A Three-Mode Erasure Code for Continuous Variable Quantum Communications Eduardo Villase nor and Robert Malaney

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A Three-Mode Erasure Code for Continuous

Variable Quantum Communications

Eduardo Villase˜

nor 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 re-

alistic 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 pre-

sented. 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.

Keywords—Continuous variable quantum information, quan-

tum communications, quantum error correction.

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]–[5].

However, Quantum information is fragile by nature - the

unavoidable interaction between quantum systems and their

environment introduces errors to the quantum states. Correc-

tion 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]–[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 chan-

nels.

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]–[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, perfor-

mance 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 perfor-

mance 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 workThis work has been acceped for publication in GLOBECOM 2022.

arXiv:2210.10230v1 [quant-ph] 19 Oct 2022

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 ˆρis deﬁned as

χ(λ1, λ2, ..., λn) = Tr nˆρˆ

D(λ1)ˆ

D(λ2)... ˆ

D(λn)o,(1)

where λi∈C. Here, ˆ

Dis the displacement operator,

D(λi) = eλjˆa†

j−λ∗

jˆaj,(2)

where ˆajand ˆa†

jare 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, |0i, expressed as

χ|0i(λ) = exp −|λ|2

2,(3)

and the coherent state, |αi=ˆ

D(α)|0i, expressed as

χ|αi(λ) = exp −|λ|2

2+ (λα∗−λ∗α).(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 oper-

ations, 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, χs(λ1), and the

entangled state, χAB(λ2, λ3). 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],

χsλ1+λ2

√2χAB λ1−λ2

√2, λ3.(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, ρ, returns the state,

ρ0= (1 −Pe)ρ+Pe|0ih0|,(6)

with Pebeing 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,

ρch =

j=1

Pjρj,(7)

with Pjcorresponding to the probability of each combina-

tion of modes erased. These range from (1 −Pe)3to P3

corresponding to zero erasures, and erasures in every mode,

respectively. At this point, the CF of the three mode state,

now deﬁned as χx

ch(λ1, λ2, λ3), will depend on the modes 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 20

has an erasure the corresponding CF is

χ{2}

ch (λ1, λ2, λ3) = χsλ1

√2χAB λ1

√2, λ3χ|0i(λ2).

(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.

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,

χx

ch λ1+λ2

√2,λ1−λ2

√2, λ3.(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 per-

formed. To this end, BS3 is applied, giving a CF for the three

mode state as

χx

BS3(λ1, λ2, λ3) = (10)

χx

ch λ1+λ2

√2,λ1

√2−λ2+λ3

2,λ2−λ3

√2.

Now dual homodyne measurements are performed on modes

200 and 300 (see Fig. 1). To compute the result after the mea-

surements it is convenient to represent the complex arguments

in the phase-space representation, by using two distinct real

numbers, xj=1

√2(λj+λ∗

j)and pj=i

√2(λ∗

j−λj). Then, for

the pair of measurement results ˜xand ˜pthe output CF (on

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AThree-ModeErasureCodeforContinuousVariableQuantumCommunicationsEduardoVillasenorandRobertMalaneySchoolofElectricalEngineering&Telecommunications,TheUniversityofNewSouthWales,Sydney,NSW2052,Australia.AbstractQuantumstatesoflightbeingtransmittedviare-alisticfree-spacechannelsoftensuffererasureerror...

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