# A Three-Mode Erasure Code for Continuous Variable Quantum Communications

Eduardo Villaseñor and Robert Malaney
School of Electrical Engineering & Telecommunications,
The University of New South Wales, Sydney, NSW 2052, Australia.

###### Abstract

Quantum states of light being transmitted via realistic free-space channels
often suffer erasure errors due to several factors such as coupling
inefficiencies between transmitter and receiver. In this work, an error
correction code capable of protecting a single-mode quantum state against
erasures is presented. Our three-mode code protects a single-mode Continuous
Variable (CV) state via a bipartite CV entangled state. In realistic
deployments, it can almost completely reverse a single erasure on the encoded
state, and for two erasures can it improve the fidelities of received states
relative to direct transmission. The bipartite entangled state used in the
encoding can be Gaussian or non-Gaussian, with the latter further enhancing
the performance of the code. Our new code is the simplest code known that
protects a single mode against erasures and should prove useful in the
construction of practical CV quantum networks that rely on free-space optics.

###### Index Terms:

Continuous variable quantum information, quantum communications, quantum error
correction.

††publicationid: pubid:  This work has been acceped for publication in
GLOBECOM 2022. Copyright: 978-1-6654-3540-6/22 © 2022 IEEE

##  I Introduction

Continuous variable (CV) quantum information, encoded in the quadrature
variables of electromagnetic signals, may offer several advantages over
discrete variable (DV) quantum information in the context of reliable-state
transfer over free space [1, 2]. The free-space transmission of CV quantum
states using weak laser pulses via satellites in low-Earth-orbit (LEO), could
potentially represent a viable path toward achieving global quantum
communications [3, 4, 5].

However, Quantum information is fragile by nature - the unavoidable
interaction between quantum systems and their environment introduces errors to
the quantum states. Correction of these errors requires the use of additional
quantum resources to construct a larger quantum system in which the
deterministic identification and correction of the errors is possible [6].
Attempts at error correction in many contexts for CV states have been
attempted, e.g. [7, 8, 9, 10, 11, 12, 13, 14, 15]. Here we focus on a type of
error that can affect CV states in practical scenarios, especially with the
advent of new communications platforms such as quantum communications via
satellite; erasure channels.

In the context of free-space quantum communications, the erasure channel
corresponds to the beam being completely lost during transmission. This can
happen as a consequence of the beam wandering effects caused by the turbulence
in the atmosphere [16, 17, 18]. The uplink transmission of quantum states from
the ground to a LEO satellite is an example where erasures are prominent due
to the prevailing beam wandering [15].

To achieve the correction of erasures on quantum states a new quantum erasure
code is presented. The code considers a single-mode quantum state as an input
and encodes it with a bipartite entangled state. Our code is different from
erasure codes previously constructed. The most similar code to that presented
here is the code of [19, 20] in which two input states are protected through
the use of four transmission channels. In contrast, our code protects fewer
states (one) but with the benefit of reduced complexity (use of three
transmission channels). As such, our code offers a pathway to more pragmatic
deployments. In addition, due to its relative simplicity, it becomes possible
to optimize faster the free parameters of our code relative to other codes -
an issue of particular importance when multiple erasures occur on the encoded
state. The novel contributions of this work are summarized as follows:

  * •

We present an erasure code for CV states and analyze in detail its
performance.

  * •

A detailed optimization procedure is presented for our code when erasures are
present. Additionally, performance with a simpler deployment where this
optimization is neglected is compared.

  * •

The use of Gaussian and non-Gaussian states for the input entangled state is
considered and their performance is compared. It is shown that, in combination
with the optimization process, the use of non-Gaussian states further
increases the performance of the code.

In section II we introduce our code and analyze its performance via the Wigner
Characteristic Function (CF) formalism. We present detailed results from the
code which detail its performance under different combinations of erasures and
different assumptions regarding the input entangled state used for encoding in
section III. Finally, we draw our conclusions in section IV.

##  II Erasure error correction code

The Wigner CF formalism will be used to study the error correction code
presented here. This is motivated by the results presented in [21] that show
that the output state of CV quantum teleportation can be easily computed from
the CFs of the quantum states involved in the protocol. In this work a similar
result is obtained, the CF of the output state after error correction
corresponds to a product of the CFs of the input state and the entangled
state.

The CF of any $n$-mode quantum state $\hat{\rho}$ is defined as

$\displaystyle\makebox{\large$\chi$}(\lambda_{1},\lambda_{2},...,\lambda_{n})=\text{Tr}\left\\{\hat{\rho}\hat{D}(\lambda_{1})\hat{D}(\lambda_{2})...\hat{D}(\lambda_{n})\right\\},$
(1)

where $\lambda_{i}\in\mathbb{C}$. Here, $\hat{D}$ is the displacement
operator,

$\displaystyle\hat{D}(\lambda_{i})=e^{\lambda_{j}\hat{a}^{{\dagger}}_{j}-\lambda_{j}^{*}\hat{a}_{j}},$
(2)

where $\hat{a}_{j}$ and $\hat{a}^{{\dagger}}_{j}$ are the annihilation and
creation operators of mode $j$, and ∗ represents the complex conjugate.
Conveniently, linear optics operations can be expressed in the CF formalism by
simply transforming the arguments of the CF, whilst leaving the functions
themselves unchanged. Relevant CFs for this work include the vacuum state,
$\ket{0}$, expressed as

$\displaystyle\makebox{\large$\chi$}_{\ket{0}}(\lambda)=\exp\left[-\frac{|\lambda|^{2}}{2}\right],$
(3)

and the coherent state, $\ket{\alpha}=\hat{D}(\alpha)\ket{0}$, expressed as

$\displaystyle\makebox{\large$\chi$}_{\ket{\alpha}}(\lambda)=\exp\left[-\frac{|\lambda|^{2}}{2}+(\lambda\alpha^{*}-\lambda^{*}\alpha)\right].$
(4)

The other important CFs we will utilize involve those of the different
entangled states used in the encoding, which we present later.

In Fig. 1 our error correction code is presented. The resources required for
the implementation and optimization of this code include entangled-bipartite
states, linear-optics operations, and classical processing. The deployment of
our error correction scheme can be divided into four steps: encoding,
decoding, syndrome measurements, and correction.

As shown in Fig. 1, Alice starts with a single-mode quantum state, which we
will refer to as the “quantum signal,” that she wishes to transmit through the
channel to Bob, and prepares an entangled bipartite state. The initial CF
corresponds to a product of the CFs of the quantum signal,
$\makebox{\large$\chi$}_{\mathrm{s}}(\lambda_{1})$, and the entangled state,
$\makebox{\large$\chi$}_{\mathrm{AB}}(\lambda_{2},\lambda_{3})$. The encoding
of the quantum signal is done via a balanced beam splitter (BS1), described by
a transformation of the CF arguments as follows [10],

$\displaystyle\makebox{\large$\chi$}_{\mathrm{s}}\left(\frac{\lambda_{1}+\lambda_{2}}{\sqrt{2}}\right)\makebox{\large$\chi$}_{\mathrm{AB}}\left(\frac{\lambda_{1}-\lambda_{2}}{\sqrt{2}},\lambda_{3}\right).$
(5)

Thereafter, the encoded state is transmitted from Alice to Bob through the
channel. In general, the erasure channel acting on a single-mode state,
$\rho$, returns the state,

$\displaystyle\rho^{\prime}=(1-P_{e})\rho+P_{e}\ket{0}\bra{0},$ (6)

with $P_{e}$ being the probability of an erasure. If the three modes of the
encoded state were sent concomitantly through the channel the result would be
either an unchanged state or a three-mode vacuum state from which no
information can be recovered. Therefore, a mechanism that transmits each mode
independently must be used. An example of such a mechanism would be one that
time multiplexes each mode using delay lines. When the three modes are sent
independently through the channel the result is a mixed state corresponding to
all of the eight combinations of modes erased,

$\displaystyle\rho_{\mathrm{ch}}=\sum_{j=1}^{8}P_{j}\rho_{j},$ (7)

with $P_{j}$ corresponding to the probability of each combination of modes
erased. These range from $(1-P_{e})^{3}$ to $P_{e}^{3}$, corresponding to zero
erasures, and erasures in every mode, respectively. At this point, the CF of
the three mode state, now defined as
$\makebox{\large$\chi$}_{\mathrm{ch}}^{\mathbf{x}}(\lambda_{1},\lambda_{2},\lambda_{3})$,
will depend on the modes $\mathbf{x}$ that suffer an erasure. This means that
for the erased modes their arguments in the CF in Eq. 5 will be set to zero
[22], while vacuum CFs are added as a product with arguments corresponding to
the erased modes. For example, if mode $2^{\prime}$ has an erasure the
corresponding CF is

$\displaystyle\makebox{\large$\chi$}_{\mathrm{ch}}^{\mathrm{\\{2\\}}}(\lambda_{1},\lambda_{2},\lambda_{3})=\makebox{\large$\chi$}_{\mathrm{s}}\left(\frac{\lambda_{1}}{\sqrt{2}}\right)\makebox{\large$\chi$}_{\mathrm{AB}}\left(\frac{\lambda_{1}}{\sqrt{2}},\lambda_{3}\right)\makebox{\large$\chi$}_{\ket{0}}(\lambda_{2}).$
(8)

Bob monitors and identifies which of the modes suffered an erasure during
transmission via the channel, and will use that information in combination
with the syndrome measurement to apply the correction. Note, our erasure code
implementation resembles that utilized in the context of secret sharing [23].
Apart from the application context, a key difference in our implementation is
the use of the erasure monitoring function and its mapping to an erasure
correction protocol. We can consider this mapping to be the following: for any
monitoring measurement that indicates a non-unity channel transmissivity, we
set that channel to be in ‘erasure’. This logic is then used to set the gains
needed to adjust the output quantum state.

Figure 1: The code to correct the erasures introduced by the quantum erasure
channel. The code protects a single-mode quantum state against erasures by
combining it with an entangled bipartite state. The encoding and decoding are
made using beam splitters. Delay lines are introduced such that each mode is
sent independently through the channel. The syndrome measurement corresponds
to a dual homodyne measurement. Using the syndrome result a correction is
applied to the remaining mode of the state to recover the original state. The
error code is optimized by monitoring another (classical) beam in the same
channels (red BSs) and preparing the entangled state accordingly. Dashed lines
represent the transmission of classical information.

As the modes are received by Bob, he applies corresponding delays such that by
the time he has received all the modes, they are all temporally coincident. To
decode the quantum state, he applies BS2 that transforms the arguments as,

$\displaystyle\makebox{\large$\chi$}_{\mathrm{ch}}^{\mathbf{x}}\left(\frac{\lambda_{1}+\lambda_{2}}{\sqrt{2}},\frac{\lambda_{1}-\lambda_{2}}{\sqrt{2}},\lambda_{3}\right).$
(9)

In the case when the channel acts as an identity (no erasures) the application
of BS2 effectively cancels the effects of BS1.

During the final step, syndrome measurements are performed. To this end, BS3
is applied, giving a CF for the three mode state as

$\displaystyle\makebox{\large$\chi$}_{\mathrm{BS3}}^{\mathbf{x}}$
$\displaystyle(\lambda_{1},\lambda_{2},\lambda_{3})=$ (10)
$\displaystyle\makebox{\large$\chi$}_{\mathrm{ch}}^{\mathbf{x}}\left(\frac{\lambda_{1}+\lambda_{2}}{\sqrt{2}},\frac{\lambda_{1}}{\sqrt{2}}-\frac{\lambda_{2}+\lambda_{3}}{2},\frac{\lambda_{2}-\lambda_{3}}{\sqrt{2}}\right).$

Now dual homodyne measurements are performed on modes $2^{\prime\prime}$ and
$3^{\prime\prime}$ (see Fig. 1). To compute the result after the measurements
it is convenient to represent the complex arguments in the phase-space
representation, by using two distinct real numbers,
$x_{j}{=}\frac{1}{\sqrt{2}}(\lambda_{j}+\lambda_{j}^{*})$ and
$p_{j}{=}\frac{i}{\sqrt{2}}(\lambda_{j}^{*}-\lambda_{j})$. Then, for the pair
of measurement results $\tilde{x}$ and $\tilde{p}$ the output CF (on mode
$1^{\prime\prime}$) is obtained by the integration [21],

$\displaystyle\makebox{\large$\chi$}_{\mathrm{m}}^{\mathbf{x}}(x,p)$
$\displaystyle=\frac{\mathcal{P}(\tilde{x},\tilde{p})^{-1}}{(2\pi)^{2}}\int
dx_{2}dp_{3}\makebox{\large$\chi$}_{\mathrm{BS3}}^{\mathbf{x}}\left(x,p,x_{2},0,0,p_{3}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times
e^{-i\tilde{x}p_{3}+i\tilde{p}x_{2}},$ (11)

with $\mathcal{P}(\tilde{x},\tilde{p})$ the probability distribution of any
pair of measurement results, given by

$\displaystyle\mathcal{P}(\tilde{x},\tilde{p})=\frac{1}{(2\pi)^{2}}\int
dx_{2}dp_{3}\makebox{\large$\chi$}_{\mathrm{BS3}}^{\mathbf{x}}\left(0,0,x_{2},0,0,p_{3}\right)$
$\displaystyle\times e^{-i\tilde{x}p_{3}+i\tilde{p}x_{2}}.$ (12)

The extra exponential term in Eq. 11 indicates that the collapsed state after
the measurement requires a corrective displacement to be recovered. The
corrective displacement is,

$\displaystyle\hat{U}_{\mathrm{corr}}(\tilde{x},\tilde{p})=\exp\left[\sqrt{2}\tilde{x}pg_{p}-i\sqrt{2}\tilde{p}xg_{x}\right],$
(13)

where $(g_{x},g_{p})$ are the gains of the correction. Here the correction
includes a factor of $\sqrt{2}$ to compensate for the global factor that
appears in the arguments of
$\makebox{\large$\chi$}_{\mathrm{ch}}^{\mathbf{x}}$ in Eq. 10.

Finally, since the output CF will depend on a specific set of measurement
results, the mean output over all possible measurement outcomes weighted with
the corresponding probability distribution must be considered, that is

$\displaystyle\makebox{\large$\chi$}_{\mathrm{out}}^{\mathbf{x}}(x,p)$
$\displaystyle=\int
d\tilde{x}d\tilde{p}\mathcal{P}(\tilde{x},\tilde{p})\makebox{\large$\chi$}_{\mathrm{m}}^{\mathbf{x}}(x,p)e^{i\sqrt{2}\tilde{x}pg_{p}-i\sqrt{2}\tilde{p}xg_{x}}$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int
d\tilde{x}d\tilde{p}dx_{2}dp_{3}\makebox{\large$\chi$}_{\mathrm{BS3}}^{\mathbf{x}}(x,p,x_{2},0,0,p_{3})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times
e^{i\tilde{x}(\sqrt{2}pg_{p}-p_{3})-i\tilde{p}(\sqrt{2}xg_{x}-x_{2})}.$ (14)

At this point the definition of the Dirac delta function, $\frac{1}{2\pi}\iint
e^{i\beta x-i\beta\alpha}f(\alpha)d\beta
d\alpha=\int\delta(x-\alpha)f(\alpha)d\alpha$, can be used twice to finally
obtain

$\displaystyle\makebox{\large$\chi$}_{\mathrm{out}}^{\mathbf{x}}(x,p)$
$\displaystyle=\int
dx_{2}dp_{3}\delta(p_{3}-\sqrt{2}g_{p}p)\delta(x_{2}-\sqrt{2}g_{x}x)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\times\makebox{\large$\chi$}_{\mathrm{BS3}}^{\mathbf{x}}(x,p,x_{2},0,0,p_{3}).$
(15)

Therefore, the resulting CF will correspond to a product of the initial CFs
with a transformation of the parameters that will depend on the specific
erasures $\mathbf{x}$ plus a vacuum contribution. The resulting expressions
for $\makebox{\large$\chi$}_{\mathrm{out}}^{\mathbf{x}}$ are summarized in
Table I. The CF of the output state depends on the erasures, which can be
divided into two cases:

####  II-1 Single erasure

If there is only a single erasure on the encoded state, the signal is
recovered by maximizing the amount of entanglement in
$\ket{\rho_{\mathrm{AB}}}$, with perfect recovery corresponding to the limit
of infinite entanglement. To this end, see that under the correct choice of
$(g_{x},g_{p})$ (e.g. if erasure on mode $2^{\prime}$, then
$g_{x}{=}g_{p}{=}1$) the CF is reduced to a product of the CFs of the quantum
signal and the entangled state. The signal is recovered since
$V{\rightarrow}\infty$ implies that
$\makebox{\large$\chi$}_{\mathrm{AB}}(\lambda,\lambda^{*}){\rightarrow}1$.
When the erasure is on mode $3^{\prime}$, the signal can always be recovered
fully independently of the entanglement used by setting $g_{x}{=}g_{p}{=}0$.

####  II-2 Two erasures

When two modes suffer an erasure, any amount of entanglement in
$\ket{\rho_{\mathrm{AB}}}$ will ultimately become added noise in the output
state. This can be seen from the fact that the CF of the entangled state
$\makebox{\large$\chi$}_{\mathrm{AB}}$ has one mode traced out, meaning that
the entangled state has now been reduced to a thermal state. Remarkably, some
information from the signal can still be recovered in this case if the
entangled state is instead replaced by two vacuum states [19].

Ultimately, in a realistic scenario, there might be a single or two erasures.
In such a scenario there is an optimal amount of entanglement in the state
$\ket{\rho_{\mathrm{AB}}}$, that depends on $P_{e}$, that will balance the
trade-off between recovering the signal when there is a single erasure, and
reducing the noise when there are two erasures.

Table I - Output state CF

Erasure on mode | $\makebox{\large$\chi$}_{\mathrm{out}}^{\mathbf{x}}$
---|---
none | $\makebox{\large$\chi$}_{\mathrm{s}}(x,p)$
$1^{\prime}$ | $\makebox{\large$\chi$}_{\mathrm{s}}\left(\frac{1-g_{x}}{2}x,\frac{1-g_{p}}{2}p\right)\makebox{\large$\chi$}_{\mathrm{AB}}\left(-\frac{1-g_{x}}{2}x,-\frac{1-g_{p}}{2}p,g_{x}x,-g_{p}p\right)\makebox{\large$\chi$}_{\ket{0}}\left(\frac{1+g_{x}}{\sqrt{2}}x,\frac{1+g_{p}}{\sqrt{2}}p\right)$
$2^{\prime}$ | $\makebox{\large$\chi$}_{\mathrm{s}}\left(\frac{1+g_{x}}{2}x,\frac{1+g_{p}}{2}p\right)\makebox{\large$\chi$}_{\mathrm{AB}}\left(\frac{1+g_{x}}{2}x,\frac{1+g_{p}}{2}p,g_{x}x,-g_{p}p\right)\makebox{\large$\chi$}_{\ket{0}}\left(\frac{1-g_{x}}{\sqrt{2}}x,\frac{1-g_{p}}{\sqrt{2}}p\right)$
$3^{\prime}$ | $\makebox{\large$\chi$}_{\mathrm{s}}(x,p)\makebox{\large$\chi$}_{\mathrm{AB}}\left(g_{x}x,g_{p}p,0,0\right)\makebox{\large$\chi$}_{\ket{0}}\left(g_{x}x,-g_{p}p\right)$
$1^{\prime}$ & $2^{\prime}$ | $\makebox{\large$\chi$}_{\mathrm{AB}}\left(0,0,g_{x}x,-g_{p}p\right)\makebox{\large$\chi$}_{\ket{0}}\left(\frac{1+g_{x}}{\sqrt{2}}x,\frac{1+g_{p}}{\sqrt{2}}p\right)\makebox{\large$\chi$}_{\ket{0}}\left(\frac{1-g_{x}}{\sqrt{2}}x,\frac{1-g_{p}}{\sqrt{2}}p\right)$
$1^{\prime}$ & $3^{\prime}$ | $\makebox{\large$\chi$}_{\mathrm{s}}\left(\frac{1-g_{x}}{2}x,\frac{1-g_{p}}{2}p\right)\makebox{\large$\chi$}_{\mathrm{AB}}\left(-\frac{1-g_{x}}{2}x,-\frac{1-g_{p}}{2}p,0,0\right)\makebox{\large$\chi$}_{\ket{0}}\left(\frac{1+g_{x}}{\sqrt{2}}x,\frac{1+g_{p}}{\sqrt{2}}p\right)\makebox{\large$\chi$}_{\ket{0}}\left(\frac{1+g_{x}}{\sqrt{2}}x,\frac{1-g_{p}}{\sqrt{2}}p\right)$
$2^{\prime}$ & $3^{\prime}$ | $\makebox{\large$\chi$}_{\mathrm{s}}\left(\frac{1+g_{x}}{2}x,\frac{1+g_{p}}{2}p\right)\makebox{\large$\chi$}_{\mathrm{AB}}\left(\frac{1+g_{x}}{2}x,\frac{1+g_{p}}{2}p,0,0\right)\makebox{\large$\chi$}_{\ket{0}}\left(\frac{1-g_{x}}{\sqrt{2}}x,\frac{1-g_{p}}{\sqrt{2}}p\right)\makebox{\large$\chi$}_{\ket{0}}\left(g_{x}x,-g_{p}p\right)$
$1^{\prime}$ & $2^{\prime}$ & $3^{\prime}$ | $\makebox{\large$\chi$}_{\ket{0}}(x,p)$

###  II-A Optimizing the correction

As discussed, the correction $\hat{U}_{\mathrm{corr}}$ and the characteristics
of the entangled state used during encoding can be optimized to enhance the
effectiveness of the code. The process of optimizing the error correction code
involves two distinct steps. One step is optimizing the entangled state, while
the other is optimizing the parameters in the correction, $(g_{x},g_{p})$, for
each of the mode-erasure combinations. Note that the optimal values
$(g_{x},g_{p})$, do not depend on the erasure probability $P_{e}$, however,
they do depend on the entangled state. On the other hand, the optimal
properties of the entangled state do depend on $P_{e}$. Thus, full
optimization of the code requires a nested optimization problem. This nested
optimization problem can be avoided if Alice and Bob prepare beforehand a
dictionary with the optimal values $(g_{x},g_{p})$ as a function of the
properties of the entangled state. In this case, the optimization process is
as follows. First, Alice monitors the erasure channel to determine the value
of $P_{e}$. Using this value she then prepares an optimal entangled state.
Alice shares the properties of the entangled state with Bob via a classical
channel. After Bob has received the three modes of the code he identifies the
erasures on the encoded state and performs syndrome measurements. With these
three pieces of information (the entangled state properties, the erasures, and
the syndrome), he then uses the dictionary to obtain the optimal values of
$(g_{x},g_{p})$ and utilizes them in the correction.

###  II-B Entangled resources

In CV quantum protocols, the two-mode squeezed vacuum state (TMSV) represents
the most accessible bipartite entangled state. States such as the TMSV state
are commonly referred to as Gaussian since they can be described fully by the
first two statistical moments of the quadratures [1]. The CF of a TMSV state
can be conveniently written using the following Bogoliuvov transformation
[10]:

$\displaystyle\hat{S}_{12}(\varrho)\hat{D}(\lambda_{1})\hat{D}(\lambda_{2})\hat{S}^{\dagger}_{12}(\varrho)=\hat{D}(\lambda_{1}^{\prime})\hat{D}(\lambda_{2}^{\prime}),$
(16)
$\displaystyle\lambda_{j}^{\prime}=\cosh(r)\lambda_{j}+e^{i\phi}\sinh(r)\lambda_{k}^{*}~{}~{}~{}~{}j,k=1,2;~{}~{}j\neq
k.$

where $\hat{S}_{12}$ is the two-mode squeezing operator with
$\varrho{=}re^{i\phi}$, throughout this work, the value $\phi{=}\pi$ is set.
The squeezing magnitude $r$ is directly proportional to the entanglement of
the TMSV state, which can be quantified by the variance $V$, which relates to
the squeezing as $V{=}\cosh(2r)$. Using the transformation in Eq. 16 the CF of
a TMSV can be written as,

$\displaystyle\makebox{\large$\chi$}_{\mathrm{TMSV}}(\lambda_{\mathrm{A}},\lambda_{\mathrm{B}})=\exp\left[-\frac{1}{2}\Big{(}|\lambda_{\mathrm{A}}^{\prime}|^{2}+|\lambda_{\mathrm{B}}^{\prime}|^{2}\Big{)}\right].$
(17)

In contrast to Gaussian states, non-Gaussian states cannot be fully described
using statistical moments, instead, they require a full description in the
form of a density operator or a Wigner function. In this work, a non-Gaussian
entangled state that is obtained by the application of the squeezing operator
to a Bell state is considered. The normalized CF of a squeezed Bell (SB) state
is [10],

$\displaystyle\makebox{\large$\chi$}_{\mathrm{SB}}(\lambda_{\mathrm{A}},\lambda_{\mathrm{B}})=(\cos^{2}(\delta)+\sin^{2}(\delta))^{-1/2}$
$\displaystyle\times\exp\left[-\frac{1}{2}\left(|\lambda_{\mathrm{A}}^{\prime}|^{2}+|\lambda_{\mathrm{B}}^{\prime}|^{2}\right)\right]\Big{[}\cos^{2}(\delta)+2\cos(\delta)\sin(\delta)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\times\Re\\{\lambda_{\mathrm{A}}^{\prime}\lambda_{\mathrm{B}}^{\prime}\\}+\sin^{2}(\delta)(1-|\lambda_{\mathrm{A}}^{\prime}|^{2})(1-|\lambda_{\mathrm{B}}^{\prime}|^{2})\Big{]},$
(18)

where $\Re\\{z\\}$ is the real part of $z$, and Eq. 16 is used. The
optimization of the properties of the entangled state, mentioned in Section
II.A, is done in terms of the amount of entanglement of the states, that is,
on the variance $V$ for both TMSV and SB states. Additionally, in the SB
state, the parameter $\delta$ is also optimized.

###  II-C Fidelity of error correction

To quantify the effectiveness of the error correction the fidelity of
transmitted coherent states is used. The fidelity corresponds to a measurement
of the closeness between the initial coherent states and the output states. In
the CF formalism this is computed as,

$\displaystyle\mathcal{F}(\alpha)=\frac{1}{\pi}\int
d^{2}\lambda\makebox{\large$\chi$}_{\mathrm{\ket{\alpha}}}(\lambda)\makebox{\large$\chi$}_{\mathrm{out}}(-\lambda).$
(19)

Note that in this case, the fidelity will be dependent on the value of
$\alpha$. Therefore, to accurately assess the effectiveness of the error
correction code, a mean fidelity over an ensemble of coherent states must be
considered. The ensemble is specified by the following distribution,

$\displaystyle
P(\alpha)=\frac{1}{\sigma\pi}\exp\left[-\frac{|\alpha|^{2}}{\sigma}\right],$
(20)

with $\sigma$ the variance of the distribution. Therefore, the mean fidelity
over the ensemble will be used, defined as

$\displaystyle\bar{\mathcal{F}}=\int
d{\alpha}^{2}P(\alpha)\mathcal{F}(\alpha).$ (21)

##  III Results

The fidelities are computed using Table I, in combination with Eq. 19. The
mean fidelities over the distribution of coherent states are computed using
Eq. 21, where the value $\sigma{=}10$ is fixed throughout this work. In the
calculations presented here, the value of $V$ has the upper limit of
$V_{\mathrm{max}}{=}9$.

To understand the output of the code, the fidelities corresponding to each
case of the mode-erasure combinations are shown in Fig. 2. The fidelities are
presented as a function of $V$ of the entangled states, while for the SB
state, the parameter $\delta$ is optimized for each value of $V$. In each
case, the parameters $(g_{x},g_{p})$ are optimized. While in principle for any
given coherent state the optimal values of $(g_{x},g_{p})$ are not equal, we
find that when the optimization is done over the distribution of states given
by Eq. 20 the optimal values of both parameters coincide, that is
$g_{x}{=}g_{p}{=}g_{\mathrm{opt}}$. See that the fidelities for the cases when
modes $1^{\prime}$ or $2^{\prime}$ suffer an erasure tend to unity as $V$
increases. The particular case when mode $3^{\prime}$ is lost is not shown in
the figure, since the fidelity is always 1 when $g_{\mathrm{opt}}{=}0$, as
discussed above.

Remarkably, in the cases where two modes are erased a higher fidelity than the
classical limit of 0.5 can still be obtained if the entangled state is
replaced by a vacuum, as discussed above. The combinations of modes erased
that reduce the output to the vacuum state are omitted (simultaneous erasures
in modes $1^{\prime}$ and $2^{\prime}$, and simultaneous erasures in all the
modes).

Figure 2: Fidelities corresponding to each possible combination of modes
erased on the encoded state during transmission (see Table I). The fidelities
are computed using a TMSV and a SB quantum state in the encoding. The optimal
correction parameter $g_{\mathrm{opt}}$ is used in the calculations of the
fidelities.

In a realistic scenario, where any number of erasures may affect the encoded
state, the optimal value $V_{\mathrm{opt}}$ must be used to maximize the total
fidelity obtained. The total fidelity is obtained from the individual
fidelities corresponding to each combination of modes erased, weighted by
their respective probability of occurrence, as

$\displaystyle\mathcal{F}_{\mathrm{total}}$
$\displaystyle=(1-P_{e})^{3}+P_{e}(1-P_{e})^{2}(\mathcal{F}_{1}+\mathcal{F}_{2}+1)$
(22)
$\displaystyle~{}~{}+P_{e}^{2}(1-P_{e})(\mathcal{F}_{13}+\mathcal{F}_{23}+\mathcal{F}_{\ket{0}})+P_{e}^{3}\mathcal{F}_{\ket{0}}$

where $\mathcal{F}_{\mathbf{x}}$, corresponding to the mean fidelity (Eq. 21)
with $\mathbf{x}$ representing the erased modes, and $\mathcal{F}_{\ket{0}}$
corresponds to the mean fidelity between input states and the vacuum. In Fig.
3 the fidelities after the error correction are presented. The results are
compared with the fidelities obtained via direct transmission through the
channel. The fidelity is optimized following the procedure discussed in
Section II.A. We see that the error correction code increases the fidelity of
transmitted coherent states for erasure error rates up to 70%. Moreover, using
the non-Gaussian state improves the fidelities over those acquired using the
TMSV state. Remarkably, once optimized, the non-Gaussian state requires a
lower amount of entanglement (lower value of $V$) compared to the TMSV (Fig. 3
inset). This could prove advantageous in a scenario where the ability to
produce highly-squeezed states is limited.

Figure 3: Fidelites obtained for coherent states transmitted through an
erasure channel using the error correction code, with TMSV and SB states used
as the entangled state. (inset) The optimal value of $V$ used in both
entangled states.

###  III-A Optimization vs post-selection

It is also worth considering a simpler protocol that does not involve
optimizing the system. A valid protocol would be to post-select the output
states to only use those that present only a single erasure, or no erasure at
all. In this case, no optimization of the entangled state is required, since
the best strategy to recover the signal is to maximize $V$ within device
limitations.

To perform a fair comparison between the post-selection and the optimization
protocol, the mean fidelities are compared. This means that in the post-
selection protocol the mean fidelity obtained is weighted by the probability
that at most only one erasure occurs,
$p_{\mathrm{success}}{=}(1-P_{e})^{3}+3P_{e}(1-P_{e})^{2}$. The results are
presented in Fig. 4. The fidelities shown in this figure are computed for the
post-processing protocol using a TMSV state with different values of $V$. The
fidelities obtained from the post-selection protocol do not exceed the
fidelity obtained by the optimized protocol. Nonetheless, we see that for low
erasure rates the results of the post-selection protocol are considerably
closer to those obtained from the optimized protocol. This means that in a
practical implementation of the erasure code the post-selection protocol could
still be used effectively if the erasure rate is low.

Figure 4: Total fidelities obtained for the post-selection protocol. A TMSV
state is used as the entangled state. For comparison, the total fidelities
obtained using the optimized protocol with a TMSV state are also presented.

##  IV Conclusions

The error correction code studied in this paper represents a practical
solution to erasures that could be implemented in future implementations -
especially satellite-based implementations in which erasures (and partial
erasures) are not uncommon. We showed that there existed several free
parameters associated with our erasure code that could be optimized. After
such optimization, the fidelities of transmitted coherent states over the
erasure channel showed a considerable increase over direct transmission. A
simpler protocol was investigated in which the parameters of the code were not
optimized, instead, the results were post-selected to consider only when a
single (or no) erasure occurs. For low error rates, the post-selected protocol
provided similar performance to the optimized protocol while being
considerably less complex. Our erasure code was further enhanced by
considering non-Gaussian states in the encoding process. Typically,
enhancements in the fidelity of $20\%$ were found if non-Gaussian states were
used in the encoding process.

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# On a generalization of quasi-metric space

Sugata Adhya and A. Deb Ray  Department of Mathematics, The Bhawanipur
Education Society College. 5, Lala Lajpat Rai Sarani, Kolkata 700020, West
Bengal, India<EMAIL_ADDRESS>Department of Pure Mathematics,
University of Calcutta. 35, Ballygunge Circular Road, Kolkata 700019, West
Bengal, India<EMAIL_ADDRESS>
###### Abstract.

We find an extension of the quasi-metric (to be called $g$-quasi metric) such
that the induced generalized topology may fail to form a topology. We show
that $g$-quasi metrizability is a $g$-topologically invariant property of
generalized topological spaces. Extending metric product and uniform
continuity for $g$-quasi metric spaces, we note that a $g$-quasi metric may
fail to be uniformly continuous in the extended sense unlike usual metric.
Finally, we extend the study of completeness, Lebesgue property and weak
$G$-completeness for $g$-quasi metric spaces.

AMS Subject Classification: 54A05, 54C08, 54E15.
Keywords: $g$-quasi metric, generalized topology, Lebesgue property, weak
$G$-complete.

##  1\. Introduction

Császár [5] proposed a notion of generalized topology by taking into account
the idea of monotone mappings. It accommodates various open-like sets existed
in literature [4, 12, 14, 15]. Given a nonempty set $X,$ it is defined as a
subcollection of $\mathcal{P}(X)$ which contains $\emptyset$ and is closed
under arbitrary union. Considering the members of a generalized topology as
open, it then became natural to study the usual topological notions for
generalized topology. Accordingly, analogoues of closed set, closure,
interior, product and subspaces, continuous functions, countability and
separation axioms, compactness and connectedness have been studied for
generalized topological spaces. Additionally, multiple weaker versions of the
above notions have also been investigated in the light of weaker forms of open
sets in the context of generalized topology. The interested readers may
consult [18] and references therein.

In view of the facts that the associated open balls in any metric space forms
a base for a natural topology and generalized topology is a generalization of
usual topology, it is natural to search for a generalization of metric
structure that under standard approach induces a generalized topology and
fails to produce a topology, in general. This paper addressed this question.

Here, in Section 3, we obtain the related spaces (termed as $g$-quasi metric
spaces) by generalizing Wilson’s widely studied notion of the quasi-metric
structure [17]. Subsequently, we discuss certain separation properties of the
generalized topology induced by a $g$-quasi metric. We demonstrate that
$g$-quasi metrizability is an invariant property of the generalized topology.
Next, in Section 4, we propose the natural extensions of the notions of metric
product and uniform continuity in the context of $g$-quasi metric spaces. It
is noted that, unlike usual metric, a $g$-quasi metric may fail to be
uniformly continuous in the extended sense while considered as a mapping from
the product space to $\mathbb{R}$.

The study made in Section 3 and Section 4 pave the way for the related
investigations on Cauchyness and completeness. Accordingly, in Section 5, we
take up the task of extending them in $g$-quasi metric spaces. Apart from the
usual completeness, we introduce two stronger forms of completenesses viz.
Lebesgue property [2, 11, 16] and weak $G$-completeness [1, 9, 10] in
$g$-quasi metric spaces. It is known that both Lebesgue property and weak
$G$-completeness are intermediate metric properties between compactness and
completeness that can be characterized in terms of pseudo-Cauchy [16] and
$G$-Cauchy [9] sequences respectively. In what follows, we explore the mutual
dependence of those completenesses for $g$-quasi metrics and enquire their
behaviours in the product spaces through Cauchy, pseudo-Cauchy and $G$-Cauchy
sequences.

##  2\. Preliminaries

This section discusses the prerequisites that will be required subsequently.

######  Definition 2.1.

[5, 3, 13] A generalized topology $\mu$ on a nonempty set $X$ is a collection
of its subsets such that $\emptyset\in\mu$ and $\mu$ is closed under arbitrary
union. The pair $(X,\mu)$ is called a generalized topological space. Moreover,
if $X\in\mu$ then $\mu$ is called a supratopology or a strong generalized
topology.

Given a generalized topological space $(X,\mu),$ the elements of $\mu$ are
called generalized open sets (or $\mu$-open sets) and their complements are
called generalized closed sets (or $\mu$-closed sets) in $(X,\mu)$.

######  Definition 2.2.

[6] Given a nonempty set $X$ and $\mathcal{B}\subset\mathcal{P}(X)$ with
$\emptyset\in\mathcal{B},$ all possible unions of elements of $\mathcal{B}$
form a generalized topology $\mu(\mathcal{B})$ on $X.$ Here $\mathcal{B}$ is
called a base for $\mu(\mathcal{B}).$ Equivalently, $\mu(\mathcal{B})$ is said
to be generated by the base $\mathcal{B}.$

######  Definition 2.3.

[5, 7] Let $(X,\mu)$ and $(Y,\mu^{\prime})$ be two generalized topological
spaces.

(a) A mapping $f:X\to Y$ is called generalized continuous or
$(\mu,\mu^{\prime})$-continuous if
$f^{-1}(G)\in\mu,~{}\forall~{}G\in\mu^{\prime}$.

(b) A mapping $f:X\to Y$ is called a generalized homeomorphism or
$(\mu,\mu^{\prime})$-homeomorphism if $f$ is bijective and $f,f^{-1}$ are
generalized continuous.

(c) A property of generalized topological spaces that is invariant under
generalized homeomorphism is called a $g$-topologically invariance.

######  Definition 2.4.

[19] Let $(X,\mu)$ be a strong generalized topological space.

(a) $(X,\mu)$ is called $\mu$-$T_{0}$ if for $x,y\in X$ with $x\neq y$ there
exists $B\in\mu$ such that exactly one of $x$ and $y$ is in $B.$

(b) $(X,\mu)$ is called $\mu$-$T_{1}$ if for $x,y\in X$ with $x\neq y$ there
exist $B_{1},B_{2}\in\mu$ such that $B_{1}$ contains $x$ but not $y$ and
$B_{2}$ contains $y$ but not $x.$

Clearly every $\mu$-$T_{1}$ strong generalized topological space is
$\mu$-$T_{0}.$

######  Theorem 2.5.

[19] Every singleton set in a $\mu$-$T_{1}$ strong generalized topological
space is $\mu$-closed.

######  Definition 2.6.

[8] Let $X$ be a nonempty set. A nonempty subset $\mathcal{U}$ of
$\mathcal{P}(X\times X)$ is called a generalized quasi-uniformity (or
$g$-quasi uniformity) if

(i) each member of $\mathcal{U}$ contains the diagonal of $X,$

(ii) $\mathcal{U}$ is closed under superset,

(iii) for $U\in\mathcal{U}$ there exists $V\in\mathcal{U}$ such that $V\circ
V\subset U.$

In this case, $(X,\mathcal{U})$ is called a generalized quasi-uniform space
(or $g$-quasi uniform space).

######  Theorem 2.7.

[8] Given a nonempty set $X$ and
$\mathcal{B}~{}(\neq\emptyset)\subset\mathcal{P}(X\times X),$ $\mathcal{B}$
forms a base for some $g$-quasi uniformity on $X$ if and only if (i)
$\Delta(X)\subset B,~{}\forall~{}B\in\mathcal{B},$ and (ii)
$B\in\mathcal{B}\implies\exists~{}V\in\mathcal{B}$ such that $V\circ V\subset
B.$

Moreover, such $\mathcal{B}$ is a base for the $g$-quasi uniformity
$\\{V\subset X:B\subset V\text{ for some }B\in\mathcal{B}\\}$ on $X.$

we finish this section by recalling certain preliminaries on Lebesgue property
and weak $G$-completeness for metric spaces.

######  Definition 2.8.

[11] A metric space on which every real-valued continuos function is uniformly
continuous is said to be Lebesgue (or Atsuji space).

######  Definition 2.9.

[11] A sequence $(x_{n})$ in a metric space $(X,d)$ is said to be pseudo-
Cauchy if given $\epsilon>0,k\in\mathbb{N}$ there exists distinct
$m,n~{}(>k)\in\mathbb{N}$ such that $d(x_{m},x_{n})<\epsilon.$

######  Theorem 2.10.

[11, 16] A metric space is Lebesgue if and only if every pseudo-Cauchy
sequence having distinct terms clusters in it.

######  Definition 2.11.

[1, 9, 10] A sequence $(x_{n})$ in a metric space $(X,d)$ is called $G$-Cauchy
if $\lim\limits_{n\to\infty}d(x_{n+p},x_{n})=0,~{}\forall~{}p\in\mathbb{N}$
(or equivalently, $\lim\limits_{n\to\infty}d(x_{n+1},x_{n})=0$). A metric
space in which every $G$-Cauchy sequence converges is said to be weak
$G$-complete.

Both Lebesgue property and weak $G$-completeness are strictly intermediate
between compactness and completeness of metric spaces [9, 11].

##  3\. $g$-Quasi Metric Spaces and the Induced Generalized Topology

######  Definition 3.1.

[17] Let $X$ be a nonempty set. A mapping $d:X\times X\to\mathbb{R}$ is called
a quasi-metric on $X,$ if:

(a) $\forall~{}x,y\in X,~{}d(x,y)\geq 0$ and $d(x,y)=0\iff x=y;$

(b) $\forall~{}x,y,z\in X,~{}d(x,y)\leq d(x,z)+d(z,y).$

Here $d$ is called a quasi-metric on $X.$ Moreover, the pair $(X,d)$ is called
a quasi-metric space.

######  Definition 3.2.

Let $X$ be a nonempty set. A mapping $d:X\times X\to\mathbb{R}$ is called a
$g$-quasi metric on $X,$ if there exists $r\geq 0$ such that

(a) $\forall~{}x,y\in X,~{}d(x,y)\geq r$ and $d(x,y)=r\iff x=y;$

(b) $\forall~{}x,y,z\in X,~{}d(x,y)\leq d(x,z)+d(z,y).$

Here $d$ is called a $g$-quasi metric on $X$ and $r,$ the index of $d.$
Moreover, the pair $(X,d)$ is called a $g$-quasi metric space (of index $r).$

Clearly a $g$-quasi metric of index $0$ is a quasi-metric and vice versa.

######  Definition 3.3.

A $g$-quasi metric $d$ on a nonempty set $X$ (and hence the related $g$-quasi
metric space $(X,d)$) is said to be symmetric if
$d(x,y)=d(y,x),~{}\forall~{}x,y\in X.$

Clearly a symmetric $g$-quasi metric of index $0$ is a metric and vice versa.

######  Theorem 3.4.

Let $(X,d)$ be a quasi-metric space. Then for $r\geq 0,~{}d^{\prime}=d+r$
forms a $g$-quasi metric on $X$ of index $r.$

###### Proof.

(a) Clearly $\forall~{}x,y\in X,~{}d^{\prime}(x,y)\geq r$ and
$d^{\prime}(x,y)=r\iff x=y;$

(b) Choose $x,y,z\in X.$ Then $d^{\prime}(x,y)=d(x,y)+r\leq
d(x,z)+d(z,y)+r\leq\\{d(x,z)+r\\}+\\{d(z,y)+r\\}\leq
d^{\prime}(x,z)+d^{\prime}(z,y).$

Hence the result follows. ∎

However given a $g$-quasi metric $d$ on a nonempty set $X,$ $d^{\prime}=d-r$
may not form a quasi-metric on it for all choices of $r\geq 0:$

######  Example 3.5.

Let $X=[2,4]$ and $d:X\times X\to\mathbb{R}$ be given by
$d(x,y)=(x-y)^{2}+100,~{}\forall~{}x,y\in X.$ Then for no values of $r\geq 0,$
$d^{\prime}=d-r$ forms a quasi-metric on $X.$ It follows by observing that for
$r=100,~{}d^{\prime}(2,4)>d^{\prime}(2,3)+d^{\prime}(3,4),$ while for all
other values of $r,~{}d^{\prime}(x,x)\neq 0,~{}\forall~{}x\in X.$

However $d$ is a $g$-quasi metric on $X$ of index $100:$

(a) $\forall~{}x,y\in X,~{}d(x,y)\geq 100$ and $d(x,y)=100\iff x=y,$

(b) $\forall~{}x,y,z\in X,~{}d(x,y)+d(y,z)\geq
100+100\geq(x-z)^{2}+100=d(x,z).$

######  Definition 3.6.

Let $(X,d)$ be a $g$-quasi metric space of index $r\geq 0.$ Given $x\in X$ and
$p>0,$ we denote the set $\\{y\in X:d(x,y)<p\\}$ by $B_{d}(x,p)$ (or simply by
$B(x,p)$). Clearly $\mathcal{B}(d)=\\{B(x,p):x\in
X,p>0\\}\bigcup\\{\emptyset\\}$ forms a base for some strong generalized
topology $\mu(d)$ on $X.$ It is called the generalized topology induced by
$d.$

######  Note 3.7.

It should be noted, at this stage, that if $(X,d)$ is a symmetric $g$-quasi
metric space of index $0$ (i.e., a metric space) then $\mathcal{B}(d),$
defined as before, forms a base for the topology induced by $d.$

In what follows, we show that for all positive values of $r,$ a symmetric
$g$-quasi metric space $(X,d)$ can be found so that (i) $d$ is of index $r,$
(ii) $\mu(d)$ does not form a topology on $X.$ We consider the following
example.

######  Example 3.8.

Let $r>0$ and $d:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ be defined by

$d(x,y)=\begin{cases}r&\text{if $x=y$}\\\ 2r&\text{if $0<|x-y|\leq r$}\\\
|x-y|&\text{if $|x-y|>r$}\end{cases}$

We show that $(\mathbb{R},d)$ is a $g$-quasi metric space of index $r$ though
$\mu(d)$ does not form a topology on $\mathbb{R}.$

(a) Clearly $d(x,y)\geq r$ and $d(x,y)=r\iff x=y,$
$\forall~{}x,y\in\mathbb{R}.$

(b) Let $x,y,z\in\mathbb{R}.$ We show that $d(x,y)\leq d(x,z)+d(z,y).$

Note for $0\leq|x-y|\leq r$ the above inequality follows from (a). So let us
assume $|x-y|>r.$

If $|x-z|=0$ or $|z-y|=0$ then the inequality is immediate.

If $|x-z|,|z-y|>r$ then it follows from the order property of $\mathbb{R}.$

If $0<|x-z|,|z-y|\leq r$ then $d(x,y)=|x-y|\leq|x-z|+|z-y|\leq
4r=d(x,z)+d(z,y).$

If $|x-z|>r$ and $0<|z-y|\leq r$ then $d(x,y)\leq d(x,z)+|z-y|\leq
d(x,z)+r\leq d(x,z)+2r=d(x,z)+d(z,y).$

If $|z-y|>r$ and $0<|x-z|\leq r$ then it follows similarly as before.

Thus $d$ forms a $g$-quasi metric on $\mathbb{R}$ of index $r.$

We now show that $\mu(d)$ does not form a topology on $\mathbb{R}.$

Since $B\left(r,2r+\frac{r}{10}\right)\bigcap$
$B\left(\frac{21r}{5},2r+\frac{r}{10}\right)$
$=\left(r-\frac{21r}{10},r+\frac{21r}{10}\right)\bigcap\left(\frac{21r}{5}-\frac{21r}{10},\frac{21r}{5}+\frac{21r}{10}\right)$
$=\left(\frac{21r}{10},r+\frac{21r}{10}\right),$ it suffices to show that
$(\frac{21r}{10},r+\frac{21r}{10})$ does not contain any nonempty set of the
form $B(x,c)$ where $x\in\mathbb{R},c>0.$

Suppose otherwise. Then
$B(x,c)\subset\left(\frac{21r}{10},r+\frac{21r}{10}\right)$ for some
$x\in\mathbb{R}$ and $c>r.$

Case I: $r<c\leq 2r.$ Then for chosen $y\in\mathbb{R}$ with $r<|x-y|<c,~{}y\in
B(x,c)$ Also $x\in B(x,c).$ Thus $x,y\in
B(x,c)\subset\left(\frac{21r}{10},r+\frac{21r}{10}\right),$ a contradiction to
$|x-y|>r.$

Case II: $c>2r.$ Then for each $y\in(x-r,x+r),~{}d(x,y)<c.$ Consequently
$(x-r,x+r)\subset
B(x,c)\implies(x-r,x+r)\subset\left(\frac{21r}{10},r+\frac{21r}{10}\right),$ a
contradiction to
$\left|\left(\frac{21r}{10},r+\frac{21r}{10}\right)\right|=r.$

The contradictions arrived at both the cases prove our claim.

######  Remark 3.9.

Let $(X,\overline{d})$ be a $g$-quasi metric space with index $r.$ For
$\epsilon>r,$ set $V_{\epsilon}=\\{(x,y)\in X\times
X:\overline{d}(x,y)<\epsilon\\}.$

If $r=0,$ then it is clear that
$\mathcal{B}_{\mathcal{U}}=\\{V_{\epsilon}:\epsilon>r\\}$ forms a base for a
$g$-quasi uniformity on $X$ alike the classical case.

However for all other non-negative values of $r,$ there is some $g$-quasi
metric space with index $r$ such that $\mathcal{B}_{\mathcal{U}}$ fails to
form a base for some $g$-quasi uniformity on $X$ as we see now.

Consider the $g$-quasi metric space $(\mathbb{R},d),$ as defined in Example
3.8, with index $r>0.$ If possible, let
$\mathcal{B}_{\mathcal{U}}=\\{V_{\epsilon}:\epsilon>r\\}$ forms a base for a
$g$-quasi uniformity on $\mathbb{R}$ where $V_{\epsilon}=\\{(x,y)\in X\times
X:d(x,y)<\epsilon\\},~{}\forall~{}\epsilon>r.$ Then there is $\delta>r$ such
that, $V_{\delta}\circ V_{\delta}\subset V_{\frac{3r}{2}}$.

Set $x=0,y=r+\frac{\delta-r}{2},z=r+\delta.$ Then
$d(x,y)=d(y,z)=r+\frac{\delta-r}{2}<\delta$ and hence $(x,y),(y,z)\in
V_{\delta}$ implies $(x,z)\in V_{\frac{3r}{2}}.$ i.e., $d(x,z)<\frac{3r}{2},$
i.e., $r+\delta<\frac{3r}{2}$ i.e., $\delta<\frac{r}{2},$ a contradiction.
Hence $\mathcal{B}_{\mathcal{U}}$ is not a base for a $g$-quasi uniformity on
$\mathbb{R}.$

######  Theorem 3.10.

Let $d$ be a $g$-quasi metric on $X.$ Then $(X,\mu(d))$ is $\mu$-$T_{1}$.

###### Proof.

Let $d$ be of index $r.$ Choose $x,y\in X$ such that $x\neq y.$

Clearly $d(x,y),~{}d(y,x)>r.$ Choose $p\in\mathbb{R}$ such that
$r<p<\min\\{d(x,y),d(y,x)\\}.$ Then $B(x,p),~{}B(y,p)$ are generalized open
sets in $(X,\mu(d))$ containing $x,~{}y$ respectively such that $x\notin
B(y,p)$ and $y\notin B(x,p).$ ∎

Let $d$ be a $g$-quasi metric on $X.$ Then we may conclude the following,
stated as corollaries:

######  Corollary 3.11.

Each singleton set in $(X,\mu(d))$ is $\mu$-closed.

######  Corollary 3.12.

$(X,\mu(d))$ is $\mu$-$T_{0}$.

######  Remark 3.13.

$g$-Quasi metrices of different indices may induce the same generalized
topology. For example, choosing $X=\\{x,y\\}$ and $d_{1},d_{2}:X\times
X\to\mathbb{R}$ as given by $d_{1}(x,x)=d_{1}(y,y)=3,d_{1}(x,y)=d_{1}(y,x)=4$
and $d_{2}(x,x)=d_{2}(y,y)=5,d_{2}(x,y)=d_{2}(y,x)=6,$ we observe that both
$d_{1},d_{2}$ induce discrete topology on $X$ though they have distinct
indices.

######  Definition 3.14.

A generalized topological space $(X,\mu)$ is called $g$-quasi metrizable if
for some $g$-quasi metric $d$ on $X,~{}\mu(d)=\mu.$

######  Theorem 3.15.

Let $(X,\mu),(Y,\mu^{\prime})$ be two generalized topological spaces and
$f:X\to Y$ be a generalized homeomorphism. If $(X,\mu)$ is $g$-quasi
metrizable, then so is $(Y,\mu^{\prime}).$

###### Proof.

Let $(X,\mu)$ be $g$-quasi metrizable and $d:X\times X\to\mathbb{R},$ a
$g$-quasi metric on $X$ of index $r\geq 0$ such that $\mu(d)=\mu.$

Define $d^{\prime}:Y\times Y\to\mathbb{R}$ by
$d^{\prime}(y_{1},y_{2})=d(f^{-1}(y_{1}),f^{-1}(y_{2})),~{}\forall~{}y_{1},y_{2}\in
Y.$ Then

(a) $\forall~{}y_{1},y_{2}\in Y,~{}d^{\prime}(y_{1},y_{2})\geq r$ and
$d^{\prime}(y_{1},y_{2})=r\iff f^{-1}(y_{1})=f^{-1}(y_{2})\iff y_{1}=y_{2};$

(b) $\forall~{}y_{1},y_{2},y_{3}\in
Y,~{}d^{\prime}(y_{1},y_{2})+d^{\prime}(y_{2},y_{3})=d(f^{-1}(y_{1}),f^{-1}(y_{2}))+d(f^{-1}(y_{2}),f^{-1}(y_{3}))\geq
d(f^{-1}(y_{1}),f^{-1}(y_{3}))=d^{\prime}(y_{1},y_{3}).$ Thus $d^{\prime}$
forms a $g$-quasi metric on $Y$ of index $r.$

Choose $V\in\mu^{\prime}$ and $y\in V.$ Then for some $x\in X,~{}p>r$ we have
$f^{-1}(y)\in B_{d}(x,p)\subset f^{-1}(V)\implies y\in f(B_{d}(x,p))\subset
V\implies y\in B_{d^{\prime}}(f(x),p)\subset V.$

Thus $\mathcal{B}(d^{\prime})$ forms a base for $\mu^{\prime}.$ Hence the
result follows. ∎

##  4\. Product of $g$-Quasi Metrics

######  Theorem 4.1.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be $g$-quasi metric spaces of the same index
$r.$ Define $d_{XY}:(X\times Y)\times(X\times Y)\to\mathbb{R}$ by

$d_{XY}((x_{1},y_{1}),(x_{2},y_{2}))=\max\\{d_{X}(x_{1},x_{2}),d_{Y}(y_{1},y_{2})\\},$

$\forall~{}x_{1},x_{2}\in X$ and $y_{1},y_{2}\in Y.$ Then $d_{XY}$ defines a
$g$-quasi metric on $X\times Y$ of index $r.$

###### Proof.

Straightforward. ∎

######  Definition 4.2.

Given two $g$-quasi metric spaces $(X,d_{X})$ and $(Y,d_{Y})$ of the same
index $r,$ $d_{XY}$ is called the $g$-quasi metric product of $d_{X}$ with
$d_{Y}$ or simply product $g$-quasi metric on $X\times Y.$

Clearly if $(X,d_{X})$ and $(Y,d_{Y})$ are metric spaces, then $d_{XY}$
defines the product metric on $X\times Y$.

######  Definition 4.3.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be $g$-quasi metric spaces of indices $r_{1}$
and $r_{2}$ respectively. A mapping $f:X\to Y$ is said to be $g$-uniformly
continuous if for $\epsilon>r_{2}$ there exists $\delta>r_{1}$ such that
$d_{X}(x_{1},x_{2})<\delta\implies d_{Y}(f(x_{1}),f(x_{2}))<\epsilon,$
$\forall~{}x_{1},x_{2}\in X.$

Clearly if $(X,d_{X})$ and $(Y,d_{Y})$ are metric spaces, then every
$g$-uniformly continuous mapping from $X$ to $Y$ is uniformly continuous as a
mapping between metric spaces.

It is known that if $(X,d)$ is a metric space, then the distance function
$d:X\times X\to\mathbb{R}$ is uniformly continuous where $X\times X$ is
equipped with the product metric and $\mathbb{R}$ with the usual metric.
However given a $g$-quasi metric space $(X,d),$ the mapping $d:X\times
X\to\mathbb{R}$ may not be $g$-uniformly continuous where $X\times X$ is
equipped with the product metric and $\mathbb{R}$ with the usual metric
(recall that, it is a $g$-quasi metric of index $0$). We consider the
following example.

######  Example 4.4.

Consider the $g$-quasi metric space $(\mathbb{R},d),$ defined in Example 3.8,
of index $r>0.$ We show that
$d:(\mathbb{R}\times\mathbb{R},d^{\prime})\to(\mathbb{R},d_{u})$ is not
$g$-uniformly continuous, where $d^{\prime}$ is the $g$-quasi metric product
of $d$ with itself on $\mathbb{R}\times\mathbb{R}$ and $d_{u}$ is the usual
metric on $\mathbb{R}$.

Suppose otherwise. Then for $\epsilon=\frac{r}{2},$ $\exists~{}\delta>r$ such
that
$d^{\prime}((x,y),(x^{\prime},y^{\prime}))<\delta\implies|d(x,y)-d(x^{\prime},y^{\prime})|<\epsilon,$
$\forall~{}(x,y),(x^{\prime},y^{\prime})\in\mathbb{R}\times\mathbb{R}.$

In particular, for $(x^{\prime},y^{\prime})=(0,0),$
$d^{\prime}((x,y),(0,0))<\delta\implies|d(x,y)-d(0,0)|<\frac{r}{2},$
$\forall~{}(x,y)\in\mathbb{R}\times\mathbb{R}.$

That is, $\max\\{d(x,0),d(y,0)\\}<\delta\implies|d(x,y)-d(0,0)|<\frac{r}{2},$
$\forall~{}(x,y)\in\mathbb{R}\times\mathbb{R}.$

Choose $n\in\mathbb{N}\backslash\\{1\\}$ such that
$\frac{\delta-r}{(n-1)^{2}}<\frac{r}{2}.$ Then
$\frac{\delta-r}{n(n-1)}<\frac{r}{2}.$

Set $x=r+\frac{\delta-r}{n}$ and $y=r+\frac{\delta-r}{n-1}.$

Then $|x-0|=r+\frac{\delta-r}{n}>r$ and $|y-0|=r+\frac{\delta-r}{n-1}>r.$

Consequently, $d(x,0)=r+\frac{\delta-r}{n}<\delta$ and
$d(y,0)=r+\frac{\delta-r}{n-1}<\delta$ whence, $\max\\{d(x,0),$ $d(y,0)\\}$
$<\delta.$

However, $|x-y|=\frac{\delta-r}{n(n-1)}<\frac{r}{2}\leq r\implies d(x,y)=2r,$
a contradiction since $|d(x,y)-d(0,0)|<\frac{r}{2}.$

Hence $d:(\mathbb{R}\times\mathbb{R},d^{\prime})\to(\mathbb{R},d_{u})$ is not
$g$-uniformly continuous.

##  5\. Completeness, Lebesgue Property and (Weak) $G$-Completeness in
$g$-Quasi Metric Spaces

In this section, we extend the study of completeness, Lebesgue property and
weak $G$-completeness for $g$-quasi metric spaces using the extended notion of
Cauchy, $G$-Cauchy and pseudo-Cauchy sequences.

######  Definition 5.1.

Let $(x_{n})$ be a sequence in a $g$-quasi metric space $(X,d)$ of index $r$
and $c\in X.$ Then

(i) $(x_{n})$ is said to be convergent to $c$ in $(X,d)$ if it is so in
$(X,\mu(d));$

(ii) $c$ is called a cluster point of $(x_{n})$ in $(X,d)$ if it is so in
$(X,\mu(d))$.

Clearly if $(x_{n})$ is convergent to $c,$ then $c$ is a cluster point of
$(x_{n})$ (in $(X,d)$).

######  Definition 5.2.

Let $(X,d)$ be a $g$-quasi metric space of index $r$ and $(x_{n})$ be a
sequence in $X.$ Then

(i) $(x_{n})$ is called Cauchy if given $\epsilon>r$ there exists
$k\in\mathbb{N}$ such that $d(x_{m},x_{n})<\epsilon,~{}\forall~{}m,n\geq k;$

(ii) $(x_{n})$ is called $G$-Cauchy if given $\epsilon>r$ there exists
$k\in\mathbb{N}$ such that $d(x_{n},x_{n+1})<\epsilon,~{}\forall~{}n\geq k;$

(iii) $(x_{n})$ is called pseudo-Cauchy if given $\epsilon>r$ and
$k\in\mathbb{N}$ there exist $p,q~{}(p\neq q)\in\mathbb{N}$ with $p,q\geq k$
such that $d(x_{p},x_{q})<\epsilon.$

######  Definition 5.3.

A $g$-quasi metric space $(X,d)$ is said to be

(i) complete if every Cauchy sequence converges to some point in it;

(ii) $G$-complete if every $G$-Cauchy sequence converges to some point in it;

(iii) weak $G$-complete if every $G$-Cauchy sequence has a cluster point in
it;

(iv) Lebesgue if every pseudo-Cauchy sequence having distinct terms has a
cluster point in it;

(v) strongly Lebesgue if every pseudo-Cauchy sequence has a cluster point in
it.

Clearly for $g$-quasi metric spaces we have the following chain of
implications:

Strongly LebesgueLebesgue${\text{Weak
}G\text{-completeness}}$Completeness${G\text{-completeness}}$

In what follows, we show that for each $r>0,$ $(\mathbb{R},d)$ of index $r,$
as defined in Example 3.8, is not weak $G$-complete.

######  Example 5.4.

Consider the sequence $(x_{n})$ in $(\mathbb{R},d)$ where
$x_{n}=rn-\frac{r}{n},~{}\forall~{}n\in\mathbb{N}.$

Then $\forall~{}n\in\mathbb{N},$
$|x_{n+1}-x_{n}|=r+r\left(\frac{1}{n}-\frac{1}{n+1}\right)>r\implies
d(x_{n+1},x_{n})=r+r\left(\frac{1}{n}-\frac{1}{n+1}\right)\implies
d(x_{n},x_{n+1})=r+r\left(\frac{1}{n}-\frac{1}{n+1}\right).$

Choose $\epsilon>r.$ Then there is $k\in\mathbb{N}$ such that
$\frac{r}{n(n+1)}<\epsilon-r,~{}\forall~{}n\geq k$ and hence,
$d(x_{n},x_{n+1})<\epsilon,~{}\forall~{}n\geq k.$ Thus $(x_{n})$ is $G$-Cauchy
in $(\mathbb{R},d).$

If possible, let there is a cluster point $c$ of $(x_{n})$ in
$(\mathbb{R},d).$

Since $(x_{n})$ is a sequence of distinct terms, $B(c,\frac{3r}{2})$ contains
infinitely many elements of $(x_{n}).$

However
$B(c,\frac{3r}{2})=\\{y\in\mathbb{R}:d(c,y)<\frac{3r}{2}\\}=\left(c-\frac{3r}{2},c-r\right)\bigcup\left(c+r,c+\frac{3r}{2}\right)\bigcup\\{c\\}$
which clearly contains finitely many elements of $(x_{n}),$ a contradiction.

Hence $(\mathbb{R},d)$ is not weak $G$-complete.

######  Lemma 5.5.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be $g$-quasi metric spaces of the same index
$r.$ A sequence $((x_{n},y_{n}))$ is Cauchy in $(X\times Y,d_{XY})$ if and
only if $(x_{n})$ and $(y_{n})$ are Cauchy in $(X,d_{X})$ and $(Y,d_{Y})$
respectively.

###### Proof.

Let $((x_{n},y_{n}))$ be Cauchy in $(X\times Y,d_{XY}).$ Choose $\epsilon>r.$
Then $\exists~{}k\in\mathbb{N}$ such that
$d_{XY}((x_{m},y_{m}),(x_{n},y_{n}))<\epsilon,~{}\forall~{}m,n\geq k.$ That
is, $d_{X}(x_{m},x_{n}),d_{Y}(y_{m},y_{n})<\epsilon,~{}\forall~{}m,n\geq k.$
Then $(x_{n})$ and $(y_{n})$ are Cauchy in $(X,d_{X})$ and $(Y,d_{Y})$
respectively.

Conversely, let $(x_{n})$ and $(y_{n})$ be Cauchy in $(X,d_{X})$ and
$(Y,d_{Y})$ respectively. Choose $\epsilon>r.$ Then
$\exists~{}p,q\in\mathbb{N}$ such that
$d_{X}(x_{m},x_{n})<\epsilon,~{}\forall~{}m,n\geq p$ and
$d_{Y}(y_{m},y_{n})<\epsilon,~{}\forall~{}m,n\geq q.$

Set $r=\max\\{p,q\\}.$ Then
$d_{XY}((x_{m},y_{m}),(x_{n},y_{n}))<\epsilon,~{}\forall~{}m,n\geq r.$

Hence $((x_{n},y_{n}))$ is Cauchy in $(X\times Y,d_{XY}).$ ∎

Similar chain of arguments yield the following results that we state without
proof.

######  Lemma 5.6.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be $g$-quasi metric spaces of the same index
$r.$ A sequence $((x_{n},y_{n}))$ is $G$-Cauchy in $(X\times Y,d_{XY})$ if and
only if $(x_{n})$ and $(y_{n})$ are $G$-Cauchy in $(X,d_{X})$ and $(Y,d_{Y})$
respectively.

######  Lemma 5.7.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be $g$-quasi metric spaces of the same index
$r.$ If $((x_{n},y_{n}))$ is a pseudo-Cauchy sequence in $(X\times Y,d_{XY})$
then $(x_{n})$ and $(y_{n})$ are pseudo-Cauchy in $(X,d_{X})$ and $(Y,d_{Y})$
respectively.

Then converse of Lemma 5.7 is not true. In support, we produce the following
example.

######  Example 5.8.

Consider the $g$-quasi metric space $(\mathbb{R},d),$ as defined in Example
3.8, of index $1.$ Then the sequences $(x_{n})$ and $(y_{n})$ in $\mathbb{R},$
defined by

$x_{n}=\begin{cases}1&\text{if }n\text{ is odd}\\\ 10^{n}&\text{if }n\text{ is
even}\end{cases}$

and

$y_{n}=\begin{cases}10^{n}&\text{if }n\text{ is odd}\\\ 1&\text{if }n\text{ is
even}\end{cases}$

$\forall~{}n\in\mathbb{N},$ are pseudo-Cauchy in $(\mathbb{R},d).$

However $((x_{n},y_{n}))$ is not pseudo-Cauchy in $\mathbb{R}\times\mathbb{R}$
(where $\mathbb{R}\times\mathbb{R}$ is equipped with $d^{\prime},$ the
$g$-quasi metric product of $d$ with itself). In fact, for any pair of
positive integers $m,q~{}(m\neq q)$ with $m,q\geq 1,$ we obtain
$d^{\prime}((x_{m},y_{m}),(x_{q},y_{q}))>2,$ by considering even and odd cases
separately for $m$ and $q.$ Thus $((x_{n},y_{n}))$ is not pseudo-Cauchy in
$(\mathbb{R}\times\mathbb{R},d^{\prime}).$

######  Theorem 5.9.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be $g$-quasi metric spaces of the same index
$r.$ Then $(X\times Y,d_{XY})$ is complete if and only if $(X,d_{X})$ and
$(Y,d_{Y})$ are complete.

###### Proof.

Let $(X\times Y,d_{XY})$ be complete.

We first show that $(X,d_{X})$ is complete. Choose a Cauchy sequence $(x_{n})$
in $(X,d_{X})$ and fix $y\in Y.$ Then by Lemma 5.5, $((x_{n},y))$ is Cauchy in
$(X\times Y,d_{XY}).$ Since $(X\times Y,d_{XY})$ is complete, $((x_{n},y))$
converges to a point $(a,b)$ in $(X\times Y,d_{XY}).$

We claim that $(x_{n})$ is convergent to $a$ in $(X,d_{X}).$

Let $V$ be a generalized open set in $(X,\mu(d_{X}))$ containing $a.$ Then
there exist $p\in X,\delta>r$ such that $a\in B_{d_{X}}(p,\delta)\subset V.$

Since $B_{d_{XY}}((p,b),\delta)$ is an open set in $(X\times Y,\mu(d_{XY}))$
containing $(a,b),~{}\exists~{}k\in\mathbb{N}$ such that $(x_{n},y)\in
B_{d_{XY}}((p,b),\delta),$ $\forall~{}n\geq k.$

Thus $\max\\{d_{X}(p,x_{n}),d_{Y}(b,y)\\}<\delta,~{}\forall~{}n\geq k\implies
d_{X}(p,x_{n})<\delta,~{}\forall~{}n\geq k.$

i.e., $x_{n}\in B_{d_{X}}(p,\delta)\subset V,~{}\forall~{}n\geq k.$ Hence
$(x_{n})$ converges to $a$ in $(X,d_{X})$ and so, $(X,d_{X})$ is complete.
Similarly $(Y,d_{Y})$ is complete.

Conversely, let $(X,d_{X})$ and $(Y,d_{Y})$ be complete.

Choose a Cauchy sequence $((x_{n},y_{n}))$ in $(X\times Y,d_{XY}).$ By Lemma
5.5, $(x_{n})$ and $(y_{n})$ are Cauchy in $(X,d_{X})$ and $(Y,d_{Y})$
respectively. So by hypothesis, there exist $a\in X,b\in Y$ such that
$(x_{n})$ converges to $a$ in $(X,d_{X})$ and $(y_{n})$ to $b$ in $(Y,d_{Y}).$

We show that $((x_{n},y_{n}))$ converges to $(a,b)$ in $(X\times Y,d_{XY}).$

Let $W$ be a generalized open set in $(X\times Y,\mu(d_{XY}))$ containing
$(a,b).$ Then there exist $(p,q)\in X\times Y,\delta>r$ such that $(a,b)\in
B_{d_{XY}}((p,q),\delta)\subset W.$ Consequently $a\in B_{d_{X}}(p,\delta)$
and $b\in B_{d_{Y}}(q,\delta).$

Since $(x_{n})$ converges to $a$ and $(y_{n})$ to $b,$ there exist
$k_{1},k_{2}\in\mathbb{N}$ such that $x_{n}\in
B_{d_{X}}(p,\delta),~{}\forall~{}n\geq k_{1}$ and $y_{n}\in
B_{d_{Y}}(q,\delta),~{}\forall~{}n\geq k_{2}.$

Set $k=\max\\{k_{1},k_{2}\\}.$ Then $(x_{n},y_{n})\in
B_{d_{XY}}((p,q),\delta)\subset W,~{}\forall~{}n\geq k.$ Thus
$((x_{n},y_{n}))$ converges to $(a,b)$ in $(X\times Y,d_{XY}).$

Thus $(X\times Y,d_{XY})$ is complete. ∎

Similar chain of arguments yield the following results that we state without
proof.

######  Theorem 5.10.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be $g$-quasi metric spaces of the same index
$r.$ Then $(X\times Y,d_{XY})$ is $G$-complete if and only if $(X,d_{X})$ and
$(Y,d_{Y})$ are $G$-complete.

######  Theorem 5.11.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be $g$-quasi metric spaces of the same index
$r.$ If $(X\times Y,d_{XY})$ is weak $G$-complete then $(X,d_{X})$ and
$(Y,d_{Y})$ are weak $G$-complete.

######  Theorem 5.12.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be $g$-quasi metric spaces of the same index
$r.$ If $(X\times Y,d_{XY})$ is (strongly) Lebesgue then $(X,d_{X})$ and
$(Y,d_{Y})$ are (strongly) Lebesgue.

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