# The Effect of Alignment Objectives on Code-Switching Translation 111

Mohamed Anwar
African Institute for Mathematical Sciences (AIMS)
<EMAIL_ADDRESS>

###### Abstract

One of the things that need to change when it comes to machine translation is
the models’ ability to translate code-switching content, especially with the
rise of social media and user-generated content. In this paper, we are
proposing a way of training a single machine translation model that is able to
translate monolingual sentences from one language to another, along with
translating code-switched sentences to either language. This model can be
considered a bilingual model in the human sense. For better use of parallel
data, we generated synthetic code-switched (CSW) data along with an alignment
loss on the encoder to align representations across languages. Using the WMT14
English-French (En-Fr) dataset, the trained model strongly outperforms
bidirectional baselines on code-switched translation while maintaining quality
for non-code-switched (monolingual) data.

##  1 Introduction

It’s very difficult to say exactly how many people in the world speak two
languages fluently. However, some estimates predict that around 60% of people
worldwide are at least bilingual. And according to Valihanov (1990), Africa is
the most multilingual continent in the world, which requires rethinking what
we expect from our machine translation models when they’re deployed especially
with the rise of social media and user-generated content worldwide. Rijhwani
et al. (2017) has collected around 50 million unique tweets and found out that
English-Spanish, English-French, and English-Portuguese were the three most
commonly mixed pairs accounting for 21.5%, 20.8% and 18.4% of all collected
code-switched tweets respectively.

One of the things that need to change when it comes to machine translation is
the models’ ability to translate code-switched content which remains
understudied in natural language processing (NLP) in general and machine
translation (MT) in particular. Code-switched (CSW) denotes the alternation of
two languages within a single utterance (Poplack (1980). It is a common
communicative phenomenon that occurs in multilingual communities during spoken
and written interactions.

The Matrix Language Frame (MLF) theory (Myers-Scotton (1997)) defines the
concept of matrix and embedded languages where the matrix language is the main
language that the sentence structure should conform to and notably provides
the syntactic morphemes, while the influence of the embedded language is
lesser and is mostly manifested in the insertion of content morphemes.

Neural machine translation (NMT) has significantly improved the quality of
machine translation in recent years (Sutskever et al. (2014); Bahdanau et al.
(2014); Zhang et al. (2015); Wu et al. (2016); Gehring et al. (2017); Vaswani
et al. (2017)). In this paper, we aim to build a bidirectional neural machine
translation (NMT), which is a single model that is able to translate in both
directions of the two languages it was trained on and is capable of
translating code-switched sentences. In a sense, we can call this model a
"Bilingual Translation Model" in the way humans are bilingual. When a human is
bilingual, it means he/she is able to comprehend two languages and any
combination of them. So, in a sense, this model is one step closer to having a
true bilingual machine translation model (Anwar et al. (2022)).

Training machine translation models on code-switched data is difficult due to
a lack of data and a complete understanding of how humans code-switch. In this
paper, we are proposing a method of generating code-switched sentences by
leveraging parallel data. This method was highly inspired by the Masked
Language Modeling (MLM) pre-training task in BERT (Devlin et al. (2019)) and
adapted from the work of Xu and Yvon (2021).

Using only parallel data accompanied by synthetic code-switched data, the NMT
model learns how to combine words from parallel sentences and identifies when
to switch from one language to the other. Moreover, it captures code-switched
constraints by attending and aligning the words in inputs, without requiring
any external knowledge. To improve the performance even more, we propose an
alignment objective applied only on the encoder side to incentivize the
encoder to create language-agnostic representations making the decoder’s job a
little bit easier based on our experiments.

Our main contribution to this paper can be summarized into the following:

  * •

A new method of generating code-switched data using only parallel data
(Section 3.1).

  * •

An analysis of the generated code-switched data (Section 3.2).

  * •

An alignment objective implemented on the encoder-side to motivate the encoder
to generate language-agnostic representations leveraging parallel & code-
switched data (Section 4).

  * •

A bilingual neural machine translation model that is almost as good as our
bidirectional baseline model, while achieving better performance than our
code-switched baseline model (Section 5).

##  2 Related Work

Research in the area of NLP on code-switching (CSW) has mostly focused on
Language Modeling, especially for Automatic Speech Recognition (ASR) (Pratapa
et al. (2018); Garg et al. (2018); Gonen and Goldberg (2018); Winata et al.
(2019); Lee and Li (2020)). Evaluation tasks and benchmarks have also been
prepared for LID in user-generated CSW content (Zubiaga et al. (2016); Molina
et al. (2019)), Named Entity Recognition (Aguilar et al. (2019)), Part-of-
Speech tagging (Ball and Garrette (2018); Aguilar et al. (2020); Khanuja et
al. (2020)) and Sentiment Analysis (Patwa et al. (2020)). CSW was also found
useful in foreign language teaching: Renduchintala et al. (2019) showed that
replacing words with their counterparts in a foreign language helps to learn
foreign language vocabulary.

Regarding Machine Translation, most past work has focused on using synthetic
code-switched (CSW) data to help conventional translation systems. Huang and
Yates (2014) used CSW corpus to improve word alignment and statistical MT.
Dinu et al. (2019) experienced replacing and concatenating source terminology
constraints by the corresponding translation(s) to boost the accuracy of term
translations. Song et al. (2019) shared the same idea by replacing phrases
with pre-specified translation to perform "soft" constraint decoding. A
different line of research is in Bulte and Tezcan (2019), which explores ways
to combine a source sentence with similar translations extracted from
translation memories.

Similar to our work, Yang et al. (2020) proposed a new technique for NMT pre-
training, which makes full use of the cross-lingual alignment information
contained in source and target monolingual corpus. The idea is to randomly
replace some words from the source language with their translations in the
target language. This was done by using unsupervised word embedding mapping.
Similar to that work, Lin et al. (2020) and Pan et al. (2021) create code-
switching sentences by randomly replacing a word in the source language with
another word in a different random language using MUSE (Conneau et al. (2017))
as a pre-training step for big multilingual machine translation model. Also,
Gautam et al. (2021) tried to translate from English to Hinglish (code-
switched sentences between Hindi and English) by fine-tuning mBART (Liu et al.
(2020)). Since mBART uses Hindi in Devanagari scripts, they converted Hindi
from Roman scripts to Devanagari scripts. After getting a code-mixture of
English-Hindu, they transliterated the Hindu from Devanagari to Roman scripts.

When it comes to generating code-switching data, Gupta et al. (2020) proposed
a new way to generate synthetic code-switching data from parallel corpus by
using a word-alignment such as fast-align (Dyer et al. (2013)) algorithm to
obtain the alignment matrix. Then, they used the Stanford library Stanza (Qi
et al. (2020)) to extract Part-of-Speech of the sentence (mainly adjective),
Named entity (NE) of types: "Person", "location", and "organization", and Noun
phrase (NP). After extracting, they inserted them into the appropriate places
of the sentences in the other language.

Very similar to our way of generating code-switched data, Xu and Yvon (2021)
proposed a way of generating code-switched sentences using only fast-align
(Dyer et al. (2013)) algorithm to obtain the alignment matrix which can be
used later for the generation. More details regarding this will be discussed
later in this paper (Section 3.1).

##  3 Data

Since English and French are among the most high-resource spoken languages in
the world and especially in Africa, we used the WMT14 English-French benchmark
for training, newstest2008-2013 for validation, and newstest2014 for testing.
To handle out-of-vocabulary words, we used a shared vocabulary of 40K Byte-
Pair-Encoding (BPE) Sennrich et al. (2016) sub-words. We removed any sentences
whose lengths are less than $1$ tokens and more than $250$ tokens.

Also, we prepended a target-language tag to the sentences as shown in Table 2
to teach the model to only care for the target language. All data were
tokenized and normalized using Moses SMT toolkit (Koehn et al. (2007)). The
WMT14 sentence length distribution before cleaning can be found in Figure 1,
and the stats of the data after cleaning can be found in Table 1.

Figure 1: Histogram of the sentence length of the WMT14 corpus. Table 1: WMT-14 French-English data stats used for training, validation, and test. Dataset | Train Size | Valid Size | Test Size | Total
---|---|---|---|---
Fr-En | 35,789,717 | 15,827 | 3,003 | 35,808,547
Bidirectional | 71,579,434 | 31,654 | 6,006 | 71,617,094
CSW | 77,198,306 | 32,581 | 6,006 | 77,236,893

###  3.1 Code-switched (CSW) Data

Parallel corpora for code-switched data are very scarce (Menacer et al.
(2019)). However, there have been previous works on generating synthetic code-
switched data. Similar to Song et al. (2019) and Xu and Yvon (2021), we
created code-switched data by modifying the steps of Xu and Yvon (2021). Since
Xu and Yvon (2021) used only code-switched (CSW) data to train their models,
they followed an exponential distribution 222The number of replacements $r$ to
appear in a derived code-switched sentence follows the distribution defined by
this equation: $P(r=k)=\frac{1}{2^{k+1}}$ where $k\in\\{1,...,rep\\}$ such
that $rep$ is a predefined hyper-parameter that represents the maximum number
of replacements. to generate code-switched data that prefers smaller number of
replacements when generating. Using this exponential distribution, they would
generate code-switched sentences with 0 replacements around 50% of the time,
and with 1 replacement around 25%, …etc.

In our case, we decided to do things a little differently. We would rather
have a separate group of code-switched sentences and mix it with another group
of monolingual data when needed. To generate the code-switched data, we were
looking for a distribution that increases the number of replacements with
longer sentences, which is what humans tend to do. We can achieve that using
linear distribution, but what is the reasonable slope for that?

We found the answer in the Masked Language Modeling (MLM) pre-training task
used by BERT (Devlin et al. (2019)), where code-switching will be used instead
of masking. In other words, we are going to code-switch the matrix language
text with the embedded language text randomly around 15% tokens of the whole
input sentence. In that way, we can study the effect of code-switching and
also can increase the number of replacements with longer sentences.

The following are the steps -in more detail- that we followed to generate the
code-switched (CSW) data; a sample of the generated code-switching data can be
found in Table 3.

  1. 1.

Data Pre-processing: First, we cleaned, tokenized, and normalized the text
data using _clean-corpus-n.perl_ script from the moses SMT (Koehn et al.
(2007)) toolkit.

  2. 2.

Word Aligning: Then, we started extracting word alignment using _fast-align_
toolkit (Dyer et al. (2013)) with _gdfa_ (grow-diag-final-and).

  3. 3.

Minimal Aligned Units: Then, we extracted minimal alignment units following
the approach of Song et al. (2019) and Xu and Yvon (2021). These correspond to
word segments $(a,b)$ extracted from the word alignments in the previous step;
such that all alignment links outgoing from words in $a$ reach a word in $b$,
and vice-versa.

  4. 4.

Random Replacement: Finally, aligned segments were replaced by under the
following conditions:

    1. (a)

Matrix (dominant) Language — defined by the Matrix Language Frame (MLF) theory
(Myers-Scotton (1997)) — is chosen randomly (50-50)%.

    2. (b)

Similar to MLM pre-training (Devlin et al. (2019)), we randomly replaced
around 15% tokens of the sentence length in the matrix language with its
aligned segments in the embedded language.

    3. (c)

Short sequences (less than 7 tokens) will have just one replacement.

    4. (d)

Positions of aligned segments were chosen uniformly.

We combined the code-switched data generated with the parallel data after
prepending a target language token (Johnson et al. (2017)). Table 2 shows an
example of the formulation of our data, where the bold words are in French and
the normal words are in English. The first two sentences are monolingual while
the last two examples are code-switching sentences. The first and last
sentences are being translated to French, hence the _< 2fr>_ language tag. The
second and third sentences are being translated to English, hence the _< 2en>_
language tag.

Table 2: Example of how data looks like when mixing parallel data with generated code-switching data. Source | Target
---|---
$<$2fr$>$ The weather today is nice. | Il fait beau aujourd’hui .
$<$2en$>$ Il fait beau aujourd’hui . | The weather today is nice .
$<$2en$>$ Il fait nice today . | The weather today is nice .
$<$2fr$>$ Il fait nice today . | Il fait beau aujourd’hui .
Table 3: A sample of the code-switched data generated from the newstest2014 dataset. Bold words are French while normal ones are English. CSW Sentence |  English Translation |  French Translation |  Matrix Language
---|---|---|---
Difficult Year pour les Pharmacists . |  Difficult Year for Pharmacists . |  Année difficile pour les pharmaciens . |  English
Il ne believe pas que l’Ontario emboîtera le pas . |  He does not believe that Ontario will follow suit . |  Il ne croit pas que l’Ontario emboîtera le pas . |  French
Asked how he had developed his character , the acteur and singer Justin Timberlake avait rappelé how he " grandi dans Tennessee , bathed in the blues and country music " . |  When asked how he came up with his character, actor and singer Justin Timberlake recalled that he " grew up in Tennessee, surrounded by blues and country music ". |  Interrogé sur la façon dont il a composé son personnage , l’acteur et chanteur Justin Timberlake avait rappelé avoir " grandi dans le Tennessee, baigné par le blues et la country " . |  English
Mes camarades cried with joie et mes parents ont conservé every journaux qu’ils ont trouvés . |  My classmates cried with joy , and my parents saved every newspaper they could find . |  Mes camarades de classe ont pleuré de joie, et mes parents ont gardé tous les journaux qu’ils ont pu trouver . |  French

###  3.2 Analysis

We used WMT-14 data for code-switching data generation and for training our
machine translation systems. As a cleaning step, we removed all English and
French sentences that were shorter than or equal to 1 token-long before using
the proposed method (Section 3.1) to generate code-switching (CSW) data. After
generation, we discarded sentence pairs with a source/target ratio higher than
1.5, with a maximum sentence length of 250. We performed the same procedure
over the training, validation, and test sets generating around 77 million
sentences, 32 thousand sentences, and 6 thousand sentences respectively. Exact
numbers can be found in Table 1.

The replacement distribution for the validation set and test set can be seen
in Figure 3. As we can see, most of the replacements were around 15% of the
input monolingual sentences. Some of the generated sentences are between
15%-20% and that’s because of the span of the minimal aligned units. As shown
in Figure 2, the span of replaced units can be longer than one token, which
counts for more percentage.

Figure 2: Histogram of the span of the aligned units that were used for
replacement when generating code-switching sentences of the validation set
(left) and test set (right).

For example, let’s assume that we have a sentence of 100 tokens, replacing 15%
means replacing around 15 tokens. But what if some of these replacements are
more than one token long, this will count for more percentage. That’s why some
of the sentences were between 15% and 20%. Sentences below 15% or above 20%
are usually short sentences. As seen from the data distribution shown in
Figure 1, a big portion of the sentences in both English and French languages
are shorter than 10-tokens long. Replacing just two tokens in these accounts
for more than 20%, while replacing one counts for less than 10%.

Figure 3: Histogram of the generated code-switching sentences of the
validation set (left) and test set (right).

##  4 Latent Alignment Objectives

Arivazhagan et al. (2019) adapted an alignment objective to help with zero-
shot translation by minimizing the discrepancy between the feature
distributions of the source and target domains improving the machine
translation model’s performance on zero-shot translation. Inspired by that
work, we decided to use the objective function with a slightly different goal
in mind.

As shown in Table 2, when mixing parallel data with code-switching data, we
will have four sentence pairs that have the same meaning despite belonging to
different distributions. To make the best use of that, we decided to use the
alignment objective to improve bidirectional machine translation models’
performance on code-switching translation by aligning the encoder’s latent
representation. In other words, we use the combination of parallel data and
code-switching data and enforce the encoder to make language-agnostic
representations about the input sentences that have the same meaning. Unlike
Arivazhagan et al. (2019) where the whole model’s parameters were updated, we
update only the encoder parameters, as shown in Figure 4.

Figure 4: shows the loss functions used with the encoder-decoder Transformer
model, where $CE$ is the cross entropy, $\Omega$ is the alignment objective
loss.

Below is an overview of the different objective functions that can be used to
achieve that goal. Originally, these objectives were used to achieve different
goals and fix different problems. However, they fit perfectly into our use
case. Then in Section 5, we are going to discuss how we are going to use them
to achieve our goal and explore their impact on the code-switching translation
task, or at least some of them.

###  4.1 Pool-Cosine Similarity

To improve the performance of multilingual NMT models on the zero-shot
translation task, Arivazhagan et al. (2019) proposed viewing the zero-shot
translation as a domain adaptation problem and use English - the language with
which we always have parallel data - as the source domain, and the other
languages collectively as the target domain. Then, train the model using
formula 1 to increase the cosine similarity between the latent representations
of the English sentences and all other translations in the other languages,
which eventually improves the model’s performance on zero-shot translation.

$\Omega=-\mathop{\mathbb{E}}_{x_{src},x_{tgt}\sim
D_{(en,T)}}[sim(Enc(x_{src}),Enc(x_{tgt}))]$ (1)

Where $\Omega$ is the alignment objective function, $D_{(en,T)}$ is the
training data containing parallel language pairs of English and other target
languages, $x_{src}$ is the source sentence while $x_{tgt}$ is the target,
$Enc(x)$ is the max-pooled encoder representation of a sentence $x$ similar to
Gouws et al. (2016) and Coulmance et al. (2016), and $sim$ is the cosine-
similarity.

We can benefit from this objective function by combining our parallel data and
mixing them with the generated code-switching data and using them as training
data. In that case, we are going to train the encoder to maximize the cosine
similarity between monolingual or code-switched sentences that have the same
translations as monolingual ones found in the parallel data

###  4.2 Cosine Similarity with Negative Sampling

Very similar to Pool-Cosine Similarity (4.1), Wieting et al. (2019) proposed
an objective function where for each sentence pair $(x_{i},y_{i})$ during
training, they randomly chose a negative target sentence $y_{i}^{\prime}$ that
is not a translation of $x_{i}$ and then train the model to ensure that source
and target sentences would be more similar than source and negative target
examples by a margin $\delta$ as shown in the following equation:

$\Omega=\mathbb{E}[\delta-
sim(Enc(x_{i}),Enc(y_{i}))+sim(Enc(x_{i}),Enc(y^{\prime}_{i}))]$ (2)

Unlike Pool-Cosine Similarity (4.1), they used different encoder for each
language, also the sentence representation $Enc()$ was achieved by averaging
the embeddings of the subword units instead of max-pooling the last layer of
the encoder representations.

###  4.3  Translation Ranking Objective

Given a sentence-pair $(x,y)$ that are translations of each other, the
translation candidate ranking task attempts to rank the true translation $y$
over all other target sentences $Y$. Guo et al. (2018) introduced a scoring
function $\phi$ that assesses the compatibility between $x$ and $y$, and then
trained the model to maximize the similarity between the correct sentence
pairs normalized by summation of the similarity scores between the source
sentence and the wrong $K$ target sentences, as shown in the following
formula:

$\Omega=-\mathbb{E}\left[\frac{e^{\phi(x,y)}}{\sum_{k=1}^{K}e^{\phi(x,y)}}\right]$
(3)

Where they defined $\phi$ to be the dot product of sentence embeddings for the
source $u$ and the target $v$, with $\phi(x,u)=u^{\top}\cdot v$.

When adapting this function to our use case, we can use this similarity metric
to make monolingual and code-switched sentences more similar to their correct
translations, normalized by other wrong translations.

###  4.4  AMS objective with Dual Encoder

Yang et al. (2019) extend Guo et al. (2018) approach -explained in section
4.3\- by using a bidirectional dual-encoder with additive margin Softmax (Wang
et al. (2018)) objective introducing a large margin, $m$, around positive
pairs, which significantly improves the model performance as shown in the
following figure:

$\Omega=-\mathbb{E}\left[\frac{e^{\phi(x,y)-m}}{e^{\phi(x,y)-m}+\sum_{k=1}^{K}e^{\phi(x,y)}}\right]$
(4)

###  4.5  Sentence Alignment Objective

Hu et al. (2020) proposed an objective to encourage cross-lingual alignment of
sentence representations in language modeling. For a source-target sentence
pair $(x,y)$ in the parallel corpus, they separately calculated sentence
embeddings denoted as $c_{x}$, $c_{y}$ by averaging the embeddings in the
final layer as the sentence embeddings. Then, they encourage the model to
predict the correct translation $y$ given a source sentence $x$. To do so,
they define the sentence alignment loss as the average negative log-likelihood
of the conditional probability of a candidate sentence $y$ being the correct
translation of a source sentence $x$; as shown in the following formula:

$\Omega=-\mathbb{E}\left[log\frac{e^{c_{x}^{\top}\cdot
c_{y}}}{\sum_{y^{\prime}\in\mathcal{M}\cup\mathcal{P}}e^{c_{x}^{\top}\cdot
c_{y}^{\prime}}}\right]$ (5)

Where $y^{\prime}$ can be any sentence in any language that is sampled within
a mini-batch of either monolingual data $\mathcal{M}$ or parallel data
$\mathcal{P}$.

Same as earlier objectives, this can be suitable for our use-case when using
our code-switching data instead of the monolingual data.

Since these different objective functions look very similar when considering
our goal and since we have limited computation resources, we decided to
experiment with only two of these objectives; namely the "Pool-Cosine
Similarity" and the "Sentence Alignment Objective". In the next section 5, we
are going to do so and discuss their impact on code-switching translation.

##  5 Experiments and Results

In all of our experiments, we used Transformer-Base (Vaswani et al. (2017))
configuration implemented in the Fairseq (Ott et al. (2019)) framework with 6
encoder layers and 6 decoder layers, each layer has a hidden size of $512$, an
$8$ attention heads, where each attention head has a size of $64$, and a feed-
forward hidden size of $2048$.

All models used in our experiments were trained using Adam optimizer (Kingma
and Ba (2014)) with $\beta_{1}=0.9,\beta_{2}=0.999$ and $\epsilon=10^{-9}$. We
started with a learning rate of $5e-4$, and warmed up the learning rate over
the first $4000$ steps using inverse square root decay. For regularization, we
used three different techniques:

  * •

Dropout (Srivastava et al. (2014)) of $0.1$.

  * •

Weight decay or L2 regularization (Cortes et al. (2012)) of $0.0001$.

  * •

Label smoothing (Szegedy et al. (2016)) of $0.1$.

For training these models, we used four Tesla T400 GPUs. For handling out-of-
vocabulary (OOV) tokens, we used a shared vocabulary of 40K BPE Sennrich et
al. (2016) sub-words as mentioned earlier, and a batch size of $4096$ tokens
per batch. The full list of the model’s hyper-parameters can be found in Table
4.

Table 4: The hyperparameter values setting for training. Hyper-parameter | Value
---|---
Number of Layers | 6
Hidden size | 512
FFN inner hidden size | 2048
Attention heads | 8
Attention head size | 64
Dropout | 0.1
Attention Dropout | 0.0
Warmup Steps | 4000
Learning Rate | 5e-4
Learning Rate Decay | inverse_sqrt
Batch Size | 4096 tokens
Label Smoothing | 0.1
Weight Decay | 0.0001
Adam $\epsilon$ | $10^{-9}$
Adam $\beta_{1}$ | 0.9
Adam $\beta_{2}$ | 0.98
Alignment Objective Weight | 1 (for SA) and 10 (for cosine)

For our experimentation, we created two baselines using the same training
hyper-parameters detailed in Table 4:

  1. 1.

Bidirectional: A $En\leftrightarrow Fr$ bidirectional model trained only on
parallel data.

  2. 2.

csw: A $En\leftrightarrow Fr$ bidirectional model trained only on synthetic
code-switched data.

Then, we trained three other models:

  1. 1.

bi+csw: A $En\leftrightarrow Fr$ bidirectional model that was trained on both
parallel data and code-switched data without any alignment objectives.

  2. 2.

bi+csw+cosine: A $En\leftrightarrow Fr$ bidirectional model that was trained
on both parallel data and code-switched data with the Pool-Cosine Similarity
(4.1) alignment objective.

  3. 3.

bi+csw+sa: A $En\leftrightarrow Fr$ bidirectional model that was trained on
both parallel data and code-switched data with the Sentiment Alignment (4.5)
objective.

Table 5 shows the results of all models trained using the same parameters seen
in Table 4 where all models were trained till convergence with patience = 10
and the reported results are case-sensitive detokenized 4-gram BLEU (Papineni
et al. (2002)) Scores with a beam size = 5.

Table 5: Case-sensitive detokenized 4-gram BLEU Score on unidirectional and CSW data from newstest2014 using SacreBLEU (Post (2018)) with beam-size of 5. Model | Steps | Unidirectional | Code-switching
---|---|---|---
En $\rightarrow$ Fr | Fr $\rightarrow$ En | CSW $\rightarrow$ Fr | CSW $\rightarrow$ En
Bidirectional | 642K | 39.57 | 36.17 | 57.86 | 60.77
csw | 594K | 8.38 | 13.66 | 68.49 | 66.65
bi+csw | 420k | 38.57 | 34.75 | 68.38 | 66.31
bi+csw+cosine | 612k | 39.18 | 35.43 | 68.69 | 66.96
bi+csw+sa | 660k | 39.00 | 35.68 | 68.58 | 66.88

From Table 5, we see that the Bidirectional baseline performs well for bi-
directional translation. However, csw baseline performs well only on CSW
translation.

Our trained models bi+csw, bi+csw+cosine and bi+csw+sa work well across the
board where bi+csw+cosine has the best performance on CSW data while achieving
competitive results on bi-directional translation compared to other baselines.

To further understand the effect of these alignment objectives on the code-
switching translation, we’ve decided to split the code-switched test data into
four groups based on the target language and whether it’s a matrix or embedded
language. So, these four groups can be seen in Table 6, and they are:

  1. 1.

CSW $\rightarrow$ Fr (Matrix): Target is French and Matrix language is French.

  2. 2.

CSW $\rightarrow$ Fr (Embedded): Target is French and Embedded language is
French.

  3. 3.

CSW $\rightarrow$ En (Matrix): Target is English and Matrix language is
English.

  4. 4.

CSW $\rightarrow$ En (Embedded): Target is English and Embedded language is
English.

Table 6: Case-sensitive detokenized 4-gram BLEU Score on code-switching data, generated from newstest2014 and scored with (Post (2018)) with beam-size of 5. Model | Steps | CSW $\rightarrow$ Fr | CSW $\rightarrow$ En
---|---|---|---
Matrix | Embedded | Matrix | Embedded
copying | - | 65.34 | 7.91 | 66.80 | 7.66
Bidirectional | 642K | 71.67 | 42.39 | 81.04 | 40.43
csw | 594K | 88.99 | 45.26 | 89.03 | 43.82
bi+csw | 420k | 88.45 | 45.65 | 88.63 | 43.52
bi+csw+cosine | 612k | 88.85 | 45.98 | 88.93 | 44.63
bi+csw+sa | 660k | 88.84 | 45.62 | 89.12 | 44.20

Table 6 shows that models trained with the alignment objectives scores very
similar scores to the csw baseline on the matrix-language side, while
achieving better scores on the embedded-language side. Also, we can see that
these models do more than just copying since the scores for copying, which
copies the code-switched source sentence and use it as the hypothesis, are
incredibly bad especially on the embedded-language side.

##  6 Conclusion & Future Work

In this paper, we introduced two ways to make best use of parallel data that
can improve the model’s performance on both unidirectional and code-switched
data. First, a statistical way to generate code-switched data that can be
aggregated with parallel data for training. Second, a loss function that
trains the encoder to generate language-independent representations. We show
that these two techniques boosted our model’s performance on both
unidirectional and code-switched translation.

This is still a work in progress, and we are exploring new ways to improve our
model even more; and more importantly experimenting with another language
family like Arabic, one of the most spoken languages in Africa.

## Acknowledgments and Disclosure of Funding

I would like to express my deepest appreciation to the "Google Cloud credits
for Academic Research" grant which provided the computational resources needed
to perform our experimentation needed for this paper.

Also, I want to extend my appreciation to Jitao Xu and Yvon Francis for
sharing their code, which we tried to improve upon to generate our synthetic
CSW data.

Finally, I’m deeply indebted to my two supervisors: Julia Kreutzer
<EMAIL_ADDRESS>and Melvin Johnson<EMAIL_ADDRESS>from Google
Research who kept me sharp, motivated, and hungry for more.

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# Weak-Valued Correlation Functions: Insights and Precise Readout Strategies

Yuan Feng<EMAIL_ADDRESS>International Center of Quantum Artificial
Intelligence for Science and Technology (QuArtist)
and Department of Physics, Shanghai University, 200444 Shanghai, China
QianWeiChang College, Shanghai University, 200444 Shanghai, China  Xi Chen
<EMAIL_ADDRESS>Department of Physical Chemistry, University of the
Basque Country UPV/EHU, Apartado 644, 48080 Bilbao, Spain  EHU Quantum Center,
University of the Basque Country UPV/EHU, 48940 Leioa, Spain  Yongcheng Ding
<EMAIL_ADDRESS>International Center of Quantum Artificial Intelligence
for Science and Technology (QuArtist)
and Department of Physics, Shanghai University, 200444 Shanghai, China
Department of Physical Chemistry, University of the Basque Country UPV/EHU,
Apartado 644, 48080 Bilbao, Spain

###### Abstract

The study of correlation functions in quantum systems plays a vital role in
decoding their properties and gaining insights into physical phenomena. In
this context, the Gell-Mann and Low theorem have been employed to simplify
computations by canceling connected vacuum diagrams. Building upon the essence
of this theorem, we propose a modification to the adiabatic evolution process
by adopting the two-state vector formalism with time symmetry. This novel
perspective reveals correlation functions as weak values, offering a universal
method for recording them on the apparatus through weak measurement. To
illustrate the effectiveness of our approach, we present numerical simulations
of perturbed quantum harmonic oscillators, addressing the intricate interplay
between the coupling coefficient and the number of ensemble copies.
Additionally, we extend our protocol to the domain of quantum field theory,
where joint weak values encode crucial information about the correlation
function. This comprehensive investigation significantly advances our
understanding of the fundamental nature of correlation functions and weak
measurements in quantum theories.

Introduction.– The correlation function is an indispensable aspect of quantum
theory that furnishes a comprehensive characterization of the properties of
quantum systems GFJS . By unifying essential concept, it serves as a poweful
tool for investigating various quantities in quantum field theory, including
scattering amplitudes, transition probabilities, and vertex amplitudes. Also,
by employing Wick’s trick, correlation functions provide insights into
statistical behaviors in many-body physics QSM . Among different types of
correlators, the out-of-time-correlator has proven valuable in exploring
intriguing phenomena such as quantum chaos and information scrambling, which
remain active frontiers of research JHEPOTOC ; JEOTOC ; scrambling ; OTOCtopo
. Before embarking on Feynman diagram calculations, it is customary to invoke
the Gell-Mann and Low theorem GL . This theorem allows one to recast the
$n$-point correlator
$\langle\Omega|\mathcal{T}[\hat{\phi}(\textbf{x}_{1})\ldots\hat{\phi}(\textbf{x}_{n})]|\Omega\rangle$
as a sum of connected diagrams with $n$ external lines. This physical
interpretation entails the evolution of two vacuum states $\langle
0||0\rangle$ from the infinite past and future to the state
$\langle\Omega^{+}||\Omega^{-}\rangle$ in the present. This evolution is
achieved by gradually switching on the interacting part of the field.
Therefore, understanding the modification or optimization of this theorem is
of great importance to unveil the underlying physics and its implications for
the correlator. One well-known approach to address this intriguing question is
the Keldysh formalism keldysh , which extends backward propagation in the
Gell-Mann and Low theorem, and provides a systematic framework for
investigating non-equilibrium systems noneqprb2003 ; noneqprd2014 ;
noneqprb2016 . This formalism offers a generalized perspective on the theorem
and allows for a comprehensive analysis of the dynamics of quantum systems in
non-equilibrium conditions.

By adopting a time-symmetric perspective, we introduce an alternative
modification to the Gell-Mann and Low theorem. This is accomplished by
leveraging the two-state vector formalism (TSVF) tsvf ; tsvfreview , which
provides a time-symmetric description of the process, ensuring consistency.
Unlike the adiabatic evolution in the original theorem, the TSVF allows for
arbitrary Hamiltonian to guide the evolution of eigenstates towards the pre-
selection or post-selection state. This departure from adiabaticity opens up
new possibilities in understanding weak measurement and the interpretation of
weak values as outcomes AAV . Furthermore, the TSVF also offers a unique
perspective on weak-valued correlation function sagawa . The insights lead to
a range of cutting-edge experiment proposals bomb ; cheshire  and advancements
dondeestabas ; foundation , using this technique. These experiments include
the record of physical quantities geometric ; eraser ; discord , quantum
tomography tomo ; tomosc , quantum steering steering ; gefen ; sqc , and
transition detection w2s ; topological . Such success and advancements inspire
our study of the weak measurement of the Aharonov, Albert, and Vaidman (AAV)
formalism AAV , as a means to read out the weak-valued correlation functions.

In this Letter, we examine the pivotal role of the Gell-Mann and Low theorem
and the TSVF in weak-valued correlation functions. Using the standard AAV
formalism, we extract weak-valued correlators of the perturbed quantum
harmonic oscillator from the apparatus. By measuring eigenvalues, we estimate
the final apparatus state, yielding a complex readout jozsa . Our findings
reveal the trade-off between measurement strength and the number of
experimental copies. This trade-off determines an optimal system-apparatus
coupling coefficient, allowing the closest estimation of the weak value with
limited experimental copies. We further extend our analysis to quantum field
theory utilizing the framework proposed by Dressel et al. dressel , which
presents an alternative paradigm for implementing weak measurement. We discuss
experimental realization and explore potential applications, providing a
valuable tool for further investigating correlators and quantum foundations.

Preliminary theory.– The Gell-Mann and Low Theorem, well-known from textbooks,
provides a method to derive the eigenstates of an interacting system by
adiabatically switching on the interaction. For instance, consider a solvable
system described by the Hamiltonian $H_{0}$ and its ground state $|0\rangle$
in the infinite past and future, denoted as $t\rightarrow\pm\infty$.
Introducing a perturbation $\lambda V$ with a time-dependent interaction term
$e^{-\epsilon|t|}$, where $\epsilon>0$, the total Hamiltonian is given by
$H_{\epsilon}(t)=H_{0}+e^{-\epsilon|t|}\lambda V$. By gradually turning on the
perturbation $\lambda V$, the ground state $|\Omega^{\pm}\rangle$ is prepared
at the present moment $t\rightarrow 0^{\pm}$ in the adiabatic limit
$\epsilon\rightarrow 0^{+}$. Capturing the evolution of the two vacuum states
$|0\rangle$ from the infinite past and future to the present, we express the
ground state of an interacting field in term of the Møller operator as

$\displaystyle\langle\Omega^{+}|=\lim_{\epsilon\rightarrow 0}\frac{\langle
0|U^{\dagger}_{\epsilon I}(\infty,0)}{\langle 0|U^{\dagger}_{\epsilon
I}(\infty,0)|0\rangle},~{}|\Omega^{-}\rangle=\lim_{\epsilon\rightarrow
0}\frac{U_{\epsilon I}(0,-\infty)|0\rangle}{\langle 0|U_{\epsilon
I}(0,-\infty)|0\rangle},~{}~{}~{}$ (1)

where $U_{\epsilon I}$ is the Møller operator with unique properties. One of
the most famous applications of the theorem is in relating the $n$-point
correlator of an interacting theory to its solvable part:

$G^{(n)}(\textbf{x}_{1},\cdots,\textbf{x}_{n})=\frac{\langle
0|\mathcal{T}[\hat{\phi}_{I}(\textbf{x}_{1})...\hat{\phi}_{I}(\textbf{x}_{n})\mathcal{S}]|0\rangle}{\langle
0|\mathcal{S}|0\rangle},$ (2)

where $\mathcal{T}$ is the time-ordering operator,
$\hat{\phi}_{I}(\textbf{x})$ represents the interacting field operator, and
$\mathcal{S}=U_{\epsilon I}(\infty,-\infty)$ is the scattering matrix
containing all information about the interaction. This relation simplifies the
calculation of correlation functions by connecting them to the solvable part
of the theory.

Meanwhile, in TSVF, we no longer require adiabaticity but instead consider the
eigenstates $|a\rangle$ and $|b\rangle$ at the past $t=t_{1}$ and future
$t=t_{2}$ after performing strong measurements of observables $\hat{A}$ and
$\hat{B}$. We then evolve these states forward and backward to the pre-
selected state $|\Omega^{-}\rangle=U(t_{now},t_{1})|a\rangle$ and the post-
selected state $\langle\Omega^{+}|=\langle b|U^{\dagger}(t_{2},t_{now})$ at a
certain time $t_{now}$. Accordingly, one can introduce an auxiliary potential
in the total Hamiltonian to compensate for excitations or design a schedule
for turning on the perturbation. As a result, the propagators
$U(t_{now},t_{1})$ and $U^{\dagger}(t_{2},t_{now})$ serve as sh to
adiabaticity rev ; adolfo , allowing for more flexibility in the evolution of
the system.

After establishing the equivalence, we recognize that TSVF provides an
explanation for weak values. In other words, we can interpret the correlation
function (2) as a weak value of an operator. This insight inspires us to
propose a design for a weak measurement to extract the weak-valued correlation
function using the AAV-type measurement scheme, as the extension to von
Neumann measurement. We focus on this type of measurement and refrain from
discussing the detailed implementation on a specific quantum platform at this
stage. Although the operator being measured may be a product of local
operators, the response obtained through weak measurements aligns with the
established theory of weak values nonlocal . Moreover, the theory remains
valid even in cases where nonlocal interactions are necessary for certain
tasks, enabling experimental devices to realize joint weak values jwv .

Perturbed Quantum Harmonic Oscillator.– Next, we exemplify our proposal in
quantum mechanics, by considering the perturbed quantum harmonic oscillator
(PQHO). The Hamiltonian reads $H=p^{2}/2m+m\omega x^{2}/2+\lambda x^{4}$,
where the anharmonic perturbation $\lambda x^{4}$ is treated as the self-
interaction term. Following the Gell-Mann and Low theorem, we adiabatically
turn on the self-interaction, evolving the Gaussian wave function to the
perturbed ground state $|\Omega\rangle$. By keeping the Dyson series up to the
first order, we approximate the perturbed ground state as
$|\Omega\rangle\simeq|0\rangle+[(-3/2\sqrt{2})|2\rangle-(\sqrt{6}/8)|4\rangle]\lambda$
in the eigenbasis of the harmonic oscillator $|n\rangle$, see the details in
Supplementary Material supp ). With the perturbed ground state
$|\Omega\rangle$, we can construct the two-point correlator in the Heisenberg
picture as $G^{\left(2\right)}(t,0)=\langle\Omega|x(t)x(0)|\Omega\rangle$.
Accordingly, we propose a weak measurement, to read out the value of the
correlator at an arbitrary time $t\geq 0$, without loos of generality. We
represent the operator $\hat{A}=x(t)x(0)=e^{iHt}xe^{-iHt}x$ in the basis of
$|\Omega\rangle$ as a 6-dimensional operator. The weak value $A_{w}(t)=\langle
f|\hat{A}|i\rangle/\langle f|i\rangle$ corresponds to the correlator with pre-
selection and post-selection states
$|i\rangle=|f\rangle=|\Omega^{\mp}\rangle$, where $|\Omega^{\mp}\rangle$
denotes the pre-selected and post-selected states.

In order to perform a weak measurement, we beigin by preparing the system wave
function in the pre-selection state $|i\rangle$, and the apparatus wave
function in an arbitrary state. For the apparatus, we use a qubit and prepare
the seperable initial state as follows:

$|i\rangle\otimes|\phi\rangle=|\Omega\rangle\otimes\left[\cos\left(\frac{\theta_{0}}{2}\right)|\uparrow\rangle+\sin\left(\frac{\theta_{0}}{2}\right)e^{i\varphi}|\downarrow\rangle\right],$
(3)

where $|\uparrow\rangle$ and $|\downarrow\rangle$ are the eigenbasis of
$\sigma_{z}$. Then, we couple the system and apparatus using an impulsive
interaction

$H_{\text{int}}=g(t)\hat{A}\otimes\hat{\sigma}_{y},~{}g(t)=g\delta(t-t_{0}),$
(4)

which entangles their wave functions of the system and the apparatus. After
the interaction, the system and apparatus evolve to a seperable final state
given by

$\displaystyle|f\rangle\otimes|\alpha\rangle~{}$ $\displaystyle\simeq$
$\displaystyle\langle f|i\rangle\;\left[|f\rangle\otimes
e^{-igA_{w}\hat{\sigma}_{y}}|\phi\rangle\right],$ (5)

in the weak coupling regime $g\ll 1$ after the post-selction $|f\rangle$ on
the system. Here, $A_{w}$ represents the weak value of the operator $\hat{A}$.
The weak value accumulates on the final state of the apparatus, given by
$|\alpha\rangle=e^{-igA_{w}\hat{\sigma}_{y}}|\phi\rangle$. The physical
interpretation of the weak value corresponds to a rotation through a small
angle $2(\theta-\theta_{0})=2gA_{w}$ around the $Y-$axis, assuming the weak
value is real.

In the case of a complex weak value $A_{w}=a+bi$ jozsa , we need to analyze
the effect of the non-unitary operator $e^{-igA_{w}\hat{\sigma}_{y}}$ on the
apparatus, to retrieve the value of correlator at arbitrary time. In the weak
coupling regime, the expectation values of the qubit along different
directions satisfy the following equations:

$\displaystyle\langle\hat{\sigma}_{y}\rangle_{f}$
$\displaystyle=\langle\hat{\sigma}_{y}\rangle_{i}+2gb\left[1-\langle\hat{\sigma}_{y}\rangle_{i}^{2}\right],$
$\displaystyle\langle\hat{\sigma}_{x}\rangle_{f}$
$\displaystyle=\langle\hat{\sigma}_{x}\rangle_{i}+2ga\langle\hat{\sigma}_{z}\rangle_{i}-2gb\langle\hat{\sigma}_{x}\rangle_{i}\langle\hat{\sigma}_{y}\rangle_{i},$
$\displaystyle\langle\hat{\sigma}_{z}\rangle_{f}$
$\displaystyle=\langle\hat{\sigma}_{z}\rangle_{i}-2ga\langle\hat{\sigma}_{x}\rangle_{i}-2gb\langle\hat{\sigma}_{z}\rangle_{i}\langle\hat{\sigma}_{y}\rangle_{i}.$
(6)

Here,
$\langle\hat{\sigma}_{j}\rangle_{i}=\langle\phi|\hat{\sigma}_{j}|\phi\rangle$
represents the expectation value of the spin component $\hat{\sigma}_{j}$ in
the initial apparatus state $|\phi\rangle$, and
$\langle\hat{\sigma}_{j}\rangle_{f}=\langle\alpha|\hat{\sigma}_{j}|\alpha\rangle/\langle\alpha|\alpha\rangle$
represents the expectation value of $\hat{\sigma}_{j}$ in the final apparatus
state $|\alpha\rangle$. By measuring the expectation values of the spin
components, we can access the value of correlator. With the estimations of
$\langle\hat{\sigma}_{y}\rangle_{f}$ and $\langle\hat{\sigma}_{x}\rangle_{f}$,
we can determine the imaginary and real part of the weak value as follows:

$\displaystyle
b=\frac{\langle\hat{\sigma}_{y}\rangle_{f}-\langle\hat{\sigma}_{y}\rangle_{i}}{2g\left[1-\langle\hat{\sigma}_{y}\rangle_{i}^{2}\right]},~{}a=\frac{b\langle\hat{\sigma}_{x}\rangle_{i}\langle\hat{\sigma}_{y}\rangle_{i}}{\langle\hat{\sigma}_{z}\rangle_{i}}+\frac{\langle\hat{\sigma}_{x}\rangle_{f}-\langle\hat{\sigma}_{x}\rangle_{i}}{2g\langle\hat{\sigma}_{z}\rangle_{i}}.$
(7)

It is worth noting that in the limit of $g\rightarrow 0$, they converge to the
true weak value.

Figure 1:  Weak-valued correlation function determined from (Weak-Valued
Correlation Functions: Insights and Precise Readout Strategies) for different
measurement strength $g$ during the time $t\in[0,30)$, plotted in (a) real
part and (b) imaginary part, respectively. The true values of the correlation
function are represented by a dark solid line, while the weak values are
depicted as a rainbow series of dotted lines.

Figure 1 provides valuable insights into the behavior of our protocol in the
PQHO model, highlighting the importance of operating within the weak coupling
regime to ensure accurate readout of the weak-valued correlation function. The
readout of our protocol is calculated through numerical simulations for
different coupling strengths $g$. We assume that the first-order approximation
adequately characterizes the ground state of the PQHO and consider the
calculated results as the true values. By observing the expectation values
$\langle\hat{\sigma}_{x}\rangle_{f}$ and $\langle\hat{\sigma}_{y}\rangle_{f}$
and solving Eqs. (Weak-Valued Correlation Functions: Insights and Precise
Readout Strategies), which hold only in the weak coupling regime, we retrieve
the weak-valued correlation function. Our simulations demonstrate that the
readouts deviate significantly from the true values at larger values of $g$,
indicating the occurrence of the weak-to-strong transition w2s . In this
regime, the higher-order terms in the expansion of the evolving operator
$\exp(-ig\hat{A}\otimes\hat{\sigma})$ introduce non-negligible effects on the
apparatus. However, we establish the validity of our protocol by showing that
the readouts at $g=0.1$ exhibit good agreement with the true values,
confirming the accuracy of our approach. s underscore the significance of
adhering to the weak coupling regime for reliable and precise readout of the
weak-valued correlation function.

It is important to note that the simulations presented in Fig. 1 are
idealized, assuming perfect estimations of the expectation values
$\langle\hat{\sigma_{i}}\rangle_{f}$, i.e., requiring infinite copies of the
final state (5). In practical implementations, the accuracy of our readout
depends on the coupling strength and the number of available copies. Let us
consider that we have access to $2N$ copies of the final state in a single
experiment, allowing us to perform $N$ shots of measurement on the operators
$\hat{\sigma}_{x}$ and $\hat{\sigma}_{y}$, respectively. By averaging the
eigenvalues $s_{i}=\pm 1$ as the measurement outcomes, we obtain the imperfect
estimations of $\langle\hat{\sigma_{i}}\rangle_{f}$ as
$\langle\hat{\sigma_{i}}\rangle_{E}=\sum_{i=1}^{N}s_{i}$. The discrepancy
between $\langle\hat{\sigma_{i}}\rangle_{f}$ and
$\langle\hat{\sigma_{i}}\rangle_{E}$ is bounded by
$|\langle\hat{\sigma_{i}}\rangle_{f}-\langle\hat{\sigma_{i}}\rangle_{E}|\leq\varepsilon(N,\delta)$,
where $\varepsilon(N,\delta)$ represents the upper bound on the measurement-
induced deviation with a probability of $1-\delta$. Several explicit bounds
can be used, such as the empirical Bernstein bound (EBB) bernstein , which is
applicable in various cases. Moreover, in practical scenarios, the optimal
coupling strength is not necessarily the weakest possible. When the signal-to-
noise ratio decreases, extracting meaningful information from the apparatus
becomes more challenging, leading to less precise retrieval of the correlator
value with a limited number of copies $N$. Conversely, a stronger coupling
enables a more accurate readout, albeit with a larger deviation between the
weak value and the true value. Thus, a trade-off exists between the coupling
strength g and the number of copies $2N$. Depending on specific criteria, one
can select an appropriate coupling strength for a given number of copies,
aiming to achieve a readout that sufficiently approximates the true value.

Figure 2:  Gaussian kernel density distribution of relative radical vector
$z_{RV}=\left(z_{R}-z_{TV}\right)/|z_{TV}|$ is presented in the complex plane,
for various numbers of copies $2N$ and coupling coefficient $g$. The marginal
density distribution are shown on the the up and right sides, providing
additional information about the distribution characteristics. In the upper
right corner of each subfigure, we indicate the optimal value of the coupling
coefficient $g$ for each corresponding $N$.

Following this spirit, we design a numerical experiment to retrieve the weak-
valued correlation function. The main results of this experiment are shown in
Figure 2, where we analyze the distribution of the relative error radical
vector $z_{RV}=\left(z_{R}-z_{TV}\right)/|z_{TV}|$. Here, $z_{R}$ is the
readout of weak value based on the estimations
$\langle\hat{\sigma_{i}}\rangle_{E}$ and $z_{TV}$ is the true value of the
correlator. We vary the coupling strength from weak $(g=0.1)$ to strong
$(g=1)$ with an interval of $0.1$, and examine the distribution of the radical
vector $z_{RV}$ by increasing the number of copies $2N$. It is observed that
with increasing $2N$, the distribution changes from a multiple-peak-like shape
to a Gaussian distribution on the complex plane. In the few-shot cases
($N=5,50$), the discrete nature of the values of possible readouts, resulting
from the estimations $\langle\hat{\sigma_{i}}\rangle_{E}$, is overcome by
performing Gaussian interpolation before plotting them as dots for
illustrative purposes. As $N$ increases significantly, the readout values tend
to converge to a continuum limit, approximating a complex normal distribution.
To ensure reliable distributions, we perform $M=10000$ individual simulations
with specific settings ${g_{i},N_{j}}$ to minimize statistical fluctuations.
In terms of the trade-off, our objective is to identify the coupling strength
that maximizes the likelihood of minimal deviation between the readout and the
true value in a single experiment. After analyzing the raw data, we indicate
the optimal coupling strength on each subfigure and provide the methodology
and additional information in the Supplementary Material supp . This numerical
experiment yields valuable insights into the behavior of our protocol, aiding
in the determination of the appropriate coupling strength and number of copies
for accurately reading the weak-valued correlation function.

Extension to quantum field theory.– After the pedagogical example in quantum
mechanics, we hereby extend the concept of weak-valued correlation function to
quantum field theory. In this context, the weak-valued correlation function
can be obtained from the n-point correlators of local field operators, which
represents the propagation of particle excitations in space-time and is
directly related to the weak value of the corresponding field operators. The
Lagrangian of the $\phi^{4}$ theory is written by
$\mathcal{L}=\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}-\frac{\lambda}{4!}\phi^{4}$,
from which $G^{(2)}(\textbf{x}_{1},\textbf{x}_{2})$, representing the two-
point correlator of the scalar field operator $\hat{\phi}$, is defined as
$G^{(2)}(\textbf{x}_{1},\textbf{x}_{2})=\langle\Omega|\mathcal{T}[\hat{\phi}(\textbf{x}_{1})\hat{\phi}(\textbf{x}_{2})]|\Omega\rangle$,
with $\langle\Omega|$ being the ground state of the self-interacting scalar
field. To circumvent the non-Hermiticity of the operator, we perform a polar
decomposition on the operator by $\hat{A}=\hat{U}\hat{R}$, where $\hat{U}$ is
a unitary operator and $\hat{R}$ is a Hermitian positive semidefinite operator
pati . The correlator can then be expressed as
$G^{(2)}(\textbf{x}_{1},\textbf{x}_{2})=R_{w}\langle\Theta|\Omega\rangle$,
where $R_{w}=\langle\Theta|\hat{R}|\Omega\rangle/\langle\Theta|\Omega\rangle$
is the weak value of $\hat{R}$, and
$|\Theta\rangle=\hat{U}^{\dagger}|\Omega\rangle$. In the framework of quantum
weak measurements in field theory dressel , we introduce an interaction
Hamiltonian of the AAV type,
$\hat{H}_{\text{int}}=g\delta(t-t_{0})\hat{R}\otimes\hat{\sigma}$, that
couples two degrees of local fields, where $\hat{\sigma}$ is the spin
operator. By applying the the quantum action principle GFJS ; QAP , we have
the joint weak value of the the Hermitian variation of the quantum action
$\delta\hat{S}=-\delta g\hat{R}\otimes\hat{\sigma}$, which encodes the
amplitude $a=\langle
F|e^{-i\int\hat{H}_{int}dt}|I\rangle=\langle\Theta|\otimes\langle
F_{a}|e^{-ig\hat{R}\otimes\hat{\sigma}}|\Omega\rangle\otimes|I_{a}\rangle$ as
a weak value $S_{w}=\langle F|\delta\hat{S}|I\rangle/\langle
F|I\rangle=-i\hbar\delta\ln a$, where
$|I\rangle=|\Omega\rangle\otimes|I_{a}\rangle$,
$|F\rangle=|\Theta\rangle\otimes|F_{a}\rangle$ are the initial and final
states of the field, respectively. Accordingly, we express the joint weak
values as the sum of the real and imaginary parts of weak values $R_{w}$ and
$\sigma_{w}$,

$\delta\ln p=2\delta
g\left(\text{Re}R_{w}\text{Im}\sigma_{w}+\text{Im}R_{w}\text{Re}\sigma_{w}\right),$
(8)

where $p=|a|^{2}$ is the detection probabilities and $\delta g$ is the tunable
coupling strength. To extract the real and imaginary parts of the weak value
$R_{w}$, we perform measurements on the left side of Eq. (8), utilizing
suitable pre-selection $|I_{a}\rangle$ and post-selection $|F_{a}\rangle$
states for solving the corresponding linear equations. By obtaining the real
and imaginary parts of $R_{w}$, we can retrieve the correlator by multiplying
it with $\langle\Theta|\Omega\rangle$, which interestingly coincides with the
amplitude between the selection on the first degree of the local field. To
demonstrate the protocol, we simulate the weak-valued two-time correlation
function of a $\phi^{4}$ lattice field theory with $1+1$ dimensions in Fig. 3.
We emphasize that weak measurement in quantum field theory, based on the
quantum action principle, provides an alternative paradigm for weak
measurement, differing from the standard AAV formalism. As an infinitesimal
variation, this paradigm is closely related to the path integral formalism and
can be applied in quantum mechanics. Refer to the Supplementary Material supp
for all simulation details.

Figure 3:  Weak-valued correlation function of a $\phi^{4}$ lattice field
theory are presented, determined from (8) for different measurement strength
$g$ during the time $t\in\\{-2,-1,...,2\\}$, plotted in (a) real part and (b)
imaginary part, respectively. We plot the true values of the correlation
function as dark dots, while weal values as crosses colored in rainbow.

Possible experimental implementation.– In our pursuit of implementing
protocols on real-world quantum devices, we turn to universal quantum
simulation, a flexible method for exploring quantum dynamics of both quantum
mechanics and quantum field theory dirac ; adscft . This approach allows us to
easily achieve entanglement and selections with qubits.

Alternatively, one also has the standard technique in quantum field theory by
introducing weak source currents linearly coupled to the field in the
Lagrangian, perturbing the field evolution within local apparatuses. The joint
averaged response of these apparatuses reveals the desired correlators,
shedding light on the dynamics and properties of the investigated quantum
field system.

Conclusion.– In summary, we redefine the Gell-Mann and Low theorem to develop
a method for calculating correlation functions. By employing TSVF to
accelerate adiabatic evolution, we establish an equivalent theory where
correlation functions are interpreted as weak values. To demonstrate this
concept, we propose a standard weak measurement of the AAV type to read out
the weak-valued correlation function of the PQHO. Our results reveal the
interplay between the coupling coefficient and the number of copies in the
readout. Furthermore, we extend our framework to quantum field theory, where
correlator calculations are crucial. Our versatile approach applies to various
models through universal quantum simulation. An intriguing application lies in
reading out OTOCs using weak measurement, where operators are non-time-
ordered. Additionally, we can explore scattering matrix modulation with
different pre- and post-selections to study non-equilibrium physics and
quantum nonlocality through propagators.

###### Acknowledgements.

Acknowledgements.– Discussions with Justin Dressel, Yiming Pan, and Jie Lu are
appreciated. This work is supported by NSFC (12075145), STCSM (Grants No.
2019SHZDZX01-ZX04), EU FET Open Grant EPIQUS (899368), HORIZON-
CL4-2022-QUANTUM-01-SGA project 101113946 OpenSuperQPlus100 of the EU Flagship
on Quantum Technologies, the Basque Government through Grant No. IT1470-22,
the project grant PID2021-126273NB-I00 funded by
MCIN/AEI/10.13039/501100011033 and by ”ERDFA way of making Europe” and ”ERDF
Invest in your Future”, and Nanoscale NMR and complex systems
(PID2021-126694NB-C21). X.C. acknowledges ayudas para contratos Ramón y Cajal
2015-2020 (RYC-2017-22482).

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Supplemental Material:
Weak-Valued Correlation Functions: Insights and Precise Readout Strategies

##  .1 Ground state calculation from Gellmann-Low Theorem

To calculate the ground state $|\Omega\rangle$ of the perturbed quantum
harmonic oscillator (PQHO) Hamiltonian using the Gell-Mann and Low theorem, we
consider the adiabatic Hamiltonian given by:

$H_{\epsilon}(t)=\frac{p^{2}}{2m}+\frac{1}{2}m\omega x^{2}+\lambda
e^{-\epsilon|t|}x^{4}.$ (S1)

We begin by focusing on the ground state $|\Omega^{-}\rangle$. Expanding the
state in terms of a perturbation series, we express the target state as the
ratio of two sets of infinite series:

$\displaystyle|\Omega^{-}\rangle=\lim_{\epsilon\rightarrow 0}\frac{U_{\epsilon
I}(0,-\infty)|0\rangle}{\langle 0|U_{\epsilon
I}(0,-\infty)|0\rangle}=\lim_{\epsilon\rightarrow
0}\frac{\sum_{n=0}^{\infty}c_{n}\lambda^{n}}{\sum_{n=0}^{\infty}b_{n}\lambda^{n}}=\sum_{n=0}^{\infty}a_{n}\lambda^{n},$
(S2)

where $U_{\epsilon I}$ is the adiabatic time evolution operator in the
interacting picture. The series division on the right-hand side is implemented
as:

$a_{n}={1}/{b_{0}}\left[c_{n}-a_{0}b_{n}-...-a_{n-1}b_{1}\right],~{}\left(n=0,1,2...\right).$

Here, $a_{n}$ represents the coefficients of the perturbation series expansion
of the ground state, which we aim to determine. The calculation involves
solving a recursive equation to obtain the values of $a_{n}$. By following
this procedure, we can determine the ground state of the PQHO Hamiltonian
using the Gell-Mann and Low theorem.

Indeed, complex analysis reveals that the series representation
$|\Omega^{-}\rangle=\sum_{n=0}^{\infty}a_{n}^{(\epsilon)}\lambda^{n}$ is a
divergent series, with an asymptotic expansion only valid to the first order.
In the Argand plane, considering $|\Omega^{-}\rangle$ as a complex function of
$\lambda$, it is non-analytic everywhere except at the origin. This can be
understood by considering the case where $\lambda<0$. In this scenario, there
is no minimum energy for the potential, resulting in the non-existence of the
ground state $|\Omega^{-}\rangle$. Consequently, there is no circle of
convergence outside the origin point.

The order of asymptotic expansion is determined in the following way. In terms
of Dyson expansion, time evolution operator is written as:

$\displaystyle U_{\epsilon I}(t,t_{0})=1+(-i)\int_{t_{0}}^{t}dt_{1}H_{\epsilon
I}(t_{1})+(-i)^{2}\int_{t_{0}}^{t}dt_{1}\int_{t_{0}}^{t_{1}}dt_{2}H_{\epsilon
I}(t_{1})H_{\epsilon I}(t_{2})$
$\displaystyle+(-i)^{3}\int_{t_{0}}^{t}dt_{1}\int_{t_{0}}^{t_{1}}dt_{2}\int_{t_{0}}^{t_{2}}dt_{3}H_{\epsilon
I}(t_{1})H_{\epsilon I}(t_{2})H_{\epsilon I}(t_{3})+...,$ (S3)

where $H_{\epsilon I}(t)=\lambda
e^{iH_{0}t}e^{-\epsilon\left|t\right|}x^{4}e^{-iH_{0}t}$ is the the adiabatic
Hamiltonian in the interaction picture. Acting on the $|0\rangle$ and
inserting identity operator $1=\sum_{n}|n\rangle\langle n|$, it leads to:

$\displaystyle U_{\epsilon I}(t,t_{0})~{}|0\rangle$ $\displaystyle=$
$\displaystyle|0\rangle+(-i)\lambda\sum_{n}|n\rangle\int_{-\infty}^{0}dt_{1}e^{\left(in+\epsilon\right)t_{1}}\langle
n|x^{4}|0\rangle$ (S4) $\displaystyle+$
$\displaystyle(-i)^{2}\lambda^{2}\sum_{n,m}|n\rangle\int_{-\infty}^{0}dt_{1}\int_{-\infty}^{t_{1}}dt_{2}~{}e^{\left[i\left(n-m\right)+\epsilon\right]t_{1}}e^{\left(im+\epsilon\right)t_{2}}\langle
n|x^{4}|m\rangle\langle m|x^{4}|0\rangle+...$ $\displaystyle=$
$\displaystyle|0\rangle+(-i)\lambda\sum_{n}\frac{\langle
n|x^{4}|0\rangle}{in+\epsilon}|n\rangle+(-i)^{2}\lambda^{2}\sum_{n,m}\frac{\langle
n|x^{4}|m\rangle\langle
m|x^{4}|0\rangle}{\left(in+2\epsilon\right)\left(im+\epsilon\right)}|n\rangle+...,$

where $|n\rangle$ is the eigenbasis of harmonic oscillator satisfying
$\hat{H_{0}}|n\rangle=E_{n}|0\rangle$ and $E_{n}=n+\frac{1}{2}$. The ground
state could be derived by putting this into (S2) and calculating the series
division:

$\displaystyle\mid\Omega^{-}\rangle=\sum_{n}a_{n}\lambda^{n}$
$\displaystyle=a_{0}+a_{1}\lambda+a_{2}\lambda^{2}+...,$ $\displaystyle
a_{0}=\langle x\mid
0\rangle,~{}a_{1}=-\frac{3}{2\sqrt{2}}|2\rangle-\frac{\sqrt{6}}{8}|4\rangle$
$\displaystyle,~{}a_{2}=\frac{75}{8\sqrt{2}}|2\rangle+\frac{9\sqrt{6}}{4}|4\rangle+\frac{17}{16}\sqrt{5}|6\rangle+\frac{3\sqrt{70}}{64}|8\rangle.$
(S5)

To find the optimal order for asymptotic expansion, we study correlation
function at $t=0$:
$G^{(2)}(0,0)={\langle\Omega|x^{2}|\Omega\rangle}/{\langle\Omega|\Omega\rangle}$.
Our simulation result of imaginary-time evolution gives $0.3701$. Reserving
(.1) to 1-order gives 0.3570, which is close to simulation result. However,
when reserved to 2-order, it becomes 0.8484, with more deviation. As expected,
the higher order it is, the greater deviation we get. Likewise,
$|\Omega^{+}\rangle$ could be derived in a similar way and it has been proven
that $|\Omega^{+}\rangle=|\Omega^{-}\rangle=|\Omega\rangle$. As a result,
1-order is the optimal asymptotic series for $|\Omega\rangle$:

$\displaystyle|\Omega\rangle\simeq|0\rangle+\left[-\frac{3}{2\sqrt{2}}|2\rangle-\frac{\sqrt{6}}{8}|4\rangle\right]\lambda.$
(S6)

Figure S1:  Correlation function simulated under different cutoff on energy
level $N_{\text{cutoff}}$. The red-blue curves show the result calculated from
(.1) under the $N_{\text{cutoff}}$ shown in the subfigure above. The black
curves are used to show numerical simulation of solving the eigenequations of
the Hamiltonians under a certain large enough cutoff energy level
$N_{\text{cutoff}}=100$.

##  .2 Truncated eigenbasis of harmonic oscillator

In order to address the challenge of infinite dimensionality in the
calculation of the operator $\hat{A}=x(t)x(0)=e^{i\hat{H}t}xe^{-i\hat{H}t}x$,
a truncation approach is employed. This approach involves selecting a cutoff
energy level $N_{\text{cutoff}}$ that ranges from a sufficiently large value
down to zero. For the sake of better illustration, we simulate the
eigenequations of the Hamiltonians under the sufficient large cutoff on energy
level $N_{\text{cutoff}}=100$.

By expressing the operator $\hat{A}$ in terms of the eigenbasis of the
harmonic oscillator, the cutoff energy level determines the range of energy
levels considered in the calculation. As shown in Fig. S1, the behavior of the
correlation function $G^{(2)}(t,0)$ is observed as the cutoff energy level
decreases. It is noted that as $N_{\text{cutoff}}$ is reduced to
$N_{\text{cutoff}}=6$, the result from our calculation (.1) matches with the
simulation result and the value of $G^{(2)}(0,0)$ remains constant at
$0.35697259$. However, a turning point is reached when $N_{\text{cutoff}}$
reaches $N_{\text{cutoff}}=5$, resulting in obvious deviation from simulation
result and $G^{(2)}(0,0)=0.34803337$. This observation can be understood by
considering that the calculated ground state is only relevant to the first
five energy levels.

Based on these observations, a decision is made to choose
$N_{\text{cutoff}}=6$ as the truncated energy level for the calculation. This
choice ensures that the relevant energy levels are included while controlling
the computational complexity associated with the infinite-dimensional space.

##  .3 Complex weak value through the spin of the pointer

For a complex weak value $A_{w}=a+bi$ measured through interaction
$\hat{A}\otimes\hat{\sigma}_{y}$, any observable $\hat{M}$ of apparatus
satisfies:

$\displaystyle\left\langle\hat{M}\right\rangle_{f}=\left\langle\hat{M}\right\rangle_{i}+iga\left\langle\hat{\sigma}_{y}\hat{M}-\hat{M}\hat{\sigma}_{y}\right\rangle_{i}+gb\left[\left\langle\hat{\sigma}_{y}\hat{M}+\hat{M}\hat{\sigma}_{y}\right\rangle_{i}-2\left\langle\hat{M}\right\rangle_{i}\left\langle\hat{\sigma}_{y}\right\rangle_{i}\right],$
(S7)

This equation is derived as follows. The initial and final state of apparatus
is $|\phi\rangle$ and
$|\alpha\rangle=e^{-igA_{w}\hat{\sigma}_{y}}|\phi\rangle\simeq\left(1-igA_{w}\hat{\sigma}_{y}\right)|\phi\rangle$,
where $|\phi\rangle$ is normalized but $|\alpha\rangle$ is not. The
expectation of any operator $\hat{M}$ in the final state is, reserved to
1-order with the help of series division:

$\displaystyle\left\langle\hat{M}\right\rangle_{f}$
$\displaystyle=\frac{\langle\alpha|\hat{M}|\alpha\rangle}{\langle\alpha|\alpha\rangle},$
(S8)
$\displaystyle\simeq\frac{\langle\phi|\left(\hat{M}-igA_{w}\hat{M}\hat{\sigma}_{y}+ig\overline{A}_{w}\hat{\sigma}_{y}\hat{M}\right)|\phi\rangle}{\langle\phi|\left(1-igA_{w}\hat{\sigma}_{y}+ig\overline{A}_{w}\hat{\sigma}_{y}\right)|\phi\rangle},$
(S9)
$\displaystyle=\frac{\langle\hat{M}\rangle+\left[ia\langle\hat{\sigma}_{y}\hat{M}-\hat{M}\hat{\sigma}_{y}\rangle_{i}+b\langle\hat{\sigma}_{y}\hat{M}+\hat{M}\hat{\sigma}_{y}\rangle_{i}\right]g}{1+\left[2b\langle\hat{\sigma}_{y}\rangle_{i}\right]g},$
(S10)
$\displaystyle\simeq\langle\hat{M}\rangle_{i}+\left[ia\left\langle\hat{\sigma}_{y}\hat{M}-\hat{M}\hat{\sigma}_{y}\right\rangle_{i}+b\left\langle\hat{\sigma}_{y}\hat{M}+\hat{M}\hat{\sigma}_{y}\right\rangle_{i}-2b\left\langle\hat{M}\right\rangle_{i}\left\langle\hat{\sigma}_{y}\right\rangle_{i}\right]g,$
(S11)

which, after appropriate arrangement, gives (S7). Putting spin components into
$M$ in (S7), we derive

$\displaystyle\langle\hat{\sigma}_{y}\rangle_{f}$
$\displaystyle=\langle\hat{\sigma}_{y}\rangle_{i}+2gb\left[1-\langle\hat{\sigma}_{y}\rangle_{i}^{2}\right],$
(S12) $\displaystyle\langle\hat{\sigma}_{x}\rangle_{f}$
$\displaystyle=\langle\hat{\sigma}_{x}\rangle_{i}+2ga\langle\hat{\sigma}_{z}\rangle_{i}-2gb\langle\hat{\sigma}_{x}\rangle_{i}\langle\hat{\sigma}_{y}\rangle_{i},$
(S13) $\displaystyle\langle\hat{\sigma}_{z}\rangle_{f}$
$\displaystyle=\langle\hat{\sigma}_{z}\rangle_{i}-2ga\langle\hat{\sigma}_{x}\rangle_{i}-2gb\langle\hat{\sigma}_{z}\rangle_{i}\langle\hat{\sigma}_{y}\rangle_{i}.$
(S14)

Figure S2:  On the upper panels are the absolute mean values for $M=10,000$
sampling points of (a) complex value
$z_{RV}=\left(z_{R}-z_{TV}\right)/|z_{TV}|$, (b) module
$|z_{RV}|=|z_{R}-z_{TV}|/|z_{TV}|$, (c) real part
$\text{Re}\left\\{z_{RV}\right\\}=\text{Re}\left\\{z_{R}-z_{TV}\right\\}/|z_{TV}|$
and (d) imaginary part
$\text{Im}\left\\{z_{RV}\right\\}=\text{Im}\left\\{z_{R}-z_{TV}\right\\}/|z_{TV}|$.
Error bars denote confidence intervals of 0.95. On the lower panels are
uncertainty versus the number of copies: $1/\delta^{2}-N$ for $g\in
G=\left\\{0.1,0.2,...,1\right\\}$, over $M=10,000$ sampling points of (e)
complex value $z_{RV}=\left(z_{md}-z_{tv}\right)/|z_{tv}|$, (f) module
$|z_{RV}|=|z_{md}-z_{tv}|/|z_{tv}|$, (g) real part
$\text{Re}\left\\{z_{RV}\right\\}=\text{Re}\left\\{z_{md}-z_{tv}\right\\}/|z_{tv}|$
and (h) imaginary part
$\text{Im}\left\\{z_{RV}\right\\}=\text{Im}\left\\{z_{md}-z_{tv}\right\\}/|z_{tv}|$.
The relation $1/\delta^{2}-N$ proves to be linear in all the four cases.

##  .4 Numerical simulation and detailed trade-off data

In the numerical simulation, we focus on a specific time slice at $t=5.1$ and
consider a set of coupling constants $g\in G=\left\\{0.1,0.2,...,1\right\\}$
for investigation. The numbers of copies $N$ and repeated experiments $M$ are
both integers, with $N$ ranging from 1 to 500,000, and $M$ fixed at 10,000.
Each repeated experiment is assigned a unique random seed ranging from 10 to
10,010.

For each experiment, we generate $N$ random numbers following a uniform
distribution over the interval $[0,1)$. These random numbers are used to
determine the measured values of $\langle\hat{\sigma}_{x}\rangle_{f}$ and
$\langle\hat{\sigma}_{y}\rangle_{f}$. To calculate
$\langle\hat{\sigma}_{x}\rangle_{f}$, we assign a value of $-1$ to random
numbers that are smaller than
$|\langle\sigma_{x}=-1~{}|\alpha\rangle|^{2}/\langle\alpha|\alpha\rangle$, and
$+1$ otherwise. Similarly, for $\langle\hat{\sigma}_{y}\rangle_{f}$, we use
$|\langle\sigma_{y}=-1~{}|\alpha\rangle|^{2}/\langle\alpha|\alpha\rangle$ as
the threshold to assign $\pm 1$ to the random numbers.

To evaluate the accuracy of the measured results, we focus on the relative
deviation from the true value, given by $z_{RV}=(z_{R}-z_{TV})/|z_{TV}|$.
Utilizing equation (7), we obtain $M=10,000$ sampling points in the complex
plane, denoted as ${z_{RV}(m;~{}g,N),~{}m\in[0,~{}M)}$, for each combination
of $g\in G$ and $N\in[1,~{}500,000]$.

In the initial step, we study several important statistical quantities to gain
a global understanding of the dataset. Specifically, we calculate the mean
value and variance for the complex value, module, real part, and imaginary
part of ${z_{RV}}$. The upper panels of Fig. S2 summarize the results for
$g\in G$ and $N\in[1,~{}5000]$. These panels clearly illustrate the trade-off
between $g$ and $N$ as the transition from $N\sim 10$ to $N\sim 1,000$ becomes
apparent.

Additionally, we explore the uncertainty properties, paying particular
attention to their dependence on $N$ and $g$. The lower panels of Fig. S2
depict the variance versus the number of copies $1/\delta^{2}-N$ for the
complex value, module, real part, and imaginary part of ${z_{RV}}$.
Horizontally, the perfect linearity in all four cases verifies our prediction
that $\delta\sim 1/\sqrt{N}$. Vertically, the consistent ordering of $g\in G$
for a given $N$ demonstrates that the uncertainty increases with smaller
values of $g$.

Finally, we investigate the trade-off between $g$ and $N$ by determining the
optimal value of $g\in G$ for different ranges of $N$. Fig. S3 presents the
optimal $g\in G$ for each $N\in[1,~{}10,000]$. Our criteria for optimality are
as follows: for a given $N$, we compare the distances to the true value
${z_{RV}(m;~{}g_{i},N)-z_{RV}(m;~{}g_{j},N),~{}m\in[0,~{}M)}$ for any two
$g_{i},~{}g_{j}\in G$. If the total number of cases where
$\left[z_{RV}(m;~{}g_{i},N)-z_{RV}(m;~{}g_{j},N)<0\right]$ is greater than
half of the total sampling $M=10,000$, we consider $g_{i}$ to be dominant. The
optimal value is the one that dominates over all other $g\in G$. In the case
of a draw, we choose the one with the smallest value of $g$.

Figure S3:  Detailed trade-off data of optimal $g\in G$ for each
$N\in\left[1,10000\right]$.

##  .5 Lattice simulation of $\phi^{4}$ field

We carry out the lattice simulation of scalar $\phi^{4}$ quantum field in a
$(1+1)-d$ discretized Minkovski space-time, with periodic boundary conditions
for both space and time. The space-time lattices are taken as integers given
by: $x^{\mu}=(t,x)=-l,-(l-1),\ldots,l-1,l$ with the lattice spacing $a=1$. We
set the initial configuration of $\phi^{4}$ as:

$\displaystyle\phi_{0}(x)=$ $\displaystyle\sqrt{L}\sum_{p}\frac{\Delta
p}{2\pi}\frac{1}{\sqrt{2E_{P}}}\left(\hat{a}_{P}e^{-iPx}+\hat{a}^{\dagger}_{P}e^{iPx}\right),$
(S15) $\displaystyle\pi_{0}(x)=$ $\displaystyle\sqrt{L}\sum_{p}\frac{-i\Delta
p}{2\pi}\sqrt{\frac{E_{P}}{2}}\left(\hat{a}_{P}e^{-iPx}-\hat{a}^{\dagger}_{P}e^{iPx}\right).$
(S16)

where the energy-momentum $E_{P}={(m^{2}+P^{2})}^{1/2}$, $P=({2\pi}/{L})x$,
the commutation relations $\left[\phi_{0}(x),\pi_{0}(y)\right]=i\delta_{xy}$,
$\left[\hat{a}_{P},\hat{a}^{\dagger}_{P}\right]=\delta_{nm}$, and $L=2l+1$.
The field operator of $\phi^{4}$ could be obtained by switching on time
evolution:

$\displaystyle\phi(t,x)=e^{i\hat{H}t}\phi(0,x)e^{-i\hat{H}t},$ (S17)

where the Hamiltonian is given by $\hat{H}=1/2\sum_{x}\Delta
x\left[{\pi_{0}}^{2}+(\nabla\phi_{0})^{2}+m^{2}\phi_{0}^{2}+\lambda/{4!}\phi_{0}^{4}\right]$.

In our numerical simulation, we consider $L=5$, with a truncated Hilbert
subspace $N_{\text{cutoff}}=5$ for each individual lattice. The total
dimension of the Fock space is $N=5^{5}=3125$. The ground state
$|\Omega\rangle$ is obtained by numerically solving the eigenequations of
Hamiltonian. We define the measured operator as

$\displaystyle\hat{A}=\mathcal{T}\phi(t,0)\phi(0,0),$ (S18)

and perform polar decomposition with the SVD method:

$\displaystyle\hat{A}=\hat{u}\hat{s}\hat{v}=(\hat{u}\hat{v})(\hat{v}^{\dagger}sv)=\hat{U}\hat{R}.$
(S19)

To make the WM of $\hat{R}$, we choose spin-x as the apparatus. The initial
and final spin states are taken as $|I_{a}\rangle_{1}=|\uparrow\rangle$,
$|F_{a}\rangle_{1}=1/\sqrt{2}\left(|\uparrow\rangle+|\downarrow\rangle\right)$
and
$|I_{a}\rangle_{2}=1/\sqrt{2}\left(i|\uparrow\rangle+|\downarrow\rangle\right)$,
$|F_{a}\rangle_{2}=1/\sqrt{2}\left(|\uparrow\rangle+|\downarrow\rangle\right)$,
resulting in a direct expression of $R_{w}$ through $\delta p$:

$\displaystyle\text{Re}\left\\{R_{w}\right\\}$ $\displaystyle=\delta\ln
p_{2}/{2\delta g},$ $\displaystyle\text{Im}\left\\{R_{w}\right\\}$
$\displaystyle=\delta\ln p_{1}/{2\delta g}.$ (S20)

Here the variation of detection probabilty is simulated by the difference
between the perturbed and the unperturbed $\delta
p=p^{\prime}-p=|\langle\Theta|\otimes\langle
F_{a}|~{}e^{-ig\hat{R}\otimes\hat{\sigma}}~{}|\Omega\rangle\otimes|I_{a}\rangle|^{2}-|\langle\Theta|\otimes\langle
F_{a}|\Omega\rangle\otimes|I_{a}\rangle|^{2}$. Our simulation result of
$R_{w}$ is shown in Fig. S4. Similar to its quantum counterpart, the accuracy
of QFT weak measurement increases with small coupling $g$.

In order to explore the connection between weak measurements in quantum field
theory (QFT) and quantum mechanics (QM), we consider the one-lattice limit
where the behavior of the $\phi^{4}$ field theory resembles that of the PQHO.
By numerically solving the eigenequations of the Hamiltonians, we obtain the
ground states for both systems. The results, shown in Fig. S5, reveal a good
agreement in terms of the true values and trends for different coupling
strengths between weak measurements in QFT and QM. However, it is worth noting
that weak measurements in QFT, based on the quantum variation principle,
exhibit less accuracy compared to weak measurements in QM under the same
conditions. This discrepancy arises due to the approximation $\delta
p=p^{\prime}-p$, which introduces uncertainty in momentum $\delta p$. In the
QFT context, this approximation may not hold as well, leading to less precise
results in weak measurements compared to QM. Further investigation and
analysis are necessary to understand the specific factors contributing to the
differences between weak measurements in QFT and QM. By examining the
limitations and approximations involved, we can gain insights into the nature
of weak measurements in both frameworks.

Figure S4:  Numerical simulation of a $\phi^{4}$ lattice field theory. On the
upper panels are the weak-valued effective hermitian operators $R_{w}$ of a
$\phi^{4}$ lattice field theory determined from (.5) for different measurement
strength $g$ during the time $t\in\\{-2,-1,...,2\\}$, plotted in (a) real
part, (b) imaginary part, and (c) module. We plot the true values of the
correlation function as dark dots, while weal values as crosses colored in
rainbow. On the lower panels are the corresponding weak-valued correlation
functions obtained from $G(t,0)=R_{w}\langle\Theta|\Omega\rangle$, plotted in
(d) real part, (e) imaginary part, and (f) module. Figure S5:  Comparison of
one-lattice $\phi^{4}$ theory with PQHO weak-valued correlation function for
different measurement strength $g$ during the continuous time
$t\in\left[0,30\right)$. On the upper panels are the simulations of weak
measurement in QFT with (a) real part and (c) imaginary part, while on the
lower panels are the simulations of weak measurement in QM with (b) real part
and (d) imaginary part.