# SuMe: A Dataset Towards Summarizing Biomedical Mechanisms

###### Abstract

Can language models read biomedical texts and explain the biomedical
mechanisms discussed? In this work we introduce a biomedical mechanism
summarization task. Biomedical studies often investigate the mechanisms behind
how one entity (e.g., a protein or a chemical) affects another in a biological
context. The abstracts of these publications often include a focused set of
sentences that present relevant supporting statements regarding such
relationships, associated experimental evidence, and a concluding sentence
that summarizes the mechanism underlying the relationship. We leverage this
structure and create a summarization task, where the input is a collection of
sentences and the main entities in an abstract, and the output includes the
relationship and a sentence that summarizes the mechanism. Using a small
amount of manually labeled mechanism sentences, we train a mechanism sentence
classifier to filter a large biomedical abstract collection and create a
summarization dataset with 22k instances. We also introduce conclusion
sentence generation as a pretraining task with 611k instances. We benchmark
the performance of large bio-domain language models. We find that while the
pretraining task help improves performance, the best model produces acceptable
mechanism outputs in only 32% of the instances, which shows the task presents
significant challenges in biomedical language understanding and summarization.

Keywords: Explanation Generation, Text Generation, Summarization, Biomedical
NLP, Relation Extraction

languageresourceLanguage Resources

SuMe: A Dataset Towards Summarizing Biomedical Mechanisms

Mohaddeseh Bastan$\spadesuit$, Nishant Shankar$\clubsuit$, Mihai
Surdeanu$\heartsuit$,
---
Niranjan Balasubramanian$\spadesuit$
$\spadesuit$ Stony Brook University $\clubsuit$Delft University of Technology
$\heartsuit$ University of Arizona
{mbastan<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>

Abstract content

## 1\. Introduction

Understanding biochemical mechanisms such as protein signaling pathways is one
of the central pursuits of biomedical research [Strötgen and Gertz, 2012]
[Arighi et al., 2011, Krallinger et al., 2017, Demner-Fushman et al., 2020].
Biomedical research has advanced tremendously in the past few decades, to the
point where we now suffer from “an embarrassment of riches”. Publications are
generated at such a rapid pace (PubMed 111https://pubmed.ncbi.nlm.nih.gov has
indexed more than 1 million publications per year in the past 8 years!) that
we need information access applications which can help extract and organize
biomedical relations and summarize the biomedical mechanisms underlying them.
Developing models that can read biomedical texts and reason about these
mechanisms is an important step towards this.

Figure 1: Biomedical Mechanism Summarization Task: Example of an entry from
the SuMe dataset. Some supporting text was removed to save space. The input is
the supporting sentences with the main two entities. The output is the
relation type and a sentence concluding the mechanism underlying the
relationship. Figure 2: Overview of the semi-automatic bootstrapping process
for SuMe creation. We use a mechanism classifier trained with small amount of
labeled data to produce weakly-labeled training data for mechanism
summarization.

In this paper, we introduce a mechanism summarization task which couples text
that discusses elements of biomedical mechanisms with their summaries. The
task requires models to read text that presents information about the
connection between two target entities and generate a summary sentence that
explains the underlying mechanism and the relation between the entities. We
see this task from two perspectives. First, summarizing biomedical mechanisms
can be seen as part of the broader efforts in extracting [Czarnecki et al.,
2012], organizing [Kemper et al., 2010, Kemper et al., 2010, Miwa et al.,
2013, Subramani et al., 2015, Poon et al., 2014], and summarizing [Azadani et
al., 2018] biomedical literature that are aimed at providing information
access tools for domain experts.

Second, from an NLP perspective this task can be seen as an explainable
relation extraction in a biomedical context, where the explanation is the
mechanism that provides information about why the relation holds or how it
comes about.

A key challenge in addressing such a task lies in creating a large scale
dataset necessary for training large neural models. However, building such a
dataset manually is a laborious process and requires deep biomedical
expertise. To address this, we turn to the structure that exists in biomedical
abstracts, make use of related datasets, and devise a semi-automatic
bootstrapping process that builds on a relatively small amount of labeling
effort from domain experts.

We introduce SuMe, a large scale dataset that we construct from abstracts of
papers that report on biomedical mechanisms. For a given abstract, we create a
task instance that consists of a pair of biochemical entities (regulated and
regulator), the relationship between them (positive/negative activation), and
supporting sentences that provide information about this relationship, and a
sentence that summarizes the mechanism underlying the relation (see Figure 1).
Creating such an instance would require a domain expert to read through an
abstract and assess if it contains a biomedical mechanism and locate it if so.
This process is difficult to scale.

To address this issue, we introduce a semi-automated annotation process to
create a large-scale set for development and automatic evaluation purposes and
a clean small-scale manually curated subset of instances for manual
evaluation. In particular, the necessary entities and relations are extracted
using an existing biomedical information extraction system [Valenzuela-
Escárcega et al., 2018]. To extract mechanism summaries we first collected a
small set of mechanism sentences with the help of domain experts. We use this
to bootstrap a larger sample by training a mechanism sentence classifier with
a biomedical language model (LM) [Kanakarajan et al., 2021] and apply it to a
large collection of about 611K abstracts that contained a conclusion sentence
about the relationship between a pair of entities. The subset that the
classifier identifies as containing mechanism sentences is used to create 22K
mechanism summarization instances. Five domain experts manually analyzed a
dataset sample of 125 instances to construct a clean partition for manual
evaluation purposes. The experts also concluded that the generated dataset has
reasonable quality, i.e., 84%. Note that it is common to tolerate some level
of noise in the training partitions of automatically constructed NLP datasets.
As an example among many, the popular relation extraction dataset by ?)
contains over 20% noise. The overall pipeline is demonstrated in Figure 2. In
summary, the contributions of this paper are the following:

  * •

We introduce the SuMe dataset, the first dataset towards summarizing
biomedical mechanisms and the underlying relations between entities. The
dataset contains 22K mechanism summarization instances collected semi-
automatically, an evaluation partition of 125 instances that were corrected by
domain experts. We also create a conclusion generation task from the larger
set of 611K abstracts which we use as a pretraining task for mechanism
generation models.

  * •

We benchmark several state-of-the-art language models for the task of
generating the underlying biochemical relations and the corresponding
mechanism sentences. We train general domain LMs (GPT2 [Radford et al., 2019],
T5 [Raffel et al., 2020a], BART [Lewis et al., 2019]), as well as science
domain adapted versions(scientific GPT2 [Papanikolaou and Pierleoni, 2020],
and SciFive [Phan et al., 2021]) and benchmark their performance through both
automatic evaluation and manual evaluation on curated evaluation samples.

  * •

The evaluation by domain experts suggests that this is a high quality dataset
coupled with a challenging task, which deserves further investigation.

  * •

To encourage reproducibility and further research, we release the dataset and
the code used during its creation. Both are available at SuMe webpage.

## 2\. Related Work

Deep learning models have been widely used in different NLP applications
[Gaonkar et al., 2020, Bastan et al., 2020, Keymanesh et al., 2021, Heidari et
al., 2021]. Amongst these applications, biomedical NLP is using these models
that looks at extracting [Alam et al., 2018, Mulyar et al., 2021, Giorgi and
Bader, 2020], organizing [Yuan et al., 2020, Zhao et al., 2020, Lauriola et
al., 2021], and summarizing information [Cohan et al., 2018] from scientific
literature.

Within this broad context, the mechanism summarization task we introduce
broadly relates to previous work in reading and generating information from
scientific texts. Most work in this area focus on generating summaries using
scientific publication and some times in combination with external information
[Yasunaga et al., 2019, DeYoung et al., 2020, Collins et al., 2017]
Some works even seek to generate part of the scientific papers. For example,
TLDR [Cachola et al., 2020] introduces a task and a dataset to generate TLDRs
(Too Long; Didn’t Read) for papers. They exploit titles and an auxiliary
training signal in their model. ScisummNet [Yasunaga et al., 2019] introduces
a large manually annotated dataset for generating paper summaries by utilizing
their abstracts and citations. TalkSumm [Lev et al., 2019] generates summaries
for scientific papers by utilizing videos of talks at scientific conferences.
PaperRobot [Wang et al., 2019] generates a paper’s abstract, title, and
conclusion using a knowledge graph. FacetSum [Cohan et al., 2018] used Emerald
journal articles to generate 4 different abstractive summaries, each targeted
at specific sections of scientific documents.

In addition to the specifics of the output that we target, our work is
different from all these other works because our proposed summarization task
is grounded with the underlying biomedical event discussed, rather than
focusing on generic summarization, which may lose the connection to the
underlying biology that is the core material discussed in these papers. We
address mechanism generation, which can be seen as a combination of
explainable relation extraction and summarization. There is a huge body of
work that addresses explainable methods (e.g., relation extraction [Shahbazi
et al., 2020] or explainable QA [Thayaparan et al., 2020]). Many prior works
in relation and event extraction treat explanations as the task of selecting
or ranking sentences that support a relation (e.g., [Shahbazi et al., 2020,
Çano and Bojar, 2020, Yasunaga et al., 2019]). Our work differs from these in
that it focuses on generating mechanisms underlying a relation from supporting
sentences, rather than identifying existing sentences.

## 3\. Mechanism Summarization

Our goal is to develop a task and a dataset that pushes models towards
distilling the mechanisms that underlie the relationships between entities
from biomedical literature. From a language processing perspective, we can
view mechanisms as a form of explanation that justifies the relationship or
connection between entities. From a biomedical science perspective, a
mechanism provides two types of explanatory information, which we use to
characterize mechanism sentences:
Why is the relation true? A sentence can be a mechanism, if it explains why
the relation exists between the two main entities. For example, one protein
(say A) might be up-regulate another (say B), which in turn inhibits yet
another protein (say C). This provides the causal reasoning to conclude the
relation that protein A inhibits protein C.

How does the relation come about? Another kind of explanatory information is
the one that describes the process or manner in which the relation exists
between the pair of entities. For example, one protein (say A) may activate
another protein (say B) via a specific process.

These provide a way to specify what constitutes a mechanism sentence and help
us to locate mechanism sentences in the literature. In particular, we consider
abstracts which discuss studies that lead to conclusions about such
mechanisms. Typically, these abstracts provide a short set of sentences that
describe the goals of the study, the methods used, the experimental
observations, the findings, which can be used to substantiate the conclusions
that establish the relation of interest, and the mechanism underlying the
relation. This suggests a language processing task that tests for ability to
understand biomedical mechanisms: given the preceding sentences in the
abstract can a model accurately generate the underlying mechanism?

### 3.1. Task Definition

Given a set of sentences from a scientific abstract (referred to as supporting
sentences) and a pair of entities $(e_{i},e_{j})$ that are the focus of the
abstract (referred to as focus entities), generate the conclusion sentence
that explains the mechanism behind the pair entities and output a relation
that connects these entities (e.g., positive_activation$(e_{i},e_{j})$).
Figure 1 shows an example of such a tuple of supporting sentences, focus
entities, relation, and mechanism sentence. As the example illustrates,
mechanism sentences describe some pathway often involving another entity or a
process (e.g., dopaminergic mechanism), require identifying and combining
information from multiple relevant sentences, and non-trivial inferences
regarding the relationship between the entities (e.g., recognizing that the
different effects on _prepulse inhibition_ imply differential involvement).

The task definition suggests what we need to build a dataset. Given an
abstract of a scientific literature we need four pieces of information: (1)
the two focus entities of the abstract; (2) the relation between entities; (3)
sentences from the abstract in support of this relation; and (4) the
conclusion sentence where the mechanism underlying the relation is summarized.

## 4\. SuMe Dataset

We aim to create a large scale dataset for the mechanism summarization task
defined above. However, identifying instances for this task requires domain
expertise and cannot be easily done at scale. Instead, here we employ a
bootstrapping process, where we first annotate a small amount of data to build
a mechanism sentence classifier that can then helps us collect a large scale
dataset for mechanism summarization. The key observation here is that
identifying sentences that express a mechanism is a simpler task than the
targeted mechanism summarization task, and, thus, should be learnable from
smaller amounts of data. We outline the process we use for creating our
mechanism summarization dataset, SuMe, and an expert evaluation of its quality
next.

### 4.1. SuMe Construction Process

We construct SuMe using biomedical abstracts from the PubMed open access
subset222https://pubmed.ncbi.nlm.nih.gov. Starting from 1.1M scientific papers
333We used all papers available in NIH active directory, we use the following
sequence of bootstrapping steps to construct SuMe. The following steps are
also elaborated in Figure 3.

Figure 3: The bootstrapping pipeline for SuMe collection and human evaluation.
The main idea behind the pipeline is to collect relatively easy to acquire
judgments from domain experts to then bootstrap and generate a weakly-labeled
large training corpus. We further assess the quality of the resulting dataset
through another round of human evaluation, which also yields a smaller curated
evaluation dataset.

1\. Finding Conclusion Sentences: First, we use simple lexical patterns to
find abstracts with a clearly specified conclusion sentence. All abstracts
which has any form of conclude word (conclusion, concluded, concluding,
concludes, etc.) at the very end of the text are extracted here. We use this
matching process to also split the abstracts into the set of supporting
sentences (the ones that lead up to the conclusion) and one conclusion
sentence (the one that includes the conclude word).

2\. Extracting Main Entities & Relation Starting with the abstracts which are
now in the form of (supporting sentences, conclusion sentence), we then run a
biomedical relation extractor, REACH [Valenzuela-Escárcega et al., 2018],
which can identify protein-protein and chemical-protein relations between
entities. In this work, we focus on the relations where one entity is the
controller and another entity is the controlled entity and the relation
between them is either positive/negative activation or  positive/negative
regulation. If an abstract does not contain any such relation, we keep it for
the pretraining step (as described in Section 5.3); otherwise we use it for
the main task.

3\. Filtering for Mechanism Sentences:

We then filter out the instances to only retain those whose conclusion
sentences are indeed a mechanism sentence. To this end, we devised a
bootstrapping process where we first collect supervised data to train a
classifier. To collect likely mechanism sentences we made use of the ChemProt
[Peng et al., 2019] relation extraction dataset which contains sentences
annotated with positive and negative regulation relations between entities.
However, not all of these sentences necessarily explain the mechanism behind
these relations. We asked 21 experts (grad students in a biomedical
department) to inspect each sentence and rate whether it explains the
mechanism behind the ChemProt annotated relation on a four-point Likert scale.
For each sentence, an annotator can select between Clearly a Mechanism,
Plausibly a Mechanism, Clearly not a Mechanism, and Not Sure. Each sentence is
annotated by three experts and we find the inter-annotator agreement between
users to be $\kappa=73\%$ (Fleiss Kappa [Landis and Koch, 1977]). The final
label for a sentence is selected based on the majority voting after combining
Clearly a Mechanism and Plausible a Mechanism labels. Finally, each sentence
is labeled as a Mechanism, or Non-Mechanism. The resulting dataset contained
439 Mechanism sentences (264 Clearly, 175 Plausibly) and 447 Non-Mechanism
sentences.

Using this small scale mechanism sentence dataset, we train binary classifiers
to identify mechanism sentences, where the positive label indicates that the
underlying sentence is a mechanism sentence. We fine-tuned multiple
transformer-based models: BioBERT [Lee et al., 2020], SciBERT [Beltagy et al.,
2019], BiomedNLP [Gu et al., 2020], and BioELECTRA [Kanakarajan et al., 2021]
models. Each model is fitted with a non-linear classification layer that takes
the output representation for the [CLS] token. The classification layer and
top three layers of the transformer are finetuned using the annotated
data444All models used are base versions with 768 hidden size and 12 layers.
We set the learning rate to be $2e-4$ with a decay of $0.001$. We used 80%-20%
split for train-test. BioELECTRA performed the best with 74% macro F1 for
mechanism sentence classification.

We use this trained mechanism sentence classifier to label all conclusion
sentences from the previous step and instances which are predicted to be
mechanism sentences are used to create the mechanism generation of the SuMe
dataset.

We separate out the abstracts which are predicted to have non-mechanism
sentences as additional data. We can define a broader conclusion generation
task, which can be be used as a pre-training task for the generative models
that eventually use for the mechanism summarization task (as we describe in
Section 5.3).

The above procedure results in a dataset that allows us to define the
following mechanism summarization task: Given a set of supporting sentences
from an abstract and a pair of entities $(e_{i},e_{j})$, generate a relation
that connects these entities and a sentence that explains the mechanism that
was the focus of the study. The statistics of the dataset are shown in Table
1. The dataset consists of three subsets, the training set with about 20k
instances which the parameters of the model are trained with, the validation
set (Dev) for tuning hyper parameters and choosing the best model, and the
test set which is not used until the final evaluation. There is also a small
set of 125 instances which is curated by experts and is used as another test
set but is not reported in this table.

Dataset | Train | Dev | Test
---|---|---|---
Abstracts | 20765 | 1000 | 1000
Avg. #words in conc. | 33.7 | 34.9 | 33.5
Avg. #words in supp. | 187.5 | 187.9 | 186.7
Avg. #sent. in supp. | 12.15 | 12.44 | 12.33
#Unique controller | 8094 | 759 | 777
#Unique controlled | 6684 | 717 | 687
#Unique pair entities | 19229 | 988 | 989
#Unique entities | 12685 | 1357 | 1364
Table 1: Dataset Statistics: Each dataset contains a number of unique
abstracts, a supporting set (supp.), a mechanism sentence (conc.) a pair of
entities. The first entity is called the regulator entity (regulator) and the
second one is called the regulated entity (regulated)

### 4.2. SuMe Quality

Our process creates a large scale, albeit, a bootstrapped dataset that can be
used to train large language generation models. What is the quality of this
dataset? To assess this we asked five biomedical experts to evaluate a random
sample of 125 sentences from the dataset. The experts were given the set of
input supporting sentences, the potential mechanism sentence, and the relation
between main entities. Our aim is two fold, first to evaluate the quality of
the data collection process, second to collect a clean human evaluated dataset
which can be used as an extra test set.

The experts were asked to assess errors in the relation label, mechanism, and
the need for background knowledge:

  1. 1.

Relation Errors: Is the expected output relation associated with the instance
valid?

  2. 2.

Mechanism vs. Non-mechanism: Is the output sentence expected for this sentence
an actual mechanism sentence?

  3. 3.

Background Knowledge:  Can the information in the output sentence concluded
from the information in the input supporting sentences?

The results of the dataset evaluation are shown in Table 2. Only 16% of the
data has some error either from relation extraction (question 1) or contains a
non-mechanism output sentence (question 2).This evaluation shows that the
generated dataset is of reasonable quality, and can serve as a meaningful
resource for training models for mechanism summarization. The clean subset
that has no relation or mechanism errors is used as an extra test for
evaluation. Last, the experts also rated 15% of the instances to require
background knowledge (question 3)indicating the fraction of hard instances.

Quality | Correct
---|---
Entities & Relation Extraction | 90%
Mechanism Sentence Classifier | 85%
Instances w/o noise | 84%
Table 2: Dataset Quality: We asked three main questions. This table shows what
percentage of each category is acceptable. The last question shows what
percentage of the sentences are approved in all questions.

## 5\. Evaluation

Model | RG (F1) | BLEURT | Rouge-1 | Rouge-2 | Rouge-L
---|---|---|---|---|---
BART | 76 | 42.49 | 46.54 | 25.92 | 35.34
GPT2 | 74 | 44.19 | 46.54 | 28.32 | 38.78
T5 | 72 | 44.41 | 48.26 | 27.63 | 38.77
GPT2-Pubmed | 78 | 46.33 | 48.37 | 29.55 | 40.19
SciFive | 79 | 47.81 | 52.10 | 32.62 | 43.31
Table 3: Benchmarking performance of strong language generation models and
some domain-adapted models. We present standard automatic evaluations measures
for the mechanism sentence generation task along with F1 for the generated
relations. The science domain versions of both GPT2 and T5 work better than
the original versions.

Our evaluation focuses on the following questions:

  1. 1.

Benchmarking: What is the performance of generic and domain-adapted large
scale language generation models on SuMe?

  2. 2.

Effect of pretraining: What is the impact of using the additional data via
pretraining?

  3. 3.

Effect of modeling supporting sentences: What is the impact of selecting a
subset of supporting sentences?

  4. 4.

Error analysis: What are the main failure modes of language generation models?

### 5.1. Experimental Setup

We use SuMe to benchmark language generation models and measure their ability
to correctly identify the relation between the focus entities and to summarize
the mechanism behind the relation based on the input sentences from the
abstract.

Models: We compare pretrained GPT-2 [Radford et al., 2019], T5 [Raffel et al.,
2020b], BART [Lewis et al., 2019] models and two domain-adapted models,
GPT2-Pubmed [Papanikolaou and Pierleoni, 2020], and SciFive [Phan et al.,
2021], which were trained on scientific literature.

Evaluation Metrics: We conduct both automatic and manual evaluation of the
model outputs.

Relation Generation (RG): The models are supposed to first generate the
relation type (positive or negative regulation) and then generate the
mechanism that underlies this relation. We evaluate the model’s output as we
would for a corresponding classification task, i.e., the generated relation is
deemed correct if it exactly matches the correct relation name. We report F1
numbers for this binary classification task.

Mechanism Generation: We evaluate the quality of the generated explanations
using two language generation metrics: the widely-used ROUGE [Lin, 2004]
scores that rely on lexical overlap, and BLEURT scores [Sellam et al., 2020]
which aim to capture semantic similarity between the generated and the gold
reference.

We use a recent version, the BLEURT-20 model that has been shown to be more
effective [Pu et al., 2021] . We compare the generated text as the hypothesis
against the actual text as the reference.

Fine-tuning and Training Details: All models are original base models
published by HuggingFace that were fine-tuned on the training portion of SuMe
for 20 epochs. For each model, we evaluate the average of BLEURT and Rouge-L
score on the validation (Dev) set and the one with the highest average is
chosen for prediction. The learning rate is set to 6e-5, we use AdamW
[Loshchilov and Hutter, 2017] optimizer with $\epsilon=1e-8$. The input token
is limited to 512 tokens, and the generated token is maxed out at 128. We
select batch size of 8 with gradient accumulation steps of two.

### 5.2. Automatic Evaluation Results

Table 3 compares the performance of the five language generation models on
both the relation generation (RG) and mechanism generation tasks.

The domain-adapted models, GPT2-Pubmed and SciFive, fare better than fine-
tuning the standard pre-trained models for both relation and mechanism
generation tasks. SciFive achieves the best performance with more than a
$7.5\%$ increase in BLEURT score and more than $9.7\%$ increase in RG F1 over
the standard T5 model, highlighting the importance of domain adaptation for
the SuMe tasks defined over scientific literature.

The overall numbers (coupled with the human evaluation in Section 5.5) suggest
that mechanism generation is a difficult and challenging task.

The models achieve better performance on the relation generation task but
there is still a substantial room for improvement here with the best model
achieving an F1 of 79. If the model is unable to generate the relation
correctly, then the mechanism it generates is not useful. Ideally we want
models to correctly generate both the relation and the mechanism that
underlies it. We also evaluated the correlation between BLEURT score and
relation generation classification score. Our analysis shows that when the
model generates an accurate relation, it gets higher BLEURT score while when
it generates an incorrect relation, its gets a 10% lower BLEURT score (50.02
vs 45.08)

### 5.3. Pretraining with Conclusion Generation

Next we analyze the impact of pre-training the models on the related task of
generating conclusion (instead of mechanism) sentences, for which we can
obtain data at scale without any manual labeling effort. We collected all
abstracts from PubMed that ended with a conclusion sentence. We can create
training instances on these abstracts in the same format as we did for the
mechanism generation instances. The only difference here is the output
sentences are conclusion sentences and not necessarily mechanisms. We call
this the conclusion generation task. SuMe includes 611K instances of this kind
which is an order of magnitude larger than the mechanism summarization
instances and can be seen as a form of data augmentation.

We study the effect of this pretraining task by varying the amount of
pretraining data. We analyze the impact in terms of the overall effectiveness
and the amount of fine-tuning (number of epochs) needed to converge when
finetuning.

Pretraining Data Size:

Figure 4: Comparison of relation generation (RG) F1 (left y-axis/blue bars)
and the mechanism generation measures (right y-axis/teal+blue curves) against
the amount of pretraining. As we increase the size of the pretraining data,
the model performance improves.

We pretrain the SciFive model on the conclusion generation task with
increasing amount of data (100K increments), and measure the performance of
finetuning the pretrained models on the mechanism summarization task. Figure 4
shows that performance increases with more data available for pretraining,
suggesting that pretraining is beneficial for learning to generate mechanisms.

Number of Epochs: We also compare the impact of the amount of pretraining on
the number of epochs needed for convergence in fine-tuning. Figure 5 compares
pretrained models with different number of pretraining epochs (x-axis) in
terms of their overall effectiveness (BLEURT score bars) and the number of
epochs to convergence (Finetuning epochs curve). The figure shows that when we
continue pretraining, not only does the resulting model performs better, but
it also converges sooner taking fewer number of epochs to reach higher
effectiveness. Together these results suggest potential for the auxiliary data
available in the SuMe dataset.

Figure 5: Number of pretraining epochs vs. fine-tuning epochs for each
pretrained model until convergence.

### 5.4. Modeling Supporting Sentences

Will it help to model the subset of sentences within the inputs sentences that
provide the best support for generating the mechanism sentence? This kind of
an extractive step has been used previously in summarization tasks to reduce
the amount of irrelevant information in the input [Narayan et al., 2018, Liu
and Lapata, 2019]. To understand the utility of this, we built a pseudo-oracle
that finds the sentences that have the best overlap (measured via BLEURT score
[Sellam et al., 2020]) with the output mechanism sentence. Then, we trained
the SciFive model and pretrained version to only use the top few sentences
according to BLEURT score such that input size is now half of the original
input size. Using this subset instead of the entire subset provides BLEURT
score improvements only for the basic SciFive model and the gains reduce when
we use the pretrained model. Unlike standard summarization tasks there are
fewer completely unrelated sentences in the abstracts and generating the
mechanism sentences remains challenging even when we are able to identify the
most relevant sentences within this set. This suggests that the task remains
hard even when the most important sentences are somehow known to the model.

Supporting Set | BLEURT | Rouge-L
---|---|---
SciFive | 47.81 | 43.31
+Oracle | 49 | 43.07
+Pretraining | 49.05 | 43.72
+Pretraining+Oracle | 49.64 | 43.81
Table 4: The effect of selecting supporting sentences with highest BLEURT
score.

### 5.5. Manual Evaluation

We also conduct a manual evaluation of the outputs from the best model — the
SciFive model that was pretrained with the conclusion generation task. We
asked three biomedical experts to evaluate output sentences for 100 instances
and answer three questions (It took $\sim 5$ minutes per expert per instance):

  1. 1.

Does the generated sentence contain a mechanism that explains the relation
between the two main entities?

  2. 2.

Is the information in the generated mechanism supported by the information
contained in the supporting set of sentences?

  3. 3.

Is the generated mechanism factually correct?

The results show that the model learns to generate some mechanism sentence
that connects the main entities for most of the instances (79%). It also
produces output for which there is support in the input sentences in a
majority of the cases (53%). Last, experts found that the output statements to
be scientifically correct in many cases (58%). In summary, however, only 32%
of the outputs were acceptable for all questions, and were deemed to be good
mechanism sentences. This again highlights the significant challenge posed by
this task.

### 5.6. Error Analysis

Gold Mechanism |  Generated Mechanism
---|---
On the basis of these observations, we conclude that IL-18 induces MCP-1 production through the PI3K/Akt and MEK/ERK1/2 pathways in macrophages. |  In conclusion, IL-18  induces  MCP-1 production in macrophages through the PI3K/Akt and MEK/ERK1/2 pathways.
Taken together, we conclude that DeltaNp73 negatively regulates  NGF-mediated neuronal differentiation by transrepressing TrkA. |  In conclusion, our results indicate that DeltaNp73  negatively regulates NGF-mediated neuronal differentiation by transcriptionally repressing the expression of TrkA.
In conclusion, the increase in SNGFR caused by ANF is associated with an increase in glomerular capillary hydraulic pressure and with a blunted maximal tubuloglomerular feedback response. |  We conclude that ANF inhibits maximal tubuloglomerular feedback by increasing glomerular capillary hydraulic pressure and by decreasing  SNGFR.
We conclude that, without modulatory factors which play a role in vivo,  NGF can enhance the synthesis of tyrosine hydroxylase n sympathetic ganglia in vitro, provided organ culture conditions which permit optimal survival of adrenergic neurons are selected. |  This conclusion was further supported by pulse-labeling of tyrosine hydroxylase with [3H]leucine, which showed that NGF  increased synthesis of tyrosine in sympathetic ganglia by approximately 50%.
We conclude that 20 mm alcohol/submaximal CCK as well supramaximal CCK stimulation can trigger pathologic basolateral exocytosis in pancreatic acinar cells via PKC alpha-mediated activation of Munc18c , which enables Syntaxin-4 to become receptive in forming a SNARE complex in the BPM. |  We conclude that alcohol can induce a clinically relevant form of pancreatitis by blocking apical exocytosis and redirecting exocytosis to less efficient BPM, mimicking supramaximal CCK stimulation.
We conclude that in the presence of high doses of insulin, FSH  decreases aromatase activity, and an uncoupling of P450  aromatase mRNA and aromatase activity occurs. |  In conclusion, insulin stimulates aromatase activity in bovine granulosa cells at low doses but fails to stimulate activity at higher doses of insulin.
Table 5: Examples of the generated outputs by the model. The first three are
good outputs where the mechanism is a simple paraphrase of the expected gold
mechanism, while the next three illustrate the types of semantic errors we
observe. The main entities are makred in Italics. The phrase explaining the
mechanism in gold data is in blue, in good generation is in green, and in bad
generation is in red.

To understand the frequent failure modes of the model, we manually categorized
the errors in 100 outputs that had the worst BLEURT scores with the reference
mechanism sentences.

  1. 1.

Missing Entities (35%) – The most prevalent issue is the absence of one of the
main entities in the generated sentence. Despite this being a necessary
feature in all of the mechanism sentences in the training data, the prevalence
of this error shows that models find it difficult to track the main entities
during generation.

  2. 2.

Incorrect Mechanism (24%) – The model is unable to generate the correct
mechanism even though it is able to identify the correct relation and fills in
some information that is either unrelated to or unsupported by the input
sentences.

  3. 3.

Flipped Relation (19%) – The model predicts the incorrect relation and
generates a mechanism that is faithful to this incorrect relation. Improving
relation generation is thus an important step for improving mechanism
generation.

  4. 4.

Non Mechanisms (11%) – While the model learns to generate mechanism like
sentences for the most part, it sometimes still fails to produce sentences
that contain any mechanism at all.

  5. 5.

Multiple pieces of information (11%) – Some complex mechanisms require
combining bits of information from different input sentences. The model
generates only a part of such mechanisms.

### 5.7. Word Analysis

We further analyzed the unigrams of the supporting sentences corresponding to
the instances where the model was most confident in its generated mechanism
and where it was least confident. The analysis shows that when the words
’binding’, ’caused’, ’demonstrated’, ’dose dependent’, ’investigated’,
’result’, and ’performed’ are available in the supporting sentences the model
can generate explanation sentences with higher quality. This shows that when
the supporting sentences convey causal relation and reasoning the model is
most confident about generating mechanisms.

Table 5 shows example generated mechanisms. The first three showcase good
outputs whereas the next three are examples of incorrect ones. In the good
ones, the first is a generated mechanism that is almost identical to the gold
mechanism with only a slight syntactic change. The second is a generated
mechanism which also conveys the gold mechanism accurately but with a
paraphrasing that expands the technical term transpressing. In the last three
examples with incorrect information, the first shows a bad output which
contains a mechanism but not of the relation connecting the main entities. The
next is a case where the information is correct but it does not even mention
the main entities. The last one is an example one of the entities are missing
(FSH) and the generated text is about another relation.

## 6\. Conclusions

We introduced SuMe, a dataset for biomedical mechanism summarization. This
dataset is coupled with a challenging summarization task, which requires the
generation of the relation between main entities as well as a textual summary
of the mechanism which explains the reason behind the underlying relation.
This dataset is collected using the sentences from actual publication
abstracts. We also introduce an easier and scalable pretraining task which
improves the baselines by augmenting a larger set of sentences to the main
dataset. We evaluated the complexity of the task using multiple state-of-the-
art transformer based models. Our evaluation suggests that the proposed task
is learnable, but we are far from solving it. The expert analysis also
suggests the difficulty and importance of the task.

All in all, we believe that SuMe dataset and associated task are a useful step
towards building true information-access applications for the biomedical
literature.

### Acknowledgments

This work was supported in part by the National Science Foundation under
grants IIS-1815358 and IIS-1815948.

## 7\. Bibliographical References

## References

  * Alam et al., 2018 Alam, F., Joty, S., and Imran, M.  (2018).  Domain adaptation with adversarial training and graph embeddings.  arXiv preprint arXiv:1805.05151.
  * Arighi et al., 2011 Arighi, C. N., Lu, Z., Krallinger, M., Cohen, K. B., Wilbur, W. J., Valencia, A., Hirschman, L., and Wu, C. H.  (2011).  Overview of the biocreative iii workshop.  BMC bioinformatics, 12(8):1–9.
  * Azadani et al., 2018 Azadani, M. N., Ghadiri, N., and Davoodijam, E.  (2018).  Graph-based biomedical text summarization: An itemset mining and sentence clustering approach.  Journal of biomedical informatics, 84:42–58.
  * Bastan et al., 2020 Bastan, M., Koupaee, M., Son, Y., Sicoli, R., and Balasubramanian, N.  (2020).  Author’s sentiment prediction.  In Proceedings of the 28th International Conference on Computational Linguistics, pages 604–615, Barcelona, Spain (Online), December. International Committee on Computational Linguistics.
  * Beltagy et al., 2019 Beltagy, I., Lo, K., and Cohan, A.  (2019).  Scibert: Pretrained language model for scientific text.  In EMNLP.
  * Cachola et al., 2020 Cachola, I., Lo, K., Cohan, A., and Weld, D. S.  (2020).  Tldr: Extreme summarization of scientific documents.  arXiv preprint arXiv:2004.15011.
  * Çano and Bojar, 2020 Çano, E. and Bojar, O.  (2020).  Two huge title and keyword generation corpora of research articles.  arXiv preprint arXiv:2002.04689.
  * Cohan et al., 2018 Cohan, A., Dernoncourt, F., Kim, D. S., Bui, T., Kim, S., Chang, W., and Goharian, N.  (2018).  A discourse-aware attention model for abstractive summarization of long documents.  CoRR, abs/1804.05685.
  * Collins et al., 2017 Collins, E., Augenstein, I., and Riedel, S.  (2017).  A supervised approach to extractive summarisation of scientific papers.  arXiv preprint arXiv:1706.03946.
  * Czarnecki et al., 2012 Czarnecki, J., Nobeli, I., Smith, A. M., and Shepherd, A. J.  (2012).  A text-mining system for extracting metabolic reactions from full-text articles.  BMC bioinformatics, 13(1):1–14.
  * Demner-Fushman et al., 2020 Dina Demner-Fushman, et al., editors.  (2020).  Proceedings of the 19th SIGBioMed Workshop on Biomedical Language Processing, Online, July. Association for Computational Linguistics.
  * DeYoung et al., 2020 DeYoung, J., Lehman, E., Nye, B., Marshall, I. J., and Wallace, B. C.  (2020).  Evidence inference 2.0: More data, better models.  arXiv preprint arXiv:2005.04177.
  * Gaonkar et al., 2020 Gaonkar, R., Kwon, H., Bastan, M., Balasubramanian, N., and Chambers, N.  (2020).  Modeling label semantics for predicting emotional reactions.  In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pages 4687–4692, Online, July. Association for Computational Linguistics.
  * Giorgi and Bader, 2020 Giorgi, J. M. and Bader, G. D.  (2020).  Towards reliable named entity recognition in the biomedical domain.  Bioinformatics, 36(1):280–286.
  * Gu et al., 2020 Gu, Y., Tinn, R., Cheng, H., Lucas, M., Usuyama, N., Liu, X., Naumann, T., Gao, J., and Poon, H.  (2020).  Domain-specific language model pretraining for biomedical natural language processing.
  * Heidari et al., 2021 Heidari, M., Zad, S., Hajibabaee, P., Malekzadeh, M., HekmatiAthar, S., Uzuner, O., and Jones, J. H.  (2021).  Bert model for fake news detection based on social bot activities in the covid-19 pandemic.  In 2021 IEEE 12th Annual Ubiquitous Computing, Electronics Mobile Communication Conference (UEMCON), pages 0103–0109.
  * Kanakarajan et al., 2021 Kanakarajan, K. r., Kundumani, B., and Sankarasubbu, M.  (2021).  BioELECTRA:pretrained biomedical text encoder using discriminators.  In Proceedings of the 20th Workshop on Biomedical Language Processing, pages 143–154, Online, June. Association for Computational Linguistics.
  * Kemper et al., 2010 Kemper, B., Matsuzaki, T., Matsuoka, Y., Tsuruoka, Y., Kitano, H., Ananiadou, S., and Tsujii, J.  (2010).  Pathtext: a text mining integrator for biological pathway visualizations.  Bioinformatics, 26(12):i374–i381.
  * Keymanesh et al., 2021 Keymanesh, M., Elsner, M., and Parthasarathy, S.  (2021).  Privacy policy question answering assistant: A query-guided extractive summarization approach.  CoRR, abs/2109.14638.
  * Krallinger et al., 2017 Krallinger, M., Pérez-Pérez, M., Pérez-Rodríguez, G., Blanco-Míguez, A., Fdez-Riverola, F., Capella-Gutierrez, S., Lourenço, A., and Valencia, A.  (2017).  The biocreative v. 5 evaluation workshop: tasks, organization, sessions and topics.
  * Landis and Koch, 1977 Landis, J. R. and Koch, G. G.  (1977).  The measurement of observer agreement for categorical data.  biometrics, pages 159–174.
  * Lauriola et al., 2021 Lauriola, I., Aiolli, F., Lavelli, A., and Rinaldi, F.  (2021).  Learning adaptive representations for entity recognition in the biomedical domain.  Journal of biomedical semantics, 12(1):1–13.
  * Lee et al., 2020 Lee, J., Yoon, W., Kim, S., Kim, D., Kim, S., So, C. H., and Kang, J.  (2020).  Biobert: a pre-trained biomedical language representation model for biomedical text mining.  Bioinformatics, 36(4):1234–1240.
  * Lev et al., 2019 Lev, G., Shmueli-Scheuer, M., Herzig, J., Jerbi, A., and Konopnicki, D.  (2019).  Talksumm: A dataset and scalable annotation method for scientific paper summarization based on conference talks.  arXiv preprint arXiv:1906.01351.
  * Lewis et al., 2019 Lewis, M., Liu, Y., Goyal, N., Ghazvininejad, M., Mohamed, A., Levy, O., Stoyanov, V., and Zettlemoyer, L.  (2019).  Bart: Denoising sequence-to-sequence pre-training for natural language generation, translation, and comprehension.  arXiv preprint arXiv:1910.13461.
  * Lin, 2004 Lin, C.-Y.  (2004).  Rouge: A package for automatic evaluation of summaries.  In Text summarization branches out, pages 74–81.
  * Liu and Lapata, 2019 Liu, Y. and Lapata, M.  (2019).  Text summarization with pretrained encoders.  arXiv preprint arXiv:1908.08345.
  * Loshchilov and Hutter, 2017 Loshchilov, I. and Hutter, F.  (2017).  Fixing weight decay regularization in adam.  CoRR, abs/1711.05101.
  * Miwa et al., 2013 Miwa, M., Ohta, T., Rak, R., Rowley, A., Kell, D. B., Pyysalo, S., and Ananiadou, S.  (2013).  A method for integrating and ranking the evidence for biochemical pathways by mining reactions from text.  Bioinformatics, 29(13):i44–i52.
  * Mulyar et al., 2021 Mulyar, A., Uzuner, O., and McInnes, B.  (2021).  Mt-clinical bert: scaling clinical information extraction with multitask learning.  Journal of the American Medical Informatics Association, 28(10):2108–2115.
  * Narayan et al., 2018 Narayan, S., Cohen, S. B., and Lapata, M.  (2018).  Don’t give me the details, just the summary! topic-aware convolutional neural networks for extreme summarization.  arXiv preprint arXiv:1808.08745.
  * Papanikolaou and Pierleoni, 2020 Papanikolaou, Y. and Pierleoni, A.  (2020).  Dare: Data augmented relation extraction with gpt-2.  arXiv preprint arXiv:2004.13845.
  * Peng et al., 2019 Peng, Y., Yan, S., and Lu, Z.  (2019).  Transfer learning in biomedical natural language processing: an evaluation of bert and elmo on ten benchmarking datasets.  arXiv preprint arXiv:1906.05474.
  * Phan et al., 2021 Phan, L. N., Anibal, J. T., Tran, H., Chanana, S., Bahadroglu, E., Peltekian, A., and Altan-Bonnet, G.  (2021).  Scifive: a text-to-text transformer model for biomedical literature.  arXiv preprint arXiv:2106.03598.
  * Poon et al., 2014 Poon, H., Quirk, C., DeZiel, C., and Heckerman, D.  (2014).  Literome: Pubmed-scale genomic knowledge base in the cloud.  Bioinformatics, 30(19):2840–2842.
  * Pu et al., 2021 Pu, A., Chung, H. W., Parikh, A. P., Gehrmann, S., and Sellam, T.  (2021).  Learning compact metrics for mt.  In EMNLP.
  * Radford et al., 2019 Radford, A., Wu, J., Child, R., Luan, D., Amodei, D., and Sutskever, I.  (2019).  Language models are unsupervised multitask learners.
  * Raffel et al., 2020a Raffel, C., Shazeer, N., Roberts, A., Lee, K., Narang, S., Matena, M., Zhou, Y., Li, W., and Liu, P. J.  (2020a).  Exploring the limits of transfer learning with a unified text-to-text transformer.  Journal of Machine Learning Research, 21(140):1–67.
  * Raffel et al., 2020b Raffel, C., Shazeer, N., Roberts, A., Lee, K., Narang, S., Matena, M., Zhou, Y., Li, W., and Liu, P. J.  (2020b).  Exploring the limits of transfer learning with a unified text-to-text transformer.  Journal of Machine Learning Research, 21:1–67.
  * Sellam et al., 2020 Sellam, T., Das, D., and Parikh, A. P.  (2020).  Bleurt: Learning robust metrics for text generation.  arXiv preprint arXiv:2004.04696.
  * Shahbazi et al., 2020 Shahbazi, H., Fern, X., Ghaeini, R., and Tadepalli, P.  (2020).  Relation extraction with explanation.  In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pages 6488–6494.
  * Strötgen and Gertz, 2012 Strötgen, J. and Gertz, M.  (2012).  Temporal tagging on different domains: Challenges, strategies, and gold standards.  In Nicoletta Calzolari (Conference Chair), et al., editors, Proceedings of the Eight International Conference on Language Resources and Evaluation (LREC’12), pages 3746–3753, Istanbul, Turkey, may. European Language Resource Association (ELRA).
  * Subramani et al., 2015 Subramani, S., Kalpana, R., Monickaraj, P. M., and Natarajan, J.  (2015).  Hpiminer: A text mining system for building and visualizing human protein interaction networks and pathways.  Journal of Biomedical Informatics, 54:121–131.
  * Thayaparan et al., 2020 Thayaparan, M., Valentino, M., and Freitas, A.  (2020).  A survey on explainability in machine reading comprehension.  arXiv preprint arXiv:2010.00389.
  * Valenzuela-Escárcega et al., 2018 Valenzuela-Escárcega, M. A., Babur, Ö., Hahn-Powell, G., Bell, D., Hicks, T., Noriega-Atala, E., Wang, X., Surdeanu, M., Demir, E., and Morrison, C. T.  (2018).  Large-scale automated machine reading discovers new cancer driving mechanisms.  Database: The Journal of Biological Databases and Curation.
  * Wang et al., 2019 Wang, Q., Huang, L., Jiang, Z., Knight, K., Ji, H., Bansal, M., and Luan, Y.  (2019).  Paperrobot: Incremental draft generation of scientific ideas.  arXiv preprint arXiv:1905.07870.
  * Yao et al., 2010 Yao, L., Riedel, S., and McCallum, A.  (2010).  Collective cross-document relation extraction without labelled data.  In Proceedings of the 2010 Conference on Empirical Methods in Natural Language Processing, pages 1013–1023.
  * Yasunaga et al., 2019 Yasunaga, M., Kasai, J., Zhang, R., Fabbri, A. R., Li, I., Friedman, D., and Radev, D. R.  (2019).  Scisummnet: A large annotated corpus and content-impact models for scientific paper summarization with citation networks.  In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 7386–7393.
  * Yuan et al., 2020 Yuan, J., Jin, Z., Guo, H., Jin, H., Zhang, X., Smith, T., and Luo, J.  (2020).  Constructing biomedical domain-specific knowledge graph with minimum supervision.  Knowledge and Information Systems, 62(1):317–336.
  * Zhao et al., 2020 Zhao, L., Wang, J., Cheng, L., and Wang, C.  (2020).  Ontosem: an ontology semantic representation methodology for biomedical domain.  In 2020 IEEE International Conference on Bioinformatics and Biomedicine (BIBM), pages 523–527. IEEE.


# Back to sources—the role of coherence in super-resolution imaging revisited

Stanisław Kurdziałek<EMAIL_ADDRESS>Centre for Quantum
Optical Technologies, Centre of New Technologies, University of Warsaw,
Banacha 2c, 02-097 Warszawa, Poland  Faculty of Physics, University of Warsaw,
Pasteura 5, 02-093 Warszawa, Poland

###### Abstract

The fundamental limit on the precision of the separation estimation between
two partially coherent sources is derived. Photon losses, inevitably
associated with imperfect imaging, are properly taken into account. We begin
our analysis with a simple and common imaging model based on the 4f system. We
then show that our results are valid for any passive, linear, translationally
invariant optical imaging system. We obtain the expression for the Quantum
Fisher Information, which is consistent with the classical Fisher Information
calculated in [1] and latest experimental work [2]. We also identify the
source of discrepancies in the results obtained earlier and show how to
correctly interpret them.

##  I Introduction

The rapid development of super-resolution optical imaging prompted physicists
to rethink the way in which the performance of different imaging methods is
quantified. The most promising approach is based on (quantum) estimation
theory [3, 4, 5, 6, 7, 8], where the attainable estimation precision of object
parameters is used to assess the quality of a given imaging scheme. In the
simplest and most commonly examined scenario, imaging of two equally bright
point sources is considered (see [9, 10] for notable exceptions). For a given
measurement, the variance of any locally unbiased estimator of the separation
($s$) between the sources is lower-bounded by the inverse of Fisher
Information (FI). When $s$ is smaller than the width of the system Point
Spread Function (PSF), it’s hard to distinguish the images of two sources,
which is reflected by the FI vanishing for $s\rightarrow 0$. This fact, true
for standard measurement of the light intensity in the image plane, is known
as the Rayleigh curse. Surprisingly, the Rayleigh curse can be omitted, if the
measurement of light in the image plane can be freely chosen [5]. Quantum
Fisher Information (QFI) associated with the estimation of $s$, which is the
classical FI maximized over all possible measurements, doesn’t depend on $s$,
and doesn’t vanish for $s\rightarrow 0$. The above statement is valid for
incoherent, Poissonian sources whose centroid position is known, when no
external noise apart from shot noise is present. Breaking any of the above
assumptions usually leads to the QFI which vanishes for $s\rightarrow 0$ [6,
11, 12].

The influence of (partial) coherence between the sources is particularly
interesting. The debate about its role was started in [13], where authors
compute the QFI for partially coherent sources and claim, that even a small
degree of coherence leads to the resurgence of the Rayleigh curse. Tsang and
Nair replied [1] by showing that the SPADE measurement, which is optimal for
incoherent case [5, 14], allows to obtain classical FI larger than the QFI
predicted in [13], which is obviously a contradiction. Different approaches,
resulting in contradictory results, were presented in a further discussion
[15, 16, 17, 6, 18, 2]. It’s assumed in [13, 15, 18], that the total signal
power in the image plane depends neither on the sources separation nor on
their degree of coherence, and therefore it’s enough to consider single-photon
states to obtain the total value of QFI. At the same time, the model used in
[1, 17] to compute classical FI for SPADE and direct imaging implicitly
assumes photon losses, which depend on object parameters. The latter approach
is also supported by experimental work [2], but is not supplemented with the
computation of the QFI, which leaves the question about the optimality of
SPADE measurement open. Another approach is presented in [16], where the cost
of creating a coherent state from an incoherent one is studied in order to
model photon losses. These results, undoubtedly very interesting from a
theoretical point of view, don’t necessarily correspond to real optical
imaging systems, as they are inconsistent with [1, 17, 2].

It’s clear, that photon losses are the bone of contention, and therefore their
origin and characteristics need to be carefully studied. Let $E(\bm{r})$ be
the electric field amplitude distribution in the object plane ($E$ is a
complex scalar because it contains phase information, and polarization is
neglected). For linear, translationally invariant, diffraction-limited imaging
system, the field distribution observed in the image plane is $E(\bm{r})\ast
u(\bm{r})$, where $\ast$ denotes the convolution, $|u(\bm{r})|^{2}$ is a PSF,
and $u(\bm{r})$ itself is called the coherent Point Spread Function (cPSF)
[19]. The Fourier transform of the initial distribution,

$\hat{E}(\bm{k})\equiv\int E(\bm{r})e^{-i2\pi\bm{k}\bm{r}}\textrm{d}\bm{r},$
(1)

becomes $\hat{E}(\bm{k})\cdot\hat{u}(\bm{k})$ after imaging system
transformation. That means, that the signal power associated with different
spatial frequencies must be modified when the resulting PSF is wider than the
Dirac delta. For a passive system, no spatial frequency of $E(\bm{r})$ can be
strengthened ($|\hat{u}(\bm{k})|\leq 1$), and therefore the described
modification can be achieved by photon losses only. Consequently, photon
losses are always present if the observed image is blurred, and the loss ratio
strictly depends on $u(\bm{r})$ and $E(\bm{r})$. These losses spoil the
statistics of the collected signal, but on the other hand, the varying total
power can be used as an extra resource in the estimation procedure.

##  II 4f imaging model

In order to properly take photon losses into account, let’s consider imaging
by the so-called 4f system [19]. Such a system consists of two lenses with
equal focal lengths ($f$), the setup geometry is shown in Fig. 1.

Figure 1: The diagram shows the $4f$ imaging system, a well-known model of
e.g. optical microscope. Lenses are ideal, so the complex field amplitude in
the right focal plane of each lens is the Fourier transform of the field in
its left focal plane [light propagates from the object plane (I) to the image
plane (IV)]. All system imperfections are modeled by the aperture with
transmission coefficient $A(x)$, which is the source of photon losses and
image blurring.

From now on, 1D imaging is considered, which is enough to show the crucial
aspects of the studied problem. The aperture with transmission coefficient
$A(x)$, placed between the two ideal lenses, is the only source of image
imperfections, and determines both the shape of the PSF and photon losses (we
also assume that $A(x)=A(-x)$). The magnification of the considered system is
1, but all results are valid for any magnification, which is generally the
ratio of the focal lengths of two lenses. We use this specific model to make
the connection between the PSF shape and the loss ratio clearer, but this
connection remains the same for other systems because the relation between
spatial frequencies attenuation and the PSF is general. Another, more
fundamental argument, showing that losses are strictly connected with image
blurring, is presented in Sec. IV. Let’s now move to the description of a
binary object consisting of two small rectangular sources of width $\delta$
(the limit $\delta\rightarrow 0$ will be considered in order to examine point
sources), separated by the distance $s\gg\delta$, with the centroid at $x=0$.
When the two sources are coherent, and the relative phase between them is
$\phi$, the electric field distribution in the object plane (denoted by (I) in
Fig. 1) equals to

$E_{\textrm{I}}(x)=\textrm{rect}\left(\frac{x-s/2}{\delta}\right)e^{-i\phi/2}+\textrm{rect}\left(\frac{x+s/2}{\delta}\right)e^{i\phi/2}.$
(2)

The field in the plane (II) just before the aperture is [19]

$E_{\textrm{II}}(x)=\frac{1}{\sqrt{f\lambda}}\hat{E}_{\textrm{I}}\left(\frac{x}{f\lambda}\right),$
(3)

where $\lambda$ is the light wavelength, $\hat{h}$ generally denotes the
Fourier transform of $h$ (convention from (1) is adopted). The aperture
modifies the field such that it becomes
$E_{\textrm{III}}(x)=E_{\textrm{II}}(x)\cdot A(x)$ in plane (III). The ratio
of the signal power transmitted through the aperture, which is also the
probability of photon transmission, is equal to

$p(\phi,s)=\frac{\int_{-\infty}^{\infty}|E_{\textrm{II}}(x)A(x)|^{2}\textrm{d}x}{\int_{-\infty}^{\infty}|E_{\textrm{II}}(x)|^{2}\textrm{d}x}.$
(4)

The field observed in the image plane (IV), which is a Fourier transformed
$E_{\textrm{III}}$ is

$E_{\textrm{IV}}(x)=\frac{1}{f\lambda}E_{\textrm{I}}(-x)\ast\hat{A}\left(\frac{x}{f\lambda}\right),$
(5)

which means that $\hat{A}\left(x/(f\lambda)\right)$ is the system cPSF. From
now on, the assumption $f\lambda=1$ will be made for simplicity.

In order to use quantum estimation theory tools, we should change the
description of our classical problem, such that normalized electric field
$E(x)$ becomes wave function $\psi(x)$ (as in e.g. [16]). Such a description
is complete when no correlations between subsequent photons are present. Let’s
consider the time of observation in which the object emits a photon with a
probability $p_{\textrm{em}}\ll 1$. The probability of emission of two photons
is negligible, and subsequent emissions are independent. The quantum state of
light in the plane (III) is

$\rho_{\textrm{coh}}=p_{1}\ket{\psi_{\phi}}\bra{\psi_{\phi}}+(1-p_{1})\ket{0}\bra{0},$
(6)

where $\ket{0}$, $\ket{\psi_{\phi}}$ denote zero- and one-photon states
respectively, $\ket{\psi_{\phi}}$ in the position basis $\\{\ket{x}\\}$ is
proportional to $E_{\textrm{III}}(x)$ and satisfies

$\braket{x}{\psi_{\phi}}=\mathcal{N}(\phi,s)A(x)\leavevmode\nobreak\
\textrm{sinc}(x\delta)\cos(\pi xs+\phi/2),$ (7)

where normalization constant $\mathcal{N}(\phi,s)$ is chosen such that
$\braket{\psi_{\phi}}{\psi_{\phi}}=1$. The probability of one-photon state is

$p_{1}=p_{\textrm{em}}p(\phi,s),$ (8)

and after using (4) and normalization condition it turns out that

$p(\phi,s)=\frac{2\delta}{\pi|\mathcal{N}(\phi,s)|^{2}}.$ (9)

The probability of photon transmission vanishes when $\delta\rightarrow 0$ for
all reasonable apertures, and that’s the reason why non-zero source size
$\delta$ is present in the model.

Our goal is to compute the QFI associated with $s$ estimation. The
transformation between planes (III) and (IV), performed by an ideal lens, is
unitary, and therefore doesn’t affect the QFI—it justifies our choice to
compute the QFI for the state in the plane (III).

The state defined in (6) represents the degree of coherence between the
sources $\gamma=e^{i\phi}$. The most general output state for a classical
object consisting of two partially coherent, non-fluctuating sources is
described by the degree of coherence between the sources $\gamma=re^{i\phi}$
($0\leq r\leq 1$) and its density matrix is

$\rho_{\textrm{p}}=p_{1}\ket{\psi_{\phi}}\bra{\psi_{\phi}}+p_{2}\ket{\psi_{\phi+\pi}}\bra{\psi_{\phi+\pi}}+p_{3}\ket{0}\bra{0},$
(10)

where

$p_{1}=p_{\textrm{em}}\leavevmode\nobreak\ p(\phi,s)\leavevmode\nobreak\
\frac{1+r}{2},\leavevmode\nobreak\ \leavevmode\nobreak\
p_{2}=p_{\textrm{em}}\leavevmode\nobreak\ p(\phi+\pi,s)\leavevmode\nobreak\
\frac{1-r}{2},$ (11)

$p_{3}=1-p_{1}-p_{2}$, and $\ket{\psi_{\phi}}$, $p(\phi,s)$ are defined in (7)
and (9) respectively.

##  III Results and discussion

The general formula for the QFI associated with the quantum states family
$\rho(s)$ is

$\mathcal{F}[\rho(s)]=\textrm{Tr}(\rho\Lambda^{2}),$ (12)

where $\Lambda$ matrix is defined by the equation

$\frac{\textrm{d}\rho}{\textrm{d}s}=\frac{1}{2}\left(\Lambda\rho+\rho\Lambda\right).$
(13)

In order to simplify further computations, let’s introduce the notation for
$\ket{\psi_{\phi}}$ without the normalization factor,

$\ket{\tilde{\psi}_{\phi}}=\mathcal{N}(\phi,s)^{-1}\ket{\psi_{\phi}},$ (14)

and write the quantum state $\rho_{\textrm{p}}$ in a form

$\rho_{\textrm{p}}=\sum_{i,j\in\\{0,\pi\\}}B_{ij}\ket{\tilde{\psi}_{i}}\bra{\tilde{\psi}_{j}}+p_{3}\ket{0}\bra{0},$
(15)

where

$\begin{bmatrix}B_{00}&B_{0\pi}\\\ B_{\pi
0}&B_{\pi\pi}\end{bmatrix}=\frac{\delta
p_{\textrm{em}}}{\pi}\begin{bmatrix}1+\textrm{Re}(\gamma)&\textrm{Im}(\gamma)\\\
\textrm{Im}(\gamma)&1-\textrm{Re}(\gamma)\end{bmatrix}.$ (16)

Coefficients $B_{ij}$ don’t depend on $s$, which, supplemented by some
orthogonality relations between states $\ket{\tilde{\psi}_{i}}$ and their
derivatives (specified in appendix A), allows us to provide a simple formula
for $\mathcal{F}$,

$\mathcal{F}[\rho_{\textrm{p}}(s)]=4\sum_{i\in\\{0,\pi\\}}B_{ii}\braket{\dot{\tilde{\psi}}_{i}}{\dot{\tilde{\psi}}_{i}}+\frac{1}{p_{3}}\left(\frac{\textrm{d}p_{3}}{\textrm{d}s}\right)^{2},$
(17)

where $\ket{\dot{\tilde{\psi}}_{i}}\equiv\partial_{s}\ket{\tilde{\psi}_{i}}$,
and the last term is of the order of $\delta^{2}$, and can be neglected when
$\delta\rightarrow 0$. From (16) and (17) follows that the QFI doesn’t depend
on $\textrm{Im}(\gamma)$. Probability $p_{\textrm{em}}$ depends on an
arbitrary observation time, we can get rid of this non-physical parameter by
computing QFI per emitted photon, which simplifies to (see appendix A)

$\mathcal{F}_{\textrm{em}}[\rho_{\textrm{p}}(s)]=\frac{8\delta}{\pi}\braket{\dot{\tilde{\psi}}_{\chi}}{\dot{\tilde{\psi}}_{\chi}}+\mathcal{O}(\delta^{2}),$
(18)

where $\cos\chi=\textrm{Re}(\gamma)$. $\mathcal{F}_{\textrm{em}}$ drops down
to $0$ for $\delta\rightarrow 0$, as all photons are absorbed by the aperture
when sources are infinitesimally small. To obtain the quantity which doesn’t
vanish for point sources, one can compute the QFI per detected photon,

$\mathcal{F}_{\textrm{det}}=\frac{\mathcal{F}}{p_{1}+p_{2}}=\frac{\mathcal{F}_{\textrm{em}}}{p(\chi,s)},$
(19)

where $p_{1},p_{2}$ are defined in (11), and relation
$p_{1}+p_{2}=p_{\textrm{em}}\leavevmode\nobreak\ p(\chi,s)$ is proven in
appendix A. However, it’s better physically justified to treat the number of
emitted photons as a resource, as, for example, it is directly connected with
the measurement time. $\mathcal{F}_{\textrm{det}}$ and
$\mathcal{F}_{\textrm{em}}$ differ by a non-trivial factor, which generally
depends on $s$ (notable exception is $\textrm{Re}(\gamma)=0$). Let’s see what
happens if the state $\rho_{\textrm{p}}$, defined in (10), is replaced with
its normalized one-photon part,

$\rho_{\textrm{p}}^{(1)}=\frac{p_{1}}{p_{1}+p_{2}}\ket{\psi_{\phi}}\bra{\psi_{\phi}}+\frac{p_{2}}{p_{1}+p_{2}}\ket{\psi_{\phi+\pi}}\bra{\psi_{\phi+\pi}}.$
(20)

The QFI computed for this single-photon state is already QFI per detected
photon, and is equal to (see appendix A for the derivation)

$\mathcal{F}_{\textrm{det}}[\rho_{\textrm{p}}^{(1)}(s)]=4\left(\braket{\dot{\psi}_{\chi}}{\dot{\psi}_{\chi}}-|\braket{\dot{\psi}_{\chi}}{\psi_{\chi}}|^{2}\right).$
(21)

When $|\gamma|=1$, then $\chi=\phi$, and (21) is just a well known formula for
QFI for a pure state $\ket{\psi_{\phi}}$. The QFI per emitted photon,
$\mathcal{F}_{\textrm{em}}[\rho_{\textrm{p}}^{(1)}]$, obtained with the help
of (19), is smaller than $\mathcal{F}_{\textrm{em}}[\rho_{\textrm{p}}]$, as
the following relation holds:

$\mathcal{F}_{\textrm{em}}[\rho_{\textrm{p}}^{(1)}(s)]=\mathcal{F}_{\textrm{em}}[\rho_{\textrm{p}}(s)]-\frac{8\delta}{\pi}|\braket{\psi_{\chi}}{\dot{\tilde{\psi}}_{\chi}}|^{2}.$
(22)

This reflects the fact, that some information about $s$ is encoded in the
total power of observed signal.

Figure 2: The QFI for Gaussian aperture,
$A(x)=\exp\left(-\frac{4\pi^{2}\sigma^{2}x^{2}}{\lambda^{2}f^{2}}\right)$,
which leads to a Gaussian PSF of width $\sigma$, is shown for different values
of $\textrm{Re}(\gamma)$. Full quantum state ($\rho_{\textrm{p}}$), and its
normalized single-photon part ($\rho_{\textrm{p}}^{(1)}$) are considered, two
possible normalization conventions are shown. Part (a) is consistent with
classical FI computed in [1], whereas part (d) is consistent with the QFI
obtained in [15, 16]. Notice the substantial difference between
$\textrm{Re}(\gamma)=-1$ and $\textrm{Re}(\gamma)=-0.98$ for cases (b) and
(d), and the difference of dimensions on $y$ axis between
$\mathcal{F}_{\textrm{det}}$ and $\mathcal{F}_{\textrm{em}}$.

To conclude, we obtained four different results—QFI per detected or emitted
photon, for states $\rho_{\textrm{p}}$ and $\rho_{\textrm{p}}^{(1)}$. It’s
more natural to use QFI per emitted photon, but the problem with this quantity
is that it vanishes for $\delta\rightarrow 0$. QFI for
$\rho_{\textrm{p}}^{(1)}$ was computed in [13, 15, 18], but in the light of
presented arguments (including the argument from Sec. IV), it’s clear that the
real physical situation is properly described by the state
$\rho_{\textrm{p}}$. However, in many practical situations, the power of light
emitted by the object isn’t known, and then it’s problematic to extract any
information from the total power of the observed signal—in such a case, the
QFI associated with $\rho_{\textrm{p}}^{(1)}$ can serve as a reasonable upper-
bound for the precision of estimation of $s$.

We studied the dependence of the QFI on $s$ and $\textrm{Re}(\gamma)$ for
Gaussian aperture $A(x)$ (which leads to a Gaussian PSF with standard
deviation $\sigma$). The results are shown in Fig. 2 (see appendix B for
analytical expressions). One can observe, that curves shown in part (a) of
Fig. 2, which correspond to $\mathcal{F}_{\textrm{em}}[\rho(s)]$, coincide
with those presented in Fig. 1 in [1], where the classical FI for SPADE
measurement is computed. Therefore, our analysis indicates, that SPADE
measurement remains optimal if partial coherence between sources arises. By
comparing the first and second column of Fig. 2, we can figure out, how
important the information hidden in the total signal power variability
is—generally speaking, this source of information is crucial for
$\textrm{Re}(\gamma)<0$, and that’s the reason why the discrepancies in the
previous works were manifested mainly for negative degrees of coherence.
Notice that the Rayleigh curse is really inevitable only for $\gamma=1$,
whereas for $\gamma=-1$, it can be eluded if the full, non-normalized signal
is properly used. What may surprise the reader, is that
$\mathcal{F}_{\textrm{em}}$ for a fixed $s/\sigma$ is proportional to
$\delta\sigma^{-3}$, not to $\sigma^{-2}$ as in [5, 1] and other references.
This discrepancy is caused by the fact, that the number of photons detected
from a single source depends on the aperture size, and consequently on
$\sigma$. This effect is often hidden in a normalization factor, which in fact
depends on $\sigma$ (e.g. factor $N$ in [5], factor $N_{0}$ in [1]).

##  IV Losses vs PSF

We would like to provide another argument why losses are inevitably connected
with the occurrence of the PSF. Let’s consider a passive, linear optical
imaging system, and denote annihilation operators of its input modes by
$\hat{a}_{1},\hat{a}_{2},...\hat{a}_{N}$. Some of these modes
($\hat{a}_{1},\hat{a}_{2},...\hat{a}_{M}$) correspond to the spatially
localized rectangular modes of width $\delta$, $i$-th mode center is placed at
the position $x_{i}=i\delta$ in the object plane. Other modes correspond
either to the remaining area of the object plane or to uncontrolled modes
(which are always in a vacuum state). Notice that $\delta$ can be arbitrarily
small, and $M$ can also be arbitrary, as long as $M\delta$ is smaller than the
total imaged area. The output modes of the considered system are
$\hat{a}^{\prime}_{1},a^{\prime}_{2},...,a^{\prime}_{N}$. Some of these modes
($\hat{a}^{\prime}_{1},a^{\prime}_{2},...,a^{\prime}_{K}$) can be observed in
the image plane, others correspond to lost photons.

States $\ket{\psi_{i}}=a_{i}^{\dagger}\ket{0}$ form o.-n. basis of single-
photon input states. Each state $\ket{\psi_{i}}$ is transformed by an imaging
system to a one-photon output state $\ket{\phi_{i}}$. From unitarity of the
transformation, we know that
$\braket{\psi_{i}}{\psi_{j}}=\braket{\phi_{i}}{\phi_{j}}=\delta_{ij}$.

The one-photon output Hilbert space ($\mathcal{H}$) can be decomposed as
$\mathcal{H}=\mathcal{H}_{0}\oplus\mathcal{H}_{1}$, where $\mathcal{H}_{0}$
and $\mathcal{H}_{1}$ correspond to subspaces of lost and observed photons
respectively. $\mathcal{H}_{1}$ is spanned by
$\\{a_{1}^{\dagger}\ket{0},a_{2}^{\dagger}\ket{0},...,a_{K}^{\dagger}\ket{0}\\}$,
$\mathcal{H}_{0}$ is spanned by
$\\{a_{K+1}^{\dagger}\ket{0},a_{K+2}^{\dagger}\ket{0},...,a_{N}^{\dagger}\ket{0}\\}$,
$\mathcal{H}_{0}$ is orthogonal to $\mathcal{H}_{1}$. Each state
$\ket{\phi_{i}}$ can be written as

$\ket{\phi_{i}}=\sqrt{1-p_{i}}\ket{\phi_{i0}}+\sqrt{p_{i}}\ket{\phi_{i1}},$
(23)

where normalized vectors $\ket{\phi_{i0}}$ and $\ket{\phi_{i1}}$ belong to
$\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ respectively, and $p_{i}$ corresponds
to the probability of photon detection in the image plane, given the initial
state was a single photon in mode $a_{i}$. We assume that the system is
translationally invariant, and therefore losses are the same for each
controlled mode, hence $p_{1}=p_{2}=...=p_{M}=p$. Even though states
$\ket{\phi_{i}}$ are orthogonal, it’s not usually true for their observable
parts, $\ket{\phi_{i1}}$—non-zero width of the PSF results in non-zero overlap
of the images of initially separated objects. Consequently, the assumption
$p=1$ is contradictory with (23) for any realistic imaging system. Moreover,
as we show in appendix C, orthonormality of $\\{\ket{\phi_{i}}\\}$ is only
possible when

$p\leq\frac{M}{\sum_{i,j=1}^{M}|\braket{\phi_{i1}}{\phi_{j1}}|}.$ (24)

The knowledge of the PSF of the system allows computing the denominator of the
RHS of the above inequality. Let’s consider Gaussian PSF, with standard
deviation $\sigma$ satisfying $\delta\ll\sigma\ll M\delta$. The first part of
the inequality can be satisfied with a proper choice of $\delta$, the second
one tells us that the total available imaged area is much larger than the PSF
size. In such a limit the condition for $p$ simplifies to

$p\leq\sqrt{8\pi}\frac{\delta}{\sigma}.$ (25)

In fact, condition (24) generally takes the form $p\leq A\delta$, where
constant $A$ depends on the PSF, and the total size of the object plane. This
is in agreement with our reasoning from Sec. II, where it was shown, that
photon emitted from point sources of width $\delta$ is transmitted through the
4f imaging system with probability proportional to $\delta$, see equation (9).

For a given value of $p$ and arbitrary input state,

$\ket{\psi}=\sum_{i=1}^{M}\alpha_{i}\ket{\psi_{i}},$ (26)

the probability of measuring a photon in one of the observed modes is the
squared norm of the observable part of the output states, namely

$P=p\sum_{i,j=1}^{M}\alpha^{*}_{i}\alpha_{j}\braket{\phi_{i1}}{\phi_{j1}}.$
(27)

Let’s now assume (motivated by $\eqref{pdelta}$), that $p=A\delta$, and
compute $P$ in a limit $\delta\rightarrow 0$. In such a limit, each
$\alpha_{i}$ should be proportional to $\sqrt{\delta}$ because of the
normalization condition for the state $\ket{\phi}$. Therefore, we can write
$\alpha_{i}=E(x_{i})\sqrt{\delta}$ where $E(x_{i})$ can be interpreted as the
electric field amplitude at point $x_{i}$. Moreover,

$\braket{\phi_{i1}}{\phi_{j1}}=\int u^{*}(x-x_{i})u(x-x_{j})\textrm{d}x,$ (28)

summation over $i$ and $j$ in (27) can be replaced with integration over
$x_{i}$ and $x_{j}$, and then we obtain

$P=A\int\left|\left(E\ast
u\right)(x)\right|^{2}\textrm{d}x=A\int\left|\hat{E}(k)\hat{u}(k)\right|^{2}\textrm{d}k,$
(29)

where convolution theorem and Parseval’s theorem were used. We see that losses
are exactly the same as those obtained in the $4f$ system (up to a
multiplicative constant), and $\hat{u}(k)$ can be interpreted as the
attenuation factor of a given spatial frequency $k$. It shows, that the
analysis performed in Sections II and III for the $4f$ system can be
generalized to any passive, linear, optical imaging system.

##  V Summary

In order to clarify the issues associated with the behavior of the QFI for the
separation between two partially coherent sources, we have carefully studied
the imaging process using the well-established $4f$ system-based model. We
have satisfactorily explained the discrepancies in previous works that
resulted from a misunderstanding of the role of photon losses in imaging. From
our analysis follows, that losses must also be present for non-coherent
sources (to be more specific, the probability of photon transmission,
$p(\chi,s)$, grows with $\textrm{Re}(\gamma)$). This contradicts the
assumptions made in [16, 20], where it is assumed that the cost of any
superposition is always greater than the cost of an incoherent mixture.
Properly computed QFI coincides with the previously calculated classical FI
for SPADE measurement [1], which indicates the optimality of this measurement.
In the last section, we show, that our results are valid for any passive,
linear optical imaging system. In [15], authors claim that they study passive,
lossless imaging system represented by a unitary transformation, but as we
proved in Sec.IV, such systems don’t exist for PSFs wider than Dirac delta.
However, the results obtained in works where such a system is examined [13,
15, 18] are not meaningless—they can be interpreted as the appropriate limits
of the estimation precision, when only the normalized signal can be used in
the estimation procedure. In fact, this is a fairly common case as the total
power of the sources is usually not known a priori.

In an independent work [21] Tsang provides a more general formula for the QFI,
which can be applied for partially coherent sources, and properly takes photon
losses into account. In this work, we provide different physical arguments
supporting the same thesis, and study the specific case of weak, partially
coherent sources in more detail.

The total signal power variations as a source of information are also studied
in another parallel work [20]. However, as we discussed above, the connection
between the model proposed by the authors and the performance of a realistic
imaging system isn’t clear—in particular, the losses associated with
$\gamma=0$ shouldn’t be neglected, as they are higher than e.g. losses for
$\gamma=1$.

##  VI Acknowledgments

First of all, I would like to thank Rafał Demkowicz-Dobrzański for very
fruitful discussions. I also thank Konrad Banaszek for sharing his experience
and valuable remarks, and Francesco Albarelli for his comments and suggestions
concerning the latest literature. This work is a part of the project ”Quantum
Optical Technologies” carried out within the International Research Agendas
programme of the Foundation for Polish Science cofinanced by the European
Union under the European Regional Development Fund.

## References

  * [1] Mankei Tsang and Ranjith Nair.  Resurgence of rayleigh’s curse in the presence of partial coherence: comment.  Optica, 6(4):400–401, 2019.
  * [2] SA Wadood, Yiyu Zhou, Jing Yang, Kevin Liang, MA Alonso, X-F Qian, T Malhotra, SM Rafsanjani, Andrew N Jordan, Robert W Boyd, et al.  Superresolution of partially coherent light sources using parity sorting.  arXiv preprint arXiv:2102.01603, 2021.
  * [3] E Bettens, D Van Dyck, AJ Den Dekker, J Sijbers, and A Van den Bos.  Model-based two-object resolution from observations having counting statistics.  Ultramicroscopy, 77(1-2):37–48, 1999.
  * [4] Sripad Ram, E. Sally Ward, and Raimund J. Ober.  Beyond rayleigh’s criterion: A resolution measure with application to single-molecule microscopy.  Proceedings of the National Academy of Sciences, 103(12):4457–4462, 2006.
  * [5] Mankei Tsang, Ranjith Nair, and Xiao-Ming Lu.  Quantum theory of superresolution for two incoherent optical point sources.  Physical Review X, 6(3):031033, 2016.
  * [6] Mankei Tsang.  Resolving starlight: a quantum perspective.  Contemporary Physics, 60(4):279–298, 2019.
  * [7] Cosmo Lupo and Stefano Pirandola.  Ultimate precision bound of quantum and subwavelength imaging.  Physical review letters, 117(19):190802, 2016.
  * [8] Ranjith Nair and Mankei Tsang.  Far-field superresolution of thermal electromagnetic sources at the quantum limit.  Physical review letters, 117(19):190801, 2016.
  * [9] Sisi Zhou and Liang Jiang.  Modern description of rayleigh’s criterion.  Phys. Rev. A, 99:013808, Jan 2019.
  * [10] F. Albarelli, M. Barbieri, M.G. Genoni, and I. Gianani.  A perspective on multiparameter quantum metrology: From theoretical tools to applications in quantum imaging.  Physics Letters A, 384(12):126311, 2020.
  * [11] Yink Loong Len, Chandan Datta, Michał Parniak, and Konrad Banaszek.  Resolution limits of spatial mode demultiplexing with noisy detection.  International Journal of Quantum Information, 18(01):1941015, 2020\.
  * [12] Changhun Oh, Sisi Zhou, Yat Wong, and Liang Jiang.  Quantum limits of superresolution in noisy environment, 2020.
  * [13] Walker Larson and Bahaa EA Saleh.  Resurgence of rayleigh’s curse in the presence of partial coherence.  Optica, 5(11):1382–1389, 2018.
  * [14] Mankei Tsang.  Subdiffraction incoherent optical imaging via spatial-mode demultiplexing.  New Journal of Physics, 19(2):023054, 2017.
  * [15] Walker Larson and Bahaa EA Saleh.  Resurgence of rayleigh’s curse in the presence of partial coherence: reply.  Optica, 6(4):402–403, 2019.
  * [16] Zdeněk Hradil, Jaroslav Řeháček, Luis Sánchez-Soto, and Berthold-Georg Englert.  Quantum fisher information with coherence.  Optica, 6(11):1437–1440, 2019.
  * [17] Kwan Kit Lee and Amit Ashok.  Surpassing rayleigh limit: Fisher information analysis of partially coherent source (s).  In Optics and Photonics for Information Processing XIII, volume 11136, page 111360H. International Society for Optics and Photonics, 2019.
  * [18] Kevin Liang, SA Wadood, and AN Vamivakas.  Coherence effects on estimating two-point separation.  arXiv preprint arXiv:2012.05311, 2020.
  * [19] Harrison H Barrett and Kyle J Myers.  Foundations of image science.  John Wiley & Sons, 2013.
  * [20] Zdeněk Hradil, Dominik Koutný, and Jaroslav Řeháček.  Exploring the ultimate limits: Super-resolution enhanced by partial coherence, 2021.
  * [21] Mankei Tsang.  Quantum and classical fisher information for weak thermal optical sources, 2021.

##  Appendix A The details of QFI calculations

In this appendix the details concerning the derivation of equations (17),
(18), (19), (21), (22) are provided. Let’s start with the lemma, which
simplifies all calculations significantly.

Lemma.— Let’s define the family of quantum states indexed by a real parameter
$s$,

$\rho(s)=\sum_{i,j=1}^{N}B_{ij}\ket{\phi_{i}(s)}\bra{\phi_{j}(s)},$ (30)

where coefficients $B_{ij}$ don’t depend on $s$, and vectors $\ket{\phi_{i}}$
and their derivatives $\ket{\dot{\phi}_{i}}\equiv\partial_{s}\ket{\phi_{i}}$
satisfy

$\braket{\phi_{i}}{\phi_{j}}=\braket{\dot{\phi}_{i}}{\phi_{j}}=\braket{\dot{\phi}_{i}}{\dot{\phi}_{j}}=0\leavevmode\nobreak\
\leavevmode\nobreak\ \textrm{for}\leavevmode\nobreak\ \leavevmode\nobreak\
i\neq j,$ (31)

and moreover $\braket{\dot{\phi}_{i}}{\phi_{i}}\in\mathbb{R}.$ Then the QFI
associated with the estimation of $s$ is

$\mathcal{F}\left[\rho(s)\right]=4\sum_{i=1}^{N}B_{ii}\braket{\dot{\phi}_{i}}{\dot{\phi}_{i}}$
(32)

Proof.— We are going to compute the QFI from its definition (12). It’s easy to
check, that the following ansatz,

$\Lambda=2\sum_{i=1}^{N}\frac{\ket{\dot{\phi}_{i}}\bra{\dot{\phi}_{i}}}{\braket{\phi_{i}}{\dot{\phi}_{i}}},$
(33)

makes the condition (13) satisfied. Indeed,

$\frac{1}{2}\left(\rho\Lambda+\Lambda\rho\right)=\sum_{i,j=1}^{N}B_{ij}\ket{\phi_{i}}\bra{\phi_{j}}\sum_{k=1}^{N}\frac{\ket{\dot{\phi}_{k}}\bra{\dot{\phi}_{k}}}{\braket{\phi_{k}}{\dot{\phi}_{k}}}+\sum_{k=1}^{N}\frac{\ket{\dot{\phi}_{k}}\bra{\dot{\phi}_{k}}}{\braket{\dot{\phi}_{k}}{\phi_{k}}}\sum_{i,j=1}^{N}B_{ij}\ket{\phi_{i}}\bra{\phi_{j}}=\sum_{i,j=1}^{N}B_{ij}\left(\ket{\dot{\phi}_{i}}\bra{\phi_{j}}+\ket{\phi_{i}}\bra{\dot{\phi}_{j}}\right),$
(34)

and the last expression is equal to $\frac{\textrm{d}\rho}{\textrm{d}s}$.
Furthermore, we have

$\Lambda^{2}=4\sum_{i=1}^{N}\frac{\braket{\dot{\phi}_{i}}{\dot{\phi}_{i}}}{(\braket{\phi_{i}}{\dot{\phi}_{i}})^{2}}\ket{\dot{\phi}_{i}}\bra{\dot{\phi}_{i}},$
(35)

and straightforward calculations show that

$\mathcal{F}=\textrm{Tr}(\rho\Lambda^{2})=4\sum_{i=1}^{N}B_{ii}\braket{\dot{\phi}_{i}}{\dot{\phi}_{i}},$
(36)

Q.E.D.

Before applying the above lemma, let’s use (7) and (14) to provide an explicit
formula for $\ket{\tilde{\psi}_{\phi}}$, which reads

$\ket{\tilde{\psi}_{\phi}}=\int_{-\infty}^{\infty}A(x)\leavevmode\nobreak\
\textrm{sinc}(x\delta)\cos(\pi xs+\phi/2)\ket{x}\textrm{d}x.$ (37)

From normalization condition,
$\braket{\psi_{\phi}}{\psi_{\phi}}=|\mathcal{N}(\phi,s)|^{2}\braket{\tilde{\psi}_{\phi}}{\tilde{\psi}_{\phi}}=1$,
the normalization constant satisfies

$|\mathcal{N}(\phi,s)|^{-2}=\int_{-\infty}^{\infty}|A(x)|^{2}\leavevmode\nobreak\
\textrm{sinc}^{2}(x\delta)\cos^{2}(\pi xs+\phi/2)\textrm{d}x.$ (38)

From (37) follows that

$\ket{\tilde{\psi}_{\phi}}=\cos\frac{\phi}{2}\ket{\tilde{\psi}_{0}}+\sin\frac{\phi}{2}\ket{\tilde{\psi}_{\pi}},$
(39)

whereas (38), together with the parity condition, $A(x)=A(-x)$, gives us the
formula

$|\mathcal{N}(\phi,s)|^{-2}=\cos^{2}\frac{\phi}{2}\leavevmode\nobreak\
|\mathcal{N}(0,s)|^{-2}+\sin^{2}\frac{\phi}{2}\leavevmode\nobreak\
|\mathcal{N}(\pi,s)|^{-2}.$ (40)

Equation (39), together with (9) and (14) is used to write the state
$\rho_{\textrm{p}}$ defined in (10), in a form presented in (15). Now, our
lemma can be applied to compute the QFI, where we substitute
$\ket{\phi_{1}}=\ket{\tilde{\psi}_{0}}$,
$\ket{\phi_{2}}=\ket{\tilde{\psi}_{\pi}}$,
$\ket{\phi_{3}}=\sqrt{p_{3}}\ket{0}$. With the help of (37) and parity
condition for $A(x)$, it’s straightforward to check that the assumptions of
the lemma (31) are satisfied for $\ket{\phi_{i}}$ defined as above. Then,
equation (17) directly follows from the thesis of our lemma (36). Equation
(18) is obtained from (17) with the help of relation

$\ket{\dot{\tilde{\psi}}_{\chi}}=\cos\frac{\chi}{2}\ket{\dot{\tilde{\psi}}_{0}}+\sin\frac{\chi}{2}\ket{\dot{\tilde{\psi}}_{\pi}}=\sqrt{\frac{1+\textrm{Re}(\gamma)}{2}}\ket{\dot{\tilde{\psi}}_{0}}+\sqrt{\frac{1-\textrm{Re}(\gamma)}{2}}\ket{\dot{\tilde{\psi}}_{\pi}},$
(41)

where $\chi$ was defined such that $\cos\chi=\textrm{Re}(\gamma)$,
$\chi\in[0,\pi]$.

Relation $p_{1}+p_{2}=p_{\textrm{em}}p(\chi,s)$, used to prove (19), follows
from (11) and (9), where normalization constants $|\mathcal{N}(\phi,s)|^{-2}$
and $|\mathcal{N}(\chi,s)|^{-2}$ are expanded with the help of (40). This
relation, together with the $\rho_{\textrm{p}}$ definition (10), shows that
$p(\chi,s)$ may be interpreted as the probability of photon transmission
through the aperture for an arbitrary $\gamma$. At this point, it’s worth
mentioning that for $\textrm{Re}(\gamma)=0$ we have $\chi=\pi/2$, and

$p(\chi,s)=\frac{2\delta}{\pi}\left|\mathcal{N}\left(\frac{\pi}{2},s\right)\right|^{-2}=\frac{\delta}{\pi}\left(|\mathcal{N}(0,s)|^{-2}+|\mathcal{N}(\pi,s)|^{-2}\right)=\frac{\delta}{\pi}\int_{-\infty}^{\infty}\left|A(x)\right|^{2}\textrm{sinc}^{2}(x\delta)\textrm{d}x=\frac{\delta}{\pi}+\mathcal{O}(\delta^{3}),$
(42)

and consequently total signal power doesn’t depend on $s$ in this specific
case.

The computation for $\mathcal{F}_{\textrm{det}}[\rho_{\textrm{p}}^{(1)}]$ are
analogous to those for $\mathcal{F}[\rho_{\textrm{p}}]$. We start with writing
down the formula for $\rho_{\textrm{p}}^{(1)}$ in a form analogous to (15),

$\rho_{\textrm{p}}^{(1)}=\sum_{i,j=1}^{2}B_{ij}\ket{\phi_{i}}\bra{\phi_{j}},$
(43)

where

$\begin{bmatrix}B_{11}&B_{12}\\\
B_{21}&B_{22}\end{bmatrix}=\frac{\delta}{\pi}\begin{bmatrix}1+\textrm{Re}(\gamma)&\textrm{Im}(\gamma)\\\
\textrm{Im}(\gamma)&1-\textrm{Re}(\gamma)\end{bmatrix},$ (44)

and vectors $\ket{\phi_{1}}=p(\chi,s)^{-1}\ket{\tilde{\psi}_{0}}$,
$\ket{\phi_{2}}=p(\chi,s)^{-1}\ket{\tilde{\psi}_{\pi}}$ satisfy the lemma
assumption. Therefore, from (36), we have

$\mathcal{F}_{\textrm{det}}\left[\rho^{(1)}(s)\right]=\frac{4\delta}{\pi}\left[(1+\textrm{Re}(\gamma))\braket{\dot{\phi}_{1}}{\dot{\phi}_{1}}+(1-\textrm{Re}(\gamma))\braket{\dot{\phi}_{2}}{\dot{\phi}_{2}}\right],$
(45)

where

$\ket{\dot{\phi}_{1/2}}=-\frac{1}{p(\chi,s)^{2}}\frac{\textrm{d}p(\chi,s)}{\textrm{d}s}\ket{\tilde{\psi}_{0/\pi}}+p(\chi,s)^{-1}\ket{\dot{\tilde{\psi}}_{0/\pi}}.$
(46)

Equations (21) and (45) are equivalent, which can be shown by inserting (46)
into (45), and expanding the RHS of (21) with the help of previously derived
relations (39), (40), (41). Relation (22) is derived analogously—LHS and RHS
are equal to each other after expressing everything in terms of
$\ket{\tilde{\psi}_{0/\pi}}$, $\ket{\dot{\tilde{\psi}}_{0/\pi}}$,
$\mathcal{N}(0/\pi,s)$.

##  Appendix B Analytical expressions for Gaussian aperture

In this section we provide analytical expressions for different types of QFI
for the case
$A(x)=\exp\left(-\frac{4\pi^{2}\sigma^{2}x^{2}}{\lambda^{2}f^{2}}\right)$,
which is studied in Fig. 2. Such an aperture leads to the Gaussian PSF with
width $\sigma$. This specific form of $A(x)$ was inserted into equations (17),
(19), (21), (22), and the following results were obtained:

$\mathcal{F}_{\textrm{em}}[\rho_{\textrm{p}}(s)]=\frac{\delta}{8\sqrt{2}\pi^{3/2}\sigma^{3}}\left(1-\left(1-\frac{s^{2}}{4\sigma^{2}}\right)\textrm{Re}(\gamma)e^{-\frac{s^{2}}{8\sigma^{2}}}\right)+\mathcal{O}(\delta^{2})$
(47)

$\mathcal{F}_{\textrm{det}}[\rho_{\textrm{p}}(s)]=\frac{1}{4\sigma^{2}}\left(1+\frac{2\textrm{Re}(\gamma)e^{-\frac{s^{2}}{8\sigma^{2}}}\left(\frac{s^{2}}{8\sigma^{2}}-1\right)}{1+\textrm{Re}(\gamma)e^{-\frac{s^{2}}{8\sigma^{2}}}}\right)+\mathcal{O}(\delta)$
(48)

$\mathcal{F}_{\textrm{em}}[\rho^{(1)}_{\textrm{p}}(s)]=\frac{\delta}{8\sqrt{2}\pi^{3/2}\sigma^{3}}\frac{1-\left(\textrm{Re}(\gamma)e^{-\frac{s^{2}}{8\sigma^{2}}}-\frac{s^{2}}{4\sigma^{2}}\right)\textrm{Re}(\gamma)e^{-\frac{s^{2}}{8\sigma^{2}}}}{1+\textrm{Re}(\gamma)e^{-\frac{s^{2}}{8\sigma^{2}}}}+\mathcal{O}(\delta^{2})$
(49)

$\mathcal{F}_{\textrm{det}}[\rho^{(1)}_{\textrm{p}}(s)]=\frac{1}{4\sigma^{2}}\frac{1-\left(\textrm{Re}(\gamma)e^{-\frac{s^{2}}{8\sigma^{2}}}-\frac{s^{2}}{4\sigma^{2}}\right)\textrm{Re}(\gamma)e^{-\frac{s^{2}}{8\sigma^{2}}}}{\left(1+\textrm{Re}(\gamma)e^{-\frac{s^{2}}{8\sigma^{2}}}\right)^{2}}+\mathcal{O}(\delta)$
(50)

##  Appendix C Proof of inequality (24) from Sec. IV

Let’s define the phase-shifted output states as

$\ket{\bar{\phi}_{i}}=e^{i\varphi_{i}}\ket{\phi_{i}}=\sqrt{1-p}\ket{\bar{\phi}_{i0}}+\sqrt{p}\ket{\bar{\phi}_{i1}},$
(51)

where $\ket{\bar{\phi}_{i0}}=e^{i\varphi_{i}}\ket{\phi_{i0}}$,
$\ket{\bar{\phi}_{i1}}=e^{i\varphi_{i}}\ket{\phi_{i1}}$, and phases
$\varphi_{i}$ are chosen such that

$\forall_{i,j\in\\{1,2,...M\\}}\leavevmode\nobreak\ \leavevmode\nobreak\
\braket{\bar{\phi}_{i1}}{\bar{\phi}_{j1}}=\left|\braket{\phi_{i1}}{\phi_{j1}}\right|.$
(52)

From the o.-n. condition $\braket{\phi_{i}}{\phi_{j}}=\delta_{ij}$ follows
that $\braket{\bar{\phi}_{i}}{\bar{\phi}_{j}}=\delta_{ij}$, and consequently

$\delta_{ij}=\braket{\bar{\phi}_{i}}{\bar{\phi}_{j}}=p\braket{\bar{\phi}_{i1}}{\bar{\phi}_{j1}}+(1-p)\braket{\bar{\phi}_{i0}}{\bar{\phi}_{j0}},$
(53)

hence we can conclude that

$M=\sum_{i,j=1}^{M}\delta_{ij}=p\sum_{i,j=1}^{M}\left|\braket{\phi_{i1}}{\phi_{j1}}\right|+(1-p)\left(\sum_{i=1}^{M}\bra{\bar{\phi}_{i0}}\right)\left(\sum_{j=1}^{M}\ket{\bar{\phi}_{j0}}\right).$
(54)

The last term of the RHS of the above equation is always non-negative, so the
following inequality for p holds:

$p\leq\frac{M}{\sum_{i,j=1}^{M}\left|\braket{\phi_{i1}}{\phi_{j1}}\right|}$
(55)