# Parity-Swap State Comparison Amplifier for Schrödinger Cat States

Gioan Tatsi, Luca Mazzarella and John Jeffers  Department of Physics,
University of Strathclyde, John Anderson Building, 107 Rottenrow, Glasgow G4
0NG, U.K.

(August 29, 2024)

###### Abstract

We propose a postselecting parity-swap amplifier for Schrödinger cat states
that does not require the amplified state to be known a priori. The device is
based on a previously-implemented state comparison amplifier for coherent
states. It consumes only Gaussian resource states, which provides an advantage
over some cat state amplifiers. It requires simple Geiger-mode photodetectors
and works with high fidelity and approximately twofold gain.

##  I Introduction

Schrödinger cat states, superpositions of two coherent states with coherent
amplitudes of the same magnitude but different phases, have been widely
studied for the significant role they could play in quantum information
Cochrane ; Nielsen ; Ralph1 ; Gilchrist ; Tipsmark , computation Lund ; Lobino
and in fundamental tests Jeong ; Strobinska ; Lee ; McKeon  as resource
states. For example Ralph et al. Ralph1  showed that cat states can be used to
implement qubit gates in an all optical quantum computation scheme, where the
logical qubits are encoded in the phase of the coherent state complex
amplitude and Jeong Jeong  considered the possibility of testing the Bell
inequalities using cat-like states as resources.

The nongaussian character of cat states renders them both challenging to
engineer and fragile. Traditionally they have been generated using a
combination of linear optics and measurement postselection techniques Gerrits
; Wakui ; Ourjoumtsev ; Nielsen2 ; Takeoka ; Marek ; Lee2 ; Kim . One simple
implementation relies on subtracting a photon from a squeezed vacuum state.
This generates a state close to an odd cat state, but only for small coherent
amplitudes ($\alpha\leq 1.2$)Laghaout . However, some quantum computation
schemes require as resources cat states of higher coherent amplitude Ralph1 ;
Lund . One way of increasing the amplitude to meet this need relies on the
process of cat state “breeding”Lund2 ; Takeoka2 ; Laghaout ; Synchev , whereby
small amplitude “kittens” are mixed coherently at a beam splitter and a
postselection measurement projects one of the two outputs into an increased
amplitude state based on the constructive interference of the two input modes.
This can be done recursively and, in principle, it can produce cat states of
arbitrary size. There is, however, an unfavourable scaling of both the output
quality and the success probability with increased “breeding seasons”. Also,
the smaller amplitude nongaussian cat states are themselves consumed at each
recursion to make larger amplitude states. Given that cat states are expensive
quantum resources this is not a particularly desirable state of affairs.

Recently, there has been a renewed interest in the amplification of cat states
Evgeny ; Lidal . Here we introduce an optical amplifier for Schrödinger cat
states that works without requiring prior knowledge of the input state, that
relies on Gaussian resources, beamsplitters and Geiger mode detectors and
offers a reasonably high gain and high output quality for a range of low
amplitude cat states. The proposed scheme is inspired by the so called state
comparison amplifier (SCAMP) Electra ; Ross ; Ross2 ; Ross3 ; David ; Luca  a
non-deterministic amplifier that has been shown to achieve high gain and high
output quality for discrete sets of input coherent states of light.

The article is organised as follows: in Section II we introduce the
Schrödinger cat states, in particular the even and odd cats. Section III
reviews the state comparison amplifier for coherent states. In Section IV we
introduce the Parity-Swap Amplifier for cat states and we benchmark the
performance of the scheme based on the fidelity and the success probability.

##  II Schrödinger Cat States

Optical coherent states are superpositions of all photon numbers,

$\displaystyle|\alpha\rangle=e^{-|\alpha|^{2}/2}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle,$
(1)

where $\alpha$ is a complex number known as the coherent amplitude Glauber .
The states therefore have photon number probabilities that satisfy a Poisson
distribution with mean $|\alpha|^{2}$. Different coherent states are not
orthogonal and so cannot be distinguished perfectly and deterministically by
any measurement . They approach orthogonality if their coherent amplitudes are
sufficiently far apart and the states can be distinguished in this limit.
Coherent states of low mean photon number naturally do not have very different
amplitudes and their indistinguishability is the only quantum property that
can be measured. For this reason coherent states are often described as the
most classical of quantum states.

Optical Schrödinger cat states are superpositions of two coherent states with
coherent amplitudes of the same magnitude but different phases, typically
opposite,

$\displaystyle|\alpha_{\theta}\rangle$
$\displaystyle=\frac{1}{\sqrt{2+2\cos{\theta}e^{-2|\alpha|^{2}}}}(|+\alpha\rangle+e^{i\theta}|-\alpha\rangle).$
(2)

For high values of the mean photon number $|\alpha|^{2}$ cat states are
superpositions of states that can be distinguished macroscopically. This is in
analogy with Schrödinger’s original gedanken experiment, which was introduced
Schrodinger  in order to advocate the difficulties in applying quantum
mechanics to the macroscopic world.

Typically we set $\theta=\\{0,\pi\\}$, obtaining the so called even and odd
cat states

$\displaystyle|\alpha_{\theta}\rangle$
$\displaystyle=\frac{1}{\sqrt{2+2\cos{\theta}e^{-2|\alpha|^{2}}}}(|\alpha\rangle+e^{i\theta}|-\alpha\rangle)$
$\displaystyle=\frac{e^{\frac{-|\alpha|^{2}}{2}}}{\sqrt{2+2\cos{\theta}e^{-2|\alpha|^{2}}}}\sum_{n=0}^{\infty}\frac{\alpha^{n}+e^{i\theta}(-\alpha)^{n}}{\sqrt{n!}}|n\rangle$
$\displaystyle=\begin{cases}\frac{1}{\sqrt{\cosh|\alpha|^{2}}}\sum_{n=0}^{\infty}\frac{\alpha^{2n}}{\sqrt{(2n)!}}|2n\rangle&\theta=0\\\
\frac{1}{\sqrt{\sinh|\alpha|^{2}}}\sum_{n=0}^{\infty}\frac{\alpha^{2n+1}}{\sqrt{(2n+1)!}}|2n+1\rangle&\theta=\pi\end{cases},$
(3)

so the even cat state $|\alpha_{0}\rangle=|\alpha_{+}\rangle$ contains only
even photon numbers and conversely the odd state
$|\alpha_{\pi}\rangle=|\alpha_{-}\rangle$ only has odd photon numbers. The
even and odd states are mutually orthogonal and, despite the fact that they
are superpostions of almost classical states, the gaps in the photon number
distribution are a signature of nonclassicality, whatever the value of
$|\alpha|$. They are also nongaussian, as attested by the negativity of their
Wigner function, another widely accepted measure of nonclassicality Hillery .

##  III State Comparison Amplifier

As previously mentioned, the state comparison amplifier (SCAMP) is a
nondeterministic amplifier for coherent states that works with high gain,
provides high-quality output and requires only classical resources. The
backbone of the amplifier (Fig. 1) is the mature technique of state comparison
Erika , which has been used in the setting of multiple phase encoding quantum
receivers Kennedy ; Dolinar ; Bondurant . In our scheme the coherent state to
be amplified is drawn uniformly at random from a discrete set, say
$\\{|+\alpha\rangle,|-\alpha\rangle\\}$ and is mixed with a guess coherent
state drawn from the set $\\{|+\beta\rangle,|-\beta\rangle\\}$ on a
beamsplitter with transmission and reflection coefficients $t_{1}$ and $r_{1}$
respectively. Here we assume that the reflection and transmission coefficients
are real, so the light incident on the lower input arm picks up a $\pi$ phase
shift upon reflection. The beamsplitter relates the output mode annihilation
operators to those for the input mode as follows

$\displaystyle\begin{pmatrix}\hat{a}_{out}\\\
\hat{b}_{out}\end{pmatrix}=\hat{U}_{BS}\begin{pmatrix}\hat{a}_{in}\\\
\hat{b}_{in}\end{pmatrix}$ (4)

where

$\hat{U}_{BS}=\begin{pmatrix}t&-r\\\ r&t\end{pmatrix}$ (5)

is the beamsplitter transformation matrix with real transmission and
reflection coefficients.

Then if the input states chosen are $|\alpha\rangle$ and $|\beta\rangle$ the
state after the beamsplitter is

$\hat{U}_{BS}|\alpha,\beta\rangle\rightarrow|t_{1}\alpha-
r_{1}\beta,t_{1}\beta+r_{1}\alpha\rangle$ (6)

Figure 1:  State Comparison Amplifier comprised of a comparison stage and a
subtraction stage.

From eq. 6 it is not difficult to see that the amplitude in the mode in the
detector arm vanishes if $t_{1}\alpha=r_{1}\beta$, i.e when the guess state
amplitude is $\beta=\frac{t_{1}\alpha}{r_{1}}$. The detector cannot fire. We
choose this amplitude for $\beta$ and use the non-firing of the detector as an
indication of the success of the amplification process and accept the output
whenever it occurs.

If, however, the input and guess states are oppositely-phased, say $\alpha$
and $-\frac{t_{1}\alpha}{r_{1}}$, light leaks into the detector arm and the
detector can fire, in which case the output state is rejected. The detector
may not fire, however, because the coherent state is not orthogonal to the
vacuum,
$\langle\alpha|0\rangle=\langle-\alpha|0\rangle=e^{-\frac{|\alpha|^{2}}{2}}$.
The detector not firing is therefore an imperfect indication that the correct
guess state has been chosen. This can limit how well the output state mimics
to the nominal amplified output. Conditioned on the detector not registering
an event, the output of SCAMP after the comparison state is a mixture of the
nominal correctly amplified state $|g\alpha\rangle=|\alpha/r_{1}\rangle$ and
another, lower amplitude coherent state, with their respective probabilities
biased towards the amplified state.

We use the quantum fidelity, F, of the output state to the nominal amplified
state as a measure of the output quality Barnett2 . As one of the states is
pure, here the fidelity is given by

$\displaystyle\mbox{F}=$ $\displaystyle\mathrm{Tr}[|g\alpha\rangle\langle
g\alpha|\rho_{\mbox{\tiny{out}}}]$ (7) $\displaystyle=$
$\displaystyle\int_{\mathbb{C}}\frac{d^{2}\xi}{\pi}\chi_{g\alpha}(-\xi)\chi_{out}(+\xi)$
(8)

where in the second line the fidelity is written in terms of the
characteristic functions of the nominal and actual output states. The
symmetrically ordered characteristic function of a general state $\rho$ is
defined by

$\displaystyle\chi_{\rho}(\xi)=\mathrm{Tr}[\rho\hat{D}(\xi)]$ (9)

in which $\xi$ is a complex parameter,
$\hat{D}(\xi)=\exp(\xi\hat{a}^{\dagger}-\xi^{*}\hat{a})$ is the displacement
operator Barnett3 . The characteristic function is in one-to-one
correspondence with the density operator as it contains all of the information
necessary to reconstruct the state. We employ this formalism throughout the
paper leaving calculational details for the Appendix.

The fidelity can be increased after the comparison stage by using another
mature technique known as photon subtraction Ban , a filtration process that
is implemented by allowing the output from the comparison beamsplitter to fall
on a second highly transmitting ($t_{2}\approx 1$) beamsplitter, with vacuum
input in the second mode. The reflected mode contains a Geiger mode detector
and this time the output is accepted whenever this second detector fires, an
event signifying that the signal contained at least one photon. The higher the
mean photon number of the input the more likely the detector is to fire, so
this filters out states with lower mean photon numbers. The improvement in the
output quality comes at the cost of a lower success probability of the overall
scheme. If the comparison beamsplitter is 50:50 the incorrect output of the
comparison stage is the vacuum, so all of the incorrect output is filtered by
subtraction leaving only the amplified state $|\pm
g\alpha\rangle=|\pm\sqrt{2}\alpha\rangle$, a perfect twofold gain.

##  IV SCAMP for Schrödinger Cat States

Motivated by the high gain and fidelity of the SCAMP for sets of individual
coherent states, we investigate a modified SCAMP system for the amplification
of cat states. In the first subsection we will introduce an updated SCAMP
scheme for the amplification of cat states which only utilises Gaussian
resources, beamsplitters and Geiger-mode detectors. In the second we will
present the results.

###  IV.1 Amplification Scheme

The amplification scheme, as one can see in Fig. 2 is an updated SCAMP in
which the input state is a cat state chosen uniformly at random from the set
of states $\\{|\alpha_{+}\rangle,|\alpha_{-}\rangle\\}$ while the guess state
this time is the squeezed vacuum $|\zeta\rangle$ state given by:

$\displaystyle|\zeta\rangle=\sqrt{\operatorname{sech}{s}}\sum_{m=0}^{\infty}\frac{\sqrt{(2m!)}}{m!}\left(-\frac{1}{2}\exp{[i\theta]}\tanh{s}\right)^{m}|2m\rangle$
(10)

where $\zeta=e^{i\theta}s$, $0\leq s\leq\infty$ and $0\leq\theta\leq 2\pi$.

Figure 2:  State Comparison Amplifier for Cat States comprising of a
comparison stage ($50:50$ beamsplitter) and a subtraction stage.

Note that the “guess” state is always $|\zeta\rangle$ irrespective of the
input state. This particular choice is motivated by the fact that the squeezed
vacuum state $|\zeta\rangle$, similarly to an even cat state, is a
superposition of even photon numbers only, and has a high overlap with
$|\alpha_{+}\rangle$, given by

$\displaystyle|\langle\alpha_{+}|\zeta\rangle|^{2}=\frac{\exp{\big{[}-(\alpha^{2}+\alpha^{*2})\tanh{s}/2\big{]}}}{\cosh{s}\cosh{|\alpha|^{2}}}$
(11)

for values of coherent amplitude up to $\alpha\approx 1$ (See Fig.3 (b)).

On the other hand, the overlap of $|\zeta\rangle$ with $|\alpha_{-}\rangle$ is
zero as the latter is a state that contains only superpositions of odd photon
numbers in the photon number basis. One perhaps would expect that such a
mixing of odd and even photon numbers in a beamsplitter would cause the
quality of the output state to deteriorate but, as we show in the appendix, in
this case such a mixing process, in conjunction with postselection in the
detector arm on no counts, acts as a quantum channel that evolves the input
states into squeezed states at the output. In Fig. 3 we plot the squeezing
magnitudes that maximise the fidelity of $|\alpha_{+}\rangle$ to
$|\zeta\rangle$ for different values of coherent amplitude $\alpha$ and also
the fidelity of $|\zeta\rangle$ with $|\alpha_{+}\rangle$. The squeezing
values that maximise the fidelity for a given coherent amplitude admit a
simple expression when we consider the coherent amplitude $\alpha$ to be real,

$\displaystyle s=-\frac{\sinh^{-1}(2\alpha^{2})}{2}.$ (12)

(a)

(b) Figure 3:  (a) Optimal squeezing in dB as a function of input coherent
amplitude $\alpha$, ($s_{db}=-10\mbox{log}[e^{2s}]$) . The inset plot shows
the absolute value of the optimal squeezing in terms of the squeezing
parameter.
(b) Corresponding maximum fidelity as a function of input $\alpha$.

The amplification protocol then proceeds as follows

  1. 1.

The input cat state to be amplified and the squeezed vacuum state are mixed at
a $50:50$ beamsplitter and a Geiger mode detector monitors the presence or
absence of light in one of the output modes. If no event is registered, the
output of this comparison stage is passed to the next stage.

  2. 2.

The output mode impinges on a second highly transmissive beamsplitter whose
other input is the vacuum state. If the second detector registers a click the
amplification is considered successful and the output is accepted. Otherwise
it is discarded.

The condition for successful amplification then can be formally given by

$P_{S}=P(D_{1}=\text{\char 55},D_{2}=\checkmark)$ (13)

which is the joint probability of the second detector clicking and the first
detector not registering a click. We show in the appendix that the comparison
stage acts as a quantum channel that evolves input states to squeezed states.
In the case of input cat states the output is squeezed cat states,i.e states
of the form $\hat{S}(\zeta)|\alpha_{\pm}\rangle$ where $\hat{S}(\zeta)$ is the
squeezing operator. The photon subtraction stage is needed to amplify the cat
states. The details of the computation have been left for the appendix.

Fig. 3 shows that the amount of squeezing required to optimise the fidelity
between the output and an ideal cat state for a wide range of input cat state
sizes is moderate. For example, if $|\alpha|^{2}=1$ the amount of squeezing
required is 6 dB - a factor of 4. Even for $|\alpha|^{2}=4$ a squeezing of 12
db ($|s|\leq 1.4$) is required, which has been experimentally generated in a
doubly resonant, type I optical parametric amplifier (OPA) operated below
threshold Vahlbruch .

###  IV.2 Results

Here we present the results of fidelity, gain and probability of success to
benchmark the amplifier. We provide the derivation of these results in the
Appendix. We have assumed that the coherent amplitudes of both the output
states and the nominal states are real, without loss of generality.

The performance of the amplifier is benchmarked using the fidelity F between
the output state and an ideal amplified cat state
$\\{|g\alpha_{+}\rangle,|g\alpha_{-}\rangle\\}$. We note that photon
subtraction changes the parity of the cat state Dakna . To see why that is the
case, consider applying the annihilation operator, $\hat{a}$, to the even cat
state, $|\alpha_{+}\rangle\sim|+\alpha\rangle+|-\alpha\rangle$; coherent
states are eigenstates of $\hat{a}$ satisfying
$|\alpha\rangle=\alpha|\alpha\rangle$ and thus we get

$\displaystyle\hat{a}|\alpha_{+}\rangle\sim\hat{a}|+\alpha\rangle+\hat{a}|-\alpha\rangle=\alpha(|+\alpha\rangle-|-\alpha\rangle)$
(14)

The right hand side of the equation is an (un-normalised) odd cat state
$|\alpha_{-}\rangle\sim|+\alpha\rangle-|-\alpha\rangle$. So for example if we
start with an even cat state, after photon subtraction we obtain an odd cat
state and this subtraction occurs in the limit $t_{2}\approx 1$. The
application of $\hat{a}$ to states of even photon number produces states of
odd photon number and vice versa.

In Fig. 4-6 we summarise results for the gain, maximum fidelity and
probability of success for the output state after the amplification process.
In the following we have assumed that the dark counts in the detection process
are negligible (they can be made so in pulsed systems by time filtering around
the pulse centre) and we have considered two scenarios where the quantum
efficiencies for the two detectors are both ideal ($\eta_{1}=\eta_{2}=1$) or
both detectors are $80\%$ efficient($\eta_{1}=\eta_{2}=0.8$).

(a)

(b) Figure 4: (a) Optimal gain g as a function of the size $\alpha$ of the
input even cat state. (b) Same plot for an odd cat state. The green line
indicates $\sqrt{2}$ gain.

Fig. 4 is a plot of the amplitude gain for different values of input cat state
size. The curves shown are for ideal detector efficiency as quantum efficiency
does not affect the gain of the amplifier. The gain does depend on the
transmission coefficient of the second beamsplitter as this quantifies the
quality of the photon subtraction. A higher transmission coefficient leads to
a higher gain as the photon subtraction stage is more faithfully approximated
as $t_{2}\approx 1$; this comes at the expense of the success probability as
in this limit there is hardly any light reflected to trigger the second
detector. One can see that there is an approximately twofold intensity gain
for both input cat states and for all values of coherent amplitude considered
here. This may seem counter-intuitive for the case where the input is an odd
cat state as the state has vanishing overlap with the squeezed vacuum state,
but we should remark that the gain in this case, contrary to the original
proposal of SCAMP, is not provided by the nullification of the signal in one
output mode after the comparison stage but is a combination of effects from
both the comparison and the photon subtraction stages (see next section for
details).

(a)

(b) Figure 5: (a) Optimal fidelity as a function of the size $\alpha$ of the
input even cat state.(b) Same plot for an odd cat state.

(a)

(b) Figure 6: (a) Corresponding success probability as a function of the size
$\alpha$ of the input even cat state. (b) Same plot for an odd cat state.

The fidelity (Fig. 5) depends both on the transmission coefficient and the
quantum efficiency of the detectors. The higher both quantities are the better
the fidelity of the output with the corresponding ideal output cat state. The
fidelity also depends on the input cat state size and it is near unity for
small values of the input coherent amplitude $\alpha$. It is greater than 80 %
for input $\alpha$ up to $\alpha\approx 1.5$ which corresponds to an output
cat state of amplitude $\beta=1.95$ if the transmission coefficient is
$t_{2}=\sqrt{0.95}$ and the detector efficiencies are $\eta=0.8$.

Not surprisingly, the lower transmission coefficient of the second
beamsplitter (higher reflection coefficient) increases the probability of
success as the probability of the second detector clicking depends on the
amplitude of the reflected light that impinges the detector. For the same
reason the success probability increases with increasing cat input amplitude.
Overall, the amplifier works with a high fidelity and approximately two-fold
intensity gain for a range of input cat state sizes up to $\alpha\approx 1.5$
corresponding to an output cat state of size $\beta=1.95$. Considering, the
most realistic implementation where ($t_{2}=\sqrt{0.95},\eta=0.8$) and keeping
in mind the practical requirement that input cat states of size $\alpha\geq
1.2$ are necessary, the Parity Swap SCAMP for Cat States can provide an output
state that has a fidelity of $87\%$ with an ideal cat state of size
$\beta=1.5$, requiring an input state of size $\alpha=1.1$ with an overall
success probability of $\approx 3\%$ without prior knowledge of the parity of
the input state. In other words, the amplifier transforms an input cat state
to an amplified output cat state of opposite parity of approximately double
the mean photon number of the input and works symmetrically on both even and
odd cat states.

The fidelity is one indication of the output quality but it does not contain
any information about certain features of the quantum states, such as those
contained in the form of the Wigner function, of which one example might be
negativity - a signature of nonclassicality. It is therefore instructive to
plot the contour plots of the output states after the amplification process
and of the equivalent ideal amplified cat states. In Figs. 7, 8 we plot the
Wigner functions of the output amplified state when the inputs are an even and
an odd cat state of size $\alpha=1$ and alongside the Wigner functions of the
ideal amplified cat states. We choose the experimentally feasible parameters
of $t_{2}=\sqrt{0.95},\eta=0.8$ where the fidelity between the output and the
ideal cat state in both cases is $91\%$.

(a)

(b) Figure 7: (a) Contour plot of the Wigner function of an even cat state of
size $\alpha=1.31$ (b) Same plot for the output state of the swap parity cat
scamp given an odd cat state input of size $\alpha=1$.

(a)

(b) Figure 8: (a) Contour plot of the Wigner function of an odd cat state of
size $\alpha=1.42$ (b) Same plot for the output state of the swap parity cat
scamp given an even cat state input of size $\alpha=1$.

The main features of the cat states in phase space, such as the
distinguishability of the constituent coherent states and the interference
features of the superposition are faithfully preserved. This provides a more
qualitative assessment of the output states which compliments the quantitative
benchmark of the fidelity metric. When the input state to the device is an odd
cat state, the negativity of its Wigner function, compared to the negativity
of the Wigner function of the ideal amplified even cat state, is lower (by
approximately a factor of 2), but for an even input cat state it is
comparable.

##  V Theoretical Considerations

To understand how SCAMP can be used to amplify Schrödinger cat states we need
to consider how photon subtraction affects a squeezed cat state. A squeezed
cat state can be generated by applying the squeezing operator $\hat{S}(\zeta)$
to a cat state $|\alpha_{\pm}\rangle$. Photon subtraction then leads to

$\displaystyle\hat{a}\hat{S}(\zeta)|\alpha_{\pm}\rangle=$
$\displaystyle\hat{S}(\zeta)\hat{S}^{\dagger}(\zeta)\hat{a}\hat{S}(\zeta)|\alpha_{\pm}\rangle$
$\displaystyle=$ $\displaystyle\hat{S}(\zeta)(\cosh{s}\hskip
2.84544pt\hat{a}|\alpha_{\pm}\rangle-\sinh{s}\hskip
2.84544pt\hat{a}^{\dagger}|\alpha_{\pm}\rangle),$ (15)

which is a squeezed superposition of a photon subtracted and a photon added
cat state. While photon subtraction simply swaps the parity of the cat state,
the photon addition swaps the parity and increases the amplitude of the state
Barnett . One may wonder whether photon addition alone would work but we
should note that a photon added even cat state is a vacuum removed state.
Therefore, the fidelity between an ideal even cat state and the photon added
cat state would be lower than the fidelity between the output state of the cat
SCAMP, which produces a state that contains a vacuum component. The combined
effect of photon addition and photon subtraction, rather than state
comparison, is the main gain mechanism of SCAMP for cat states. In principle
we could undo the squeezing by a local unitary operation and we would only be
left with a superposition of a photon subtracted and a photon added cat state,
but it may be challenging to implement experimentally.

The overlap between an ideal cat state and the output of the amplification
scheme is

$\displaystyle\langle\beta_{\mp}|\hat{a}\hat{S}(\zeta)|\alpha_{\pm}\rangle=$
$\displaystyle N_{\pm}^{-\frac{1}{2}}\Big{[}2\sqrt{\operatorname{sech}{s}}$
$\displaystyle
e^{-|\beta-\gamma|^{2}/2+(-\beta\gamma^{*}+\beta^{*}\gamma)/2-\tanh{s}(\beta^{*}-\gamma^{*})^{2}/2}$
$\displaystyle(\gamma-(\beta^{*}-\gamma^{*})\tanh{s})$
$\displaystyle+2\sqrt{\operatorname{sech}{s}}$ $\displaystyle
e^{-|\beta+\gamma|^{2}/2+(\beta\gamma^{*}-\beta^{*}\gamma)/2-\tanh{s}(\beta^{*}+\gamma^{*})^{2}/2}$
$\displaystyle(\pm\gamma\pm(\beta^{*}+\gamma^{*})\tanh{s})\Big{]}$ (16)

where $\gamma=\alpha\cosh{s}-\alpha^{*}\sinh{s}$ and $N_{\pm}$ is given by

$\displaystyle N_{\pm}=$ $\displaystyle(2\mp 2e^{-2|\beta|^{2}})$
$\displaystyle\big{\\{}2|\alpha|^{2}(1\mp
e^{-2|\alpha|^{2}})(\cosh{s}^{2}+\sinh{s}^{2})$ $\displaystyle+(2\pm
2e^{-2|\alpha|^{2}})$
$\displaystyle(\sinh{s}^{2}-2\cosh{s}\sinh{s}Re[\alpha]^{2})\big{\\}}$ (17)

The fidelity is then simply
$\mbox{F}=|\langle\beta_{\mp}|\hat{a}\hat{S}(\zeta)|\alpha_{\pm}\rangle|^{2}$.

The state after the comparison stage and upon postselection on zero clicks,
assuming a perfect detector, is a squeezed cat state of a scaled squeezing
parameter and a scaled coherent amplitude, both smaller than the equivalent
quantities of the input states. The squeezing parameter $s^{\prime}$ is given
by

$\displaystyle
s^{\prime}=\ln{\sqrt{\frac{\cosh{s}+(1-r_{1}^{2})\sinh{s}}{\cosh{s}-(1-r_{1}^{2})\sinh{s}}}}$
(18)

where s is the squeezing parameter of the input squeezed vacuum state and
$r_{1}$ the reflection coefficient of the beamsplitter of the comparison
stage.

The scaled coherent amplitude, $\alpha^{\prime}$ of the state after the
comparison stage is given by

$\displaystyle\alpha^{\prime}=\frac{r_{1}\alpha\cosh{s}}{\sqrt{(\cosh{s})^{2}-(1-r_{1}^{2})^{2}(\sinh{s})^{2}}}$
(19)

Then we can plot the ratio of the ideal cat size $\beta$, which maximises the
fidelity to the photon subtracted squeezed cat state, to the input cat size
$\alpha$ as a function of the input cat size $\alpha$.

Figure 9:  Ideal gain given by subtracting a photon from a squeezed even cat
state as a function of input cat size.

One can indeed verify that Fig. 4 (a) and Fig. 9 differ marginally due to the
fact that photon subtraction can only be implemented approximately and is
limited by the transmission coefficient of the beamsplitter used to perform
the photon subtraction (which is always less than unity, $t_{2}=\sqrt{0.99}$
in this case).

In this section we have described the theoretical mechanism for the gain of
our device. Physically it is based on mean photon number matching between a
resource state (squeezed vacuum) and the input cat state after photon
subtraction conditioning. If the cat state has a significantly higher photon
number than the resource state the output will effectively be a photon
subtracted cat state. If the resource state has a significantly higher photon
number than the cat state the output will be a photon subtracted resource
state. Only if they are comparable in mean photon number does the significant
state overlap between the two give an amplified cat. The approximate photon
number matching of the two states is also the reason for the approximate
twofold gain.

##  VI Other Schemes for Amplification of Schrödinger Cats

Figure 9 suggests that we could amplify a Schrödinger cat state reasonably
well by first squeezing the cat state and then subtracting a photon. The
process amounts to a slightly simpler theoretical scheme than ours. However,
squeezing low amplitude non-vacuum states in a controlled fashion is more
difficult experimentally, particularly in the pulsed domain. Optical
nonlinearities are small and require high pump fields with a good mode overlap
with the signal. For these reasons we believe that such a direct-drive scheme
is a little less experimentally practical than ours.

The state of the art experimental scheme of Synchev  which utilises the cat
“breeding” method produces an amplified cat state with amplitude $\alpha=1.85$
while consuming two cat states of amplitude $\alpha=1.15$ as resources. The
probability of success is $P_{S}=0.2$ and the fidelity of the produced state
with an ideal cat state is $\mbox{F}=0.77$. Seemingly, SCAMP is on par with
cat “breeding” schemes in terms of fidelity and gain while it offers a lower
probability of success. One has though to take into account the fact that cat
“breeding” schemes consume two nongaussian resource states, as opposed to just
one for SCAMP, that have a low success probability of production currently and
therefore offset this advantage. If push-button quantum state generation
becomes a reality then the intrinsic advantages of cat “breeding” may be more
easily realised.

##  VII Conclusions

In summary, we have presented a theoretical scheme for the amplification of
input Schrödinger cat states based on a nondeterministic amplifier (SCAMP)
comprised of the two mature techniques of state comparison and photon
subtraction. The device both amplifies and swaps the parity of a state chosen
from the set of odd and even Schrodinger cat states. Hence we call it the
parity-swap cat state comparison amplifier.

The implementation of SCAMP for Schrödinger cat states would require only a
Gaussian resource state (squeezed vacuum), linear optical components and
Geiger mode detectors, making it experimentally feasible. The resource state
is not randomly chosen, as it is in the standard SCAMP for coherent state
amplification. This, coupled with the fact that it is Gaussian gives a
significant advantage over other schemes. The SCAMP itself provides the photon
subtraction required to render the output nongaussian. The parity-swap cat
SCAMP can work almost symmetrically for both even and odd cat states without
prior knowledge of the input state and offers high fidelity and reasonably
high gain for the range of input cat state sizes of interest.

We have characterised the performance of the amplifier via gain, fidelity and
success probability. The intensity gain is shown to be approximately twofold
for both the even and odd cat states for a wide range of mean input photon
numbers, reflecting the fact that the gain of a standard SCAMP is
approximately two for a 50/50 comparison beamsplitter. The fidelity of the
output with an ideal amplified cat state $|g\alpha_{\pm}\rangle$ depends on
the transmission coefficient, $t_{2}$, of the beamsplitter used to perform the
photon subtraction stage. The fidelity remains high ($F\geq 81\%$)) for values
of input $\alpha$ up to $\alpha\approx 1.5$ in realistic parameter ranges. As
an example, one can amplify an input cat state of size $\alpha=1.1$ which,
after the amplification, has a fidelity of $87\%$ with a cat state of size
$\beta=1.5$ and the overall success probability is approximately $3\%$. The
probability of success is comparable with schemes that consume non-Gaussian
resources. However, when one takes into account the overall experimental
probability of producing the non-Gaussian resources, which is significantly
lower compared to Gaussian ones, the scheme presented here has a clear
advantage.

There are a couple of limitations to the usefulness of the scheme. The low
success probability is a common problem in the quantum regime for all schemes
that rely on photon subtraction. The original SCAMP for coherent states has a
similar success probability in this regime. This is compensated for by running
the experiment at a high rate and the same technique would work here, although
the limiting speed may be the cat-state production rate for the input states.
Secondly, the resource state, the squeezed vacuum, contains only even photon
numbers and so the parity-swap cat SCAMP can only work for cat states of even
or odd photon number. States such as

$\displaystyle|\alpha_{\pm i}\rangle=N\left(|\alpha\rangle\pm
i|-\alpha\rangle\right)$ (20)

do not satisfy this criterion and cannot be amplified in this fashion.
However, such states have never been produced in the laboratory and so are of
limited interest.

##  VIII Acknowlegments

The authors would like to acknowledge financial support from the UK National
Quantum Technology Programme via EPSRC and the Quantum Technology Hub in
Quantum Communications (Grant EP/M013472/1). GT would also like to acknowledge
EPSRC for a partial studentship from the Doctoral Training Grant to the
University of Strathclyde.

## Appendix

In this appendix section we will use the formalism of characteristic functions
to sketch the method used to derive all of the required results. We will see
how the states and operators can be represented under the formalism and show
how one can compute the benchmark metrics considered in the main body of the
text.

### Characteristic Functions

A useful feature of the characteristic function formalism is that the trace of
operators can be evaluated as an integral in phase space. More formally, the
trace rule for two operators $\hat{O}_{1}$ and $\hat{O}_{2}$ is given by
Ferraro

$\mbox{Tr}[\hat{O}_{1}\hat{O}_{2}]=\int_{\mathbb{C}}\frac{d^{2}\xi}{\pi}\chi_{\hat{O}_{1}}(\xi)\chi_{\hat{O}_{2}}(-\xi)$
(21)

and this formula gives us the fidelity F. States whose characteristic function
is a Gaussian are known as Gaussian states and one such example we have
already seen; namely the coherent state $|\alpha\rangle$ whose characteristic
function is

$\displaystyle\chi_{\alpha}(\xi)=\exp(\xi\alpha^{*}-\xi^{*}\alpha-|\xi|^{2}/2)$
(22)

and the squeezed vacuum state whose characteristic function is

$\displaystyle\chi_{\zeta}(\xi)=$
$\displaystyle\exp\Big{(}-\xi^{2}_{r}\exp[2s]/2+\xi^{2}_{i}\exp[-2s]\Big{)}$
(23)

where $\xi_{r}$ and $\xi_{i}$ are the real and imaginary parts of the complex
variable $\xi$.

The cat states on the other hand are nongaussian states as their
characteristic functions are given by sums of Gaussians:

$\displaystyle\chi_{\alpha_{+}}(\xi)=$ $\displaystyle
N_{+}^{2}\Big{[}\exp\left(\xi\alpha^{*}-\alpha\xi^{*}-|\xi|^{2}/2\right)$
$\displaystyle+\exp\left(-\xi\alpha^{*}+\alpha\xi^{*}-|\xi|^{2}/2\right)$
$\displaystyle+\exp\left(-\xi\alpha^{*}-\alpha\xi^{*}-2|\alpha|^{2}-|\xi|^{2}/2\right)$
$\displaystyle+\exp\left(\xi\alpha^{*}+\alpha\xi^{*}-2|\alpha|^{2}-|\xi|^{2}/2\right)\Big{]}$
(24)

$\displaystyle\chi_{\alpha_{-}}(\xi)=$ $\displaystyle
N_{-}^{2}\Big{[}\exp\left(\xi\alpha^{*}-\alpha\xi^{*}-|\xi|^{2}/2\right)$
$\displaystyle+\exp\left(-\xi\alpha^{*}+\alpha\xi^{*}-|\xi|^{2}/2\right)$
$\displaystyle-\exp\left(-\xi\alpha^{*}-\alpha\xi^{*}-2|\alpha|^{2}-|\xi|^{2}/2\right)$
$\displaystyle-\exp\left(\xi\alpha^{*}+\alpha\xi^{*}-2|\alpha|^{2}-|\xi|^{2}/2\right)\Big{]}$
(25)

where $N_{\pm}^{2}=\frac{1}{2\pm 2e^{-2|\alpha|^{2}}}$.

### Performance Benchmark

Typically the performance of amplifiers can be characterised by the fidelity,
F, and the probability of success, $P_{s}$ of the amplifier. The fidelity
metric quantifies the closeness of two quantum states. In this case it
quantifies the closeness of the output state after the amplification to an
ideal output state. If the ideal output state is pure then F has a simple form
given by

$\mbox{F}=\mathrm{Tr}[\rho_{1}\rho_{2}]$ (26)

where $\rho_{1},\rho_{2}$ are some general single mode states and at least one
of them is pure. In terms of characteristic functions, as we have already
seen, the fidelity is given by

$\mbox{F}=\int\frac{d^{2}\xi}{\pi}\chi_{1}(\xi)\chi_{2}(-\xi)$ (27)

The probability of success metric on the other hand depends on the working
details of the amplifier. For the SCAMP the probability of success is defined
as the joint probability of the second detector clicking given that the first
detector did not register a click, $P_{s}=P(D_{1}=\text{\char
55},D_{2}=\checkmark)$. The probability for a detector on Geiger mode to not
register a click is given by the Kelley-Kleiner formula Kelley

$P_{\text{\char 55}}=\mathrm{Tr}[\rho\hat{\pi}_{\text{\char
55}}]=\mathrm{Tr}[\rho:\exp(-\eta\hat{a}^{\dagger}\hat{a}):]$ (28)

where $\eta$ is the quantum efficiency of the detector and the colons indicate
normal ordering of the operators. In terms of the characteristic function the
no counts operator is given by

$\chi_{\hat{\pi}_{\text{\char
55}}}(\xi)=\frac{1}{\eta}\exp(-\frac{2-\eta}{2\eta}|\xi|^{2})$ (29)

and the probability of success is given by

$P_{\text{\char
55}}=\int\frac{d^{2}\xi}{\pi}\chi(\xi)\chi_{\hat{\pi}_{\text{\char 55}}}(\xi)$
(30)

The probability of success if a detector clicks on the other hand is give by

$P_{\checkmark}=\mathrm{Tr}[\rho\hat{\pi}_{\checkmark}]=\mathrm{Tr}[\rho:\mathds{1}-\exp(-\eta\hat{a}^{\dagger}\hat{a}):]$
(31)

The click operator, $\hat{\pi}_{\checkmark}$, in the characteristic function
formalism is given by

$x_{\hat{\pi}_{\checkmark}}(\xi)=\pi\delta^{(2)}(\xi)-x_{\hat{\pi}_{\text{\char
55}}}(\xi)$ (32)

and the probability of success is given by

$P_{\checkmark}=\int\frac{d^{2}\xi}{\pi}\chi(\xi)\chi_{\hat{\pi}_{\checkmark}}(\xi)$
(33)

### Output of Comparison Stage

Now that we have introduced the formalism of the characteristic functions we
can have a close look at what happens when a coherent state $|\alpha\rangle$
and a squeezed vacuum $|\zeta\rangle$ are incident on a beamsplitter and one
of the two output modes is projected onto the vacuum state.

The input to the beamsplitter is given by

$\displaystyle\chi_{in}(\xi_{1},\xi_{2})=\chi_{\alpha}(\xi_{1})\chi_{\zeta}(\xi_{2})$
(34)

The output (global) state after the beamsplitter can be found by transforming
the arguments of the characteristic function of the input as follows

$\displaystyle\chi_{out}(\xi_{3},\xi_{4})=\chi_{\alpha}(t_{1}\xi_{3}+r_{1}\xi_{4})\chi_{\zeta}(t_{1}\xi_{4}-r_{1}\xi_{3})$
(35)

Postselection then on zero clicks, assuming a perfect detector, leads to

$\chi_{4}(\xi_{4})=\frac{1}{\pi P_{\text{\char
55}}}\int\chi_{out}(\xi_{3},\xi_{4})\chi_{\hat{\pi}_{\text{\char
55}}}(\xi_{3})d^{2}\xi_{3}\\\
=\exp\Bigg{[}-\frac{x^{2}\\{\cosh{s}+(1-r_{1}^{2})\sinh{s}\\}}{2(\cosh{s}-(1-r_{1}^{2})^{2}\sinh{s}^{2})}\\\
-\frac{y^{2}\\{\cosh{s}-(1-r_{1}^{2})\sinh{s}\\}}{2(\cosh{s}-(1-r_{1}^{2})^{2}\sinh{s}^{2})}\\\
-2ir_{1}xIm[\alpha]\cosh{s}\frac{(\cosh{s}+(1-r_{1}^{2})\sinh{s})}{\cosh{s}^{2}-(1-r_{1}^{2})^{2}\sinh{s}^{2}}\\\
+2ir_{1}yRe[\alpha]\cosh{s}\frac{(\cosh{s}-(1-r_{1}^{2})\sinh{s})}{\cosh{s}^{2}-(1-r_{1}^{2})^{2}\sinh{s}^{2}}\Bigg{]}$
(36)

where $P_{\text{\char 55}}$ is given by

$P_{\text{\char
55}}=\frac{1}{\pi}\iint\chi_{out}(\xi_{3},\xi_{4})\chi_{\hat{\pi}_{\text{\char
55}}}(\xi_{3})d^{2}\xi_{3}d^{2}\xi_{4}\\\
=\sqrt{\frac{1}{(e^{-s}+r_{1}^{2}\sinh{s})(e^{s}-r_{1}^{2}\sinh{s})}}*\\\
\exp\Bigg{[}-\frac{e^{s}(1-r_{1}^{2})\sinh{s}\mbox{Re}[\alpha]^{2}}{e^{s}-r_{1}^{2}\sinh{s}}\\\
-\frac{e^{-s}(1-r_{1}^{2})\sinh{s}\mbox{Im}[\alpha]^{2}}{e^{-s}+r_{1}^{2}\sinh{s}}\Bigg{]}$
(37)

The characteristic function of a squeezed coherent state,
$\hat{S}(\zeta)|\alpha\rangle$ is given by

$\chi_{\zeta,\alpha}(\xi)=\exp\big{(}-x^{2}e^{2s}/2-y^{2}e^{-2s}/2\\\
-2ie^{s}x\mbox{Im}[\alpha]+2ie^{-s}\mbox{Re}[\alpha]\big{)}$ (38)

where $\xi=x+iy$.

If we compare Eq. 36 to Eq.38 we can see that they both represent the same
state with the transformations $s\rightarrow s^{\prime}$, and
$\alpha\rightarrow\alpha^{\prime}$ given by

$\displaystyle s^{\prime}$
$\displaystyle=\ln{\sqrt{\frac{\cosh{s}+(1-r_{1}^{2})\sinh{s}}{\cosh{s}-(1-r_{1}^{2})\sinh{s}}}}$
(39) $\displaystyle\alpha^{\prime}$
$\displaystyle=\frac{r_{1}\alpha\cosh{s}}{\sqrt{(\cosh{s})^{2}-(1-r_{1}^{2})^{2}(\sinh{s})^{2}}}$
(40)

Then we can think of the comparison stage as a quantum channel that transforms
a coherent state to a squeezed coherent state of squeezing parameter
$s^{\prime}$ and scaled coherent amplitude $\alpha^{\prime}$. The linearity of
quantum mechanics then implies that a linear superposition of coherent states,
such as cat states, are transformed in a similar manner. Therefore, the
comparison stage transforms cat states into squeezed cat states of squeezing
parameter $s^{\prime}$ and cat size $\alpha^{\prime}$.

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