# A Morphologically-Aware Dictionary-based Data Augmentation Technique for
Machine Translation of Under-Represented Languages

Md Mahfuz Ibn Alamα Sina Ahmadiα,β Antonios Anastasopoulosα,γ
αDepartment of Computer Science, George Mason University βUniversity of Zurich
γArchimedes AI Research Unit, RC Athena, Greece
<EMAIL_ADDRESS>
###### Abstract

The availability of parallel texts is crucial to the performance of machine
translation models. However, most of the world’s languages face the
predominant challenge of data scarcity. In this paper, we propose strategies
to synthesize parallel data relying on morpho-syntactic information and using
bilingual lexicons along with a small amount of _seed_ parallel data. Our
methodology adheres to a realistic scenario backed by the small parallel seed
data. It is linguistically informed, as it aims to create augmented data that
is more likely to be grammatically correct. We analyze how our synthetic data
can be combined with raw parallel data and demonstrate a consistent
improvement in performance in our experiments on 14 languages (28
English$\leftrightarrow$X pairs) ranging from well- to very low-resource ones.
Our method leads to improvements even when using only five seed sentences and
a bilingual lexicon.111Data and code will be publicly released upon
acceptance.

A Morphologically-Aware Dictionary-based Data Augmentation Technique for
Machine Translation of Under-Represented Languages

Md Mahfuz Ibn Alamα Sina Ahmadiα,β Antonios Anastasopoulosα,γ αDepartment of
Computer Science, George Mason University βUniversity of Zurich γArchimedes AI
Research Unit, RC Athena, Greece<EMAIL_ADDRESS>

##  1 Introduction

Ew gîtarê pir baş lê dideHe plays the guitar very well

Analyze: guitar = [n;acc;sg, lemma=guitar]

Replace: guitar $\leftarrow$ flower, flower= gul

Generate: [n;acc;sg, lemma=gul]$\rightarrow$ gulê

He plays the flower very wellEw gulê pir baş lê dide

Figure 1: A schema of our approach. After aligning ‘guitar’ (in English) and
‘gîtarê’ (in Kurmanji Kurdish), the new word ‘flower’ is randomly selected to
replace ‘guitar’ and its translation ‘gul’ in a bilingual dictionary is
inflected according to its morphological features as ‘gulê’. Small caps refer
to lemmata.

One of the major challenges in machine translation (MT) is the lack of
parallel data for most of the world’s languages. Traditional approaches Wu et
al. (2008); Mikolov et al. (2013) used to rely on dictionaries and linguistic
knowledge for MT. One of the naive ways to use dictionaries for MT is to
translate by looking up words of a source sentence in a bilingual lexicon and
replacing their corresponding translations in the target language. However,
this approach has certain shortcomings Wang et al. (2022a). Firstly, the
coverage of translations depends on the size and comprehensiveness of the
lexicon, which can result in incomplete translations and code-mixed versions
of the source and target languages. The translated sentences may also not
adhere to the target language’s grammatical rules or word order. Furthermore,
most dictionaries operate at the lemma level, posing challenges for
morphologically-rich languages. Therefore, solely relying on dictionaries is
not a viable solution for low-resource languages.

In recent approaches to MT that mainly rely on encoder-decoder networks like
transformers Vaswani et al. (2017), the ideal scenario is to train an MT model
on a large parallel corpus. Creating a parallel corpus for a given language,
however, requires linguistic and technical expertise lacking for under-
resourced languages and is also a costly and time-consuming task. To remedy
this, recent studies in natural language processing (NLP) focus on
unsupervised methods based on monolingual data Sennrich et al. (2016a); Lample
et al. (2018a), back-translation Edunov et al. (2018a, b), other data
augmentation techniques Sánchez-Cartagena et al. (2021), or fine-tune pre-
trained models to adapt to a different language, domain, or dialect Bapna and
Firat (2019). Therefore, the usage of dictionaries is largely under-studied,
even though they are still practically in use Peng et al. (2020); Sennrich et
al. (2016b).

In this paper, we put forward a dictionary-based approach akin to early
dictionary-based MT systems Tyers (2009); Koehn and Knight (2002, 2001);
Sánchez-Cartagena et al. (2011) yet more sophisticated as it relies on the
morpho-syntactic analysis of words to generate a parallel corpus
synthetically. As illustrated in Figure 1, our approach consists of four
components: alignment, analysis, replacement, and generation. Given a small
set of parallel text as seed data, we first retrieve possible word-level
translation pairs in the source and target languages as in ‘guitar’ and
‘gîtar’ in English and Kurmanji Kurdish, respectively. We then morphologically
analyze the source words in the translation pairs, e.g., ‘guitar’ is a
singular noun in the accusative case in the example. With the morphological
features of a word in the source sentence, we can now sample a word from a
bilingual dictionary with the same morphological features, e.g., gulê, and
"plug" it into our sentence to generate a new sentence pair synthetically. As
such, the synthetically-generated sentences are likely to create new
grammatically-sound translations.

To summarize, the contributions of our work are three-fold:

  * •

We propose a morphologically-informed replacement method to create a new
synthetic sentence.

  * •

We show that this synthetic parallel data helps improve the MT system’s
quality when mixed with real parallel data.

  * •

We also demonstrate the effectiveness of our method in extremely data-scarce
scenarios, where as little as five parallel seed sentences are rendered useful
with our approach.

Note that we will interchange the terms “dictionary” and “bilingual lexicon”
throughout the paper for readability reasons.

##  2 Method

Our method requires a small parallel dataset called seed data containing
sentences in the source and target languages to create synthetic parallel
data. Our approach consists of three components. We first prepare data by
tokenizing sentences and obtaining word-level alignment between the parallel
sentences. This step is completed by morphologically analyzing aligned word
pairs. Then, we replace words considering the morphological features in the
augmentation component and filter the synthetic sentences using language
models. Finally, we build MT systems in different settings varying the number
of synthetic sentences.

###  2.1 Analysis

#### Alignment

We perform word alignment to our seed data, identifying the relationship
between words in the seed sentence. This is necessary for knowing which words
are translations of each other. If we replace a word in the source sentence,
the aligned target word of the target sentence must also be replaced to
reflect the changes.

#### Morphological Tagging

We analyze the entries in the bilingual lexicon’s source side words to
facilitate the data augmentation process. This way, we can categorize entries
based on morphological features and find the part-of-speech (POS) tags, e.g.,
adj, of our bilingual lexicon’s source side words.

#### Word-pair Selection

We randomly choose words from the source side, here in English, for each seed
sentence. We refer to Figure 1 as our example where we generate the
morphological feature and POS tag for the given word “guitar”. We also find
the translation of “guitar” in the seed sentence’s target side. Here, that
word is “gîtarê", which we get from the alignment in §2.1. We find the
morphological feature and POS tag of “gîtarê” too.

###  2.2 Augmentation

We introduce two different approaches for the augmentation of the seed
sentences:

#### Morphologically-Informed

  1. 1.

Referring to Figure 1, we first replace “guitar” with another random word,
e.g. “flower”, having identical morphological features created in §2.1. As
such, a new sentence is synthetically created as “He plays the flower very
well”. It should be noted that this procedure does not consider the semantic
relevance of the candidate word. In other words, it may yield nonsensical
sentences yet morpho-syntactically valid.

  2. 2.

Then, we replace “gîtarê” with the translation of “flower” being “gul” in
Kurmanji Kurdish from the bilingual lexicon. It is worth mentioning that we
use PanLex dictionaries222https://panlex.org/snapshot where some of the
entries are not in the lemma form. Therefore, we also lemmatize the retrieved
word form in the dictionary, i.e., “gul” to mitigate the impact of the
inaccuracy of the lexicographic data.

  3. 3.

Last, we perform morphological inflection where a lemma is inflected based on
morphological features of the word that will be substituted, i.e., ‘gîtar’.
Doing this, we create a new sentence where the randomly selected word in the
dictionary ‘gul’ appears grammatically and morphologically correct as ‘gulê’.
We do this to guarantee that the new word follows the correct morphological
features. Thus creating a synthetic target translation “Ew gulê pir baş lê
dide” of the synthetic source sentence “He plays the flower very well”.

#### Naive

In contrast to the previous augmentation technique where we consider the
morphological features, we carry out a naive random word replacement approach
where only the POS tag is identical, without lemmatizing or inflecting the
word based on the sentence. For instance in Figure 1, a synthetic sentence
created this way would be “He plays the flower very well” and its generated
translation “Ew gul pir baş lê dide”. Here “gul” has not been converted into
‘gulê’. In the _Morphologically-Informed_ setup, we preserve the morphological
information of the word we change, thus making the synthetic data more likely
to be grammatically correct. In this _naive_ approach, on the other hand, we
lose this information.

###  2.3 Filtering with LMs

We create synthetic sentences for each seed sentence with the above
augmentation approaches. Given that the synthetic sentences may not be
meaningful, e.g., “He plays the flower very well”, we also incorporate
information from a language model (LM) by estimating the perplexity ($ppl$) of
the synthetic sentences:

$ppl(x)=exp\\{-1/t\sum_{i}^{t}logp_{\theta}(x_{i}|x_{<i})\\},$

Where the probability of a sentence of length $t$ containing words $x$
existing in the LM. The lower the perplexity is, the more natural the sentence
is. We filter the augmented sentences using the LM and rank them based on the
perplexity scores to pick the sentences with the correct context. This step
yields sentences more likely to appear with the lowest perplexity.

###  2.4 Neural Machine Translation

Using the synthetic data, we build neural MT systems for each language pair in
one direction. To do so, one of the approaches is to train a transformer-based
encoder-decoder model from scratch with random weights only on the parallel
data. This model type excels in high-resource settings but hardly reaches up
to the mark performance for low-resource languages Duh et al. (2020). Another
approach is to fine-tune a model based on a pre-trained model. Instead of
initializing with random weights, the training is carried out on a previously-
trained transformer model. The pre-trained model can be either monolingual or
multilingual and can be pre-trained on any task, normally on denoising ones.
This approach Alabi et al. (2022) is promising to improve low-resource
languages as the model does not need to learn all language components from
scratch. If the pre-trained model is multilingual, the model can leverage
resources from other high-resource languages.

##  3 Experimental Setup

###  3.1 Dataset

#### Parallel Data

To create synthetic data, we use the parallel sentences in the OPUS-100 Zhang
et al. (2020) corpus333https://data.statmt.org/opus-100-corpus with English as
the source language and other languages as target languages. We use this
training set as our parallel seed data for training. For testing and
validation, we use the devtest and dev sets of the
FLORES-200444https://github.com/facebookresearch/flores/tree/main/flores200
benchmark Team et al. (2022) respectively. Table 1 summarizes the statistics
of our datasets. We divide the languages into four categories according to
their data availability: extremely low-resource, low-resource, well-resourced,
and high-resource.

Language (code) | # Seed | # Entries
---|---|---
Armenian (hye) | 7,059 | 161,798
Wolof (wol) | 7,918 | 4,971
Kurmanji (kmr) | 8,199 | 47,461
Scottish Gaelic (gla) | 16,316 | 51,416
Marathi (mar) | 27,007 | 65,559
Uyghur (uig) | 72,170 | 9,054
Kazakh (kaz) | 79,927 | 40,516
Tamil (tam) | 227,014 | 230,882
Irish (gle) | 289,524 | 71,436
Galician (glg) | 515,344 | 185,946
Hindi (hin) | 534,319 | 409,076
Urdu (urd) | 753,913 | 86,695
Greek (ell) | 1,000,000 | 407,311
Maltese (mlt) | 1,000,000 | 33,131
Table 1: Statistics of our datasets (seed parallel data and dictionary
entries). Sorted according to the number of available training sentences.

#### Bilingual Dictionaries

We extract dictionaries between English and each target language from the
PanLex database containing 25 million words in 2,500 dictionaries of 5,700
languages.

###  3.2 Tools

We use Stanza555https://stanfordnlp.github.io/stanza/ for tokenization,
morphological feature tagging, POS tagging, and lemmatization. Stanza uses
different models for different languages. For word alignment we use
fast$\\_$align666https://github.com/clab/fast_align Dyer et al. (2013), and we
use pyinflect777https://pypi.org/project/pyinflect/0.2.0/ for morphological
inflection. Note that pyinflect only supports English, but in this work, we
only do inflection on the English side for our synthetic data creation
framework. We use the HuggingFace Wolf et al. (2020) toolkit for training the
language models.

###  3.3 Implementation Details

#### Language Model

To construct the language model (LM), we adopt the methodology outlined in the
GPT-2 recipe provided by HuggingFace Radford et al. (2019). We utilize the
monolingual side of the parallel data specific to each language as the
training dataset. Given that many of the languages involved in our experiment
are not highly resourced, we make certain modifications to the GPT-2 model. We
employ only six layers instead of the original 12 to mitigate resource
limitations. Additionally, we decrease the vocabulary size to 5000. These
adjustments help tailor the model to our experiment’s specific requirements of
low-resource languages.

#### Seed Data

We do not use all the available seed data for creating synthetic sentences.
Short sentences with less than seven tokens are not used as seed sentences. As
Stanza uses context to generate morphological features, short sentences seem
not to provide enough context for the model to produce reasonable
annotations.888This was based on preliminary experiments and manual inspection
of Stanza’s outputs.

#### Lexicons

The bilingual lexicons often provide several translations for one source word.
We organize the lexicon so that only one translation is available for each
source word. We do so naively, only taking the first translation of a word and
discarding the rest. We also ensure that the source and target have the same
POS tag. To produce the morphological features of the dictionary entries, we
rely not only on Stanza999As there is no sentential context, Stanza is bound
to be error-prone. but we also perform lookups on _Unimorph_
101010https://unimorph.github.io/ Batsuren et al. (2022), which provides
morphological inflection paradigms for dozens of languages (including the ones
we work on) annotated with POS tags and morphological features. For this work,
we only work with augmentation, focusing on nouns, adjectives, and verbs.

#### Synthetic Data

We create five sizes of synthetic data: 5K, 10K, 50K, 100K, and 200K for each
language pair. In each set, the previous set’s data is used. That means that
when compiling the 10K synthetic dataset, we create new 5K data to add to the
previous 5K data, and so on. This ensures that our experiments only vary based
on the newly added synthetic data (and not due to additional randomness).

For each sentence, we randomly choose at most two words for replacement. As
the word replacement is random, getting the exact number of sentences for each
set is not guaranteed, and there may be duplicates. From each seed sentence,
$M$ number of synthetic sentences are created. Let’s say we want to make a
total of 5,000 seed sentences. Then, $M$ is chosen to get barely more than
5,000 sentences. After that, we sort with the perplexity of the LM and select
the sentences with lower perplexity to create that set.

#### Model Details

We fine-tune DeltaLM Ma et al. (2021), a large pre-trained multilingual
encoder-decoder model that regards the decoder as the task layer of off-the-
shelf pre-trained encoders. This is done separately for each language, not
multilingually. The baseline system is the one that uses only the available
real parallel data. Throughout the paper, we refer to the baseline as the _0K
(untagged)_ model, as it has seen 0 synthetic data. The rest of the models are
use tags _< clean>_ and _< noisy>_ at the beginning of the sentences to
distinguish between real and synthetic data. The model’s name (e.g. _5K_)
indicates how much synthetic data has been added to the seed data during
training.

5K10K50K100K200K$4$$4.5$$5$$5.5$$6$4.21baseline (no augmentation)$5.2$$5.1$$5.5$$5.3$$4.8$$4.8$$5.2$$5.6$$5.1$$5$BLEUenglish$\rightarrow$scottish gaelic | 5K10K50K100K200K$19$$20$$21$$22$19.13baseline (no augmentation)$19.9$$20$$20.3$$20.6$$21.5$$20.3$$19.9$$20.4$$20.9$$22.3$english$\rightarrow$irish | 5K10K50K100K200K$10.5$$11$$11.5$$12$$12.5$11.26baseline (no augmentation)$12.1$$12.3$$12$$12$$11.1$$12.2$$12.4$$11.5$$12$$11.1$english$\rightarrow$urdu
---|---|---
5K10K50K100K$30$$31$$32$29.86baseline (no augmentation)$31.4$$31.4$$31.3$$31.7$$31$$31.3$$31.1$$31.5$Synthetic Data AmountBLEUenglish$\rightarrow$galician | 5K10K50K100K200K$38$$39$$40$38.03baseline (no augmentation)$39.9$$39.8$$39.6$$39.4$$39.2$$40.1$$39.1$$39.5$$38.7$$39.4$Synthetic Data Amountenglish$\rightarrow$maltesenaiveours | 5K10K50K100K200K$21$$22$$23$$24$22.48baseline (no augmentation)$23.2$$22.5$$22.7$$22.4$$22.1$$23.3$$23.4$$22.4$$22.2$$21.2$Synthetic Data Amountenglish$\rightarrow$hindi
Figure 2: BLEU scores on the test sets for six languages in the english-x direction. X-axis indicates the amount of synthetic parallel data we use along with seed data. The baseline uses no synthetic data. Except for Irish and Galician, all the other languages do not benefit from the increasing amounts of synthetic data. It seems like Irish has even room for more improvement. ours is the morphologically-informed method. 5K10K50K100K200K$10$$12$$14$9.08baseline (no augmentation)$13.1$$11.8$$13$$12.4$$11.9$$13.3$$12.8$$12.3$$13.1$$12.4$BLEUscottish gaelic$\rightarrow$english | 5K10K50K100K200K$29$$30$$31$29.09baseline (no augmentation)$30.9$$30.3$$30.8$$30.5$$29.7$$31.1$$30.7$$30.6$$29.6$$29.6$hindi$\rightarrow$english | 5K10K50K100K200K$16$$18$$20$19.58baseline (no augmentation)$21$$20.7$$18.9$$18$$16.6$$20.8$$21.3$$20.5$$19.3$$18.5$kazakh$\rightarrow$english
---|---|---
5K10K50K100K200K$9$$10$$11$$12$9.73baseline (no augmentation)$11.6$$11.1$$11$$9.5$$9.2$$12$$12$$10.9$$10.4$$9.4$Synthetic Data AmountBLEUkurmanji$\rightarrow$english | 5K10K50K100K200K$18$$19$$20$$21$19.83baseline (no augmentation)$20.7$$20.6$$19.9$$19$$18.1$$21.3$$20.7$$19.8$$19.3$$18.6$Synthetic Data Amounttamil$\rightarrow$englishnaiveours | 5K10K50K100K200K$18$$19$$20$18.63baseline (no augmentation)$20$$19.8$$19.8$$19$$18.1$$20$$19.4$$19.2$$19.4$$18.5$Synthetic Data Amounturdu$\rightarrow$english
Figure 3: BLEU scores on the test sets for six languages in the x-English
direction. ours is the morphologically-informed method. Morphologically-
informed approach outperforms the naive approach in all the six language
pairs. X-axis indicates the amount of synthetic parallel data we use along
with seed data. The baseline uses no synthetic data.

##  4 Results

#### From English

In 11 out of the 14 Eng-X language pairs, our approaches yield improvements
randgng from 0.4 points to more than 3 BLEU Papineni et al. (2002) points.

Due to space constraint, we show all results for the six language pairs with
the largest improvement over the baseline in Figure 2. Comparing the data
augmentation methods, our morphologically aware approach yields a better score
than the naive one in all cases except for Galician. We find that the
augmentation is consistently beneficial for Irish and Galician, regardless of
how much data we add. But for other pairs adding more synthetic data does not
lead to sustained improvements.

Table A.1 in the Appendix shows our experimental results on all 14 language
pairs from English. We use two pre-trained models: _DeltaLM-Base_ and
_DeltaLM-Large_. We tried to use DeltaLM-Large for all language pairs, but
low-resource language pairs quickly overfit on the large model and do not
generalize well. Apart from Armenian (hye), we get a higher BLEU score with
our settings for all other language pairs. The improvement margin is negligent
in languages where the baseline system is already very bad. Languages like
Wolof (wol) and Uyghur (uig) have baseline BLEU scores of less than 2, showing
us that our parallel seed data is not of good quality. For languages, Kazakh
(kaz), Marathi (mar), and Tamil (tam), all rather low-resource languages, the
improvement is less than 0.5 BLEU points, but it ranges from 0.79 to 3.21 BLEU
score for all other languages. We also observe a similar trend of improvement
in the case of adding more noisy data. The score improves to the highest
point, but as more synthetic data is added the system gets worse.

#### To English

As before we show the best-performing six language pairs in the X-English
direction in Figure 3. Unlike English-X, the patterns here are the same for
all language pairs. In all 14 languages except for Armenian our approach
improves upon the baseline, and the morphologically-informed method is better
than the naive approach. In every case adding more synthetic data after a
while does not lead to more improvements.

Table A.2 in the Appendix lists our results on all 14 language pairs in the
to-English direction. The BLEU scores are generally higher in this setting, as
the pre-trained model has seen a lot of English data on the target side. Apart
from Armenian (hye), we get a higher BLEU score with our settings for all
other language pairs, the same as before. The improvement margin is negligent
for Greek (ell) and Maltese (mlt), showing that the language has no room for
improvement through this type of augmentation, as the models are already
fairly good. For Wolof (wol) and Uyghur (uig), our improvement is less than 1
BLEU score. This is also the same as before, showing that the parallel data
for these languages is not of high quality. The improvement ranges from 1.12
to 4.24 BLEU scores for all other languages. We also see a similar trend as in
the From-English direction; after some point, the more synthetic data we add,
the system worsens; most improvements are obtained with 5K and 10K synthetic
examples.

##  5 Analysis

#### Is the performance tied to any single component?

We perform an ablation study to find out which component of the model is
responsible for the performance boost. We work on this experiment with five
thousand synthetic data and the scottish gaelic-english direction. We compare
three different components of our pipeline:

  * •

Does the number of the generated synthetic data matter? For this setup,
instead of creating three synthetic sentences from each seed sentence, we
create 30 synthetic sentences from each seed sentence. We refer to this setup
as a “5K Number”.

  * •

Does the length of the seed sentences matter? We create synthetic sentences
from seed sentences with less than seven tokens for this setup. We refer to
this setup as a “5K Length”.

  * •

Does the choice of alignment model play any role? Instead of aligning the seed
sentences with fast$\\_$align, we use awesome-align using “bert-base-
multilingual-cased” for this setup. We refer to this setup as a “5K Align”.

Ablations on Scottish Gaelic-English
---
| Ours | Naive
5K | 13.32 | 13.05
5K Number | 12.76 | 13.13
5K Length | 12.58 | 12.28
5K Align | 12.14 | 12.62
Table 2: Ablation result of the importance of different components of our
method. If we don’t use one of the components, the BLEU score drops
significantly.

Table 2 shows the results of these experiments. The main takeaway is that
every component of our method is necessary to boost scores: scores decrease
when we replace one component. The most important for low-resource languages
is to use a compatible alignment model. As large multilingual pre-trained
models do not represent them very well, and awesome_align relies on such a
model, we are better off using fast_align. The number of generated synthetic
data also matters as we anticipated. The reason is that when we create a huge
sentence pool and sample a small number of sentences from there, the number of
unique seed sentences that contribute to the synthetic data is reduced. We
also confirm that the length of the sentence matters for Stanza: the shorter
the sentence is, the less context it has, thus reducing the quality of the
morphological analysis and consequently of our synthetic sentences.

#### Does the number of seed sentences or the amount of new vocabulary matter?

To do this experiment, we work again on the GLA-ENG direction to create five
thousand synthetic data. We build four different models:

  * •

5K: This is the original five thousand-size synthetic dataset we created. We
create three sentences from each seed sentence and randomly choose words from
all candidate words for replacement.

  * •

5K (one): In this setup, we try to create 5000 sentences from only one seed
sentence and randomly choose words from all candidate words for replacement.
However, our process could not generate 5,000 unique synthetic sentences from
one seed sentence; instead, it took five seed sentences to generate 5,000
synthetic sentences.

  * •

5K (half): In this setup, we create three sentences from each seed sentence
and randomly choose words from the first half of the candidate words for
replacement.

  * •

5K (remove): In this setup, we create ten sentences from each seed sentence
and randomly choose words from all candidate words for replacement, but when
we choose a word, we remove that word as a candidate so that it is not chosen
again. Ideally, we would have sentences of the same amount of the lexicon
vocabulary.

| Scottish Gaelic-English Ours | Scottish Gaelic-English Naive
---|---|---
|  | # ENG | # GLA | # seed |  | # ENG | # GLA | # seed
| BLEU | types | types | sentences | BLEU | types | types | sentences
0K | 9.08 | 11077 | 13826 | 0 | 9.08 | 11077 | 13836 | 0
5K (one) | 12.79 | 11269 | 14020 | 5 | 12.58 | 12588 | 15883 | 5
5K | 13.32 | 12763 | 15847 | 1511 | 13.05 | 12057 | 15082 | 1512
5K (half) | 13.13 | 12367 | 15216 | 1527 | 12.57 | 11811 | 14740 | 1508
5K (remove) | 12.91 | 12528 | 15702 | 1909 | 13.29 | 12511 | 15694 | 1909
Table 3: Ablation about the number of new vocabulary introduced and the number
of seed sentences used to create five thousand synthetic data. Using as little
as five seed sentences boosts the 3.71 BLEU score.

Table 3 shows these experiments’ results. An exciting result is the one for 5K
(one), where we use only five seed sentences to create five thousand synthetic
sentences. In doing so, we introduce 200 new words, but we get a substantial
jump of 3.71 BLEU score, which shows the promise of our method. Even if we
have few high-quality parallel sentences and a good-quality lexicon, our
method is bound to boost MT system quality.

#### How much does filtering with LM help?

To do this experiment, we perform a control experiment to contrast with our
filtering-with-LM-perplexity approach. In the control setting, we choose
sentences randomly from the pool of synthetic sentences. We randomly select a
subset of a hundred thousand seed sentences from the OPUS-100 ENG, ELL dataset
and do ablation on both ENG-ELL and ELL-ENG directions.

Pairs | 0K | 5K | 10K | 50K | 100K | 200K
---|---|---|---|---|---|---
English-Modern Greek (random) | 14.24 | 13.98 | 5.01 | 15.7 | 14.92 | 4.23
English-Modern Greek (filtered) | 14.62 | 15.14 | 15.2 | 14.99 | 15.54
Modern Greek-English (random) | 11.6 | 12.2 | 10.24 | 18.34 | 9.21 | 12.42
Modern Greek-English (filtered) | 16.91 | 17.25 | 17.05 | 19.01 | 17.64
Table 4: Filtering the synthetic data leads to consistent improvements, but
random data sampling leads to unstable results. Instead, the BLEU score drops
randomly for a random approach.

Table 4 shows these experiments’ results. In the random setting, the results
are rather unstable, with very low BLEU scores for some settings. This could
be because we might be randomly choosing bad sentences from the pool. The
results with informed sentence selection (through perplexity), instead, are
stable and consistently improving.

##  6 Related Work

Dictionaries have been and are indispensable resources in various applications
in NLP Wilson et al. (2020); Wang et al. (2019); Xiao and Guo (2014). More
specifically, many previous studies use dictionaries in MT to improve
translation quality for low-resource languages with or without monolingual or
parallel corpora. A closely related task is bilingual lexicon induction that
departs from an unsupervised MT task where no parallel resources, including
the ground-truth bilingual lexicon, are incorporated Artetxe et al. (2017);
Lample et al. (2018b). The bilingual lexicon is often utilized as a seed in
bilingual lexicon induction that aims to induce more word pairs within the
language pair Mikolov et al. (2013). Another utilization of the bilingual
lexicon is for translating low-frequency words in supervised neural MT Arthur
et al. (2016); Zhang and Zong (2016).

On the usage of dictionaries in MT, Peng et al. (2020) employ dictionaries for
cross-lingual MT, Fadaee et al. (2017) propose a data augmentation approach to
target low-frequency words by generating sentence pairs containing rare words,
Duan et al. (2020) use dictionaries to drive the semantic spaces of the source
and target languages becoming closer in MT training without parallel sentences
and Wang et al. (2022b) explore the utilization of dictionaries for
synthesizing textual or labeled data, focusing on tasks such as named entity
recognition and part-of-speech tagging.

Unlike many of the previous approaches that are fixated on only monolingual
data, our approach considers using a bilingual lexicon and maintaining
morphology in augmentation. Our approach is similar in spirit to Fadaee et al.
(2017) technique with additional consideration of morphological complexity in
the synthetic data augmentation process. Also, inspired by Wang et al.
(2022b)’s approach, our research shares a common thread by using different
strategies for synthesizing data using lexicons and integrating such data with
monolingual or parallel text when accessible. Both studies aim to leverage
lexicons to enhance various NLP tasks, albeit in different contexts.

##  7 Conclusion

Our approaches have proven beneficial for most of the 14 languages under
investigation, except for Armenian. Even if the improvements in BLEU scores
may be small for some languages, there is a noticeable boost in most.
Interestingly, we observed improvements even in language pairs (e.g., wol-eng,
eng-kmr, eng-gla) with unsatisfactory initial baseline scores. This
observation suggests our approach can enhance performance even in more
challenging scenarios. The results also highlight the importance of obtaining
high-quality seed sentences. We found that as few as five good-quality seed
data points can contribute to creating five thousand synthetic data samples of
good quality that would boost performance. This data augmentation process
could play a vital role in improving the overall performance of machine
translation systems and be combined with other augmentation techniques (e.g.,
back-translation) as we deem it orthogonal to them.111111We note that back-
translation is rarely effective in most extremely low-resource languages due
to the abysmal quality of the initial systems.

#### Future Work

In our current work, we focused on conducting morphological inflection
exclusively on the English side of the translation task. The main reason for
this choice was the availability of a reliable morphological inflector
specifically designed for English. However, we encountered challenges when
applying the same approach to other languages. We lacked suitable
morphological inflection tools for those languages, or the accuracy of the
available tools did not meet our requirements. Incorporating these tools would
have posed a significant bottleneck to the effectiveness of our approach. For
future research, we aim to explore how our approach can be extended by
performing morphological inflection in other languages. This involves
developing or obtaining accurate and reliable morphological inflection tools.

##  8 Limitations

One of the limitations of our current approach is the use of the Stanza model.
Since bilingual lexicons have no context, relying solely on morphological
features obtained from the lexicon results in more general features. This can
be particularly challenging in morphologically rich languages, where a single
word can have multiple meanings depending on the sentence context. Another
limitation is the language support provided by the Stanza model, which is
currently limited to 60 languages. This constraint restricts the applicability
of our approach to only those languages supported by Stanza. To expand our
work to languages not supported by Stanza, it is necessary to create custom
Stanza models specifically tailored for those languages. This process requires
additional time and effort to develop and validate the models for each
language of interest.

## References

  * Alabi et al. (2022) Jesujoba O. Alabi, David Ifeoluwa Adelani, Marius Mosbach, and Dietrich Klakow. 2022\.  Adapting pre-trained language models to African languages via multilingual adaptive fine-tuning.  In _Proceedings of the 29th International Conference on Computational Linguistics_ , pages 4336–4349, Gyeongju, Republic of Korea. International Committee on Computational Linguistics.
  * Artetxe et al. (2017) Mikel Artetxe, Gorka Labaka, Eneko Agirre, and Kyunghyun Cho. 2017.  Unsupervised neural machine translation.  _CoRR_ , abs/1710.11041.
  * Arthur et al. (2016) Philip Arthur, Graham Neubig, and Satoshi Nakamura. 2016.  Incorporating discrete translation lexicons into neural machine translation.  In _Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing_ , pages 1557–1567, Austin, Texas. Association for Computational Linguistics.
  * Bapna and Firat (2019) Ankur Bapna and Orhan Firat. 2019.  Simple, scalable adaptation for neural machine translation.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing, EMNLP-IJCNLP 2019, Hong Kong, China, November 3-7, 2019_ , pages 1538–1548. Association for Computational Linguistics.
  * Batsuren et al. (2022) Khuyagbaatar Batsuren, Omer Goldman, Salam Khalifa, Nizar Habash, Witold Kieraś, Gábor Bella, Brian Leonard, Garrett Nicolai, Kyle Gorman, Yustinus Ghanggo Ate, Maria Ryskina, Sabrina Mielke, Elena Budianskaya, Charbel El-Khaissi, Tiago Pimentel, Michael Gasser, William Abbott Lane, Mohit Raj, Matt Coler, Jaime Rafael Montoya Samame, Delio Siticonatzi Camaiteri, Esaú Zumaeta Rojas, Didier López Francis, Arturo Oncevay, Juan López Bautista, Gema Celeste Silva Villegas, Lucas Torroba Hennigen, Adam Ek, David Guriel, Peter Dirix, Jean-Philippe Bernardy, Andrey Scherbakov, Aziyana Bayyr-ool, Antonios Anastasopoulos, Roberto Zariquiey, Karina Sheifer, Sofya Ganieva, Hilaria Cruz, Ritván Karahóǧa, Stella Markantonatou, George Pavlidis, Matvey Plugaryov, Elena Klyachko, Ali Salehi, Candy Angulo, Jatayu Baxi, Andrew Krizhanovsky, Natalia Krizhanovskaya, Elizabeth Salesky, Clara Vania, Sardana Ivanova, Jennifer White, Rowan Hall Maudslay, Josef Valvoda, Ran Zmigrod, Paula Czarnowska, Irene Nikkarinen, Aelita Salchak, Brijesh Bhatt, Christopher Straughn, Zoey Liu, Jonathan North Washington, Yuval Pinter, Duygu Ataman, Marcin Wolinski, Totok Suhardijanto, Anna Yablonskaya, Niklas Stoehr, Hossep Dolatian, Zahroh Nuriah, Shyam Ratan, Francis M. Tyers, Edoardo M. Ponti, Grant Aiton, Aryaman Arora, Richard J. Hatcher, Ritesh Kumar, Jeremiah Young, Daria Rodionova, Anastasia Yemelina, Taras Andrushko, Igor Marchenko, Polina Mashkovtseva, Alexandra Serova, Emily Prud’hommeaux, Maria Nepomniashchaya, Fausto Giunchiglia, Eleanor Chodroff, Mans Hulden, Miikka Silfverberg, Arya D. McCarthy, David Yarowsky, Ryan Cotterell, Reut Tsarfaty, and Ekaterina Vylomova. 2022.  UniMorph 4.0: Universal Morphology.  In _Proceedings of the Thirteenth Language Resources and Evaluation Conference_ , pages 840–855, Marseille, France. European Language Resources Association.
  * Duan et al. (2020) Xiangyu Duan, Baijun Ji, Hao Jia, Min Tan, Min Zhang, Boxing Chen, Weihua Luo, and Yue Zhang. 2020.  Bilingual dictionary based neural machine translation without using parallel sentences.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, ACL 2020, Online, July 5-10, 2020_ , pages 1570–1579. Association for Computational Linguistics.
  * Duh et al. (2020) Kevin Duh, Paul McNamee, Matt Post, and Brian Thompson. 2020.  Benchmarking neural and statistical machine translation on low-resource African languages.  In _Proceedings of the Twelfth Language Resources and Evaluation Conference_ , pages 2667–2675, Marseille, France. European Language Resources Association.
  * Dyer et al. (2013) Chris Dyer, Victor Chahuneau, and Noah A. Smith. 2013.  A simple, fast, and effective reparameterization of IBM model 2.  In _Proceedings of the 2013 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 644–648, Atlanta, Georgia. Association for Computational Linguistics.
  * Edunov et al. (2018a) Sergey Edunov, Myle Ott, Michael Auli, and David Grangier. 2018a.  Understanding back-translation at scale.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, Brussels, Belgium, October 31 - November 4, 2018_ , pages 489–500. Association for Computational Linguistics.
  * Edunov et al. (2018b) Sergey Edunov, Myle Ott, Michael Auli, and David Grangier. 2018b.  Understanding back-translation at scale.
  * Fadaee et al. (2017) Marzieh Fadaee, Arianna Bisazza, and Christof Monz. 2017.  Data augmentation for low-resource neural machine translation.  In _Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)_ , pages 567–573, Vancouver, Canada. Association for Computational Linguistics.
  * Koehn and Knight (2001) Philipp Koehn and Kevin Knight. 2001.  Knowledge sources for word-level translation models.  In _Proceedings of the 2001 Conference on Empirical Methods in Natural Language Processing_.
  * Koehn and Knight (2002) Philipp Koehn and Kevin Knight. 2002.  Learning a translation lexicon from monolingual corpora.  In _Proceedings of the ACL-02 Workshop on Unsupervised Lexical Acquisition_ , pages 9–16, Philadelphia, Pennsylvania, USA. Association for Computational Linguistics.
  * Lample et al. (2018a) Guillaume Lample, Alexis Conneau, Ludovic Denoyer, and Marc’Aurelio Ranzato. 2018a.  Unsupervised machine translation using monolingual corpora only.  In _6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Conference Track Proceedings_. OpenReview.net.
  * Lample et al. (2018b) Guillaume Lample, Myle Ott, Alexis Conneau, Ludovic Denoyer, and Marc’Aurelio Ranzato. 2018b.  Phrase-based & neural unsupervised machine translation.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 5039–5049, Brussels, Belgium. Association for Computational Linguistics.
  * Ma et al. (2021) Shuming Ma, Li Dong, Shaohan Huang, Dongdong Zhang, Alexandre Muzio, Saksham Singhal, Hany Hassan Awadalla, Xia Song, and Furu Wei. 2021.  Deltalm: Encoder-decoder pre-training for language generation and translation by augmenting pretrained multilingual encoders.  _CoRR_ , abs/2106.13736.
  * Mikolov et al. (2013) Tomás Mikolov, Quoc V. Le, and Ilya Sutskever. 2013.  Exploiting similarities among languages for machine translation.  _CoRR_ , abs/1309.4168.
  * Papineni et al. (2002) Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. 2002.  Bleu: a method for automatic evaluation of machine translation.  In _Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics_ , pages 311–318, Philadelphia, Pennsylvania, USA. Association for Computational Linguistics.
  * Peng et al. (2020) Wei Peng, Chongxuan Huang, Tianhao Li, Yun Chen, and Qun Liu. 2020.  Dictionary-based data augmentation for cross-domain neural machine translation.  _CoRR_ , abs/2004.02577.
  * Radford et al. (2019) Alec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. 2019.  Language models are unsupervised multitask learners.
  * Sánchez-Cartagena et al. (2021) Víctor M. Sánchez-Cartagena, Miquel Esplà-Gomis, Juan Antonio Pérez-Ortiz, and Felipe Sánchez-Martínez. 2021\.  Rethinking data augmentation for low-resource neural machine translation: A multi-task learning approach.  In _Proceedings of the 2021 Conference on Empirical Methods in Natural Language Processing, EMNLP 2021, Virtual Event / Punta Cana, Dominican Republic, 7-11 November, 2021_ , pages 8502–8516. Association for Computational Linguistics.
  * Sánchez-Cartagena et al. (2011) Víctor M. Sánchez-Cartagena, Felipe Sánchez-Martínez, and Juan Antonio Pérez-Ortiz. 2011.  Integrating shallow-transfer rules into phrase-based statistical machine translation.  In _Proceedings of Machine Translation Summit XIII: Papers_ , Xiamen, China.
  * Sennrich et al. (2016a) Rico Sennrich, Barry Haddow, and Alexandra Birch. 2016a.  Improving neural machine translation models with monolingual data.  In _Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics, ACL 2016, August 7-12, 2016, Berlin, Germany, Volume 1: Long Papers_. The Association for Computer Linguistics.
  * Sennrich et al. (2016b) Rico Sennrich, Barry Haddow, and Alexandra Birch. 2016b.  Neural machine translation of rare words with subword units.  In _Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics, ACL 2016, August 7-12, 2016, Berlin, Germany, Volume 1: Long Papers_. The Association for Computer Linguistics.
  * Team et al. (2022) NLLB Team, Marta R. Costa-jussà, James Cross, Onur Çelebi, Maha Elbayad, Kenneth Heafield, Kevin Heffernan, Elahe Kalbassi, Janice Lam, Daniel Licht, Jean Maillard, Anna Sun, Skyler Wang, Guillaume Wenzek, Al Youngblood, Bapi Akula, Loic Barrault, Gabriel Mejia Gonzalez, Prangthip Hansanti, John Hoffman, Semarley Jarrett, Kaushik Ram Sadagopan, Dirk Rowe, Shannon Spruit, Chau Tran, Pierre Andrews, Necip Fazil Ayan, Shruti Bhosale, Sergey Edunov, Angela Fan, Cynthia Gao, Vedanuj Goswami, Francisco Guzmán, Philipp Koehn, Alexandre Mourachko, Christophe Ropers, Safiyyah Saleem, Holger Schwenk, and Jeff Wang. 2022.  No language left behind: Scaling human-centered machine translation.
  * Tyers (2009) Francis M. Tyers. 2009.  Rule-based augmentation of training data in Breton-French statistical machine translation.  In _Proceedings of the 13th Annual Conference of the European Association for Machine Translation_ , Barcelona, Spain. European Association for Machine Translation.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017.  Attention is all you need.  _CoRR_ , abs/1706.03762.
  * Wang et al. (2019) Qi Wang, Yangming Zhou, Tong Ruan, Daqi Gao, Yuhang Xia, and Ping He. 2019.  Incorporating dictionaries into deep neural networks for the Chinese clinical named entity recognition.  _Journal of biomedical informatics_ , 92:103133.
  * Wang et al. (2022a) Xinyi Wang, Sebastian Ruder, and Graham Neubig. 2022a.  Expanding pretrained models to thousands more languages via lexicon-based adaptation.  In _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 863–877, Dublin, Ireland. Association for Computational Linguistics.
  * Wang et al. (2022b) Xinyi Wang, Sebastian Ruder, and Graham Neubig. 2022b.  Expanding pretrained models to thousands more languages via lexicon-based adaptation.
  * Wilson et al. (2020) Steven Wilson, Walid Magdy, Barbara McGillivray, Kiran Garimella, and Gareth Tyson. 2020.  Urban dictionary embeddings for slang NLP applications.  In _Proceedings of the Twelfth Language Resources and Evaluation Conference_ , pages 4764–4773.
  * Wolf et al. (2020) Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander M. Rush. 2020.  Huggingface’s transformers: State-of-the-art natural language processing.
  * Wu et al. (2008) Hua Wu, Haifeng Wang, and Chengqing Zong. 2008.  Domain adaptation for statistical machine translation with domain dictionary and monolingual corpora.  In _Proceedings of the 22nd International Conference on Computational Linguistics (Coling 2008)_ , pages 993–1000, Manchester, UK. Coling 2008 Organizing Committee.
  * Xiao and Guo (2014) Min Xiao and Yuhong Guo. 2014.  Distributed word representation learning for cross-lingual dependency parsing.  In _Proceedings of the Eighteenth Conference on Computational Natural Language Learning, CoNLL 2014, Baltimore, Maryland, USA, June 26-27, 2014_ , pages 119–129. ACL.
  * Zhang et al. (2020) Biao Zhang, Philip Williams, Ivan Titov, and Rico Sennrich. 2020.  Improving massively multilingual neural machine translation and zero-shot translation.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 1628–1639, Online. Association for Computational Linguistics.
  * Zhang and Zong (2016) Jiajun Zhang and Chengqing Zong. 2016.  Bridging neural machine translation and bilingual dictionaries.  _CoRR_ , abs/1610.07272.

##  Appendix A Appendix

Pairs | 0K (untagged) | 5K (tagged) | 10K (tagged) | 50K (tagged) | 100K (tagged) | 200K (tagged) | $\Delta$
---|---|---|---|---|---|---|---
DeltaLM-Base |
ENG-GLA Naive | 4.21 | 5.16 | 5.07 | 5.5 | 5.31 | 4.78 | 1.41
ENG-GLA Ours | 4.75 | 5.22 | 5.62 | 5.13 | 4.97
ENG-HYE Naive | 5.61 | 4.86 | 4.65 | 2.91 | 2.75 | 2.11 | 0.0
ENG-HYE Ours | 4.28 | 4.52 | 3.3 | 2.97 | 2.83
ENG-KAZ Naive | 6.13 | 6.34 | 5.93 | 5.16 | 5.39 | 4.58 | 0.37
ENG-KAZ Ours | 6.28 | 6.5 | 5.27 | 5.68 | 4.79
ENG-KMR Naive | 1.66 | 2.36 | 2.05 | 2.12 | 1.66 | 1.82 | 0.79
ENG-KMR Ours | 2.45 | 2.27 | 2.32 | 1.89 | 1.78
ENG-WOL Naive | 1.08 | 0.78 | 1.03 | 1.19 | 1.05 | 0.94 | 0.12
ENG-WOL Ours | 1.08 | 1.13 | 1.11 | 1.2 | 1.02
DeltaLM-Large |
ENG-ELL Naive | 23.06 | 22.2 | 23.09 | 22.17 | 22.55 | 23.01 | 0.03
ENG-ELL Ours | 22.17 | 21.93 | 22.24 | 22.64 | 23.07
ENG-GLE Naive | 19.13 | 19.89 | 19.97 | 20.3 | 20.58 | 21.52 | 3.21
ENG-GLE Ours | 20.3 | 19.94 | 20.44 | 20.9 | 22.34
ENG-GLG Naive | 29.86 | 31.42 | 31.41 | 31.34 | 31.68 |  | 1.82
ENG-GLG Ours | 30.97 | 31.27 | 31.13 | 31.46 |
ENG-HIN Naive | 22.48 | 23.2 | 22.53 | 22.72 | 22.35 | 22.05 | 0.96
ENG-HIN Ours | 23.29 | 23.44 | 22.42 | 22.17 | 21.17
ENG-MAR Naive | 5.03 | 4.86 | 4.64 | 4.55 | 4.54 | 4.04 | 0.44
ENG-MAR Ours | 5.47 | 5.35 | 4.76 | 4.72 | 4.34
ENG-MLT Naive | 38.03 | 39.86 | 39.77 | 39.64 | 39.41 | 39.24 | 2.08
ENG-MLT Ours | 40.11 | 39.08 | 39.52 | 38.69 | 39.43
ENG-TAM Naive | 5.3 | 5.64 | 5.33 | 5.35 | 5.10 | 5.58 | 0.46
ENG-TAM Ours | 5.44 | 5.76 | 5.28 | 5.48 | 5.34
ENG-URD Naive | 11.26 | 12.06 | 12.31 | 12.03 | 11.98 | 11.12 | 1.11
ENG-URD Ours | 12.24 | 12.37 | 11.45 | 11.95 | 11.07
ENG-UIG Naive | 1.24 | 0.7 | 0.88 | 0.76 | 0.87 |  | 0.04
ENG-UIG Ours | 0.63 | 1.06 | 1.17 | 1.28 |
Table A.1: BLEU score of 9 languages from ENG-X direction. Columns indicate the amount of synthetic parallel data we use. The 0K (untagged) column is our baseline. The rows indicating Naive is the approach where we replace words of the same POS tag. The rows indicating Ours is the approach where we replace words of the same morphological feature. $\Delta$ is the difference between the baseline and the best model’s score. $\Delta$ is 0.0 if the baseline is the best model.  Pairs | 0K (untagged) | 5K (tagged) | 10K (tagged) | 50K (tagged) | 100K (tagged) | 200K (tagged) | $\Delta$
---|---|---|---|---|---|---|---
DeltaLM-Base |
HYE-ENG Naive | 16.04 | 15.78 | 13.84 | 9.71 | 8.39 | 7.79 | 0.0
HYE-ENG Ours | 15.59 | 14.63 | 10.11 | 9.24 | 8.72
WOL-ENG Naive | 1.83 | 2.59 | 2.24 | 2.09 | 1.53 | 1.21 | 0.76
WOL-ENG Ours | 2.31 | 2.22 | 1.71 | 1.55 | 1.1
DeltaLM-Large |
GLA-ENG Naive | 9.08 | 13.05 | 11.81 | 12.99 | 12.36 | 11.86 | 4.24
GLA-ENG Ours | 13.32 | 12.76 | 12.32 | 13.08 | 12.35
KAZ-ENG Naive | 19.58 | 20.97 | 20.74 | 18.86 | 17.99 | 16.59 | 1.72
KAZ-ENG Ours | 20.78 | 21.3 | 20.52 | 19.28 | 18.45
KMR-ENG Naive | 9.73 | 11.63 | 11.08 | 11.04 | 9.45 | 9.19 | 2.28
KMR-ENG Ours | 11.97 | 12.01 | 10.87 | 10.35 | 9.43
ELL-ENG Naive | 31.94 | 32.33 | 32.32 | 32.34 | 32.35 | 31.98 | 0.42
ELL-ENG Ours | 32.36 | 31.88 | 32.34 | 32.23 | 32.1
GLE-ENG Naive | 28.71 | 30.03 | 29.54 | 28.65 | 28.73 | 28.57 | 1.32
GLE-ENG Ours | 29.96 | 29.7 | 30.03 | 28.99 | 29.36
GLG-ENG Naive | 37.07 | 37.89 | 38.01 | 37.89 | 38.02 |  | 1.12
GLG-ENG Ours | 37.84 | 38.19 | 38.02 | 37.61 |
HIN-ENG Naive | 29.09 | 30.89 | 30.34 | 30.81 | 30.46 | 29.71 | 1.96
HIN-ENG Ours | 31.05 | 30.74 | 30.58 | 29.6 | 29.62
MAR-ENG Naive | 22.96 | 23.66 | 23.36 | 22.1 | 21.22 | 20 | 1.15
MAR-ENG Ours | 24.11 | 23.54 | 22.05 | 21.78 | 19.92
MLT-ENG Naive | 45.21 | 45.2 | 45.43 | 45.23 | 45.23 | 44.9 | 0.44
MLT-ENG Ours | 45.65 | 45.07 | 44.87 | 44.94 | 44.7
TAM-ENG Naive | 19.83 | 20.72 | 20.61 | 19.9 | 18.98 | 18.11 | 1.47
TAM-ENG Ours | 21.3 | 20.69 | 19.75 | 19.25 | 18.63
URD-ENG Naive | 18.63 | 19.95 | 19.81 | 19.75 | 19 | 18.12 | 1.33
URD-ENG Ours | 19.96 | 19.39 | 19.15 | 19.44 | 18.54
UIG-ENG Naive | 10.49 | 11.08 | 11.27 | 10.67 | 9.56 | 8.77 | 0.83
UIG-ENG Ours | 11.32 | 11.31 | 10.71 | 9.76 | 8.18
Table A.2: BLEU score of 9 languages from X-ENG direction. Columns indicate
the amount of synthetic parallel data we use. The 0K (untagged) column is our
baseline. The rows indicating Naive is the approach where we replace words of
the same POS tag. The rows indicating Ours is the approach where we replace
words of the same morphological features. $\Delta$ is the difference between
the baseline and the best model’s score. $\Delta$ is 0.0 if the baseline is
the best model.


# Laser-free method for creation of two-mode squeezed state and beam-splitter
transformation with trapped ions

Bogomila S. Nikolova  Department of Physics, St. Kliment Ohridski University
of Sofia, James Bourchier 5 blvd, 1164 Sofia, Bulgaria  Peter A. Ivanov
Department of Physics, St. Kliment Ohridski University of Sofia, James
Bourchier 5 blvd, 1164 Sofia, Bulgaria

###### Abstract

We propose a laser-free method for creation of a phonon two-mode squeezed
state and a beam-splitter transformation, using time-varying electric fields
and non-linear couplings between the normal modes in a linear ion crystal.
Such non-linear Coulomb-mediated interactions between the collective
vibrational modes arise under specific trap-frequency conditions in an ion
trap. We study the quantum metrological capability for parameter estimation of
the two quantum states and show that a Heisenberg limit of precision can be
achieved when the initial state with $n$ phonons evolves under the action of
the beam-splitter transformation. Furthermore, we show that the phonon non-
linearity and the spin-dependent force can be used for creation of a three-
qubit Fredkin gate.

##  I Introduction

One of the most promising quantum platforms with applications in the
realization of elementary quantum processors and high-precision quantum
metrology is the system of laser-cooled trapped ions Blatt2008 ; Leibfried2003
. The balance between the Coulomb repulsion and the harmonic confinement leads
to the formation of an ion crystal, where the ions undergo small oscillations
around their equilibrium positions. Within the harmonic approximation, the
ion’s vibration can be expressed in terms of uncoupled collective vibrational
modes James1998 . However, under a given resonance trap-frequency condition,
the higher-order terms, that arise due to the Coulomb interaction, cannot be
neglected, which causes coupling between the different collective vibrational
modes Marquet2003 . Such a non-linear cross-Kerr coupling between the
collective modes was experimentally observed in a system of two trapped ions
Roos2008 ; Nie2009 . Recently, a quantum simulation of a trilinear Hamiltonian
was demonstrated with two and three trapped ions Ding2018 ; Maslennikov ;
Ding2017 ; Ding2017_1 .

Here, we propose a _laser-free_ method for the creation of two-mode squeezed
state by using non-linear interaction between the vibrational modes, in which
one phonon in the axial $a$-mode is converted into one phonon in the radial
$b$\- and $c$-modes, and vice versa. We show that, by applying a time-varying
electric field on $a$-mode, the energy exchange between the latter mode and
the other two radial modes can be suppressed, which causes two-mode squeezing
between $b$\- and $c$\- modes. The essence of this technique is that it relies
only on using electric field, instead of laser fields with specific laser
frequency conditions Cardoso2021 , which thus simplifies the physical
realization. Furthermore, we show that the same technique can be used for the
creation of a beam-splitter transformation. In that case the time-varying
electric field is applied on the $b$-mode and causes an effective beam-
splitter transformation between $a$\- and $c$-modes.

We discuss the metrological properties of the two-mode squeezed state and the
beam-splitter transformation. Recently, a parameter estimation of the non-
linear coupling was studied in Ivanov2022 . Here, we show that the magnitude
of the two-mode squeezed state can be estimated by measuring the phonon state
probabilities. We show that, using beam-splitter transformation, one can
significantly enhance the parameter estimation to a sub-Heisenberg limit of
precision. In that case, the initial state, which contains $n$ phonons in each
mode, evolves unitarily under the action of the beam-splitter transformation.
We quantify the estimation precision in terms of quantum Fisher information
(QFI). We show that the measurement uncertainty can reach a sub-Heisenberg
limit of sensitivity, where the QFI scales as $n^{2}+n$. We discuss the
optimal measurement basis in which the quantum Cramér-Rao bound is saturated,
and show that the measurement of the phonon state probabilities of one of the
vibrational modes leads to equality between the classical and quantum Fisher
information.

Finally, we extend the discussion to the creation of conditional beam-splitter
transformation. In that case a spin-dependent force acts either on the radial
$b$\- or the radial $c$\- mode. As an example, we propose physical realization
of a quantum Fredkin gate. Such a three-qubit quantum gate was recently
experimentally implemented with trapped ions Gan2020 . We show that the phonon
non-linear coupling and the spin-dependent force can be used for the
realization of a Fredkin gate, where the control qubit is composed of the
internal states of the trapped ion, and the target qubits are the Fock states
of the $a$\- and $c$\- vibrational modes. We show that our method can be used
for the creation of high-fidelity spin-dependent N00N states.

The paper is organized as follows: In Section II we provide theoretical
framework for non-linear phonon-phonon interaction that can arise due to the
higher-order terms in the expansion of the Coulomb potential. In Sections III
and IV we propose physical realization of a two-mode squeezed state and a
beam-splitter transformation based on the combination of time-varying electric
fields and phonon-phonon coupling. The quantum metrological properties of the
two-mode squeezed state and the beam-splitter transformation are discussed in
Sec. V. We show that, for initial state with $n$ phonons in each mode, one can
achieve a sub-Heisenberg limit of precision. In Sec. VI the realization of a
quantum Fredkin gate, based on a spin-dependent force and non-linear phonon
interaction is discussed. Finally, the conclusions are presented in Sec. VII.

##  II Non-linear phonon-phonon interaction

Here, we briefly discuss the intrinsic motional non-linearity which arises due
to the long-range Coulomb interaction between the trapped ions. We consider a
system of $N$ ions with electric charge $e$ and mass $m$, confined in a linear
Paul trap. The total potential energy of the system includes the harmonic
confinement and mutual Coulomb repulsion. It reads Leibfried2003 ; James1998 ;
Marquet2003

$\hat{V}=\frac{m}{2}\sum_{\chi=x,y,z}\sum_{l=1}^{N}\omega_{\chi}^{2}\hat{r}^{2}_{\chi,l}+\sum_{l>k}\frac{e^{2}}{4\pi\epsilon_{0}|\hat{\textbf{r}}_{l}-\hat{\textbf{r}}_{k}|}.$
(1)

Here $\hat{\textbf{r}}_{l}=(\hat{r}_{x,l},\hat{r}_{y,l},\hat{r}_{z,l})$ is the
position vector for ion $l$ and $\epsilon_{0}$ is the permittivity of free
space. The trap frequencies are $\omega_{\chi}$ ($\chi=x,y,z$), where we
assume that the transverse trap frequencies $\omega_{x,y}$ are much larger
than the axial trap frequency $\omega_{z}$. This condition ensures that each
ion occupies equilibrium position
$\hat{\textbf{r}}^{(0)}_{l}=(0,0,z^{(0)}_{l})$ along the $z$ trap axis. The
small displacement around an ion’s equilibrium position is denoted by
$\delta\hat{r}_{\chi,l}$. Within the harmonic approximation, the Hamiltonian
for the ion’s vibration is given by

$\hat{H}_{0}=\sum_{\chi=x,y,z}\sum_{n=1}^{N}\left\\{\frac{\hat{P}^{2}_{\chi,n}}{2m}+\frac{m\omega^{2}_{\chi,n}}{2}\hat{Q}^{2}_{\chi,n}\right\\},$
(2)

where $\omega_{\chi,n}$ is the collective vibration frequency along the
spatial direction $\chi$. Here we have introduced the normal-mode coordinate
operators
$\hat{Q}_{\chi,n}=\sum_{l=1}^{N}M^{\chi}_{l,n}\delta\hat{r}_{\chi,l}$ and the
corresponding conjugate momentum operators
$\hat{P}_{\chi,n}=\sum_{l=1}^{N}M^{\chi}_{l,n}\hat{p}_{\chi,l}$, with
$M^{\chi}_{l,n}$ being the amplitude of the normal mode $n$ on ion $l$. The
Hamiltonian (2) indicates that, up to lowest-order harmonic approximation, the
vibrational modes are decoupled.

Higher-order terms in the Taylor-expansion of the Coulomb potential give rise
to the coupling between different vibration modes. The non-linear Hamiltonian
reads Marquet2003 ,

$\displaystyle\hat{H}_{\rm nl}$ $\displaystyle=$
$\displaystyle\frac{M\omega^{2}_{z}}{l}\sum_{p,m,n=1}^{N}D_{mnp}\hat{Q}_{z,p}(2\hat{Q}_{z,m}\hat{Q}_{z,n}-3\hat{Q}_{x,m}\hat{Q}_{x,n}$
(3) $\displaystyle-3\hat{Q}_{y,m}\hat{Q}_{y,n})+O(\hat{Q}^{4}_{\chi,p}),$

where $D_{p,m,n}$ are the coupling coefficients which depend on the number of
ions. It is convenient to introduce a phonon creation and annihilation
operators $\hat{a}^{{\dagger}}_{\chi,p}$ and $\hat{a}_{\chi,p}$ of the $p$-th
collective vibration mode, so that
$\hat{Q}_{\chi,p}=\sqrt{\hbar/2m\omega_{\chi,p}}(\hat{a}^{{\dagger}}_{\chi,p}+\hat{a}_{\chi,p})$
and $\hat{P}_{\chi,p}=i\sqrt{\hbar
m\omega_{\chi,p}/2}(\hat{a}^{{\dagger}}_{\chi,p}-\hat{a}_{\chi,p})$. Then, the
Hamiltonian (2) can be written as
$\hat{H}_{0}=\sum_{\chi}\sum_{n}\hbar\omega_{\chi,n}(\hat{a}^{{\dagger}}_{\chi,n}\hat{a}_{\chi,n}+1/2)$.
Moving to rotating frame with respect to $\hat{H}_{0},$ the non-linear
Hamiltonian rapidly oscillates and thus can be neglected in the vibrational
rotating-wave approximation. Crucially, for a given resonance trap-frequency
condition, the non-linear phonon-phonon interaction can not be neglected. For
concreteness, we consider here an ion string with three ions. Then, imposing
the resonance condition $\omega_{a}\approx\omega_{b}+\omega_{c},$ where
$\omega_{a}$ is the zigzag axial normal-mode frequency and, respectively,
$\omega_{b}$ and $\omega_{c}$ are the transverse rocking and zigzag mode
frequencies, and neglecting the fast-rotating terms, we obtain Ding2018 ;
Maslennikov

$\hat{H}_{\rm
v}=\hbar\omega_{a}\hat{a}^{{\dagger}}\hat{a}+\hbar\omega_{b}\hat{b}^{{\dagger}}\hat{b}+\hbar\omega_{c}\hat{c}^{{\dagger}}\hat{c}+\hbar\xi(\hat{a}^{{\dagger}}\hat{b}\hat{c}+\hat{a}\hat{b}^{{\dagger}}\hat{c}^{{\dagger}}),$
(4)

where
$\xi=(9\omega^{2}_{z}/5z_{0})\sqrt{\hbar/m\omega_{a}\omega_{b}\omega_{c}}$ is
the non-linear phonon-phonon coupling and
$z_{0}=(5e^{2}/16\pi\epsilon_{0}m\omega^{2}_{z})^{1/3}$ is the distance
between neighbouring ions. The Hamiltonian describes processes of non-linear
coherent energy exchange between the three collective vibrational modes, in
which one phonon from the axial zigzag mode ($a$-mode) is converted into one
phonon in the radial rocking mode ($b$-mode) and one in the radial zigzag mode
($c$-mode), and vice versa.

##  III Laser-free creation of two-mode squeezed state

In the following, we propose a laser-free creation of the two-mode squeezed
state, which relies on the non-linear phonon-phonon interaction, described by
Hamiltonian (4).

A two-mode squeezed state is defined as a coherent superposition of twin Fock
states Gerry2005

$|\psi_{\rm tmss}\rangle=\hat{S}(\zeta)|0,0\rangle=\frac{1}{\cosh
r}\sum_{n=0}^{\infty}(e^{i\theta}\tanh r)^{n}|n,n\rangle,$ (5)

where
$\hat{S}(\zeta)=e^{\zeta\hat{a}^{{\dagger}}_{A}\hat{a}^{{\dagger}}_{B}-\zeta^{*}\hat{a}_{A}\hat{a}_{B}}$
is the two-mode squeezing operator with $\zeta=re^{i\theta},$ where $r$ is the
squeezing amplitude and $\theta$ is the phase. Here
$|n,n\rangle=|n\rangle_{A}|n\rangle_{B}$ is the twin Fock state with $n$
bosons (phonons) in each mode.

Let us now assume that a time-dependent oscillating electric field is applied,
therefore displaces the motional excitation of axial $a$-mode. Its effect is
described by the term
$\hat{H}_{F_{a}}(t)=F_{a}z_{a}\cos(\omega_{d}t+\varphi)(\hat{a}^{{\dagger}}+\hat{a})$,
where $F_{a}$ is the magnitude of the electric field, $\omega_{d}$ is the
driving frequency, $\varphi$ is the phase, and
$z_{a}=\sqrt{\hbar/2m\omega_{a}}$ is the spread of the zero-point wave
function along the axial direction. We assume that the trap frequencies
satisfy the condition

$\omega_{a}=\omega_{b}+\omega_{c}+\omega$ (6)

and the driving frequency $\omega_{d}=\omega_{a}-\omega$ is shifted from the
$a$-mode frequency $\omega_{a}$ by a small detuning $\omega$
($\omega_{a}\gg\omega$). Moving into a rotating frame with respect to
$\hat{U}_{a}(t)=e^{-i(\omega_{a}-\omega)t\hat{a}^{{\dagger}}\hat{a}-i\omega_{b}\hat{b}^{{\dagger}}\hat{b}-i\omega_{c}\hat{c}^{{\dagger}}\hat{c}}$,
the total Hamiltonian $\hat{H}=\hat{U}_{a}^{{\dagger}}(t)(\hat{H}_{\rm
v}+\hat{H}_{F_{a}}(t))\hat{U}_{a}(t)-i\hbar\hat{U}_{a}(t)^{{\dagger}}\partial_{t}\hat{U}_{a}(t)$
becomes

$\displaystyle\hat{H}=\hbar\omega\hat{a}^{{\dagger}}\hat{a}+\hat{H}_{a},$
$\displaystyle\hat{H}_{a}=\hbar\Omega_{a}(\hat{a}^{{\dagger}}e^{i\varphi}+\hat{a}e^{-i\varphi})+\hbar\xi(\hat{a}^{{\dagger}}\hat{b}\hat{c}+\hat{a}\hat{b}^{{\dagger}}\hat{c}^{{\dagger}}),$
(7)

where $\Omega_{a}=F_{a}z_{a}/2\hbar$.

Figure 1: (Color online) (a) Time evolution of the probability $p_{n}(t)$ to
find the system in twin Fock state $|n,n\rangle$. We compare the probabilities
derived from the original Hamiltonian (7) and the analytical expression
$p_{n}(t)=\tanh^{2n}(r)/\cosh^{2}(r)$ (solid lines). We assume an initial
vacuum phonon state for all three modes. The parameters are set to
$\Omega_{a}/2\pi=3.5$ kHz, $\xi/2\pi=0.2$ kHz, and $\omega/2\pi=20$ kHz. The
Hilbert space for the three modes is truncated at $n_{\rm max}=20$. (b) Mean
phonon number in the radial $b$ mode for various $\lambda$ compared with the
analytical result $n_{b}=\sinh^{2}(r)$ (solid lines).

In the following we consider a weak-coupling regime,
$\Omega_{a},\xi\ll\omega$, in which the phonon excitations of the $a$-mode are
suppressed and thus can be eliminated from the dynamics. This can be carried
out by performing the canonical transformation $\hat{U}=e^{\hat{S}_{a}}$ to
Hamiltonian (7), such that $\hat{H}_{\rm
eff}=e^{-\hat{S}_{a}}\hat{H}e^{\hat{S}_{a}}\approx\hbar\omega\hat{a}^{{\dagger}}\hat{a}+(1/2)[\hat{H}_{a},\hat{S}_{a}]$
Ivanov2016  with
$\hat{S}_{a}=(\Omega_{a}/\omega)(\hat{a}e^{-i\varphi}-\hat{a}^{{\dagger}}e^{i\varphi})+(\xi/\omega)(\hat{a}\hat{b}^{{\dagger}}\hat{c}^{{\dagger}}-\hat{a}^{{\dagger}}\hat{b}\hat{c})$.
Keeping only the terms of order $\Omega_{a}\xi/\omega,$ we obtain the
following effective Hamiltonian:

$\hat{H}_{\rm
eff}=\hbar\omega\hat{a}^{{\dagger}}\hat{a}-\hbar\frac{\Omega_{a}\xi}{\omega}(e^{i\varphi}\hat{b}^{{\dagger}}\hat{c}^{{\dagger}}+e^{-i\varphi}\hat{b}\hat{c})+\hat{H}_{a}^{\prime}.$
(8)

The results indicate that, up to leading order, the off-resonance oscillating
electric force suppresses the phonon exchange between the axial $a$-mode and
the other two transverse $b$\- and $c$-modes. This induces an effective two-
mode squeezing between the $b$\- and $c$-modes with magnitude
$r=(\Omega_{a}\xi/\omega)t$ and phase $\theta=\varphi+\pi/2$. The residual
Hamiltonian is
$\hat{H}_{a}^{\prime}=-\Omega^{2}_{a}/\omega+(\xi^{2}/\omega)(\hat{n}_{a}+\hat{n}_{a}\hat{n}_{b}+\hat{n}_{a}\hat{n}_{c}-\hat{n}_{b}\hat{n}_{c}),$
where the first term gives rise to a global phase-shift while the second term
describes residual phonon-phonon interaction. It can, however, be neglected,
as long as $\Omega_{a}\gg\xi$.

Figure 2: (Color online) (a) Two-mode squeezed state fidelity, calculated from
the numerical simulation of Hamiltonian (7) as a function of time for various
$\omega$. The parameters are set to $\xi/2\pi=0.2$ kHz, $\Omega_{a}=3.5$ kHz,
and $\varphi=\pi/8$. (b) Analogously but now we have set $t=4$ ms and varied
$\Omega_{a}$ for different $\omega$.

In Fig. 1 we compare the exact results for the twin Fock probabilities and the
mean phonon number, with the analytical results, where very good agreement is
observed. Furthermore, we calculate the fidelity of producing the two-mode
squeezed state using $F=\langle\psi_{\rm tmss}|{\rm
Tr}_{a}\\{\hat{U}(t)|\psi_{\rm in}\rangle\langle\psi_{\rm
in}|\hat{U}^{{\dagger}}(t)\\}|\psi_{\rm tmss}\rangle$, where a partial trace
is performed over the vibrational degree of freedom of the axial $a$-mode and
$\hat{U}(t)=e^{-i\hat{H}t/\hbar}$ is the unitary operator with $\hat{H}$ given
by Eq. (7). The results displayed in Fig. 2 demonstrate that, by increasing
$\omega$, the higher order terms in (8) can be neglected, and so the fidelity
is improved. Also, for higher frequency $\Omega_{a},$ the condition for the
adiabatic elimination of $a$-mode is not satisfied, which decreases the
fidelity. For example, consider $\xi/2\pi=0.2$ kHz, $\Omega_{a}/2\pi=3.5$,
$\omega/2\pi=20$, and $t_{\rm max}=4$ ms for which the infidelity estimate is
of order $1-F(t_{\rm max})\approx 10^{-3}$.

##  IV Beam splitter transformation

We now discuss the creation of beam-splitter transformation using again the
non-linear phonon-phonon interaction (4). Consider now that the oscillating
electric field displaces a motion amplitude either of $b$\- or $c$-modes. For
concreteness, assume that the displacement term is
$\hat{H}_{b}(t)=F_{b}z_{b}\cos(\omega_{d}t+\varphi)(\hat{b}^{{\dagger}}+\hat{b})$,
where $z_{b}=\sqrt{\hbar/2m\omega_{b}}$, and the driving frequency is set to
$\omega_{d}=\omega_{b}+\omega$ with $\omega_{b}\gg\omega$. We assume that the
trap frequencies satisfy the condition (6). Then, moving into rotating frame
with respect to
$\hat{U}_{b}(t)=e^{-i\omega_{a}t\hat{a}^{{\dagger}}\hat{a}-i(\omega_{b}+\omega)\hat{b}^{{\dagger}}\hat{b}-i\omega_{c}\hat{c}^{{\dagger}}\hat{c}},$
the total Hamiltonian becomes

$\displaystyle\hat{H}=-\hbar\omega\hat{b}^{{\dagger}}\hat{b}+\hat{H}_{b},$
$\displaystyle\hat{H}_{b}=\hbar\Omega_{b}(\hat{b}^{{\dagger}}e^{i\varphi}+\hat{b}e^{-i\varphi})+\hbar\xi(\hat{a}^{{\dagger}}\hat{b}\hat{c}+\hat{a}\hat{b}^{{\dagger}}\hat{c}^{{\dagger}}),$
(9)

where $\Omega_{b}=F_{b}z_{b}/2\hbar$. Assuming that $\Omega_{b},\xi\ll\omega$
and performing a canonical transformation with
$\hat{S}_{b}=(\Omega_{b}/\omega)(b^{{\dagger}}e^{i\varphi}-\hat{b}e^{-i\varphi})+(\xi/\omega)(\hat{a}\hat{b}^{{\dagger}}\hat{c}^{{\dagger}}-\hat{a}^{{\dagger}}\hat{b}\hat{c}),$
the effective Hamiltonian $\hat{H}_{\rm
eff}=e^{-\hat{S}_{b}}\hat{H}e^{\hat{S}_{b}}\approx-\hbar\omega\hat{b}^{{\dagger}}\hat{b}+(1/2)[\hat{H}_{b},\hat{S}_{b}]$
becomes

$\hat{H}_{\rm
eff}=-\hbar\omega\hat{b}^{{\dagger}}\hat{b}+\frac{\Omega_{b}\xi}{\omega}(\hat{a}^{{\dagger}}\hat{c}e^{i\varphi}+\hat{a}\hat{c}^{{\dagger}}e^{-i\varphi})+\hat{H}_{b}^{\prime},$
(10)

where
$\hat{H}_{b}^{\prime}=\Omega^{2}_{b}/\omega-(\xi^{2}/\omega)(\hat{n}_{a}+\hat{n}_{a}\hat{n}_{b}+\hat{n}_{a}\hat{n}_{c}-\hat{n}_{b}\hat{n}_{c})$.
The result indicates that the off-resonant oscillating electric field
suppresses the energy exchange between the $b$\- mode and the other two modes,
which creates a beam-splitter transformation between the $a$\- and $c$-modes.

For arbitrary initial phonon state $|\psi_{in}\rangle=|n_{1},n_{2}\rangle$,
the beam-splitter transformation $\hat{U}_{\rm bs}(t)=e^{-i\epsilon
t(\hat{a}^{{\dagger}}\hat{c}e^{i\varphi}+\hat{a}\hat{c}^{{\dagger}}e^{-i\varphi})}$
with $\epsilon=(\Omega_{b}\xi/\omega)$ creates a final state
$|\psi_{f}\rangle=\sum_{N_{1},N_{2}}e^{-i(\varphi-\frac{\pi}{2})(n_{1}-N_{1})}C^{n_{1},n_{2}}_{N_{1},N_{2}}|N_{1},N_{2}\rangle$,
where, Kim2002 :

$\displaystyle C^{n_{1},n_{2}}_{N_{1},N_{2}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\sum_{k=0}^{n_{1}}\sum_{l=0}^{n_{2}}e^{i\pi(n_{1}-k)}\sin^{n_{1}+n_{2}-k-l}(\epsilon
t)\cos^{k+l}(\epsilon t)$ (11)
$\displaystyle\\!\\!\\!\times\frac{\sqrt{n_{1}!n_{2}!N_{1}!N_{2}!}}{k!(n_{1}-k)!l!(n_{2}-l)!}\delta_{N_{1},n_{2}+k-l}\delta_{N_{2},n_{1}-k+l}.$

In Fig. (3) we display the exact results for the probabilities to observe
$N_{1}$ phonons in $a$-mode and $N_{2}$ phonons in $c$-mode, compared to the
exact result $p_{N_{1},N_{2}}(t)=|C^{n_{1},n_{2}}_{N_{1},N_{2}}|^{2}$. We see
that the phonon dynamics of $b$-mode are suppressed due to the time-varying
oscillating force, which causes an effective beam-splitter transformation
between the other two vibrational modes.

Figure 3: (Color online) Probability to observe a state $|N_{1},N_{2}\rangle$
as a function of time. We compare the probabilities derived from the original
Hamiltonian (9) and the analytical expressions
$p_{N_{1},N_{2}}(t)=|C^{n_{1},n_{2}}_{N_{1},N_{2}}|^{2}$. Due to symmetry,
$p_{3,1}(t)=p_{1,2}(t)$ and $p_{4,0}(t)=p_{0,4}(t).$ We assume an initial
state $|\psi_{in}\rangle=|2,0,2\rangle$. The parameters are set to
$\xi/2\pi=0.2$ kHz, $\Omega_{b}=6.5$ kHz, and $\omega/2\pi=17$ kHz.

##  V Quantum Metrology

In the following section we discuss the metrological properties for parameter
estimation of the two-mode squeezed state and the beam-splitter
transformation.

###  V.1 Generalized theoretical framework for single-parameter quantum
estimation

The value of $\lambda$, the parameter we wish to estimate, can be extracted by
performing suitable discrete set of measurements, defined in terms of its
corresponding positive-operator-valued measurement (POVM),
$\\{\hat{\Pi}_{n}\\}$, with $\sum_{n}\hat{\Pi}_{n}=1$. The corresponding
classical Fisher information (CFI), quantifying the amount of information on
the unknown parameter, is given by Paris2009

$\mathcal{F}_{\rm
C}(\lambda)=\sum_{n}\frac{1}{p_{n}(\lambda)}\left(\frac{dp_{n}(\lambda)}{d\lambda}\right)^{2},$
(12)

where
$p_{n}(\lambda)=\langle\psi_{\lambda}|\hat{\Pi}_{n}|\psi_{\lambda}\rangle$ is
the probability to get outcome $n$ from the performed measurement, and
$|\psi_{\lambda}\rangle$ is the state vector of the system and depends on the
parameter $\lambda$. The statistical uncertainty in the parameter estimation
is quantified by the Cramér-Rao bound:

$\delta\lambda^{2}\geq\frac{1}{\mathcal{F}_{\rm C}(\lambda)}.$ (13)

The statistical uncertainty however depends on the choice of an observable.
Therefore, optimizing over all possible measurements allows us to determine
the unknown parameter with an ultimate precision. Indeed, the CFI is upper-
bounded by $\mathcal{F}_{\rm C}(\lambda)\leq\mathcal{F}_{\rm Q}(\lambda)$,
where $\mathcal{F}_{\rm Q}(\lambda)={\rm Tr}(\hat{\rho}_{\lambda}\hat{L}^{2})$
is the quantum Fisher information (QFI), with $\hat{L}$ being the so-called
symmetrical logarithmic derivative (SLD) operator, which satisfies the
operator equation
$2\partial_{\lambda}\hat{\rho}_{\lambda}=\hat{\rho}_{\lambda}\hat{L}+\hat{L}\hat{\rho}_{\lambda}$.
For pure states the QFI is $\mathcal{F}_{\rm
Q}(\lambda)=4(\langle\partial_{\lambda}\psi_{\lambda}|\partial_{\lambda}\psi_{\lambda}\rangle-|\langle\psi_{\lambda}|\partial_{\lambda}\psi_{\lambda}\rangle|^{2})$.
Furthermore, if $|\psi_{\lambda}\rangle=e^{i\lambda\hat{G}}|\psi_{0}\rangle$,
the QFI is equal to four times the variance of $\hat{G}$ with respect to the
initial state, namely $\mathcal{F}_{\rm
Q}(\lambda)=4\Delta\hat{G}^{2}=4(\langle\hat{G}^{2}\rangle-\langle\hat{G}\rangle^{2})$.
Finally, the ultimate precision, which can be achieved, is quantified by the
quantum Cramér-Rao bound

$\delta\lambda^{2}\geq\frac{1}{\mathcal{F}_{\rm Q}(\lambda)}.$ (14)

For a single parameter estimation there always exists a measurement basis in
which CFI is equal to QFI. Such a basis is formed by the eigenstates of the
SLD operator. In general, a set of measurements that lead to the equality
$\mathcal{F}_{\rm C}(\lambda)=\mathcal{F}_{\rm Q}(\lambda)$ is called optimal.

Figure 4: (Color online) Classical Fisher information for the probability
$p_{k}(t_{f})=\langle k|{\rm
Tr}_{a,b}\\{|\psi(t)\rangle\langle\psi(t)|\\}|k\rangle$ to observe $k$ phonons
in $c$-mode as a function of the initial number of phonons $n$. The numerical
result with Hamiltonian (9) is compared with the QFI (15) (solid line). The
parameters are set $t_{f}=1$ ms, $\Omega_{b}/2\pi=4.5$ kHz, $\omega/2\pi=20$
kHz, and $\xi/2\pi=0.2$ kHz.

For a quantum state with $n$ uncorrelated particles, the statistical
uncertainty obeys the scaling $\delta\lambda^{2}\sim 1/n$ which is the shot-
noise limit. The super-resolution and the so-called Heisenberg sensitivity can
be achieved by using entangled states where $\delta\lambda^{2}\sim 1/n^{2}$
Degen2017 ; Pezze2018 ; Giovannetti2011 ; Giovannetti2006 .

###  V.2 Two-mode squeezed state

Consider that the parameter we wish to estimate is the magnitude of the two-
mode squeezing operator, namely $\lambda=r$. Then, it is straightforward to
show that the QFI is $\mathcal{F}_{\rm Q}(r)=4$. We find that the optimal
basis consists of measurements of Fock state probabilities of one of the
vibrational modes $p_{n}=\tanh^{2n}(r)/\cosh^{2}(r)$ which leads to equality
$\mathcal{F}_{\rm C}(r)=\mathcal{F}_{\rm Q}(r)=4$. We may also consider an
estimation of the phase $\theta$ in Eq. (5). Then, we have $\mathcal{F}_{\rm
Q}(\theta)=4\bar{n}(\bar{n}+1)$, where $\bar{n}=\sinh^{2}(r)$ is the mean-
phonon number of one of the vibrational modes. Note that the Fock state
probabilities are independent on $\theta$ so that the optimal basis is given
by the eigenvectors of the SLD operator.

###  V.3 Heisenberg precision

We now discuss the quantum metrology using beam-splitter transformation. We
assume that the system is prepared initially in the Fock state with $n$
phonons in each mode, $|\psi_{in}\rangle=|n,n\rangle$. Then, the system
evolves into the final state $|\psi_{f}\rangle=e^{-i\epsilon
t(\hat{a}^{{\dagger}}\hat{c}+\hat{a}\hat{c}^{{\dagger}})}|n,n\rangle$.
Consider that the parameter we wish to estimate is $\lambda=\epsilon$, then
the QFI information is given by

$\mathcal{F}_{\rm Q}(\epsilon)=8n(n+1),$ (15)

which gives Heisenberg-limited precision. Note that, because
$(n^{2}+n)^{-1}<n^{-2},$ a sub-Heisenberg limit in the parameter estimation
can be achieved Anisimov2010 .

In Fig. 4 we show the exact result for the CFI, using the full Hamiltonian (9)
which is compared to the QFI (15). We assume that the parameter $\epsilon$ is
extracted by measuring the probabilities to observe a Fock state with $k$
phonons in one of the vibrational modes, namely $p_{k}(t)=\langle k|{\rm
Tr}_{a,b}\\{|\psi(t)\rangle\langle\psi(t)|\\}|k\rangle$, where
$|\psi(t)\rangle$ is the exact state vector of the total system. Crucially,
the measurement of the Fock state probabilities is an optimal, in sense that
the CFI is equal to the QFI, as can be seen from Fig. 4. Finally, we point out
that, recently, an efficient adiabatic technique for measuring the Fock state
probabilities with trapped ions was proposed Kirkova2021 .

Figure 5: (Color online) Conditional beam-splitter transformation. The system
is prepared initially in the state $\left|\uparrow,1,0\right\rangle$ where the
motional $a$\- and $c$-modes are swapped at the gate time $t_{g}$ to
$\left|\uparrow,0,1\right\rangle$. We show the respective probabilities
$p_{\uparrow,1,0}(t)$ and $p_{\uparrow,0,1}(t)$ which are derived from the
original Hamiltonian (16). The probability $p_{\uparrow,1,1}(t)$ when the
system is prepared initially in the state $\left|\uparrow,1,1\right\rangle$.
The parameters are set to $\xi/2\pi=0.2$ kHz, $g_{b}/2\pi=5.5$ kHz, and
$\omega/2\pi=18$ kHz, and $\eta_{b}=0.06$.

##  VI Conditional Beam-Splitter Transformation

In this Section we extend the discussion to the creation of spin-dependent
(conditional) beam-splitter transformation. Let us consider the ion’s two-
metastable levels $\left|\uparrow\right\rangle$ and
$\left|\downarrow\right\rangle$ which compose an effective spin system with
transition frequency $\omega_{0}$. Instead of applying an oscillating electric
field, one can decouple the $b$-mode from the other two $a$\- and $c$\- modes
using a laser field onto a single ion. Assume that the first ion is addressed
by pairs of laser beams in a Raman configuration with a wave-vector difference
$\Delta\vec{k}$ along the transverse $x$-direction ($|\Delta\vec{k}|=k_{x}$)
and laser frequency difference $\Delta\omega_{\rm L}=\omega_{b}+\omega$ tuned
near the $b$-mode. Such a laser configuration generates a spin-dependent
force, which provides a coupling between the spin state
$\left|\uparrow\right\rangle$ and the $b$-mode Lee2005 . The Hamiltonian
describing the laser-ion interaction is given by
$\hat{H}_{I}(t)=\hbar\Omega\\{e^{ik_{x}\delta\hat{r}_{x}-\Delta\omega_{\rm
L}t}+{\rm H.c.}\\}\left|\uparrow\right\rangle\left\langle\uparrow\right|$,
where $\Omega$ is the Rabi frequency Lee2005 ; Wineland1998 ; Haffner2008 ;
Schneider2012 . Let us define the Lamb-Dicke parameter
$\eta_{b}=k_{x}\sqrt{\frac{\hbar}{2m\omega_{b}}}M_{1,b}^{x}$. Then, expanding
the laser-ion interaction in power series of $\eta_{b}$ and neglecting the
fast rotating terms, we obtain

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle-\hbar\omega\hat{b}^{{\dagger}}\hat{b}+\hbar
g_{b}(\hat{b}^{{\dagger}}\hat{F}(\hat{n}_{b})+\hat{F}(\hat{n}_{b})\hat{b})\left|\uparrow\right\rangle\left\langle\uparrow\right|$
(16)
$\displaystyle+\hbar\xi(\hat{a}^{{\dagger}}\hat{b}\hat{c}+\hat{a}\hat{b}^{{\dagger}}\hat{c}^{{\dagger}}),$

where $g_{b}=\Omega\eta_{b}$ is the spin-phonon coupling. The non-linear
operator, which quantifies the higher-order terms in the Lamb-Dicke expansion,
is given by Vogel1995

$\hat{F}(\hat{n}_{b})=e^{-\eta^{2}_{b}/2}\sum_{n=0}^{\infty}\frac{(-\eta_{b}^{2})^{n}}{n!(n+1)!}\hat{b}^{{\dagger}n}\hat{b}^{n}.$
(17)

Note that, for sufficiently strong coupling $g_{b},$ but still
$g_{b}\ll\omega,$ the coupling strength of the higher-order terms in the Lamb-
Dicke expansion is compatible with the non-linear coupling $\xi$. In order to
account for this, we include these terms in the numerical simulations.

Figure 6: (Color online) Fidelity for the creation of spin-dependent N00N
state as a function of time for $n=2$, calculated from the numerical
simulation with Hamiltonian (16). We assume that an additional term
$\hat{H}_{\rm
AC}=-(g^{2}/\omega)\left|\uparrow\right\rangle\left\langle\right\uparrow|$ is
applied to (16), and compensates the shift of the spin frequency due to the
spin-motional interaction. The parameters are set to $g_{b}/2\pi=6.3$ kHz, and
$\omega/2\pi=15.8$ kHz, $\xi/2\pi=0.3$ kHz and $\eta_{b}=0.05$

Here, as an example, we study the efficiency of generation of Fredkin gate,
where the control qubit consists of the two internal states of the ion and the
target qubits are composed of the Fock states of $a$\- and $c$-modes. We have

$\hat{U}_{F}\left|\downarrow,n,m\right\rangle=\left|\downarrow,n,m\right\rangle,\quad\hat{U}_{F}\left|\uparrow,n,m\right\rangle=(-i)^{n+m}\left|\uparrow,m,n\right\rangle,$
(18)

where $\hat{U}_{F}$ is the Fredkin gate. In the following we show that, using
a laser-ion interaction and the non-linear phonon-phonon interaction (16), we
can implement the Fredkin gate.
Ideally, the Fredkin gate can be carried out using the conditional beam-
splitter transformation $\hat{U}_{\rm
cbs}(t)=e^{-i\epsilon_{b}t\left|\uparrow\right\rangle\left\langle\uparrow\right|(\hat{a}^{{\dagger}}\hat{c}+\hat{a}\hat{c}^{{\dagger}})}$
with $\epsilon_{b}=g_{b}\xi/\omega$ such that, for time
$t_{g}=\frac{\pi}{2\epsilon_{b}}$, we have $\hat{U}_{F}=\hat{U}_{\rm
sbm}(t_{g})$.

In Fig. 5 we show the exact result for the probabilities to observe states
$\left|\uparrow,1,0\right\rangle$ and $\left|\uparrow,0,1\right\rangle$ when
the system is prepared initially in the state
$\left|\uparrow,1,0\right\rangle$. We also show $p_{\uparrow,1,1}(t)$ when the
system is prepared in the state $\left|\uparrow,1,1\right\rangle$. At the gate
time $t_{g}$, the two motional $a$\- and $c$\- modes are swapped only if the
spin is in the excited state $\left|\uparrow\right\rangle$, according to (18).

Furthermore, we consider the generation of a spin-dependent N00N-like state
$\left|\psi_{\rm
N00N}\right\rangle=(\left|\downarrow\right\rangle\left|n,0\right\rangle+(-i)^{n}\left|\uparrow\right\rangle\left|0,n\right\rangle)/\sqrt{2}$,
using the Fredkin gate (18). We assume that the system is initially prepared
in the state $\left|\psi_{in}\right\rangle=|\psi_{\rm spin}\rangle|n,0\rangle$
with $|\psi_{\rm
spin}\rangle=(\left|\downarrow\right\rangle+\left|\uparrow\right\rangle)/\sqrt{2}$,
such that, after the application of the Fredkin gate, we create the entangled
state $\left|\psi_{\rm N00N}\right\rangle$. As a figure of merit for the
fidelity, we consider $F_{G}(t)=\langle\psi_{\rm N00N}|{\rm
Tr}_{b}\\{\hat{U}(t)|\psi_{in}\rangle\langle\psi_{in}|\hat{U}^{{\dagger}}(t)\\}|\psi_{\rm
N00N}\rangle$, where a partial trace is performed over the vibrational degree
of freedom of the radial $b$-mode, and $\hat{U}(t)=e^{-i\hat{H}t/\hbar}$ is
the unitary operator, with $\hat{H}$ given by Eq. (16). In Fig. 6 we display
the numerical result for $F_{G}(t)$. We assume that the shift of the spin
frequency, due to the spin-phonon interaction, is compensated by the term
$\hat{H}_{\rm
AC}=-(g^{2}/\omega)\left|\uparrow\right\rangle\left\langle\right\uparrow|$.
For example, consider the parameters $g_{b}/2\pi=6.3$ kHz, and
$\omega/2\pi=15.8$ kHz, for which we estimate infidelity of order of
$1-F_{G}(t_{g})\sim 6\times 10^{-3}$.

##  VII Conclusion

In conclusion, we have proposed a laser-free method for creation of two-mode
squeezed states which relies on the coupling between the axial zigzag mode and
the radial rocking and zigzag modes in a linear ion crystal. We have shown
that a weak, time-varying electric field suppresses the energy exchange
between the axial mode and the other two radial modes, and leads to an
effective two-mode squeezing of the $b$\- and $c$\- modes. The same technique
can be used for the creation of a beam-splitter transformation between the
axial $a$-mode and one of the radial $b$\- or $c$-modes. We have studied the
quantum metrological properties of the two-mode squeezed state and the beam-
splitter transformation. We have shown that, for an initial state with
$n$-phonons in each mode, one can achieve sub-Heisenberg limit of precision
using a beam-splitter transformation, where the quantum Fisher information
scales with the number of phonons as $n^{2}+n$. Finally, we have discussed the
implementation of a conditional beam-splitter-transformation. In that case a
spin-dependent force is applied to one of the radial modes. We have shown that
the combination of the spin-dependent force and phonon non-linearity leads to
the creation of conditional beam-splitter transformation. For a given gate
time, the conditional beam-splitter transformation is identical to the three-
qubit Fredkin gate, in which the target qubit consists of the internal states
of the ion and the control qubits are the phonon Fock states. Finally, we have
shown that the Fredkin gate can be used for the creation of high-fidelity
quantum N00N states.

## Acknowledgments

This research is supported by the Bulgarian national plan for recovery and
resilience, contract BG-RRP-2.004-0008-C01 (SUMMIT: Sofia University Marking
Momentum for Innovation and Technological Transfer), project number 3.1.4.

## References

  * (1) R. Blatt and D. Wineland, Nature 453, 1008 (2008).
  * (2) D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 281 (2003).
  * (3) D. F. V. James, Appl. Phys. B 66, 181 (1998).
  * (4) C. Marquet, F. Schmidt-Kaler, and D. F. V. James, App. Phys. B 76, 199 (2003).
  * (5) C. F. Roos, T. Monz, K. Kim, M. Riebe, H. Häffner, D. F. V. James, and R. Blatt, Phys. Rev. A 77, 040302(R) (2008).
  * (6) X. Nie, C. F. Roos, and D. F. V. James, Phys. Lett. A 373, 422 (2009).
  * (7) S. Ding, G. Maslennikov, R. Hablutzel, and D. Matsukevich, Phys. Rev. Lett. 121, 130502 (2018).
  * (8) G. Maslennikov, S. Ding, R. Hablutzel, J. Gan, A. Roulet, S. Nimmrichter, J. Dai, V. Scarani, and D. Matsukevich, Nat. Commun. 10, 202 (2019).
  * (9) S. Ding, G. Maslennikov, R. Hablützel, and D. Matsukevich, Phys. Rev. Lett. 119, 193602 (2017).
  * (10) S. Ding, G. Maslennikov, R. Hablützel, H. Loh, and D. Matsukevich, Phys. Rev. Lett. 119, 150404 (2017).
  * (11) F. R. Cardoso, D. Z. Rossatto, G. P. L. M. Fernandes, G. Higgins, and C. J. Villas-Boas, Phys. Rev. A 103, 062405 (2021).
  * (12) P. A. Ivanov, Phys. Rev. A 105, 032617 (2022).
  * (13) J. Gan, G. Maslennikov, K. W. Tseng, C. H. Nguyen, and D. Matsukevich, Phys. Rev. Lett. 124, 170502 (2020).
  * (14) C. Gerry and P. L. Knight, _Introductory quantum optics_ (Cambridge University, Cambridge, 2005).
  * (15) P. A. Ivanov, N. V. Vitanov, and K. Singer, Sci. Rep. 6, 28078 (2016).
  * (16) M. S. Kim, W. Son, V. Buzek, and P. L. Knight, Phys. Rev. A 65, 032323 (2002).
  * (17) M. G. A. Paris, Int. J. Quantum Inf. 07, 125 (2009).
  * (18) C. L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod. Phys. 89, 035002 (2017).
  * (19) L. Pezze, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Rev. Mod. Phys. 90, 035005 (2018).
  * (20) V. Giovannetti, S. Lloyd, and L. Maccone, Nat. Photonics 5, 222 (2011).
  * (21) V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 96, 010401 (2006).
  * (22) P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, Phys. Rev. Lett. 104, 103602 (2010).
  * (23) A. Kirkova, W. Li, and P. A. Ivanov, Phys. Rev. Research 3, 013244 (2021).
  * (24) P. J. Lee, K.-A. Brickman, L. Deslauriers, P. C. Haljan, L.-M. Duan, and C. Monroe, J. Opt. B 7, S371 (2005).
  * (25) D. J. Wineland, C. Monroe, W. M. Itano, D. Liebfried, B. E. King, and D. M. Meekhof, J. Res. Natl. Inst. Stand. Technol. 103, 259 (1998).
  * (26) H. Häffner, C. F. Roos, and R. Blatt, Phys. Rep. 469, 155 (2008).
  * (27) C. Schneider, D. Porras, and T. Schaetz, Rep. Prog. Phys. 75, 024401 (2012).
  * (28) W. Vogel and R. L. de Matos Filho, Phys. Rev. A 52, 4214 (1995).