# Optimal fidelity of teleportation with continuous-variable using three-
tunable parameters in realistic environment

Li-Yun Hu1,2∗, Zeyang Liao1, Shengli Ma1 and M. Suhail Zubairy1∗ Corresponding
author. Email<EMAIL_ADDRESS>1Institute for Quantum Science and
Engineering (IQSE) and Department of Physics and Astronomy, Texas A&M
University, College Station, TX 77843, USA 2Center for Quantum Science and
Technology, Jiangxi Normal University, Nanchang 330022, China *Corresponding
author.

###### Abstract

We introduce three tunable parameters to optimize the fidelity of quantum
teleportation with continuous-variable in nonideal scheme. Using the
characteristic function formalism, we present the condition that the
teleportation fidelity is independent of the amplitude of input coherent
states for any entangled resource. Then we investigate the effects of tunable
parameters on the fidelity with or without the presence of environment and
imperfect measurements, by analytically deriving the expression of fidelity
for three different input coherent state distributions. It is shown that, for
the linear distribution, the optimization with three tunable parameters is the
best one with respect to single- and two-parameter optimization. Our results
reveal the usefulness of tunable parameters for improving the fidelity of
teleportation and the ability against the decoherence.

PACS number(s): 03.65 -a. 42.50.Dv

##  I Introduction

Quantum teleportation has an indispensable role in the manipulation of quantum
states and processing of quantum information 1 ; 2 ; 3 ; 3a . Usually, the
two-mode squeezed vacuum is often used as the entanglement resource with
continuous-variable (CV). However, due to the limit of experiment, it is hard
to achieve high degree of squeezing which leads to a low effect of
teleportation fidelity.

In order to increase the entanglement and fidelity of teleportation, a number
of stratigies have been proposed 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 10 ; 11 ; 12 ; 13 .
Among them, non-Gaussian operations including the photon subtraction $a$ or
addition $a^{{\dagger}}$ or the superposition of both can be used to realize
this purpose for given Vaidman-Brauntein-Kimble (VBK) scheme. For example, the
superposition operator $ta^{{\dagger}}+ra$ is proposed for quantum state
engineering to transform a classical state to a nonclassical one 14 , and then
is performed on two-mode squeezed vacuum (TMSV) for enhancing quantum
entanglement as well as the fidelity of teleportation 15 . It is found that
the fidelity teleporting coherent state can be further improved by optimizing
the superposition operation compared with the other non-Gaussian states, such
as photon-subtraction TMSV. As another example, a remarkable improvement of
the teleportation fidelity with CV can be obtained by introducing another
optimal non-Gaussian resources when given the usual and (non-)ideal VBK scheme
16 ; 17 ; 18  by considering imperfect Bell measurements and damping. In Ref.
18, the “shot-fidelity” and single-gain factor are used to discuss the
performance of teleportation. In fact, these protocols above are employed to
enhance the fidelity of teleportation by changing quantum entangled resources.

Then an inverse question is that given a certain class of entangled resources
with some given properties, how we can modify the VBK scheme to improve the
fidelity of teleportation. It is interesting to notice that there is an
alternative method to improve the fidelity of teleportation by using calssical
information. For instance, the teleportation fidelity of CV can be enhanced by
tuning the gain in the measurement dependent modulation on the output field 6
; 19  in Heisenberg picture and EPR resources without loss, which has been
realized experimentally by Furusawa et al 20 . However, these two important
theoretical works are concerned with the study of the ideal protocol
implementation using Gaussian resources 6 ; 19 . In addition, there are some
other strategies of gain tuning and of optimal gain 21 ; 22 , using the
Heisenberg picture and the Wigner function, but they can not be directly
applied to more general cases. In addition, in Refs. 21 ; 23  the gain factor
is used to maximize the teleportation fidelity for the case of Gaussian
resources, but the gain-optimized fidelity of teleportation is strongly
suppressed when considered the dissipation. Recently, a hybrid entanglement
swapping protocol is proposed experimentally to transfer discrete-variable
(DV) entanglement by using continuous-variable (CV) Gaussian entangled
resources and by tuning a gain factor of the teleporter 22 ; 24 ; 25 , which
shows that DV entanglement remains present after teleportation for any
squeezing by optimal gain tuning. For more information about advances in
quantum teleportation, we refer to a recent review paper 26  and references
therein.

The fidelity of teleportation, as mentioned above, can be improved by using
tunable entangled resources or classical parameters 16 ; 17 ; 19 ; 22 . In
Ref. 19 , a three-parameter optimal stratigy is introduced to improve the
quality of teleporttaion, including unbalanced beam splitter (BS) and two non-
unity gains. However, they only considered an ideal case. Actually, the
interaction between quantum system and environment can not be avoided and Bell
measurements are usually imperfect. Thus, it would be interesting that whether
it is still effective to enhance the fidelity by using these tunable
parameters in realistic case. In this paper, using the characteristics
function (CF) formalism, we shall extend the analysis of the parameter
optimization strategy for realistic input states and non-ideal entangled
resources and investigate the nonideal quantum teleportation by deriving an
analytical expression of the teleportation fidelity. This formalism is very
convenient to discuss the teleportation for the nonideal case and any
entangled resources. As far as we know, there is no related report up to now.

This paper is arranged as following. In Sec. II, we give a description of the
characteristic function formulism for the case of nonideal parameterized
teleportation with CV scheme. In this scheme, we find the condition that the
fidelity is independent of the amplitude of input coherent states for any
entangled resource. And then we present a qualitative description about
fidelity and average fidelity. In Sec. III, we derive the analytical
expression of the fidelity of teleportation when the TMSV and coherent states
are used as entangled channel and teleported states, respectively. In Sec. IV
we study the performance of amplitude-independent optimal fidelity using the
condition found in Sec. II. It is found that the optimal condition is just
that the two-gain factors and $\theta$ are equal to unit and $\pi/4$,
respectively. Sec. V is devoted to discussing the optimal fidelity over these
three tunable parameters and three different probability distributions for the
input coherent states by deriving the analytical expression of the optimal
fidelity. Our conclusions are drawn in the last section.

##  II Mode and quantitative analysis

Here, we consider a more realistic case of teleportation scheme shown in Fig.
1(a). In this scheme, there are three tunable parameters, unbanlanced BS and
two non-unity gains ($g_{q}$ and $g_{p}$). The input state (mode 1) and the
entangled resources (shared by modes 2 and 3) are not limited to be pure
states. Considering that the mode 2 can be prepared close to the sender Alice
while the mode 3 usually has to propagate over much longer distance, we can
assume that the mode 2 is not affected by losses but the mode 3 is. In
addition, two symmetrical lossy bosonic channel have been considered before
making Bell measurements, which are simulated through an extra vacuum mode and
a beam splitter with transmission coefficient $T$. The input states of modes 4
and 5 are pure vacuum states.

Figure 1: (Colour online) Realistic schematic diagram for CV teleportation.
BS: Beam splitter.

Next, we shall give a description about the scheme in the formalism of CF
where it is very convenient to discuss the teleportation for the nonideal case
and non-Gaussian entangled resourecs 17 ; 27 .

###  II.1 The input-output relation of BS in CF formalism

In order to obtain the relation between input and output, we first calculate
the output of a beam splitter with a vacuum and an arbitrary density operator
as inputs shown in Fig. 1(b). For simplicity, we denote the vacuum and the
input state as $\left|0\right\rangle_{5}$ and $\rho_{1}$, respectively. The
output state (denoted as $\rho_{1}^{\prime}$) is given by

$\rho_{1}^{\prime}=\text{Tr}_{5}\left[B_{15}\left(T\right)\rho_{1}\otimes\left|0\right\rangle_{5,5}\left\langle
0\right|B_{15}^{{\dagger}}\left(T\right)\right],$ (1)

where Tr5 is the partial trace over the ancilla mode 5 and
$B_{kl}\left(T\right)=\exp[\varphi(a_{k}a_{l}^{{\dagger}}-a_{k}^{{\dagger}}a_{l})]$
is the beam splitter operator describing the interaction between modes 1 and 5
with $\cos\varphi=\sqrt{T}$ and $a_{k,l}$ ($k=1,l=5$) being the photon
annihilation operator of the $k(l)-$modes. Using the Weyl expansion of density
operator, we can express the density operator $\rho_{1}$ and the vacuum
projector $\left|0\right\rangle_{55}\left\langle 0\right|$ as the following
forms:

$\displaystyle\rho_{1}$
$\displaystyle={\displaystyle\int}\frac{d^{2}\alpha}{\pi}\chi_{1}\left(\alpha\right)D_{1}\left(-\alpha\right),$
$\displaystyle\left|0\right\rangle_{55}\left\langle 0\right|$
$\displaystyle={\displaystyle\int}\frac{d^{2}\beta}{\pi}e^{-\frac{1}{2}\left|\beta\right|^{2}}D_{5}\left(-\beta\right),$
(2)

where $D_{1}\left(\alpha\right)=\exp\\{\alpha
a_{1}^{{\dagger}}-\alpha^{\ast}a_{1}\\}$ is the displacement operator, and
$\chi_{1}\left(\alpha\right)$ is the CF of $\rho_{1}$. On the other hand,
using the following transformation relation

$B_{15}D_{1}\left(-\alpha\right)D_{5}\left(-\beta\right)B_{15}^{{\dagger}}=D_{1}\left(\bar{\alpha}\right)D_{5}\left(\bar{\beta}\right),$
(3)

where $R=1-T$, and
$\bar{\alpha}=\beta\sqrt{R}-\alpha\sqrt{T},\bar{\beta}=-\beta\sqrt{T}-\alpha\sqrt{R}$,
we can derive

$\displaystyle\text{Tr}_{5}\left[B_{15}D_{1}\left(-\alpha\right)D_{5}\left(-\beta\right)B_{15}^{{\dagger}}\right]$
$\displaystyle=D_{1}\left(\bar{\alpha}\right)\text{Tr}_{5}\left[D_{5}\left(\bar{\beta}\right)\right]$
$\displaystyle=D_{1}\left(\bar{\alpha}\right)\pi\delta^{(2)}(\bar{\beta}).$
(4)

Here we have used the relation
Tr${}_{5}D_{5}(\bar{\beta})=\pi\delta^{(2)}(\bar{\beta})$. Then substituting
Eqs. (2) and (4) into Eq. (1) yields

$\displaystyle\rho_{1}^{\prime}$
$\displaystyle={\displaystyle\int}\frac{d^{2}\alpha
d^{2}\beta}{\pi^{2}}e^{-\frac{1}{2}\left|\beta\right|^{2}}\chi_{1}\left(\alpha\right)\text{Tr}_{5}\left[D_{1}\left(\bar{\alpha}\right)D_{5}\left(\bar{\beta}\right)\right]$
$\displaystyle={\displaystyle\int}\frac{d^{2}\alpha
d^{2}\beta}{\pi^{2}}e^{-\frac{1}{2}\left|\beta\right|^{2}}\chi_{1}\left(\alpha\right)D_{1}\left(\bar{\alpha}\right)\pi\delta^{(2)}\left(\bar{\beta}\right)$
$\displaystyle={\displaystyle\int}\frac{d^{2}\alpha}{\pi}e^{-\frac{1}{2}R\left|\alpha\right|^{2}}\chi_{1}\left(\sqrt{T}\alpha\right)D_{1}\left(-\alpha\right),$
(5)

where $e^{-\frac{1}{2}\left|\beta\right|^{2}}$ is the CF of the vacuum state,
and in the second step in Eq. (5) the CF $\chi_{1}\left(\alpha\right)$ is
transformed to $\chi_{1}(\sqrt{T}\alpha)$ with a Gaussian term
$e^{-\frac{1}{2}R\left|\alpha\right|^{2}}$ due to the photon loss. It is then
convenient to obtain the input-output relation of the teleportation scheme
shown in Fig. 1(a) as follows.

###  II.2 The input-output relation of the teleportation scheme in CF
formalism including photon-loss or imperfect Bell measurements

Now, we consider the effect of photon-loss on the relation between input and
output of the teleportation scheme in CF formalism. Here we use BS2 and BS3
with vacuum inputs to simulate the photon loss or imperfect Bell measurements
(see Fig. 1(b)), and denote the teleported state, entangled resource, and
auxiliary vacuum as $\rho_{1},$ $\rho_{23}$ and $\left|00\right\rangle_{45}$,
respectively. In oder to realize the teleportation, Alice shall make her Bell
measurements. Before she does, the unitary state evolution can be formulated
as

$\rho_{1-5}=U\otimes\rho_{1}\otimes\rho_{23}\otimes\left|00\right\rangle_{45}\left\langle
00\right|U^{\dagger},$ (6)

where the unitary evolution operator is defined as
$U\left.=\right.B_{24}B_{15}B_{12}$ and $B_{kl}$ are the BS operator defined
before. In a similar way to deriving Eq. (5), and using the Weyl expansion for
the entangled resource, the reduced output state denoted as
$\rho_{1-3}\equiv$Tr${}_{45}\rho_{1-5}$ is given by

$\displaystyle\rho_{1-3}$
$\displaystyle=\text{Tr}_{45}\left[U\otimes\rho_{1}\otimes\rho_{23}\otimes\left|00\right\rangle_{45}\left\langle
00\right|U^{\dagger}\right]$
$\displaystyle={\displaystyle\int}\frac{d^{2}\alpha d^{2}\beta
d^{2}\gamma}{\pi^{3}}\chi_{1}\left(\alpha\right)\chi_{23}\left(\beta,\gamma\right)$
$\displaystyle\times\text{Tr}_{45}\left[UD_{1}\left(-\alpha\right)D_{2}\left(-\beta\right)D_{3}\left(-\gamma\right)\left|00\right\rangle_{45}\left\langle
00\right|U^{\dagger}\right]$
$\displaystyle={\displaystyle\int}\frac{d^{2}\alpha d^{2}\beta
d^{2}\gamma}{\pi^{3}}\chi_{1}\left(\alpha\right)\chi_{23}\left(\beta,\gamma\right)\text{Tr}_{45}[B_{24}$
$\displaystyle\times
B_{15}D_{1}(-\alpha_{1})D_{2}(-\beta_{1})D_{3}\left(-\gamma\right)\left|00\right\rangle_{45}\left\langle
00\right|B_{24}^{\dagger}B_{15}^{\dagger}],$ (7)

where
$B_{12}D_{1}\left(-\alpha\right)D_{2}\left(-\beta\right)B_{12}^{{\dagger}}=D_{1}(-\alpha_{1})D_{2}(-\beta_{1})$
with $\alpha_{1}=\alpha\cos\theta-\beta\sin\theta$,
$\beta_{1}=\beta\cos\theta+\alpha\sin\theta$ and $\cos^{2}\theta$ being the
transmission coefficient of beam splitter $B_{12}$. Using Eq. (1) and (4), we
can obtain

$\displaystyle\rho_{1-3}$ $\displaystyle={\displaystyle\int}\frac{d^{2}\alpha
d^{2}\beta
d^{2}\gamma}{\pi^{3}}\chi_{1}(\sqrt{T}\alpha)\chi_{23}(\sqrt{T}\beta,\gamma)$
$\displaystyle\times
e^{-\frac{R}{2}(\left|\alpha_{1}\right|^{2}+\left|\beta_{1}\right|^{2})}D_{1}\left(-\alpha_{1}\right)D_{2}\left(-\beta_{1}\right)D_{3}\left(-\gamma\right).$
(8)

Eq. (8) is the representation in CF of deduced density operator before Bell
measurements but after BS2 and BS3.

Then, as the first step of teleportation, Alice makes a joint measurements for
modes 1 and 2 at the output ports, i.e., measures two observables
corresponding to coordinate and momentum of modes 1 and 2. Thus after the
measurements, the outcomes $\rho_{M}$ ($M$ means measurement) in mode 3 are

$\rho_{M}\equiv\frac{1}{P\left(q,p\right)}\text{Tr}_{12}\left[\left|q\right\rangle_{11}\left\langle
q\right|\otimes\left|p\right\rangle_{22}\left\langle
p\right|\rho_{1-3}\right],$ (9)

where $P\left(q,p\right)$ is the probability distribution function of the Bell
measurement outcomes,
$P\left(q,p\right)=$Tr${}_{3}\\{$Tr${}_{12}[\left|q\right\rangle_{11}\left\langle
q\right|\otimes\left|p\right\rangle_{22}\left\langle p\right|\rho_{1-3}]\\}$,
and $\left|q\right\rangle_{1}\ $and $\left|p\right\rangle_{2}$ are the
eigenstates of coordinate and momentum operators $Q_{1}$ and $P_{2}$
corresponding to modes 1 and 3, respectively.

According to the definition of CF, and using the following relations

$\displaystyle\text{Tr}_{1}\left[\left|q\right\rangle_{1,1}\left\langle
q\right|D_{1}\left(\alpha\right)\right]$
$\displaystyle=e^{i\sqrt{2}q\operatorname{Im}\alpha}\delta\left(\sqrt{2}\operatorname{Re}\alpha\right),$
$\displaystyle\text{Tr}_{2}\left[\left|p\right\rangle_{2,2}\left\langle
p\right|D_{2}\left(\beta\right)\right]$
$\displaystyle=e^{-i\sqrt{2}p\operatorname{Re}\beta}\delta\left(\sqrt{2}\operatorname{Im}\beta\right),$
(10)

and

$\text{Tr}_{3}\left[D_{3}\left(-\gamma\right)D_{3}\left(\eta\right)\right]=\pi\delta^{(2)}\left(\eta-\gamma\right),$
(11)

the CF of $\rho_{M}$ defined as
$\chi_{M}\left(q,p;\eta\right)$=Tr${}_{3}\left[\rho_{M}D_{3}\left(\eta\right)\right]$
reads

$\displaystyle\chi_{M}\left(q,p;\eta\right)$
$\displaystyle=\frac{P^{-1}\left(q,p\right)}{\sin
2\theta}{\displaystyle\int}\frac{d^{2}\alpha}{\pi^{2}}\exp\left\\{\alpha^{\ast}\xi-\alpha\xi^{\ast}\right\\}$
$\displaystyle\times\chi_{1}(\sqrt{T}\alpha)\chi_{23}\left[\sqrt{T}\left(\alpha\cot
2\theta+\alpha^{\ast}\csc 2\theta\right),\eta\right]$
$\displaystyle\times\exp\left\\{-\frac{R}{2}[(\operatorname{Re}\alpha)^{2}\csc^{2}\theta+(\operatorname{Im}\alpha)^{2}\sec^{2}\theta]\right\\},$
(12)

where we have defined $\xi=(q/\cos\theta+ip/\sin\theta)/\sqrt{2}$. When $T=1$,
Eq. (12) just reduces to Eq.(2.9) in Ref. 27 .

After Alice informs the measured results ($q,p$) to Bob, Bob needs to make a
unitary transformation to obtain the output state at this stage. Here, we
consider the unitary transformation to be the displacement operator
$D_{3}\left(Z\right)$ with non-nunity and aysmetrical gains characteristic,
where $Z\equiv g_{q}q+ig_{p}p$ with $g_{q}$ and $g_{p}$ are two tunable gain
parameters. Thus, after the displacement, the output state can be expressed as
$\rho_{D}\equiv\int
d^{2}\eta\chi_{M}\left(q,p;\eta\right)D_{3}\left(Z\right)D_{3}\left(-\eta\right)D_{3}\left(-Z\right)/\pi$.
Usually, we do not have interest in every measurement result but the average
effect. Thus we perform an ensemble averaging over all measurement results,
then the average CF $\bar{\chi}_{out}$ is given by

$\displaystyle\bar{\chi}_{out}\left(\beta\right)$
$\displaystyle=\text{Tr}_{3}[D_{3}\left(\beta\right)\int
dqdpP\left(q,p\right)\rho_{D}]$ $\displaystyle=\chi_{1}\left(f_{1}\beta-
f_{2}\beta^{\ast}\right)\chi_{23}\left(\beta^{\ast}f_{3}-\beta\allowbreak
f_{4},\beta\right)$
$\displaystyle\times\exp\\{-R[g_{p}^{2}(\operatorname{Re}\beta)^{2}+g_{q}^{2}(\operatorname{Im}\beta)^{2}]\\},$
(13)

where

$\displaystyle f_{1}$
$\displaystyle=\sqrt{\frac{T}{2}}\left(g_{q}\cos\theta+g_{p}\sin\theta\right),$
$\displaystyle f_{2}$ $\displaystyle=\sqrt{\frac{T}{2}}\left(g_{q}\cos\theta-
g_{p}\sin\theta\right),$ $\displaystyle f_{3}$
$\displaystyle=\sqrt{\frac{T}{2}}\left(g_{q}\sin\theta+g_{p}\cos\theta\right),$
$\displaystyle f_{4}$ $\displaystyle=\sqrt{\frac{T}{2}}\left(g_{q}\sin\theta-
g_{p}\cos\theta\right),$ (14)

and $T$ and $R$ ($T+R=1$) denote, respectively, the transmissivity and
reflectivity of the BS2 and BS3 that stimulate the photon losses.

###  II.3 Relation between input and output including noise in mode 3

Next, we consider the effect in CF formalism of decoherence of environment on
the mode 3. Here we consider the case that the mode 3 propagates in a noisy
channel such as photon loss, and thermal noise after Alice’s measurement but
before its reaching Bob’s location (see Fig. 1(a)). In the interaction picture
and the Born-Markov approximation, the time evolution of the density matrix
describing the thermal environment is governed by the master equation (ME) 28
:

$\displaystyle\frac{d}{dt}\rho\left(t\right)$
$\displaystyle=\kappa\bar{n}\left(2a^{{\dagger}}\rho a-aa^{{\dagger}}\rho-\rho
aa^{{\dagger}}\right)$ $\displaystyle+\kappa\left(\bar{n}+1\right)\left(2a\rho
a^{{\dagger}}-a^{{\dagger}}a\rho-\rho a^{{\dagger}}a\right),$ (15)

where $\kappa$ and $\bar{n}$ are the dissipative coefficient and the average
thermal photon number of the environment, respectively. When $\bar{n}=0,$ Eq.
(15) reduces to the one describing the photon-loss channel. By solving the ME
in the CF form, one can find that the evolution of CF described by Eq. (15) is
given by

$\chi\left(\gamma;t\right)=\chi\left(\gamma e^{-\kappa
t};0\right)\exp\left\\{-\Gamma\left|\gamma\right|^{2}\right\\},$ (16)

where $\Gamma=\left(2\bar{n}+1\right)\left(1-e^{-2\kappa t}\right)/2$, and
$\chi\left(\gamma;0\right)$ is the CF of initial state $\rho\left(0\right)$.
In a similar way to deriving Eq. (13), at Bob’s location, the CF
$\bar{\chi}_{f}$ of final output state for the teleportation scheme can be
directly given by

$\displaystyle\bar{\chi}_{f}\left(\beta;t\right)$
$\displaystyle=e^{-\Gamma\left|\beta\right|^{2}}\chi_{1}\left(f_{1}\beta-
f_{2}\beta^{\ast}\right)$
$\displaystyle\times\chi_{23}\left(\beta^{\ast}f_{3}-\beta\allowbreak
f_{4},\beta e^{-\kappa t}\right)$
$\displaystyle\times\exp\\{-R[g_{p}^{2}(\operatorname{Re}\beta)^{2}+g_{q}^{2}(\operatorname{Im}\beta)^{2}]\\}.$
(17)

The form of Eq. (17) shows the different roles played by the noise channel
($\Gamma,\kappa$) and gain factors ($g_{p},g_{q}$) as well as unbalanced BS
($\theta$), the reflectivity $R$. The decoherence effect from the noisy
channel affects only mode 3 by means of the exponentially decreasing weight
$e^{-\kappa t}$ in the arguments of $\chi_{23}$. Eq. (17) is just the general
description of the nonideal scheme in terms of the CF, which just reduces to
the factorized form of the output CF in Eq. (9) and Eq. (4) in Ref. 17 , as
expected, when $\kappa t=0$ and $g_{q}=g_{p}=g$, $\theta=\pi/4$, respectively.
Thus, Eq. (17) is the generalized input-output relation in the CF formalism.

###  II.4 Fidelity and average fidelity

In order to measure the effectivity of the teleportation scheme, we appeal to
the fidelity of teleportation, defined by
$\mathcal{F=}$Tr$\left(\rho_{in}\rho_{out}\right)$. Within the formalism of
CF, the fidelity reads

$\mathcal{F=}\int\frac{d^{2}\lambda}{\pi}\chi_{in}(\lambda)\chi_{out}(-\lambda),$
(18)

where $\chi_{in}$ and $\chi_{out}$ are the CFs corresponding to density
operators $\rho_{in}$ and $\rho_{out}$, respectively. Eq. (18) is the
fundamental quantity that measures the performance of a CV teleportation,
which will be often used in the following calculations. On the basis of Eqs.
(17) and (18), we can examine the performance of teleportation for any input
states and any entangled resources including non-Gaussian ones.

In particular, when we specify the input teleported states at Alice’s location
to be coherent states
$\rho_{1}=\left|\epsilon\right\rangle\left\langle\epsilon\right|$ with complex
amplitude $\epsilon$, whose CF reads
$\chi_{1}\left(\lambda\right)=e^{-\frac{1}{2}\left|\lambda\right|^{2}+\lambda\epsilon^{\ast}-\epsilon\lambda^{\ast}}$,
then substituting it and Eq. (17) into Eq. (18) we can get

$\displaystyle\mathcal{F}$
$\displaystyle=\int\frac{d^{2}\lambda}{\pi}\exp\left\\{-\frac{1}{2}\left(1+f_{1}^{2}+f_{2}^{2}+2\Gamma\right)\left|\lambda\right|^{2}\right\\}$
$\displaystyle\times\exp\left\\{\frac{1}{2}f_{1}f_{2}\left(\lambda^{2}+\lambda^{\ast}{}^{2}\right)+\lambda\Delta-\lambda^{\ast}\Delta^{\ast}\right\\}$
$\displaystyle\times\chi_{23}\left(\lambda\allowbreak
f_{4}-\lambda^{\ast}f_{3},-\lambda e^{-\kappa t}\right)$
$\displaystyle\times\exp[-R(g_{p}^{2}\operatorname{Re}^{2}\lambda+g_{q}^{2}\operatorname{Im}^{2}\lambda)],$
(19)

where $\Delta=\allowbreak\left(1-f_{1}\right)\epsilon^{\ast}-\epsilon
f_{2}\allowbreak$. From Eq. (19) we can see that if we choose $\Delta=0$ then
the fidelity will be independent of $\epsilon$ for any entangled resources.
The condition of $\Delta=0$ leads to

$g_{q}=\frac{1}{\sqrt{2T}\cos\theta},g_{p}=\frac{1}{\sqrt{2T}\sin\theta}.$
(20)

This is the only choice making the fidelity independent of $\epsilon$, which
allows us to have no information about the input coherent states. The
condition (20) depends on $T$ and $\theta$ but is independent of the
decoherence involved in mode 3. This is true for any entangled resources. In
particular, when $\theta=\pi/4$, Eq. (20) just reduces to the case in Ref. 17
; while for $T=1$ and $\theta=\pi/4,$ this result is just the case discussed
in Ref. 28a .

Generally speaking, the fidelity in Eq. (19) depends on the teleported input
states which are usually unknown by the sender and the receiver. Here, in
order to further describe the fidelity, we assume a partial knowledge of the
input states about a probability distribution $P\left(\mu\right)$ satisfying
the normalization condition, i.e., $\int P\left(\mu\right)d\mu=1$ where the
integral is taken over all possible values of $\mu$. For a given distribution
$P\left(\mu\right)$, the average fidelity is

$\bar{F}=\int\mathcal{F}\left(\mu\right)P\left(\mu\right)d\mu.$ (21)

In the following, we will take three probability distributions into account
for input coherent states, such as line-, circle- and 2D-Gaussian-distribution
19 .

##  III Two-mode squeezed vacuum as entangled resources

In this section, we use the usual TMSV as entangled resources to analyze the
performance of these three tunable parameters for improving the fidelity of
teleportation. The TMSV entangled resource, most commonly used in continuous-
variable teleportation, can be generated by the parametric down-conversion
(PDC) process, and theoretically can be defined as

$\displaystyle\left|\Phi\right\rangle_{sv}$
$\displaystyle=S\left(r\right)\left|00\right\rangle$
$\displaystyle=\text{sech}r\exp\left(a^{\dagger}b^{\dagger}\tanh
r\right)\left|00\right\rangle,$ (22)

where $S\left(r\right)=\exp\left\\{r(a^{\dagger}b^{\dagger}-ab)\right\\}$ is
the two-mode squeezing opertor with $r$ being the squeezing parameter, and
$a^{\dagger}\left(a\right)$ and $b^{\dagger}\left(b\right)$ are photon
creation (annihilation) operators. According to the definition of CF, the CF
of the TMSV is given by

$\displaystyle\chi_{sv}\left(\alpha,\beta\right)$
$\displaystyle=\exp\left\\{-\frac{1}{2}\left(\left|\alpha\right|^{2}+\left|\beta\right|^{2}\right)\cosh
2r\allowbreak\right\\}$
$\displaystyle\times\exp\left\\{\frac{1}{2}\left(\alpha\beta+\alpha^{\ast}\allowbreak\beta^{\ast}\right)\sinh
2r\right\\}.$ (23)

In particular, for the largest entangled resource with $r\rightarrow\infty$,
and ideal measurements with $T=1$ and $R=0$, as well as banlanced BS, we have
$\lim_{r\rightarrow\infty}\chi_{sv}\left(\beta^{\ast},\beta\right)=1$.
Substituting it into Eq. (13) yields
$\lim_{r\rightarrow\infty}\bar{\chi}_{out}\left(\beta\right)=\chi_{1}\left(\beta\right)$,
i.e., a perfect teleportation, as expected.

When Alice use the TMSV as entangled resource to teleport the coherent states,
then the fidelity in Eq. (19) can be calculated as

$\mathcal{F}=\frac{1}{\sqrt{G}}\exp\left\\{\frac{-K_{1}\left|\Delta\right|^{2}+K_{2}\left(\Delta^{2}+\Delta^{\ast
2}\right)}{G}\right\\},$ (24)

where we have defined $G=K_{1}^{2}-4K_{2}^{2}$ and

$\displaystyle K_{1}$
$\displaystyle=\frac{1}{2}\left(1+f_{1}^{2}+f_{2}^{2}+2\Gamma\right)+\frac{R}{2}\left(g_{p}^{2}+g_{q}^{2}\right)$
$\displaystyle+\frac{1}{2}\left(f_{3}^{2}+f_{4}^{2}+e^{-2\kappa t}\right)\cosh
2r-f_{3}e^{-\kappa t}\sinh 2r,$ $\displaystyle K_{2}$
$\displaystyle=\frac{1}{2}\left\\{f_{1}f_{2}-R\left(g_{p}^{2}-g_{q}^{2}\right)/2\right.$
$\displaystyle+\left.f_{3}f_{4}\cosh 2r-f_{4}e^{-\kappa t}\sinh 2r\right\\},$
(25)

and used the following integration formula

$\displaystyle\int\frac{d^{2}z}{\pi}\exp\left(\zeta\left|z\right|^{2}+\xi
z+\eta z^{\ast}+fz^{2}+gz^{\ast 2}\right)$
$\displaystyle=\frac{1}{\sqrt{\zeta^{2}-4fg}}\exp\left\\{\frac{-\zeta\xi\eta+\xi^{2}g+\eta^{2}f}{\zeta^{2}-4fg}\right\\}.$
(26)

From Eq. (24) one can see that the fidelity $\mathcal{F}$ depends on the
amplitude of the teleported coherent states. In the next sections, we shall
consider two kinds of cases: one is independent of the amplitude by the choice
in Eq. (20) and the other is not, but partial information about the input
state distribution is known.

##  IV $\epsilon$-independent optimal fidelity

In this section, we examine the fidelity for teleporting coherent state with
two gain factors fixed to be $g_{q}=1/(\sqrt{2T}\cos\theta)$,
$g_{p}=1/(\sqrt{2T}\sin\theta)$. This choice allows us to have no information
about the amplitude of coherent states. Noticing that $f_{1}=1$, $f_{2}=0$,
$f_{3}=\csc 2\theta$ and $f_{4}=-\cot 2\theta$, then from Eq. (24) we can get

$\mathcal{F}_{\epsilon}=\left\\{H[1/(\sqrt{2T}c_{1}),c_{2}]H[1/(\sqrt{2T}c_{2}),c_{1}]\right\\}^{-1/2},$
(27)

where $c_{1}=\cos\theta,c_{2}=\sin\theta$ and we defined the function
$H\left(x,y\right)$ as

$\displaystyle H\left(x,y\right)$
$\displaystyle=\frac{1}{2}+\Gamma+x^{2}\left(1+2Ty^{2}\sinh^{2}r\right)$
$\displaystyle+\frac{1}{2}e^{-2\kappa t}\cosh 2r-xye^{-\kappa t}\sqrt{2T}\sinh
2r.$ (28)

It is clear that the fidelity $\mathcal{F}_{\epsilon}$ depends on multi-
parameters such as $r,$ $\kappa t,$ $\bar{n},$ $T$ and $\theta$. At fixed $r,$
$\kappa t,$ $\bar{n}$ and $T$, the optimal fidelity of teleportation is
defined as

$\mathcal{F}_{opt}=\max_{\theta}\mathcal{F}\left(r,\theta\right).$ (29)

In order to maximize the fidelity in Eq. (27) over $\theta$, we can take
$\partial\mathcal{F}_{\epsilon}/\partial\theta=0$, which leads to the
following condition

$\cos 2\theta=0,$ (30)

or

$\displaystyle\csc 2\theta$ $\displaystyle=\frac{1}{2}\left\\{\frac{e^{-\kappa
t}\sinh 2r}{1/T+2\sinh^{2}r}\right.+$ $\displaystyle\left.\frac{e^{-\kappa
t}\sinh 2r}{1/T+1+2\Gamma+e^{-2\kappa t}\cosh 2r}\right\\}.$ (31)

It is easy to see that the first item (FI) in the right hand side (RHS) of Eq.
(31) is less than unit, and the second item (SI) satisfies (by taking $T=1$
and $\bar{n}=0$)

$\frac{e^{-\kappa t}\sinh 2r}{1/T+1+2\Gamma+e^{-2\kappa t}\cosh
2r}\leqslant\frac{e^{-\kappa t}\sinh 2r}{3+2e^{-2\kappa t}\sinh^{2}r}.$ (32)

By numerical calculation, we can find that when the squeezing parameter $r$ is
less than a threshold value of about $2.1$, the sum of $($FI$+$SI$)/2$ is
alway less than unit which will lead to an impossible case, i.e., $\csc
2\theta<1$. Thus within the region of threshold value, the optimal point is at
$\theta=\pi/4$ and $g_{q}=g_{p}=1/\sqrt{T}$ which is independent of $\bar{n}$
and $e^{-\kappa t}$. The threshold value of $r$ will increase with the
increasing $\bar{n}$ and $1/T$. Actually, the presence of threshold value
results from the decoherence through mode 3, since the SI is always less than
unit for any squeezing $r$ when $\kappa t=0$.

Substituting the above optimal condition into Eq. (27), we get the optimal
fidelity

$\mathcal{F}_{opt}=[\frac{1}{T}+\Gamma+e^{-\kappa t}(\cosh\kappa t\cosh
2r-\sinh 2r)]\allowbreak^{-1}.$ (33)

It is clear that $\mathcal{F}_{opt}$ decreases with the increasing $\bar{n}$
and $1/T$, as expected. In particular, when $\kappa t=0$ and $T=1$, Eq. (33)
just reduces to $\mathcal{F}_{opt}=(1+\tanh r)/2$, which is the best fidelity
when we use the coherent states as inputs and the TMSV as entangled resources
in the BK scheme independent of teleported coherent state amplitude. In
addition, when $\kappa t\rightarrow\infty$,
$\mathcal{F}_{opt}\rightarrow(\frac{1}{T}+\bar{n}+\cosh^{2}r)^{-1}$, which
decreases with the increasing $\bar{n},r$ and the decreasing $T$.

Figure 2: (Colour online) The optimal fidelity for teleporting coherent states
as a function of the squeezing parameter $r$ with some different values of $T$
and $\kappa t$ as well as $\bar{n}=0$. Here solid lines: the transmissivity
$T=1,0.95,0.9,0.85,0.8$ and $\kappa t=0;$ Blue-dash lines: $\kappa
t=0.1,0.2,0.3,0.4$ and $T=1$. The corresponding lines are arranged from top to
bottom with the increasing $1/T$ and $\kappa t$, respectively.

In order to examine clearly the effects of different parameters on the optimal
fidelity, we plot the optimal fidelities as a function of squeezing parameter
$r$ in Fig. 2. From Fig. 2, we can see that for $\kappa t=0$ (without
decoherence on mode 3) the optimal fidelities increase monotonically with the
increasing squeezing parameter $r$ and the transmissivity $T$. In addition,
when we consider the effects of decoherence on mode 3, the optimal fidelities
first increase and then decrease with the increasing $r$. The maximal value
and the corresponding value of $r_{\max}$ reduces as $\kappa t$ increases. In
fact, we can take $\partial\mathcal{F}_{opt}/\partial r=0$ to get a simple
relation as following ($\cosh\kappa t=\coth 2r_{\max})$

$e^{2r_{\max}}=\coth\frac{\kappa t}{2}.$ (34)

It is interesting to notice that $r_{\max}$ is independent of $T$ and
$\bar{n}$.

##  V Average optimal fidelity and the effect of tunable parameters

In the last section, we consider the $\epsilon$-independent optimal fidelity.
However, when $g_{q}\neq 1/(\sqrt{2T}\cos\theta)$ and $g_{p}\neq
1/(\sqrt{2T}\sin\theta)$, the scenario will be changed dramatically. In this
case, the fidelity in (24) will depend on the amplitude $\epsilon$ of coherent
state. In this section, we examine the average optimal fidelity for three
different probability distributions of the teleported input states in which
the partial information is known by Alice and Bob. For example, they may be
completely sure of phase of the states but the amplitude is unknown 19 .

###  V.1 Optimal fidelity for teleporting coherent states on a line

First, let us consider the teleportation of coherent states on a line. Without
losing the generality, here we assume that the phase of the teleported
coherent states is zero because we can always achieve this point by rotating
frame. Then the corresponding probability distribution can be given by
(letting $\epsilon=x+iy$)

$P\left(x,y\right)=\frac{1}{2L}\delta\left(y\right)\times\left\\{\begin{array}[c]{cc}1,&\left|x\right|\leqslant
L\\\ 0,&\text{else}\end{array}\right..$ (35)

Thus substituting Eqs. (35) and (24) into Eq. (21) yields the average fidelity

$\displaystyle\mathcal{\bar{F}}_{line}$
$\displaystyle=\frac{1}{2L\sqrt{G}}\int_{-L}^{L}dxe^{-\frac{M}{G}x^{2}}$
$\displaystyle=\frac{\sqrt{\pi}}{2L\sqrt{M}}\text{Erf}\left\\{\frac{|1-\sqrt{2T}g_{q}\cos\theta|}{[H\left(g_{q},\sin\theta\right)]^{1/2}L^{-1}}\right\\},$
(36)

where Erf$\\{a\\}=$ $1/\sqrt{\pi}\int_{-a}^{a}e^{-x^{2}}dx$ is the error
function and
$M\left.=\right.(1-\sqrt{2T}g_{q}\cos\theta)^{2}H\left(g_{p},\cos\theta\right)$.

Noticing the separability of $g_{q}$ and $g_{p}$ in
$\mathcal{\bar{F}}_{line}$, the optimal value of $g_{p}$ can be obtained by
$\partial\mathcal{\bar{F}}_{line}/\partial g_{p}=0$ equivalent to $\partial
H\left(g_{p},\cos\theta\right)/\partial g_{p}=0,$ which leads to

$g_{p}^{opt}=\frac{e^{-\kappa t}\sqrt{2T}\cos\theta^{opt}\sinh
2r}{2(1+2T\cos^{2}\theta^{opt}\sinh^{2}r)}.$ (37)

It is interesting to notice that the optimal value of $g_{p}^{opt}$ is related
to $e^{-\kappa t}$ and $T$ but independent of the average thermal photon-
number.

Next, we will maximize the fidelity by numerical calculation. At fixed $r$,
$\kappa t$, $\bar{n}$ and $T$, the optimal fidelity of teleportation can be
defined as

$\mathcal{\bar{F}}_{line}^{opt}=\max_{g_{q},g_{p},\theta}\mathcal{\bar{F}}_{line}\left(r,g_{q},g_{p},\theta\right).$
(38)

In Fig. 3 we plot the optimal fidelity as a function of squeezing parameter
$r$ for some different values of parameters $\kappa t$, $\bar{n}$ and $T$. In
Fig. 3(a), we consider the optimal fidelities with some different values of
$L$ and $T=1,\bar{n}=0$ as well as $\kappa t=0.2$ (for comparison, the case of
$\kappa t=0$ is also plotted as dash lines). From Fig. 3(a), we can see that
the optimal fidelities grow with increasing $r$ and $1/L$. The fidelities can
be greatly optimized with respect to the standard teleportation scheme lines
(STS with $g_{q}=g_{p}=1$ and $\theta=\pi/4$, see short dash-dot-(dot) lines).
Especially for a smaller $L$ ($L=0.1$), the optimal fidelity can almost access
to unit. While for a larger $L$ (say $L=300$), the fidelity achieves a
limitation (still over 0.8) which is still superior to that in the STS. In
Fig. 3 (b), we consider the effect of different values of $T$ on the optimal
fidelity at $\kappa t=0,0.2$. It is shown that the optimal values decease with
the increasing $1/T$ for a given $\kappa t$; while for the case of $T\neq 1$,
by comparing the fidelities at $\kappa t=0,0.2$ for a given $T$, it is found
that the optimal fidelities first increase and then decrease with the
increasing $r$, and it is interesting to notice that the optimal fidelity with
$\kappa t=0.2$ is superior to that with $\kappa t=0$ when $r$ exceeds a
certain cross-point.

Figure 3: The optimal fidelity for teleporting CSs as a function of $r$ with
$\bar{n}=0$, $\kappa t=0,0.2$ corresponding to solid and dash lines,
respectively. (a) $L=0.1,0.5,1,3,300$ and $T=1;$ for comparison, the
teleportation in the STS is also plotted as dash-dot and dash-dot-dot lines
for $\kappa t=0,0.2$; (b) $T=1,0.95,0.9,0.85,0.8$ and $L=1$. The corresponding
lines are arranged from top to bottom with the increasing $L$ and $1/T$ for a
given $\kappa t$.

Furthermore, in order to clearly see whether there is the ability against the
decoherence by using these tunable parameters, here we plot the optimied
fidelity as a function of the evolution time $\kappa t$ for some fixed values.
For this purpose, we fix the squeezing parameter at the intermediate value
$r=0.8$ with $T=0.9$, $\bar{n}=0$. Experimentally, the attainable squeezing
degree is about $1.5$. Fig. 4 shows that the optimal fidelity remains above
0.8 which exceeds the classical threshold up to significantly large values of
$\kappa t$. This case is true even for the limitation of $L\rightarrow\infty$.
These results indicate that the optimizal fidelities by three parameters
present superior behavior to and higher ability against the decoherence than
those in the standard teleportation scheme (with $g_{q}=g_{p}=1$ and
$\theta=\pi/4$, also see dash lines in Fig. 4).

Figure 4: (Colour online) The optimal fidelity for teleporting CSs as a
function of $\kappa t$ with $\bar{n}=0$, $r=0.8,$ $T=0.9$ and
$L=0.1,0.5,1,3,300$ from top to bottom, respectively.

In addition, we make a comparison among the optimal effects of different
optimal parameters. In Fig. 5, we plot the optimized fidelity over different
tunable parameters as the function of squeezing parameter $r$ for a given
$L=1$ with $T=1$, $\bar{n}=0$ and $\kappa t=0.2$. It is found that the
optimization by three tunable parameters is the best when compared to single-
and two-parameter optimization, especially in the region of small
entanglement, which indicates that the role of parameter is different from
each other. Thus, it is necessary to perform a simultaneous balanced
optimization over these three parameters to obtain a maximization of
teleportation fidelity for the probability distribution in Eq. (35).

Figure 5: (Colour online) The optimal fidelity for teleporting CSs as a
function of $r$ with $\bar{n}=0$, $\kappa t=0.2,$ $T=L=1$ for several
different optimal parameters.

###  V.2 Optimal fidelity for teleporting CSs on a circle

In this subsection, we consider the optimal fidelity for teleporting CSs on a
circle, which means $\epsilon=\left|\epsilon\right|e^{i\varphi}\equiv
Ae^{i\varphi}$ with a known amplitude $\left|\epsilon\right|=A$ and unknown
phase $\varphi$. In this case, the distribution function is
$P\left(A,\varphi\right)=\delta\left(\left|\epsilon\right|-A\right)/2\pi$
satisfying
$\int_{0}^{\infty}\int_{0}^{2\pi}P\left(A,\varphi\right)d\left|\epsilon\right|d\varphi=1$,
then the average fidelity can be calculated as

$\mathcal{\bar{F}}_{circle}=\frac{e^{-R_{1}}}{\sqrt{G}}\sum_{k=0}^{\infty}\frac{\left(R_{2}\right)^{2k}}{k!k!},$
(39)

where we have set
$R_{1}=A^{2}\\{K_{1}[\left(1-f_{1}\right)^{2}+f_{2}^{2}]+4K_{2}\left(1-f_{1}\right)f_{2}\\}/G$,
$R_{2}=A^{2}\\{K_{1}\left(1-f_{1}\right)f_{2}+K_{2}[\left(1-f_{1}\right)^{2}+f_{2}^{2}]\\}/G$.
Maximizing $\mathcal{\bar{F}}_{circle}$ over these three parameters, we can
get the optimized fidelity
$\mathcal{\bar{F}}_{circle}^{opt}=\max_{g_{q},g_{p},\theta}\mathcal{\bar{F}}_{circle}\left(r,g_{q},g_{p},\theta\right)$.
Our random numerical calculations show that, for the probability distribution,
the maximum value of fidelity can be achieved at the point with
$g_{q}=g_{p}=g$ and $\theta=\pi/4$, which is different from the case in
subsection A. Under this condition we have $f_{1}=f_{3}=g\sqrt{T}$ and
$f_{2}=f_{4}=0$, as well as $R_{2}=K_{2}=0$. Thus the optimized fidelity can
be given by

$\mathcal{\bar{F}}_{circle}^{opt}=\frac{1}{\Theta}\exp\left\\{-\frac{A^{2}}{\Theta}(1-g\sqrt{T})^{2}\right\\},$
(40)

where we have set $\Theta=\Gamma+[\allowbreak
g^{2}(R+1)+1+(g\sqrt{T}-e^{-\kappa t})^{2}\cosh 2r+2g\sqrt{T}e^{-\kappa
t}e^{-2r}]/2$. It is obvious that $\Theta>0$.

Using Eq. (39) or (40), we have plotted the optimal fidelity as a function of
squeezing parameter $r$ for some different values of $A$ and $T$ in Fig. 6. In
Fig. 6(a), we consider the optimal fidelities with some different values of
$A$ with $T=1,\bar{n}=0$ as well as $\kappa t=0,0.2$. From Fig. 6(a), we can
see that the optimal fidelities grow monotonously with increasing $r$ for
$\kappa t=0,$ but for $\kappa t=0.2$ the optimal fidelities first increase and
then decrease with increasing $r$ especially for a large $A$ (say $A=3$). In
addition, for a small $A$, the optimal fidelity almost access to unit. In Fig.
6(b), we also examine the effect of different $T$ on the fidelity. It is
interesting to notice that the optimal fidelity with $\kappa t=0.2$ can be
better than that with $\kappa t=0$ when the squeezing $r$ exceeds a certain
value. This case is similar to that in Fig. 3(b). In Fig. 6, the point
$r_{\max}$ corresponding to the maximum fidelity depends on $\kappa t,$ and
for given $\kappa t$, $A$ and $T$, the value of $r_{\max}$ can be determined
by taking $\partial\mathcal{\bar{F}}_{circle}^{opt}/\partial r=0,$ which leads
to

$\left\\{\Theta-A^{2}(1-g\sqrt{T})^{2}\right\\}\frac{\partial\Theta}{\partial
r}=0.$ (41)

After a straightforward calculation, we can abtain

$\tanh 2r_{\max}=\frac{2g\sqrt{T}e^{-\kappa t}}{g^{2}T+e^{-2\kappa t}}.$ (42)

or

$e^{4r_{\max}}-be^{2r_{\max}}+c^{2}=0,$ (43)

where $b=4[A^{2}(1-g\sqrt{T})^{2}-\Gamma]/(g\sqrt{T}-e^{-\kappa t})^{2},$ and
$c=(g\sqrt{T}+e^{-\kappa t})/(g\sqrt{T}-e^{-\kappa t})$. In particular, when
$g=1/\sqrt{T}$ the fidelity is independent of the amplitude $A$ and since that
$\Theta>0$, the value of $r_{\max}$ in Eq. (42) reduces to that in Eq. (34).

Figure 6: (Colour online) The optimal fidelity for teleporting CSs as a
function of $r$ with $\bar{n}=0$, $\kappa t=0,0.2$ (a) $A=0.1,0.5,1,3,300$ and
$T=1;$ (b) $T=1,0.95,0.9,0.85,0.8$ and $A=1$. For each optimized case
(associated with a special plot style), the corresponding lines are arranged
from top to bottom with the increasing $A$ and $1/T$ at the point $r=0$,
respectively.

In Fig. 7, choosing the same values of parameters $T,$ $r$ and $\bar{n}$ as in
Fig. 4, we plot the optimal fidelity as a function of $\kappa t$. For
comparison, we also plot the fidelity without the optimization, i.e.,
$g_{q}=g_{p}=1$ and $\theta=\pi/4$ (see dash lines). Fig. 7 shows that the
teleportation fidelity can be always above the classical limitation 0.5 up to
significantly large values of $\kappa t\left(\leqslant 2\right)$ when the
amplitude $A$ is less than about $1.7$, and while it can go below the
limitation $0.5$ for $A>1.7$ when $\kappa t$ exceeds a certain threshold
value. The optimal fidelities with $A=15,300$ are indistinguishable. In the
STS, the fidelity is less than 0.5 when $A>15$. Comparing the fidelities with
and without optimization, it is shown that the former have better
teleportation performance than the latter. However, this improvement is
inferior to that shown in Fig. 4 where the optimal fidelities are over 0.8 for
any $L$ and $\kappa t$.

Figure 7: (Colour online) The optimal fidelity for teleporting CSs as a
function of $\kappa t$ with $\bar{n}=0$, $r=0.8,$ $T=0.9$ and
$A=0.1,0.5,1,1.7,3,300$ from top to bottom, respectively.

###  V.3 Optimal fidelity for teleporting CSs by 2D Gaussian distribution

In the last subsection, we consider another simple probability
distribution—–two-dimensional (2D) Gaussian distribution. The corresponding
distribution is given by
$P\left(\alpha\right)=1/(\pi\chi)\exp[-\left|\alpha\right|^{2}/\chi]$
satisfying $\int P\left(\alpha\right)d^{2}\alpha=1$ 17 ; 19 ; 29 , where the
variance parameter $\chi$ determines the cutoff of the amplitude $\alpha$.
Thus, using Eqs. (21) and (24), the averaged fidelity can be calculated as

$\displaystyle\mathcal{\bar{F}}_{G}$
$\displaystyle=\frac{1}{\sqrt{H(g_{p},\cos\theta)+\chi(1-\sqrt{2T}g_{p}\sin\theta)^{2}}}$
$\displaystyle\times\frac{1}{\sqrt{H(g_{q},\sin\theta)+\chi(1-\sqrt{2T}g_{q}\cos\theta)^{2}}},$
(44)

where the function $H(x,y)$ is defined in Eq. (28)
[$\left(K_{1}+2K_{2}\right)=H\left(g_{q},\sin\theta\right)$,
$\left(K_{1}-2K_{2}\right)=H\left(g_{p},\cos\theta\right)$]. Noticing that the
parameters $g_{q}$ and $g_{p}$ are independent from each other in Eq. (44),
thus it is not hard to obtain the optimal point by
$\partial\mathcal{\bar{F}}_{G}/\partial
g_{q}=\partial\mathcal{\bar{F}}_{G}/\partial
g_{p}=\partial\mathcal{\bar{F}}_{G}/\partial\theta=0$, i.e., $\theta=\pi/4$
and $g_{q}=g_{p}=g$, where$\ g$ is given by

$g=\frac{\sqrt{T}\left(e^{-\kappa t}\sinh
2r+2\chi\right)}{2[1+T(\sinh^{2}r+\chi)]}.$ (45)

At this optimal point, the average optimized fidelity can be expressed as

$\mathcal{\bar{F}}_{G}^{opt}=\frac{1}{H(g,1/\sqrt{2})+\chi(1-g\sqrt{T})^{2}}.$
(46)

It is clearly seen that the optimal factor $g$ depends not only on $T$, but
also on the evolution time $\kappa t$ in a different form. In particular, when
$\allowbreak T\rightarrow 1$ and $\kappa t\rightarrow 0$, Eq. (45) reduces to
the result in Ref. 30 . In addition, in the limitation case of
$\chi\rightarrow\infty$ which implies that the probability distribution
includes the whole complex plane, then we have $g\rightarrow 1/\sqrt{T}$,
which just corresponds to the fidelity independent of $\epsilon$.

Using Eq. (44) or (46), we have plotted the optimal fidelity as a function of
squeezing parameter $r$ and $\kappa t$ for some different values of $\chi$ in
Fig. 8 and Fig. 9, respectively. From Fig. 8, we can see that the smaller the
distribution $\chi$ is, the higher the optimal fidelity is. As $\chi$
increases which implies that we have less knowledge of the amplitude of the
teleported states, the optimal fidelity approaches to that in the standard
scheme ($g=1$). In addition, as $r$ increases, the fidelity first increases up
to a $\kappa t$-dependent maximum $r_{\max}$, and then decreases for a larger
values of $r$ for a given big $T\chi$. Actually, using
$\partial\mathcal{\bar{F}}_{G}^{opt}/\partial r=0$, we can get

$e^{2r_{\max}}=\frac{\left(e^{\kappa t}+1\right)T\chi+1}{\left(e^{\kappa
t}-1\right)T\chi-1}.$ (47)

For instance, when $T=1,\chi=300$ and $\kappa t=0.2$, then $r_{\max}\simeq
1.\,\allowbreak 16$, which is in agreement with the numerical result in Fig.
8. In Fig. 9, we also consider the effect of decoherence on the fidelity. We
can see the similar results to the case of circle distribution. Among these
three distributions used above, the line distribution presents the most
improvement for fidelity, but the Gaussian distribution presents the lowest
improvement. However, a common advantage is that the fidelity with CV can be
improved by using the tunable parameters even in the environments when
compared with the standard teleportation scheme.

Figure 8: (Colour online) The optimal fidelity for teleporting CSs as a
function of $r$ with $\bar{n}=0$, $T=1,$ $\chi=0.1,0.5,1,3,300$ and $T=1.$ For
each optimized case (associated with a special plot style), the corresponding
lines are arranged from top to bottom with the increasing $\chi$. $\kappa
t=0,0.2$ correspond to solid and dash lines, respectively.  Figure 9: (Colour
online) The optimal fidelity for teleporting CSs as a function of $\kappa t$
with $\bar{n}=0$, $r=0.8,$ $T=0.9$ and $\chi=0.1,0.5,1,3,300$ from top to
bottom, respectively. For comparison, the fidelities with g=1 are plotted here
(dash lines).

##  VI Conclusions

In this paper, we have examined the performance of three-tunable parameters in
realistic scheme of CV quantum teleportation with input coherent states and
the TMSV entangled resources. For our purpose, we have appealed to the input
and output relation in the CF formalism, which is convinent for nonideal
inputs and any entangled resources. In this realistic scheme, we have derived
the condition that the fidelity is independent of the amplitude of input
coherent states for any entangled resource. In order to investigate the effect
of three-tunable parameters on the fidelity of teleportation in the nonideal
scheme, we have derived the analytical expressions of the optimal fidelity for
input coherent states with three different prabability distributions and
investigated the performance of optimal fidelity. It is theoretically shown
that the usefulness of tunable parameters for improving the fidelity of
teleportation with or without the effect of environment and imperfect
measurements. In particular, for the input coherent states with a linear
distribution, the optimization with three tunable parameters is the best one
with respect to single- and two-parameter optimization, especially in the
region of small squeezing.

It would be interesting to extend the present analysis to teleport two-mode
states (ideal or nonideal cases) using multipartite (non-)Gaussian entangled
resources in the formalism of CF. In addition, a recent comparison between the
well-known CV VBK scheme and the recently proposed hybrid one by AR has been
made 31 ; 32 . It is shown that the VBK teleportation is actually inferior to
the AR teleportation within a certain range, even when considering a gain
tuning and an optimized non-Gaussian resource. Thus it may be worthy of
considering whether these three-parameter optimization can further improve the
fidelity in VBK scheme over that in AR scheme especially in the non-ideal
scheme.

ACKNOWLEDGEMENTS: This work is supported by a grant from the Qatar National
Research Fund (QNRF) under the NPRP project 7-210-1-032. L. Y. Hu is supported
by the China Scholarship Council (CSC) and the National Natural Science
Foundation of China (No.11264018), as well as the Natural Science Foundation
of Jiangxi Province (No. 20151BAB212006).

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[1]Joshua G. Albert

[1]Leiden Observatory, Leiden University, PO Box 9513, Leiden, 2300, The
Netherlands

# Phantom-Powered Nested Sampling

<EMAIL_ADDRESS>*

###### Abstract

We introduce a novel technique within the Nested Sampling framework to enhance
efficiency of the computation of Bayesian evidence, a critical component in
scientific data analysis. In higher dimensions, Nested Sampling relies on
Markov Chain-based likelihood-constrained prior samplers, which generate
numerous ’phantom points’ during parameter space exploration. These points are
too auto-correlated to be used in the standard Nested Sampling scheme and so
are conventionally discarded, leading to waste. Our approach discovers a way
to integrate these phantom points into the evidence calculation, thereby
improving the efficiency of Nested Sampling without sacrificing accuracy. This
is achieved by ensuring the points within the live set remain asymptotically
i.i.d. uniformly distributed, allowing these points to contribute meaningfully
to the final evidence estimation. We apply our method on several models,
demonstrating substantial enhancements in sampling efficiency, that scales
well in high-dimension. Our findings suggest that this approach can reduce the
number of required likelihood evaluations by at least a factor of 5. This
advancement holds considerable promise for improving the robustness and speed
of statistical analyses over a wide range of fields, from astrophysics and
cosmology to climate modelling.

###### keywords:

Nested Sampling, High-dimensional statistics, Computational Efficiency,
Markov-chain

##  1 Introduction

Nested sampling [NS; 20] is a Bayesian method of Lebesgue integration,
enabling practical computation of the Bayesian evidence,

$\displaystyle Z=\int_{\mathcal{A}}L(x)\,\mathrm{d}p(x),$ (1)

where $p(x)$ is a probability measure of a random variable, $x\in\mathcal{A}$,
and $L(x)$ is a likelihood function. More generally, NS applies to scenarios
where $L$ is any measurable function defined on a measurable space
$(\mathcal{A},\Sigma,p)$. A naive quadrature approach to evaluating Eq. 1,
even when $L$ and $p$ are well-behaved, will quickly fail as the resolution of
such a quadrature grid would quickly exceed hardware limits as the dimension
of the problem grows. Nested sampling was introduced by John Skilling as a way
to overcome this challenge, and does so by reformulating the problem of
integration into a problem of compression.

Firstly, we define the likelihood constrained subset
$\mathcal{A}(\lambda)=\\{x:L(x)>\lambda,x\in\mathcal{A}\\}$, and its
associated measure,

$\displaystyle X_{\lambda}\triangleq
p(\mathcal{A}(\lambda))=\int_{\mathcal{A}(\lambda)}\mathrm{d}p(x),$ (2)

which we refer to as the enclosed prior volume. Intuitively, one can see that
as $\lambda$ increases the associated sub-space, $\mathcal{A}(\lambda)$,
shrinks, and therefore $p(\mathcal{A}(\lambda))$ shrinks monotonically. Now,
consider any partitioning of likelihood values
$(\lambda_{0},\ldots,\lambda_{n})$, where $\lambda_{0}$ is the infimum of
likelihood values, and $\lambda_{n}$ is the supremum of likelihood values.
Associated with this partitioning is a sequence of sub-spaces
$\mathcal{A}(\lambda_{0})\subseteq\ldots\subseteq\mathcal{A}(\lambda_{n})$.
Since the likelihood values partition the full range of the likelihood
function, collectively the subsets cover the entire set,

$\displaystyle\mathcal{A}=$
$\displaystyle\bigcup_{i=0}^{n}\mathcal{A}(\lambda_{i})$ (3) $\displaystyle=$
$\displaystyle\bigcup_{i=1}^{n}\mathcal{A}(\lambda_{i})\setminus\mathcal{A}(\lambda_{i-1}).$
(4)

This last line writes the entire set in terms of the union of non-intersecting
sets, and this then allows us to rewrite Eq. 1 as a sum,

$\displaystyle
Z=\lim_{n\to\infty}\sum_{i=1}^{n}\lambda_{i}(X_{\lambda_{i}}-X_{\lambda_{i-1}}).$
(5)

Equation 5 is incredible, as it has turned a difficult high-dimensional
integral into a simple 1-dimensional sum. The challenge and crux of NS then
becomes in how to approximate the sequence
$((\lambda_{i},X_{\lambda_{i}}))_{i=0}^{n}$. A natural Bayesian method for
this falls from the construction above.

Starting from $i=0$ and $\lambda_{0}=0$ and $X_{\lambda_{0}}=1$ repeat the
following:

  1. 1.

Draw: Uniformly draw $x\sim p(\mathcal{A}(\lambda_{i}))$, and label
$\lambda_{i+1}=L(x)$.

  2. 2.

Shrink: The corresponding subset measure will decrease by the order statistics
of the uniform distribution, $t\sim\mathcal{U}[0,1]$, so that
$X_{\lambda_{i+1}}=tX_{\lambda_{i}}$.

  3. 3.

Increment: Stop if $\lambda_{i+1}$ is close to the supremum, otherwise $i\to
i+1$.

This is the so-called shrinkage procedure, and is the general building block
upon which all NS variations are built. In the draw step, instead of drawing
only a single sample, $x\sim p(\mathcal{A}(\lambda_{i}))$, a set of $m$
identically and independently uniformly distributed points can be drawn, and
the smallest likelihood point selected, leading to a generalised order
statistic $t\sim\mathrm{Beta}[m,1]$. In the shrink step, typically
implementations keep track of expectations of $X_{\lambda}$, otherwise it can
be Monte-Carlo simulated easily. Finally, typically implementations have more
nuanced stopping conditions than sensing if the likelihood supremum is
reached, indeed, likelihoods need not be bounded.

Since its inception in Skilling [20] numerous studies have improved all
aspects of NS, from efficiently drawing from the likelihood-constrained prior
[10, 5, 11, 8, 24], dynamically choosing the number, $m$, of points to draw
[13], parallel-computation [12], and much more. A wonderful review of the
field is given by Buchner [6]. Nested sampling has become an important tool in
many fields, including cosmology [16, 9, 14, 19], compact high-energy
astrophysics [1, 21], spectra modelling [2, 7], gravitational lensing [23, 4],
and more [22, 15].

The present paper introduces a methodology for vastly increasing the
efficiency of nested sampling, i.e. decreasing the overall number of
likelihood evaluations required to reach a target evidence accuracy. The paper
is structured as follows: in Section 2 we introduce a NS formulation that
illuminates the opportunity for our proposed efficiency improvement; in
Section 3 we explain how we empirically validate our method, and in Section 4
we show empirical results demonstrating the efficiency improvement; then in
Section 5 we discuss when this improvement can be employed, and its
limitations.

##  2 Formulating the opportunity for improving efficiency

One of the most challenging aspects of NS is sampling from the likelihood-
constrained prior. Many solutions to this problem maintain a set of
simultaneously evolved samples within the likelihood constraint, which is
called the ‘live set’. For models of low dimension often rejection samplers,
e.g. MultiNest [10], are the most efficient as there are enough samples in the
live set at step $i$ to reconstruct convenient bounding representations of the
subsets $\mathcal{A}(\lambda_{i})$. However, as model dimension grows, the
samples will cluster near the outer shell of the subsets, leading to
exponentially decaying sample efficiency. This is known as the curse of
dimensionality. Therefore, in higher dimensions Markov-chain samplers are
used, e.g. PolyChord [11], which have polynomial decaying sample efficiency.
Often these methods also require some analysis of the samples in the live set.

Markov-chain samplers work by creating a sequence of proposal samples which
satisfy the likelihood-constraint, with each sample conditionally dependent on
the preceding sample. As the number of steps increase the proposed samples
become more independent from the initial point of the chain, and after a
certain number of steps the chain can stop. The last proposed sample becomes
the accepted sample. The only necessary criterion for Markov-chain samplers is
erogodicity, i.e. that eventually all possible points in the model space can
be reached in a finite number of steps, and that the sampler cannot get stuck
in periodic cycles.

The discarded samples, i.e. all-but-the-last proposed sample, are called the
phantom samples. We note that discarding the phantom samples seems wasteful,
as each proposal required hard work to generate a sample that satisfies the
likelihood-constraint. Therefore, we dig deeper into why we discard phantom
samples.

Recall that NS works by iteratively shrinking the enclosed prior volume,
$X_{\lambda}$, and that this process depends on the order statistics of a set
of $m$ likelihoods $\mathcal{S}=\\{\lambda_{i}\\}_{i=1}^{m}$. The order
statistic for shrinkage corresponding to the lowest likelihood is precisely
described by $t\sim\mathrm{Beta}[m,1]$ if all orderings of likelihoods have
equal probability. If the orderings of likelihoods in $\mathcal{S}$ are not
all equally likely, then the order statistics will deviate and the resulting
evidence calculation will become biased.

So, the reason why phantom samples are discarded is to ensure the ordering of
the resulting set $\mathcal{S}$ remains uniform. The question that then arises
is, is discarding all phantom samples the best we can do, i.e. can we use some
phantom samples and still keep $\mathcal{S}$ uniform?

To illuminate the answer to this question we introduce a useful representation
of the shrinkage process, which we call a sample tree. A sample tree
represents the samples produced by NS as a tree, where samples are nodes, and
an edge connects each node to the node with likelihood label equal to the
likelihood constraint that the sample was generated within. The tree can then
be drawn with horizontal position of each node equal to the likelihood label.
This is illustrated in Figure 1 for a single Markov-chain. A useful property
of a sample tree, so arranged, is that $\mathcal{S}$ can be dynamically
constructed for any likelihood constraint by drawing a vertical line and
collecting all samples whose in-edges are crossed. Thus, the size of
$\mathcal{S}$, $m$, for any likelihood constraint is given by counting edge
crossings. It is irrelevant if the sample tree is constructed by
simultaneously evolving a set of ‘live points’, or many parallel independent
Markov-chains, or indeed if samples are added to it via any other means. A
similar representation was coincidentally introduced in Buchner [6], however
our sample tree was developed independently.

A sample tree is useful for visualising how phantom samples are related to
accepted samples. An accepted sample and all its phantom samples (shown in
green in Figure 1) share the same likelihood constraint, thus all have edges
connected to the same sender node. Suppose there are $k$ phantom samples and
$c$ parallel Markov-chains, as visualised Figure 5 in Appendix A. We make two
observations. Firstly, the last few phantom samples before the accepted sample
are nearly identically sampled from the same distribution as the accepted
sample, however they are not independent from one another. This is because
after enough proposal steps from an initial point, all further proposals can
be considered uniformly sampled within the likelihood constraint, despite
being correlated among themselves. Secondly, the likelihoods in a sorted
ordering of $\mathcal{S}$ will asymptotically approach uniformity as $c$
increases. This is proven in Appendix A.

Based on these two observations we propose a method of boosting NS sampling
efficiency when using a Markov-chain sampler:

  1. 1.

During each shrink step, take a large enough number of proposal steps, $i_{\rm
min}+k$, so that samples from all steps $i>i_{\rm min}$ are identically
sampled within the same likelihood constraint.

  2. 2.

Keep the last $k$ phantom samples, in addition to the accepted sample, by
attaching them to your sample tree as shown in Figure 1.

  3. 3.

Choose a large enough number of parallel Markov-chain samplers $c$, so that
the ordering of $\mathcal{S}$ approaches uniformity at all likelihoods.

This essentially says that, while it is crucial that all points in
$\mathcal{S}$ are identically and independently uniformly drawn from within a
likelihood constraint, we can ensure they are identically drawn, and
asymptotically ensure they are independently drawn by following this simple
procedure.

root
---
$m=1$
---
$m=3$
---
$m=2$
---
$m=4$
---
$m=3$
---
$m=2$
---
$m=1$
---
Figure 1: A sample tree representations of samples produced from shrinkage
process with a single Markov-chain, with phantom points in green and accepted
samples in black, with horizontal position given by the likelihood of the
sample. Edges connect samples to the likelihood constraint they were sampled
within. Note, accepted samples connect only to other accepted samples. The
size of $\mathcal{S}$, $m$, at each point is found by counting the number of
edges crossed by vertical lines, shown by the grey dashed lines.

##  3 Method

We empirically validate the method on a highly-correlated multivariate
Gaussian likelihood with Gaussian prior model using the probabilistic
programming framework JAXNS [3] to implement the model and the proposed
method. For the prior we consider a standard centred D-dimensional Gaussian of
unit variance. For the likelihood we consider a mean offset by 15 standard
deviations from the prior, and a highly correlated covariance matrix with unit
variance, and 99% positive correlation between all dimensions. The evidence
for this model is analytically known. We evaluate the method over two model
dimension sizes of $D=8$ and $D=16$.

For our Markov-chain likelihood-constrained sampler we use a standard
1-dimensional slice sampler [17], with a few modifications. Firstly, in place
of a step-out procedure, we initialise the bounding interval end points at the
boundary of the model space. While initially seeming naive, this avoids
spending likelihood evaluations during the step out procedure, and perfectly
satisfies detailed balance – a stronger condition than ergodicity. Secondly,
during the contraction step, when a point along the slice is observed to fall
outside the likelihood contour we shrink to the mid-point with the slice
origin. This increases the exponential contraction rate, while not impacting
autocorrelation much [18]. In sum, these two modifications result in a simple
1-dimensional slice sampler that quickly explores parameter space, while
respecting the ergodicity requirement precisely. It also lacks any additional
analysis or meta-computation, e.g. clustering, required by some other
implementations.

In order to probe the impact of Markov-chain autocorrelation, and determine
$i_{\rm min}$, we explore the slice factor, $s$, from which the number of
proposal steps between accepted samples is $sD$. Smaller values of $s$ will
result in accepted sample samples that are less independent from the initial
point of the chain, whereas higher values of $s$ will be more independent. We
explore values $s\in\mathcal{P}_{s}\triangleq\\{1,2,3,4,5,6\\}$.

We retain the last $k$ phantom samples per accepted sample. We explore values
$k\in\mathcal{P}_{k}\triangleq\\{0,1,2,3,4,5\\}$, where $k=0$ is standard
nested sampling.

In order to study the convergence of the proposed method we explore number of
parallel Markov-chains, $c\in\mathcal{P}_{c}\triangleq\\{10D,20D,30D,40D\\}$.
In order to compare this with other standard nested sampling implementations
this would be approximately equivalent to simultaneously evolving a set of
$m=c(1+k)$ live points.

For each combination of $s$, $k$ and $c$ we run the nested sampling from 100
different random seeds. We compute ensemble averages of the log-evidence
statistics, and all other run diagnostics.

Our primary aim is to measure the asymptotic convergence to standard nested
sample, $k=0$. Therefore, we define the target distribution as the run with
$s_{\rm target}=\sup_{s}{\mathcal{P}_{s}}$, $c_{\rm
target}=\sup_{c}{\mathcal{P}_{c}}$, and $k=0$. Our primary measurement is bias
from this target log-evidence distribution.

We note, there is a degeneracy in how to attribute likelihood evaluations per
sample for $k>0$. In our implementations, we do this by counting the total
number of likelihood evaluations per accepted sample and equally distributing
them among the accepted sample and $k$ retained phantom samples.

##  4 Results

Figure 2: The log-evidence bias as a function of slice factor, $s$, for the
$D=8$ model, coloured by the number of retained phantom samples, $k$ for
different values of $c$. The dashed line represents the target log-evidence
distribution for standard nested sampling. Each point and error bar is
ensemble averaged over 100 runs.

We discuss here the results for the $D=8$ model, with the same results holding
for the $D=16$ model which can be found in Appendix B.

In Figure 2 we can see the log-evidence bias as a function of slice factor $s$
for the $D=8$ model, for different numbers of parallel Markov-chains, $c$, and
retained phantom points, $k$. We observe two well-known facts for standard
nested sampling $k=0$ [e.g. 6]. Firstly, that runs with lower $s$ tend to be
negatively biased due to autocorrelation of accepted samples, and as $s$
increases the log-evidence distribution asymptotes to a single distribution.
Secondly, as $c$ increases the width of the log-evidence distribution becomes
more narrow, i.e. evidence estimations become more precise.

In addition we observe for fixed values of $s$ and $c$ a variation between
log-evidence distributions over values of $k$. This implies that retaining
phantom samples introduces an inaccuracy into the evidence calculation, as
expected. However, as $s$ and $c$ increase we see this variation shrink.
Visually we note that the log-evidence statistics appear consistent for all
runs with $s\geq 4$ and $c\geq 20D$. Since all runs within this region have
consistent log-evidence bias statistics, we label this the consistent region.
We note this region may be different for different models.

We then explore the efficiency and cost per run in the consistent region.
Figure 3 shows relative efficiency improvements, relative to $k=0$, of all
runs in the consistent region, against the fraction of totally random pairwise
orderings, defined in Appendix A. As expected, as $k$ increases the sample
efficiency goes up and the runs complete faster leading to an overall speed up
relative to $k=0$. An maximal efficiency improvement of 5 is observed for
$k=5$, as expected. In terms of number of likelihood evaluations, we observe
that the lowest number of likelihood evaluations is observed for largest $k$,
and minimal $s$ and $c$ within the consistent region. Importantly, since all
runs in the consistent region have consistent log-evidence distributions, we
are at liberty to choose the run setting with the smallest required likelihood
evaluations. This result empirically supports the validity of the proposed
method.

Figure 3: Top panel: The efficiency improvement vs $f$ for the $D=8$ model for
runs in the consistent region, $s\geq 4$ and $c\geq 20D$, relative to standard
nested sampling, $k=0$. Note, runs with the same $s$ have nearly identical
efficiency improvement and so are indistinguishably plotted on top of each
other. The dashed line is equality with $k=0$. Bottom panel: The total number
of likelihood evaluations vs $f$. Each point is ensemble averaged over 100
runs.

##  5 Discussion

Taken together these results suggest that there is a natural sweet spot for
accelerating NS, where minimal sufficiently large $s$ and $c$ should be
selected and the highest possible $k>0$ number of phantom points retained.
Based on the results, we see no reason why higher values of $k$, than those
explored here, could be used for even greater speed ups. However, as $k\to
sD-1$ we expect autocorrelation to creep back into the samples, and the
resulting evidence calculations to become biased. To explore this matter
further we perform a quick ablation study to understand the impact of choosing
larger values of $k$. We run the method for the $D=8$ model with $s=6$ and
$c=40D$. These high values of $s$ and $c$ are well within the consistency
region, and ensures that accepted samples are independent. We then explore
values of $k\in\\{0,\ldots,sD-1\\}$, again taking ensemble averages over 100
random seeds.

Figure 4 shows the resulting log-evidence bias as a function $k$. Indeed we
observe that for low enough values of $k$ there is negligable impact on
evidence calculations, however starting from around $k=D$ (the dashed red
line) we see bias start to grow. This confirms our expectation that there is a
limit to the speed up attainable from this method. A tentative guess at the
maximal number of retainable phantom points would be $k_{\rm max}=D$, which
would lead to a maximal relative efficiency improvement of $D$, i.e. an
efficiency improvement that grows with model dimension. It is unclear here if
the bias increases due to breakage of the identicality or independence
requirements of $\mathcal{S}$. This finding should be verified with numerous
other models, as well as larger values of $s$, before being trusted. In
particular, using larger values of $s$ may allow larger values of $k$ in a
problem-dependent manner.

A question arises as to whether this method can be used for arbitrarily
complex likelihood models. If we look at the basic assumptions of the proposed
method then there are no aspects that are dependent on the complexity of the
likelihood. In fact all the hard work of dealing with difficult likelihoods
has already been put into creating good Markov-chain samplers. The only
aspects which require model-dependent consideration are choosing minimally
sufficiently larger $s$ and $c$. This method takes advantage of a general
property of all Markov-chain samplers, which is that after enough proposal
samples all further samples are identically uniformly distributed within the
likelihood constraint. It then takes advantage of a scaling relationship that
asymptotically ameliorates the impact of inter-sample dependence during the
shrinkage process. So we do not have any reason to suppose this might not be
effective for models of any complexity.

Figure 4: The bias for $s=6$ and $c=40D$ as we vary $k$ from low values to its
maximal value of $sD-1$, for the $D=8$ model. At $k=0$ we have standard nested
sampling. The red vertical line is $k=D$. We observed that for $k\lesssim D$
the evidence statistics appear consistent, and for $k\gtrsim D$ they become
biased as $k$ increases. Each point and error bar is ensemble averaged over
100 runs.

##  6 Conclusions

This paper introduces an enhancement to the nested sampling algorithm,
particularly addressing the challenge of computational inefficiency in high-
dimensional Bayesian evidence calculations. The crux of our method lies in the
integration of ‘phantom points’ – previously disgarded due to their high
autocorrelation – into the evidence computation process. By ensuring these
points remain asymptotically i.i.d. uniformly distributed we propose a method
for them to be utilised effectively, contributing to the accuracy of the final
evidence estimate while simultaneously boosting sampling efficiency.

Our theoretical argument, empirical evaluation, and ablation study have
demonstrated that:

  1. 1.

sampling efficiency can be increased without degrading evidence accuracy;

  2. 2.

any ergodic Markov-chain likelihood-constrained sampler can be easily adapted
to this method; and

  3. 3.

the method scales well with dimension.

The reduction in likelihood evaluations is a critical advancement,
particularly in fields like astrophysics, cosmology, and climate modelling,
where computational resources are often a limiting factor. Based on our
discussion, a tentative maximal achievable speed up could be $D$, implying a
potential efficiency that increases with model dimension. However, this
maximal speed is currently only weakly supported, and should be verified on a
wider variety of models. In the models we explored, we observed an efficiency
improvement factor of 5. Since the efficiency gains scale well with model
dimension, our method helps opens the door to Bayesian analysis of larger data
sets and models.

In addition, it should be noted that nested sampling has been applied outside
the realm of Bayesian analysis, particularly in material sciences [e.g. 25],
and it would be interesting to explore if the proposed method can be applied
there.

Supplementary information

All code and data required to reproduce the results of this study are provided
in a git repository https://www.github.com/joshuaalbert/jaxns.

Acknowledgments

JA would like to thank Do Kester and Romke Bontekoe for pointing out
clarifications, and John Skilling for discussions around nested sampling, as
well as Aleksandrina Skvortsova for helpful discussions around general
compression.

##  Appendix A Proving asymptotic i.i.d. uniformity of $\mathcal{S}$

root
---
$m=6$
---
$\mathbb{E}[m]=c(k+1)$
---
Figure 5: A sample tree representation of $c=2$ parallel Markov-chains with
phantom samples in green, and accepted samples in black. A vertical line at
any likelihood value will cross on average $c(k+1)$ edges.

Given $k$ phantom samples and $c$ parallel Markov-chains, we will prove that
the likelihoods in a sorted ordering of $\mathcal{S}$ will asymptotically
approach total randomness as $c$ increases.

We consider the situation where there are $c>1$ parallel Markov-chains, for
which the sample tree is helpful for visualising. Figure 5 shows two parallel
Markov-chains. We retain $k$ phantom samples for every accepted sample. We
assume that each accepted sample and its $k$ phantom samples are identically,
but not independently, uniformly distributed within the same likelihood
constraint. Note, it can easily be seen that we have $m=c(k+1)$ crossed edges
on average at every likelihood value along the sample tree.

Consider a likelihood set, $\mathcal{S}=\\{\lambda_{i}\\}_{i=1}^{m}$, and
define the sorted ordering of likelihoods,
$\sigma(\mathcal{S})=(\sigma(1),\ldots,\sigma(m)\\}$. Initially, suppose $k=0$
so that all likelihoods in $\mathcal{S}$ are identically and independently
uniformly distributed. Thus, for any two selected indices $j,j^{\prime}$ we
have $\mathrm{Prob}(\sigma(j)<\sigma(j^{\prime}))=1/2$, i.e. we have total
randomness. Total randomness is sufficient to show that the probabilities for
any sample, $j$, to have any sorted index, $i$, are all equal, i.e.
$\mathrm{Prob}(\sigma(j)=i)=1/m$.

Now, suppose $k>0$ so that accepted samples and their phantom samples are no
longer independently distributed, but still identically distributed. Recognise
that accepted samples and their phantom samples, form cliques where – due to
their non-independence – for all $j,j^{\prime}$ in a clique
$\mathrm{Prob}(\sigma(j)<\sigma(j^{\prime}))\neq 1/2$. However, we still have
independence between cliques, so for $j,j^{\prime}$ not in the same clique
$\mathrm{Prob}(\sigma(j)<\sigma(j^{\prime}))=1/2$, i.e. we have total
randomness for inter-clique orderings. We can think of this as relative
pairwise ordering being totally random between cliques, and not so for
pairwise orderings within a clique.

Let us now count favourable outcomes of pairwise orderings which are totally
random within $\mathcal{S}$. Each likelihood is within a clique of size $k+1$
and forms an inter-clique pair with $m-(k+1)$ other likelihoods. It follows
that number of favourable outcomes is $m(m-(k+1))$, and the total number of
pairwise orderings is $m(m-1)$. Thus, using $m\approx c(k+1)$, the fraction of
totally random pairwise orderings is,

$\displaystyle f\triangleq$ $\displaystyle\frac{m-(k+1)}{m-1}$ (6)
$\displaystyle=$ $\displaystyle\frac{c-1}{c-\frac{1}{k+1}}$ (7)

Clearly, $f\to 1$ as $c\to\infty$. Since $f=1$ is a sufficient condition for
uniform likelihood orderings, therefore the likelihoods in a sorted ordering
of $\mathcal{S}$ will asymptotically approach total randomness as $c$
increases. $\blacksquare$

##  Appendix B 16-Dimensional Results

Figure 6: The log-evidence bias as a function of slice factor, $s$, for the
$D=16$ model, coloured by the number of retained phantom samples, $k$ for
different values of $c$. The dashed line represents the target log-evidence
distribution for standard nested sampling. Each point and error bar is
ensemble averaged over 100 runs.

Results for the $D=16$ model are provided here, with log-evidence bias as a
function of $s$ for different values of $c$ and $k$ shown in Figure 6. We
visually see log-evidence distributions appear consistency for all $k$ for all
runs in $s\geq 3$ and $c\geq 20D$, which is the same consistency region
observed for the $D=8$ model.

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