS protocol
$(C_{N}=1)$, and each noise term will be weighted by a factor $\nu_{k}^{2j}$.
Alternatively, noiseless attenuation can be omitted with all $\nu_{k}=1$, and
noise terms will still be suppressed by a factor $C_{N}^{-2j}$ relative to the
pure NOON state term. Fig. 8(a) shows that when plotted as a function of gain,
both fidelity expressions approach unity in the high-gain limit, and in fact
are equivalent. However, noiseless attenuation is clearly detrimental since
high-fidelity transmission will be performed with a much lower probability of
success, as seen in Fig. 8(b). A more explicit comparison to the similarly-
behaved $W$ state can be seen in Appendix B.

Figure 7: $N=3$ NOON state fidelity and probability of success after NLS with
$(\nu=\frac{1}{g\tau})$ and without $(\nu=1)$ noiseless attenuation, as a
function of gain. Channel transmittance $\tau^{2}=0.25$ for all modes. Note
that the regime $g<1/\tau=2$ is unphysical for the case that includes
noiseless attenuation.

##  VIII On the necessity of Noiseless Attenuation

As we have seen, the naive application of the original NLS protocol to each
mode of an M-mode entangled state independently will result in the desired end
state. However, as noted in sections IV and VII, the noiseless attenuation is,
for some states, redundant and only reduces the overall probability of
success. Here we provide a criteria for when states do or do not require the
noiseless attenuation step of NLS. Intuitively, the result indicates that for
states that encode information in the vacuum, meaning one term in the
superposition includes no photons, the inability to amplify this term means
that noiseless attenuation is required to balance all coefficients and recover
the initial state.

We begin by returning to the general treatment of an M-mode NLS scheme and the
resulting state Eq. 7. Consider the form of each operator
$\hat{B}^{(k)}_{j_{k}}$ in the Fock basis (reprinted from Section II for ease
of reference):

$\hat{B}^{(k)}_{j_{k}}=\sum_{m=j_{k}}^{N}\sqrt{\binom{m}{j_{k}}}(1-\tau_{k}^{2})^{j_{k}/2}(\nu_{k}\tau_{k}g_{k})^{m-j_{k}}\ket{m-j_{k}}\bra{m}$
(63)

Following the original NLS protocol, we can set $\nu_{k}\tau_{k}g_{k}=1$ such
that in the case of zero photon loss $j_{k}=0$, each of the
$\hat{B}^{(k)}_{0}$ operators becomes the identity operator. This leaves all
input states unchanged, while also leaving the values of gain $g_{k}$ free to
increase and reduce the relative size of the noise terms. If we adjust the NLS
protocol by omitting all noiseless attenuation $(\nu_{k}=1)$, inspection of
Eq. 63 reveals that noise reduction is still possible: $j_{k}>0$ terms are
suppressed by a factor of $g_{k}^{-2j_{k}}$ relative to the signal,
disappearing in the limit $g_{k}\rightarrow\infty$ for all $k$.

However, we also require that increasing the gains past $g_{k}=1/\tau_{k}$
leaves the input state unchanged, which is now only possible for a specific
class of input states. Consider a pure M-mode input state of the general form:

$\ket{\psi}=\sum_{l=1}^{L}c_{l}\ket{n_{1}\cdots n_{M}}^{(l)}$ (64)

where each of the $L$ terms in the superposition has a unique set of
occupation numbers $\\{n_{k}^{(l)}\\}$ across $M$ modes. Applying the NLS
protocol to this state, the term corresponding to zero photon losses
$(j_{k}=0)$ is given by:

$\ket{\tilde{\psi}}\equiv\left(\prod_{k=1}^{M}{(\nu_{k}g_{k}\tau_{k})^{\hat{n}}}\right)\ket{\psi}$
(65)

where we have assumed the $N$-photon truncation for amplification is large
enough that it can be safely ignored.

For some states, we know it’s possible to make
$\ket{\tilde{\psi}}\propto\ket{\psi}$ while omitting all noiseless attenuation
$(\nu_{k}=1)$ and simply increasing the gain in every channel (e.g., sections
IV and VII). To maintain the structure of $|\psi\rangle$ in these cases, we
hence require that it is an eigenstate of the “lossless transmission” operator
$\prod_{k=1}^{M}{(g_{k}\tau_{k})^{\hat{n}}}$ with a constant eigenvalue which
can be factored out of every term in the superposition, for $l=1,...,L$:

$\prod_{k=1}^{M}(g_{k}\tau_{k})^{n_{k}^{(l)}}=\text{const.}$ (66)

This factoring condition for eigenstates can be equivalently expressed as a
set of $L$ linear equations:

$\sum_{k=1}^{M}n_{k}^{(l)}x_{k}=\text{const.}$ (67)

with $x_{k}\equiv\log(g_{k}\tau_{k})$. We require solutions with all
$x_{k}>0$, such that both sides of the equations can be multiplied by an
arbitrarily large constant and correspond to increasing gain in all modes.
This system of equations can also be expressed with a $L\times M$ matrix
multiplying the vector of $x_{k}$.

Note that $\ket{\psi}$ is an eigenstate of the operator
$\prod_{k=1}^{M}{(g_{k}\tau_{k})^{\hat{n}}}$ if and only if it is also an
eigenstate of the combined noiseless attenuation operators, which have the
same form: $\prod_{k=1}^{M}{(\nu^{\prime}_{k})^{\hat{n}}}$. In other words, if
we can choose a set of $\nu^{\prime}_{k}<1$ for all noiseless attenuators that
leaves the input state unchanged, then there also exists a set of amplifier
gains $g_{k}>1/\tau_{k}$ that can restore the state without a need for
noiseless attenuation. For example, we saw that noiseless attenuation had no
effect on the path-entangled single-photon state when $\nu_{1}=\nu_{2}$, or
similarly for the W state, when all $\nu_{k}=\nu$. For those states, we could
just set $\nu_{k}=1$ and $g_{k}=1/(\nu\tau_{k})$. This simple construction
also works more generally. We showed that typical NLS with
$g_{k}=1/(\nu_{k}\tau_{k})$ works for all input states, so if there is a set
of $\nu^{\prime}_{k}<1$ which leaves the input state unchanged, we can
equivalently replace the attenuators with open channels $\nu_{k}=1$ and keep
the noiseless amplifiers with $g_{k}=1/(\nu^{\prime}_{k}\tau_{k})$. It still
follows from Eq. 67 that we can continue to increase all gains while keeping
all modes appropriately balanced, simply by scaling all
$x_{k}=\log(g_{k}\tau_{k})$ by a constant factor.

For the W and NOON states, the values of $n_{k}$ can be arranged as a diagonal
matrix, and hence we can always find a positive solution. For these states,
the NLS protocol can be completed without noiseless attenuation. This result
also follows for other superpositions of a fixed number of photons $N$
distributed across $M$ modes. In contrast, any state with a vacuum term will
not have a solution, since one of the rows of this matrix will be the null
vector $(0,...,0)$. Similarly, states such as the TMSV with more superposition
terms than modes ($L>M$) will likely form an overdetermined set of equations
with no solution. For these states, noiseless attenuation is necessary for the
NLS protocol.

##  IX Discussion and conclusions

Here, we have extended the analysis of Mičuda et al. to various entangled
quantum states, including bipartite squeezed vacuum (TMSV), NOON states, and
M-partite W and GHZ states. Our results derive the specific NLS conditions
required to recover the initial state form supplemented by vacuum terms,
analogous to the results in the original single-mode treatment. In particular,
we found that for both GHZ and TMSV states NLS can be performed by applying
the single-mode NLS protocol to each mode of the entangled state
independently. However, in a departure from the single-mode results, we showed
that noiseless attenuation is not essential for suppressing vacuum terms for
both W and NOON states. This discovery underscores the nuanced role of
noiseless attenuation when combined with noiseless amplification and offers a
practical example of how GHZ and W state entanglement differ.

The role of noiseless attenuation and NLA in NLS is analogous to that of
filters in Procrustean filtering in that noiseless attenuation and NLS each
adjust the coefficients of terms in the superposition in order to recover the
form of the initial state. Similarly, previous work Ulanov  _et al._ (2015),
has shown that amplification alone can recover the form of the initial state
with additional vacuum noise terms (though, these noise terms cannot be
reduced to zero, as in NLS). What we found is that when a single term in the
superposition consists of only the vacuum, NLA alone cannot adjust the weight
of this term, as there is nothing to amplify, and hence, in general, for these
states (e.g., GHZ and TMSV) the additional degree of freedom to reduce the
magnitude of coefficients with noiseless attenuation is required in order to
recover the initial state form, but at the cost of a reduced probability of
success.

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##  Appendix A Simplified forms of expressions

We include in this appendix simplifications of the general fidelity and
probability of success expressions presented in the main text. In particular,
we consider the special case where all channels have the same parameters,
including those for noiselesss attenuation and amplification. While the
expressions here do not strictly include new information, the simplification
can often be dramatic and provide a more clear and intuitive picture of the
NLS protocol.

###  A.1 W state

Output fidelity for a $W$ state with $C_{W}=\nu_{l}\tau_{l}g_{l}$ for all
modes $l=1,...,M$:

$\displaystyle F=$
$\displaystyle\frac{\bra{W}\hat{\rho}\ket{W}}{\text{Tr}\\{\hat{\rho}\\}}=\frac{C_{W}^{2}}{C_{W}^{2}+\frac{1}{M}\sum_{l=1}^{M}\left(1-\tau_{l}^{2}\right)\nu_{l}^{2}}$
(68) $\displaystyle=$
$\displaystyle\frac{C_{W}^{2}}{C_{W}^{2}+(1-\tau^{2})\nu^{2}}\quad\text{with
all channels equal}.$

The corresponding probability of success:

$\displaystyle P_{s}=$
$\displaystyle\text{Tr}\\{\hat{\rho}\\}=\left(\prod_{k=1}^{M}g_{k}^{-2}\right)\left(C_{W}^{2}+\frac{1}{M}\sum_{l=1}^{M}\left(1-\tau_{l}^{2}\right)\nu_{l}^{2}\right)$
(69) $\displaystyle=$ $\displaystyle
g^{-2M}\left[C_{W}^{2}+(1-\tau^{2})\nu^{2}\right]\quad\text{with all channels
equal.}$

###  A.2 GHZ state

Output fidelity for a GHZ state with $g_{k}=1/(\nu_{k}\tau_{k})$ for each mode
$k=1,...,M$:

$\displaystyle F=$
$\displaystyle\frac{\bra{\text{GHZ}}\hat{\rho}\ket{\text{GHZ}}}{\text{Tr}\\{\hat{\rho}\\}}=\frac{1+\frac{1}{4}\prod_{k=1}^{M}(1-\tau_{k}^{2})\nu_{k}^{2}}{1+\frac{1}{2}\left(\sum_{\mathbf{j}\in
B^{M}\setminus{1^{M}}}\left[\prod_{k=1}^{M}\left((1-\tau_{k}^{2})\nu_{k}^{2}\right)^{\neg
j_{k}}\right]\right)}$ (70) $\displaystyle=$
$\displaystyle\frac{1+\frac{1}{4}[(1-\tau^{2})\nu^{2}]^{M}}{1+\frac{1}{2}\left(\sum_{\mathbf{j}\in
B^{M}\setminus{1^{M}}}\left[\prod_{k=1}^{M}\left((1-\tau^{2})\nu^{2}\right)^{\neg
j_{k}}\right]\right)}\quad\text{with all channels equal}$ (71)
$\displaystyle=$
$\displaystyle\frac{1+\frac{1}{4}[(1-\tau^{2})\nu^{2}]^{M}}{1+\frac{1}{2}\left[\sum_{j_{tot}=1}^{M}\binom{M}{j_{tot}}\left(\nu^{2}(1-\tau^{2})\right)^{j_{tot}}\right]}$
$\displaystyle=$
$\displaystyle\frac{1+\frac{1}{4}[(1-\tau^{2})\nu^{2}]^{M}}{1+\frac{1}{2}\left[\left(1+\nu^{2}(1-\tau^{2})\right)^{M}-1\right]}$

where for a binary string $\bm{j}=(j_{1},j_{2},j_{3},...,j_{M})$,
$j_{tot}=\sum_{k}{\neg j_{k}}$ and $j_{tot}$ corresponds to the total number
of lost photons. When summing over all the strings $\bm{j}$, there are
$\binom{M}{j_{tot}}$ different strings with the same number of lost photons
$j_{tot}$, so we can write it as a sum over $j_{tot}$. Note, the sum starts at
$j_{tot}=1$ instead of $0$ because string $1^{M}$ was previously factored out.

The corresponding probability of success:

$\displaystyle P_{s}=$
$\displaystyle\left(\prod_{k=1}^{M}g_{k}^{-2}\right)\left[1+\frac{1}{2}\left(\sum_{\mathbf{j}\in
B^{M}\setminus{1^{M}}}\left[\prod_{k=1}^{M}\left((1-\tau_{k}^{2})\nu_{k}^{2}\right)^{\neg
j_{k}}\right]\right)\right]$ (72) $\displaystyle=$ $\displaystyle
g^{-2M}\left[1+\frac{1}{2}\left[\left(1+\nu^{2}(1-\tau^{2})\right)^{M}-1\right]\right]\quad\text{with
all channels equal.}$

Without noiseless attenuation $(\nu_{k}=1)$ and setting
$C_{GHZ}=\tau_{k}g_{k}$ for $k=1,...M$, the fidelity is given by:

$\displaystyle F=$
$\displaystyle\frac{\bra{\text{GHZ}}\hat{\rho}\ket{\text{GHZ}}}{\text{Tr}\\{\hat{\rho}\\}}=\frac{\frac{(1+C_{\text{GHZ}}^{M})^{2}}{4}+\frac{1}{4}\prod_{k=1}^{M}(1-\tau_{k}^{2})}{\frac{1+C_{\text{GHZ}}^{2M}}{2}+\frac{1}{2}\left(\sum_{\mathbf{j}\in
B^{M}\setminus{1^{M}}}\left[\prod_{k=1}^{M}C_{\text{GHZ}}^{2j_{k}}\left(1-\tau_{k}^{2}\right)^{\neg
j_{k}}\right]\right)}$ (73) $\displaystyle=$
$\displaystyle\frac{\frac{(1+C_{\text{GHZ}}^{M})^{2}}{4}+\frac{1}{4}(1-\tau^{2})^{M}}{\frac{1+C_{\text{GHZ}}^{2M}}{2}+\frac{1}{2}\left[\left(C_{\text{GHZ}}^{2}+1-\tau^{2}\right)^{M}-C_{\text{GHZ}}^{2M}\right]}\quad\text{with
all channels equal}$

with a probability of success:

$\displaystyle P_{s}=$
$\displaystyle\left(\prod_{k=1}^{M}g_{k}^{-2}\right)\left[\frac{1+C_{\text{GHZ}}^{2M}}{2}+\frac{1}{2}\left(\sum_{\mathbf{j}\in
B^{M}\setminus{1^{M}}}\left[\prod_{k=1}^{M}C_{\text{GHZ}}^{j_{k}}\left(1-\tau_{k}^{2}\right)^{\neg
j_{k}}\right]\right)\right]$ (74) $\displaystyle=$
$\displaystyle\frac{g^{-2M}}{2}\left[1+\left(C_{\text{GHZ}}^{2}+1-\tau^{2}\right)^{M}\right]\quad\text{with
all channels equal.}$

###  A.3 NOON state

Output fidelity for a NOON state with $C_{N}=\nu_{l}\tau_{l}g_{l}$ for all
modes $l=1,...,M$:

$\displaystyle F=$
$\displaystyle\frac{C_{N}^{2n}}{C_{N}^{2n}+\frac{1}{2}\sum_{l=1}^{2}\sum_{j=1}^{n}\binom{n}{j}C_{N}^{2(n-j)}\left[(1-\tau_{l}^{2})\nu_{j}^{2}\right]^{j}}$
(75) $\displaystyle=$
$\displaystyle\frac{C_{N}^{2n}}{[C_{N}^{2}+\nu^{2}(1-\tau^{2})]^{n}}\quad\text{with
all channels equal}$

where the sum in the denominator has been simplified with the binomial
theorem. The probability of success is given by:

$\displaystyle P_{s}=$
$\displaystyle\left(\frac{1}{g_{1}g_{2}}\right)^{2n}\left(C_{N}^{2n}+\frac{1}{2}\sum_{l=1}^{2}\sum_{j=1}^{n}\binom{n}{j}C_{N}^{2(n-j)}\left[(1-\tau_{l}^{2})\nu_{j}^{2}\right]^{j}\right)$
(76) $\displaystyle=$ $\displaystyle
g^{-4n}\left[C_{N}^{2}+\nu^{2}(1-\tau^{2})\right]^{n}\quad\text{with all
channels equal.}$

##  Appendix B Comparing the fidelity and probability of success for different
states

Here we provide plots of the fidelity and probability of success for all four
types of states discussed in the paper. These plots show the same results as
Figs. 4-8 but are arranged for an easier comparison of how the NLS protocol
performs for each state.

Figure 8: Plots of (a) fidelity and (b) probability of success after NLS with
($\nu=\frac{1}{g\tau}$, solid) and without ($\nu=1$, dashed) noiseless
attenuation as a function of gain for all four states discussed in the paper.
The darker dotted lines show the initial fidelity of the GHZ state case (dark
red) and TMSV case (dark cyan) without applying the NLS protocol. Channel
transmittance $\tau^{2}=0.25$ for all modes. Note that the regime $g<1/\tau=2$
is unphysical for the case that includes noiseless attenuation.


(1.2) stays unchanged since the derivative in $t$ of order $n-1$ of the
solution is continuous at $t=0$ and consequently no terms of the kind of
supported at $\\{t=0\\}$ distributions emerge in the equation while
calculating the highest $t$-derivative of order $n$ \- completely similar to
the case $n=1$. We also cut the right hand side off to a finite in $x$
function and apply the smoothing to obtain the right hand side from
$C^{\infty}(R^{N}\times(-\infty,\infty))$, exactly as it was done in sections
9.4 and 10.2.

On the next step we obtain solvability and estimates of the solution to the
problem with a finite right hand side of the class
$C^{\infty}(R^{N}\times(-\infty,\infty))$. As it was in section 10.3, we apply
the Fourier transform with respect to all variables and obtain the
representation for the highest derivatives of the solution

$\frac{\widehat{\partial^{n}u_{m,\varepsilon}}}{\partial
t^{n}}(\xi,\xi_{0})=\frac{(i\xi_{0})^{n}}{(i\xi_{0})^{n}+{\sum_{k=1}^{r}}|\zeta_{k}|^{\sigma_{k}}}\widehat{f_{m,\varepsilon}}(\xi,\xi_{0})\equiv\widehat{m}_{0}(\xi,\xi_{0})\widehat{f_{m,\varepsilon}}(\xi,\xi_{0}),$
(14.1)
$\widehat{(-\Delta_{z_{i}})^{\frac{\sigma_{i}}{2}}u_{m,\varepsilon}}(\xi,\xi_{0})=\frac{|\zeta_{i}|^{\sigma_{i}}}{(i\xi_{0})^{n}+{\sum_{k=1}^{r}}|\zeta_{k}|^{\sigma_{k}}}\widehat{f_{m,\varepsilon}}(\xi,\xi_{0})\equiv\widehat{m}_{i}(\xi,\xi_{0})\widehat{f_{m,\varepsilon}}(\xi,\xi_{0}).$
(14.2)

Note that the right hand side $f(x,t)$ in the extended to the domain
$\\{t<0\\}$ problem is not continuous at $t=0$ in general since the derivative
in $t$ of order $n$ in the equation is not generally continuous at $t=0$.
Correspondingly, the smoothed right hand side $f_{m,\varepsilon}(x,t)$ in
(14.1) and in (14.2), similar to section 10.3, does not have uniformly bounded
in mollifying parameter $\varepsilon$ Hölder seminorm in $t$. Consequently,
similar to section 10.3, we apply to the multipliers
$\widehat{m}_{0}(\xi,\xi_{0})$ and $\widehat{m}_{i}(\xi,\xi_{0})$ not only
Theorem 8.1 to obtain estimate (10.26), but also Theorem 8.2, which permits to
obtain uniform in $\varepsilon$ estimate of the highest Hölder seminorms of
the solution in the space variables $x$. At that, since by assumption $n$ is
not equal to a number of the form $4j+2$, $j=0,1,...$, then $(i\xi_{0})^{n}$
is not equal to $-|\xi_{0}|^{n}$ for all nonzero $\xi_{0}$. Consequently, the
modulus of denominator in the expressions for $\widehat{m}_{0}(\xi,\xi_{0})$
and $\widehat{m}_{i}(\xi,\xi_{0})$ is separated from zero on the sets
$B_{\nu}$ from (8.9). Therefore, similar to section 10.3, it is not difficult
to verify conditions of theorems 8.1 and 8.2. Application of these theorems
leads, under definite smallness of the exponent $\alpha$, to the estimate of
the highest seminorms of the solution $u_{m,\varepsilon}(x,t)$ for the problem
with the smoothed finite right hand side $f_{m,\varepsilon}(x,t)$. In
particular, for the highest derivative in $t$, based also on Lemma 10.3, we
obtain the estimate in the whole space $R^{N+1}$

$\left\langle\frac{\partial^{n}u_{m,\varepsilon}(x,t)}{\partial
t^{n}}\right\rangle_{t,R^{N+1}}^{(n\alpha)}\leq
C(\overline{\sigma})\left\langle
f_{m,\varepsilon}(x,t)\right\rangle_{R^{N+1}}^{(\overline{\sigma}\alpha,n\alpha)}\leq
C(\overline{\sigma},\alpha)\varepsilon^{-n\alpha}|f|_{\overline{R_{T}^{N}}}^{(\overline{\sigma}\alpha,n\alpha)},$
(14.3)

which is an analog of (10.26). And similar to (10.30), based also on the
equation, we obtain the uniform in $\varepsilon$ estimate in the domain
$\overline{R_{T}^{N}}$

$\left\langle
u_{m,\varepsilon}\right\rangle_{\overline{R_{T}^{N}}}^{(\overline{\sigma}+\overline{\sigma}\alpha,n+n\alpha)}\leq
C(\alpha,\overline{\sigma},\\{N_{k}\\})\left|f(x,t)\right|_{\overline{R_{T}^{N}}}^{(\overline{\sigma}\alpha,n\alpha)},\quad
m=1,2,...,\varepsilon\in(0,1).$ (14.4)

To obtain an analog of (10.32) and (10.33) we first make use of (14.3) to
obtain

$\left|\frac{\partial^{n}u_{m,\varepsilon}(x,t)}{\partial
t^{n}}\right|=\left|\frac{\partial^{n}u_{m,\varepsilon}(x,t)}{\partial
t^{n}}-\frac{\partial^{n}u_{m,\varepsilon}(x,-2\varepsilon)}{\partial
t^{n}}\right|\leq$ $\leq\left\langle
u_{m,\varepsilon}\right\rangle_{t,R^{N+1}}^{(n+n\alpha)}|-2\varepsilon|^{n\alpha}\leq
C(n,\overline{\sigma},\alpha)|f|_{\overline{R_{T}^{N}}}^{(\overline{\sigma}\alpha,n\alpha)},\quad
t\in[-2\varepsilon,0].$ (14.5)

And then, analogously to (10.32),

$\left|u_{m,\varepsilon}(x,0)\right|\leq
C(n){\int\limits_{-2\varepsilon}^{0}}\left|\frac{\partial^{n}u_{m,\varepsilon}(x,t)}{\partial
t^{n}}\right|t^{n-1}dt\leq
C(n,\overline{\sigma},\alpha)|f|_{\overline{R_{T}^{N}}}^{(\overline{\sigma}\alpha,\alpha)}\varepsilon^{n}.$
(14.6)

Eventually, analogously to the obtaining (10.33), we successively obtain on
$\overline{R_{T}^{N}}$

$\left|\frac{\partial^{n}u_{m,\varepsilon}(x,t)}{\partial
t^{n}}\right|\leq\left|\frac{\partial^{n}u_{m,\varepsilon}(x,0)}{\partial
t^{n}}\right|_{R^{N}}^{(0)}+\left\langle\frac{\partial^{n}u_{m,\varepsilon}(x,t)}{\partial
t^{n}}\right\rangle_{t,\overline{R_{T}^{N}}}^{(n\alpha)}t^{n\alpha}\leq$ $\leq
C(n,\overline{\sigma},\alpha)|f|_{\overline{R_{T}^{N}}}^{(\overline{\sigma}\alpha,n\alpha)}\left(1+T^{n\alpha}\right)$

and consequently

$\left|u_{m,\varepsilon}(x,t)\right|\leq\left|u_{m,\varepsilon}(x,0)\right|+{C(n)\int\limits_{0}^{t}}\left|\frac{\partial^{n}u_{m,\varepsilon}(x,\tau)}{\partial\tau^{n}}\right|(t-\tau)^{n-1}d\tau\leq$
$\leq
C(n,\overline{\sigma},\alpha)|f|_{\overline{R_{T}^{N}}}^{(\overline{\sigma}\alpha,n\alpha)}(\varepsilon^{n}+t^{n}\left(1+T^{n\alpha}\right)),$

that is

$\left|u_{m,\varepsilon}(x,t)\right|_{\overline{R_{T}^{N}}}^{(0)}\leq
C(n,\overline{\sigma},\alpha)|f|_{\overline{R_{T}^{N}}}^{(\overline{\sigma}\alpha,n\alpha)}(1+T^{n+n\alpha}).$
(14.7)

Further, completely similar to section 10.3 and analogously to reasoning of
section 9.7 at the proving of Proposition 9.2, based on estimates (14.4) and
(14.7), we make the transition to the limit on the parameters of smoothing and
cutting-off under $\varepsilon\rightarrow 0$, $m\rightarrow\infty$ on the set
$\overline{R_{T}^{N}}$ in the sequence of the problems with smooth finite
data. This results in a solution $u(x,t)$ of the original problem with the
estimates

$\left\langle
u\right\rangle_{\overline{R_{T}^{N}}}^{(\overline{\sigma}(1+\alpha),n+n\alpha)}\leq
C(\overline{\sigma},n,\alpha)\left(|f|_{\overline{R_{\infty}^{N}}}^{(\overline{\sigma}\alpha,n\alpha)}+{\sum\limits_{i=0}^{n-1}}|u_{i}|_{R^{N}}^{(\overline{\sigma}(1+\alpha)-\frac{i}{n}\overline{\sigma})}\right),$
(14.8) $\left|u\right|_{\overline{R_{\widetilde{T}}^{N}}}^{(0)}\leq
C(\overline{\sigma},\alpha,n)\left(|f|_{\overline{R_{\infty}^{N}}}^{(\overline{\sigma}\alpha,n\alpha)}+{\sum\limits_{i=0}^{n-1}}|u_{i}|_{R^{N}}^{(\overline{\sigma}(1+\alpha)-\frac{i}{n}\overline{\sigma})}\right)\times$
(14.9)
$\times(1+\widetilde{T}^{n+\alpha})+|u_{0}|_{R^{N}}^{(0)},\quad\widetilde{T}\leq
T,$

where $T$ can be equal to $\infty$ if the original function $f(x,t)$ was
defined on $R^{N}\times[0,\infty)$ \- analogously to Proposition 10.5 in
section 10.4.

The uniqueness of the solution is obtained completely similar to section 10.4
by the extension of the solution and applying the Fourier transform in the
space of distributions, which completes the proof of the solvability and the
estimates for problem (1.2), (1.3) under a sufficiently small $\alpha>0$.

On the last step, repeating verbatim first reasonings from section 10.5 and
then from section 11, we prove the solvability and the estimates of problem
(1.2), (1.3) for an arbitrary positive $\alpha$.

Thus, as a result, we obtain the existence and the boundedness of the inverse
operator $\emph{L}^{\emph{-1}}$ to the operator L, which completes the proof
of Theorem 2.1.

###  14.2 The proof of Theorem 2.3.

In the case $\theta\in(0,1)$ this theorem, evidently, coincides with Theorem
12.3. In the case of an arbitrary noninteger $\theta>0$ the proof step by step
follows the proofs of Theorems 9.1 and 12.3 with some minor obvious changes.
We will just highlight some of them.

Firstly, the reducing of the problem to zero initial data is made not just by
substraction of the initial function similar to section 9.1, but by
substraction of a function $w(x,t)$ from the appropriate space, which
satisfies the initial conditions. Such function was constructed in Theorem
13.1 for an arbitrary noninteger $\theta>0$ with the restriction
$\\{\theta\\}+\theta\alpha>1$ on the exponent $\alpha$, and also in Theorem
13.2 for a noninteger $\theta\in(0,2)$ and for an arbitrary $\alpha>0$.

Further, after the extension of the unknown function by zero to the domain
$\\{t<0\\}$, in view of zero initial data, the derivatives in $t$ up to the
order $[\theta]$ stays continuous, and consequently, the same is valid for the
fractional derivative $D_{\ast}^{\theta}u(x,t)$ itself. Therefore, after
applying, similar to section 9.5, the Fourier transform to obtain the
representation of the solution in terms of it’s Fourier image we can use
Theorem 8.1. At that the representations for the highest derivatives have the
form

$\widehat{D_{\ast}^{\theta}u(x,t)u_{m,\varepsilon}}(\xi,\xi_{0})=\frac{(i\xi_{0})^{\theta}}{(i\xi_{0})^{\theta}+{\sum_{k=1}^{r}}|\zeta_{k}|^{\sigma_{k}}}\widehat{f_{m,\varepsilon}}(\xi,\xi_{0})\equiv\widehat{m}_{0}(\xi,\xi_{0})\widehat{f_{m,\varepsilon}}(\xi,\xi_{0}),$
(14.10)
$\widehat{(-\Delta_{z_{i}})^{\frac{\sigma_{i}}{2}}u_{m,\varepsilon}}(\xi,\xi_{0})=\frac{|\zeta_{i}|^{\sigma_{i}}}{(i\xi_{0})^{\theta}+{\sum_{k=1}^{r}}|\zeta_{k}|^{\sigma_{k}}}\widehat{f_{m,\varepsilon}}(\xi,\xi_{0})\equiv\widehat{m}_{i}(\xi,\xi_{0})\widehat{f_{m,\varepsilon}}(\xi,\xi_{0}).$
(14.11)

Since $\theta$ is a noninteger, then in the first term of the denominator
$(i\xi_{0})^{\theta}=(\pm i)^{\theta}|\xi_{0}|^{\theta}$ the numerical
coefficient $(\pm i)^{\theta}$ has a non-zero imaginary part. Therefore on the
annulus $B_{\nu}$ from (8.9) the denominator of the multipliers
$\widehat{m}_{0}(\xi,\xi_{0})$ and $\widehat{m}_{i}(\xi,\xi_{0})$ is strictly
separated from zero. This permits to verify the conditions of Theorem 8.1 \-
completely similar to section 9.5.

Highlight, finally, one more very simple alteration in the proof, caused by
the high order of the derivative in $t$. To obtain the corresponding analog of
Proposition 9.5 on the extension of the solution to the whole time interval
$t\in(0,\infty)$ one should use the Taylor polynomial of degree $[\theta]$ on
the role of the function $\widetilde{u}(x,t)$ from (9.77) and,
correspondingly, to use Lemma 9.4, which is proved for an arbitrary noninteger
$\theta>0$.

Besides the pointed above simplest alterations all the others steps in the
proof of Theorem 2.3 coincide with the corresponding steps in the proofs first
of Theorem 9.1 and then of Theorem 12.3, which leads to the assertion of
Theorem 2.3.

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