# On the Teleportation of Discrete Variable Qubits Via Continuous Variable
Lossy Channels

Mingjian He<EMAIL_ADDRESS>Robert Malaney<EMAIL_ADDRESS>School of Electrical Engineering and Telecommunications,
The University of New South Wales, Sydney, NSW 2052, Australia.  Ryan
Aguinaldo<EMAIL_ADDRESS>Northrop Grumman Corporation, San Diego, CA
92128, USA.

###### Abstract

The Continuous-Variable (CV) quantum state of light allows for the
teleportation of a Discrete-Variable (DV) photonic qubit. Such an operation is
useful in the realm of hybrid quantum networks. However, it is known that the
teleportation of a DV qubit via a Gaussian CV resource channel is severely
limited under channel loss, with a teleportation fidelity beyond the classical
limit unattainable for losses exceeding a small threshold of 0.5 dB. In this
work, we present a new non-deterministic teleportation protocol that combines
a Gaussian CV resource channel with a modified form of the Bell State
Measurement that accommodates a DV-qubit fidelity beyond the classical limit
for channel losses up to 20 dB. Beyond this orders of magnitude improvement,
we also show how the use of non-Gaussian operations on the CV resource channel
can lead to a DV-qubit fidelity approaching unity for any channel loss.

##  I Introduction

Quantum teleportation is the process of transmitting unknown quantum states
from one location to another. At its core, is the consumption of entanglement
shared between the locations. Teleportation can be classified into three
categories depending on whether Discrete-Variable (DV) entanglement [1, 2],
Continuous-Variable (CV) entanglement [3, 4], or a combination (hybrid) of DV
and CV entanglement is used as the resource channel for the teleportation [5,
6, 7, 8].

A non-Gaussian resource channel built on DV entanglement was originally
proposed for the teleportation of DV qubits [1]. However, in the context of
optical communications such a teleportation scheme cannot be implemented in a
deterministic manner using only linear optics [9]. In principle, more
sophisticated DV-only schemes, which approach unit fidelity, are available for
deterministic DV-qubit teleportation, albeit with no clear practical route to
deployment [10]. In contrast, it was proposed in [11] and demonstrated in
[12], that the deterministic teleportation of DV qubits is possible with an
alternative entanglement resource, the Gaussian Two-Mode Squeezed-Vacuum
(TMSV) state, albeit at the cost of non-unit fidelity (except in the
unattainable infinite-squeezing limit).

However, recent work shows that the teleportation fidelity of DV qubits via a
TMSV resource channel is severely limited under a lossy environment – fidelity
cannot reach the classical limit when the resource channel loss exceeds 11%
(0.5 dB) even with infinite squeezing [13]. Such sensitivity to loss greatly
limits the potential of DV-qubit teleportation via a TMSV resource - an
outcome that inhibits the useful operation of heterogeneous quantum networks
in which quantum information will be exchanged between DV-enabled and CV-
enabled devices [14, 15, 16]. This is somewhat disappointing as it is believed
that TMSV resource channels between the different types of quantum devices
could be easily delivered and utilized as the “inter-connect channel” in
emerging hybrid quantum networks e.g., [17, 18, 19, 20], and play an important
role in hybrid quantum information processors - particularly in the realm of
optical measurement-based quantum computation [21].

Regarding the teleportation of a DV qubit via a TMSV resource channel, it is
therefore interesting to ask: Do improved loss-resistant schemes that retain
reasonable success probabilities exist? If in the affirmative, such an outcome
could have profound implications for a range of quantum information protocols.
In this work, we indeed answer this question in the affirmative by presenting
a new non-deterministic teleportation protocol that combines a TMSV resource
channel with a hybrid form of a Bell State Measurement (BSM). Henceforth, we
refer to this measurement as the H-BSM, and its use in teleportation as the
H-BSM protocol. As we shall see, the replacement of the CV-BSM with our H-BSM,
leads to dramatic improvements in DV-qubit teleportation fidelity for a range
of lossy-TMSV resource channels, with the classical limit being surpassed even
when the resource-channel losses exceed 99% (20 dB).

Beyond the use of our H-BSM, for comparison, we also study the use of
distillation operations on the resource channel via Quantum Scissors (QS) and
Photon Catalysis (PC)111We focus on QS and PC since both have been shown to
function as a noiseless linear amplifier for states with low photon number
[22, 23, 24, 25]., as a means to improve the traditional CV teleportation
scheme, as applied to DV qubits. Such non-Gaussian operations have been shown
to improve the teleportation of CV states (e.g., [26, 27, 28, 29, 30, 31, 32,
33, 34]). However, it remains unclear whether such distillation operations
improve the teleportation of DV qubits. As we shall see, when applied to the
TMSV resource channel, both QS and PC can further improve the teleportation
fidelity for DV qubits. More specifically, we will see that a teleportation
fidelity approaching unity is possible with QS or PC.

Our teleportation protocols are non-Gaussian in nature and require the use of
number-resolving single-photon detectors. However, the implementation of
single-photon detection with perfect efficiency is still challenging [35, 36,
37, 38]. To determine the performance of our protocols under less challenging
settings, we also study their use with non-ideal single-photon detectors,
showing that teleportation fidelity outcomes above the classical limit are
still achievable under a range of detector-efficiency conditions.

The rest of the paper is organized as follows: In Section II, we compare
existing teleportation protocols with our new protocol. The use of additional
non-Gaussian operations in our protocol is studied in Section III. The impact
of single-photon detectors with non-unity efficiency is studied in Section IV.
Section V concludes our work.

##  II Teleportation of DV Qubits with TMSV States

Figure 1: (a) Distribution of a TMSV state with modes $1$ and $2$ over lossy
channels with transmissivities $T_{1}$ and $T_{2}$. The distributed TMSV state
can be distilled at the receivers by QS or PC. (b) The CV-BSM protocol. The
distributed TMSV state with modes $1^{\prime}$ and $2^{\prime}$ is used as the
teleportation resource channel for an input DV qubit. (c) The H-BSM protocol,
which also uses the TMSV teleportation resource channel (HOM: homodyne
detector.)

First, consider the use of the protocol proposed by [3] in the teleportation
of DV qubits expressed in the Fock basis as
$\ket{\mathrm{in}}=\cos(\theta/2)\ket{0}+\exp{(i\phi)}\sin(\theta/2)\ket{1}$.222The
term “DV” is used to discriminate between the particle-like qubit and the
wave-like qubit [39]. In this work, we only consider the particle-like qubit
that resides in a finite-dimensional space spanned by the absence or presence
of a single photon. For conciseness, we will drop the term “DV” in the rest of
the paper. In this protocol, a TMSV state is used as the teleportation
resource channel. An input qubit (the state being teleported) is coupled with
one mode of the resource channel at a 50:50 beam-splitter. A CV-BSM with
homodyne detectors is then performed on the coupled modes. Based on the
measurement result, a displacement operation is applied to the other mode of
the resource channel. In the limit of infinite squeezing, the multi-photon
components of the displaced mode (the output mode) vanish and it approaches
the input qubit. We refer to this protocol as the CV-BSM protocol.

The relation between the input qubit and the output mode for the CV-BSM
protocol can be concisely described using the characteristic function
formalism. Consider a realistic scenario where two locations do not share any
entangled state before teleportation, and a TMSV state is first prepared at a
middle station (see Fig. 1a). The characteristic function for the TMSV state
can be written as

$\displaystyle\makebox{\large$\chi$}_{\mathrm{TMSV}}(\xi_{1},\xi_{2})=$
$\displaystyle\exp\Big{\\{}-\frac{1}{2}\Big{[}\frac{1+\lambda^{2}}{1-\lambda^{2}}(|\xi_{1}|^{2}+|\xi_{2}|^{2})$
(1)
$\displaystyle-\frac{2\lambda}{1-\lambda^{2}}(\xi_{1}\xi_{2}+\xi_{1}^{*}\xi_{2}^{*})\Big{]}\Big{\\}},$

where $\xi_{1}$ and $\xi_{2}$ are complex variables, $\lambda=\tanh{r}$, and
$r>0$ is the squeezing parameter for the TMSV state. The two modes of the TMSV
state are then sent through two independent lossy channels characterized by
the transmissivities $T_{1}$ and $T_{2}$, respectively. The characteristic
function for the state after the channel transmission can be written as [40]

$\displaystyle\makebox{\large$\chi$}_{\mathrm{TMSV}}^{\prime}(\xi_{1},\xi_{2})=$
$\displaystyle\exp{\left\\{-\frac{1}{2}\left[(1-T_{1})|\xi_{1}|^{2}+(1-T_{2})|\xi_{2}|^{2}\right]\right\\}}$
(2)
$\displaystyle\times\makebox{\large$\chi$}_{\mathrm{TMSV}}(\sqrt{T_{1}}\xi_{1},\sqrt{T_{2}}\xi_{2}).$

As shown in Fig. 1b, the entangled state with the above characteristic
function is then used as the teleportation resource channel. Let
$\makebox{\large$\chi$}_{\mathrm{in}}(\xi)$ and
$\makebox{\large$\chi$}_{\mathrm{out}}(\xi)$ be the characteristic functions
of the input qubit and the output mode, respectively. The latter can then be
written as [28]

$\makebox{\large$\chi$}_{\mathrm{out}}(\xi)=\makebox{\large$\chi$}_{\mathrm{in}}(g\xi)\makebox{\large$\chi$}_{\mathrm{TMSV}}^{\prime}(\xi,g\xi^{*}),$
(3)

where $g$ is the displacement gain factor. Assume the input qubit is pure. For
this CV-BSM protocol, the teleportation fidelity, which measures the closeness
between the characteristic functions of the input qubit and the output mode,
can then be written as

$\mathcal{F}=\frac{1}{\pi}\int
d^{2}\xi\makebox{\large$\chi$}_{\mathrm{in}}(\xi)\makebox{\large$\chi$}_{\mathrm{out}}(-\xi).$
(4)

Next, consider our new protocol, the H-BSM protocol of Fig. 1c. This uses the
same teleportation resource channel as before, but with a different strategy
on the measurement of the input qubit and the correction of the output mode.
As shown in Fig. 1c, the H-BSM is performed on the input qubit and one mode of
the resource channel. Note, our hybrid qubit-mode H-BSM is different from the
usual DV-BSM considered for a qubit-qubit system. The different input states
to the H-BSM lead to a range of alternate outcomes, in terms of success
probabilities, dependent on the setup of the experiment. Depending on the
results from the H-BSM, a unitary transformation is then applied to the output
mode to recover the input qubit. For a given input qubit
$\hat{\rho}_{\mathrm{in}}=\ket{\mathrm{in}}\bra{\mathrm{in}}$, the output mode
for our H-BSM protocol can be obtained as follows.333 Different from the CV-
BSM protocol, the bra-ket formalism is adopted since the characteristic
function formalism does not provide a concise description of the relation
between the input and output for this new protocol.

In the Fock basis, the initial TMSV state prepared at the middle station can
be written as

$\ket{\mathrm{TMSV}}=\sqrt{1-\lambda^{2}}\sum_{n=0}^{\infty}\lambda^{n}\ket{nn}_{12}.$
(5)

The two lossy channels alter the TMSV state to [41]

$\hat{\rho}_{\mathrm{TMSV}}^{\prime}=\mathcal{T}_{1}\circ\mathcal{T}_{2}(\hat{\rho}_{\mathrm{TMSV}}),$
(6)

where $\hat{\rho}_{\mathrm{TMSV}}=\ket{\mathrm{TMSV}}\bra{\mathrm{TMSV}}$,
$\circ$ represents the composition of transformations, and the transformation
$\mathcal{T}_{k}$ ($k=1,2$) is defined as

$\mathcal{T}_{k}(\hat{\rho}):=\sum_{l=0}^{\infty}\hat{G}^{(l)}_{k}\hat{\rho}\hat{G}^{(l)\dagger}_{k},$
(7)

with

$\hat{G}_{k}^{(l)}=\frac{\hat{a}^{l}_{k}\sqrt{T_{k}}^{\hat{a}_{k}^{\dagger}\hat{a}_{k}}}{\sqrt{l!}}\sqrt{\frac{1-T_{k}}{T_{k}}}^{l},$
(8)

and $\hat{a}_{k}$ the annihilation operator of mode $k$. For clarity, the two
modes of the TMSV state, after passing through the channels, are labeled as
$1^{\prime}$ and $2^{\prime}$, respectively. An H-BSM, which detects the four
Bell states,

$\displaystyle\ket{\Phi^{\pm}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{00}\pm\ket{11}\right)$ (9)
$\displaystyle\ket{\Psi^{\pm}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{01}\pm\ket{10}\right),$

is then performed on the input qubit and mode $1^{\prime}$. Depending on the
result of the H-BSM, a unitary transformation chosen from

$\displaystyle\hat{\sigma}_{I}$ $\displaystyle=\ket{0}\bra{0}+\ket{1}\bra{1}\
(\mathrm{for}\ket{\Phi^{+}})$ (10) $\displaystyle\hat{\sigma}_{Z}$
$\displaystyle=\ket{0}\bra{0}-\ket{1}\bra{1}\ (\mathrm{for}\ket{\Phi^{-}})$
$\displaystyle\hat{\sigma}_{X}$ $\displaystyle=\ket{0}\bra{1}+\ket{1}\bra{0}\
(\mathrm{for}\ket{\Psi^{+}})$ $\displaystyle\hat{\sigma}_{ZX}$
$\displaystyle=\ket{0}\bra{1}-\ket{1}\bra{0}\ (\mathrm{for}\ket{\Psi^{-}})$

is applied to mode $2^{\prime}$ of the teleportation resource channel. On the
detection of $\ket{\Phi^{+}}$ for the H-BSM, the output mode of our protocol
can be written as

$\hat{\rho}_{\mathrm{out}}=\hat{\sigma}_{I}\frac{1}{P_{\Phi^{+}}}\left(\bra{\Phi^{+}}\hat{\rho}_{\mathrm{in}}\otimes\hat{\rho}_{\mathrm{TMSV}}^{\prime}\ket{\Phi^{+}}\right)\hat{\sigma}_{I}^{\dagger},$
(11)

where $P_{\Phi^{+}}$ is the probability for the detection of $\ket{\Phi^{+}}$.
Similar definitions of $P_{\Phi^{-}}$, $P_{\Psi^{+}}$, and $P_{\Psi^{-}}$ are
in place for the detection of $\ket{\Phi^{-}}$, $\ket{\Psi^{+}}$, and
$\ket{\Psi^{-}}$, respectively.

The measurement basis of our H-BSM is not complete in the infinite-dimensional
space of mode $1^{\prime}$. Consequently, the measurement is non-deterministic
and its success probability decreases with increasing mean photon number in
mode $1^{\prime}$. In this section, we assume the H-BSM that only
discriminates the Bell states $\ket{\Psi^{\pm}}$ is adopted. This is the
simplest measurement possible, and one that is readily implemented using
current technology. An implementation of the H-BSM that discriminates the four
Bell states, and one which requires additional non-Gaussian operations, will
be discussed in the next section. Details on the implementation of both forms
of the H-BSM can be found in Appendix A.

We define the total success probability for the H-BSM protocol as

$P_{\mathrm{total}}=P_{\mathrm{BSM}}P_{\mathrm{operation}},$ (12)

where

$P_{\mathrm{BSM}}=P_{\Phi^{+}}+P_{\Phi^{-}}+P_{\Psi^{+}}+P_{\Psi^{-}},$ (13)

is the success probability for the H-BSM, and $P_{\mathrm{operation}}$ is the
success probability for additional non-Gaussian operations (beyond photon
detection). Mode $2^{\prime}$ will be discarded when the protocol fails. For
the H-BSM considered in this section, $P_{\Phi^{+}}=P_{\Phi^{-}}=0$, and
$P_{\mathrm{total}}=P_{\mathrm{BSM}}$ because no additional operation is
adopted.

For a pure qubit as the input (the state being teleported) to our protocol,
the teleportation fidelity can be written as

$\mathcal{F}=\mathrm{tr}\left\\{\hat{\rho}_{\mathrm{in}}\hat{\rho}_{\mathrm{out}}\right\\},$
(14)

which is equivalent to the fidelity defined in Eq. (4). Assume the input qubit
is uniformly distributed on the Bloch sphere. Then the joint probability
density function for the parameters $\theta$ and $\phi$ can be written as
$P(\theta,\phi)=(\sin\theta)/4\pi$. The averaged fidelity,
$\bar{\mathcal{F}}$, will be used as the performance metric. For the H-BSM
protocol, $\bar{\mathcal{F}}$ is obtained by first averaging the $\mathcal{F}$
in Eq. (14) over $\theta$, $\phi$, and the two possible outcomes of the H-BSM.
The averaged fidelity is then normalized by $P_{\mathrm{BSM}}$. For the CV-BSM
protocol, $\bar{\mathcal{F}}$ is obtained by averaging the $\mathcal{F}$ in
Eq. (4) over $\theta$ and $\phi$.

First, consider the scenario where the lossy channels are symmetric, i.e.,
$T_{1}=T_{2}=T$. In the following figures, the channel loss in dB units is
determined via $T\mathrm{[dB]}=-10\log_{10}\left(T\right)$, and the initial
squeezing of the TMSV states in dB units is determined via
$r\mathrm{[dB]}=10\log_{10}[\exp(2r)]$. In all calculations, the TMSV state is
truncated to a finite dimension such that
$\mathrm{tr}(\hat{\rho}_{\mathrm{TMSV}})>0.95$ is satisfied.

Fig. 2a compares the $\bar{\mathcal{F}}$ for the CV-BSM protocol and the H-BSM
protocol. For the CV-BSM protocol, the displacement gain $g$ is set to
maximize $\bar{\mathcal{F}}$. The CV-BSM protocol is not effective over most
regions of the parameter space since it only provides a narrow region where
the teleportation channel provides $\bar{\mathcal{F}}$ higher than the
classical limit of $2/3$ [42] (see Appendix B for a discussion on the
classical limit). The H-BSM protocol provides $\bar{\mathcal{F}}$ higher than
$2/3$ for a region larger than the CV-BSM protocol. Average fidelity higher
than the classical limit is attainable with the H-BSM protocol when the
channel loss is 4.7 dB. The initial squeezing of the TMSV state required to
provide such an $\bar{\mathcal{F}}$ is around 7 dB.

(a)

(b)

Figure 2: (a) Average teleportation fidelity $\bar{\mathcal{F}}$ for the CV-
BSM protocol (blue surface) and the H-BSM protocol (red surface). The gray
plane represents the classical limit for the teleportation of qubits. (b) The
total success probability for the H-BSM protocol averaged over the parameters
of the input qubit ($\theta$ and $\phi$). The numbers below each curve
represents the squeezing (in dB) of the initial TMSV state.

Fig. 2b shows the averaged total success probability for the H-BSM protocol
against the channel loss for various values of the initial squeezing of the
TMSV resource states. The average is taken over the parameters $\theta$ and
$\phi$ of the input qubit. The H-BSM protocol provides a success probability
higher than 40% when the classical limit is surpassed. Note, the TMSV state
approaches two vacuum modes as the channel loss increases (in dB), and the
total success probability for the H-BSM protocol approaches 50% as the multi-
photon components vanish.

For the scenario of asymmetric lossy channels, the H-BSM protocol still
provides $\bar{\mathcal{F}}$ higher than the classical limit for a region
larger than the CV-BSM protocol. For the scenario where $T_{1}=T$ and
$T_{2}=1$, $\bar{\mathcal{F}}$ for both protocols decreases monotonously as
the channel loss increases (in dB).

##  III Teleportation with Additional Operations

###  III.1 The H-BSM Protocol without Entanglement Distillation

Figure 3: (a) The diagram for the QS operation on one mode of a state
$\hat{\rho}$. The operation has been successful if a single photon is detected
in either of the single-photon detectors. (b) The diagram for the PC
operation. The operation has been successful if a single photon is detected in
the single-photon detector. In both diagrams, the input and output to the
operation are denoted by $\hat{\rho}$ and $\hat{\rho}^{\prime}$, respectively.

Non-Gaussian operations are defined as operations that map a quantum state to
a non-Gaussian state. In this section, we consider the use of additional non-
Gaussian operations in the H-BSM protocol, beyond the non-Gaussian photon
detection utilized in the previous section.

Assume the TMSV resource state is first distributed to the two locations
through lossy channels. and similar to before, assume the two channels have
the same transmissivity. In the first instance, let us also assume an H-BSM
that can discriminate the four Bell states is adopted, and that
$P_{\mathrm{BSM}}=1$. A specific experimental route to the implementation of
this form of the H-BSM can be found in Appendix A. The implementation
discussed there requires the truncation of the mode of the resource channel
that couples the input qubit. The truncation can be realized by a QS operation
(see Fig. 3a), which implements the transformation [43]

$\hat{\rho}_{\mathrm{trun}}=\frac{1}{P_{\mathrm{trun}}}\hat{M}_{\mathrm{trun}}\hat{\rho}_{\mathrm{TMSV}}^{\prime}\hat{M}_{\mathrm{trun}}^{\dagger},$
(15)

where

$\hat{M}_{\mathrm{trun}}=\frac{1}{\sqrt{2}}(\ket{0}_{1}\bra{0}+\ket{1}_{1}\bra{1}),$
(16)

$P_{\mathrm{trun}}=\operatorname{tr}\\{(\ket{0}_{1}\bra{0}+\ket{1}_{1}\bra{1})\hat{\rho}^{\prime}_{\mathrm{TMSV}}/2\\}$
is the success probability for the truncation, and
$\hat{\rho}_{\mathrm{TMSV}}^{\prime}$ is the density operator given by Eq.
(6). The truncated TMSV state, $\hat{\rho}_{\mathrm{trun}}$, is then used as
the teleportation resource channel. The total success probability for the
H-BSM protocol can then be written as $P_{\mathrm{total}}=P_{\mathrm{trun}}$.

###  III.2 The H-BSM Protocol with Entanglement Distillation

Next, consider using QS and PC to distill the entanglement of the distributed
TMSV state before teleportation. On performing the same QS operation on both
modes of the distributed TMSV state, the density operator for the state can be
written as [22, 25]

$\hat{\rho}_{\mathrm{QS}}=\frac{1}{P_{\mathrm{QS}}}\hat{M}_{2}\hat{M}_{1}\hat{\rho}_{\mathrm{TMSV}}^{\prime}\hat{M}_{1}^{\dagger}\hat{M}_{2}^{\dagger},$
(17)

where $P_{\mathrm{QS}}$ is the overall-success probability for the two QS
operations,444The overall-success probability for the two QS operations is the
product of the success probabilities for each QS operation. A similar
definition is used for the overall-success probability for the PC operations.

$\hat{M}_{i}=\sqrt{T_{\mathrm{s}}}\ket{0}_{k}\bra{0}+\sqrt{1-T_{\mathrm{s}}}\ket{1}_{k}\bra{1},k=1,2$
(18)

is the operator for the QS operation, $T_{\mathrm{s}}<1/2$ is the
transmissivity of the beam-splitter used in the QS operation, and again
$\hat{\rho}_{\mathrm{TMSV}}^{\prime}$ is the density operator given by Eq.
(6).

On performing the same PC operation on both modes of the distributed TMSV
state, the density operator for the resultant state can be written as [44]

$\hat{\rho}_{\mathrm{PC}}=\frac{1}{P_{\mathrm{PC}}}\hat{R}_{2}\hat{R}_{1}\hat{\rho}_{\mathrm{TMSV}}^{\prime}\hat{R}_{1}^{\dagger}\hat{R}_{2}^{\dagger},$
(19)

where $P_{\mathrm{PC}}$ is the overall-success probability for the two PC
operations,

$\hat{R}_{k}=\sqrt{T_{\mathrm{c}}}\left(\frac{T_{\mathrm{c}}-1}{T_{\mathrm{c}}}\hat{a}_{k}^{\dagger}\hat{a}_{k}+1\right)\sqrt{T_{\mathrm{c}}}^{\hat{a}_{k}^{\dagger}\hat{a}_{k}},$
(20)

is the operator for the PC operation, $T_{\mathrm{c}}<1/4$ is the
transmissivity for the beam-splitter in PC (see Fig. 3b), and
$\hat{a}_{k}^{\dagger}$ is the creation operator of mode $k$.

The distilled TMSV state is then used as the teleportation resource channel.
Again we assume the form of the H-BSM that can discriminate all four Bell
states is adopted. Note, the truncation previously used is not required for a
resource channel that has been distilled by QS. In terms of total success
probabilities we have the following two outcomes: (1)
$P_{\mathrm{total}}=P_{\mathrm{QS}}$, if the TMSV state has been distilled by
QS before it is used as the resource channel. (2)
$P_{\mathrm{total}}=P_{\mathrm{PC}}P^{\prime}_{\mathrm{trun}}$, if the TMSV
state has been distilled by PC before it is used as the resource channel,
where
$P^{\prime}_{\mathrm{trun}}=\operatorname{tr}\\{(\ket{0}_{1}\bra{0}+\ket{1}_{1}\bra{1})\hat{\rho}_{\mathrm{PC}}/2\\}$.

(a)

(b)

(c)

Figure 4: (a) and (b) The average fidelity of the H-BSM protocol with QS or PC
used for entanglement distillation. The red surface represents the average
fidelity for the case without entanglement distillation. The purple (green)
surface represents the average fidelity for the case with QS (PC). The gray
plane represents the classical limit. (c) The corresponding total success
probability for the H-BSM protocol. Curves with different markers represent
different initial squeezing of the resource channel. The red curves represent
the case without entanglement distillation.

Fig. 4 compares the performance of the H-BSM protocol with entanglement
distillation implemented by QS or PC. In Fig. 4a and b, the red surface
corresponds to the case without entanglement distillation. When all four Bell
states can be discriminated, the H-BSM protocol provides $\bar{\mathcal{F}}$
higher than the CV-BSM protocol over the entire region of parameter space.
Average fidelity higher than the classical limit is attainable with the H-BSM
protocol even when the channel loss is 20 dB. The purple and green surfaces
correspond to the cases with QS and PC, respectively. For QS and PC the
transmissivities $T_{\mathrm{s}}$ and $T_{\mathrm{c}}$ are set to maximize
$\bar{\mathcal{F}}$. In comparison to not performing any distillation before
teleportation, both QS and PC can improve the average fidelity over some
parameter spaces. QS provides the largest $\bar{\mathcal{F}}$ over the entire
parameter space. The average fidelity provided by QS is always larger than the
classical limit. The total success probability for the H-BSM protocol with
additional non-Gaussian operations is shown in Fig. 4c. When the initial
squeezing of the TMSV resource state approaches zero (curves marked with
diamonds), both QS and PC provide for an average fidelity approaching unity -
at the cost of vanishing success probability for the operations. The total
success probability for the protocol is reduced significantly when PC is
applied.

For the scenario of asymmetric lossy channels, parameters for the non-Gaussian
operations performed on the two modes of the resource channel can be set
independently to maximize $\bar{\mathcal{F}}$. QS provides the largest
$\bar{\mathcal{F}}$ over the entire parameter space of $\\{r,T_{1},T_{2}\\}$.
For both QS and PC, their $P_{\mathrm{total}}$ for the case of $T_{1}>T,\
T_{2}=T$ is always higher than the case of $T_{1}=T_{2}=T$.

###  III.3 The CV-BSM Protocol

For completeness, the use of non-Gaussian operations in the CV-BSM
teleportation protocol is also studied. To determine fidelities, the
characteristic functions for the non-Gaussified resource states are useful.
The characteristic function for the resource state after the symmetric QS
operations can be directly obtained from

$\makebox{\large$\chi$}_{\mathrm{QS}}(\xi_{1},\xi_{2})=\mathrm{tr}\left\\{\hat{D}(\xi_{1})\hat{D}(\xi_{2})\hat{\rho}_{\mathrm{QS}}\right\\},$
(21)

where $\hat{D}(\xi)$ is the displacement operator, and
$\hat{\rho}_{\mathrm{QS}}$ is the density operator given by Eq. (17).

Finding the characteristic function for the resultant state after the
symmetric PC operations is more involved [45] \- it is given by

$\displaystyle\makebox{\large$\chi$}_{\mathrm{PC}}(\xi_{1},\xi_{2})=$
$\displaystyle\int\frac{d^{2}\gamma_{1}d^{2}\gamma_{2}}{\pi^{2}(1-T_{\mathrm{c}})^{2}}\makebox{\large$\chi$}_{\mathrm{TMSV}}^{\prime}(\gamma_{1},\gamma_{2})$
(22) $\displaystyle\times
k(\xi_{1},\gamma_{1})k(\gamma_{1},\xi_{1})k(\xi_{2},\gamma_{2})k(\gamma_{2},\xi_{2}),$

where $\makebox{\large$\chi$}_{\mathrm{TMSV}}^{\prime}(\gamma_{1},\gamma_{2})$
is the characteristic function given by Eq. (2), $T_{\mathrm{c}}$ is the
transmissivity for the beam-splitter in PC, and

$k(\xi,\gamma)=\frac{\xi-\gamma\sqrt{T_{\mathrm{c}}}}{\sqrt{1-T_{\mathrm{c}}}}.$
(23)

For a given resource channel, depending on which non-Gaussian operations is
performed, the characteristic function for the distilled resource channel can
be obtained by putting Eq. (21) or Eq. (22) into Eq. (3).

Fig. 5 compares the performance of the CV-BSM protocol with QS or PC.
Different from Fig. 4, for added clarity, the difference between the average
fidelity for the cases with QS or PC and the average fidelity for the case
without entanglement distillation is shown. Both QS and PC can improve the
average fidelity over a certain range of the parameter space. The improvement
is the most significant when the initial squeezing of the TMSV state
approaches zero. However, at such squeezing this improvement is meaningless as
none of the non-Gaussian operations can provide average fidelity larger than
the classical limit.

Figure 5: The difference of the average fidelity of the CV-BSM protocol with
QS or PC. The gray surface represents the difference between the classical
limit $2/3$ and the average fidelity for the case without entanglement
distillation. The purple (green) surface represents the difference between the
average fidelity for the case with QS (PC) and the case without entanglement
distillation.

##  IV The Impact of Imperfect Single-Photon Detectors

Consider the use of single-photon detectors with detection efficiency
$\eta$.555The detection efficiency of the single-photon detector, which is
defined as the probability of registering a count if a photon arrives at the
detector, is considered as the product of coupling losses and the intrinsic
quantum efficiency of the detector [46]. To bring focus to this discussion, as
shown in Fig. 6a, we will consider the photon detectors used in the QS
operation and the H-BSM (recall from the previous section, QS outperforms PC
in terms of fidelity and success probability). Considering that an imperfect
single-photon detector, i.e., a detector with $\eta<1$, is equivalent to a
detector with unity efficiency preceded by a beam-splitter with transmissivity
$\eta$ [47], we will use such an equivalent form and model the detection
efficiency by an additional beam-splitter with transmissivity $\eta$. A mode
to be detected first interacts with a vacuum mode at the beam-splitter.

Note, the impact the detection efficiency has on the H-BSM depends on the
implementation of the measurement [48]. For simplicity, we revert in this
section back to the form of the H-BSM that only discriminates the Bell states
$\ket{\Psi^{\pm}}$.

Figure 6: (a) The H-BSM protocol with QS operations and the H-BSM that only
discriminates the Bell states $\ket{\Psi^{\pm}}$. (b) The QS operation with
imperfect single-photon detectors. (c) The H-BSM that only discriminates the
Bell states $\ket{\Psi^{\pm}}$ with imperfect single-photon detectors.

The QS operation with imperfect single-photon detectors is shown in Fig. 6b.
The teleportation resource channel after the two symmetric QS operations with
the same imperfect detectors can be written as

$\hat{\rho}_{\mathrm{QS}}=\mathcal{Q}_{1^{\prime}}\circ\mathcal{Q}_{2^{\prime}}(\hat{\rho}_{\mathrm{TMSV}}^{\prime}),$
(24)

where the transformation for the QS operation on a mode of a state
$\hat{\rho}$ can be written as [49]

$\displaystyle\hat{\rho}^{\prime}$ $\displaystyle=\mathcal{Q}(\hat{\rho})$
(25) $\displaystyle:=\frac{1}{P_{\eta}}\sum_{n=1,\
n^{\prime}=0}^{\infty}n\eta(1-\eta)^{n^{\prime}+n-1}\hat{M}_{n,n^{\prime}}\hat{\rho}\hat{M}_{n,n^{\prime}}^{\dagger},$

where $P_{\eta}$ is a normalization constant, and

$\displaystyle\hat{M}_{n,n^{\prime}}=$
$\displaystyle\left(-1\right)^{n^{\prime}}{2}^{-\frac{n+n^{\prime}-1}{2}}(n-n^{\prime})\sqrt{\frac{\left(n+n^{\prime}-1\right)!}{n!n^{\prime}!}}$
(26) $\displaystyle\times\sqrt{T_{\mathrm{s}}}\ket{0}\bra{n+n^{\prime}-1}$
$\displaystyle+\left(-1\right)^{n^{\prime}}{2}^{-\frac{n+n^{\prime}-1}{2}}\sqrt{\frac{(n+n^{\prime})!}{n!n^{\prime}!}}\sqrt{1-T_{\mathrm{s}}}\ket{1}\bra{n+n^{\prime}},$

which reduces to the operator given by Eq. (18) when $\eta=1$. We note, four
imperfect single-photon detectors are used in the two QS operations.

As shown in Fig. 6c, suppose the H-BSM adopts single-photon detectors with the
same detection efficiency as the QS operations. Let
$\hat{\rho}_{\mathrm{det}}$ be the full description of the system before the
H-BSM with imperfect detectors,

$\hat{\rho}_{\mathrm{det}}=\hat{\rho}_{\mathrm{in}}\otimes\hat{\rho}_{\mathrm{QS}},$
(27)

where $\hat{\rho}_{\mathrm{in}}$ is the qubit to be teleported, and
$\hat{{\rho}}_{\mathrm{QS}}$ is the density operator given by Eq. (24). We
note that the beam-splitters modeling the detection efficiency commute with
the 50:50 beam-splitter in the H-BSM [47]. To simplify the analysis we swap
the order of the beam-splitters with transmissivity $\eta$ and the 50:50 beam-
splitter. The beam-splitters with transmissivity $\eta$ alter the state
$\hat{\rho}_{\mathrm{det}}$ to

$\hat{\rho}_{\mathrm{eq}}=\mathcal{E}_{\mathrm{in}}\circ\mathcal{E}_{1^{\prime}}(\hat{\rho}_{\mathrm{det}}),$
(28)

where the transformation $\mathcal{E}$ is similar to the transformation given
by Eq. (7), but with $T_{i}$ being replaced by $\eta$. An H-BSM using
efficient single-photon detectors ($\eta=1$) is then performed on mode ‘in’
and mode $1^{\prime}$ of $\hat{\rho}_{\mathrm{eq}}$.

(a)

(b)

Figure 7: (a) The average fidelity for the H-BSM protocol with imperfect
single-photon detectors for different initial squeezing ($r$) and channel loss
($T$). The red curves represent the case of no entanglement distillation (QS)
adopted. The blue curves represent the case with QS. The black dashed line
represents the classical limit $\bar{\mathcal{F}}_{\mathrm{lim}}$. (b) The
corresponding total success probability for the H-BSM protocol.

Consider the scenario of symmetric lossy channels. The average fidelity for
the H-BSM protocol with imperfect single-photon detectors is shown in Fig. 7a.
For the case of no QS adopted, The average fidelity is not sensitive to the
decrease of $\eta$ when the initial squeezing is small. With QS the H-BSM
protocol is more robust against the decrease of $\eta$. Average fidelity
higher than the classical limit, $\bar{\mathcal{F}}_{\mathrm{lim}}$, is
attainable even when $\eta$ drops to 0.2 (see Appendix B for a discussion on
the classical limit). When single-photon detectors with $\eta\approx 0.9$
(e.g. [50]) are used, the H-BSM protocol with QS provides average fidelity
higher than the classical limit over the entire channel loss region considered
(up to 20dB loss, which is similar to the H-BSM protocol discussed in Section.
III.1).

Fig. 7b shows the corresponding total success probability for the H-BSM
protocol. For both cases, $P_{\mathrm{total}}$ decreases monotonously as
$\eta$ decreases, independent of the channel loss level and the initial
squeezing.

##  V Conclusion

We have proposed a new hybrid teleportation protocol for DV qubits that
combines a CV teleportation resource channel and a new form of a hybrid Bell
state measurement. We discussed two forms of this measurement, one in which
only two of the Bell states are discriminated, and a more complex one in which
all Bell states are discriminated. We found that the loss tolerance in our
protocol was significantly improved relative to known teleportation protocols
and that high fidelity and success probability were both possible. The use of
non-Gaussian operations on the resource channel was also studied. With such
operations, a teleportation fidelity approaching unity was found possible at
any channel loss level, albeit at vanishing success probability. Finally, we
studied the impact of single-photon detection with non-unity efficiency in our
protocol, finding that it can accommodate significant channel losses using
current single-photon detectors.

##  Appendix A Implementation of the H-BSM

Here, we discuss the implementation of the H-BSM. First, consider the form of
the H-BSM that can only discriminate the Bell States $\ket{\Psi^{\pm}}$ [51],
which contains a beam-splitter and two single-photon detectors. We define
$n_{i}$ as the number of photons detected at the $i$-th detector,
$\hat{a}_{1}^{\dagger}$ as the creation operator for the qubit to be
teleported, and $\hat{a}_{2}^{\dagger}$ as the creation operator for the mode
of the teleportation resource channel that couples the qubit. Modes $1$ and
$2$ are mixed at the beam-splitter, which implements the transformation,

$\begin{pmatrix}\hat{b}_{1}^{\dagger}\\\
\hat{b}_{2}^{\dagger}\end{pmatrix}=\mathbf{S}^{(1)}\begin{pmatrix}\hat{a}_{1}^{\dagger}\\\
\hat{a}_{2}^{\dagger}\end{pmatrix},$ (29)

where

$\mathbf{S}^{(1)}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ -1&1\end{pmatrix},$
(30)

and $\hat{b}_{1}^{\dagger}$ and $\hat{b}_{2}^{\dagger}$ are the creation
operators for the output modes of the beam-splitter.

The H-BSM is successful when one photon is detected by one of the detectors.
For $n_{1}=1$ and $n_{2}=0$, the input modes are projected into

$\hat{b}_{1}^{\dagger}\ket{00}=\frac{1}{\sqrt{2}}\left(\hat{a}_{1}^{\dagger}+\hat{a}_{2}^{\dagger}\right)\ket{00}=\frac{1}{\sqrt{2}}\left(\ket{10}+\ket{01}\right).$
(31)

For $n_{1}=0$ and $n_{2}=1$, the input modes are projected into
$\frac{1}{\sqrt{2}}\left(\ket{10}-\ket{01}\right)$. The probabilities for the
projection of $\ket{\Psi^{+}}$ and $\ket{\Psi^{-}}$ are, respectively,

$\displaystyle
P_{\Psi^{+}}=\operatorname{tr}\\{\ket{\Psi^{+}}\bra{\Psi^{+}}\hat{\rho}_{12}\\},$
(32) $\displaystyle
P_{\Psi^{-}}=\operatorname{tr}\\{\ket{\Psi^{-}}\bra{\Psi^{-}}\hat{\rho}_{12}\\},$

where $\hat{\rho}_{12}$ is the density operator for modes 1 and 2. When no
other non-Gaussian operations are adopted, the total success probability for
the H-BSM protocol can then be written as
$P_{\mathrm{total}}=P_{\Psi^{+}}+P_{\Psi^{-}}$.

Next, built upon [52], we present a variation of the H-BSM that can
discriminate the four Bell states expressed in the Fock basis (see also [53]).
We note that this variation is only one of the many routes possible to such a
full-discrimination measurement. As shown in Fig. 8, this H-BSM consists of an
array of beam-splitters ($2^{2N-2}$ in total), $2^{N}-2$ ancillary modes, and
$2^{N}$ single-photon detectors ($N>1$). Same as before, the creation operator
for the qubit to be teleported is denoted by $\hat{a}_{1}^{\dagger}$ and the
creation operator for the mode of the teleportation resource channel that
couples the qubit is denoted by $\hat{a}_{2}^{\dagger}$. The creation
operators for the ancillary modes are denoted by
$\hat{a}_{3}^{\dagger},\hat{a}_{4}^{\dagger},\dots,\hat{a}_{2^{N}}^{\dagger}$.
Mode $2$ should be truncated to the space spanned by $\\{\ket{0},\ket{1}\\}$
before it enters the beam-splitter array. This can be implemented by
performing a QS operation with $T_{\mathrm{s}}=0.5$ on mode $2$. Let
$\hat{\rho}_{2}$ be the density operator of mode $2$. The success probability
for such a QS operation can then be written as

$P_{\mathrm{trun}}=\frac{1}{2}\operatorname{tr}\\{\left(\ket{0}\bra{0}+\ket{1}\bra{1}\right)\hat{\rho}_{2}\\}.$
(33)

The beam-splitter array implements the transformation,

$\begin{pmatrix}\hat{b}_{1}^{\dagger}\\\ \hat{b}_{2}^{\dagger}\\\ \vdots\\\
\hat{b}_{2^{N}}\end{pmatrix}=\mathbf{S}^{(N)}\begin{pmatrix}\hat{a}_{1}^{\dagger}\\\
\hat{a}_{2}^{\dagger}\\\ \vdots\\\ \hat{a}_{2^{N}}\end{pmatrix},$ (34)

where

$\mathbf{S}^{(N)}=\mathbf{S}^{(1)}\otimes\mathbf{S}^{(N-1)},$ (35)

and
$\hat{b}_{1}^{\dagger},\hat{b}_{2}^{\dagger},\dots,\hat{b}_{2^{N}}^{\dagger}$
are the creation operators for the output modes of the beam-splitter array.

Before entering the beam-splitter array, the $2^{N}-2$ ancillary modes are in
the state

$\ket{\Xi^{(N)}}=\bigotimes_{j=1}^{N-1}\ket{\xi^{(j)}},$ (36)

which is a product state of $N-1$ different entangled states,

$\ket{\xi^{(j)}}=\frac{1}{\sqrt{2}}\left(\underbrace{\ket{00\dots
0}}_{2^{j}}+\underbrace{\ket{11\dots 1}}_{2^{j}}\right).$ (37)

Figure 8: The diagram for the H-BSM, where $\hat{a}_{1}^{\dagger}$ and
$\hat{a}_{2}^{\dagger}$ are the creation operators for the qubit to be
teleported and the mode of the resource channel that couples the qubit,
respectively,
$\hat{a}_{3}^{\dagger},\hat{a}_{4}^{\dagger},\dots,\hat{a}_{2^{N}}^{\dagger}$
are the creation operators for the ancillary modes, $T_{\mathrm{s}}$ is a
parameter associated with the QS, and $n_{i}$ is the photon number detected at
the $i$-th single-photon detector. The mode of the resource channel is first
truncated by a QS operation before entering the beam-splitter array.

We label the summation of the numbers of photons detected by the detectors as
$n_{\mathrm{sum}}=\sum_{i=1}^{2^{N}}n_{i}\leq 2^{N}$. On the detection of
$\bigotimes_{i=1}^{2^{N}}\ket{n_{i}}$ at the output of the beam-splitter
array, modes $1$ and $2$ are projected into the state

$\hat{\rho}_{12}^{\prime}=\frac{\hat{\Pi}\hat{\rho}_{12}\hat{\Pi}^{\dagger}}{\operatorname{tr}\\{\hat{\Pi}\hat{\rho}_{12}\\}},$
(38)

where $\hat{\Pi}=\ket{\zeta}\bra{\zeta}$, and

$\ket{\zeta}=\bra{\Xi^{(N)}}\frac{1}{\sqrt{2}^{n_{\mathrm{sum}}}}\prod_{i=1}^{2^{N}}\frac{\left(\sum_{j=1}^{2^{N}}s^{(N)}_{i,j}\hat{a}_{j}^{\dagger}\right)^{n_{i}}}{\sqrt{n_{i}!}}\ket{0},$
(39)

where $s^{(N)}_{i,j}$ are elements of the matrix $\mathbf{S}^{(N)}$.

Depending on the value of $n_{\mathrm{sum}}$, the detection outcomes can be
categorized into four mutually exclusive groups:

(1) When $n_{\mathrm{sum}}=0$, modes 1 and 2 are projected into $\ket{00}$.
The probability for this group is

$P_{00}=\frac{\operatorname{tr}\\{\ket{00}\bra{00}\hat{\rho}_{12}\\}}{2^{N-1}}.$
(40)

(2) When $n_{\mathrm{sum}}=2^{N}$, independent of the permutation of
$\\{n_{i}\\}$, modes 1 and 2 are projected into $\ket{11}$. The probability
for this group is

$P_{11}=\frac{\operatorname{tr}\\{\ket{11}\bra{11}\hat{\rho}_{12}\\}}{2^{N-1}}.$
(41)

(3) When $n_{\mathrm{sum}}$ is even (but not equal to $0$ and $2^{N}$),
depending on the permutation of $\\{n_{i}\\}$, modes 1 and 2 are projected
into $\ket{\Phi^{+}}$ or $\ket{\Phi^{-}}$. For a given permutation of
$\\{n_{i}\\}$, the states $\ket{\Phi^{+}}$ and $\ket{\Phi^{-}}$ can be
determined by using Eq. (38). The probabilities for the projection of
$\ket{\Phi^{+}}$ and $\ket{\Phi^{-}}$ are, respectively,

$\displaystyle
P_{\Phi^{+}}=\left(1-\frac{1}{2^{N-1}}\right)\operatorname{tr}\\{\ket{\Phi^{+}}\bra{\Phi^{+}}\hat{\rho}_{12}\\},$
(42) $\displaystyle
P_{\Phi^{-}}=\left(1-\frac{1}{2^{N-1}}\right)\operatorname{tr}\\{\ket{\Phi^{-}}\bra{\Phi^{-}}\hat{\rho}_{12}\\}.$

(4) When $n_{\mathrm{sum}}$ is odd, depending on the permutation of
$\\{n_{i}\\}$, modes 1 and 2 are projected into $\ket{\Psi^{+}}$ or
$\ket{\Psi^{-}}$. For a given permutation of $\\{n_{i}\\}$, the states
$\ket{\Psi^{+}}$ and $\ket{\Psi^{-}}$ can be determined by using Eq. (38). The
probabilities for the projection of $\ket{\Psi^{+}}$ and $\ket{\Psi^{-}}$ are,
respectively,

$\displaystyle
P_{\Psi^{+}}=\operatorname{tr}\\{\ket{\Psi^{+}}\bra{\Psi^{+}}\hat{\rho}_{12}\\},$
(43) $\displaystyle
P_{\Psi^{-}}=\operatorname{tr}\\{\ket{\Psi^{-}}\bra{\Psi^{-}}\hat{\rho}_{12}\\}.$

The H-BSM is successful when modes 1 and 2 are projected into one of the four
Bell states. The corresponding success probability can be written as

$P_{\mathrm{BSM}}=P_{\Phi^{+}}+P_{\Phi^{-}}+P_{\Psi^{+}}+P_{\Psi^{-}}=1-\frac{P_{\mathrm{id}}}{2^{N-1}},$
(44)

where
$P_{\mathrm{id}}=\operatorname{tr}\\{\left(\ket{00}\bra{00}+\ket{11}\bra{11}\right)\hat{\rho}_{12}\\}$.
When no other non-Gaussian operations are adopted, the total success
probability for the H-BSM protocol can then be written as
$P_{\mathrm{total}}=P_{\mathrm{trun}}P_{\mathrm{BSM}}$, which approaches
$P_{\mathrm{trun}}$ as $N$ increases.

##  Appendix B The Classical Limit for the Teleportation of Qubits

Consider a qubit expressed in the Fock basis as
$\ket{\mathrm{in}}=\cos(\theta/2)\ket{0}+\exp{(i\phi)}\sin(\theta/2)\ket{1}$.
Alice is given $\ket{\mathrm{in}}$, unknown to her, and Bob is required to
prepare a qubit $\ket{\mathrm{out}}$ as close as possible to
$\ket{\mathrm{in}}$. Alice and Bob are connected only by a classical channel
and their goal is to maximize the fidelity
$|\braket{\mathrm{out}}{\mathrm{in}}|^{2}$ averaged over the parameters
$\theta$ and $\phi$. We denote the classical limit, which is the upper limit
for the average fidelity, as $\bar{\mathcal{F}}_{\mathrm{lim}}$. One strategy
for reaching $\bar{\mathcal{F}}_{\mathrm{lim}}$ is as follows [42]. Alice
measures $\ket{\mathrm{in}}$ by a single-photon detector, of which the result
can be $\ket{0}$ or $\ket{1}$. Alice informs Bob about her measurement result,
and Bob prepares a $\ket{0}$ (or $\ket{1}$) if the result is $\ket{0}$ (or
$\ket{1}$).

Now consider the use of a single-photon detector with non-unity efficiency.
Again, we use a beam-splitter with transmissivity $\eta$ to model the
efficiency in the detector. For a given $\ket{\mathrm{in}}$, using such a
detector the probabilities for the detection of $\ket{0}$ and $\ket{1}$ are,
respectively,

$\displaystyle P_{0}$
$\displaystyle=\operatorname{tr}\\{\ket{0}\bra{0}\mathcal{E}(\ket{\mathrm{in}}\bra{\mathrm{in}})\\}=\cos^{2}\frac{\theta}{2}+(1-\eta)\sin^{2}\frac{\theta}{2},$
(45) $\displaystyle P_{1}$
$\displaystyle=\operatorname{tr}\\{\ket{1}\bra{1}\mathcal{E}(\ket{\mathrm{in}}\bra{\mathrm{in}})\\}=\eta\sin^{2}\frac{\theta}{2},$

where again the transformation $\mathcal{E}$ is similar to the transformation
given by Eq. (7), but with $T_{i}$ being replaced by $\eta$. The classical
limit for the average fidelity can then be written as

$\displaystyle\bar{\mathcal{F}}_{\mathrm{lim}}=$
$\displaystyle\int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi\left(\frac{\sin\theta}{4\pi}\right)\left(\cos^{2}\frac{\theta}{2}P_{0}+\sin^{2}\frac{\theta}{2}P_{1}\right)$
(46) $\displaystyle=$ $\displaystyle\frac{3+\eta}{6},$

The classical limit is $2/3$ when an efficient single-photon detector
($\eta=1$) is used. When $\eta=0$, Bob will prepare a qubit in the state of
$\ket{0}$ independent of the state of Alice’s initial qubit
$\ket{\mathrm{in}}$. The corresponding classical limit is $1/2$.

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