# CLEAN–EVAL: Clean Evaluation on Contaminated Large Language Models

Wenhong Zhu1 Hongkun Hao1 Zhiwei He1 Yunze Song2
Yumeng Zhang3 Hanxu Hu4 Yiran Wei2 Rui Wang122footnotemark: 2 Hongyuan Lu2
1Shanghai Jiao Tong University 2FaceMind Corporation
3Tsinghua University 4University of Edinburgh
{zwhong714, haohongkun, zwhe.cs<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Partial work was done when Wenhong Zhu was interning at FaceMind Corporation.
Hongyuan Lu and Rui Wang are corresponding authors.

###### Abstract

We are currently in an era of fierce competition among various large language
models (LLMs) continuously pushing the boundaries of benchmark performance.
However, genuinely assessing the capabilities of these LLMs has become a
challenging and critical issue due to potential data contamination, and it
wastes dozens of time and effort for researchers and engineers to download and
try those contaminated models. To save our precious time, we propose a novel
and useful method, Clean-Eval, which mitigates the issue of data contamination
and evaluates the LLMs in a cleaner manner. Clean-Eval employs an LLM to
paraphrase and back-translate the contaminated data into a candidate set,
generating expressions with the same meaning but in different surface forms. A
semantic detector is then used to filter the generated low-quality samples to
narrow down this candidate set. The best candidate is finally selected from
this set based on the BLEURT score. According to human assessment, this best
candidate is semantically similar to the original contamination data but
expressed differently. All candidates can form a new benchmark to evaluate the
model. Our experiments illustrate that Clean-Eval substantially restores the
actual evaluation results on contaminated LLMs under both few-shot learning
and fine-tuning scenarios.

##  1 Introduction

In recent years, LLMs have made breakthroughs in handling complex and nuanced
scenarios, achieved superior performance in some professional and academic
benchmarks, and attracted a lot of resources from industry and academia OpenAI
(2023); Touvron et al. (2023); Golchin and Surdeanu (2023). This subsequently
opened the arms race era of LLMs, and various LLMs were continuously launched,
such as GPT-4 OpenAI (2023), LLama2 Touvron et al. (2023), Baichuan2 Yang et
al. (2023) and other LLMs, which have refreshed various evaluation benchmarks
continuously.

Figure 1: The motivation for this paper. Today’s LLMs may have intentionally
or unintentionally included benchmark data in the training set, and we hope
that the data set we calibrated can truly evaluate the capability of the LLMs.

There is room for doubt regarding the potential overestimation of these
benchmark measurements. One of the reasons is that LLMs are trained on data
extracted from websites and publicly accessible datasets OpenAI (2023);
Touvron et al. (2023). Therefore, it becomes challenging to guarantee the
purity of the benchmark datasets depicted in Figure 1. This, in turn,
introduces a significant concern: the risk of data contamination.

Data contamination refers to the model’s pre-training data containing the test
data, resulting in slightly higher test performance Magar and Schwartz (2022);
Golchin and Surdeanu (2023). Currently, most models do not disclose their
training sets in their technical reports, leading to questions about whether
some benchmark datasets are included in their training sets. It is an urgent
problem that needs to be solved Wei et al. (2023), as it wastes dozens of time
and effort for both researchers and engineers to download and try those
contaminated models. We raise an urgent call to evaluate LLMs in a cleaner
manner. If the real power of LLMs can be evaluated, it can better promote the
development of the community of LLMs. Currently, the most effective way to
tackle this problem is to re-label a new data set after the new model is
introduced to evaluate its capabilities. Unfortunately, it is time-consuming
and laborious.

In this paper, we entirely use the previously proposed data sets to construct
a new evaluation data set, that is, use the LLM to calibrate these data sets
to reduce the phenomenon of data contamination. Our method is called Clean-
Eval, and the primary objective is to assess LLMs’ capabilities accurately.
Leveraging these models’ remarkable creative generation abilities, we engage
in diverse levels of paraphrasing and back-translation in multiple language
directions on data that may have been contaminated. The outcome yields a
candidate pool of calibrated data. Employing semantic detectors, we filter out
low-quality samples and determine the best items according to the BLEURT score
between the calibration and contamination data. Experimental analysis revealed
that a lower BLEURT score achieved through back-translation and paraphrasing
produces noticeable calibrated effects. Clean-Eval, a blend of paraphrasing
and back-translation, demonstrates the potential to enhance sentence
structure, grammar, and linguistic diversity while preserving the underlying
semantics, as confirmed through human evaluation results. Given our inability
to ascertain whether the model has been tainted within a specific dataset, we
employ a simulated data contamination method. For in-context learning, we
control the contaminated data in the construction of the demonstration. For
fine-tuning we utilize the contaminated data to fine-tune the model. Our
experiments have demonstrated that the calibrated data processed through our
method effectively restores the model’s actual performance.

(a) Data collection.

(b) Paraphrase and Back-translation.

(c) Filter.

Figure 2: An overview of our method.

##  2 Related Work

###  2.1 Data Contamination Detection

Data contamination refers to the scenario in which test data becomes
inadvertently incorporated into the model’s training data. Model trainers and
model users have done much work on data contamination detection.

##### Model Trainers.

Brown et al. (2020) conducted experiments addressing data contamination in
their report. They employed an n-gram overlap metric to assess the level of
duplication between the training and test sets and subsequently removed such
duplications from the training dataset. Similarly, Dodge et al. (2021)
measured exact matches, considering capitalization and punctuation
normalization, which means evaluating whether the entire input of an
evaluation text exists in the training data. However, Touvron et al. (2023)
argued that previous researchers’ high-order n-gram-based detection method
could not precisely determine the extent of contamination within a given
sample. Instead, they proposed identifying contamination at the token level
and allowing matched spans between an evaluation sample and the training data
to differ in a few positions to accommodate minor variations in overlap. Wei
et al. (2023) compared the LM loss across the training split, test split, and
a specifically curated reference set generated by GPT-4 to mimic all the
datasets; a lower value indicates potential contamination.

##### Model Users.

Carlini et al. (2023) construct a set of prompts using the model’s training
data. They investigated by supplying prefixes of these prompts to the trained
model to assess the model’s capacity to complete the remaining portion of the
example verbatim. Their study revealed that as the model’s capacity,
duplicated numbers, and context length increased, the models would be more
proficient in memorizing data. Meanwhile, Golchin and Surdeanu (2023)
introduced a groundbreaking approach involving the development of guided
instructions that include the initial segment of a data instance and its
corresponding partition name. These guided instructions are subsequently
utilized to induce the model to generate the second part of the data, based on
a provided prompt. Rouge, BLEURT, and GPT4 auto evaluation determine whether
the model had data contamination. Furthermore, Li (2023) analyzed six
prominent multi-choice QA benchmarks, quantifying their overlap with the
training dataset of Llama to detect potential data contamination.

###  2.2 Existing Benchmark

Many benchmarks have been proposed, including MMLULi et al. (2023a),
CEVALHuang et al. (2023), etc., to measure the capability of LLMs
comprehensively. However, labeling a large data set is time-consuming and
laborious, and ensuring no overlap with the model’s training set is often
challenging. There is also work to reformulate existing benchmarks to build
new ones. Li et al. (2023b) propose ReForm-Eval to reformulate existing
benchmarks into unified large vision-language model compatible formats.

Nevertheless, based on our knowledge, there is no proposed solution to the
problem of data contamination causing excessive model evaluation performance.
In this paper, we propose an effective method to mitigate this problem.
Experiments demonstrate that our methods work in evaluating both closed and
open LLMs.

##  3 Clean-Eval

The framework of our method is shown in Figure 2. Our methodology comprises
three primary stages. Initially, we concentrate on gathering established
benchmarks to assess LLMs, as shown in Figure 2(a). In the subsequent phase,
we meticulously cleaned contamination in the collected benchmarks. This
involves refining samples using the creative capacities of the LLM and
performing multi-language back translation on the data as shown in Figure
2(b). In the final phase, we filter the outcomes of the contamination cleanup,
eliminating subpar results and selecting the ultimate results based on the
BlEURT score.

###  3.1 Back-translation

Back-translation (BT) involves retranslating content from the target language
into its source language using literal terms (Sennrich et al., 2016). In this
process, slight differences can be introduced, such as replacing synonyms.
Therefore, we translate the raw data into various language orientations and
then revert to the original language to compose our candidate set of
contamination cleanup data. We aim to achieve a distinct expression from the
original sample while preserving the semantics.

###  3.2 Paraphrase

There are three levels of paraphrase: sentence simplification, sentence
complexification, and the same level, which all depend on syntactic and
lexical complexity Chi et al. (2023). In recent years, LLMs have shown great
promise in some professional fields such as creative writing Touvron et al.
(2023). We utilize the creative power of large language models to paraphrase
the raw data multiple times but keep differences, which likewise constitutes a
candidate set of contamination cleanup data. These candidate sets contain more
syntactic and lexical diversity.

###  3.3 Filter

However, these candidate sets might need to be further examined to ensure
their quality. As shown in Figure 2(c), we use a detector to judge whether the
content in the candidate set is semantically similar to the original content
to reduce the set of candidate sets further and select the candidate according
to the BlEURT score as the final result.111This detector is optional. Removing
the detector saves computational and token costs, but can potentially degrade
the quality of the selected candidates.

With these essential steps, we have achieved greater efficiency in harnessing
existing datasets, mitigated data contamination concerns, and furnished high-
calibrated new data suitable for evaluating model performance. This strategy
optimizes resource allocation and lays a robust foundation for the model’s
ongoing advancement.

##  4 Evaluation Setting

(a) In-context Learning

(b) Fine-tuning

Figure 3: An overview of our evaluation setting.

Nearly all large language models operate with proprietary training datasets,
making it challenging to ascertain whether the data being tested is free from
contamination. To address this issue, we focus on evaluating the impact of
contamination data before and after calibration, and to facilitate this, we
introduce an experimental framework for simulating data contamination.

Evaluation Type | RTE | QQP | QNLI | MRPC | MNLI | WNLI
---|---|---|---|---|---|---
Contamination | 96.39 | 95.00 | 90.67 | 93.67 | 80.00 | 95.78
Clean | 84.12 | 83.33 | 80.00 | 68.33 | 71.00 | 54.93
Calibration | 76.90 $\Downarrow$ | 80.67 $\Downarrow$ | 78.00 $\Downarrow$ | 67.00 $\Downarrow$ | 71.67 $\Downarrow$ | 71.83 $\Downarrow$
Evaluation Type | SNLI | IMDB | PIQA | COPA | CB | BOOLQ
Contamination | 94.00 | 95.67 | 86.33 | 92.00 | 98.21 | 87.33
Clean | 73.67 | 89.00 | 80.33 | 90.00 | 82.14 | 81.33
Calibration | 62.00 $\Downarrow$ | 91.00 $\Downarrow$ | 76.33 $\Downarrow$ | 73.00 $\Downarrow$ | 69.64 $\Downarrow$ | 71.67$\Downarrow$
Evaluation Type | SST2 | AG News | GSM8K | MultiArith | MMLU | CEVAL
Contamination | 90.67 | 55.00 | 64.33 | 65.00 | 73.67 | 66.33
Clean | 80.00 | 43.00 | 12.33 | 35.00 | 59.00 | 41.00
Calibration | 78.00 $\Downarrow$ | 54.00 $\Downarrow$ | 50.67 $\Downarrow$ | 60.00 $\Downarrow$ | 57.00 $\Downarrow$ | 38.33 $\Downarrow$
Table 1: ICL experiments and metrics is accuracy. $\Downarrow$ is compared to
the contamination dataset. The optimal candidate is chosen according to the
lowest BLEURT score.

###  4.1 In-context Learning

In-context learning (ICL) is a specific technique for presenting a task
demonstration to the model as part of a natural language prompt. Brown et al.
(2020) has proposed that LLMs fall under the category of few-shot learners.
Therefore, we have adopted ICL to simulate the data contamination issue. Under
the ICL scenario, we propose and compare three kinds of evaluation settings
including contamination, clean, and calibration for any given benchmark.

##### Contamination Setting.

For each benchmark, we create a contamination dataset $\mathcal{X}$, which
consists of $n$ samples $s_{1},s_{2},\cdots,s_{n}$. Each sample $s_{i}$ is a
tuple consisting of $(\mathrm{demonstration},\mathrm{tested\quad sample})$.
The demonstration is chosen from the benchmark’s validation or testing set,
and the tested sample is the same as the demonstration, as shown in the left
part of Figure 3(a). In this setting, we aim to simulate the scenario where
LLM is contaminated.

##### Clean Setting.

Similarly, we create a clean dataset $\mathcal{X}_{\text{clean}}$ for each
benchmark. The only difference is that the demonstration of each sample is
randomly chosen from the benchmark’s training set, as shown in the middle part
of Figure 3(a). In this setting, we aim to simulate the scenario where LLM is
not contaminated since the demonstration has nothing to do with the
corresponding tested sample.

##### Calibration Setting.

Similarly, we also create a calibrated dataset $\mathcal{X}_{\text{cal}}$ for
each benchmark. For each sample in $\mathcal{X}_{\text{cal}}$, the
demonstration is chosen from the benchmark’s validation or testing set, and
the tested sample is the calibrated one of the demonstration processed by
Clean-Eval, as shown in the right part of Figure 3(a). In this setting, we aim
to test the performance of the contaminated model on the calibrated benchmark.

###  4.2 Fine-tuning

Fine-tuning refers to further optimization adjustments for a specific task or
dataset based on a pre-trained LLM. As shown in Figure 3(b), for any given
benchmark, we specifically fine-tune two models, one is a contaminated model
and another is a clean model.

For the contaminated model, we fine-tune it using all demonstrations in the
contamination dataset $\mathcal{X}$ in §4.1, which consists of $n$ samples
from the benchmark’s validation or test set. Similarly, for the clean model,
we fine-tune it using all demonstrations in the clean dataset
$\mathcal{X}_{\text{clean}}$ in §4.1, which are $n$ samples randomly chosen
from the benchmarks’s training set. 222We cannot deny that the pre-trained
model may have been pre-trained using this dataset.

##### Contamination Setting.

Under this setting, we test the performance of the contaminated model using
all tested samples in the contamination dataset $\mathcal{X}$ in §4.1, which
are exactly the same as the fine-tuning data for the contaminated model.

##### Clean Setting.

Similarly, under this setting, we test the performance of the clean model
using all tested samples in the contamination dataset $\mathcal{X}$, which are
also the tested samples in the clean dataset $\mathcal{X}_{\text{clean}}$ in
§4.1.

##### Calibration Setting.

Under this setting, we test the performance of the contaminated model using
all tested samples in the calibrated dataset $\mathcal{X}_{\text{cal}}$, in
order to verify the effectiveness of our calibration method in measuring the
real capacity of contaminated models.

##  5 Experiments

##### Datasets

We have meticulously curated 20 datasets, spanning a wide array of tasks.
These tasks encompass text implication, problem pair matching, natural
language reasoning, semantic similarity, sentiment analysis, common sense
reasoning, text classification, mathematical reasoning, examinations, and even
some natural language generation tasks. This classification provides valuable
insights into the performance of various task types concerning data
contamination. Below is the comprehensive list of datasets we have utilized:

  * •

Nature Language Inference. GLUE dataset Wang et al. (2019b) that includes
QNLI, MNLI, SNLI, WNLI, RTE, QQP, MRPC, SST2; IMDB Maas et al. (2011); BOOLQ
Clark et al. (2019); Super-GlUE dataset Wang et al. (2019a) that includes
COPA, CB; Suqad-v2 Rajpurkar et al. (2016), Ag News Zhang et al. (2015).

  * •

Nature Language Generation. CNN_Dailymail See et al. (2017), BBC_XSUM Narayan
et al. (2018).

  * •

Arithmetic Reasoning. GSM8K Cobbe et al. (2021), MultiArith

  * •

Examination. MMLU Hendrycks et al. (2021), CEVAL Huang et al. (2023).

###  5.1 Metrics

##### ROUGE & BLEURT.

To measure the degree of overlap between a generated instance and a reference,
we utilize both ROUGE Lin (2004) and BLEURT scores Sellam et al. (2020). ROUGE
evaluates lexical similarity, focusing on shared words and phrases, while
BLEURT assesses the semantic relevance and fluency of the generated sequence
concerning the reference instance.

##### Equivalence.

We employed the text-davinci-003 model Brown et al. (2020) to assess
equivalence before and after the processing of contaminated data by Clean-
Eval. Details of the prompt design can be found in Appendix A.1.

###  5.2 Contamination Cleanup.

##### Models.

We utilize the text-davinci-003 model Brown et al. (2020) as our paraphrase,
back-translation and semantic model. We also employ the BLEURT-20 model Sellam
et al. (2020) to calculate the BLEURT score and choose the optimal one.

##### Process.

Because the format and content of each dataset are different, our criteria for
processing each data set are other. Firstly, due to resource constraints, our
method, Clean-eval, does not comprehensively process every aspect of the data.
We thoroughly deal with all its contents in cases like the SNLI paired
dataset. However, with question-options-answer or question-answer datasets, we
focus solely on the questions. Furthermore, we limit our analysis to the
initial three sentences or less when dealing with lengthy text content. This
approach helps mitigate the model’s limitations in handling long text.
Secondly, All generated samples are passed through the semantic detector, and
if they do not pass the semantic detection, the original sample will be
output.

##### Results.

The results are shown in Table 3. Following our method’s processing, the
surface form of the newly generated sample notably differs from the original
sample, particularly in terms of n-gram variations. However, the presence of
the semantic detector ensures the quality and fidelity of the generated
results, assuring their reliability despite these surface-level alterations.

###  5.3 In-context Learning

Data Type | CNN/Daily-Mail | BBC-XSUM
---|---|---
| rouge-1 | rouge-2 | rouge-L | rouge-1 | rouge-2 | rouge-L
Contamination | 23.38 | 9.45 | 21.69 | 33.64 | 18.56 | 29.21
Clean | 21.18 | 7.18 | 19.57 | 22.97 | 7.78 | 19.08
Calibration | 23.14 $\Downarrow$ | 9.35 $\Downarrow$ | 21.41 $\Downarrow$ | 33.09$\Downarrow$ | 17.92 $\Downarrow$ | 28.90 $\Downarrow$
Table 2: ICL experiments and metrics is Rouge. $\Downarrow$ is compared to the
contamination dataset. The optimal candidate is chosen according to the lowest
BLEURT score.

(a) SST2 Datasets.

(b) QNLI Datasets.

(c) RTE Datasets.

(d) MNLI Datasets.

(e) SNLI Datasets.

(f) MRPC Datasets.

Figure 4: Fine-tuning Experiments. For SST2, QNLI, and RTE datasets, the
candidate data consists of samples with the highest BLEURT score. Conversely,
MNLI, SNLI, and MRPC datasets utilize samples with the lowest BLEURT score as
candidate data.

##### Model.

We utilize the text-davinci-003 model Brown et al. (2020) for conducting ICL
experiments.

##### Implementation Details.

Each tested use case is provided with task-specific instructions. For
instance, a dataset attributed to CNN/Dailymail would receive a prompt such as
“The task is to summarize this article:”. Detailed designs for all prompts are
in Appendix A.2.

##### Results and Analysis

The results, presented in Table 1 and Table 2, demonstrate consistently higher
performance scores across all tasks when exposed to data contamination
compared to both contamination calibration and clean data. This indicates a
tangible performance benefit for models in specific tasks due to data
contamination, improving their overall efficacy.

Notably, for straightforward tasks like RTE, IMDB, and COPA, the model
showcases robust generalization abilities, as evidenced by the strong
performance on clean datasets 333Since we can not access the training dataset
of text-davinci-003, we can not exclude the potential contamination in these
datasets.. And when contamination is present on these tasks, the model returns
the answer almost right.

However, in more complex tasks such as GSM8K and MultiArith in mathematical
reasoning, and MMLU and CEVAL in Exams, a significant performance gap between
Contamination and Clean is observed. Here, the model’s reduced performance on
clean datasets may be attributed to the absence of a generated chain of
thought, leading to performance degradation. Additionally, as depicted in
Table 2, even when limiting the processing to the first three sentences or
fewer of an article, our method effectively reduces the extent of data
contamination. All the results demonstrate that processing with our Clean-Eval
method causes a gradual performance decline, aligning more closely with the
performance in a clean setting.

###  5.4 Fine-tuning

##### Models.

For fine-tuning, we employ the LLama2-7b-chat model Touvron et al. (2023).

##### Implementation Details

As the model parameters continue to grow in size, achieving full fine-tuning
becomes increasingly challenging. In such cases, we adopt LoRA’s method for
fine-tuning Hu et al. (2021). The process begins by transforming original data
into instructional data. Each piece of data now comprises three essential
components: an instruction, an input, and an output. Subsequently, we proceed
with single-instruction fine-tuning. Besides, given the extensive collection
of datasets, conducting in-depth fine-tuning for every model to achieve
optimal performance would be impractical and time-consuming. Instead, we
preserve model checkpoints after more than 30 training rounds and test
performance on these checkpoints.

##### Results and Analysis

From the insights gained in Figure 4, it’s evident that models fine-tuned on
$\mathcal{X}$ achieve almost perfect accuracy with increasing training rounds.
Notably, when choosing the highest BLEURT score, the models’ performance on
the calibration setting lies between that on the clean setting and
contamination setting. However, opting for the lowest BLEURT score indicates a
slower model performance on the calibration data than the clean setting. This
underlines how the BLEURT score can regulate the extent of calibration for
contaminated data. It signifies that models lacking data contamination
calibration might exhibit exaggerated evaluation metric values, whereas data
processed through Clean-Eval effectively mitigates such discrepancies.

##  6 Analysis

##### Performance Gap.

To compare the degree of impact of the contaminated data processed by our
method on the model performance, we designed the following indicators. This
indicator can be used as a reference to select the best candidate.

$\displaystyle\mathrm{PG}(\mathcal{X},\mathcal{X}_{\text{cal}})$
$\displaystyle=|\mathrm{Perf}(\mathcal{X})-\mathrm{Perf}_{\mathcal{X}}(\mathcal{X}_{\text{cal}})|$
(1)
$\displaystyle\mathrm{PG}(\mathcal{X}_{\text{cal}},\mathcal{X}_{\text{clean}})$
$\displaystyle=|\mathrm{Perf}(\mathcal{X}_{\text{cal}})-\mathrm{Perf}(\mathcal{X})|$
$\displaystyle\mathrm{PG}$
$\displaystyle=\mathrm{PG}(\mathcal{X},\mathcal{X}_{\text{cal}})-\mathrm{PG}(\mathcal{X}_{\text{cal}},\mathcal{X}_{\text{clean}})$

Where $\mathrm{Perf}(\cdot)$ represents the performance of an LLM on a
specific dataset. $\mathrm{PG}(\mathcal{X},\mathcal{X}_{\text{cal}})$ measures
the performance gap between the contamination setting and calibration setting,
while $\mathrm{PG}(\mathcal{X}_{\text{cal}},\mathcal{X}_{\text{clean}})$
quantifies the performance difference between the calibration setting and
clean setting. When $\mathrm{PG}$ is negative, the model’s performance on the
calibrated data is akin to that on the contaminated data. And when
$\mathrm{PG}$ is positive, the model’s performance on the calibrated data is
comparable to that on the clean setting.

###  6.1 Ablation Study

In Table 3, we conducted an ablation study comparing three methods including
back-translation, paraphrase, and Clean-Eval across three chosen datasets.
Back-translation consistently yields higher Rouge and Bleurt scores than other
methods across all datasets. This suggests that back-translation, by itself,
is effective in maintaining lexical and semantic features from the original
text. Paraphrase introduces variations in content expression, showcasing the
ability to offer alternative ways of expressing the same semantic content.
Clean-Eval, which combines Back-translation and Paraphrase, emerges as a
comprehensive approach. It not only maintains semantic equivalence, as
indicated by the Equivalence score but also enhances the diversity of content
expression.

Data | Method | Rouge-1 | Rouge-2 | Rouge-L | BlEURT | Equivalence
---|---|---|---|---|---|---
QNLI | Back-translation | 54.08 | 29.80 | 50.47 | 63.44 | 100.00
Paraphrase | 48.50 | 26.02 | 43.28 | 63.19 | 100.00
Clean-Eval | 46.85 | 22.90 | 42.53 | 60.21 | 100.00
SST2 | Back-translation | 52.35 | 32.05 | 51.01 | 59.94 | 100.00
Paraphrase | 30.39 | 9.98 | 27.77 | 42.96 | 100.00
Clean-Eval | 26.66 | 7.55 | 23.64 | 40.90 | 100.00
MMLU | Back-translation | 52.85 | 30.15 | 48.68 | 57.91 | 100.00
Paraphrase | 45.71 | 23.10 | 40.79 | 55.46 | 100.00
Clean-Eval | 42.42 | 19.70 | 38.32 | 51.99 | 100.00
Table 3: The difference between the sample processed with different methods
and the original sample. And we choose the lowest BLEURT score as our optimal
candidate.

####  6.1.1 BLEURT Score

In this part, we explore whether the selection based on the BLEURT score
impacts the model performance.

| Score | QNLI | SST2 | MMLU
---|---|---|---|---
BT | worst | 7.33 | 6.00 | 14.67
middle | -5.99 | 6.00 | 3.33
best | -10.67 | 6.00 | 4.01
| Score | QNLI | SST2 | MMLU
Para | worst | -8.67 | 6.00 | 8.01
middle | 4.01 | 4.66 | 5.33
best | 0.01 | 6.00 | 8.67
Table 4: Performance gap on different Bleurt-score.

##### Results.

As seen from Table 4, for Back-Translation, the selection based on the lowest
bleurt score in the candidate set can make the performance after contamination
cleaning equal to that of the clean data set. However, the Paraphrase method
does not seem obvious for the QNLI dataset. Therefore, we can choose the
degree of calibration according to the BLEURT score.

####  6.1.2 Combination sequence

We compared the performance impact of the order of the two methods.

Order | QNLI | SST2 | MMLU
---|---|---|---
Para + BT | 10.67 | 6.00 | 14.67
BT + Para | 10.67 | 6.00 | 12.67
Table 5: Performance gap on different orders.

##### Results.

From Table 5, it demonstrates that the order of paraphrasing and back-
translation methods can have varying impacts on different tasks. While QNLI
and SST2 tasks are less sensitive to method order, the MMLU task shows slight
differences. Therefore, we can tailor the order based on task requirements and
we choose to first paraphrase and then back-translation in Clean-Eval.

####  6.1.3 Equivalence Detector

Continuous use of back translation would end up with a string that differs
markedly from that which you started Way (2013). A combination of paraphrase
and back-translation can also cause this problem, so a semantic detector is
very important.

Method | QNLI | SST2 | MMLU
---|---|---|---
BT | 74.17 | 86.33 | 73.33
Para | 91.67 | 82.67 | 73.33
Clean-Eval | 72.17 $\Downarrow$ | 56.34$\Downarrow$ | 60.34$\Downarrow$
Table 6: Model performance on calibrated dataset without equivalence detector.

##### Results.

As we can see from Table 6, across all three datasets, the paraphrasing method
demonstrates relatively high performance, especially in QNLI and SST2. In the
absence of a semantic detector, results generated through Clean-Eval that
first paraphrase and then back-translation exhibit a general decline in
performance. This suggests the possibility of introducing semantic errors or
inaccuracies during the generation process.

###  6.2 Human Evaluation

We performed human evaluations of the generated output to assess potential
changes in the calibrated data, including alterations in the question’s
meaning, generation quality and so on.

Type | SST2
---|---
Equivalent | 97%
Not Equivalent | 3%
Table 7: Human Evaluation on SST2 datasets.

##### Results.

The human evaluation results for the SST2 dataset are presented in Table 7.
The human evaluation results indicate a high level of equivalence between the
calibrated output and the original data, suggesting that the model generally
maintains the intended meaning of the questions. This is a positive outcome as
it demonstrates the effectiveness of the generated output in preserving the
semantics of the input.

##  7 Case Study

Figure 5: A case study from SST2 dataset.

In this case, the paraphrased sentence successfully conveys the essence of the
original while introducing some variation. The transformation maintains the
positive sentiment, emphasizing the documentary’s coolness and appeal to the
audience. Back-translation aims to ensure that the paraphrased sentence
retains its intended meaning. The back-translated sentence aligns well with
the paraphrased version. The key elements, such as the documentary’s
uniqueness, coolness, and audience appeal, are preserved. The combined
approach of paraphrasing and backtranslation proves effective in enhancing the
original sentence. The paraphrased version introduces a nuanced expression,
and the subsequent back-translation successfully captures the intended
meaning. The final output maintains a positive tone and successfully
communicates the documentary’s appeal.

##  8 Conclusion

Data contamination is an urgent problem for the development of LLMs society.
Downloading and trying contaminated models can be a waste of time for both
researchers and developers. To save their time, this paper intends to mitigate
the issue of data contamination in LLMs through the introduction of the Clean-
Eval method. This approach leverages existing datasets to create a new
evaluation dataset, effectively mitigating the impact of contamination.
Experimental results demonstrate the method’s success in accurately assessing
model capabilities and restoring real performance. Clean-Eval holds promise in
enhancing transparency and reliability in the evaluation of LLMs.

## References

  * Brown et al. (2020) Tom B. Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, Sandhini Agarwal, Ariel Herbert-Voss, Gretchen Krueger, Tom Henighan, Rewon Child, Aditya Ramesh, Daniel M. Ziegler, Jeffrey Wu, Clemens Winter, Christopher Hesse, Mark Chen, Eric Sigler, Mateusz Litwin, Scott Gray, Benjamin Chess, Jack Clark, Christopher Berner, Sam McCandlish, Alec Radford, Ilya Sutskever, and Dario Amodei. 2020.  Language models are few-shot learners.
  * Carlini et al. (2023) Nicholas Carlini, Daphne Ippolito, Matthew Jagielski, Katherine Lee, Florian Tramer, and Chiyuan Zhang. 2023.  Quantifying memorization across neural language models.
  * Chi et al. (2023) Alison Chi, Li-Kuang Chen, Yi-Chen Chang, Shu-Hui Lee, and Jason S Chang. 2023.  Learning to paraphrase sentences to different complexity levels.  _arXiv preprint arXiv:2308.02226_.
  * Clark et al. (2019) Christopher Clark, Kenton Lee, Ming-Wei Chang, Tom Kwiatkowski, Michael Collins, and Kristina Toutanova. 2019.  Boolq: Exploring the surprising difficulty of natural yes/no questions.
  * Cobbe et al. (2021) Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, Christopher Hesse, and John Schulman. 2021.  Training verifiers to solve math word problems.  _arXiv preprint arXiv:2110.14168_.
  * Dodge et al. (2021) Jesse Dodge, Maarten Sap, Ana Marasović, William Agnew, Gabriel Ilharco, Dirk Groeneveld, Margaret Mitchell, and Matt Gardner. 2021.  Documenting large webtext corpora: A case study on the colossal clean crawled corpus.
  * Golchin and Surdeanu (2023) Shahriar Golchin and Mihai Surdeanu. 2023.  Time travel in llms: Tracing data contamination in large language models.
  * Hendrycks et al. (2021) Dan Hendrycks, Collin Burns, Steven Basart, Andy Zou, Mantas Mazeika, Dawn Song, and Jacob Steinhardt. 2021.  Measuring massive multitask language understanding.  _Proceedings of the International Conference on Learning Representations (ICLR)_.
  * Hu et al. (2021) Edward J. Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen. 2021.  Lora: Low-rank adaptation of large language models.
  * Huang et al. (2023) Yuzhen Huang, Yuzhuo Bai, Zhihao Zhu, Junlei Zhang, Jinghan Zhang, Tangjun Su, Junteng Liu, Chuancheng Lv, Yikai Zhang, Jiayi Lei, Yao Fu, Maosong Sun, and Junxian He. 2023.  C-eval: A multi-level multi-discipline chinese evaluation suite for foundation models.  In _Advances in Neural Information Processing Systems_.
  * Li et al. (2023a) Haonan Li, Yixuan Zhang, Fajri Koto, Yifei Yang, Hai Zhao, Yeyun Gong, Nan Duan, and Timothy Baldwin. 2023a.  Cmmlu: Measuring massive multitask language understanding in chinese.
  * Li (2023) Yucheng Li. 2023.  An open source data contamination report for llama series models.
  * Li et al. (2023b) Zejun Li, Ye Wang, Mengfei Du, Qingwen Liu, Binhao Wu, Jiwen Zhang, Chengxing Zhou, Zhihao Fan, Jie Fu, Jingjing Chen, Xuanjing Huang, and Zhongyu Wei. 2023b.  Reform-eval: Evaluating large vision language models via unified re-formulation of task-oriented benchmarks.
  * Lin (2004) Chin-Yew Lin. 2004.  ROUGE: A package for automatic evaluation of summaries.  In _Text Summarization Branches Out_ , pages 74–81, Barcelona, Spain. Association for Computational Linguistics.
  * Maas et al. (2011) Andrew L. Maas, Raymond E. Daly, Peter T. Pham, Dan Huang, Andrew Y. Ng, and Christopher Potts. 2011.  Learning word vectors for sentiment analysis.  In _Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies_ , pages 142–150, Portland, Oregon, USA. Association for Computational Linguistics.
  * Magar and Schwartz (2022) Inbal Magar and Roy Schwartz. 2022.  Data contamination: From memorization to exploitation.
  * Narayan et al. (2018) Shashi Narayan, Shay B. Cohen, and Mirella Lapata. 2018.  Don’t give me the details, just the summary! topic-aware convolutional neural networks for extreme summarization.  _ArXiv_ , abs/1808.08745.
  * OpenAI (2023) OpenAI. 2023.  Gpt-4 technical report.
  * Rajpurkar et al. (2016) Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang. 2016.  SQuAD: 100,000+ Questions for Machine Comprehension of Text.  _arXiv e-prints_ , page arXiv:1606.05250.
  * See et al. (2017) Abigail See, Peter J. Liu, and Christopher D. Manning. 2017.  Get to the point: Summarization with pointer-generator networks.  In _Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 1073–1083, Vancouver, Canada. Association for Computational Linguistics.
  * Sellam et al. (2020) Thibault Sellam, Dipanjan Das, and Ankur Parikh. 2020.  BLEURT: Learning robust metrics for text generation.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 7881–7892, Online. Association for Computational Linguistics.
  * Sennrich et al. (2016) Rico Sennrich, Barry Haddow, and Alexandra Birch. 2016.  Improving neural machine translation models with monolingual data.  In _Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 86–96, Berlin, Germany. Association for Computational Linguistics.
  * Touvron et al. (2023) Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, Dan Bikel, Lukas Blecher, Cristian Canton Ferrer, Moya Chen, Guillem Cucurull, David Esiobu, Jude Fernandes, Jeremy Fu, Wenyin Fu, Brian Fuller, Cynthia Gao, Vedanuj Goswami, Naman Goyal, Anthony Hartshorn, Saghar Hosseini, Rui Hou, Hakan Inan, Marcin Kardas, Viktor Kerkez, Madian Khabsa, Isabel Kloumann, Artem Korenev, Punit Singh Koura, Marie-Anne Lachaux, Thibaut Lavril, Jenya Lee, Diana Liskovich, Yinghai Lu, Yuning Mao, Xavier Martinet, Todor Mihaylov, Pushkar Mishra, Igor Molybog, Yixin Nie, Andrew Poulton, Jeremy Reizenstein, Rashi Rungta, Kalyan Saladi, Alan Schelten, Ruan Silva, Eric Michael Smith, Ranjan Subramanian, Xiaoqing Ellen Tan, Binh Tang, Ross Taylor, Adina Williams, Jian Xiang Kuan, Puxin Xu, Zheng Yan, Iliyan Zarov, Yuchen Zhang, Angela Fan, Melanie Kambadur, Sharan Narang, Aurelien Rodriguez, Robert Stojnic, Sergey Edunov, and Thomas Scialom. 2023.  Llama 2: Open foundation and fine-tuned chat models.
  * Wang et al. (2019a) Alex Wang, Yada Pruksachatkun, Nikita Nangia, Amanpreet Singh, Julian Michael, Felix Hill, Omer Levy, and Samuel R Bowman. 2019a.  Superglue: A stickier benchmark for general-purpose language understanding systems.  _arXiv preprint arXiv:1905.00537_.
  * Wang et al. (2019b) Alex Wang, Amanpreet Singh, Julian Michael, Felix Hill, Omer Levy, and Samuel R. Bowman. 2019b.  Glue: A multi-task benchmark and analysis platform for natural language understanding.
  * Way (2013) Andy Way. 2013.  Emerging use-cases for machine translation.  In _Proceedings of Translating and the Computer 35_.
  * Wei et al. (2023) Tianwen Wei, Liang Zhao, Lichang Zhang, Bo Zhu, Lijie Wang, Haihua Yang, Biye Li, Cheng Cheng, Weiwei Lü, Rui Hu, Chenxia Li, Liu Yang, Xilin Luo, Xuejie Wu, Lunan Liu, Wenjun Cheng, Peng Cheng, Jianhao Zhang, Xiaoyu Zhang, Lei Lin, Xiaokun Wang, Yutuan Ma, Chuanhai Dong, Yanqi Sun, Yifu Chen, Yongyi Peng, Xiaojuan Liang, Shuicheng Yan, Han Fang, and Yahui Zhou. 2023.  Skywork: A more open bilingual foundation model.
  * Yang et al. (2023) Aiyuan Yang, Bin Xiao, Bingning Wang, Borong Zhang, Ce Bian, Chao Yin, Chenxu Lv, Da Pan, Dian Wang, Dong Yan, Fan Yang, Fei Deng, Feng Wang, Feng Liu, Guangwei Ai, Guosheng Dong, Haizhou Zhao, Hang Xu, Haoze Sun, Hongda Zhang, Hui Liu, Jiaming Ji, Jian Xie, JunTao Dai, Kun Fang, Lei Su, Liang Song, Lifeng Liu, Liyun Ru, Luyao Ma, Mang Wang, Mickel Liu, MingAn Lin, Nuolan Nie, Peidong Guo, Ruiyang Sun, Tao Zhang, Tianpeng Li, Tianyu Li, Wei Cheng, Weipeng Chen, Xiangrong Zeng, Xiaochuan Wang, Xiaoxi Chen, Xin Men, Xin Yu, Xuehai Pan, Yanjun Shen, Yiding Wang, Yiyu Li, Youxin Jiang, Yuchen Gao, Yupeng Zhang, Zenan Zhou, and Zhiying Wu. 2023.  Baichuan 2: Open large-scale language models.
  * Zhang et al. (2015) Xiang Zhang, Junbo Jake Zhao, and Yann LeCun. 2015.  Character-level convolutional networks for text classification.  In _NIPS_.

##  Appendix A Prompt Design

###  A.1 Equivalence Detector

Prompt: Please determine whether the following sentences are equivalent.

###  A.2 Instruction for Each Dataset

RTE: The task is to determine whether a pair of sentences are entailed by each
other. Just return entailment or not_entailment.
QQP, MRPC: The task is to determine whether a pair of questions are
semantically equivalent. Just return equivalent or not_equivalent.
QNLI: The task is to determine whether the context sentence contains the
answer to the question. Just return entailment or not_entailment.
MNLI, CB: The task is to predict whether the premise entails the hypothesis,
contradicts the hypothesis, or neither. Just return entailment, contradiction,
or neutral.
WNLI: The task is to predict if the sentence with the pronoun substituted is
entailed by the original sentence. Just return entailment or not_entailment.
SNLI:The task is to determine whether a pair of sentences are entailed,
contradicted or neutral each other. Just return entailment, contradiction, or
neutral.
IMDB: The task is to determine whether the sentiment of the text is positive
or negative. Just return positive or negative.
PIQA: The task is to select the best solution to the question. Just return
solution1 or solution2.
COPA: Given a premise, choose one of the following two choices that express
the sample["question"] relationship. Just return choice1 or choice2.
BOOLQ: The task is to answer true or false given the question. Just return
true or false.
SST2: The task is to determine whether the sentiment of the sentence is
positive or negative. Just return positive or negative.
AG News: The task is to classify the article into sports, world, business, or
sci/tech. Just return sports, world, business, or sci/tech.
GSM8K, MultiArith:The task is to answer a given mathematical question. Just
directly return the final number answer.
MMLU, CEVAL: Please select the best answer from the options according to the
question. Just return one answer with A, B, C, or D.
CNN_Dailymail, BBC_XSUM: Please summarize this article.


# Enhancement of Entanglement via Josephson Parametric Amplifier in a Dual
Cavity-Magnon System

Abdelkader Hidki  LPTHE, Department of Physics, Faculty of Sciences, Ibn Zohr
University, Agadir, Morocco  Abderrahim Lakhfif  LPHE-Modeling and
Simulations, Faculty of Science, Mohammed V University in Rabat, Rabat,
Morocco  Mostafa Nassik  LPTHE, Department of Physics, Faculty of Sciences,
Ibn Zohr University, Agadir, Morocco  Rizwan Ahmed  Physics Division, Pakistan
Institute of Nuclear Science and Technology (PINSTECH), P. O. Nilore,
Islamabad 45650, Pakistan.  Amjad Sohail<EMAIL_ADDRESS>Department
of Physics, Government College University, Allama Iqbal Road, Faisalabad
38000, Pakistan

###### Abstract

In the two microwave (MW) cross-shaped cavity magnon system, we describe a
method to produce multipartite entanglement and quantum steering. To achieve
squeezed states of the magnons, a Josephson parametric amplifier (JPA) creates
a squeezed vacuum field that drives the two cavities. We theoretically
demonstrate that the cavity-cavity entanglement can be generated at the
resonance point, however, increasing the cavity and magnon decay rates
generate the cavity-magnon entanglement. By changing the squeezing parameter
and increasing the decay rates, we can transfer the cavity-cavity entanglement
to cavity-magnon entanglement. Furthermore, the cavity-cavity entanglement
(survive up to 2.8K) not only found to be much stronger but also more robust
as compared to cavity-magnon entanglement (survive up to 0.4K). More
importantly, the genuine photon-magnon-photon tripartite entanglement could be
achieved, which is robust against the thermal fluctuations and depends
strongly on squeezing parameter. Furthermore, for current dual cavity-magnon
system, two-way quantum steering is found when the optomagnonical couplings
are equal. The current study offers a straightforward and practical method for
achieving multipartite quantum correlations.

##  I Introduction

The investigation of macroscopic quantum states has received increasing
attention due to significant advances in experimental technology, especially
since the proposal of Schrödinger’s cat states1 . Notably, cavity
optomechanics (COM) 2 ; 3 , which highlights the complex interaction between
electromagnetic fields and mechanical motion through radiation pressure, and
thereby presents a possibility of creating macroscopic quantum states. Over
the past decade, considerable advancement have been carried out in
optomechanical systems to generate macroscopic quantum states in massive
mechanical oscillators 4 ; 5 . These achievements incorporate the successful
realization of finding entangled states involving both mechanical oscillators
and electromagnetic fields 6 , as well as entangled states between two
mechanical oscillators 7 ; 8 ; 9 ; 10 . Moreover, researchers have
successfully produced superposition states 11 ; 12  and squeezed states 13
for mechanical motion, etc. In addition, COM has enabled the generation of
nonclassical states, such as Fock states 14 , superposition states 15 , cat
states 16  and entangled states 17 ; 18 , in macroscopic mechanical resonators
by controlled coupling with superconducting qubits.

Recently, the cavity-magnonics system, with the ferrimagnetic materials such
as the famous yttrium-iron-garnet (YIG) sphere 19 , has emerged as a prominent
area of research 20 . In these quantum systems, which are based on cavity
quantum electrodynamics (QED), basic excitations brought on by uniform
magnetic fields lead to the formation of magnons (Kittel mode) in a YIG
crystal 21 . These magnons can be directly coupled to MW photons 22 ; 23 ; 24
, optical photons 25 ; 26 , and mechanical phonons 27  via the processes of
magnetic-dipole interaction, magneto-optical effects, and magnetostrictive
interaction, respectively. This compelling area of research has uncovered a
multitude of intriguing phenomena, including the exceptional points 28 , Kerr
effect 29 , non-reciprocity transmission 30 , magnon-induced absorption 31 ,
magnon-induced transparency 32 , coherent MW-to-optical conversion 33 , magnon
squeezing by two-tone driving of a qubit 34 , and quantum entanglement of
magnons 35 ; 36 ; 37 ; 38 .

Along with the coupling between MW and magnons, the latter can also interact
with the deformation vibration phonons of the YIG sphere through nonlinear
magnetostrictive interaction, giving rise to what is called cavity
magnomechanics (CMM) 39 . This interaction opens up various fascinating
possibilities, such as photon-magnon-phonon entanglement, steering and magnon-
squeezing states 40 ; 41 ; 42 ; 43 ; 44 ; 45 ; 46 ; 47 ; 48 ; 49 ; 50 ,
generation of magnon laser or chaos 51 ; 52 , ultra-slow light engineering 32
; 53 , use of coherent feedback loop 54 ; 55 , realization of ground-state
cooling of mechanical vibrations mode 56 ; 57 , magnon-assisted photon-phonon
conversion 58  and quantum state retrieval and storage 59 , among several
other applications. An important observation is is how frequently squeezed-
reservoir engineering has been used to examine entanglement and squeezing in
different optomechanical and magnomechanical systems 60 . This involves
utilizing broadband squeezed light as a squeezed reservoir to drive the
cavity, and it has found extensive application across different systems 10 ;
61 ; 62 . Particularly, single-mode squeezed light is employed to generate
Einstein-Podolsky-Rosen (EPR) quantum steering 63 , and magnon-magnon
correlation 35  in the cavity-magnon system. Furthermore, two-mode squeezed
microwave fields are also used to observe a strong correlation between two
magnon modes 36 , the vibrational modes of two YIG spheres 44 , and manipulate
one-way Gaussian steering 64 .

In this manuscript, we propose an efficient scheme to achieve bi-(tri)partite
entanglement and quantum steering among a magnon and two MW fields. Several
suggestions have been made previously, proposing distinct mechanisms for
preparing entangled states involving two MW cavities, including the
nonlinearity of magnetostrictive force 43 ; 45 ; 46 , the Kerr nonlinearity 65
, and the optical parametric amplifier 66 . Here, we propose a different
approach where squeezed light is used to enhance genuine bi-(tri-)partite
entanglement between a magnon and two MW fields in a cavity-magnon system.

It is worth noting that the first experimental demonstration of spatially
distant two-mode squeezed states of MW light was achieved in Ref. 67  by E.
Flurin et. al.. Using a flux-driven Josephson parametric amplifier (JPA) to
prompt the squeezed vacuum MW fields 68 ; 69 , the two MW cavity photons and
the magnon can be entangled simultaneously. In addition, a variety of factors
employed in the current system can be changed to affect multipartite
entanglement and quantum steering. Moreover, a true photon-magnon-photon
tripartite entanglement is present in the current dual cavity-magnon system,
and it can be realized with experimentally realizable parameters. We show that
for the current cavity-magnon system, the squeezing parameter is cruicial for
manipulating and controlling both multipartite entanglement and quantum
steering within the system. Additionally, the generated entanglement exhibit
more robustness against changes in the environmental temperature.

The remainder of the article is structured as follows. The model Hamiltonian
of the current cavity-magnon system is described in Section 2. We use the
standard Langevin approach in Section 3 to examine the system’s dynamics and
obtain a drift matrix. Additionally, in Section 4, we analytically examine
bipartite, tripartite, and quantum steering while we present conclusion in
Sect. 5.

##  II The Model

As shown in Fig. 1, the cavity-magnon system comprise a magnon $m$ and the two
MW cavity modes, $c_{1}$ and $c_{2}$. The two cavities are arranged in a
cross-shaped and simultaneously pumped by a two-mode squeezed vacuum MW field
induced by JPA 69 . The collective excitations of many spins inside a YIG
sphere serve as the physical representation of a quasiparticle known as a
magnon 41 . The magnetic dipole interactions strengthen the optomagnonic
coupling strength between the two MW cavity modes and the magnon because of
the high spin density in the YIG sphere 22 ; 23 ; 70 . The YIG sphere is
positioned inside the cross-shaped cavities where there exist maximum magnetic
fields of the two MW cavity modes. The magnetic field of the cavities $c_{1}$
($c_{2}$) is along x-(y-)directions. Furthermore, a bias magnetic field is
also applied which is perpendicular to both the magnetic field of the cavity
modes. In addition, we also consider a very small sized YIG sphere as compared
to wavelengths of the MW cavities, therefore, we can safely neglect the
effects of radiation pressure induced by MW fields 41 . The Hamiltonian of the
current cavity-magnon system reads as:

Figure 1: (a) Graphical representation of the proposed system: A magnon in a
YIG sphere is positioned inside two cross-shaped MW cavities near the maximum
magnetic fields of the cavity modes and simultaneously in a uniform bias
magnetic field. The magnetic dipole interaction is responsible to couple two
MW fields $c_{1}$ and $c_{2}$ with a magnon mode $m$ in the YIG sphere. JPA
induces a two-mode squeezed vacuum to simultaneously drive the two cross-
shaped cavities (b). The $k$th MW cavity mode, with dissipation rate
$\kappa_{k}$ ($k=1,2$), couples with the magnon mode $m$, with decay rate
$\kappa_{m}$, by the coupling strength $\Gamma_{k}$.

$\displaystyle H$ $\displaystyle=$
$\displaystyle\hbar\sum_{k=1}^{2}\omega_{k}c_{k}^{{\dagger}}c_{k}+\hbar\omega_{m}m^{{\dagger}}m+\hbar\sum_{k=1}^{2}\Gamma_{k}\left(c_{k}m^{{\dagger}}+c_{k}^{{\dagger}}m\right),$
(1)

where $c_{k}$ and $\left(c_{k}^{{\dagger}}\right)$ are annihilation and
creation of the $k$th cavity mode while $m$ and $\left(m^{{\dagger}}\right)$
represent the annihilation creation operator for the magnon mode. In addition,
both cavity and magnon modes satisfy the following relation,
$[z,z^{{\dagger}}]=1$ ($z=m,c_{k}$). In addition, $\omega_{k}$ ($\omega_{m}$)
denotes the resonance frequency of the $k$th cavity (magnon) mode.
$\omega_{m}$ can be tuned via $\omega_{m}=\gamma_{0}B$, where $B$ is the
external bias magnetic field and $\gamma_{0}$ denoted the gyromagnetic ratio.
In addition, $\Gamma_{k}$ defines the optomagnonical coupling between the
magnon mod and the $k$th cavity. Furthermore, $\Gamma_{k}$ is taken much
larger than the cavity decay rate $\kappa_{k}$ and magnon decay rates
$\kappa_{m}$, i.e., $\Gamma_{k}>\kappa_{k}$, $\kappa_{m}$, leading to strong
coupling regime 22 ; 23 ; 44 ; 70 .

In the rotating frame at the drive frequency $\omega_{l}$ (the frequency of
the output of the two-mode squeezed field generated by the JPA), the
Hamiltonian of the current cavity-magnon system takes the form:

$\displaystyle H/\hbar$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{2}\Delta_{k}c_{k}^{{\dagger}}c_{k}+\Delta_{m}m^{{\dagger}}m+\sum_{k=1}^{2}\Gamma_{k}\left(c_{k}m^{{\dagger}}+c_{k}^{{\dagger}}m\right),$
(2)

where $\Delta_{k}$ and $\Delta_{m}$ are the cavity and magnon detunings, and
are $\Delta_{k}=\omega_{k}-\omega_{l}$ ($k=1,2$) and
$\Delta_{m}=\omega_{m}-\omega_{l}$.

##  III Dynamics of the system

We utilize the standard quantum Langevin formalism to describe the system’s
dynamics. By incorporating the dissipation’s of the cavity-magnon system, we
arrive to the following set of quantum Langevin equations (QLEs) in the
rotating frame:

$\displaystyle\dot{c}_{k}$ $\displaystyle=$
$\displaystyle-\left(\kappa_{k}+i\Delta_{k}\right)c_{k}-i\Gamma_{k}m+\sqrt{2\kappa_{k}}c_{k}^{in},\;\;k=1,2$
(3) $\displaystyle\dot{m}$ $\displaystyle=$
$\displaystyle-\left(\kappa_{m}+i\Delta_{m}\right)m-i\sum_{k=1}^{2}\Gamma_{k}c_{k}+\sqrt{2\kappa_{m}}m^{in},$
(4)

where $\kappa_{k}$ and $\kappa_{m}$ are the dissipation rates of the $k$th
cavity mode and magnon mode, respectively. Additionally, the zero mean input
noise operators for the MW cavities and the magnon modes, respectively, are
$c_{k}^{in}$ and $m^{in}$. These noise operators incorporate the squeezed
field’s driving of the two cavities and meet the following time-domain
correlation functions 71 :

$\displaystyle\left\langle
c_{k}^{in{\dagger}}(t)c_{k}^{in}(t^{\prime})\right\rangle$ $\displaystyle=$
$\displaystyle N\delta(t-t^{\prime}),\quad k=1,2$ (5)
$\displaystyle\left\langle
c_{k}^{in}(t)c_{k}^{in{\dagger}}(t^{\prime})\right\rangle$ $\displaystyle=$
$\displaystyle(N+1)\delta(t-t^{\prime}),$ (6) $\displaystyle\left\langle
c_{k}^{in}(t)c_{l}^{in}(t^{\prime})\right\rangle$ $\displaystyle=$
$\displaystyle\left\langle
c_{k}^{in{\dagger}}(t)c_{l}^{in{\dagger}}(t^{\prime})\right\rangle=M\delta(t-t^{\prime}),\quad
k\neq l=1,2$ (7) $\displaystyle\left\langle
m^{in{\dagger}}(t)m^{in}(t^{\prime})\right\rangle$ $\displaystyle=$
$\displaystyle n_{m}\delta(t-t^{\prime}),\ \ \ \ \left\langle
m^{in}(t)m^{in{\dagger}}(t^{\prime})\right\rangle=(n_{m}+1)\delta(t-t^{\prime}),$
(8)

where $M={\rm sinh}(r){\rm cosh}(r)$ and $N={\rm sinh}^{2}(r)$, $r$ being the
squeezing parameter of the two-mode squeezed vacuum field, generated by JPA.
$n_{m}=[\exp(\frac{\hbar\omega_{m}}{k_{B}T})-1]^{-1}$ is the equilibrium mean
thermal magnon number, with $k_{B}$ is the Boltzmann constant and $T$ the
environmental temperature 40 .

To investigate the quantum correlations in the cavity-magnon system, we first
derive the quantum fluctuations of the magnon and MW cavity modes. We then
decompose each operator as: $z=\left\langle z\right\rangle+\delta z$
$(z=c_{k},m)$, where $\left\langle z\right\rangle$ ($\delta z$) is the average
value (quantum fluctuation). In this way, we obtain the linearized set of QLEs
explaining the dynamics of the quantum fluctuations:

$\displaystyle\delta\dot{c}_{k}$ $\displaystyle=$
$\displaystyle-\left(\kappa_{k}+i\Delta_{k}\right)\delta
c_{k}-i\Gamma_{k}\delta m+\sqrt{2\kappa_{k}}c_{k}^{in},$ (9)
$\displaystyle\delta\dot{m}$ $\displaystyle=$
$\displaystyle-\left(\kappa_{m}+i\Delta_{m}\right)\delta
m-i\sum_{k=1}^{2}\Gamma_{k}\delta c_{k}+\sqrt{2\kappa_{m}}m^{in}.$ (10)

In addition, the average value of each dynamical operator is given by:

$\displaystyle\left\langle c_{k}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{-i\Gamma_{k}\left\langle
m\right\rangle}{\left(\kappa_{k}+i\Delta_{k}\right)},$ (11)
$\displaystyle\left\langle m\right\rangle$ $\displaystyle=$
$\displaystyle\frac{-i\sum_{k=1}^{2}\Gamma_{k}\left\langle
c_{k}\right\rangle}{\left(\kappa_{m}+i\Delta_{m}\right)}$ (12)

We now switch to the quadrature form of the QLEs by introducing the
quadratures of the quantum fluctuations defined as $\delta
m=\frac{1}{\sqrt{2}}(\delta x+i\delta y)$ and $\delta
c_{k}==\frac{1}{\sqrt{2}}(\delta X_{k}+i\delta Y_{k})$ Therefore, Eqs. (9) and
(10) can be written in the following compact matrix form:

$\dot{\mathcal{B}}(t)=\mathcal{M}\mathcal{B}(t)+\mathcal{N}_{{noise}}(t),$
(13)

where quantum fluctuation $\mathcal{B}(t)$ and input noise
$\mathcal{N}_{{noise}}(t)$ vectors are given by:

$\displaystyle\mathcal{B}(t)$ $\displaystyle=$ $\displaystyle[\delta
x(t),\delta y(t),\delta X_{1}(t),\delta Y_{1}(t),\delta X_{2}(t),\delta
Y_{2}(t)]^{\rm T},$ (14) $\displaystyle\mathcal{N}_{{noise}}(t)$
$\displaystyle=$
$\displaystyle[\sqrt{2\kappa_{m}}x^{in}(t),\sqrt{2\kappa_{m}}y^{in}(t),\sqrt{2\kappa_{1}}X_{1}^{in}(t),\sqrt{2\kappa_{1}}Y_{1}^{in}(t),\sqrt{2\kappa_{2}}X_{2}^{in}(t),\sqrt{2\kappa_{2}}Y_{2}^{in}(t)]^{\rm
T}.$ (15)

Furthermore, the drift matrix $\mathcal{M}$ is given by:

$\mathcal{M}=\left(\begin{array}[]{cccccc}-\kappa_{m}&\Delta_{m}&0&\Gamma_{1}&0&\Gamma_{2}\\\
-\Delta_{m}&-\kappa_{m}&-\Gamma_{1}&0&-\Gamma_{2}&0\\\
0&\Gamma_{1}&-\kappa_{1}&\Delta_{1}&0&0\\\
-\Gamma_{1}&0&-\Delta_{1}&-\kappa_{1}&0&0\\\
0&\Gamma_{2}&0&0&-\kappa_{2}&\Delta_{2}\\\
-\Gamma_{2}&0&0&0&-\Delta_{2}&-\kappa_{2}\end{array}\right).$ (16)

The Gaussian nature of the system is preserved thanks to the linear dynamics
and the Gaussian input noises. Therefore, the steady state of the quantum
fluctuations of the system defines the continuous-variable (CV) three-mode
Gaussian state. A $6\times 6$ matrix (CM) V, also known as a covariance
matrix, perfectly describes such a state. Its matrix components are as
follows:

$V_{ij}(t)=\frac{1}{2}\left\langle\mathcal{F}_{i}(t)\mathcal{F}_{j}(t^{\prime})+\mathcal{F}_{j}(t^{\prime})\mathcal{F}_{i}(t)\right\rangle,\quad
i,j=1,2,\cdots,6$ (17)

where the stability conditions of the system derived by using the Routh-
Hurwitz criterion are fulfilled (see Appendix A) 72 , the elements of $V$ can
directly be achieved by numerically solving the following Lyapunov equation 73
:

$\mathcal{M}V+V\mathcal{M}^{T}=-\mathcal{D},$ (18)

where $\mathcal{D}$ is the diffusion matrix that describes the noise
correlations. Its elements are defined as:

$\mathcal{D}_{ij}(t)\delta(t-t^{\prime})=\frac{1}{2}\left\langle\mathcal{N}_{i}(t)\mathcal{N}_{j}(t^{\prime})+\mathcal{N}_{j}(t^{\prime})\mathcal{N}_{i}(t)\right\rangle,\quad
i,j=1,2,\cdots,6$ (19)

By using Eqs. (5)-(8), we obtain the explicit expression of the matrix
$\mathcal{D}$:

$\mathcal{D}=\left(\begin{array}[]{cccccc}\kappa_{m}(2n_{m}+1)&0&0&0&0&0\\\
0&\kappa_{m}(2n_{m}+1)&0&0&0&0\\\
0&0&\kappa_{1}(2N+1)&0&2M\sqrt{\kappa_{1}\kappa_{2}}&0\\\
0&0&0&\kappa_{1}(2N+1)&0&-2M\sqrt{\kappa_{1}\kappa_{2}}\\\
0&0&2M\sqrt{\kappa_{1}\kappa_{2}}&0&\kappa_{2}(2N+1)&0\\\
0&0&0&-2M\sqrt{\kappa_{1}\kappa_{2}}&0&\kappa_{2}(2N+1)\end{array}\right).$
(20)

Since the Lyapunov equation (18) is a linear equation , it has a
straightforward solution. After acquiring the system’s CM, it is simple to
extract the reduced CMs of the bipartite subsystems (cavity1-cavity2,
cavity1-magnon and cavity2-magnon) by neglecting the unwanted columns and rows
in the global CM $V$.

In order to quantify the bipartite entanglement in the dual cavity-magnon
system, we employ the logarithmic negativity, which is a quantitative measure
that captures the Simon criterion 74 . It is defined by 75 :

$E^{N}_{\alpha-\beta}=\max[0,-\ln 2\eta^{-}],$ (21)

where $\alpha$ and $\beta$ are any two modes, $\eta^{-}=$min
eig$|i\Omega_{2}\widetilde{{V}_{4}}|$ is the minimum symplectic eigenvalue of
the partially transposed CM
$\widetilde{\mathcal{V}_{4}}=\varrho_{1|2}\mathcal{V}_{4}\varrho_{1|2}$, where
$\mathcal{V}_{4}$ is a $4\times 4$ matrix representing any bipartite subsystem
(cavity1-cavity2, cavity1-magnon and cavity2-magnon) and $\varrho_{1|2}={\rm
diag}(1,-1,1,1)$ is the $4\times 4$ matrix that defines the partial
transposition at the covariance matrices level. Furthermore,
$\Omega_{2}=\bigoplus^{2}_{j=1}i\sigma_{y}$ being a $4\times 4$ symplectic
matrix with $\sigma_{y}$ is the $y$-Pauli matrix.

On the other hand, the tripartite entanglement can be described by the minimal
residual contangle. The tripartite entanglement remains invariant of under all
possible permutations of the modes. It is defined as 75 ; 76 ; 77 :

$\mathcal{R}_{\tau}^{min}\equiv\min[\mathcal{R}_{\tau}^{m|c1c2},\mathcal{R}_{\tau}^{c2|mc1},\mathcal{R}_{\tau}^{c1|c2m}],$
(22)

where $\mathcal{R}_{\tau}^{u|vw}$ is the the residual contangle given by:

$\mathcal{R}_{\tau}^{u|vw}\equiv C_{u|vw}-C_{u|v}-C_{u|w}\geq 0,\ \
(u,v,w=m,c_{1},c_{2}).$ (23)

The condition $C_{u|vw}\geq C_{u|v}+C_{u|w}$ is similar to the Coffman-Kundu-
Wootters monogamy inequality achieved for the system of three qubits 77 .
$C_{x|y}$ is the contagle, which is a proper entanglement monotone, of
subsystems $u$ and $v$ ($u$ can contain just one mode, while $y$ can contain
one or two modes). The contangle is related to logarithmic negativity and can
be expressed as the square of the logarithmic negativity:
$C_{x|y}=E_{N,x|y}^{2}$. Noteworthy, for computing the one-mode-vs-two-modes
logarithmic negativity $E_{u|vw}$, one must adopt the same definition of Eq.
(21), however, the smallest symplectic eigenvalue $\eta^{-}$ must be modified
as:

$\eta_{u|vw}^{-}=\min eig|i\Omega_{3}\varrho_{u|vw}{V}\varrho_{u|vw}|,$ (24)

where $V$ is the global tripartite covariance matrix,
$\Omega_{3}=\bigoplus^{3}_{j=1}i\sigma_{y}$ is the $6\times 6$ symplectic
matrix and $\varrho_{u|vw}$ are the partial transposition matrices given by:

$\displaystyle\varrho_{1|23}$ $\displaystyle=$
$\displaystyle\text{diag}(1,-1,1,1,1,1),$ $\displaystyle\varrho_{2|31}$
$\displaystyle=$ $\displaystyle\text{diag}(1,1,1,-1,1,1),$
$\displaystyle\varrho_{3|12}$ $\displaystyle=$
$\displaystyle\text{diag}(1,1,1,1,1,-1).$

Owing to the asymmetric nature, the gaussian steering which have differing
behaviour than that for entanglement. It is widely known that entanglement
between two modes is a symmetric feature and the entangled state is
symmetrically shared between Alice and Bob without specifying the direction,
i.e., if Alice is entangled with Bob, Bob is essentially entangled with Alice.
However, this is not true in case of quantum steering due to its asymmetric
nature, i.e., Alice may steers Bob but vice versa may or may not possible
Based on quantum coherent information, the quantum steerability in different
directions, for any two interacting mode Gaussian state, was first introduced
by 78

$\displaystyle\zeta_{\alpha|\beta}$ $\displaystyle=$
$\displaystyle\max\\{0,\mathcal{S}(2\mathcal{V}_{f})-\mathcal{S}(2\mathcal{V}_{in})\\},$
(25) $\displaystyle\zeta_{\beta|\alpha}$ $\displaystyle=$
$\displaystyle\max\\{0,\mathcal{S}(2\mathcal{V}_{g})-\mathcal{S}(2\mathcal{V}_{in})\\},$
(26)

where

$\mathcal{V}_{in}=\left[\begin{array}[]{cc}\mathcal{V}_{f}&\mathcal{V}_{fg}\\\
\mathcal{V}_{fg}^{T}&\mathcal{V}_{g}\end{array}\right].$ (27)

Furthermore, the positivity of the Rényi-2 entropy determines the degree of
steering, is defined as $\mathcal{S}(\nu)=\frac{1}{2}\ln\det(\nu)$. Each entry
of the above matrix (Eq. (27)) is a $2\times 2$ matrix. Furthermore, the
diagonal elements $\mathcal{V}_{f}$ and $\mathcal{V}_{g}$ stands for the
reduced state of two modes $f$ and $g$, respectively, while the off-diagonal
entries define the correlation between $f$ and $g$. $\zeta_{\alpha|\beta}$
represents the directional EPR steering from mode $\alpha$ to mode $\beta$,
while $\zeta_{\beta|\alpha}$ denotes the EPR steering in swaped direction. No-
way steering is observed if $\zeta_{\alpha|\beta}=0$ and
$\zeta_{\beta|\alpha}=0$. In addition, the quantification for one-way steering
is to have either $\zeta_{\alpha|\beta}>0$ and $\zeta_{\beta|\alpha}=0$ or
$\zeta_{\alpha|\beta}=0$ and $\zeta_{\beta|\alpha}>0$. Finally, if the system
has two-way steering, then $\zeta_{\alpha|\beta}>0$ and
$\zeta_{\beta|\alpha}>0$. In addition, asymmetric steerability between any two
modes of the Gaussian states can be manipulated by looking at the the steering
asymmetry, specified as

$\zeta_{S}=|\zeta_{\alpha|\beta}-\zeta_{\beta|\alpha}|.$ (28)

Note that we get one-way or two way gaussian steering as long as steering
asymmetry remains positive, i.e., $\zeta_{S}>0$ and no-way gaussian steering
if $\zeta_{S}=0$ 79 .

Figure 2:  Density plot of the bipartite entanglement: (a) $E^{N}_{c-c}$ and
(c) $E^{N}_{c-m}$ as a function of $\Delta_{1}/\kappa_{c}$ and
$\Delta_{2}/\kappa_{c}$ with $\Delta_{m}=0$ in (a) and
$\Delta_{m}=2\kappa_{c}$ in (c). Density plot of the bipartite entanglement:
(b) $E^{N}_{c-c}$ and (d) $E^{N}_{c-m}$ as a function of
$\Delta_{1}/\kappa_{c}$ and $\Delta_{m}/\kappa_{c}$ with $\Delta_{2}=0$ in (b)
and $\Delta_{2}=2\kappa_{c}$ in (d).  Figure 3:  Density plot of the bipartite
entanglement: (a) $E^{N}_{c-c}$ and (b) $E^{N}_{c-m}$ as a functions of $r$
and $\Gamma_{2}/\Gamma_{1}$ ($\Gamma_{1}$ is fixed). The parameters are:
$\Delta_{m}=\Delta_{1}=\Delta_{2}=0$ in (a) and
$\Delta_{m}=-\Delta_{1}=\Delta_{2}=2\kappa_{c}$ in (b).  Figure 4:  Density
plot of the bipartite entanglement: (a) $E^{N}_{c-c}$ and (b) $E^{N}_{c-m}$ as
a functions of $r$ and $T$. The parameters are:
$\Delta_{m}=\Delta_{1}=\Delta_{2}=0$ in (a) and
$\Delta_{m}=-\Delta_{1}=\Delta_{2}=2\kappa_{c}$ in (b).

Figure 5: Tripartite entanglement $\mathcal{R}^{min}_{\tau}$ between
cavity1-cavity2-magnon modes as a function of (a) $\Delta_{1}/\kappa_{c}$ and
$\Delta_{2}/\kappa_{c}$, (b) $\Delta_{1}/\kappa_{c}$ and
$\Delta_{m}/\kappa_{c}$, (c) $r$ and $T$, and (d) $r$ and
$\Gamma_{2}/\Gamma_{1}$. The parameters are: $\Delta_{m}=2\kappa_{c}$ in (a),
$\Delta_{2}=2\kappa_{c}$ in (b), and
$\Delta_{m}=-\Delta_{1}=\Delta_{2}=2\kappa_{c}$ in (c-d).

Figure 6: Density plots of the gaussian steering $\zeta_{c|c}$, versus
detunings (a) $\Delta_{1}/\kappa_{c}$ and $\Delta_{m}/\kappa_{c}$ with
$r=0.4$, (b) $r$ and $T$ (c) $r$ and $\Gamma_{2}/\Gamma_{1}$. The parameters
are same as Fig. 5.

Figure 7: Plot of the Gaussian steering $\zeta_{c_{2}}|\zeta_{c_{1}}$ (black
dashed), $\zeta_{c_{1}}|\zeta_{c_{2}}$ (yellow dot-dashed black dashed),
$\zeta_{S}$ (magenta solid) and $E_{c-c}^{N}$ (cyan solid in insets) versus
detunings (a) $\Gamma_{2}/\Gamma_{1}$ with $\kappa_{1}=\kappa_{2}$ and (b)
$\kappa_{2}/\kappa_{1}$ with $\Gamma_{2}=\Gamma_{1}$. In addition, we take
$\Delta_{1}=\Delta_{2}=\Delta_{m}=0$ and $r=0.4$. The other optimal parameters
are as given in text.

##  IV Results and Discussion

Following, we present our main findings with special emphasis on the bipartite
entanglements between two microwave cavities and between cavity and magnon. We
consider recent experimentally feasible parameters 39 ; 41 :
$\omega_{c}/2\pi=\omega_{m}/2\pi=10$ GHz,
$\kappa_{c}/2\pi\equiv\kappa_{1}/2\pi=\kappa_{2}/2\pi=5$ MHz ,
$\kappa_{m}/2\pi=1$ MHz, $r=0.4$,
$\Gamma\equiv\Gamma_{1}=\Gamma_{2}=4\kappa_{c}$ and $T=20$ mK.

Figures 2(a-d) show the bipartite entanglements between the two microwave
cavity modes. We observe that the two cavity modes are maximally entangled
either the two cavity modes are completely resonant with the magnon mode,
i.e., $\Delta_{1}=\Delta_{2}=\Delta_{m}=0$ or at
$\Delta_{1}=\Delta_{2}=\Delta_{m}=4\kappa_{c}$ as shown by Fig. 2(a-b).
However, we obtained concentrated cavity-magnon entanglement if we increase
the cavity and magnon detunings ($\Delta_{m}=2\kappa_{c}$ in Fig. 2(c) and
$\Delta_{2}=2\kappa_{c}$ in Fig. 2(d)) as shown by Fig. 2(c-d). Note that in
this case, the two cavity modes must be in resonance with the two sidebands of
the magnon mode, respectively, i.e., $-2\Delta_{1}=2\Delta_{2}=2\Delta_{m}$.

Next, we discuss the effect of the mismatch of the two couplings on the
current scheme Nair, et. al. 35  recently proposed a system to give rise
entanglement between two magnon modes using a single-mode squeezed field. They
have demonstrated that the optimal entanglement between two magnon modes can
be established only if one of the collective quadrature of two magnon modes is
squeezed while taking equal amount of photon-magnon coupling strengths.
However, we use a different scheme to entangle two cavity modes and cavity-
magnon modes using the quantum correlation transferred from a squeezed field.
Note that Fig. 3 (a) shows that, with the rise of the squeezing coefficient
$r$, considerable entanglement between two cavity modes $E_{c-c}^{N}$ is
produced for a broad range of mismatch of the two photon-magnon coupling
strengths. Moreover, it can be seen that smaller value of squeezing parameter
$r$ is much more permissive to the mismatch. However, the entanglement between
cavity and magnon $E_{c-m}^{N}$ is concentrated for a specific range of
squeezing parameter and optimal entanglement is achieved around $r=0.45$ as
exhibited by Fig. 3 (b).

It is vital to mention that we have considered the cavity linewidths is
smaller than bandwidths of input squeezed fields. In Ref. 67 , logarithmic
negativity $E^{N}=0.8$ which corresponds to the bandwidth of 12.5 MHz and
squeezing r = 0.4 has been generated for a two-mode squeezed field. Employing
the same experimental parameters, we have achieve cavity-cavity entanglement
$E_{c-c}^{N}=0.7$ and cavity-magnon entanglement $E_{c-m}^{N}=0.18$ at
$T=20$mK as shown by Fig. 4 (a-b). Furthermore, it is vital to note that
$E_{c-c}^{N}$ ($E_{c-m}^{N}$) survives upto larger temperature than our
previous findings 45 , i.e., upto $T\sim 2.8$K ($T\sim 0.4$K) which turns out
to be much simpler and better than the systems based on nonlinearities 49 .
Figure 4 (a-b) demonstrates how increasing the squeezing parameter $r$ causes
an increase in the average photon number in the squeezed field, which
initially improves cavity-magnon entanglement before being linearly shifted to
enhance the entanglement between two MW cavities. However, this enhancement is
finite to specific region. Furthermore, the bipartitions $E_{c-c}^{N}$ and
$E_{c-m}^{N}$ decreases with the enhancement of temperature.

The creation of photon-magnon-photon genuine tripartite entanglement is a
eminent feature in the cavity-magnon system. To measure genuine tripartite
entanglement, we have considered the minimum residual cotangle criterion. In
Fig. 5, we demonstrate genuine tripartite entanglement (photon-magnon-photon)
by the nonzero (positive) minimum residual contangle
$\mathcal{R}_{\tau}^{min}$. It clear from Fig. 5 (a-b) that the genuine
tripartite photon-magnon-photon entanglement is maximally found when the two
cavity modes are almost resonant with the magnon mode sidebands, respectively,
i.e., at $-\Delta_{1}=\Delta_{2}=\Delta_{m}$ or
$\Delta_{1}=-\Delta_{2}=\Delta_{m}$. It can be seen in Fig. 5 (c-d) that the
genuine tripartite entanglement has been significantly altered by the
squeezing parameter $r$. In addition, Fig. 5 (c) shows the genuine photon-
magnon-photon tripartite entanglement versus temperature and squeezing
parameter, clearly show that it exist for a specific range of squeezing
parameter i.e., $0.1\leq r\leq 0.6$. In addition, 5 (c) clearly showing that
tripartite entanglement cease to exist around $380$mK for $r=0.4$, which is
even larger than the previous findings of generating tripartite entanglement
without JPA 65 . Hence, JPA in present scheme plays a key role in enhancing
photon-magnon-photon tripartite entanglement.

In the conversation that follows is to discuss quantum steering in detail. It
is well known that steerable states between any two modes are always
entangled, but, the converse is not generally legitimate, that is, there is no
restriction on entangled states to be steerable. This feature endorse the idea
that there must be stronger correlations to realize the Gaussian steering. In
Fig. 6, we display an important resulting showing two-way steering between
indirectly coupled cavity modes, however, we have not observe the steering
between magnon and cavity modes. It is important to mention here that as we
have taken the same parameters for the two cavities, therefore, the two
identical cavity modes equally steer each other, i.e.,
$\zeta_{c_{{}_{1}}|c_{{}_{2}}}=\zeta_{c_{{}_{2}}|c_{{}_{1}}}=\zeta_{c|c}$. It
is shown that both cavity modes maximally steer each other either at the
resonance point $\Delta_{1}=\Delta_{2}=\Delta_{m}=0$ or far from the resonance
point $\Delta_{1}=\Delta_{2}=0$, $\Delta_{m}=\pm 4\kappa_{c}$ as shown in Fig.
6 (a). The two-way steering implies that Alice and Bob, as the two parties,
can assure one another that they are sharing an entangled state. It is also
shown in Fig. 6 (b) that not only the steerability but also the survival of
steerability increase with the enhancement of squeezing parameter $r$, which
suggests that Gaussian steering is noticeably sensitive to thermal noise. In
addition, we obtained optimal directional steerability when the two
optomagnonical couplings become equal for the large value of squeezing
parameter $r$ as shown in Fig. 6 (c).

In Fig. 7 (a-b), we plot the directional steering
$\zeta_{c_{1}}|\zeta_{c_{2}}$ (red solid), $\zeta_{c_{2}}|\zeta_{c_{1}}$ (blue
dot-dashed), steering symmetry $\zeta_{S}$ (yellow dashed) and entanglement
$E_{c-c}^{N}$ (cyan solid) to discuss the effect of unequal magnomechanical
coplings and decay rates on steerability and entanglement between the two
cavity modes. Since steerable states are always entangled, therefore, strong
quantum correlations are obligatory for realizing the quantum steering than
that for the entanglement. It can be seen from Fig. 7(a) that
$\zeta_{c_{2}|c_{1}}>\zeta_{c_{1}|c_{2}}$ for $\Gamma_{2}<\Gamma_{1}$ and
$\zeta_{c_{2}|c_{1}}<\zeta_{c_{1}|c_{2}}$ for $\Gamma_{2}>\Gamma_{1}$,
however, $\zeta_{c_{2}|c_{1}}=\zeta_{c_{1}|c_{2}}$ for
$\Gamma_{2}=\Gamma_{1}$. In addition, Fig. 7(b) shows that
$\zeta_{c_{2}|c_{1}}>\zeta_{c_{1}|c_{2}}$ for $\kappa_{2}>\kappa_{1}$ and
$\zeta_{c_{2}|c_{1}}<\zeta_{c1|c2}$ for $\kappa_{2}<\kappa_{1}$, however,
$\zeta_{c_{2}|c_{1}}=\zeta_{c_{1}|c_{2}}$ for $\kappa_{2}=\kappa_{1}$. It is
important to mention here that a crucial and significant tool for safe quantum
teleportation of continuous-variable systems is two-way Gaussian steering.

##  V Conclusions

We have presented a protocol to investigate the enhancement of entanglement of
mainly two kinds bipartitions via JPA, namely cavity-cavity and cavity-magnon,
in a dual cavity magnon system. A magnon mode is simultaneously coupled with
the two cavity modes via magnetic dipole interaction. In this paper, we have
theoretically studied that the steady-state entanglements in the current
system can be achieved within the experimentally feasible parameters. The
entanglements $E^{N}_{c-c}$ and $E^{N}_{m-c}$ were found to be robust against
ambient temperature and survive up to $T\sim 2.8$K and $T\sim 0.4$K,
respectively. In addition, we have also examined the effectiveness of two
entanglements under the situation of mismatch of optomagnonical couplings of
magnon with both cavities and found that cavity-cavity entanglement transfer
efficiency can be attained by substantial squeezing parameter and
optomagnonical coupling strengths. Furthermore, the correlation degree of
tripartite entanglement not only found to be controllable via adjusting the
system parameters such as cavity and magnon detunings, the cavity-magnon
optomagnonical coupling strengths, and squeezing parameter, but also robust
against the temperature. Our proposal for generating entanglement between two
indirectly cavity modes does not demand any nonlinearities and, therefore,
goes against the conventional sense of improving entanglement. Finally, the
quantum steering between two indirectly coupled mode is achieved at the
resonance point with equal magnomechanical couplings and decay rates. Our
scheme broadens the horizons to manipulate the quantum correlation in hybrid
quantum systems.

Figure 8: The maximum of the real parts of the eigenvalues obtained from the
reduced drift matrix for (a) cavity-cavity (b) cavity-magnon vs normalized
cavity detunings.

## Appendix A. Stability of the system

Discussing the stability of the existing system is essential since stability
in the cavity-magnon system is challenging to accomplish. In general, the
system only demonstrates stability when all of the negative real components
eigenvalues of the drift matrix $\mathcal{M}$ are obtained. However, a system
only exhibits instability if any real part of the eigenvalue has a positive
sign. The Routh-Hurwitz criterion, which completely ensures the stability of
any system, is therefore adopted in this context 80 . In Fig. 8, we show the
graphical depiction of the eigenvalues (maximum of the real parts)
$\lambda_{\mathcal{M}}$ ($|\mathcal{M}-\lambda_{\mathcal{M}}\mathbb{1}|=0$) of
the drift matrix $\mathcal{M}$ vs. the normalized detunings. By employing the
chosen parameters for the current system, Fig. 8 unequivocally shows that the
system is stable because the maximum of the real parts of the eigenvalues
remains negative. Furthermore, It can be seen in Fig. 8 (a-b) that the system
is more stable around the resonance points than surrounding region.
Conclusively, the experimental parameters utilized throughout the simulations
fulfills the stability conditions required by the Routh-Hurwitz criterion and
therefore, the working regime is the regime of stability.

## Data availability

All data used during this study are available within the article.

## References

  * (1) Schrödinger E 1935 Naturwissenschaften 23 844
  * (2) Aspelmeyer M, Kippenberg T J and Marquardt F 2014 Reviews of Modern Physics 86 1391
  * (3) Singh S, Mazaheri M, Peng J X, Sohail A, Khalid M and Asjad M 2023 Frontiers in Physics 11 245
  * (4) Singh S, Mazaheri M, Peng J X, Sohail A, Gu Z and Asjad M 2023 Quantum Information Processing 22 198
  * (5) Lakhfif A, Hidki A, El Qars J and Nassik M 2022 Physics Letters A 445 128247
  * (6) Palomaki T, Teufel J, Simmonds R and Lehnert K W 2013 Science 342 710-713
  * (7) Riedinger R, Wallucks A, Marinkovíc I, L¨schnauer C, Aspelmeyer M, Hong S and Gr¨blacher S 2018 Nature 556 473-477
  * (8) Ockeloen-Korppi C, Damskägg E, Pirkkalainen J M, Asjad M, Clerk A, Massel F, Woolley M and Sillanpä M 2018 Nature 556 478-482
  * (9) Amazioug M, Nassik M and Habiballah N 2019 Chinese Journal of Physics 58 1-7
  * (10) Lakhfif A, Hidki A, El Qars J and Nassik M 2022 Journal of Physics B: Atomic, Molecular and Optical Physics 55 115501.
  * (11) Wallucks A, Marinkovíc I, Hensen B, Stockill R and Gr¨blacher S 2020 Nature Physics 16 772-777
  * (12) Fiaschi N, Hensen B, Wallucks A, Benevides R, Li J, Alegre T P M and Gröblacher S 2021 Nature Photonics 15 817-821
  * (13) Wollman E E, Lei C, Weinstein A, Suh J, Kronwald A, Marquardt F, Clerk A A and Schwab K 2015 Science 349 952-955
  * (14) Chu Y, Kharel P, Yoon T, Frunzio L, Rakich P T and Schoelkopf R J 2018 Nature 563 666-670
  * (15) Satzinger K J, Zhong Y, Chang H S, Peairs G A, Bienfait A, Chou M H, Cleland A, Conner C R, Dumur E, Grebel J et al. 2018 Nature 563 661-665.
  * (16) Bild M, Fadel M, Yang Y, von Lüpke U, Martin P, Bruno A and Chu Y 2023 Science 380 274-278
  * (17) Wollack E A, Cleland A Y, Gruenke R G, Wang Z, Arrangoiz-Arriola P and Safavi-Naeini A H 2022 Nature 604 463-467
  * (18) El Qars J, Daoud M and Laamara R A 2018 Physical Review A 98 042115
  * (19) Cherepanov V, Kolokolov I and Lov V 1993 Physics reports 229 81-144
  * (20) Lachance-Quirion D, Tabuchi Y, Gloppe A, Usami K and Nakamura Y 2019 Applied Physics Express 12 070101
  * (21) Kittel C 1948 Physical review 73 155
  * (22) Zhang X, Zou C L, Jiang L and Tang H X 2014 Physical review letters 113 156401
  * (23) Tabuchi Y, Ishino S, Ishikawa T, Yamazaki R, Usami K and Nakamura Y 2014 Physical review letters 113 083603
  * (24) Hidki A, Lakhfif A, El Qars J and Nassik M 2022 The European Physical Journal D 76 64
  * (25) Hisatomi R, Osada A, Tabuchi Y, Ishikawa T, Noguchi A, Yamazaki R, Usami K and Nakamura Y 2016 Physical Review B 93 174427
  * (26) Ihn Y S, Lee S Y, Kim D, Yim S H and Kim Z 2020 Physical Review B 102 064418
  * (27) Tan H and Li J 2021 Physical Review Research 3 013192
  * (28) Zhang D, Luo X Q, Wang Y P, Li T F and You J 2017 Nature communications 8 1368
  * (29) Zhang G, Wang Y and You J 2019 Science China Physics, Mechanics & Astronomy 62 1-11
  * (30) Chai C Z, Zhao H Q, Tang H X, Guo G C, Zou C L and Dong C H 2020 Laser & Photonics Reviews 14 1900252.
  * (31) Wang F and Gou C 2023 Optics Letters 48 1164-1167
  * (32) Ullah K, Naseem M T and Mustecaploğlu Ö E 2020 Physical Review A 102 033721
  * (33) Chai C Z, Shen Z, Zhang Y L, Zhao H Q, Guo G C, Zou C L and Dong C H 2022 Photonics Research 10 820-827
  * (34) Guo Q, Xu D, Cheng J, Tan H and Li J 2023 arXiv preprint arXiv:2304.10760
  * (35) Nair J M and Agarwal G 2020 Applied Physics Letters 117
  * (36) Yu M, Zhu S Y and Li J 2020 Journal of Physics B: Atomic, Molecular and Optical Physics 53 065402
  * (37) Sohail A, Qasymeh M and Eleuch H 2023 Physical Review Applied 20 054062
  * (38) Sohail A, Peng J X, Hidki A, Khalid M and Singh S 2023 Scientific Reports 13 21840
  * (39) Zhang X, Zou C L, Jiang L and Tang H X 2016 Science advances 2 e1501286
  * (40) Li J, Zhu S Y and Agarwal G 2019 Physical Review A 99 021801
  * (41) Li J, Zhu S Y and Agarwal G 2018 Physical review letters 121 203601
  * (42) Hidki A, Ren Y l, Lakhfif A, El Qars J and Nassik M 2023 Journal of Physics B: Atomic, Molecular and Optical Physics 56 115401
  * (43) Yu M, Shen H and Li J 2020 Physical Review Letters 124 213604
  * (44) Li J and Gr¨blacher S 2021 Quantum Science and Technology 6 024005
  * (45) Sohail A, Hassan A, Ahmed R and Yu C s 2022 Quantum Information Processing 21 207
  * (46) Sohail A, Ahmed R, Zainab R and Yu C s 2022 Physica Scripta 97 075102
  * (47) Sohail A, Ahmed R, Shahzad A and Khan M A 2022 International Journal of Theoretical Physics 61 174
  * (48) Hidki A, Lakhfif A, El Qars J and Nassik M 2023 Journal of Modern Optics 1-8
  * (49) Sohail A, Ahmed R, Peng J X, Shahzad A and Singh S 2023 JOSA B 40 1359-1366
  * (50) Sohail A, Ahmed R, Peng J X, Munir T, Shahzad A, Singh S and de Oliveira M C 2023 Journal of Applied Physics 133
  * (51) Wang B, Jia X, Lu X H and Xiong H 2022 Physical Review A 105 053705
  * (52) Liu Z X and Xiong H 2020 Optics Letters 45 5452-5455
  * (53) Kong C, Wang B, Liu Z X, Xiong H and Wu Y 2019 Optics express 27 5544-5556
  * (54) Amazioug M, Teklu B and Asjad M 2023 Scientific Reports 13 3833
  * (55) Zheng Q, Zhong W, Cheng G and Chen A 2023 Results in Physics 48 106422
  * (56) Kani A, Sarma B and Twamley J 2022 Physical Review Letters 128 013602
  * (57) Ding M S, Zheng L and Li C 2020 JOSA B 37 627-634
  * (58) Qi S f and Jing J 2021 Physical Review A 103 043704
  * (59) Sarma B, Busch T and Twamley J 2021 New Journal of Physics 23 043041
  * (60) Zhong W, Zhao D, Cheng G and Chen A 2021 Optics Communications 497 127138
  * (61) Qin W, Miranowicz A, Li P B, Lü X Y, You J Q and Nori F 2018 Physical Review Letters 120 093601
  * (62) Cao W, Lu X, Meng X, Sun J, Shen H and Xiao Y 2020 Physical Review Letters 124 030401
  * (63) Yang Z B, Liu X D, Yin X Y, Ming Y, Liu H Y and Yang R C 2021 Physical Review Applied 15 024042
  * (64) Yang Z B, Liu H Y and Yang R C 2021 Annalen der Physik 533 2100156
  * (65) Yang Z B, Liu J S, Jin H, Zhu Q H, Zhu A D, Liu H Y, Ming Y and Yang R C 2020 Optics Express 28 31862-31871
  * (66) Hidki A, Ren Y L, Lakhfif A, El Qars J and Nassik M 2023 Physics Letters A 463 128667
  * (67) Flurin E, Roch N, Mallet F, Devoret M H and Huard B 2012 Physical review letters 109 183901
  * (68) Jähne K, Genes C, Hammerer K, Wallquist M, Polzik E and Zoller P 2009 Physical Review A 79 063819
  * (69) Eichler C, Bozyigit D, Lang C, Baur M, Steffen L, Fink J, Filipp S and Wallraff A 2011 Physical Review Letters 107 113601
  * (70) Huebl H, Zollitsch C W, Lotze J, Hocke F, Greifenstein M, Marx A, Gross R and Goennenwein S T 2013 Physical Review Letters 111 127003
  * (71) Gardiner C 1986 Physical review letters 56 1917
  * (72) DeJesus E X and Kaufman C 1987 Physical Review A 35 5288
  * (73) La Salle J P 1976 The stability of dynamical systems (SIAM)
  * (74) Simon R 2000 Physical Review Letters 84 2726
  * (75) Adesso G and Illuminati F 2007 Journal of Physics A: Mathematical and Theoretical 40 7821
  * (76) Adesso G and Illuminati F 2006 New Journal of Physics 8 15
  * (77) Coffman V, Kundu J and Wootters W K 2000 Physical Review A 61 052306
  * (78) Kogias I, Lee A R, Ragy S and Adesso G 2015 Physical review letters 114 060403
  * (79) Sohail A, Abbas Z, Ahmed R, Shahzad A, Akhtar N and Peng J X 2023 Annalen der Physik 2300087
  * (80) DeJesus E X and Kaufman C 1987 Physical Review A 35 5288