# Quantum repeaters and teleportation via entangled phase-modulated multimode
coherent states

R. Goncharov<EMAIL_ADDRESS>Quantum Information Laboratory, ITMO
University, Kadetskaya Line, 3, Saint Petersburg, 199034, Russia  Leading
Research Center "National Center for Quantum Internet", ITMO University,
Birzhevaya Line, 16, Saint Petersburg, 199034, Russia  SMARTS-Quanttelecom
LLC, 6th Vasilyevskogo Ostrova Line, 59, Saint Petersburg, 199178, Russia
Alexei D. Kiselev<EMAIL_ADDRESS>Laboratory of Quantum Processes
and Measurements, ITMO University,199034 Kadetskaya Line 3b, Saint Petersburg,
Russia  Leading Research Center "National Center for Quantum Internet", ITMO
University, Birzhevaya Line, 16, Saint Petersburg, 199034, Russia  Quantum
Information Laboratory, ITMO University, Kadetskaya Line, 3, Saint Petersburg,
199034, Russia  E. S. Moiseev  Kazan Quantum Center, Kazan National Research
Technical University, 18a Chetaeva str., Kazan, 420111, Russia  E. Samsonov
Quantum Information Laboratory, ITMO University, Kadetskaya Line, 3, Saint
Petersburg, 199034, Russia  SMARTS-Quanttelecom LLC, 6th Vasilyevskogo Ostrova
Line, 59, Saint Petersburg, 199178, Russia  Leading Research Center "National
Center for Quantum Internet", ITMO University, Birzhevaya Line, 16, Saint
Petersburg, 199034, Russia  S. A. Moiseev  Kazan Quantum Center, Kazan
National Research Technical University, 18a Chetaeva str., Kazan, 420111,
Russia  F. Kiselev  Quantum Information Laboratory, ITMO University,
Kadetskaya Line, 3, Saint Petersburg, 199034, Russia  SMARTS-Quanttelecom LLC,
6th Vasilyevskogo Ostrova Line, 59, Saint Petersburg, 199178, Russia  Leading
Research Center "National Center for Quantum Internet", ITMO University,
Birzhevaya Line, 16, Saint Petersburg, 199034, Russia  V. Egorov  Quantum
Information Laboratory, ITMO University, Kadetskaya Line, 3, Saint Petersburg,
199034, Russia  SMARTS-Quanttelecom LLC, 6th Vasilyevskogo Ostrova Line, 59,
Saint Petersburg, 199178, Russia  Leading Research Center "National Center for
Quantum Internet", ITMO University, Birzhevaya Line, 16, Saint Petersburg,
199034, Russia

(August 27, 2024)

###### Abstract

We present a scheme of quantum repeater that uses entangled multimode coherent
states which are obtained by electro-optic modulation of symmetric and
antisymmetric Schrödinger cat states. Part of generated entangled frequency
modes are sent to a symmetric beam splitter at the central node, while the
remaining modes are stored locally in quantum memories. The entangled coherent
states between remote quantum memories are conditionally prepared by photon
counting measurements at the output channels of the beam splitter. We study
how the effects of decoherence in the quantum channel affect statistics of
photocounts and, for the heralding outcomes determined by the parity of
photocounts, evaluate the probability of success and the fidelity of the
prepared entanglement depending on the symmetry of input cat states. It is
demonstrated that the generated entanglement can be employed for teleportation
of the phase information from the modulated states which are utilized in
quantum key distributions with subcarrier wave encoding.

##  I Introduction

Transferring a quantum state between remote parties is the primary purpose of
quantum communication [1, 2]. It lies at the heart of a variety of
applications that include secure transfer of classical messages using quantum
key distribution (QKD) [3, 4], quantum metrology [5, 6, 7] and distributed
computations [8, 9].

One of the key problems of quantum communication is the generation of high-
fidelity quantum states entangled between distant sites [1]. In the majority
of approaches to this problem, photons represent information carriers that can
function as flying qubits and the fundamental difficulty is that the photons
are subject to losses (optical absorption) and other noise-induced
perturbations present in photonic channels such as optical fibers and
turbulent atmosphere. This noise has a detrimental effect on the quality of
entanglement generated between two remote parties leading to exponential decay
of the entanglement degree with the channel length. Owing to the exponential
losses, for the photons propagating in optical fibers, the achievable
distances are limited to about $200$ km and the transmission of entanglement
over global distances (thousands of kilometers) becomes a challenging task.
The concept of quantum repeaters (QRs) was put forward in [10] as the method
to overcome this limitation.

The mode of operation of QRs assumes that the transmission channel is divided
into several segments (elementary links). The first step is to prepare
entanglement between the two nodes (at the ends) of each link. Then, at the
next step, entanglement swapping between neighboring links is used to transfer
entanglement over significant distances to the target points of a quantum
network.

There is a number of reviews focusing on different aspects of QRs [11, 12, 13,
14]. For instance, the review [11] focuses on the so-called DLCZ protocol (for
Duan, Lukin, Cirac, and Zoller) developed in [15] and its improvements. In
this protocol, linear optics is combined with photon counting to perform
needed operations and atomic ensembles are used as quantum memories (see [16]
for a review). The primitives and fundamental components needed for QRs along
with classification of QR protocols into three relevant generations are
reviewed in [12]. Advantages and challenges of each generation of QRs
determined by the methods utilized to suppress loss and operation errors are
analyzed in [13]. A more recent review [14] additionally discusses newly
emerging classes of repeaters such as memoryless, error-corrected, and all-
photonic repeaters [17, 18, 19] and put particular emphasis on increasingly
important role of QRs for development of long-distance quantum networks
(quantum internet) [20, 21, 22]. For such entanglement-assisted networks, the
primary challenge is to go beyond the limit of point-to-point quantum
communication, achieving high-rate secure communication without using trusted
relay nodes.

Among a variety of photonic quantum states used in QR protocols the coherent
states have been attracted considerable attention [23, 24, 25] as the states
that are relatively easy to produce and control. The QR protocol analyzed in
[23] uses hybrid entanglement where the coherent states are entangled with
atomic (spin) qubits. On the other hand, the entanglement generation and
swapping protocols studied in [24, 25] are based on the entangled coherent
states.

These states were originally introduced in [26] (see Ref. [27] for a review)
and there is a number of quantum information processing tasks that can be
performed using the entangled coherent states [28, 29, 30, 31, 32]. Multimode
coherent states also provide a potentially promising source of multipartite
entanglement [33, 34] required by the quantum networks. In this paper, the
approach to QRs based on such states will be our primary concern.

More specifically, we present and theoretically study the scheme of a quantum
repeater that uses multimode coherent states generated by electro-optic
modulation of Schrödinger cat states. This is the method that produces phase-
coded multimode signals to perform the subcarrier wave (SCW) encoding which is
proved to be useful in point-to-point [35, 36, 37, 38], plug-and-play (P&P)
[39], continuous-variable (CV) quantum key distribution (QKD) [40, 41, 42] and
twin-field QKD [43]. The scope of this method is not limited to the above
protocols and, owing to its robustness to environmental distortion of fiber
line, interferometer-free scheme and multiplexing capacity, the SCW encoding
is one of promising approaches for quantum communication.

For such electro-optic phase modulation-based method, we examine both the
heralded entanglement generation in elementary links and the entanglement
swapping procedures utilized for the creation and distribution of entanglement
between broadband quantum memories (QMs) (reports on recent experimental
developments in the rapidly developing field of the QMs can be found in, e.g.,
Refs. [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55] ). As compared to QR
schemes based on controlled beam splitters (see e.g. [24]), the electro-optic
phase modulator provides a more flexible tool. In practical implementation,
using such tool opens up new possibilities such as ability to control
entanglement swapping by choosing different sidebands and quick adjustment of
the mutual phase between Alice and Bob. An important feature of our QR model
is that it can be associated with a real-life experimental detection scheme
and its applicability goes beyond the scope of QR protocols. In order to
demonstrate the latter, we additionally discuss the task of teleportation of
the coherent state phase.

The paper is organized as follows. In Section II we present our modulator-
based scheme of the elementary link and analytical model for the heralded
entanglement generation procedure employed to produce coherent cat states
entangled between remote nodes of the link. In Section III we shall analyze
performance of the proposed scheme and quality of the generated entanglement
by evaluating the fidelity and probabilities of photocounts for heralding
events. In this analysis, we consider nonideal photodetectors and take into
account effects of decoherence using the decoherence model which is typical
for CV quantum channels with energy transfer to environmental modes. In
Section IV we discuss the entanglement swapping technique used to create long-
distance entanglement and its performance. Section V presents a teleportation
scheme that employs created entangled coherent states to transfer phase
information encoded into SCW states between remote parties Finally, we discuss
and summarize our results in Section VI.

##  II Elementary link

Figure 1 presents the optical scheme of an elementary link that creates
entanglement between two remote nodes (Alice and Bob) using the phase
modulation method. In this method, Alice and Bob utilize electro-optic phase
modulators to produce local multimode entangled states by modulating
superpositions of one-mode coherent states known as the Schrödinger cat
states. An important advantage of using fast phase modulators is that Alice
and Bob can actively control the output states by changing the modulation
index and phase.

Figure 1: Optical schematic of the subcarrier wave quantum repeater elementary
link. Source is a source of Schrödinger cat states, PM is the electro-optic
modulator, Circ is the circulator, SF is the spectral filter, QM is the
quantum memory, BS is the beam splitter, PNRD is the photon-number-resolving
detector. Diagrams in circles show the absolute value of signal spectrum
taking into account only the first-order sidebands.

Referring to Fig. 1, after generation of the local modulated states, one of
the frequency modes is sent to the central relay via spectral filtering while
keeping other modes stored in quantum memories. In this section, we perform
our analysis assuming that the input (non-modulated) state of the (central)
carrier wave mode for Alice’s and Bob’s modulators, $|{S_{A}}\rangle$ and
$|{S_{B}}\rangle$, are generally two different one-mode Schrödinger cat states
of the following form:

$\displaystyle|{S_{A}}\rangle=|{\Psi_{\nu^{\prime}}(\alpha)}\rangle,\>|{S_{B}}\rangle=|{\Psi_{\nu}(\beta)}\rangle,\>\nu,\nu^{\prime}\in\\{+,-\\},$
(1)

where $|{\Psi_{+}(\alpha)}\rangle$ and $|{\Psi_{-}(\alpha)}\rangle$ are the
symmetric (even) and antisymmetric (odd) Schrödinger cats, respectively. These
states were originally introduced in [56] as even and odd coherent states
given by

$\displaystyle|{\Psi_{\pm}(\alpha)}\rangle=\frac{1}{\sqrt{M_{\pm}(\alpha)}}|{\alpha^{(\pm)}}\rangle,\quad|{\alpha^{(\pm)}}\rangle\equiv|{\alpha}\rangle\pm|{-\alpha}\rangle$
(2)

where

$\displaystyle
M_{\pm}(\alpha)=\langle{\alpha^{(\pm)}|\alpha^{(\pm)}}\rangle=2(1\pm\exp(-2|\alpha|^{2})).$
(3)

There is a variety of experimental techniques used to generate optical
Schrödinger cats [57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67]. It includes the
method based on photon subtraction from the squeezed vacuum state [57, 58, 59,
60, 61], the protocol that uses homodyne detection and photon number states
[62], reservoir engineering [63, 64, 65] and the methods that involve making
quadrature measurements of one of the modes of a biphoton NOON state [66] and
reflecting coherent-state photons from a microwave cavity containing a
superconducting qubit [67].

According to the model of electro-optic modulator [68], for the input states
$|{\pm\alpha}\rangle_{A}$ and $|{\pm\beta}\rangle_{B}$, the modulated states
also known as the SCW states can be described as the multimode coherent states
given by

$\displaystyle|{\pm\alpha}\rangle_{A}\to|{\pm\bm{\alpha}}\rangle_{A}=\otimes_{\mu=-S}^{S}|{\pm\alpha_{\mu}}\rangle_{A},$
$\displaystyle|{\pm\beta}\rangle_{B}\to|{\pm\bm{\beta}}\rangle_{B}=\otimes_{\mu=-S}^{S}|{\pm\beta_{\mu}}\rangle_{B},$
(4) $\displaystyle\alpha_{\mu}={U_{\mu
0}^{(A)}}^{\ast}\alpha,\quad\beta_{\mu}={U_{\mu 0}^{(B)}}^{\ast}\beta,$ (5)

where $U_{\mu 0}^{(A,B)}$ are the elements of the evolution matrix. In the
large $S$ limit, these elements can be approximated as follows [68]

$\displaystyle U_{\mu
0}^{(A,B)}\approx\mathrm{e}^{-i\mu\phi_{A,B}}J_{\mu}(m_{A,B}),$ (6)

where $\phi_{A,B}$ and $m_{A,B}$ are the phases and the modulation indices,
respectively.

From Eq. (4) it can be readily seen that the electro-optic modulator acts like
a multiport beam splitter that transforms the one-mode cat states (1) into the
multimode Schrödinger cat states of the following form:

$\displaystyle|{S_{A}}\rangle\to|{\Psi_{A}}\rangle=|{\Psi_{\nu^{\prime}}^{(A)}(\bm{\alpha})}\rangle,\>|{S_{B}}\rangle\to|{\Psi_{B}}\rangle=|{\Psi_{\nu}^{(B)}(\bm{\beta})}\rangle,$
(7)

where

$\displaystyle|{\Psi_{\pm}^{(A)}(\bm{\alpha})}\rangle=\frac{1}{\sqrt{M_{\pm}(\bm{\alpha})}}|{\bm{\alpha}^{(\pm)}}\rangle_{A},$
$\displaystyle|{\Psi_{\pm}^{(B)}(\bm{\beta})}\rangle=\frac{1}{\sqrt{M_{\pm}(\bm{\beta})}}\\{|{\bm{\beta}}\rangle_{B}\pm|{-\bm{\beta}}\rangle_{B}\\}$
(8)
$\displaystyle|{\bm{\alpha}^{(\pm)}}\rangle_{A}=|{\bm{\alpha}}\rangle_{A}\pm|{-\bm{\alpha}}\rangle_{A},$
(9) $\displaystyle
M_{\pm}(\bm{\alpha})=2(1\pm\exp(-2|\bm{\alpha}|^{2})),\>|\bm{\alpha}|^{2}=\sum_{\mu=-S}^{S}|\alpha_{\mu}|^{2}.$
(10)

Thus the modulated state shared by Alice and Bob, $|{\Psi_{AB}}\rangle$, is
the tensor product of two multimode Schrödinger cat states after the Alice’s
and Bob’s modulators:
$|{\Psi}\rangle_{AB}=|{\Psi_{\nu^{\prime}}^{(A)}(\bm{\alpha})}\rangle\otimes|{\Psi_{\nu}^{(B)}(\bm{\beta})}\rangle$.
This state can be written in the form

$\displaystyle|{\Psi}\rangle_{AB}=\frac{1}{N_{AB}}\Bigl{\\{}|{{\bm{\alpha}},{\bm{\beta}}}\rangle_{AB}+\nu^{\prime}\nu|{-{\bm{\alpha}},-{\bm{\beta}}}\rangle_{AB}$
$\displaystyle+\nu|{{\bm{\alpha}},-{\bm{\beta}}}\rangle_{AB}+\nu^{\prime}|{-{\bm{\alpha}},{\bm{\beta}}}\rangle_{AB}\Bigr{\\}}=\sqrt{\frac{M_{\nu^{\prime}\nu}(\bm{\alpha},\bm{\beta})}{M_{\nu^{\prime}}(\bm{\alpha})M_{\nu}(\bm{\beta})}}$
$\displaystyle\times\bigl{\\{}|{\Psi_{\nu^{\prime}\nu}^{(AB)}(\bm{\alpha},\bm{\beta})}\rangle+\nu|{\Psi_{\nu^{\prime}\nu}^{(AB)}(\bm{\alpha},-\bm{\beta})}\rangle\bigr{\\}},$
(11) $\displaystyle
M_{\pm}({\bm{\alpha}},{\bm{\beta}})=2\left\\{1\pm\exp[-2(|\bm{\alpha}|^{2}+|\bm{\beta}|^{2})]\right\\},$
(12)

where $N_{AB}=\sqrt{M_{\nu^{\prime}}(\alpha)M_{\nu}(\beta)}$ and
$|{{\bm{\alpha}},{\bm{\beta}}}\rangle_{AB}$ represents
$|{\bm{\alpha}}\rangle_{A}\otimes|{\bm{\beta}}\rangle_{B}$ rearranged into the
tensor product of modes.

From now on we shall restrict our analysis to the important special case where
$\bm{\alpha}=\bm{\beta}$ and, following the general approach [11, 24] to
preparation of entangled states shared by Alice and Bob, we shall assume that
the modes $\bm{\alpha}$ that enter the modulated states,
$|{\Psi_{\nu}(\bm{\alpha})}\rangle\equiv|{\Psi_{\nu}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{bs}})}\rangle$,
are divided into two groups: the modes $\bm{\alpha}_{\mathrm{qm}}$ stored in
the quantum memory and the modes $\bm{\alpha}_{\mathrm{bs}}$ put to interfere
onto a symmetric $50:50$ beam splitter with the output channels $C$ and $D$.
The above modes brought into interference onto the beam splitter appear to be
transformed as follows

$\displaystyle\hat{T}_{\bm{\alpha}_{\mathrm{bs}}\bm{\alpha}_{\mathrm{bs}}\to
CD}|{\pm\bm{\alpha}_{\mathrm{bs}}}\rangle_{A}\otimes|{\pm\bm{\alpha}_{\mathrm{bs}}}\rangle_{B}=|{\pm\bm{\gamma}_{\mathrm{bs}}}\rangle_{C}\otimes|{\mathbf{0}}\rangle_{D},$
$\displaystyle\hat{T}_{\bm{\alpha}_{\mathrm{bs}}\bm{\alpha}_{\mathrm{bs}}\to
CD}|{\pm\bm{\alpha}_{\mathrm{bs}}}\rangle_{A}\otimes|{\mp\bm{\alpha}_{\mathrm{bs}}}\rangle_{B}=|{\mathbf{0}}\rangle_{C}\otimes|{\pm\bm{\gamma}_{\mathrm{bs}}}\rangle_{D},$
(13)

where $\bm{\gamma}_{\mathrm{bs}}=\sqrt{2}\bm{\alpha}_{\mathrm{bs}}$.

We can now apply the transformation (II) to the state shared by Alice and Bob
(see (II)). The result reads

$\displaystyle\hat{T}_{\bm{\alpha}_{\mathrm{bs}}\bm{\alpha}_{\mathrm{bs}}\to
CD}|{\Psi_{\nu^{\prime}}^{(A)}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{bs}})}\rangle\otimes|{\Psi_{\nu}^{(B)}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{bs}})}\rangle=\frac{1}{2{\sqrt{M_{\nu^{\prime}}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{bs}})M_{\nu}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{bs}})}}}$
$\displaystyle\times\sum_{\mu=\pm}\sqrt{M_{\mu^{\prime}}(\bm{\gamma}_{\mathrm{bs}})M_{\mu}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{qm}})}\Bigl{\\{}|{\Psi_{\mu^{\prime}}^{(C)}(\bm{\gamma}_{\mathrm{bs}})}\rangle\otimes|{\mathbf{0}}\rangle_{D}\otimes|{\Psi_{\mu}^{(AB)}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{qm}})}\rangle$
$\displaystyle+\nu|{\mathbf{0}}\rangle_{C}\otimes|{\Psi_{\mu^{\prime}}^{(D)}(\bm{\gamma}_{\mathrm{bs}})}\rangle\otimes|{\Psi_{\mu}^{(AB)}(\bm{\alpha}_{\mathrm{qm}},-\bm{\alpha}_{\mathrm{qm}})}\rangle\Bigr{\\}},\quad\mu^{\prime}=\nu\nu^{\prime}\mu,$
(14)

where $\bm{\gamma}_{\mathrm{bs}}\equiv\sqrt{2}\bm{\alpha}_{\mathrm{bs}}$.

If, for instance, we now perform a measurement on the output mode $C$ to
distinguish the states
$|{\Psi_{\nu^{\prime}\nu}^{(C)}(\bm{\gamma}_{\mathrm{bs}})}\rangle$ and
$|{\Psi_{-\nu^{\prime}\nu}^{(C)}(\bm{\gamma}_{\mathrm{bs}})}\rangle$, the
multimode state will collapse onto either
$|{\Psi_{+}^{(AB)}({\bm{\alpha}}_{\mathrm{qm}},{\bm{\alpha}}_{\mathrm{qm}})}\rangle$
or
$|{\Psi_{-}^{(AB)}({\bm{\alpha}}_{\mathrm{qm}},{\bm{\alpha}}_{\mathrm{qm}})}\rangle$,
respectively. Thus preparation of the entangled coherent cat states is
heralded by the parity of clicks of a photon-number-resolving detector placed
at the output channel $C$.

(a)

(b)

(c)

(d)

Figure 2:  Probabilities $P_{\pm}^{(-)}$ and $P_{\mp}^{(c)}$ associated with
the cat states $|{\Phi_{\pm}(\bm{\gamma}_{\mathrm{bs}})}\rangle$ at an output
port of the beam splitter computed from Eqs. (19) and (20) as a function of
$|\alpha|^{2}$ and
$r_{\mathrm{bs}}=|\bm{\alpha}_{\mathrm{bs}}|^{2}/|\bm{\alpha}|^{2}$.

Now we introduce the modified symmetric cat state

$\displaystyle|{\tilde{\Psi}_{+}(\bm{\alpha})}\rangle$
$\displaystyle=\frac{1}{\sqrt{\tilde{M}_{+}(\bm{\alpha})}}\Bigl{[}|{\tilde{\bm{\alpha}}}\rangle+|{-\tilde{\bm{\alpha}}}\rangle\Bigr{]},$
$\displaystyle|{\pm\tilde{\bm{\alpha}}}\rangle$
$\displaystyle\equiv|{\pm\bm{\alpha}}\rangle-\mathrm{e}^{-|\bm{\alpha}|^{2}/2}|{\mathbf{0}}\rangle,$
(15)

where
$\tilde{M}_{+}(\bm{\alpha})=M_{+}(\bm{\alpha})-4\exp(-|\bm{\alpha}|^{2})$,
defined in terms of the coherent states renormalized by subtracting the vacuum
contribution. Then we have a set of three orthonormal states:
$|{\Psi_{-}(\sqrt{2}\bm{\alpha}_{\mathrm{bs}})}\rangle\equiv|{\Phi_{-}(\bm{\gamma}_{\mathrm{bs}})}\rangle$,
$|{\tilde{\Psi}_{+}(\sqrt{2}\bm{\alpha}_{\mathrm{bs}})}\rangle\equiv|{\Phi_{+}(\bm{\gamma}_{\mathrm{bs}})}\rangle$,
and $|{\mathbf{0}}\rangle$ that can be conveniently used to render the state
(II) into the form

$\displaystyle\hat{T}_{\bm{\alpha}_{\mathrm{bs}}\bm{\alpha}_{\mathrm{bs}}\to
CD}|{\Psi_{\nu^{\prime}}^{(A)}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{bs}})}\rangle\otimes|{\Psi_{\nu}^{(B)}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{bs}})}\rangle$
$\displaystyle=\sqrt{P_{0}^{(\nu^{\prime}\nu)}}|{\mathbf{0},\mathbf{0}}\rangle_{CD}\otimes|{\Psi_{\nu^{\prime}}^{(A)}(\bm{\alpha}_{\mathrm{qm}})}\rangle\otimes|{\Psi_{\nu}^{(B)}(\bm{\alpha}_{\mathrm{qm}})}\rangle$
$\displaystyle+\sum_{\mu=\pm}\sqrt{P_{\mu}^{(\nu^{\prime}\nu)}}\Bigl{\\{}|{{\Phi}_{\mu^{\prime}}^{(C)}(\bm{\gamma}_{\mathrm{bs}})}\rangle\otimes|{\mathbf{0}}\rangle_{D}\otimes|{\Psi_{\mu}^{(AB)}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{qm}})}\rangle$
$\displaystyle+\nu|{\mathbf{0}}\rangle_{C}\otimes|{\Phi_{\mu^{\prime}}^{(D)}(\bm{\gamma}_{\mathrm{bs}})}\rangle\otimes|{\Psi_{\mu}^{(AB)}(\bm{\alpha}_{\mathrm{qm}},-\bm{\alpha}_{\mathrm{qm}})}\rangle\Bigr{\\}},$
(16)

where $P_{\mu}^{(\nu^{\prime}\nu)}$ is the probability for the states
$|{\Phi_{\mu^{\prime}}(\bm{\gamma}_{\mathrm{bs}})}\rangle_{C}\otimes|{\mathbf{0}}\rangle_{D}$
and
$|{\mathbf{0}}\rangle_{C}\otimes|{\Phi_{\mu^{\prime}}(\bm{\gamma}_{\mathrm{bs}})}\rangle_{D}$
to be detected at the output ports of the beam splitter, whereas
$P_{0}^{(\nu^{\prime}\nu)}$ is the probability to detect the vacuum state
$|{\mathbf{0}}\rangle_{C}\otimes|{\mathbf{0}}\rangle_{D}$. So, we have

$\displaystyle
2P_{+}^{(\nu^{\prime}\nu)}+2P_{-}^{(\nu^{\prime}\nu)}+P_{0}^{(\nu^{\prime}\nu)}=1$
(17)

and the expressions for $P_{\mu}^{(\nu^{\prime}\nu)}$ are given by

$\displaystyle P_{\mu}^{(\nu^{\prime}\nu)}=\mathrm{Prob}(\mu|\nu^{\prime}\nu)$
$\displaystyle=\frac{1}{4}\frac{\tilde{M}_{\mu^{\prime}}(\bm{\gamma}_{\mathrm{bs}})M_{\mu}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{qm}})}{M_{\nu^{\prime}}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{bs}})M_{\nu}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{bs}})},\quad\mu\in\\{+,-\\},$
(18a) $\displaystyle
P_{0}^{(\nu^{\prime}\nu)}=\mathrm{Prob}(0|\nu^{\prime}\nu)$
$\displaystyle=\frac{M_{\nu^{\prime}}(\bm{\alpha}_{\mathrm{qm}})M_{\nu}(\bm{\alpha}_{\mathrm{qm}})}{M_{\nu^{\prime}}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{bs}})M_{\nu}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{bs}})}\exp(-|\bm{\gamma}_{\mathrm{bs}}|^{2}).$
(18b)

In the case, where the modulated cat states are identical with
$\nu=\nu^{\prime}$, formulas (18) can be simplified giving the probabilities
$P_{\mu}^{(\nu\nu)}\equiv P_{\mu}^{(\nu)}$ in the form

$\displaystyle P_{\pm}^{(+)}=\tanh^{2}{|{\alpha}|^{2}}P_{\pm}^{(-)},$ (19a)
$\displaystyle|\alpha|^{2}=|\bm{\alpha}|^{2}=|\bm{\alpha}_{\mathrm{qm}}|^{2}+|\bm{\alpha}_{\mathrm{bs}}|^{2},\quad|\bm{\alpha}_{\mathrm{bs}}|^{2}=r_{\mathrm{bs}}|\bm{\alpha}|^{2}$
(19b) $\displaystyle
P_{-}^{(-)}(r_{\mathrm{bs}},|{\alpha}|^{2})=\frac{\sinh(2(1-r_{\mathrm{bs}})|{\alpha}|^{2})\sinh(2r_{\mathrm{bs}}|{\alpha}|^{2})}{4\sinh^{2}(|{\alpha}|^{2})},$
(19c) $\displaystyle
P_{+}^{(-)}(r_{\mathrm{bs}},|{\alpha}|^{2})=\frac{\cosh(2(1-r_{\mathrm{bs}})|{\alpha}|^{2})\sinh^{2}(r_{\mathrm{bs}}|{\alpha}|^{2})}{2\sinh^{2}(|{\alpha}|^{2})},$
(19d) $\displaystyle
P_{0}^{(-)}(r_{\mathrm{bs}},|{\alpha}|^{2})=\frac{\sinh^{2}((1-r_{\mathrm{bs}})|{\alpha}|^{2})}{\sinh^{2}(|{\alpha}|^{2})},$
$\displaystyle
P_{0}^{(+)}(r_{\mathrm{bs}},|{\alpha}|^{2})=\frac{\cosh^{2}((1-r_{\mathrm{bs}})|{\alpha}|^{2})}{\cosh^{2}(|{\alpha}|^{2})},$
(19e)

where the parameter $r_{\mathrm{bs}}$ is the ratio of the average number of
photons for the modes sent to the beam splitter,
$|\bm{\alpha}_{\mathrm{bs}}|^{2}/|\bm{\alpha}|^{2}$, and the input mean photon
number, $|\bm{\alpha}|^{2}=|\alpha|^{2}$. In the opposite case with modulated
cat states that differ in symmetry, the probabilities
$P_{\mu}^{(+-)}=P_{\mu}^{(-+)}\equiv P_{\mu}^{(c)}$ are given by

$\displaystyle
P_{-}^{(c)}(r_{\mathrm{bs}},|{\alpha}|^{2})=\frac{\sinh(2(1-r_{\mathrm{bs}})|{\alpha}|^{2})\sinh^{2}(r_{\mathrm{bs}}|{\alpha}|^{2})}{\sinh(2|{\alpha}|^{2})},$
(20a) $\displaystyle
P_{+}^{(c)}(r_{\mathrm{bs}},|{\alpha}|^{2})=\frac{\cosh(2(1-r_{\mathrm{bs}})|{\alpha}|^{2})\sinh(2r_{\mathrm{bs}}|{\alpha}|^{2})}{2\sinh(2|{\alpha}|^{2})},$
(20b) $\displaystyle
P_{0}^{(c)}(r_{\mathrm{bs}},|{\alpha}|^{2})=\frac{\sinh(2(1-r_{\mathrm{bs}})|{\alpha}|^{2})}{\sinh(2|{\alpha}|^{2})}.$
(20c)

From Eq. (19b), the probabilities for two antisymmetric cat states
$P_{\pm}^{(-)}$ are higher than those describing the case of two symmetric
cats, $P_{\pm}^{(+)}$. In particular, at small values of $|\alpha|$, the
latter is proportional to $|\alpha|^{2}$, whereas, for $P_{\pm}^{(-)}$, we
have

$\displaystyle P^{(-)}_{-}\approx r_{\mathrm{bs}}(1-r_{\mathrm{bs}}),\quad
P^{(-)}_{+}\approx r_{\mathrm{bs}}^{2}/2.$ (21)

Note that the relations

$\displaystyle P^{(c)}_{-}\approx
r_{\mathrm{bs}}(1-r_{\mathrm{bs}})^{2}|\alpha|^{4},\quad P^{(c)}_{+}\approx
r_{\mathrm{bs}}/2$ (22)

describe behavior of $P_{\pm}^{(c)}$ in the region of small amplitudes
$|\alpha|$.

Now we briefly discuss how the photon number ratio $r_{\mathrm{bs}}$ affects
$|{\alpha}|^{2}$-dependence of the probabilities. Figure 2 shows the surfaces
representing the probabilities $P_{\pm}^{(-)}$ and $P_{\pm}^{(c)}$ plotted in
the $r_{\mathrm{bs}}$-$|\alpha|^{2}$ plane.

At $0<r_{\mathrm{bs}}<1$, all the probabilities $P_{\pm}^{(+)}$,
$P_{\pm}^{(-)}$ and $P_{\pm}^{(c)}$ approach the value $1/4$ in the limit of
large amplitudes with $|\alpha|\to\infty$. They also vanish provided the ratio
$r_{\mathrm{bs}}$ is zero. Obviously, in this case, we cannot produce
entangled states.

From Eqs. (19) and (20), at $r_{\mathrm{bs}}=1$, the probabilities
$P_{-,0}^{(-)}$ and $P_{-,0}^{(c)}$ equal zero, whereas
$P_{+}^{(-)}=P_{+}^{(c)}=1/2$. Another case where the probabilities
$P_{-}^{(-)}$ and $P_{+}^{(c)}$ (see Eqs. (19c) and (20b)) are independent of
$|\alpha|$, $P_{-}^{(-)}=P_{+}^{(c)}=1/4$ , occurs at the ratio equal to one
$1/2$, $r_{\mathrm{bs}}=1/2$.

##  III Photodetection and decoherence

In the previous section, we have found that generation of the entangled cat
states is heralded by outcomes of measurements performed at the relay node in
the orthonormal basis
$\\{|{\Phi_{+}}\rangle,|{\Phi_{-}}\rangle,|{\mathbf{0}}\rangle\\}$. Such
measurements would perfectly discriminate between the states at the output
ports of the beam splitter producing the entangled states,
$|{\Psi^{(AB)}_{\pm}}\rangle$, shared by the remote nodes of the link with the
probabilities $P_{\pm}^{(\nu\nu^{\prime})}$ conditioned on the symmetry of the
input local states:
$|{\Psi^{(A)}_{\nu^{\prime}}}\rangle\otimes|{\Psi^{(B)}_{\nu}}\rangle$. In
this section, we extend our analysis to the cases where outcomes of the
measurements are determined by the statistics of photocounts registered by the
photodetector and decoherence-induced effects are taken into consideration.

###  III.1 Probability of photocounts

We begin with the decoherence effects for the states transmitted through the
fiber channels from Alice and Bob to the beam splitter (central) node. The
decoherence processes will affect the output states of the beam splitter,
$|{\Phi_{\pm}^{(S)}(\bm{\gamma}_{\mathrm{bs}})}\rangle=|{\Phi_{\pm}^{(S)}(\gamma)}\rangle$,
where $S$ stands for the output channel of the beam splitter ($S$ is either
$C$ or $D$), that enter the right hand side of Eq. (II). For simplicity, we
restrict ourselves to the one-mode case with
$|{\Phi_{\pm}^{(S)}(\bm{\gamma}_{\mathrm{bs}})}\rangle=|{\Phi_{\pm}^{(S)}(\gamma)}\rangle$
and assume that the noisy channels are identical and represented by the well-
known pure-loss bosonic channel that can be described by introducing
additional environmental (ancillary) mode (see, e.g., Refs. [30, 25]). An
isometry representing the Stinespring dilation of the fiber channel where the
signal mode is supplemented with an extra system $E$ (the channel’s
environment) is given by the following mapping

$\displaystyle|{\gamma}\rangle_{S}\to|{\gamma_{s}}\rangle_{S}\otimes|{\gamma_{e}}\rangle_{E}=|{\gamma_{s},\gamma_{e}}\rangle\equiv|{\bm{\gamma}}\rangle,$
$\displaystyle|\gamma|^{2}=|\gamma_{s}|^{2}+|\gamma_{e}|^{2},$ (23)

where $|\gamma_{s}|^{2}=\eta|\gamma|^{2}$ and
$|\gamma_{e}|^{2}=(1-\eta)|\gamma|^{2}$;  $\eta=\exp(-L/L_{\mathrm{att}})$ is
the fiber transmittance, $L$ is the fiber length and $L_{\mathrm{att}}$ is the
attenuation length. Note that, the pure-loss channel provides a simplified
model of light propagation in a fiber, where a number of polarization and
dispersion dependent effects are not taken in account.

One of the methods to discriminate between the symmetric and antisymmetric cat
states is to perform photocounting measurements that may distinguish the
parity of the photon number registered by a photodetector. For a photon-number
resolving photodetector, the probability to detect $k$ photons in the signal
mode conditioned on the event that the initial state
$|{\Psi_{\nu^{\prime}}^{(A)}}\rangle\otimes|{\Psi_{\nu}^{(B)}}\rangle$ is
projected onto the entangled state $|{\Psi_{\mu}^{(AB)}}\rangle$

$\displaystyle\mathrm{Prob}(k|\mu,\nu^{\prime}\nu)=P(k|\mu^{\prime}),\quad\mu^{\prime}=\nu^{\prime}\nu\mu$
(24)

that expressed in terms of the probability of clicks $P(k|\mu)$ determined by
the statistics of photocounts for the cat states
$|{\Phi_{\mu}(\bm{\gamma})}\rangle$. According to the well-known Kelley-
Kleiner formula [69, 70], this probability reads

$\displaystyle
P(k|\mu)=\langle{\Phi_{\mu}(\bm{\gamma})}|\hat{\Pi}_{k}|{\Phi_{\mu}(\bm{\gamma})}\rangle,\quad
P(k|0)=\delta_{k0},$ (25)

where $\hat{\Pi}_{k}$ is the positive-operator-valued measure for a number-
resolving detector given by

$\displaystyle\hat{\Pi}_{k}=:\frac{(\xi\hat{n}_{s})^{k}}{k!}\mathrm{e}^{-\xi\hat{n}_{s}}:,\quad\hat{n}_{s}={\hat{a}}^{\dagger}_{s}\hat{a}_{s},$
(26)

where $\xi$ is the efficiency of the detector. We can now apply the algebraic
identity

$\displaystyle\mathop{\rm
Tr}\nolimits_{E}|{\bm{\gamma}^{(\pm)}}\rangle\langle{\bm{\gamma}^{(\pm)}}|=\frac{1}{4}\left\\{M_{\pm}(\gamma_{e})|{\gamma_{s}^{(+)}}\rangle\langle{\gamma_{s}^{(+)}}|\right.$
$\displaystyle\left.+M_{\mp}(\gamma_{e})|{\gamma_{s}^{(-)}}\rangle\langle{\gamma_{s}^{(-)}}|\right\\}$
(27)

to deduce the following expressions of the probabilities (25)

$\displaystyle
P(k|+)=\frac{1}{4\tilde{M}_{+}(\bm{\gamma})}\bigl{\\{}M_{+}(\gamma_{e})C_{+}(k,\gamma_{s})$
$\displaystyle+M_{-}(\gamma_{e})C_{-}(k,\gamma_{s})\bigr{\\}},\quad k>0,$ (28)
$\displaystyle
P(k|-)=\frac{1}{4{M}_{-}(\bm{\gamma})}\bigl{\\{}M_{-}(\gamma_{e})C_{+}(k,\gamma_{s})$
$\displaystyle+M_{+}(\gamma_{e})C_{-}(k,\gamma_{s})\bigr{\\}},\quad P(k|0)=0,$
(29)

where

$\displaystyle
C_{\mu}(k,\gamma_{s})=\langle{\gamma_{s}^{(\mu)}}|\hat{\Pi}_{k}|{\gamma_{s}^{(\mu)}}\rangle=2\frac{(\xi|\gamma_{s}|^{2})^{k}}{k!}\mathrm{e}^{-|\gamma_{s}|^{2}}$
$\displaystyle\times\bigl{\\{}\mathrm{e}^{(1-\xi)|\gamma_{s}|^{2}}+(-1)^{k}\mu\mathrm{e}^{-(1-\xi)|\gamma_{s}|^{2}}\bigr{\\}}.$
(30)

In what follows, we consider the heralding outcomes described by the two
different parity of clicks, $p_{c}\in\\{\mathrm{even},\mathrm{odd}\\}$, and
thus discriminate between the cases, where the number of registered photons is
either odd or even. The sum of the corresponding probabilities

$\displaystyle P(\mathrm{even}|\pm,0)=\sum_{n=1}^{\infty}P(2n|\pm,0),$
$\displaystyle P(\mathrm{odd}|\pm,0)=\sum_{n=0}^{\infty}P(2n+1|\pm,0),$ (31)

and the no-click probability, $P(0|\mu)$, gives unity and we have the
completeness identity

$\displaystyle P(0|\pm,0)+P(\mathrm{even}|\pm,0)+P(\mathrm{odd}|\pm,0)=1.$
(32)

By substituting Eqs. (III.1)–(III.1) into relations (31) we obtain the
expressions for the probabilities of the detection outcomes determined by the
parity of clicks

$\displaystyle
P(p_{c}|+)=\frac{1}{4\tilde{M}_{+}(\bm{\gamma})}\bigl{\\{}M_{+}(\gamma_{e})C_{+}(p_{c},\gamma_{s})$
$\displaystyle+M_{-}(\gamma_{e})C_{-}(p_{c},\gamma_{s})\bigr{\\}},\quad
p_{c}\in\\{\mathrm{even},\mathrm{odd}\\},$ (33a) $\displaystyle
P(p_{c}|-)=\frac{1}{4{M}_{-}(\bm{\gamma})}\bigl{\\{}M_{-}(\gamma_{e})C_{+}(p_{c},\gamma_{s})$
$\displaystyle+M_{+}(\gamma_{e})C_{-}(p_{c},\gamma_{s})\bigr{\\}},\quad
P(p_{c}|0)=0,$ (33b)

where

$\displaystyle
C_{\pm}(\mathrm{even},\gamma_{s})=2\mathrm{e}^{-|\gamma_{s}|^{2}}(\cosh(\xi|\gamma_{s}|^{2})-1)$
$\displaystyle\times\bigl{\\{}\mathrm{e}^{(1-\xi)|\gamma_{s}|^{2}}\pm\mathrm{e}^{-(1-\xi)|\gamma_{s}|^{2}}\bigr{\\}},$
(34a) $\displaystyle
C_{\pm}(\mathrm{odd},\gamma_{s})=2\mathrm{e}^{-|\gamma_{s}|^{2}}\sinh(\xi|\gamma_{s}|^{2})$
$\displaystyle\times\bigl{\\{}\mathrm{e}^{(1-\xi)|\gamma_{s}|^{2}}\mp\mathrm{e}^{-(1-\xi)|\gamma_{s}|^{2}}\bigr{\\}}.$
(34b)

(a)

(b)

Figure 3: Conditional probabilities (a) $P(\mathrm{even}|+)$ and (b)
$P(\mathrm{odd}|-)$ plotted against the mean photon number $|\gamma_{s}|^{2}$
and the channel transmittance $\eta$ at different values of the detection
efficiency $\xi$.

It can be checked that these relations meet the algebraic identity

$\displaystyle\frac{P(\mathrm{even}|-)P_{-}^{(\pm)}}{P(\mathrm{even}|+)P_{+}^{(\pm)}}=\frac{P(\mathrm{odd}|+)P_{-}^{(c)}}{P(\mathrm{odd}|-)P_{+}^{(c)}}.$
(35)

They can also be readily generalized to the case of multimode signal, where
$|{\gamma_{s}}\rangle\to|{\bm{\gamma}_{s}}\rangle\equiv|{\gamma_{1}^{(s)},\ldots,\gamma_{N_{s}}^{(s)}}\rangle$
and $N_{s}$ is the number of signal modes registered by a broadband detector,
by replacing $|\gamma_{s}|^{2}$ and $\xi|\gamma_{s}|^{2}$ with
$|\bm{\gamma}_{s}|^{2}=\sum_{i=1}^{N_{s}}|\gamma_{i}^{(s)}|^{2}$ and
$\sum_{i=1}^{N_{s}}\xi_{i}|\gamma_{i}^{(s)}|^{2}$.

Referring to Fig. 3, it is seen that, for the limiting case of the ideal
photodetector and the lossless channel with the efficiency and the
transmittance both equal to unity, $\xi=\eta=1$, the probability of odd (even)
number of clicks vanishes for the symmetric (antisymmetric) states,
$P(\mathrm{odd}|+)=0$ ($P(\mathrm{even}|-)=0$), and thus the cat states are
perfectly distinguishable with $P(\mathrm{even}|+)=1$ ($P(\mathrm{odd}|-)=1$).
It can also be noted that low detector efficiency $\xi$ has a noticeable
detrimental effect on the conditional probabilities.

###  III.2 Performance and entanglement

We can now evaluate the probability of success for the heralding outcomes
labeled by the parity $p_{c}$. Similar to all other probabilities, this
probability is conditioned on the symmetry of the initial states and can be
written in the following general form:

$\displaystyle
P_{s}^{(\nu^{\prime}\nu)}(p_{c})=\sum_{\mu=\pm}\mathrm{Prob}(p_{c}|\mu,\nu^{\prime}\nu)\mathrm{Prob}(\mu|\nu^{\prime}\nu)$
$\displaystyle=\sum_{\mu=\pm}P(p_{c}|\mu^{\prime})P^{(\nu^{\prime}\nu)}_{\mu}.$
(36)

After substituting formulas given by Eqs. (20)–(19) and (33) into Eq. (36) and
performing cumbersome but rather straightforward algebra, it can be shown that
$P_{\pm}^{(\nu^{\prime}\nu)}$ with the photon number ratio $r_{\mathrm{bs}}$
replaced by the product $\zeta=\xi\eta r_{\mathrm{bs}}$ give explicit
expressions for the probabilities of success $P_{s}^{(\nu^{\prime}\nu)}$ as
follows

$\displaystyle P_{s}^{(+)}(p_{c})=\tanh^{2}(|\alpha|^{2})P_{s}^{(-)}(p_{c}),$
(37a) $\displaystyle
P_{s}^{(-)}(\mathrm{odd})=P_{-}^{(-)}(\zeta,|\alpha|^{2}),\>P_{s}^{(c)}(\mathrm{odd})=P_{+}^{(c)}(\zeta,|\alpha|^{2}),$
(37b) $\displaystyle
P_{s}^{(-)}(\mathrm{even})=P_{+}^{(-)}(\zeta,|\alpha|^{2}),\>P_{s}^{(c)}(\mathrm{even})=P_{-}^{(c)}(\zeta,|\alpha|^{2}),$
(37c)

where $P_{\mp}^{(-)}$ and $P_{\mp}^{(c)}$ are given by Eqs. (19c)–(19d) and
Eqs. (20a)–(20b), respectively.

(a)

(b)

(c)

(d)

Figure 4: The success probabilities, $P_{s}^{(-)}(p_{c})$ and
$P_{s}^{(c)}(p_{c})$, (the parity of photocounts is labeled by
$p_{c}\in\\{\mathrm{even,odd}\\}$), of outcomes that herald generation of
predominantly (a)-(b) symmetric and (c)-(d) antisymmetric cat states with
$F_{+}^{(\nu^{\prime}\nu)}>1/2$ and $F_{-}^{(\nu^{\prime}\nu)}>1/2$,
respectively. The curves are computed at
$r_{\mathrm{bs}}\in\\{r_{1/2},r_{1/2}\pm 0.2\\}$ with $\eta=0.95$ and
$\xi=0.9$, where $r_{1/2}\equiv(2\xi\eta)^{-1}$ ($\zeta=1/2$).

(a)

(b)

Figure 5: Fidelities of heralded symmetric and antisymmetric cat states: (a)
$F_{+}^{(\pm)}(\mathrm{even})$ and (b) $F_{-}^{(\pm)}(\mathrm{odd})$ as a
function of $|\alpha|^{2}$ computed at different values of $r_{\mathrm{bs}}$
(the parameters are listed in the caption of Fig. 4).

These relations imply that the results presented at the end of Sec. III is
directly applicable to the success probabilities. As is shown in Fig. 4, in
the limit of vanishing $\alpha$, the largest probabilities of success
$P_{s}^{(-)}(\mathrm{odd})$ and $P_{s}^{(c)}(\mathrm{odd})$ take the values
$\zeta(1-\zeta)$ and $\zeta/2$, respectively. As the amplitude $|\alpha|$
increases, the probabilities vary approaching the limiting value equal to
$1/4$.

Given the heralding event $p_{c}$, the heralded state is represented by the
density matrix

$\displaystyle\hat{\rho}_{AB}^{(\nu^{\prime}\nu)}(p_{c})=\sum_{\mu=\pm}F_{\mu}^{(\nu^{\prime}\nu)}(p_{c})$
$\displaystyle\times|{\Psi_{\mu}^{(AB)}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{qm}})}\rangle\langle{\Psi_{\mu}^{(AB)}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{qm}})}|,$
(38)

where
$F_{\mu}^{(\nu^{\prime}\nu)}=\langle{\Psi_{\mu}^{(AB)}}|\hat{\rho}_{AB}^{(\nu^{\prime}\nu)}|{\Psi_{\mu}^{(AB)}}\rangle$
are the fidelities that can be computed using the well-known Bayesian formula

$\displaystyle
F_{\mu}^{(\nu^{\prime}\nu)}(p_{c})=\mathrm{Prob}(\mu|p_{c},\nu^{\prime}\nu)=\frac{\mathrm{Prob}(p_{c}|\mu,\nu^{\prime}\nu)\mathrm{Prob}(\mu|\nu^{\prime}\nu)}{P_{s}^{(\nu^{\prime}\nu)}(p_{c})}$
$\displaystyle=\frac{P(p_{c}|\mu^{\prime})P_{\mu}^{(\nu^{\prime}\nu)}}{P_{s}^{(\nu^{\prime}\nu)}(p_{c})}$
(39)

and meet the symmetry condition

$\displaystyle
F_{\mu}^{(\pm)}(\mathrm{even}/\mathrm{odd})=F_{\mu}^{(c)}(\mathrm{odd}/\mathrm{even})$
(40)

which is a direct consequence of the identity (35). Similar to Eqs. (37a)–
(37c), after tedious algebra, we derive the following expressions for the
fidelities:

$\displaystyle F_{-}^{(\pm)}(\mathrm{odd})=F_{-}^{(c)}(\mathrm{even})$
$\displaystyle=\frac{\tanh(2(1-r_{\mathrm{bs}})|\alpha|^{2})}{\tanh(2(1-r_{\mathrm{bs}})|\alpha|^{2})+\tanh(2(r_{\mathrm{bs}}-\zeta)|\alpha|^{2})},$
(41a) $\displaystyle F_{+}^{(\pm)}(\mathrm{even})=F_{+}^{(c)}(\mathrm{odd})$
$\displaystyle=\frac{1}{1+\tanh(2(1-r_{\mathrm{bs}})|\alpha|^{2})\tanh(2(r_{\mathrm{bs}}-\zeta)|\alpha|^{2})}.$
(41b)

Note that, owing to the completeness condition
$F_{+}^{(\nu^{\prime}\nu)}(p_{c})+F_{-}^{(\nu^{\prime}\nu)}(p_{c})=1$ and the
symmetry relations (40), we need to specify only two fidelities. These
fidelities $F_{-}^{(\pm)}(\mathrm{odd})$ and $F_{+}^{(\pm)}(\mathrm{even})$
are both above 1/2 and represent the predominantly antisymmetric and symmetric
cat states, respectively. From $|\alpha|^{2}$-dependencies of
$F_{-}^{(\pm)}(\mathrm{odd})$ and $F_{+}^{(\pm)}(\mathrm{even})$ depicted Fig.
5, both the fidelities start from their zero-amplitude values

$\displaystyle F_{+}^{(\pm)}(\mathrm{even},\alpha=0)=1,\quad
F_{-}^{(\pm)}(\mathrm{odd},\alpha=0)=\frac{1-r_{\mathrm{bs}}}{1-\zeta}.$ (42)

and decay approaching $1/2$ with the amplitude $|\alpha|$.

At $r_{\mathrm{bs}}=r_{1/2}\equiv(2\eta\xi)^{-1}$, we have $\zeta=1/2$, so
that $2P_{s}^{(c)}(\mathrm{odd})$ and $2P_{s}^{(-)}(\mathrm{odd})$ equal $1/2$
(see Fig. 4) as the probabilities of the most probable outcomes heralding
generation of the symmetric and antisymmetric cat states with the fidelities
$F_{+}^{(\pm)}(\mathrm{even})$ and $F_{-}^{(\pm)}(\mathrm{odd})$,
respectively. In this case, formula (42) gives unity and $1/(2-\eta\xi)$ as
the maximum values of $F_{+}^{(\pm)}(\mathrm{even})$ and
$F_{-}^{(\pm)}(\mathrm{odd})$.

Interestingly, $F_{+}^{(\pm)}(\mathrm{even},\alpha=0)$ does not depend on the
parameters, whereas the maximum value of $2P_{s}^{(c)}(\mathrm{odd})$ is
$\zeta$ provided the photon number ratio $r_{\mathrm{bs}}$, is above
$(2\eta\xi)^{-1}$. By contrast, when $r_{\mathrm{bs}}\neq(2\eta\xi)^{-1}$, the
probability $2P_{s}^{(-)}(\mathrm{odd})$ (it equals $(1-(2\zeta-1)^{2})/2$ at
$\alpha=0$) cannot exceed one half and the fidelity
$F_{-}^{(\pm)}(\mathrm{odd},\alpha=0)$ grows as the ratio $r_{\mathrm{bs}}$
becomes smaller than $(2\eta\xi)^{-1}$.

##  IV Entanglement swapping

In this section, we shall discuss the entanglement swapping protocol that uses
the heralded entanglement generation procedure studied in the previous section
to create entanglement between remote points of two elementary links,
$A_{1}-B_{1}$ and $B_{2}-A_{2}$, depicted in Fig. 6.  In our subsequent
analysis, we consider the structure where the neighboring end-nodes, $B_{1}$
and $B_{2}$, of the adjacent links, $A_{1}-B_{1}$ and $B_{2}-A_{2}$, are
located at the repeater station (the repeater node $B_{1}B_{2}$).

According to Eq. (III.2)), the phase-modulated entangled states shared between
the nodes of the links are described by the density matrices of the form:

$\displaystyle\hat{\rho}_{A_{1}B_{1}}^{(\nu^{\prime}_{1}\nu_{1})}(p_{c})\equiv\hat{\rho}_{1}=\sum_{\mu_{1}=\pm}F_{\mu_{1}}^{(1)}$
$\displaystyle\times|{\Psi_{\mu_{1}}^{(A_{1}B_{1})}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{qm}})}\rangle\langle{\Psi_{\mu_{1}}^{(A_{1}B_{1})}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{qm}})}|,$
(43a)
$\displaystyle\hat{\rho}_{A_{2}B_{2}}^{(\nu^{\prime}_{2}\nu_{2})}(p_{c})\equiv\hat{\rho}_{2}=\sum_{\mu_{2}=\pm}F_{\mu_{2}}^{(2)}$
$\displaystyle\times|{\Psi_{\mu_{2}}^{(A_{2}B_{2})}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{qm}})}\rangle\langle{\Psi_{\mu_{1}}^{(A_{2}B_{2})}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{qm}})}|,$
(43b)

where $F_{\mu_{i}}^{(i)}\equiv F_{\mu_{i}}^{(\nu^{\prime}_{i}\nu_{i})}$, and
$\hat{\rho}_{1}\otimes\hat{\rho}_{2}$ is the initial state of the two-link
system. So, the initial state is the statistical ensemble of the quantum
states

$\displaystyle|{\Psi_{\mu_{1}\mu_{2}}^{(A_{1}B_{1},A_{2}B_{2})}}\rangle$
$\displaystyle\equiv|{\Psi_{\mu_{1}}^{(A_{1}B_{1})}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{qm}})}\rangle\otimes|{\Psi_{\mu_{2}}^{(A_{2}B_{2})}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{qm}})}\rangle$
(44)

with the probabilities $F_{\mu_{1}}^{(1)}F_{\mu_{2}}^{(2)}$.

Figure 6: Entanglement swapping scheme. $A_{1}-B_{1}$ and $B_{2}-A_{2}$ are
adjacent elementary links.

When the modes stored in the quantum memories of the sites $B_{1}$ and $B_{2}$
are released and sent to the beam splitter, the transformed state for each
member of the ensemble is given by general formula (II). Similar to Sec.
III.1, we assume that the heralding events of the photocounting measurements
are determined by the parity of photodetector clicks, so that we may closely
follow the line of reasoning presented in Sec. III.1 and III.2 to deduce the
heralded state for each state
$|{\Psi_{\mu_{1}\mu_{2}}^{(A_{1}B_{1},A_{2}B_{2})}}\rangle$ in the modified
form of Eq. (III.2) with $|{\Psi_{\mu}^{(AB)}}\rangle$ and $(\nu^{\prime}\nu)$
replaced with $|{\Psi_{\mu}^{(A_{1}A_{2})}}\rangle$ and $(\mu_{1}\mu_{2})$,
respectively.

As for elementary links, the fidelities,
$\tilde{F}^{(\mu_{1}\mu_{2})}_{\mu}(p_{c})$, and and the probabilities,
$\tilde{P}^{(\mu_{1}\mu_{2})}_{s}(p_{c})$, at the repeater node, can be
evaluated using Eq. (41) and Eq. (37), respectively. Since, at this node, the
modes are evenly partitioned with
$\bm{\alpha}_{\mathrm{bs}}=\bm{\alpha}_{\mathrm{qm}}$ and
$|\bm{\alpha}_{\mathrm{bs}}|^{2}=|\bm{\alpha}_{\mathrm{qm}}|^{2}=(1-r_{\mathrm{bs}})|\alpha|^{2}$,
the parameters $\\{r_{\mathrm{bs}},|\alpha|^{2}\\}$ should be accordingly
changed as follows:
$\\{r_{\mathrm{bs}},|\alpha|^{2}\\}\to\\{1/2,2(1-r_{\mathrm{bs}})|\alpha|^{2}\\}$.

For the measurements performed locally at the repeater node, the fiber
transmission losses are negligible and the bulk part of losses can be
attributed to inefficiency of the quantum memories. In order to take into
account such losses, we, following a widely used phenomenological approach
(see, e.g., Refs [71, 21, 72]), shall introduce the memory efficiency,
$\eta_{\mathrm{m}}$, as the parameter that replaces the transmission
coefficient $\eta$. The latter implies that, similar to the effects of the
fiber transmission losses and inefficiency of the photodetectors, decoherence
effects in quantum memories are modelled using a pure-loss bosonic channel
(see Eq. (III.1) in Sec. III.1). As a result, at the repeater node, the
parameter $\zeta$ appears to be equal to the product $\eta_{\mathrm{m}}\xi/2$:
$\zeta\to\zeta_{\mathrm{m}}=\eta_{\mathrm{m}}\xi/2$.

The final result for the entanglement swapping transformation reads

$\displaystyle\hat{\rho}_{1}\otimes\hat{\rho}_{2}\to\hat{\rho}_{12}(p_{c})=\sum_{\mu=\pm}F_{\mu}^{(12)}(p_{c})$
$\displaystyle\times|{\Psi_{\mu}^{(A_{1}A_{2})}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{qm}})}\rangle\langle{\Psi_{\mu}^{(A_{1}A_{2})}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{qm}})}|,$
(45)

where $F_{\mu}^{(12)}(p_{c})$ is the fidelity given by

$\displaystyle
F_{\mu}^{(12)}(p_{c})=\frac{1}{P_{s}^{(12)}(p_{c})}\sum_{\mu_{1},\mu_{2}=\pm}P(p_{c}|\mu_{1}\mu_{2}\mu)P^{(\mu_{1}\mu_{2})}_{\mu}F_{\mu_{1}}^{(1)}F_{\mu_{2}}^{(2)}$
$\displaystyle=\frac{1}{P_{s}^{(12)}(p_{c})}\sum_{\mu_{1},\mu_{2}=\pm}\tilde{F}^{(\mu_{1}\mu_{2})}_{\mu}(p_{c})\tilde{P}^{(\mu_{1}\mu_{2})}_{s}(p_{c})F_{\mu_{1}}^{(1)}F_{\mu_{2}}^{(2)}.$
(46)

It is rather straightforward to deduce the expression for the probabilities of
success

$\displaystyle P_{s}^{(12)}(p_{c})$
$\displaystyle=\sum_{\mu,\mu_{1},\mu_{2}=\pm}\mathrm{Prob}(p_{c}|\mu,\mu_{1}\mu_{2})\mathrm{Prob}(\mu|\mu_{1}\mu_{2})\mathrm{Prob}(\mu_{1},\mu_{2})$
$\displaystyle=\sum_{\mu,\mu_{1},\mu_{2}=\pm}P(p_{c}|\mu_{1}\mu_{2}\mu)P^{(\mu_{1}\mu_{2})}_{\mu}F_{\mu_{1}}^{(1)}F_{\mu_{2}}^{(2)}$
$\displaystyle=\sum_{\mu_{1},\mu_{2}=\pm}\tilde{P}^{(\mu_{1}\mu_{2})}_{s}(p_{c})F_{\mu_{1}}^{(1)}F_{\mu_{2}}^{(2)},$
(47)

where $\tilde{F}^{(\mu_{1}\mu_{2})}_{\mu}(p_{c})$
($\tilde{P}^{(\mu_{1}\mu_{2})}_{s}(p_{c})$) is the above-discussed fidelity
(success probability) given by Eq. (41) (Eq. (37)) with the parameters
$\\{r_{\mathrm{bs}},\zeta,|\alpha|^{2}\\}$ changed to
$\\{1/2,\eta_{\mathrm{m}}\xi/2,2(1-r_{\mathrm{bs}})|\alpha|^{2}\\}$, that
determine performance of the protocol.

Our concluding remarks deal with analysis of the performance of a repeater
with two links. To this end, we shall follow the method presented in [71] and
start with the probabilities of success for the links

$\displaystyle\color[rgb]{0,0,0}2P_{s}^{(\nu^{\prime}_{1}\nu_{1})}(p_{c})|_{A_{1}B_{1}}\equiv
p_{1},\quad 2P_{s}^{(\nu^{\prime}_{2}\nu_{2})}(p_{c})|_{A_{2}B_{2}}\equiv
p_{2}$ (48)

that give the joint probability distribution for numbers of attempts is

$\displaystyle P(n_{1},n_{2})=p_{1}q_{1}^{n_{1}-1}p_{2}q_{2}^{n_{2}-1},$ (49)

where $q_{i}=1-p_{i}$. An important point is that the entanglement swapping
can be performed only after the entanglement is established in both links.
Similar to Ref. [71], we take the assumption that the time required for each
attempt is $T=L/c$,  where $L$ is the distance between the end-nodes and the
central node of an elementary link giving the length of the fiber channel and
$c$ is the speed of light. In our case, the length of the elementary link is
$L_{0}=2L$, whereas the total repeater length is
$L_{\mathrm{tot}}=N_{L}L_{0}$, where $N_{L}$ is the number of links.

From Eq. (48), it is not difficult to obtain the probability distribution for
the magnitude of the difference in number of attempts

$\displaystyle\mathrm{Prob}(|n_{1}-n_{2}|=k)=\frac{p_{1}p_{2}(q_{1}^{k}+q_{2}^{k})}{2(1-q_{1}q_{2})}(2-\delta_{k0})$
(50)

and evaluate the expectation values

$\displaystyle\langle{n_{w}}\rangle\equiv\langle{|n_{1}-n_{2}|}\rangle=\frac{p_{2}^{2}q_{1}+p_{1}^{2}q_{2}}{p_{1}p_{2}(1-q_{1}q_{2})},$
$\displaystyle\langle{n_{t}}\rangle\equiv\langle{n_{1}+n_{2}}\rangle=\frac{p_{1}+p_{2}}{p_{1}p_{2}},$
(51)
$\displaystyle\langle{n_{\mathrm{max}}}\rangle\equiv\langle{\max(n_{1},n_{2})}\rangle=\frac{\langle{n_{t}}\rangle+\langle{n_{w}}\rangle}{2},$
$\displaystyle\langle{n_{\mathrm{min}}}\rangle\equiv\langle{\min(n_{1},n_{2})}\rangle=\frac{\langle{n_{t}}\rangle-\langle{n_{w}}\rangle}{2}$
(52)

that are related to the preparation and waiting times, $T_{\mathrm{prep}}$ and
$T_{\mathrm{w}}$, as follows [71]

$\displaystyle T_{\mathrm{prep}}=\langle{n_{\mathrm{max}}}\rangle T,\quad
T_{\mathrm{w}}=\langle{n_{w}}\rangle T,\quad T=\frac{L}{c}.$ (53)

Figure 7: Dependence of waiting times, $T_{\mathrm{w}}^{(-)}$ and
$T_{\mathrm{w}}^{(c)}$, on $|\alpha|^{2}$ computed from Eq. (53) for swapping
between identical links with $p_{1}=p_{2}=2P_{s}^{(-)}(\mathrm{odd})$ and
$p_{1}=p_{2}=2P_{s}^{(c)}(\mathrm{odd})$, respectively. The quantum channel
transmittance is taken to be $\eta(L)=10^{-\kappa L/10}$ with $\kappa=0.2$
dB/km, and the photon number ratio $r_{\mathrm{bs}}$ is $0.2$.

For successful operation of the repeater, the waiting time is required to be
shorter than  the lifetime (alternatively, this time is called either the
dephasing time or the coherence time) of the quantum memory:
$T_{\mathrm{w}}<T_{\mathrm{m}}$. Under this condition, degradation of the
generated entanglement has a negligible detrimental effect on the swapping
protocol.

Figure 7 shows the $|\alpha|^{2}$-dependencies of the waiting times evaluated
from Eqs. (53) and (IV) assuming that the success probabilities for the links
are both equal to either $P_{s}^{(-)}(\mathrm{odd})$ (the waiting time is
$T_{\mathrm{w}}^{(-)}$) or $P_{s}^{(c)}(\mathrm{odd})$ (the waiting time is
$T_{\mathrm{w}}^{(c)}$). In our estimates, in addition to
$r_{\mathrm{bs}}=0.2$ and $\xi=0.9$, we have used the transmittance
$\eta(L)=10^{-\kappa L/10}$ with the attenuation coefficient $\kappa=0.2$
dB/km describing losses in an optical fiber. It can be seen that the estimated
waiting times that fall within the range of miliseconds may favorably compare
with the lifetimes reported in [71, 72, 73].

##  V Teleportation

It is well-known that shared entanglement is a resource of vital importance in
fundamental quantum communication protocols such as quantum teleportation. The
above discussed entanglement generation procedure provides this resource
through the phase-modulated cat states entangled between the remote nodes. In
this section, we demonstrate how such entanglement can be utilized to teleport
phase information from an SCW state resulting from the output state of the
electro-optic modulator (see Eq. (4)). In order to develop some intuition on
how the teleportation procedure works, we consider the simplest case where
Alice and Bob share a single-mode entangled coherent cat state

$\displaystyle|{\Psi_{\nu}^{(AB)}(\alpha,\alpha)}\rangle=\frac{1}{\sqrt{M_{\nu}(\alpha,\alpha)}}\left\\{|{\alpha,\alpha}\rangle_{AB}+\nu|{-\alpha,-\alpha}\rangle_{AB}\right\\},$
(54)

where $M_{\pm}(\alpha,\alpha)=2(1\pm\exp[-4|\alpha|^{2}])$ is the
normalization constant. Charlie at Alice’s site possesses the SCW state (see
Eq. (4)) with the amplitude $|\alpha|$ and the phase $\phi_{c}$ of the SCW
state which is targeted for teleportation

$\displaystyle|{\Psi_{\text{SCW}}}\rangle=|{\bm{\gamma}}\rangle_{C},\quad\gamma_{\mu}={U_{\mu
0}^{(C)}}^{\ast}\alpha\approx J_{\mu}(m_{C})e^{i\mu\phi_{c}}\alpha,$ (55)

where we assume that the phase coherence between Alice and Charlie is
maintained. Thus the joint state of the tripartite system Alice, Bob, and
Charlie is

$\displaystyle|{S_{CAB}^{(\nu)}}\rangle=|{\Psi_{\text{SCW}}}\rangle\otimes|{\Psi_{\nu}^{(AB)}(\alpha,\alpha)}\rangle,\>|{\Psi_{\text{SCW}}}\rangle=|{\bm{\gamma}}\rangle_{C}$
(56)

(a)

(b)

(c)

Figure 8:  a) The teleportation scheme to transfer the SCW phase from Charlie
to Bob using entangled Schrödinger cat state. PM is the phase modulator; FF is
the frequency filter; PS is the phase shifter; $D_{1}$ and $D_{2}$ are the
photodetectors. b) Fidelity between $|{i\alpha}\rangle$ and
$\frac{1}{\sqrt{2}}\left(|{\alpha}\rangle-i|{-\alpha}\rangle\right)$ as a
function of the amplitude $|\alpha|$ computed using Eq. (V). c) Dependence of
the fidelity given by Eq. (V) between $|{\alpha e^{i\phi}}\rangle$ and
$\frac{1}{\sqrt{2}}(|{\alpha}\rangle_{B}\mp i|{-\alpha}\rangle_{B})$ on the
phase $\phi$ at $|\alpha|=0.2$.

The scheme of the teleportation from Charlie to Bob is depicted in Fig. 8a.
Transfer of phase information from Charlie’s SCW state to Bob’s state can be
implemented in the following steps:

  1. 1.

Alice uses the modulator with the modulation frequency and index identical to
Charlie’s ones to modulate her share of the state (54) at the phase
$\phi_{a}$.

  2. 2.

Charlie’s and Alice’s states are interfered on $50:50$ beam splitter.

  3. 3.

For the selected sideband, the presence of a single photon at the output
channels of the beam splitter is monitored by two single photon detectors,
$\text{D}_{1}$ and $\text{D}_{2}$.

  4. 4.

After a photon is registered by a photodetector, Alice communicates the result
to Bob and he shifts the phase of his state depending on which detector is
clicked. Charlie’s SCW phase thus appears to be transferred to the phase of
the Bob’s single-mode coherent state.

Below we describe these steps in detail. Phase modulation applied by Alice
transforms the shared entangled state (56) as follows

$\displaystyle|{S_{CAB}^{(\nu)}}\rangle\mapsto|{\Psi_{CAB}^{(\nu)}}\rangle=|{\bm{\gamma}}\rangle_{C}\otimes|{\Psi_{\nu}^{(AB)}(\bm{\alpha},\beta)}\rangle,$
$\displaystyle\alpha_{\mu}={U_{\mu 0}^{(A)}}^{\ast}\alpha\approx
J_{\mu}(m_{C})e^{i\mu\phi_{a}}\alpha,$ (57)

where the modulation indices of Charlie and Alice are assumed to be identical,
so that $|U_{\mu 0}^{(A)}|=|U_{\mu 0}^{(C)}|\equiv|U_{\mu 0}|$ and
$J_{\mu}(m_{A})=J_{\mu}(m_{C})$.

After Alice’s and Charlie’s modes are brought into interference onto the 50:50
beam splitter, the output state after the beam splitter

$\displaystyle\hat{T}_{\bm{\gamma}\bm{\alpha}\to
D_{1}D_{2}}|{\Psi_{CAB}^{(\nu)}}\rangle=|{\Psi_{D_{1}D_{2}B}^{(\nu)}}\rangle$
$\displaystyle=\frac{1}{\sqrt{M_{\nu}(\alpha,\alpha)}}\bigr{\\{}|{\bm{\gamma}_{+}}\rangle_{D_{1}}\otimes|{\bm{\gamma}_{-}}\rangle_{D_{2}}\otimes|{\alpha}\rangle_{B}$
$\displaystyle+\nu|{\bm{\gamma}_{-}}\rangle_{D_{1}}\otimes|{\bm{\gamma}_{+}}\rangle_{D_{2}}\otimes|{-\alpha}\rangle_{B}\bigr{\\}},$
(58)

where $\bm{\gamma}_{\pm}=\frac{1}{\sqrt{2}}(\bm{\gamma}\pm\bm{\alpha})$, has
the two modes $|{\bm{\gamma}_{\pm}}\rangle_{D_{1}}$ and
$|{\bm{\gamma}_{\pm}}\rangle_{D_{2}}$ monitored by the detectors
$\text{D}_{1}$ and $\text{D}_{2}$, respectively.

For the sake of simplicity, in the limit of low amplitudes $|\alpha|\ll 1$, we
truncate the Hilbert space up to a single photon. For an ideal single photon
detector that measures the sideband with index $\mu$ at $D_{1}$ ($D_{2}$)
channel, detection of a single photon will project the state (V) onto the one-
photon state
$|{\Phi_{\mu}^{(1)}}\rangle=|{1_{\mu}}\rangle_{D_{1}}\otimes|{\mathbf{0}}\rangle_{D_{2}}$
(
$|{\Phi_{\mu}^{(2)}}\rangle=|{\mathbf{0}}\rangle_{D_{1}}\otimes|{1_{\mu}}\rangle_{D_{2}}$
), where $|{1_{\mu}}\rangle\equiv|{0,\ldots,0,1_{\mu},0,\ldots,0}\rangle$ is
the $\mu$th sideband one photon state, as follows

$\displaystyle|{\Psi_{D_{1}D_{2}B}^{(\nu)}}\rangle\mapsto|{\Psi_{B}^{(\nu,\mu,i)}}\rangle=\langle{\Phi_{\mu}^{(i)}|\Psi_{D_{1}D_{2}B}^{(\nu)}}\rangle/\sqrt{P_{\mu}^{(\nu)}},$
(59) $\displaystyle
P_{\mu}^{(\nu)}=|\langle{\Phi_{\mu}^{(i)}|\Psi_{D_{1}D_{2}B}^{(\nu)}}\rangle|^{2}=|U_{\mu
0}|^{2}P^{(\nu)},$ (60) $\displaystyle
P^{(\nu)}=\frac{|\alpha|^{2}}{1+\nu\mathrm{e}^{-4|\alpha|^{2}}}\mathrm{e}^{-2|\alpha|^{2}},$
(61)

where $P_{\mu}^{(\nu)}$ is the probability to detect a single photon in the
$\mu$th sideband by a photodetector. Since
$\gamma_{\mu}\pm\alpha_{\mu}=|U_{\mu}|\alpha(\exp(i\mu\phi_{c})\pm\exp(i\mu\phi_{a}))$,
the post-measurement state of Bob (up to the global phase) can be written in
the following explicit form

$\displaystyle|{\Psi_{B}^{(\nu,\mu,1)}}\rangle\equiv|{\Psi_{B}^{(\nu\phi)}}\rangle=\cos(\nu\phi/2)|{\alpha}\rangle+i\sin(\nu\phi/2)|{-\alpha}\rangle,$
$\displaystyle\phi=\mu(\phi_{c}-\phi_{a}),$ (62)
$\displaystyle|{\Psi_{B}^{(\nu,\mu,2)}}\rangle=|{\Psi_{B}^{(\nu,\mu,1)}}\rangle\Bigr{|}_{\alpha\to-\alpha}.$
(63)

When the clicked detector is changed from $D_{1}$ to $D_{2}$, relation (63)
shows that Bob needs to apply the $\pi$ shift changing $\alpha$ to $-\alpha$
so as to have the same teleported state.

$\phi_{c}$ |  $D_{1}$ clicks |  $D_{2}$ clicks
---|---|---
+1 sideband | -1 sideband | +1 sideband | -1 sideband
$\phi_{a}$ | $|{\alpha}\rangle$ | $|{\alpha}\rangle$ | $|{-\alpha}\rangle$ | $|{-\alpha}\rangle$
$\phi_{a}+\pi$ | $|{-\alpha}\rangle$ | $|{-\alpha}\rangle$ | $|{\alpha}\rangle$ | $|{\alpha}\rangle$
$\phi_{a}+\pi/2$ | $|{-i\alpha}\rangle$ | $|{i\alpha}\rangle$ | $|{i\alpha}\rangle$ | $|{-i\alpha}\rangle$
$\phi_{a}+3\pi/2$ | $|{i\alpha}\rangle$ | $|{-i\alpha}\rangle$ | $|{-i\alpha}\rangle$ | $|{i\alpha}\rangle$
Table 1: Truth table for the heralded state at Bob’s site $|{\Psi_{B}}\rangle$
after performing teleportation with entangled coherent state
$|{\Psi_{+}^{(AB)}}\rangle$ depending on the sideband registered, the
photodetector that clicks, and the value of Charlie’s SCW phase.

(a)

(b)

Figure 9:  Dependence of the success probabilities of teleportation,
$P^{(\pm)}_{\mathrm{tlp}}$, on the squared amplitude $|\alpha|^{2}$ computed
from Eq. (70).

For the special case, where $\phi_{c}-\phi_{a}\in\\{0,\pi/2,\pi,3\pi/2\\}$ and
$\mu=\pm 1$, Table 1 summarizes how Charlie’s SCW phase is mapped into the
phase of Bob’s coherent state provided Alice’s phase vanishes ($\phi_{a}=0$)
and the cat state is symmetric ($\nu=1$). Clearly, the teleported state
exactly reproduces Charlie’s phase when it is either zero or $\pi$. In this
table, we have used the approximation
$\frac{1}{\sqrt{2}}(|{\alpha}\rangle_{B}\pm
i|{-\alpha}\rangle_{B})\approx|{\mp i\alpha}\rangle_{B}$ with the fidelity
plotted in Figs. 8b-c.

More generally, in the low amplitude (mean photon number) region where
$|\alpha|$ is small, the teleported state can be well approximated by the
coherent state

$\displaystyle|{\Psi_{B}^{(\phi)}}\rangle=\cos(\phi/2)|{\alpha}\rangle+i\sin(\phi/2)|{-\alpha}\rangle\approx|{\mathrm{e}^{-i\phi}\alpha}\rangle.$
(64)

The latter can be seen from the fidelity between $|{\Psi_{B}^{(\phi)}}\rangle$
and $|{\mathrm{e}^{-i\psi}\alpha}\rangle$ given by

$\displaystyle|\langle{\mathrm{e}^{-i\psi}\alpha|\Psi_{B}^{(\phi)}}\rangle|^{2}=\mathrm{e}^{-2|\alpha|^{2}}\bigl{|}\cos(\phi/2)\exp(\mathrm{e}^{i\psi}|\alpha|^{2})$
$\displaystyle+i\sin(\phi/2)\exp(-\mathrm{e}^{i\psi}|\alpha|^{2})\bigr{|}^{2}$
$\displaystyle\approx 1+2(\cos(\phi-\psi)-1)|\alpha|^{2}+\ldots,$ (65)

where the last equality is the fidelity expanded into a power series over
$|\alpha|$ up to second order. Figure 8b illustrates that, at $\phi=-\pi/2$,
the fidelity can be higher than 99% when the amplitude $|\alpha|$ is smaller
than 0.25.

It is rather straightforward to generalize our analysis to the non-symmetric
case, where the amplitudes, $|\gamma|$, $|\alpha|$ and $|\beta|$, that
determine Charlie’s, Alice’s, and Bob’s states, respectively, differ from each
other. In this case, Alice and Bob share the state

$\displaystyle|{S_{AB}^{(\nu)}}\rangle=|{\Psi_{\nu}^{(AB)}(\alpha,\beta)}\rangle$
$\displaystyle=\frac{1}{\sqrt{M_{\nu}(\alpha,\beta)}}\\{|{\alpha}\rangle_{A}\otimes|{\beta}\rangle_{B}+\nu|{-\alpha}\rangle_{A}\otimes|{-\beta}\rangle_{B}\\},$
(66)

where $M_{\nu}(\alpha,\beta)=2(1+\nu\exp[-2(|\alpha|^{2}+|\beta|^{2})])$,
while Charlie holds the state (55) with $\gamma_{\mu}={U_{\mu
0}^{(C)}}^{\ast}\gamma$.

Once Alice has applied the phase modulation to her mode and interfered it with
Charlie’s state on 50:50 beam splitter, a single photon detection in the
sideband $\mu$ heralds preparation of the following state at Bob’s node

$\displaystyle|{\Psi_{B}^{(\nu,\mu,1)}}\rangle=\frac{(\gamma\mathrm{e}^{i\phi_{C}}+\alpha\mathrm{e}^{i\phi_{A}})|{\beta}\rangle+\nu(\gamma\mathrm{e}^{i\phi_{C}}-\alpha\mathrm{e}^{i\phi_{A}})|{-\beta}\rangle}{\sqrt{2(|\gamma|^{2}+|\alpha|^{2}+(|\gamma|^{2}-|\alpha|^{2})\mathrm{e}^{-2|\beta|^{2}})}},$
(67)
$\displaystyle|{\Psi_{B}^{(\nu,\mu,2)}}\rangle=|{\Psi_{B}^{(\nu,\mu,1)}}\rangle\Bigr{|}_{\beta\to-\beta}$
(68)

with the probability of success given by

$\displaystyle P_{\mu,\;\mathrm{tlp}}^{(\nu)}=|U_{\mu
0}|^{2}P^{(\nu)}_{\mathrm{tlp}},$ $\displaystyle
P^{(\nu)}_{\mathrm{tlp}}=\frac{|\gamma|^{2}+|\alpha|^{2}+(|\gamma|^{2}-|\alpha|^{2})\mathrm{e}^{-2|\beta|^{2}}}{2(1+\nu\mathrm{e}^{-2(|\alpha|^{2}+|\beta|^{2})})}\mathrm{e}^{-|\alpha|^{2}-|\gamma|^{2}},$
(69)

Note that, since $P_{\mathrm{tlp}}^{(-)}>P_{\mathrm{tlp}}^{(+)}$, using the
antisymmetric entangled coherent state $|{\Psi_{-}^{(AB)}}\rangle$ generally
yields a higher teleportation rate as compared to the symmetric one
$|{\Psi_{+}^{(AB)}}\rangle$. It can be seen from the curves shown in Fig. 9.
These curves represent $|\alpha|^{2}$-dependencies of
$P_{\mathrm{tlp}}^{(\pm)}$ computed at $|\alpha|=|\gamma|$ from the formula

$\displaystyle
P_{\mathrm{tlp}}^{(\nu)}(|\alpha|,|\beta|)=\frac{|\alpha|^{2}}{2(1+\nu\mathrm{e}^{-2(|\alpha|^{2}+|\beta|^{2})})}\mathrm{e}^{-2|\alpha|^{2}}.$
(70)

Figure 9 presents the results for two cases: the curves computed at
$|\alpha|=|\beta|$ ($P_{\mathrm{tlp}}^{(\nu)}(|\alpha|,|\alpha|)$ is given by
Eq. (61)) and the ones where the amplitude of $\beta$ is fixed (see Eq. (70)).
From Eq. (70), it is not difficult to see that
$P_{\mathrm{tlp}}^{(+)}(|\alpha|,|\alpha|)>P_{\mathrm{tlp}}^{(+)}(|\alpha|,|\beta|)$
and
$P_{\mathrm{tlp}}^{(-)}(|\alpha|,|\alpha|)<P_{\mathrm{tlp}}^{(-)}(|\alpha|,|\beta|)$
at $|\alpha|>|\beta|$. As is shown in Fig. 9a, for symmetric cats, all the
curves are qualitatively similar and the maximum value of
$P_{\mathrm{tlp}}^{(+)}(|\alpha|,|\alpha|)$ reached at
$|\alpha|=|\alpha|_{\mathrm{max}}$ will give the highest probability provided
$|\beta|<|\alpha|_{\mathrm{max}}$.

For teleportation using antisymmetric cats, behavior of teleportation success
probabilities is strikingly different. Referring to Fig. 9b,
$P_{\mathrm{tlp}}^{(-)}(|\alpha|,|\alpha|)$ starts from its maximal value
$1/4$ ($P_{\mathrm{tlp}}^{(-)}(|\alpha|,|\alpha|)$ tends to $1/4$ at
$|\alpha|\to 0$) and monotonically decays zero as $|\alpha|$ increases. By
contrast, the curve for $P_{\mathrm{tlp}}^{(-)}(|\alpha|,|\beta|)$ is similar
to $P_{\mathrm{tlp}}^{(+)}$. At sufficiently small $|\beta|$, the maximum of
$P_{\mathrm{tlp}}^{(-)}(|\alpha|,|\beta|)$ is above $1/4$ and can be close to
the limiting value $1/2$. This is the case giving the highest success
probability of teleportation.

Our concluding remark concerns the small mean photon number approximation (64)
applicable to Bob’s state at $|\beta|<0.2$. Such approximation may be of
practical value as teleported Bob’s state appears to be quasi-Gaussian with
the corresponding benefits of Gaussian state control. In particular, the phase
of Bob’s state can be measured with moderate resources and designing an
apparatus for single-shot measurement of a phase for Schrödinger cat and
coherent state simultaneously is no longer required.

##  VI Conclusions and discussion

In this work, we have proposed to use phase-modulated multimode Schrödinger
cat states for producing entanglement between remote parties (Alice and Bob)
and for entanglement swapping. This approach to quantum repeaters is based on
multimode coherent states generated by an electro-optic modulator and may
reveal a number of interesting potential extensions.

Figure 10: Elementary link of an SCW quantum repeater in star topology.

For instance, as is illustrated in Fig. 10, one of the advantages of using
entangled multimode states is that the modes may, in principle, be distributed
and stored in quantum memories located at different nodes of a star topology
network. Since these modes differ in frequency, the DWDM demultiplexer can be
employed to send them to different locations.

We have analyzed the heralded entanglement generation scheme assuming that
Alice and Bob send a part of modes that enter their multimode cat states,
$|{\Psi_{\nu^{\prime}}^{(A)}}\rangle$ and $|{\Psi_{\nu}^{(B)}}\rangle$, to the
symmetric ($50:50$) beam splitter (see Fig. 1 and Eq. (II)) and have computed
the probabilities, $P_{\mu}^{(\nu^{\prime}\nu)}$, for three orthogonal states
(the vacuum state, $|{\mathbf{0}}\rangle$, the antisymmetric and modified
symmetric cat states, $|{\Phi_{-}}\rangle$ and $|{\Phi_{+}}\rangle$) to occur
at the output ports of the beam splitter for different couples
$(\nu^{\prime},\nu)$ of input cats (see Eqs. (19) and (20)).

We have modeled the optical fiber quantum channel with the transmittance
$\eta$ using the Stinesping dilation based on a beam splitter transformation
for an enlarged system supplemented with environmental (noisy) modes (see Eq.
(III.1)). For the heralding events determined by the parity of photons,
$p_{c}\in\\{\mathrm{odd},\mathrm{even}\\}$, registered by a photodetector with
the efficiency $\xi$, it was found that the conditional probabilities of
detection (33) give the probabilities of success (36) expressed in terms of
$P_{\pm}^{(\nu^{\prime}\nu)}$ with the renormalized mean photon number ratio:
$r_{\mathrm{bs}}\to\zeta=r_{\mathrm{bs}}\eta\xi$.

It is shown that the fidelities, $F_{\mu}^{(\nu^{\prime}\nu)}(p_{c})$, between
the density matrix of the heralded states (III.2) and the entangled cat
states, $\Phi_{\mu}^{(AB)}$, can be described by two analytical expressions
for the fidelities of predominantly symmetric and antisymmetric cats:
$F_{-}^{(\pm)}(\mathrm{odd})$ and $F_{+}^{(\pm)}(\mathrm{even})$ given by Eqs.
(41a) and (41b), respectively. It turned out that these fidelities
characterizing the quality of the generated entanglement are monotonically
decreasing functions of the amplitude $|\alpha|$ (see Fig. 5).

Our key analytical results are formulas (36) and (41) for the success
probabilities and the fidelities describing performance and quality of the
entanglement generation procedure. These results imply that producing high-
quality entangled states requires the amplitude $|\alpha|$ to be sufficiently
small. In this region, the highest values of the probability of success
correspond to odd number of clicks for either two antisymmetric input states
(the heralded cat state is predominantly antisymmetric) or two states of
different symmetry (the heralded cat state is predominantly symmetric). By
contrast, in the low-amplitude region, the success probability for a couple of
input symmetric cat states is negligibly small (see Eq. (37a)).

Note that, in reality, we have two detectors and each of the detectors
registers the odd (even) number of photons with the same probability of
success $P_{s}^{(\nu^{\prime}\nu)}(\mathrm{odd})$
($P_{\pm}^{(\nu^{\prime}\nu)}(\mathrm{even})$). The difference between the
corresponding heralded states,
$|{\Psi_{\mu}^{(AB)}(\bm{\alpha}_{\mathrm{qm}},\bm{\alpha}_{\mathrm{qm}})}\rangle$
and
$|{\Psi_{\mu}^{(AB)}(\bm{\alpha}_{\mathrm{qm}},-\bm{\alpha}_{\mathrm{qm}})}\rangle$,
can be corrected by applying $\pi$-shift to the phases of the modes at either
Alice’s or Bob’s site. It means that total performance of entanglement
generation can be characterized by twice the success probability.

For generation of antisymmetric cats, this probability cannot be higher than
50% because $P_{s}^{(-)}(\mathrm{odd})\leq 1/4$ (see Fig. 4). Interestingly,
$P_{s}^{(-)}(\mathrm{odd})$ is identically equal to $1/4$ when the product
$\zeta\equiv r_{\mathrm{bs}}\eta\xi$ is $1/2$ (or, equivalently,
$r_{\mathrm{bs}}=r_{1/2}=1/(2\eta\xi)$). From the other hand, according to
Fig. 5, the photon number ratio $r_{\mathrm{bs}}$ should be lowered to enhance
the quality of entanglement with the maximum value of fidelity
$F_{-}^{(\pm)}(\mathrm{odd})$ exceeding 90%.

By contrast, when the input cat states differ in summery and the heralded
states are symmetric cats, double the probability of success
$2P_{s}^{(c)}(\mathrm{odd})$ is above $1/2$ at $r_{\mathrm{bs}}>r_{1/2}$ (see
Fig. 4). In addition, as is shown in Fig. 5, the maximum value of
$F_{+}^{(c)}(\mathrm{odd})$ reached at $|\alpha|=0$ is unity. So, procedure
leading to entangled symmetric cats is superior to that for the case of
antisymmetric ones in both performance and quality.

However, given the symmetric cat states efficiently produced to create
entanglement between the nodes of elementary links, the success probability
for entanglement swapping given by Eq. (IV) will be close to zero. Therefore,
antisymmetric cat states play important part in reaching reasonably high
performance of the repeater. An important point is that there is a tradeoff
between the probability of success and the fidelity (IV) that needs to be
optimized.

Part of our considerations assumes that the modes can be stored in a multimode
quantum memory [46, 50]. For the repeater with two links, we have estimated
the so-called waiting time (53) giving the lower bound for the storage time of
the quantum memory. The results presented in Fig. 7 show that the waiting time
falls within the range of hundreds of microseconds for tens kilometer long
distances. For instance, according to Ref. [52], QMs based on atomic frequency
comb protocol with rare-earth-ion doped crystals are promising multimode QMs
that may tolerate such waiting times. Efficiency of such QMs, however, still
needs to be improved.

Alternatively, our system may operate without a multimode QM. In this
approach, the carrier mode at the central frequency is the only mode to be
stored in the QM, whereas the sidebands are distributed over the repeater
nodes. After extracting this mode from the memory, re-modulation can be
carried out and then the carrier mode can be stored again. High efficiency of
the electro-optic modulator allows such process to be recurring in cycles.

Thus, our analysis suggests feasibility of the proposed scheme for
antisymmetric cat states and relatively short distances to the central node.
In general, approaches to QRs using phase modulation to produce and control
multimode states may lead to promising methods for generation and distribution
of entanglement.

Quantum teleportation protocol that uses the entangled cat states to transfer
the phase information between remote parties discussed in Sec. V exemplifies
one of such methods. Interestingly, one of our findings is that the
antisymmetric cat states are superior to symmetric ones in teleportation
performance.

###### Acknowledgements.

The work was done by Leading Research Center "National Center for Quantum
Internet" of ITMO University by order of JSCo Russian Railways. The work of
ADK was also financially supported by the Ministry of Education and Science of
the Russian Federation (Passport No. 2019-0903). ESM and SAM appreciate
support within framework project # 00075-02-2020-051/1 from 02.03.2020.

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k_{1}k_{2}}{n}\right)\sum_{(i,j)\in\Lambda_{k_{1}}}\sum_{(s,t)\in\Lambda_{k_{2}}}\mu_{i,j}\mu_{s,t}\frac{(i-j)(t-s)}{n}=0.$
(62)

For the fourth term, we have to be a bit more careful. Let us consider

$\displaystyle\sum_{k_{1}=1}^{\tilde{k}}\sum_{k_{2}=1}^{\tilde{k}}k_{1}k_{2}\sum_{(i,j)\in\Lambda_{k_{1}}}\sum_{(s,t)\in\Lambda_{k_{2}}}\mu_{i,j}\mu_{s,t}\frac{(i-j)^{2}(t-s)^{2}}{n^{2}}$
$\displaystyle=$
$\displaystyle\left(\sum_{k_{1}=1}^{\tilde{k}}k_{1}\sum_{(i,j)\in\Lambda_{k_{1}}}\mu_{i,j}\frac{(i-j)^{2}}{n}\right)\left(\sum_{k_{2}=1}^{\tilde{k}}k_{2}\sum_{(s,t)\in\Lambda_{k_{2}}}\mu_{s,t}\frac{(t-s)^{2}}{n}\right)$
$\displaystyle=$ $\displaystyle\,S^{2},$

where

$S:=\sum_{k=1}^{\tilde{k}}k\sum_{(i,j)\in\Lambda_{k}}\mu_{i,j}\frac{(i-j)^{2}}{n}.$

Using (6), we see that, since $\tilde{k}\leq n^{\frac{2}{3}}$, then

$\displaystyle S$
$\displaystyle=O\left(\sum_{k=1}^{\tilde{k}}ke^{-\frac{\delta
k}{2}}\sum_{(i,j)\in\Lambda_{k}}\frac{(i-j)^{2}}{(ij)^{\frac{3}{2}}}\left(\frac{i}{j}\right)^{j-i}\exp\left(-\frac{(i-j)^{2}}{2n}\right)\right)=O\left(\sum_{k=1}^{\tilde{k}}ke^{-\frac{\delta
k}{2}}\sum_{i+j=k}\frac{(i-j)^{2}}{(ij)^{\frac{3}{2}}}\left(\frac{i}{j}\right)^{j-i}\exp\left(-\frac{(i-j)^{2}}{2n}\right)\right)$
$\displaystyle=O\left(\sum_{k=1}^{\tilde{k}}ke^{-\frac{\delta
k}{2}}\sum_{d=-k}^{k}\frac{d^{2}}{(k^{2}-d^{2})^{\frac{3}{2}}}\left(\frac{k-d}{k+d}\right)^{d}\exp\left(-\frac{d^{2}}{2n}\right)\right).$
(63)

Firstly, we note that for small $k$ the sum is negligible. Indeed,

$\sum_{k=1}^{\epsilon^{-\frac{2}{5}}}ke^{-\frac{\delta
k}{2}}\sum_{d=-k}^{k}\frac{d^{2}}{(k^{2}-d^{2})^{\frac{3}{2}}}\left(\frac{k-d}{k+d}\right)^{d}\exp\left(-\frac{d^{2}}{2n}\right)\leq\sum_{k=1}^{\epsilon^{-\frac{2}{5}}}k\sum_{d=-k}^{k}d^{2}\leq\epsilon^{-2}=O\left(\sqrt{\frac{n}{\epsilon}}\right).$
(64)

For $k\geq\epsilon^{-\frac{2}{5}}$, we split the inner sum up further into two
ranges

$T_{1}:=\sum_{|d|\leq
k^{\frac{3}{5}}}\frac{d^{2}}{(k^{2}-d^{2})^{\frac{3}{2}}}\left(\frac{k-d}{k+d}\right)^{d}\exp\left(-\frac{d^{2}}{2n}\right)\qquad\text{and}\qquad
T_{2}:=\sum_{k\geq|d|\geq
k^{\frac{3}{5}}}\frac{d^{2}}{(k^{2}-d^{2})^{\frac{3}{2}}}\left(\frac{k-d}{k+d}\right)^{d}\exp\left(-\frac{d^{2}}{2n}\right).$

By the same argument as in Lemma 2.2, we see that, since $k=\omega(1)$,

$T_{2}\lesssim\int_{k^{\frac{3}{5}}}^{\infty}y^{2}\exp\left(-\frac{y^{\frac{1}{5}}}{2}\right)dy=O\left(e^{-\frac{1}{2}k^{\frac{3}{25}}}k^{\frac{42}{25}}\right)=o\left(k^{-\frac{5}{4}}\right).$
(65)

Furthermore, we can bound $T_{1}$ naively, using Hölder’s inequality and Lemma
2.2, to obtain

$\displaystyle T_{1}$ $\displaystyle=\sum_{|d|\leq
k^{\frac{3}{5}}}\frac{d^{2}}{(k^{2}-d^{2})^{\frac{3}{2}}}\left(\frac{k-d}{k+d}\right)^{d}\exp\left(-\frac{d^{2}}{2n}\right)\leq\sqrt{\sum_{|d|\leq
k^{\frac{3}{5}}}d^{4}}\sqrt{\sum_{|d|\leq
k^{\frac{3}{5}}}\frac{1}{(k^{2}-d^{2})^{\frac{3}{2}}}\left(\frac{k-d}{k+d}\right)^{d}\exp\left(-\frac{d^{2}}{2n}\right)}$
$\displaystyle=O\left(\sqrt{k^{3}}\sqrt{k^{-\frac{11}{2}}}\right)=O\left(k^{-\frac{5}{4}}\right).$
(66)

Therefore, by (65) and(66), we have

$\displaystyle\sum_{k=\epsilon^{-\frac{2}{5}}}^{\tilde{k}}ke^{-\frac{\delta
k}{2}}\sum_{d=-k}^{k}\frac{d^{2}}{(k^{2}-d^{2})^{\frac{3}{2}}}\left(\frac{k-d}{k+d}\right)^{d}\exp\left(-\frac{d^{2}}{2n}\right)=O\left(\sum_{k=\epsilon^{-\frac{2}{5}}}^{\tilde{k}}k^{-\frac{1}{4}}e^{-\frac{\delta
k}{2}}\right)=O\left(\int_{y=1}^{\infty}y^{-\frac{1}{4}}e^{-\frac{\delta
y}{2}}dy\right)$
$\displaystyle=\left(\epsilon^{-\frac{3}{2}}\int_{x=\frac{\epsilon^{2}}{4}}^{\infty}x^{-\frac{1}{4}}e^{-x}dx\right)=O\left(\epsilon^{-\frac{3}{2}}\right).$
(67)

Hence, by (63), (64), and (67) we see that

$\displaystyle
S=O\left(\sqrt{\frac{n}{\epsilon}}+\epsilon^{-\frac{3}{2}}\right)=O\left(\sqrt{\frac{n}{\epsilon}}\right),$

and so

$\sum_{k_{1}=1}^{\tilde{k}}\sum_{k_{2}=1}^{\tilde{k}}k_{1}k_{2}\sum_{(i,j)\in\Lambda_{k_{1}}}\sum_{(s,t)\in\Lambda_{k_{2}}}\mu_{i,j}\mu_{s,t}\frac{(i-j)^{2}(t-s)^{2}}{n^{2}}=S^{2}=O\left(\frac{n}{\epsilon}\right).$
(68)

Hence, by (59), (60), (61), (62) and (68) we can conclude that

$\displaystyle\text{Var}(Z_{1})$
$\displaystyle\leq\mathbb{E}(Z_{2})+\frac{4\epsilon}{n}\mathbb{E}(Z_{2})^{2}+O\left(\frac{n}{\epsilon}\right).$
(69)

Using (6) and Lemma 2.2, we can bound

$\displaystyle\mathbb{E}(Z_{2})$
$\displaystyle\leq\sum_{k=1}^{n^{\frac{2}{3}}}k^{2}\sum_{(i,j)\in\Lambda_{k}}\mu_{i,j}=O\left(n\sum_{k=1}^{n^{\frac{2}{3}}}k^{2}e^{-\frac{\delta
k}{2}}\sum_{(i,j)\in\Lambda_{k}}\frac{1}{(ij)^{\frac{3}{2}}}\left(\frac{i}{j}\right)^{j-i}\exp\left(-\frac{(i-j)^{2}}{2n}\right)\right)$
$\displaystyle=O\left(n\sum_{k=1}^{n^{\frac{2}{3}}}\frac{1}{\sqrt{k}}e^{-\frac{\delta
k}{2}}\right)=O\left(n\int_{y=1}^{\infty}\frac{1}{\sqrt{y}}e^{-\frac{\delta
y}{2}}dy\right)=O\left(\frac{n}{\sqrt{\delta}}\int_{x=\frac{\delta}{2}}^{\infty}\frac{1}{\sqrt{x}}e^{-x}dx\right)=O\left(\frac{n}{\epsilon}\right).$

Finally, putting this together with (69), we can conclude that

$\text{Var}(Z_{1})\leq
O\left(\frac{n}{\epsilon}\right)+\frac{4\epsilon}{n}O\left(\frac{n^{2}}{\epsilon^{2}}\right)+O\left(\frac{n}{\epsilon}\right)=O\left(\frac{n}{\epsilon}\right).$

##  Appendix C Proof of Theorem 4.1

A standard argument tells us that, for a fixed vertex $v$ the order of the
component in $G(n,n,p)$ containing $v$ is stochastically dominated by the
order of the component of the root in a random subgraph of $T_{n}$, the
infinite $(n+1)$-regular rooted tree, where we include each edge independently
with probability $p$.

It is shown in [7, Corollary 3] that if we let $t(k,n)$ be the number of
subtrees of $T_{n}$ that contain the root and have order $k$ and
$k=\omega(1)$, then

$t(k,n)\approx\frac{1}{\sqrt{2\pi}k^{\frac{3}{2}}}n^{k-1}\left(\frac{n}{n-1}\right)^{k(n-1)+2}.$

Hence, the probability that the component of the root in a random subgraph of
$T_{n}$ has order $k$ is given by

$\displaystyle P_{k}(n,p)$
$\displaystyle=t(k,n)p^{k-1}(1-p)^{k(n-1)+2}\approx\frac{(pn)^{k-1}}{\sqrt{2\pi}k^{1.5}}\left(\frac{n(1-p)}{n-1}\right)^{k(n-1)+2}.$

It follows that, if we let $p=\frac{1-\epsilon}{n}$, then

$\displaystyle P_{k}(n,p)$
$\displaystyle\approx\frac{1}{\sqrt{2\pi}k^{1.5}}(1-\epsilon)^{k-1}\left(1+\frac{\epsilon}{n-1}\right)^{{k(n-1)+2}}\leq\frac{1}{\sqrt{2\pi}k^{1.5}}(1-\epsilon)^{k-1}\text{exp}\left(\epsilon
k+O\left(\frac{\epsilon}{n}\right)\right)\lesssim\frac{1}{\sqrt{2\pi}k^{1.5}}\left((1-\epsilon)e^{\epsilon}\right)^{k-1}.$

Furthermore, it is clear by comparison with a branching process that with
probability $1$ the component of the root is finite, and hence
$\sum_{k}P_{k}(n,p)=1$. It follows that the probability that a vertex in
$G(n,n,p)$ belongs to a component of order larger than $k_{0}\in\mathbb{N}$ is
equal to

$\sum_{k\geq k_{0}}P_{k}(n,p).$

Hence, if we let $Y_{\geq k_{0}}$ be the number of vertices in $G(n,n,p)$
which belong to a component of order larger than $k_{0}$, then we have that

$\displaystyle\mathbb{E}(Y_{\geq k_{0}})$ $\displaystyle=n\sum_{k\geq
k_{0}}P_{k}(n,p)\lesssim n\sum_{k\geq
k_{0}}\frac{1}{\sqrt{2\pi}k^{1.5}}\left((1-\epsilon)e^{\epsilon}\right)^{k-1}\lesssim
nk_{0}^{-\frac{3}{2}}\frac{((1-\epsilon)e^{\epsilon})^{k_{0}}}{1-(1-\epsilon)e^{\epsilon}}.$

However, $(1-\epsilon)e^{\epsilon}=1-\frac{\epsilon^{2}}{2}+O(\epsilon^{3})$
and so

$\mathbb{E}(Y_{\geq k_{0}})\lesssim
nk_{0}^{-\frac{3}{2}}\frac{4}{\epsilon^{2}}.$

Taking $k_{0}=\sqrt{\frac{n}{3\epsilon}}$, we see that

$\mathbb{E}\left(Y_{\geq\sqrt{\frac{n}{3\epsilon}}}\right)=O\left(\frac{n^{\frac{1}{4}}}{\epsilon^{\frac{5}{4}}}\right)=o\left(\sqrt{\frac{n}{\epsilon}}\right).$