11institutetext: United Nations Global Pulse 11email:
<EMAIL_ADDRESS>22institutetext:
Mila Québec Artificial Intelligence Institute, Université de Montréal, Canada
33institutetext: NYU Stern School of Business, New York, USA 44institutetext:
Institute for Data Science, Durham University, United Kingdom

# Ensuring the Inclusive Use of Natural Language Processing in the Global
Response to COVID-19

Alexandra Sasha Luccioni 1122 Katherine Hoffmann Pham 1133 Cynthia Sin Nga Lam
11 Joseph Aylett-Bullock 1144 Miguel Luengo-Oroz 11

###### Abstract

Natural language processing (NLP) plays a significant role in tools for the
COVID-19 pandemic response, from detecting misinformation on social media to
helping to provide accurate clinical information or summarizing scientific
research. However, the approaches developed thus far have not benefited all
populations, regions or languages equally. We discuss ways in which current
and future NLP approaches can be made more inclusive by covering low-resource
languages, including alternative modalities, leveraging out-of-the-box tools
and forming meaningful partnerships. We suggest several future directions for
researchers interested in maximizing the positive societal impacts of NLP.

##  1 Introduction and Context

The COVID-19 pandemic has changed the way we live, work, and travel. In their
response to the pandemic, governments and organizations have grappled with
pandemic response, deploying numerous tools and initiatives, many of which
leverage novel technologies to guide decision-making and build more resilient
communities. Textual resources in particular have been critical in pandemic
response, facilitating access to scientific information to guide research and
enabling the detection of misinformation in the midst of the COVID-19
‘infodemic’ [6, 14]. Going forward, there is a continuing need for the use of
NLP to facilitate the search for and dissemination of accurate information and
to counter false narratives and vaccine skepticism. However, the availability,
accessibility and use of these resources has been unequal, prioritizing
certain languages and benefiting some communities more than others,
exacerbating existing inequalities and widening the gap between those who have
access to advanced digital technologies and those who do not [4].

To ensure the inclusive use of NLP in the global response to the COVID-19
pandemic, we argue that it is paramount to develop NLP tools that are both
comprehensive and accessible, and to apply them in ways that maximize positive
impact and minimize potential risks. In this commentary, we identify four
areas that we find to be particularly important to achieving this goal:
covering low-resource languages, including alternative modalities such as
audio or video, using out-of-the-box tools where appropriate and forming
meaningful partnerships to broaden access to technical and local expertise, as
well as data and tools. We discuss each of these themes below, and conclude
with a series of observations and considerations for future research.

##  2 Low-Resource Languages

While the exact number of existing languages is debated, estimates range
anywhere from 6,000 to 7,000 [13]. However, current NLP technologies such as
language models, datasets and tool kits have predominantly been developed for
a limited number of ‘high-resource’ languages, which can leave large parts of
the world behind. In the context of the COVID-19 pandemic response, where
fighting the infodemic of misinformation and fake news has become a top
priority, low-resource languages may also suffer a disadvantage since accurate
information from verifiable sources can arrive with delays and gaps.
Furthermore, while setting up a simple pipeline for detecting misinformation
in English or French can take a few days using pre-trained models and tools,
it can quickly become a multi-month, resource-intensive endeavor for
underrepresented languages [7]. Finally although research initiatives for
misinformation detection in low-resource languages do exist (e.g. [18, 10]),
there have not been, to our knowledge, tools deployed for this purpose in the
context of the COVID-19 pandemic.

There are many challenges to developing NLP tools for low-resource languages,
ranging from regional variants that are hard to represent using standard
language models (e.g. Arabic), to languages with predominantly oral traditions
that have little or no written texts (e.g. Métis). However, the biggest
bottleneck for developing language models and tools for low-resource languages
remains data availability. For example, mainstream NLP approaches such as word
embeddings and large language models are difficult to implement since there
are few pre-compiled large text corpora available [24]. This has spurred
promising research on NLP in low-data scenarios, with approaches ranging from
few-shot and transfer learning [19, 21, 35] to data augmentation [15] and
cross-lingual language models [8]. However, many languages remain under-served
in terms of NLP resources and datasets, especially in the context of
misinformation detection, which often requires customizing neural language
models to encompass specific contexts and topics. In addition to further
research developments in this area, it is important to promote the work of
those teams compiling large datasets for previously low-resource languages
[29, 33]. Overall, the state of available resources for languages that are not
mainstream is unfortunate and makes it difficult to deploy scalable NLP
systems for these languages. Finally, more transparency should be established
between existing commercial endeavors to detect fake news (such as those
deployed by major social media websites) and researchers, since releasing
datasets that have been flagged as misinformation by human moderators can help
train NLP models to mimic this behavior.

##  3 Alternative Modalities

In addition to improving the coverage of underrepresented languages, analyzing
data from a broader range of modalities could also help to expand the scope of
NLP research. Given the need for rapid response to the COVID-19 pandemic,
there has been a bias towards analysis of text or multimedia data sources that
are easy to mine, such as Twitter. However, these types of content are created
and consumed by only a fraction of the world’s population, and are biased
towards those with ready access to the internet. In other contexts,
alternative channels such as radio, television, or messaging applications
might be more common sources of information [27].

Radio in particular is a promising channel for studying opinions and
discussions of emerging topics such as COVID-19. There are an estimated 44,000
radio stations worldwide (see e.g. http://radio.garden for a sample), and in
the developing world radio stations typically have far wider and more
consistent coverage than high-speed internet, with radios available to 75% of
households [38]. Even in the United States, radios were estimated to reach 92%
of adult listeners on a weekly basis as of 2019, exceeding the reach of any
other channel [28]. Radio shows disseminate news, discuss current events,
debate rumors, record spontaneous expressions of local opinions and capture
information-seeking behavior. In the face of crises or natural disasters,
radio may be one of the quickest and most reliable sources of information
about the situation on the ground [25].

During the COVID-19 pandemic, researchers at the United Nations (UN) have used
radio mining techniques in Uganda to monitor discussions relating to outbreaks
and health system capacity, rumors and misinformation, and the socioeconomic
impacts of COVID-19 [16]. Developing a radio-based NLP pipeline poses a number
of distinct challenges. These include engineering challenges such as capturing
and recording radio streams; efficiently managing the high volume of data
collected; and transcribing audio recordings to text for further analysis.
They also include technical challenges such as speaker disambiguation when
multiple individuals are interacting in conversation; filtering out irrelevant
content (e.g. commercials, news, and music); and identifying topic-relevant
discussions through approximate keyword searches (for example, transcription
may introduce new keyword variants – such as “Koby” and “cobi” when “COVID” is
said by Spanish speakers). Given the relative lack of prior research on radio
data, these challenges might pose fruitful areas for innovation in the NLP
community.

Radio is just one of several possible alternative modalities; as noted above,
other data sources of interest include television and informal messaging
channels or even SMS-based question-and-answer forums. In addition to original
research analyzing these channels, another potential direction for future work
involves the creation of tools to better enable on-the-ground practitioners to
engage with the NLP pipeline in order to analyze alternative data sources.
Since much of the content on these alternative channels may involve informal
language, local dialects, and context-specific references, allowing
practitioners to filter, explore, and improve the output of NLP algorithms is
particularly important in this setting. While original research is of vital
importance, the use of out-of-the-box tools in NLP pipelines can avoid the
duplication of efforts and provide operationally-ready interfaces for
practitioners to interact with, which we discuss in further detail below. Some
out-of-the box tools have already implemented such functionality for social
media sources such as Twitter, as we discuss in further detail below.

##  4 Out-of-the-Box Tools for Infodemic Management

Social listening is an important means for gaining fast insights into public
perceptions and information needs as well as detecting misinformation. This is
especially important during crises like the COVID-19 pandemic, where there is
an overwhelming volume of data, intertwined with conflicting information [32,
34]. In response to these growing needs, many commercially available machine
learning-driven monitoring tools have been developed to public health
practitioners, information and communications practitioners and researchers
[2]. These out-of-the-box tools built by different companies vary in purpose,
data input, capabilities, models, workflow management and output [36].
However, while these tools present tremendous value, they also come with novel
risks.

Crises strain resources, and sufficient financial, time and personnel
resources are not always available to local public health authorities to
invest in social media strategies to begin with [1]. Devoting staff time to
processing, analyzing and real-time monitoring of abundant social media data
from scratch can be a luxury to organizations that lack the necessary
resources and skills [22]. Out-of-the-box tools can simplify the data
gathering, cleaning, analysis and presentation processes to low- or no-code
[20], providing user-friendly interfaces that facilitate more efficient
storage and analysis of data, and offering AI-powered automation for sentiment
analysis and continuous monitoring. While in some cases these tools can
provide end-to-end insights, they can also be used by technical experts to
form one component of a custom-designed NLP pipeline.

Nevertheless, these tools are not without shortcomings. First, the integrity
of social media metrics and the ambiguity of their definitions are obstacles
for both data analysts and the audience of their reports. ‘Visible’ social
media metrics such as the numbers of likes and followers are inherently
fallible due to the lack of context (e.g. ‘liking’ a post can be driven by
many different motives and emotions) as well as the commercialization of fake
followers and metrics [3, 9]. The further processed metrics presented in
social media listening tools, such as ‘reach’ and ‘impressions’, suffer from
another layer of uncertainty. Often the tools do not fully disclose how the
metrics are being calculated, and the definition of each metric may not be
standardized across tools, making it even harder for users and readers to
contextualize the information.

Furthermore, the accuracy of built-in classifiers for sentiment analysis is
questionable [31]. Studies that compare manual sentiment coding with automatic
sentiment analysis have shown that there is not only a poor correlation
between the manual coding and the tool-generated results [11], but also a lack
of agreement between the tools [5, 41]. It is challenging for researchers to
independently investigate why this disagreement occurs, as the proprietary
nature of the out-of-the-box tools prevents any external assessment of the
integrity of their models. This is further complicated by the fact that
language models trained prior to COVID-19 might not fully reflect the current
reality because of the emergence of Out-of-Vocabulary words [42].

Some tools attempt to address these problems by implementing human-in-the-loop
machine learning, enabling users to correct wrongly classified information
[36] or train custom classifiers adapted to users’ own context. This
highlights the value of versatility in out-of-the-box tools – predicting
changes in the global landscape of communications might not be always
possible, but by making these tools more adaptable by design, it is possible
to cater to more diverse, novel use cases.

##  5 Importance of Partnerships

To unlock the full potential of NLP methodologies and tools to address crises
such as COVID-19, it is important that these techniques are appropriately
tailored to each particular application setting; this should be part of any
system design thinking. Partnerships with domain experts, end users and other
beneficiaries can serve as a crucial resource to help direct research and to
ensure meaningful and effective deployment.

In the case of using NLP for the COVID-19 response, productive partnerships
could include stakeholders with technical and contextual expertise, data
providers and tool developers. In many research settings, methods are
developed first and possible applications found later. While this approach is
useful for developing theoretical concepts and frameworks, in order to
maximize social impact in crisis response scenarios the aims of the
application should drive research, with meaningful objectives jointly
determined by researchers and implementing partners. Giving greater agency to
users of the tools will encourage long-lasting use.

Engaging with partners from different disciplines can be challenging. Often,
research agendas and funding cycles require different timelines than the
deployment of a tool, sometimes resulting in projects ending prematurely. In
the case of many recent projects to tackle COVID-19, it has been difficult to
quickly reach the level of maturity needed to operationalize projects and
systems at scale [23]. It is therefore essential to establish agendas which
accommodate the different time constraints of all stakeholders from the
beginning of any project.

In many of the examples mentioned above, partnership development can help NLP
researchers, who may be working in geographies which are removed from the
application setting of interest, to understand the questions which need to be
answered as well as the relevant modalities. For example, in the case of
Uganda discussed in Section 3, local entities could inform researchers that
radio streams are a dominant linguistic modality for reliable information and
opinion tracking, thereby redirecting effort away from developing social
media-related listening strategies. Similarly, partnerships can provide access
to relevant data sources, as well as pre-designed tools (see Section 4).

##  6 Conclusion

To advance the inclusive use of NLP approaches in the COVID-19 response, we
have identified the following focus areas:

Data – High-quality data helps build high-quality models and guides research
priorities and makes it possible to establish benchmarks. A major challenge
with many of the approaches mentioned above is the need for more inclusive
large-scale public datasets to allow NLP researchers to explore and work with
low-resource languages and alternative modalities. There are promising
endeavors to curate datasets of misinformation such as the Fakeddit dataset
[26], Poynter and COVIDLies [17], as well as as large community modeling
efforts such as the FastAI Model Zoo, which could be expanded to cover a wider
range of regions and languages. Similarly, efforts to build datasets from a
wider range of sources – e.g., via the creation of radio transcription
pipelines – could encourage researchers to work with data from alternative
modalities. Partnerships can also serve as a key source for localized and
relevant datasets. Finally, evaluation would be facilitated by the creation of
multilingual COVID-related datasets, as well as a set of benchmarks for how
well models perform on certain tasks (e.g. detecting misinformation).

Adaptation and Flexibility – There are many existing NLP tools, models and
approaches that can be (re-)used in the context of pandemic response. While
these tools bring convenience and efficiency to social listening in a crisis
like COVID-19, one size does not fit all. Different tools are designed for
different purposes and use cases, and modifications are often necessary to
adapt them to specific contexts. It is necessary to evaluate the quality of
the tools and their algorithms as much as possible to avoid making decisions
based on flawed data or analysis [30]. These tools should be seen as supports
for decision making and cannot be relied upon alone; manual validation of the
results in context is always crucial [5]. For instance, one potential strategy
for addressing the limitations of these tools involves the use of human-in-
the-loop machine learning to ensure human oversight of ultimate decision-
making.

Incentives and Frameworks – Beyond the specific context of the pandemic
response, the research opportunities described above present challenges that
are both academically well-motivated (as many understudied research questions
remain to be explored) and likely to make a real-world impact (given the
urgency and unprecedented nature of the pandemic). Even as the market for NLP
tools for high-resource languages is becoming increasingly saturated, digital
inclusion is creating new markets within communities that currently have few
or no customized NLP resources and a large demand for them. There is a need to
translate between technical and domain expertise and to bridge the gap between
current technological capabilities and the needs of stakeholders. Many
resources exist to assist in developing data-driven partnerships across
different domains [37, 12] – for instance, the UN Sustainable Development
Goals and their associated targets can act as a common framework for setting
aims and objectives, which is shared among institutions on a global level and
can guide the response to and recovery from the COVID-19 pandemic [39, 40]. As
such, they can assist in the coordination and prioritization of tasks, help
with the development of a common language between institutions, and outline
effective areas for contributions by the NLP community.

## References

  * [1] Avery, E.: Public information officers’ social media monitoring during the Zika virus crisis, a global health threat surrounded by public uncertainty. Public Relations Review 43 (03 2017). https://doi.org/10.1016/j.pubrev.2017.02.018
  * [2] Batrinca, B., Treleaven, P.: Social media analytics: A survey of techniques, tools and platforms. AI & SOCIETY 30, 89–116 (02 2014). https://doi.org/10.1007/s00146-014-0549-4
  * [3] Baym, N.: Data not seen: The uses and shortcomings of social media metrics. First Monday 18 (10 2013). https://doi.org/10.5210/fm.v18i10.4873
  * [4] Beaunoyer, E., Dupéré, S., Guitton, M.J.: COVID-19 and digital inequalities: Reciprocal impacts and mitigation strategies. Computers in Human Behavior p. 106424 (2020)
  * [5] Boukes, M., Velde, B., Araujo, T., Vliegenthart, R.: What’s the tone? Easy doesn’t do it: Analyzing performance and agreement between off-the-shelf sentiment analysis tools. Communication Methods and Measures 14, 1–22 (10 2019). https://doi.org/10.1080/19312458.2019.1671966
  * [6] Bullock, J., Luccioni, A., Pham, K.H., Lam, C.S.N., Luengo-Oroz, M., et al.: Mapping the landscape of Artificial Intelligence applications against COVID-19. Journal of Artificial Intelligence Research 69, 807–845 (2020)
  * [7] Bullock, J., Luengo-Oroz, M.: Automated speech generation from UN General Assembly statements: Mapping risks in AI generated texts. International Conference on Machine Learning AI for Social Good Workshop (2019)
  * [8] Conneau, A., Baevski, A., Collobert, R., Mohamed, A., Auli, M.: Unsupervised cross-lingual representation learning for speech recognition. arXiv preprint arXiv:2006.13979 (2020)
  * [9] Cresci, S., Di Pietro, R., Petrocchi, M., Spognardi, A., Tesconi, M.: Fame for sale: Efficient detection of fake Twitter followers. Decision Support Systems 80, 56 – 71 (2015). https://doi.org/https://doi.org/10.1016/j.dss.2015.09.003, http://www.sciencedirect.com/science/article/pii/S0167923615001803
  * [10] Cruz, J.C.B., Tan, J.A., Cheng, C.: Localization of fake news detection via multitask transfer learning. arXiv preprint arXiv:1910.09295 (2019)
  * [11] Deiner, M.S., Fathy, C., Kim, J., Niemeyer, K., Ramirez, D., Ackley, S.F., Liu, F., Lietman, T.M., Porco, T.C.: Facebook and Twitter vaccine sentiment in response to measles outbreaks. Health Informatics Journal 25(3), 1116–1132 (2019). https://doi.org/10.1177/1460458217740723, https://doi.org/10.1177/1460458217740723, pMID: 29148313
  * [12] DSEG: A framework for the ethical use of advanced data science methods in the humanitarian sector (2020)
  * [13] Eberhard, D.M., Simons, G.F., Fennig, C.D.e.: Ethnologue: Languages of the world. twenty-third edition, online version. http://www.ethnologue.com (2020)
  * [14] Eysenbach, G.: How to fight an infodemic: The four pillars of infodemic management. Journal of Medical Internet Research 22, e21820 (06 2020). https://doi.org/10.2196/21820
  * [15] Fadaee, M., Bisazza, A., Monz, C.: Data augmentation for low-resource neural machine translation. arXiv preprint arXiv:1705.00440 (2017)
  * [16] Hidalgo-Sanchis, P.: Using speech-to-text technology to support response to the COVID-19 pandemic. https://www.unglobalpulse.org/2020/05/using-speech-to-text-technology-to-support-response-to-the-covid-19-pandemic/ (May 2020)
  * [17] Hossain, T., Logan IV, R.L., Ugarte, A., Matsubara, Y., Young, S., Singh, S.: COVIDLIES: Detecting COVID-19 misinformation on social media. In: Proceedings of the 1st Workshop on NLP for COVID-19 (Part 2) at EMNLP 2020 (2020)
  * [18] Hossain, Z., Rahman, A., Islam, S., Kar, S.: BanFakeNews: A dataset for detecting fake news in Bangla. arXiv preprint arXiv:2004.08789 (2020)
  * [19] Johnson, M., Schuster, M., Le, Q.V., Krikun, M., Wu, Y., Chen, Z., Thorat, N., Viégas, F., Wattenberg, M., Corrado, G., et al.: Google’s multilingual neural machine translation system: Enabling zero-shot translation. Transactions of the Association for Computational Linguistics 5, 339–351 (2017)
  * [20] Lee, I.: Social media analytics for enterprises: Typology, methods, and processes. Business Horizons 61 (11 2017). https://doi.org/10.1016/j.bushor.2017.11.002
  * [21] Levy, S., Wang, W.Y.: Cross-lingual transfer learning for COVID-19 outbreak alignment. arXiv preprint arXiv:2006.03202 (2020)
  * [22] Lindsay, B.R.: Social media and disasters: Current uses, future options, and policy considerations (2011)
  * [23] Luengo-Oroz, M., Hoffmann Pham, K., Bullock, J., Kirkpatrick, R., Luccioni, A., Rubel, S., Wachholz, C., Chakchouk, M., Biggs, P., Nguyen, T., Purnat, T., Mariano, B.: Artificial intelligence cooperation to support the global response to COVID-19. Nature Machine Intelligence 2(6) (2020)
  * [24] Magueresse, A., Carles, V., Heetderks, E.: Low-resource languages: A review of past work and future challenges. arXiv preprint arXiv:2006.07264 (2020)
  * [25] Munro, R.: Crowdsourcing and the crisis-affected community. Information Retrieval 16(2), 210–266 (Apr 2013). https://doi.org/10.1007/s10791-012-9203-2
  * [26] Nakamura, K., Levy, S., Wang, W.Y.: r/fakeddit: A new multimodal benchmark dataset for fine-grained fake news detection. arXiv preprint arXiv:1911.03854 (2019)
  * [27] Newman, L., Hutchinson, P., Meekers, D.: Key findings of the 3-2-1 service COVID-19 surveys: Information (Wave 1). https://viamo.io/wp-content/uploads/2020/09/Updated_-3-2-1-Service-COVID-19-Survey_-Information-1.pdf (2020)
  * [28] Nielsen: The steady reach of radio: Winning consumer attention. https://www.nielsen.com/us/en/insights/article/2019/the-steady-reach-of-radio-winning-consumers-attention/ (Jun 2019)
  * [29] Orife, I., Kreutzer, J., Sibanda, B., Whitenack, D., Siminyu, K., Martinus, L., Ali, J.T., Abbott, J., Marivate, V., Kabongo, S., et al.: Masakhane – Machine translation for Africa. arXiv preprint arXiv:2003.11529 (2020)
  * [30] Rai, A.: Explainable AI: From black box to glass box. Journal of the Academy of Marketing Science 48 (12 2019). https://doi.org/10.1007/s11747-019-00710-5
  * [31] Rappaport, S.: Listening solutions: A marketer’s guide to software and services. Journal of Advertising Research - JAR 50 (06 2010). https://doi.org/10.2501/S002184991009135X
  * [32] Ruggiero, A., Vos, M.: Social media monitoring for crisis communication: Process, methods and trends in the scientific literature. Online Journal of Communication and Media Technologies 4, 103–130 (01 2014). https://doi.org/10.29333/ojcmt/2457
  * [33] Scannell, K.P.: The Crúbadán Project: Corpus building for under-resourced languages. In: Building and Exploring Web Corpora: Proceedings of the 3rd Web as Corpus Workshop. vol. 4, pp. 5–15 (2007)
  * [34] Sheppard, B.: Mitigating terror and avoidance behavior through the risk perception matrix to augment resilience. Journal of Homeland Security and Emergency Management 8 (01 2011). https://doi.org/10.2202/1547-7355.1840
  * [35] Spangher, A., Peng, N., May, J., Ferrara, E.: Enabling low-resource transfer learning across COVID-19 corpora by combining event-extraction and co-training. In: Proceedings of the 1st Workshop on NLP for COVID-19 at ACL 2020 (2020)
  * [36] Stavrakantonakis, I., Gagiu, A.E., Kasper, H., Toma, I., Thalhammer, A.: An approach for evaluation of social media monitoring tools (2012)
  * [37] UNDP, UN Global Pulse: A guide to data innovation for development: From idea to proof of concept (2016)
  * [38] UNESCO: Statistics on Radio. http://www.unesco.org/new/en/unesco/events/prizes-and-celebrations/celebrations/international-days/world-radio-day-2013/statistics-on-radio/ (2013)
  * [39] United Nations: 17 goals to transform our world. https://www.un.org/sustainabledevelopment/
  * [40] Vinuesa, R., Azizpour, H., Leite, I., Balaam, M., Dignum, V., Domisch, S., Felländer, A., Langhans, S.D., Tegmark, M., Fuso Nerini, F.: The role of Artificial Intelligence in achieving the sustainable development goals. Nature Communications 11(1) (2020)
  * [41] Young, L., Soroka, S.: Affective news: The automated coding of sentiment in political texts. Political Communication - POLIT COMMUN 29, 205–231 (04 2012). https://doi.org/10.1080/10584609.2012.671234
  * [42] Zheng, X., Liu, Y., Gunceler, D., Willett, D.: Using synthetic audio to improve the recognition of out-of-vocabulary words in end-to-end ASR systems (2020)


# Weak entanglement improves quantum communication using only product
measurements

Amélie Piveteau  Department of Physics, Stockholm University, S-10691
Stockholm, Sweden  Alastair A. Abbott  Univ. Grenoble Alpes, Inria, 38000
Grenoble, France  Sadiq Muhammad  Department of Physics, Stockholm University,
S-10691 Stockholm, Sweden  Mohamed Bourennane  Department of Physics,
Stockholm University, S-10691 Stockholm, Sweden  Armin Tavakoli  Physics
Department, Lund University, Box 118, 22100 Lund, Sweden

###### Abstract

We show that weakly entangled states can improve communication over a qubit
channel using only separate, interference-free, measurements of individual
photons. We introduce a communication task corresponding to the cryptographic
primitive known as secret sharing and show that all steerable two-qubit
isotropic states provide a quantum advantage in the success rate using only
product measurements. Furthermore, we show that such measurements can even
reveal communication advantages from noisy partially entangled states that
admit no quantum steering. We then go further and consider a stochastic
variant of secret sharing based on more sophisticated, yet standard, partial
Bell state analysers, and show that this reveals advantages also for a range
of unsteerable isotropic states. By preparing polarisation qubits in
unsteerable states, we experimentally demonstrate improved success rates of
both secret sharing tasks beyond the best entanglement-unassisted qubit
protocol. Our results reveal the capability of simple and scalable
measurements in entanglement-assisted quantum communication to overcome large
amounts of noise.

Introduction.—Entanglement is well-known to be a crucial resource for quantum
communication applications. However, not all forms of entanglement are always
useful. For instance, while entanglement that is strong enough to generate
nonlocality has been found to improve noiseless classical communication beyond
its conventional limitations (see, e.g., [1, 2, 3, 4, 5, 6]), weaker entangled
states that cannot violate any Bell inequality do not have that ability [7].
In contrast, if the system communicated is itself quantum, e.g. a qubit
instead of a bit, then some weaker forms of entanglement also become useful.
This can be seen in the celebrated dense coding protocol [8] where a maximally
entangled state, $|\phi^{+}\rangle=\frac{|00\rangle+|11\rangle}{\sqrt{2}}$, is
exploited to allow a qubit message to transmit two bits instead of one bit.
When the state is noisy, the isotropic state

$\rho_{v}=v|\phi^{+}\rangle\\!\langle\phi^{+}|+\frac{1-v}{4}\openone,$ (1)

where $v\in[0,1]$ is the visibility, achieves an advantage over an unassisted
qubit message whenever $v>\frac{1}{3}$ [9, 10]. This coincides with the
visibility at which the state becomes separable, $v_{\text{sep}}=\frac{1}{3}$
and it is considerably lower than the critical visibility for quantum
steering, $v_{\text{unsteer}}=\frac{1}{2}$ [11, 12], and far below the even
higher visibility needed for Bell nonlocality [13].

However, to harness the dense coding advantage one must measure in the Bell
basis $\\{\Phi^{+},\Phi^{-},\Psi^{+},\Psi^{-}\\}$, where
$\Phi^{\pm}=|\phi^{\pm}\rangle\\!\langle\phi^{\pm}|$ and
$\Psi^{\pm}=|\psi^{\pm}\rangle\\!\langle\psi^{\pm}|$ are the projectors onto
the states $|\phi^{\pm}\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt{2}}$ and
$|\psi^{\pm}\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt{2}}$. In optical
systems, it is impossible to implement a linear optics Bell basis measurement
on separate photons without the use of auxiliary photons [14]. While dense
coding experiments have been reported [15, 16, 17, 18, 19], implementation of
the Bell basis is not expected to be scalable in optical systems in the near
future. Nevertheless, it has recently been found that entanglement can yield
advantages in one-shot quantum communication scenarios by using much more
limited, yet much less experimentally demanding, optical measurements that are
compatible with passive linear optics or even just separate single-photon
measurements [20]. However, the schemes considered thus far come with greater
requirements on the quality of entanglement, in particular needing states that
violate a Bell inequality.

Here, we show that the simplest optical measurements are sufficient to reveal
communication advantages from highly noisy entangled states. To this end, we
introduce a version of secret sharing. In secret sharing, a secret is
distributed between two parties in such a way that they must cooperate to
reconstruct it [21, 22]. This is of considerable interest for quantum
cryptography, and has consequently received much attention (see e.g. [23, 24,
25, 26, 27]). We prove that for our secret sharing task every steerable
isotropic state enables an advantage over unassisted qubits using only product
measurements on separate photons. Furthermore, we also identify unsteerable
states for which an advantage is possible with such measurements. In addition,
we prove that, at the cost of using somewhat more sophisticated measurements,
namely partial Bell state analysers, which require optical interference but
are still ammenable to standard passive linear optics [28, 29], it is possible
to reveal advantages from unsteerable instances of the isotropic state. For
this, we use a stochastic modification of the original secret sharing task.
Using polarisation qubits generated via spontaneous parametric down-
conversion, we use both a maximally entangled state to experimentally
demonstrate optimal secret sharing protocols, and unsteerable states to
outperform the best possible entanglement-unassisted qubit protocols using
both product measurements and partial Bell state analysers.

Deterministic secret sharing.—Alice wishes to generate a random secret bit
$a\in\\{0,1\\}$ that is shared with Bob and Charlie in such a way that they
individually have no knowledge of $a$ but can learn its value if they
cooperate. To this end, we consider the scenario illustrated in Fig 1.

Figure 1: The secret scharing scenario. Bob and Charlie select private inputs
and perform separate transformations on a shared two-qubit entangled state.
They relay their respective output qubits to Alice who measures them. Her
output is the secret and her input determines how it can be reconstructed
collectively by Bob and Charlie.

Bob and Charlie each privately select two uniformly random bits
$x\equiv(x_{0},x_{1})\in\\{0,1\\}^{2}$ and
$y\equiv(y_{0},y_{1})\in\\{0,1\\}^{2}$, respectively, and each communicate a
message to Alice. Alice privately selects a binary input $z\in\\{0,1\\}$ and
accordingly decodes the two incoming messages into an output $a\in\\{0,1\\}$.
Here, $z$, which is later announced publicly, determines which bits of Bob and
Charlie hold the shared secret: the success condition is $a=x_{z}\oplus
y_{z}$. The average success probability of secret sharing becomes

$\displaystyle\mathcal{S}=\frac{1}{32}\sum_{x,y,z}p(a=x_{z}\oplus
y_{z}|x,y,z).$ (2)

Naturally, the parties must have some physical limitations. On the one hand,
we are interested in the situation in which Bob and Charlie share no prior
entanglement and simply send messages encoded in qubit states $\beta_{x}$ and
$\gamma_{y}$, respectively, while Alice can decode using a general quantum
measurement $\\{M_{a|z}\\}_{a}$. The observed correlations are then given by
$p_{\text{qubit}}(a|x,y,z)=\operatorname{tr}\left[(\beta_{x}\otimes\gamma_{y})M_{a|z}\right]$.
On the other hand, we also consider the situation illustrated in Fig. 1, where
Bob and Charlie may additionally share a two-qubit entangled state $\rho$ and
then encode their qubit messages through local quantum channels
$\Lambda_{x}^{B}$ and $\Lambda_{y}^{C}$, respectively. In this entanglement
assisted case, the correlations are given by
$p_{\text{EAqubit}}(a|x,y,z)=\operatorname{tr}\left[\big{(}\Lambda_{x}^{B}\otimes\Lambda_{y}^{C}\left(\rho\right)\big{)}M_{a|z}\right]$.
We now show that there exists an entanglement-assisted quantum protocol that
achieves a perfect secret sharing rate while requiring only separate
measurements of the incoming quantum messages.

Ideal entanglement-assisted and unassisted protocols.—Consider a maximally
entangled state $\rho=|\phi^{+}\rangle\\!\langle\phi^{+}|$ and let Bob’s and
Charlie’s local channels correspond to implementing the four unitaries
$U^{B}_{x}=\sigma_{X}^{x_{0}}\sigma_{Z}^{x_{1}}$ and
$U^{C}_{y}=\sigma_{X}^{y_{0}}\sigma_{Z}^{y_{1}}$ respectively, where
$\sigma_{X}$ and $\sigma_{Z}$ are the Pauli bit-flip and phase-flip operators.
This means that the two-qubit state arriving to Alice is one of the four Bell
states. When $z=0$, Alice measures $\sigma_{Z}\otimes\sigma_{Z}$, i.e., she
discriminates the states $|\phi^{\pm}\rangle$ from the states
$|\psi^{\pm}\rangle$. When $z=1$, she measures $\sigma_{X}\otimes\sigma_{X}$,
i.e., she discriminates the states $|\phi^{+}\rangle$ and $|\psi^{+}\rangle$
from the states $|\phi^{-}\rangle$ and $|\psi^{-}\rangle$. This gives
$\mathcal{S}=1$.

In contrast, in the scenario when entanglement is absent, it is no longer
possible to succeed deterministically. We have determined the maximum value of
$\mathcal{S}$ achievable with entanglement-unassisted qubits and general
measurements for Alice. This is achieved using a straightforward modification
of a hierarchy of semidefinite programs for bounding dimensionally-restricted
quantum correlation [30]. In order to obtain sufficiently tight upper bounds
on $\mathcal{S}_{\text{qubit}}$, we have considered terms appearing in the
first three levels of this hierarchy 111Specifically, we used what would, in
standard terminology, be called the “1+AB+AC+ABC” level, leading to a moment
matrix of size $109$.. Up to solver precision, we obtain the upper bound
$\mathcal{S}_{\text{qubit}}\leq\frac{3}{4}$. This bound holds even if all
three parties share some pre-agreed classical randomness.

In fact, a simple classical strategy can saturate
$\mathcal{S}_{\text{qubit}}$, thus showing both that the bound is tight and
that there exists no quantum-over-classical advantage without using
entanglement. To this end, consider that Bob and Charlie send $x_{0}$ and
$y_{0}$ respectively to Alice. When $z=0$ she correctly outputs $a=x_{0}\oplus
y_{0}$ and when $z=1$ she chooses $a$ at random. This leads to
$S_{\text{classical}}=\frac{3}{4}$. We conclude that any value in the range
$\frac{3}{4}<\mathcal{S}\leq 1$ implies an advantage over both the best
classical and the best entanglement-unassisted quantum protocol, and hence is
powered by the consumption of entanglement.

Advantage from noisy entanglement.—If, in the above ideal entanglement-
assisted protocol, we substitute the maximally entangled state for the
isotropic state (4), we find that $\mathcal{S}=\frac{1+v}{2}$. Hence, when
$v>\frac{1}{2}=v_{\text{unsteer}}$, product measurements reveal an
entanglement-based advantage in the task. We conclude that every steerable
isotropic state is a communication resource under product measurements.
Furthermore, while unsteerable isotropic states do not provide an advantage,
there exist other unsteerable states that are useful in the task. To this end,
consider the states
$\rho_{v}^{\theta}=v|\phi^{+}_{\theta}\rangle\\!\langle\phi^{+}_{\theta}|+\frac{1-v}{4}\openone$,
where $v\in[0,1]$ and
$|\phi^{+}_{\theta}\rangle=\cos\theta|00\rangle+\sin\theta|11\rangle$ is a
pure partially entangled state for $\theta\in(0,\pi/4]$. In [32], local hidden
state models have been computed for $\rho_{v}^{\theta}$. A particularly
sizable advantage is found at $\theta_{*}=0.2356$, for which the state is
unsteerable when $v\lesssim 0.7325$ but achieves a secret sharing success rate
$\mathcal{S}>\mathcal{S}_{\text{qubit}}$ whenenver $v\gtrsim 0.6878$. Hence, a
significant amount of additional noise can be added to the unsteerable state
before it ceases to generate a communication advantage under product
measurements. Note also that applying the protocol to an arbitrary pure
entangled state $|\phi^{+}_{\theta}\rangle$, we obtain
$\mathcal{S}=\frac{3+\sin\left(2\theta\right)}{4}$ which exceeds the
entanglement-unassisted limit if and only if the state is entangled.

Stochastic secret sharing.—We have seen that advantages in secret sharing can
be propelled by all steerable isotropic states using product measurements. Now
we will see that advantages can even be obtained with unsteerable isotropic
states, at the cost of departing from product measurements to standard, linear
optics, partial Bell state analysers. To this end, we will consider such
measurements in a stochastic version of our secret sharing task.

Consider again the scenario in Fig. 1 but now let Alice have three possible
outcomes $a\in\\{0,1,\perp\\}$, where $a\in\\{0,1\\}$ corresponds to the
secret bit stored in $x_{\bar{z}}\oplus y_{\bar{z}}$, where $\bar{z}=z\oplus
1$, and $a=\perp$ is associated to a control parameter which indicates that
the round is not used for secret sharing. We define that a round should be
used for secret sharing exclusively when $x_{z}\oplus y_{z}=1$. Hence, the
average success rate of secret sharing in the relevant rounds becomes

$\displaystyle\mathcal{R}_{\text{scrt}}=\frac{1}{16}\sum_{z}\sum_{x_{z}\oplus
y_{z}=1}p(a=x_{\bar{z}}\oplus y_{\bar{z}}|x,y,z),$ (3)

whereas the average probability of correctly identifying a round not used for
secret sharing is
$\mathcal{R}_{\text{ctrl}}=\frac{1}{16}\sum_{z}\sum_{x_{z}\oplus
y_{z}=0}p(a=\perp|x,y,z)$. This serves as a control parameter.

We are now interested in determining the largest value of
$\mathcal{R}_{\text{scrt}}$ for a given value of the control parameter. Note
that an ideal protocol would achieve perfect secret sharing in all relevant
rounds and perfectly identify all non-secret sharing rounds, i.e.
$\mathcal{R}_{\text{scrt}}=\mathcal{R}_{\text{ctrl}}=1$. This is possible with
entanglement: Bob and Charlie perform the same unitaries as before and Alice
does a three-outcome measurement corresponding to a projection onto
$\\{\Psi^{+},\Psi^{-},\Phi^{+}+\Phi^{-}\\}$ for $z=0$ and onto
$\\{\Phi^{-},\Psi^{-},\Phi^{+}+\Psi^{+}\\}$ for $z=1$. These measurements may
be seen as partial Bell state analysers, resolving two of the four Bell
states, as compatible with passive linear optics [33].

In contrast, in the scenario when entanglement is absent, the trade-off
between $\mathcal{R}_{\text{scrt}}$ and $\mathcal{R}_{\text{ctrl}}$ is non-
trivial. Numerical exploration suggests that it is characterised by
$\mathcal{R}=\frac{1}{2}\mathcal{R}_{\text{scrt}}+\frac{1}{2}\mathcal{R}_{\text{ctrl}}\leq\frac{5}{8}$
with $\mathcal{R}_{\text{ctrl}}\leq\frac{3}{4}$. Exploiting symmetries 222Note
that $\mathcal{R}$ is invariant under i) simultaneous bit-flip of $x_{0}$ and
$y_{0}$, ii) simultaneous bit-flips of $x_{1}$ and $y_{1}$ and iii)
simultaneous swaps $x_{0}\leftrightarrow x_{1}$ and $y_{0}\leftrightarrow
y_{1}$.  to make the problem more tractable, we used semidefinite relaxations
[35] to prove the upper bound $\frac{5}{8}$ up to solver precision. As in the
deterministic case, this bound can be saturated classically: Bob and Charlie
relay the bits $x_{0}$ and $y_{0}$, respectively, to Alice. If $z=1$, she
outputs $a=x_{0}\oplus y_{0}$. If $z=0$, Alice outputs $a=\perp$ if
$x_{0}\oplus y_{0}=0$ and $a=0$ otherwise. This gives
$\mathcal{R}_{\text{scrt}}=\frac{3}{4}$ and
$\mathcal{R}_{\text{ctrl}}=\frac{1}{2}$, thus saturating the entanglement-
unassisted bound.

If the above entanglement-assisted protocol is performed using the isotropic
state (4), we find that $\mathcal{R}=\frac{3+5v}{8}$. Thus, we observe an
advantage ascribed to entanglement whenever $\mathcal{R}>\frac{5}{8}$, which
occurs whenever $v>\frac{2}{5}$. Consequently, when
$v\in\left(\frac{2}{5},\frac{1}{2}\right]$, the isotropic state is both
unsteerable and a communication resource with the employed decoding resources.

Experiment.—We report here on experimental implementations of the
deterministic and stochastic secret sharing protocols. We implement both the
ideal quantum protocols using maximally entangled states encoded in the
polarisation of photons, and the protocols using weakly entangled,
unsteerable, states. We begin by presenting the deterministic secret sharing
experiment using partially entangled states $\rho_{v}^{\theta}$, before
explaining how this is modified for the stochastic protocol using the simpler
isotropic states $\rho_{v}$.

Figure 2: Experimental setup. Entangled photon pairs in spatial modes (a) and
(b) are generated through the SPDC process. For the deterministic secret
sharing experiment (Box I), entangled photon pairs are generated by the same
way but with a different ratio between the two modes. Isotropic states are
prepared by randomly transforming the maximally entangled $|\phi^{+}\rangle$
state into one of three other Bell states using two half wave plate (HWP) and
a phase shifter (PS). The unitaries of Bob and Charlie are implemented using
half wave plates (HWP) and phase shifters (PS). Alice’s projective
measurements are implemented using HWP, polarising beam splitters (PBS) and
then detected by single photon detectors (DET). For stochastic secret sharing
experiment (Box II), isotropic states are prepared by randomly transforming
the maximally entangled $|\phi^{+}\rangle$ state into one of the other Bell
states using one half wave plate (HWP) and a phase shifters (PS). The
unitaries of Bob and Charlie are implemented using half wave plates (HWP) and
phase shifters (PS). Alice’s partial Bell state measurements are implemented
using HWP, polarising beam splitters (PBS), beam splitters (BS) and then
detected by single photon detectors (DET).

To generate the entangled states, ultraviolet light centered at a wavelength
of 390 nm is focused onto two 2 mm thick $\beta$ barium borate (BBO) nonlinear
crystals placed in an interferometric configuration to produce photon pairs
emitted into two spatial modes through the second order degenerate type-I
spontaneous parametric down-conversion process (SPDC). The spectral, and
temporal distinguishability between the down-converted photons is carefully
removed by passing through narrow-bandwidth interference filters and quartz
wedges respectively (see Fig. 2). For the deterministic secret sharing
protocol, we prepare the pure partially entangled state
$|\phi_{\theta}^{+}\rangle=\cos\theta|HH\rangle+\sin\theta|VV\rangle$ by
setting the polarisation of the pump laser
($\cos\theta|H\rangle+\sin\theta|V\rangle$), where $H$ and $V$ are,
respectively, the horizontal and vertical photonic polarisation modes (a) and
(b) used to encode the basis states in the standard way as
$|0\rangle:=|H\rangle$ and $|1\rangle:=|V\rangle$. We then prepare a
depolarisation $\rho_{v}^{\theta}$ of the the state
$|\phi_{\theta}^{+}\rangle=\cos\theta|HH\rangle+\sin\theta|VV\rangle$. To
achieve the mixture with the maximally mixed state, we randomly transform,
with probability $1-v$, the pure partially entangled state into one of these
four orthogonal states: $|\phi^{+}_{\theta}\rangle$,
$|\phi_{\theta}^{-}\rangle=\sin\theta|HH\rangle-\cos\theta|VV\rangle$,
$|\psi_{\theta}^{+}\rangle=\cos\theta|HV\rangle+\sin\theta|VH\rangle$ and
$|\psi_{\theta}^{-}\rangle=\sin\theta|HV\rangle-\cos\theta|VH\rangle$. These
transformations were experimentally realised by a motorised rotation of two
half wave plates and a phase shifter (see Fig. 2). Details and experimental
settings are provided in the Supplemental Material. Alice performs product
measurements in the $H$-$V$ and diagonal bases using two HWPs placed before
the input of two PBS distributed on each of the two modes (a) and (b), see
Fig. 2, Box I.

Bob’s and Charlie’s transformations $U^{B}_{x}$ and $U^{C}_{y}$, respectively,
are performed using two HWPs and a PS to change the relative phase between the
two modes (a) and (b) (see Fig. 2 and Supplemental Material for the settings).

We chose the visibility $v=0.72$ (with $\theta=\theta_{*}=0.2356$), for which
the ideal state $\rho_{v}^{\theta_{*}}$ is unsteerable. Performing state
tomography, we obtain a state fidelity of $0.9965\pm 0.0007$ and we verified
that the experimentally realised state is indeed unsteerable. In the
implementation of the protocol the two-photon coincidences rate was one per
second. The total number of events recorded is approximately $8\times 10^{5}$,
leading to a parameter value of $\mathcal{S}=0.757\pm 0.001$ wich is
significantly above the entanglement-unassisted limit
$\mathcal{S}=\frac{3}{4}$. We also performed the protocol using the pure state
$|\phi_{\theta_{*}}^{+}\rangle=\cos\theta_{*}|HH\rangle+\sin\theta_{*}|VV\rangle$,
with a total of $7\times 10^{7}$ two-photon coincidence events. We measured
the state fidelity to be $0.990\pm 0.0001$ and found a success probability of
$\mathcal{S}=0.8519\pm 0.0002$ which can be compared to the theoretical
prediction of $\mathcal{S}\approx 0.8635$.

For the stochastic secret sharing protocol, we prepared the maximally
entangled state $|\phi^{+}\rangle$ and isotropic states $\rho_{v}$ in a
similar way (recalling that $|\phi^{+}\rangle=|\phi^{+}_{\pi/4}\rangle$). In
this case the mixing with white noise, obtained by randomly transforming, with
probability $1-v$, $|\phi^{+}\rangle$ into one of the four Bell states
$|\phi^{\pm}\rangle,|\psi^{\pm}\rangle$ is simplified, and was realised by a
motorised rotation of one half wave plate (HWP) and a phase shifter (PS)
placed on the mode (a); see Fig. 2. Bob’s and Charlie’s transformations remain
the same as in the deterministic secret sharing protocol.

The polarisation measurements are performed using HWP and polarising beam
splitters (PBS), beam splitters (BS) placed at the two output modes of the
PBS, and then by single photon detectors (actively quenched Si-avalanche
photodiodes). The partial Bell analyser (Fig. 2, Box II) is implemented
through two-photon interference, using PBS and HWPs set at $22.5^{\circ}$. The
two-photon Hong–Ou–Mandel dip visibility is $99.2\%\pm 0.6$, where the
substantial statistical error is due to the low two-photon coincidence rate
used in the experiment (one per second) and a measurement time of 2400 seconds
per point (see Supplemental Material). To switch from a Bell measurement
discriminating between $\\{\Psi^{+},\Psi^{-},\Phi^{+}+\Phi^{-}\\}$ to
$\\{\Phi^{-},\Psi^{-},\Phi^{+}+\Psi^{+}\\}$, Alice uses three HWPs placed
before the first PBS (see Supplemental Material for the HWP settings). All
single-detection events were registered using a VHDL-programmed multichannel
coincidence logic unit, with a time coincidence window of $1.7$ ns.

To perform state tomography of isotropic states $\rho_{v}$ for different $v$,
we performed tomography of the four Bell states generated in the randomisation
procedure. Measurements were made at a rate of one two-coincidence per second
over 1400 seconds for each of the nine settings needed to perform the state
tomography for each Bell state. These results were then recombined _a
posteriori_ at different ratios to establish the density matrices $\rho_{v}$.
We considered the reconstructed states $\rho_{v}$ for $v=0.4$ to $v=0.5$, with
a step-size of $0.01$, corresponding to the resourceful but unsteerable range.
Naturally, the reconstructed density matrices are not exactly isotropic
states. To ensure the unsteerability of the experimentally realised states, we
used the linear programming method of Ref. [36] which allows one to obtain a
certificate of unsteerability for an arbitrary two-qubit state. We chose to
proceed during the experiment with $v=0.47$, as this represented a good
balance between being below the steering threshold of
$v_{\text{unsteer}}=\frac{1}{2}$ while allowing for good enough statistics to
show a significant quantum advantage. The fidelity of the reconstructed state
with the target isotropic state for $v=0.47$ is $0.9983\pm 0.0004$. Detailed
tomography results for the density matrices, the state fidelities, and the
certificates of unsteerability are presented in the Supplemental Material.

The protocol was then carried out with the isotropic state at $v=0.47$ with
the noise added by randomly changing the Bell state between each two-photon
coincidence while maintaining the necessary ratio between these states. To
obtain at most one event per change of Bell state and thus ensure the
randomness and therefore unpredictability of each event, we chose to work at a
rate one two-photon detection coincidence per second. The effective
measurement time per setting was $2.8$ hours. We obtained a success
probability of $\mathcal{R}=0.6524\pm 0.0004$ which significantly goes beyond
the theoretical entanglement-unassisted limit of
$\mathcal{R}_{\text{qubit}}\leq\frac{5}{8}$, hence showing an advantage from
unsteerable states.

The same experiment was also performed for a maximally entangled state
$|\phi^{+}\rangle$, i.e. without any randomisation over Bell states. This
increased the rate to 800 two-photon detection coincidences per second, with a
measurement time per setting of 2 hours. The fidelity of the state was
measured to be $0.9947\pm 0.0009$. The observed success probability is
$\mathcal{R}=0.9748\pm 0.0001$, which is close to the ideal maximum value of
$\mathcal{R}=1$, thus showing a large advantage due to entanglement.

Discussion.—In this letter we demonstrated theoretically, and confirmed
experimentally, that one can obtain quantum communication advantages in a
secret sharing task using weakly entangled unsteerable states while
restricting only to product measurements or entangled measurements compatible
with passive linear optics. Figure 3 summarises the ranges of noise $v$ for
which quantum communication advantages are achievable with two-qubit isotropic
states and the relation to its nonlocal properties. It remains an open
question whether the visibilities can be further reduced both for standard
linear optics measurements and product measurements. Of particular relevance
is to investigate whether product measurements also can generate advantages
from highly noisy multi-qubit states, as this would pave the way for
experiments that go beyond small-scale demonstrations.

Figure 3: Nonlocal properties (red) and quantum communication advantages
(blue) of the two-qubit isotropic state. The range for quantum communication
advantages from general entangled measurements follows from dense coding-like
protocols [9, 10]. The ranges shown for quantum communication advantages with
passive linear optics measurements and product measurements are established in
this work and are upper bounds on the critical visibilities.

###### Acknowledgements.

A.A.A. and A.T. thank Anthony Martin and Alek Lagarrigue for discussions on an
early iteration of the task presented here. This work supported by the Swedish
research council, the Wenner-Gren Foundation and by the Knut and Alice
Wallenberg Foundation through the Wallenberg Center for Quantum Technology
(WACQT).

## References

  * Cleve and Buhrman [1997] R. Cleve and H. Buhrman, Substituting quantum entanglement for communication, Phys. Rev. A 56, 1201 (1997), arXiv:quant-ph/9704026 .
  * Brukner  _et al._ [2004] Č. Brukner, M. Żukowski, J.-W. Pan, and A. Zeilinger, Bell’s inequalities and quantum communication complexity, Phys. Rev. Lett. 92, 127901 (2004), arXiv:quant-ph/0210114 .
  * Cubitt  _et al._ [2010] T. S. Cubitt, D. Leung, W. Matthews, and A. Winter, Improving zero-error classical communication with entanglement, Phys. Rev. Lett. 104, 230503 (2010), arXiv:arXiv:0911.5300 [quant-ph] .
  * Brunner and Linden [2013] N. Brunner and N. Linden, Connection between Bell nonlocality and Bayesian game theory, Nat Commun 4, 2057 (2013), arXiv:1210.1173 [quant-ph] .
  * Tavakoli and Żukowski [2017] A. Tavakoli and M. Żukowski, Higher-dimensional communication complexity problems: Classical protocols versus quantum ones based on Bell’s theorem or prepare-transmit-measure schemes, Phys. Rev. A 95, 042305 (2017), arXiv:1611.00977 [quant-ph] .
  * Tavakoli  _et al._ [2020] A. Tavakoli, M. Żukowski, and Č. Brukner, Does violation of a Bell inequality always imply quantum advantage in a communication complexity problem?, Quantum 4, 316 (2020), arXiv:1907.01322 [quant-ph] .
  * Pauwels  _et al._ [2022] J. Pauwels, S. Pironio, E. Z. Cruzeiro, and A. Tavakoli, Adaptive advantage in entanglement-assisted communications, Phys. Rev. Lett. 129, 120504 (2022), arXiv:2203.05372 [quant-ph] .
  * Bennett and Wiesner [1992] C. H. Bennett and S. J. Wiesner, Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states, Phys. Rev. Lett. 69, 2881 (1992).
  * Tavakoli  _et al._ [2018] A. Tavakoli, A. A. Abbott, M.-O. Renou, N. Gisin, and N. Brunner, Semi-device-independent characterisation of multipartite entangled states and measurements, Phys. Rev. A 98, 052333 (2018), arXiv:1805.00377 [quant-ph] .
  * Moreno  _et al._ [2021] G. Moreno, R. Nery, C. de Gois, R. Rabelo, and R. Chaves, Semi-device-independent certification of entanglement in superdense coding, Phys. Rev. A 103, 022426 (2021), arXiv:2102.02709 [quant-ph] .
  * Renner [2023] M. J. Renner, Compatibility of all noisy qubit observables (2023), arXiv:2309.12290 [quant-ph] .
  * Zhang and Chitambar [2023] Y. Zhang and E. Chitambar, Exact steering bound for two-qubit werner states (2023), arXiv:2309.09960 [quant-ph] .
  * Designolle  _et al._ [2023] S. Designolle, G. Iommazzo, M. Besançon, S. Knebel, P. Gelß, and S. Pokutta, Improved local models and new Bell inequalities via Frank-Wolfe algorithms, Phys. Rev. Research 5, 043059 (2023), arXiv:2302.04721 [quant-ph] .
  * Lütkenhaus  _et al._ [1999] N. Lütkenhaus, J. Calsamiglia, and K.-A. Suominen, Bell measurements for teleportation, Phys. Rev. A 59, 3295 (1999), arXiv:quant-ph/9809063 .
  * Mattle  _et al._ [1996] K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, Dense coding in experimental quantum communication, Phys. Rev. Lett. 76, 4656 (1996).
  * Li  _et al._ [2002] X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, Quantum dense coding exploiting a bright Einstein-Podolsky-Rosen beam, Phys. Rev. Lett. 88, 047904 (2002), arXiv:quant-ph/0107068 .
  * Barreiro  _et al._ [2008] J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, Beating the channel capacity limit for linear photonic superdense coding, Nature Phys. 4, 282 (2008), arXiv:1009.5128 [quant-ph] .
  * Williams  _et al._ [2017] B. P. Williams, R. J. Sadlier, and T. S. Humble, Superdense coding over optical fiber links with complete Bell-state measurements, Phys. Rev. Lett. 118, 050501 (2017), arXiv:1609.00713 [quant-ph] .
  * Hu  _et al._ [2018] X.-M. Hu, Y. Guo, B.-H. Liu, Y.-F. Huang, C.-F. Li, and G.-C. Guo, Beating the channel capacity limit for superdense coding with entangled ququarts, Science Advances 4, eaat9304 (2018), arXiv:1807.10452 [quant-ph] .
  * Piveteau  _et al._ [2022] A. Piveteau, J. Pauwels, E. Håkansson, S. Muhammad, M. Bourennane, and A. Tavakoli, Entanglement-assisted quantum communication with simple measurements, Nature Commun. 13, 7878 (2022), arXiv:2205.09602 [quant-ph] .
  * Blakley [1979] G. R. Blakley, Safeguarding cryptographic keys, in  _Proceedings of the 1979 AFIPS National Computer Conference_ (AFIPS Press, Monval, NJ, USA, 1979) pp. 313–317.
  * Shamir [1979] A. Shamir, How to share a secret, Commun. ACM 22, 612 (1979).
  * Hillery  _et al._ [1999] M. Hillery, V. Bužek, and A. Berthiaume, Quantum secret sharing, Phys. Rev. A 59, 1829 (1999), arXiv:quant-ph/9806063 .
  * Karlsson  _et al._ [1999] A. Karlsson, M. Koashi, and N. Imoto, Quantum entanglement for secret sharing and secret splitting, Phys. Rev. A 59, 162 (1999).
  * Guo and Guo [2003] G.-P. Guo and G.-C. Guo, Quantum secret sharing without entanglement, Phys. Lett. A 310, 247 (2003), arXiv:quant-ph/0212056 .
  * Tavakoli  _et al._ [2015] A. Tavakoli, I. Herbauts, M. Żukowski, and M. Bourennane, Secret sharing with a single $d$-level quantum system, Phys. Rev. A 92, 030302 (2015), arXiv:1501.05582 [quant-ph] .
  * Karimipour and Asoudeh [2015] V. Karimipour and M. Asoudeh, Quantum secret sharing and random hopping: Using single states instead of entanglement, Phys. Rev. A 92, 030301 (2015), arXiv:1506.02966 [quant-ph] .
  * Weinfurter [1994] H. Weinfurter, Experimental Bell-state analysis, Europhys. Lett. 25, 559 (1994).
  * Braunstein and Mann [1995] S. L. Braunstein and A. Mann, Measurement of the Bell operator and quantum teleportation, Phys. Rev. A 51, R1727 (1995).
  * Navascués and Vértesi [2015] M. Navascués and T. Vértesi, Bounding the set of finite dimensional quantum correlations, Phys. Rev. Lett. 115, 020501 (2015), arXiv:1412.0924 [quant-ph] .
  * Note [1] Specifically, we used what would, in standard terminology, be called the “1+AB+AC+ABC” level, leading to a moment matrix of size $109$.
  * Fillettaz  _et al._ [2018] M. Fillettaz, F. Hirsch, S. Designolle, and N. Brunner, Algorithmic construction of local models for entangled quantum states: Optimization for two-qubit states, Phys. Rev. A 98, 022115 (2018).
  * Calsamiglia and Lütkenhaus [2001] J. Calsamiglia and N. Lütkenhaus, Maximum efficiency of a linear-optical Bell-state analyzer, Appl. Phys. B 72, 67 (2001), arXiv:quant-ph/0007058 .
  * Note [2] Note that $\mathcal{R}$ is invariant under i) simultaneous bit-flip of $x_{0}$ and $y_{0}$, ii) simultaneous bit-flips of $x_{1}$ and $y_{1}$ and iii) simultaneous swaps $x_{0}\leftrightarrow x_{1}$ and $y_{0}\leftrightarrow y_{1}$.
  * Tavakoli  _et al._ [2019] A. Tavakoli, D. Rosset, and M.-O. Renou, Enabling computation of correlation bounds for finite-dimensional quantum systems via symmetrization, Phys. Rev. Lett. 122, 070501 (2019), arXiv:1808.02412 [quant-ph] .
  * Nguyen  _et al._ [2019] H. C. Nguyen, H.-V. Nguyen, and O. Gühne, Geometry of Einstein-Podolsky-Rosen correlations, Phys. Rev. Lett. 122, 240401 (2019), arXiv:1808.09349 [quant-ph] .
  * Wiseman  _et al._ [2007] H. M. Wiseman, S. J. Jones, and A. C. Doherty, Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox, Phys. Rev. Lett. 98, 140402 (2007), arXiv:quant-ph/0612147 .
  * Anwer  _et al._ [2020] H. Anwer, S. Muhammad, W. Cherifi, N. Miklin, A. Tavakoli, and M. Bourennane, Experimental characterization of unsharp qubit observables and sequential measurement incompatibility via quantum random access codes, Phys. Rev. Lett. 125, 080403 (2020), arXiv:2001.04768 [quant-ph] .
  * Hong  _et al._ [1987] C. K. Hong, Z. Y. Ou, and L. Mandel, Measurement of subpicosecond time intervals between two photons by interference, Phys. Rev. Lett. 59, 2044 (1987).

Supplemental Material for:
Weak entanglement improves quantum communication using only product
measurements

Here we provide additional details on the theoretical analysis of the
unsteerability of the states, and on the experimental setup including details
about the state preparation, the parties’ unitary operations and how they are
implemented, experimental error estimation, and two-fold Hong-Ou-Mandel dip
visibility.

## Supplementary note 1: Unsteerability of experimental states

Recall that the isotropic state $\rho_{v}$ is defined as

$\rho_{v}=v\Phi^{+}+\frac{(1-v)}{4}\openone,$ (4)

where $\Phi^{+}=|\phi^{+}\rangle\\!\langle\phi^{+}|$. It is well known that
$\rho_{v}$ is unsteerable for $v\leq\frac{1}{2}$ and entangled for
$v>\frac{1}{3}$ [37]. The states prepared experimentally, however, are
necessarily not exactly the isotropic states. To exhibit an advantage in the
stochastic secret sharing task from entangled but unsteerable states, it is
necessary to verify that the experimentally prepared states, as reconstructed
via state tomography, are indeed unsteerable. The same is true for the noisy
partially entangled states $\rho_{v}^{\theta}$ defined as

$\rho_{v}^{\theta}=v|\phi^{+}_{\theta}\rangle\\!\langle\phi^{+}_{\theta}|+\frac{(1-v)}{4}\openone,$
(5)

where $|\phi^{+}_{\theta}\rangle=\cos\theta|00\rangle+\sin\theta|11\rangle$
used to show an advantage in the deterministic secret sharing protocol from
unsteerable states.

While two-qubit entanglement can easily be verified by calculating its
negativity, verifying unsteerability is more challenging. Here, we use a
recently developed method [36] that allows one to obtain certificates of
unsteerability via linear programming.

In particular, the approach of Ref. [36] computes, for an arbitrary two-qubit
state and a given direction of steering (say, from Bob to Charlie or Charlie
to Bob), both upper and lower bounds, $r_{\text{crit}}^{\text{upper}}$ and
$r_{\text{crit}}^{\text{lower}}$, respectively, on a “critical radius”
$r_{\text{crit}}$ satisfying
$r_{\text{crit}}^{\text{lower}}<r_{\text{crit}}<r_{\text{crit}}^{\text{upper}}$.
They show that a two-qubit state $\rho$ is steerable (in a given direction) if
and only if $r_{\text{crit}}<1$. Thus, by finding
$r_{\text{crit}}^{\text{lower}}\geq 1$ for both possible directions of
steering, we are guaranteed a state is unsteerable.

The precision of the upper and lower bounds can be increased at the cost of
computational time and memory. On a standard desktop computer, using the CPLEX
packaged provided by Ref. [36], we were typically able to obtain, for the
experimental states approximating $\rho_{v}$, a gap
$r_{\text{crit}}^{\text{upper}}-r_{\text{crit}}^{\text{lower}}\approx 0.02$
with a few minutes of computation. This can be reduced by an order of
magnitude by allocating somewhat more computational time.

For the tomographically reconstructed density matrices $\tilde{\rho}_{v}$, for
$v=0.4$ to $v=0.5$ (with a step size of $0.1$), we computed
$r_{\text{crit}}^{\text{lower}}$ and found the states to be unsteerable for
all values except $v=0.5$. However, to ensure that this result is robust
(recalling, of course, the experimental error in the description of
$\tilde{\rho}_{v}$) and ensure we choose a state providing a large enough
quantum advantage (which diminishes as $v$ decreases), we chose to perform the
experiment with $v=0.47$.

For this state, we performed a finer analysis of the unsteerability of
$\tilde{\rho}_{0.47}$, taking into account the errors in each term of the
density matrix. Adopting a conservative approach, we applied the worst case
errors to all subsets of density matrix elements (up to Hermiticity) before
calculating $r_{\text{crit}}^{\text{lower}}$ for the (renormalised) perturbed
density matrix. We found that, in all cases,
$r_{\text{crit}}^{\text{lower}}\gtrsim 1.031$ (while for the reconstructed
state $\tilde{\rho}_{0.47}$ we found $r_{\text{crit}}^{\text{lower}}\approx
1.055$), showing that the prepared state is clearly unsteerable, even when
errors are taken into account.

We likewise performed a similar analysis on the tomographically reconstructed
density matrix $\tilde{\rho}_{v}^{\theta_{*}}$ for $\theta_{*}=0.2356$ and
$v=0.72$. By applying worst case errors in the same manner we found that, in
all cases, $r^{\text{lower}}_{\text{crit}}\gtrsim 1.020$, while for the
reconstructed state $\tilde{\rho}_{v}^{\theta_{*}}$ itself we found
$r^{\text{lower}}_{\text{crit}}\approx 1.031$. As for the isotropic state, we
thus see that the experimentally prepared state is clearly unsteerable, even
when errors are taken into account.

## Supplementary note 2: State preparation, transformations and measurements
for the deterministic secret sharing experiment

### A. State implementation

The setup used generates an entangled
$|\phi^{+}_{\theta}\rangle=\cos\theta|00\rangle+\sin\theta|11\rangle$ state
with a visibility close to 100%. To create this state, we use an HWP to change
the power distribution between the H and V crystals. When the angle of this
HWP is 0, only the H-polarised crystal is pumped and when the HWP is
$\frac{\pi}{2}$, only the V-polarised crystal is pumped. When the angle is
$\frac{\pi}{4}$, the prepared state is $|\phi^{+}\rangle$. We have to select
the right angle ($\theta$) value to prepare the desired state. To create the
maximally mixed state $\frac{\openone}{4}$ we randomly switch between
$|\phi^{+}_{\theta}\rangle$,
$|\phi_{\theta}^{-}\rangle=\sin\theta|00\rangle-\cos\theta|11\rangle$,
$|\psi_{\theta}^{+}\rangle=\cos\theta|01\rangle+\sin\theta|10\rangle$ and
$|\psi_{\theta}^{-}\rangle=\sin\theta|01\rangle-\cos\theta|10\rangle$. Indeed,
these states form an orthonormal basis.

We can thereby generate $\rho_{v}^{\theta}$ for any $v$ by choosing correctly
the weights with which we randomly change $|\phi^{+}_{\theta}\rangle$ to one
of the other states. The switching between these different states is done with
the help of two half-wave plates and a phase plate; see Fig. 2 of the main
text.

These plates are motorised and switched randomly (with the desired
probability) from one state to another every second. The number of
coincidences being less than one per second, the result is to switch from one
state to another randomly and independently between each pair of successive
coincidences. The values of the angles allowing one to pass from one state to
another are given in Table 1.

Table 1:  Half wave plate rotation angles (in degrees) and phase for the preparation of the differents states. State | $\frac{\lambda_{b}}{2}$ | $\frac{\lambda_{c}}{2}$ | Phase plate
---|---|---|---
$|\phi_{\theta}^{+}\rangle$ | 0 | 0 | 0
$|\phi_{\theta}^{-}\rangle$ | 45 | 45 | $\Pi$
$|\psi_{\theta}^{+}\rangle$ | 0 | 45 | 0
$|\psi_{\theta}^{-}\rangle$ | 45 | 0 | $\Pi$

### B. Bob’s and Charlie’s rotations

As described in the main text, on inputs $x=(x_{0},x_{1})\in\\{0,1\\}^{2}$ and
$y=(y_{0},y_{1})\in\\{0,1\\}^{2}$, Bob and Charlie apply the unitaries
$U^{B}_{x}=\sigma_{X}^{y_{0}}\sigma_{Z}^{y_{1}}$ and
$U^{C}_{y}=\sigma_{X}^{y_{0}}\sigma_{Z}^{y_{1}}$, respectively. Table 2
summarises explicitly the operations that Bob and Charlie must perform in
inputs $x$ and $y$, as well as the Bell state expected to be received by Alice
in the ideal case (assuming Bob and Charlie share a $\Phi^{+}$ state). The
settings to be applied to achieve each of the four different possible
rotations are given in Table 3

Table 2:  Transformation table: the operations $\sigma^{x_{0}}_{X}\sigma^{x_{1}}_{Z}$ and $\sigma^{y_{0}}_{X}\sigma^{y_{1}}_{Z}$ that need to be performed by Bob and Charlie, respectively. The expected Bell state that Alice will receive in the ideal case that Bob and Charlie share a noiseless maximally entangled state is shown in the final column. $x_{0}$ | $x_{1}$ | $y_{0}$ | $y_{1}$ | Bob | Charlie | Alice’s expected state
---|---|---|---|---|---|---
0 | 0 | 0 | 0 | $\mathds{1}$ | $\mathds{1}$ | $\Phi^{+}$
0 | 0 | 1 | 0 | $\mathds{1}$ | $\sigma_{X}$ | $\Psi^{+}$
0 | 0 | 1 | 1 | $\mathds{1}$ | $-i\sigma_{Y}$ | $\Psi^{-}$
0 | 0 | 0 | 1 | $\mathds{1}$ | $\sigma_{Z}$ | $\Psi^{-}$
1 | 0 | 0 | 0 | $\sigma_{X}$ | $\mathds{1}$ | $\Psi^{-}$
1 | 0 | 1 | 0 | $\sigma_{X}$ | $\sigma_{X}$ | $\Phi^{+}$
1 | 0 | 1 | 1 | $\sigma_{X}$ | $-i\sigma_{Y}$ | $\Phi^{-}$
1 | 0 | 0 | 1 | $\sigma_{X}$ | $\sigma_{Z}$ | $\Psi^{-}$
1 | 1 | 0 | 0 | $-i\sigma_{Y}$ | $\mathds{1}$ | $\Psi^{-}$
1 | 1 | 1 | 0 | $-i\sigma_{Y}$ | $\sigma_{X}$ | $\Psi^{+}$
1 | 1 | 1 | 1 | $-i\sigma_{Y}$ | $-i\sigma_{Y}$ | $\Phi^{+}$
1 | 1 | 0 | 1 | $-i\sigma_{Y}$ | $\sigma_{Z}$ | $\Psi_{+}$
0 | 1 | 0 | 0 | $\sigma_{Z}$ | $\mathds{1}$ | $\Phi^{-}$
0 | 1 | 1 | 0 | $\sigma_{Z}$ | $\sigma_{X}$ | $\Psi_{-}$
0 | 1 | 1 | 1 | $\sigma_{Z}$ | $-i\sigma_{Y}$ | $\Psi_{+}$
0 | 1 | 0 | 1 | $\sigma_{Z}$ | $\sigma_{Z}$ | $\Phi_{+}$
Table 3: Rotation angles (in degrees) for the unitaries for Bob’s and Charlie’s operations. Unitary | Phase plate | HWP
---|---|---
$\mathds{1}$ | 0 | $\Pi$
$\sigma_{X}$ | 45 | 0
$-i\sigma_{Y}$ | 0 | 0
$\sigma_{Z}$ | 45 | $\Pi$

### C. Alice’s measurements

Depending of her setting $z\in\\{0,1\\}$, Alice performs a projective
measurement in the diagonal or in the $H\text{-}V$ basis. To make these
measurements, she uses two HWPs $\frac{\lambda_{A1}}{2}$ and
$\frac{\lambda_{A2}}{2}$ placed before the input ports of each PBS (see Fig. 2
of the main text). The values of the angles of these HWPs are given in Table
9.

Table 4: Rotation angles for the HWPs used to implement Alice’s parojective measurements, for $z=0$ and $z=1$. $z$ | $\frac{\lambda_{A1}}{2}$ | $\frac{\lambda_{A2}}{2}$
---|---|---
0 | 0 | 0
1 | 22.5 | 22.5

## Supplementary note 3: Experimental results and error estimation for the
deterministic secret sharing experiment

Tables 5 and 6 list our experimental results alongside the theoretical success
rates $\mathcal{S}$.

Table 5:  Theoretical and experimental success probabilities for the partially entangled state $|\phi^{+}_{\theta_{*}}\rangle$ (with $\theta_{*}=0.2356$), conditioned on $z$ and then averaged to give the overall success probability $\mathcal{S}$. Parameter | Theoretical | Experimental | error
---|---|---|---
$S^{[z=0]}$ | 0.5 | 0.4975 | 0.0005
$S^{[z=1]}$ | 0.3635 | 0.35440 | 0.00009
$\mathcal{S}$ | 0.8635 | 0.8519 | 0.0002
Table 6:  Theoretical and experimental success probabilities for $\rho_{v}^{\theta_{*}}$ with $v=0.72$, conditioned on $z$ and then averaged to give the overall success probability $\mathcal{S}$. Parameter | Theoretical | Experimental | error
---|---|---|---
$S^{[z=0]}$ | 0.43 | 0.43 | 0.001
$S^{[z=1]}$ | 0.3317 | 0.327 | 0.002
$\mathcal{S}$ | 0.7617 | 0.757 | 0.001

The errors were calculated using the same method as for the stochastic secret
sharing experiment.

## Supplementary note 4: Density matrix tomography for the deterministic
secret sharing experiment

To ensure that the prepared state is the desired one, a tomography measurement
is performed for a perfect $|\phi^{+}_{\theta_{*}}\rangle$ state and for the
noisy states $\rho_{v}^{\theta_{*}}$ with a visibility of $v=0.72$ (and with
$\theta_{*}=0.2356$), which are the two states for which the experimental
demonstration of the protocol was carried out. The reconstructed density
matrices are shown in Figures 4 and 5, respectively.

Figure 4: State tomography of $\rho_{v}^{\theta_{*}}$ for $v=1$; the real part
of the density matrix is shown on the left, and the imaginary part on the
right.

Figure 5: State tomography of $\rho_{v}^{\theta_{*}}$ for $v=0.72$; the real
part of the density matrix is shown on the left, and the imaginary part on the
right.

The fidelity of each state obtained during the tomography with the ideal state
$\rho_{v}^{\theta_{*}}$ was found to be above $0.99$. Table 7 gives the
fidelity for each of these states.

Table 7:  Fidelity of measured states with the ideal state $\rho_{v}^{\theta_{*}}$ for $v=0.72$ and $v=1$. Visibility $v$ | Fidelity with $\rho_{v}^{\theta_{*}}$ | error
---|---|---
0.72 | 0.9965 | 0.0007
1.00 | 0.990 | 0.001

## Supplementary note 5: State preparation, transformations and measurements
for the stochastic secret sharing experiment

### A. State implementation

The setup used generates an entangled $\Phi^{+}$ state with a visibility close
to 100%. In order to generate an isotropic state $\rho_{v}$
$\eqref{eq:isotropic}$, this state needs to be mixed with probability $(1-v)$
with the maximally mixed state $\frac{\openone}{4}$. We implement the
maximally mixed state by randomly switching between the four Bell states,
$\Phi^{\pm},\Psi^{\pm}$, exploiting the fact that
$\frac{\openone}{4}=\frac{1}{4}(\Phi^{+}+\Phi^{-}+\Psi^{+}+\Psi^{-})$. We can
thereby generate $\rho_{v}$ for any $v$ by choosing correctly the weights with
which we randomly change $\Phi^{+}$ to one of the other Bell states.

The switching between these different states is done with the help of one
half-wave plates and a phase plate; see Fig. 2 of the main text.

These plates are motorised and switched randomly (with the desired
probability) from one state to another every second. The number of
coincidences being less than one per second, the result is to switch from one
Bell state to another randomly and independently between each pair of
successive coincidences. The values of the angles allowing one to pass from
one state to another are given in Table 8.

Table 8:  Half wave plate rotation angles (in ∘) and phase for the preparation of the different Bell states. State | $\frac{\lambda}{2}$ | Phase plate
---|---|---
$\Phi^{+}$ | 0 | 0
$\Phi^{-}$ | 0 | $\Pi$
$\Psi^{+}$ | 45 | 0
$\Psi^{-}$ | 45 | $\Pi$

### B. Bob’s and Charlie’s rotations

Bob’s and Charlie’s rotations are the same as in the stochastic secret sharing
experiment; see Section 2.B above.

### C. Alice’s measurements

Depending of her setting $z\in\\{0,1\\}$, Alice performs one of the three-
outcome partial Bell measurement $\\{\Psi^{+},\Psi^{-},\Phi^{+}+\Phi^{-}\\}$
or $\\{\Phi^{-},\Psi^{-},\Phi^{+}+\Psi^{+}\\}$ and obtains an outcome
$c\in\\{0,1,\perp\\}$. To make these measurements, she uses three HWPs
$\frac{\lambda_{A1}}{2}$, $\frac{\lambda_{A2}}{2}$ and
$\frac{\lambda_{A3}}{2}$, with the first two ($A1$ and $A2$) placed before one
of the input ports of the PBS, and last one ($A3$) placed before the other
input port (see Fig. 2 of the main text). The values of the angles of these
HWPs are given in Table 9.

Table 9: Rotation angles for the HWPs used to implement Alice’s partial Bell state measurements, for $z=0$ and $z=1$. $z$ | $\frac{\lambda_{A1}}{2}$ | $\frac{\lambda_{A2}}{2}$ | $\frac{\lambda_{A1}}{2}$
---|---|---|---
0 | 0 | 45 | 0
1 | 0 | 22.5 | 22.5

## Supplementary note 6: Experimental results and error estimation for the
isotropic states $\rho_{v}$

Tables 10 and 11 list our experimental results alongside the theoretical
success rates $\mathcal{R}$ with the related errors.

Table 10:  Theoretical and experimental success rate for the maximally entangled state $\Phi^{+}$, conditioned on $z$ and then averaged to give the overall success rate $\mathcal{R}$. Parameter | Theoretical | Experimental | error
---|---|---|---
$R^{[z=0]}_{\text{ctrl}}$ | 1 | 0.9900 | 0.0002
$R^{[z=0]}_{\text{scrt}}$ | 1 | 0.9811 | 0.0003
$R^{[z=1]}_{\text{ctrl}}$ | 1 | 0.9795 | 0.0002
$R^{[z=1]}_{\text{scrt}}$ | 1 | 0.9485 | 0.0003
$\mathcal{R}$ | 1 | 0.9748 | 0.0001
Table 11:  Theoretical and experimental success rates for the isotropic state $\rho_{v}$ with $v=0.47$, conditioned on $z$ and then averaged to give the overall success rate $\mathcal{R}$. Parameter | Theoretical | Experimental | error
---|---|---|---
$R^{[z=0]}_{\text{ctrl}}$ | 0.735 | 0.7238 | 0.0007
$R^{[z=0]}_{\text{scrt}}$ | 0.6025 | 0.6086 | 0.0009
$R^{[z=1]}_{\text{ctrl}}$ | 0.735 | 0.6613 | 0.0007
$R^{[z=1]}_{\text{scrt}}$ | 0.603 | 0.6161 | 0.001
$\mathcal{R}$ | 0.66875 | 0.6524 | 0.0004

To calculate these errors, following [38] we consider errors originating from
the measurement side only. To reduce experimental errors in the measurements,
we used computer controlled high precision motorised rotation stages, for Bob,
Charlie and the generation of Werner states, to set the orientation of the
wave-plates with a repeatably high precision of $0.02^{\circ}$. The use of
different settings $(x,y)$ induces a systematic error, which we estimate using
Monte Carlo simulation. We assume that the wave plates’ setting errors are
normally distributed with a standard deviation of $0.02^{\circ}$. Alice’s wave
plates are not motorised. In order to reduce systematic errors in the angle
settings, all settings for $z=0$ are measured, then all settings for $z=1$, so
the results are initially presented separately for each case. We estimate the
accuracy of the angles of these WPs to be half a degree. This, together with
the Poissonian error in photon counting statistics, comprises the final error
reported here. Due to inefficiency in the single photon detectors, the photons
are detected randomly and their counting is Poissonian. To decrease Poissonian
counting error, we have chosen for $v=1$ a measurement time of two hours for
every setting and collected approximately 93 million events. To guarantee that
both parties receive single qubits, we worked at a low rate ($\approx 800$
coincidence per sec) to suppress higher order coincidences to almost $0.9$ per
sec. For $v=0.47$, we have collected around one two million two-photon
coincidences and the low rate renders the higher order coincidences completely
negligible.

## Supplementary note 7: Two-fold Hong-Ou-Mandel dip visibility

Bell state measurements are implemented through two-photon interference, using
PBS and HWP plates set at 22.5°. The photons are detected by Si avalanche
photodiodes and the coincidences are registered with an eight channel
multifold coincidence counting unit. This Bell analyser consists of coherent
interference at a polarising beam splitter (PBS). To achieve the necessary
indistinguishability of the photons, due to their arrival times, we adjusted
the path length of one of the photons using a delay line [39]. In Figure 6,
the coincidences between the detectors versus the delay path length is shown.
The zero delay corresponds to a maximal overlap (maximum
indistinguishability). The interfering photons bunch (i.e., they both exit in
the same output arm of the PBS) causing the coincidences to vanish. The
measured visibility of the two-fold Hong-Ou-Mandel dip is $99.2\%\pm 0.6$,
with a coincidence rate outside the dip around $0.9224$ coincidences per
second, identical to the rate at which we performed the experiment for
$v=0.47$.

Figure 6: Two-fold Hong-Ou-Mandel dip. The plot displays the two-fold photon
counting coincidences versus the delay (the path difference between the two
arms). The error bars indicate the Poissonian photon counting error
statistics. The data is fitted with a Gaussian curve.

## Supplementary note 8: Density matrix tomography for stochastic secret
sharing

To ensure that the prepared state is the desired one, a tomography measurement
is performed for a perfect $\Phi^{+}$ state and for isotropic states
$\rho_{v}$ with a visibility between $v=0.4$ and $v=0.5$. The reconstructed
density matrices obtained for the cases of $v=1$ (i.e., the ideal case of
$\rho_{1}=\Phi^{+}$) and $v=0.47$, which are the two states for which the
experimental demonstration of the protocol was carried out, are presented in
Figures 7 and 8, respectively.

Figure 7: State tomography of $\rho_{v}$ for $v=1$; the real part of the
density matrix is shown on the left, and the imaginary part on the right.

Figure 8: State tomography of $\rho_{v}$ for $v=0.47$; the real part of the
density matrix is shown on the left, and the imaginary part on the right.

The fidelity of each state obtained during the tomography with the ideal
isotropic state $\rho_{v}$ was found to be between $0.9947$ and $0.9983$.
Table 12 gives the fidelity for each of these states.

Table 12:  Fidelity of each measured state with the ideal isotropic state $\rho_{v}$. Visibility $v$ | Fidelity with $\rho_{v}$ | error
---|---|---
0.4 | 0.9982 | 0.0004
0.41 | 0.9982 | 0.0004
0.42 | 0.9983 | 0.0004
0.43 | 0.9983 | 0.0004
0.44 | 0.9983 | 0.0004
0.45 | 0.9983 | 0.0004
0.46 | 0.9983 | 0.0004
0.47 | 0.9983 | 0.0004
0.48 | 0.9983 | 0.0004
0.49 | 0.9983 | 0.0004
0.50 | 0.9983 | 0.0004
1.00 | 0.9947 | 0.0009

## References

  * Cleve and Buhrman [1997] R. Cleve and H. Buhrman, Substituting quantum entanglement for communication, Phys. Rev. A 56, 1201 (1997), arXiv:quant-ph/9704026 .
  * Brukner  _et al._ [2004] Č. Brukner, M. Żukowski, J.-W. Pan, and A. Zeilinger, Bell’s inequalities and quantum communication complexity, Phys. Rev. Lett. 92, 127901 (2004), arXiv:quant-ph/0210114 .
  * Cubitt  _et al._ [2010] T. S. Cubitt, D. Leung, W. Matthews, and A. Winter, Improving zero-error classical communication with entanglement, Phys. Rev. Lett. 104, 230503 (2010), arXiv:arXiv:0911.5300 [quant-ph] .
  * Brunner and Linden [2013] N. Brunner and N. Linden, Connection between Bell nonlocality and Bayesian game theory, Nat Commun 4, 2057 (2013), arXiv:1210.1173 [quant-ph] .
  * Tavakoli and Żukowski [2017] A. Tavakoli and M. Żukowski, Higher-dimensional communication complexity problems: Classical protocols versus quantum ones based on Bell’s theorem or prepare-transmit-measure schemes, Phys. Rev. A 95, 042305 (2017), arXiv:1611.00977 [quant-ph] .
  * Tavakoli  _et al._ [2020] A. Tavakoli, M. Żukowski, and Č. Brukner, Does violation of a Bell inequality always imply quantum advantage in a communication complexity problem?, Quantum 4, 316 (2020), arXiv:1907.01322 [quant-ph] .
  * Pauwels  _et al._ [2022] J. Pauwels, S. Pironio, E. Z. Cruzeiro, and A. Tavakoli, Adaptive advantage in entanglement-assisted communications, Phys. Rev. Lett. 129, 120504 (2022), arXiv:2203.05372 [quant-ph] .
  * Bennett and Wiesner [1992] C. H. Bennett and S. J. Wiesner, Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states, Phys. Rev. Lett. 69, 2881 (1992).
  * Tavakoli  _et al._ [2018] A. Tavakoli, A. A. Abbott, M.-O. Renou, N. Gisin, and N. Brunner, Semi-device-independent characterisation of multipartite entangled states and measurements, Phys. Rev. A 98, 052333 (2018), arXiv:1805.00377 [quant-ph] .
  * Moreno  _et al._ [2021] G. Moreno, R. Nery, C. de Gois, R. Rabelo, and R. Chaves, Semi-device-independent certification of entanglement in superdense coding, Phys. Rev. A 103, 022426 (2021), arXiv:2102.02709 [quant-ph] .
  * Renner [2023] M. J. Renner, Compatibility of all noisy qubit observables (2023), arXiv:2309.12290 [quant-ph] .
  * Zhang and Chitambar [2023] Y. Zhang and E. Chitambar, Exact steering bound for two-qubit werner states (2023), arXiv:2309.09960 [quant-ph] .
  * Designolle  _et al._ [2023] S. Designolle, G. Iommazzo, M. Besançon, S. Knebel, P. Gelß, and S. Pokutta, Improved local models and new Bell inequalities via Frank-Wolfe algorithms, Phys. Rev. Research 5, 043059 (2023), arXiv:2302.04721 [quant-ph] .
  * Lütkenhaus  _et al._ [1999] N. Lütkenhaus, J. Calsamiglia, and K.-A. Suominen, Bell measurements for teleportation, Phys. Rev. A 59, 3295 (1999), arXiv:quant-ph/9809063 .
  * Mattle  _et al._ [1996] K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, Dense coding in experimental quantum communication, Phys. Rev. Lett. 76, 4656 (1996).
  * Li  _et al._ [2002] X. Li, Q. Pan, J. Jing, J. Zhang, C. Xie, and K. Peng, Quantum dense coding exploiting a bright Einstein-Podolsky-Rosen beam, Phys. Rev. Lett. 88, 047904 (2002), arXiv:quant-ph/0107068 .
  * Barreiro  _et al._ [2008] J. T. Barreiro, T.-C. Wei, and P. G. Kwiat, Beating the channel capacity limit for linear photonic superdense coding, Nature Phys. 4, 282 (2008), arXiv:1009.5128 [quant-ph] .
  * Williams  _et al._ [2017] B. P. Williams, R. J. Sadlier, and T. S. Humble, Superdense coding over optical fiber links with complete Bell-state measurements, Phys. Rev. Lett. 118, 050501 (2017), arXiv:1609.00713 [quant-ph] .
  * Hu  _et al._ [2018] X.-M. Hu, Y. Guo, B.-H. Liu, Y.-F. Huang, C.-F. Li, and G.-C. Guo, Beating the channel capacity limit for superdense coding with entangled ququarts, Science Advances 4, eaat9304 (2018), arXiv:1807.10452 [quant-ph] .
  * Piveteau  _et al._ [2022] A. Piveteau, J. Pauwels, E. Håkansson, S. Muhammad, M. Bourennane, and A. Tavakoli, Entanglement-assisted quantum communication with simple measurements, Nature Commun. 13, 7878 (2022), arXiv:2205.09602 [quant-ph] .
  * Blakley [1979] G. R. Blakley, Safeguarding cryptographic keys, in  _Proceedings of the 1979 AFIPS National Computer Conference_ (AFIPS Press, Monval, NJ, USA, 1979) pp. 313–317.
  * Shamir [1979] A. Shamir, How to share a secret, Commun. ACM 22, 612 (1979).
  * Hillery  _et al._ [1999] M. Hillery, V. Bužek, and A. Berthiaume, Quantum secret sharing, Phys. Rev. A 59, 1829 (1999), arXiv:quant-ph/9806063 .
  * Karlsson  _et al._ [1999] A. Karlsson, M. Koashi, and N. Imoto, Quantum entanglement for secret sharing and secret splitting, Phys. Rev. A 59, 162 (1999).
  * Guo and Guo [2003] G.-P. Guo and G.-C. Guo, Quantum secret sharing without entanglement, Phys. Lett. A 310, 247 (2003), arXiv:quant-ph/0212056 .
  * Tavakoli  _et al._ [2015] A. Tavakoli, I. Herbauts, M. Żukowski, and M. Bourennane, Secret sharing with a single $d$-level quantum system, Phys. Rev. A 92, 030302 (2015), arXiv:1501.05582 [quant-ph] .
  * Karimipour and Asoudeh [2015] V. Karimipour and M. Asoudeh, Quantum secret sharing and random hopping: Using single states instead of entanglement, Phys. Rev. A 92, 030301 (2015), arXiv:1506.02966 [quant-ph] .
  * Weinfurter [1994] H. Weinfurter, Experimental Bell-state analysis, Europhys. Lett. 25, 559 (1994).
  * Braunstein and Mann [1995] S. L. Braunstein and A. Mann, Measurement of the Bell operator and quantum teleportation, Phys. Rev. A 51, R1727 (1995).
  * Navascués and Vértesi [2015] M. Navascués and T. Vértesi, Bounding the set of finite dimensional quantum correlations, Phys. Rev. Lett. 115, 020501 (2015), arXiv:1412.0924 [quant-ph] .
  * Note [1] Specifically, we used what would, in standard terminology, be called the “1+AB+AC+ABC” level, leading to a moment matrix of size $109$.
  * Fillettaz  _et al._ [2018] M. Fillettaz, F. Hirsch, S. Designolle, and N. Brunner, Algorithmic construction of local models for entangled quantum states: Optimization for two-qubit states, Phys. Rev. A 98, 022115 (2018).
  * Calsamiglia and Lütkenhaus [2001] J. Calsamiglia and N. Lütkenhaus, Maximum efficiency of a linear-optical Bell-state analyzer, Appl. Phys. B 72, 67 (2001), arXiv:quant-ph/0007058 .
  * Note [2] Note that $\mathcal{R}$ is invariant under i) simultaneous bit-flip of $x_{0}$ and $y_{0}$, ii) simultaneous bit-flips of $x_{1}$ and $y_{1}$ and iii) simultaneous swaps $x_{0}\leftrightarrow x_{1}$ and $y_{0}\leftrightarrow y_{1}$.
  * Tavakoli  _et al._ [2019] A. Tavakoli, D. Rosset, and M.-O. Renou, Enabling computation of correlation bounds for finite-dimensional quantum systems via symmetrization, Phys. Rev. Lett. 122, 070501 (2019), arXiv:1808.02412 [quant-ph] .
  * Nguyen  _et al._ [2019] H. C. Nguyen, H.-V. Nguyen, and O. Gühne, Geometry of Einstein-Podolsky-Rosen correlations, Phys. Rev. Lett. 122, 240401 (2019), arXiv:1808.09349 [quant-ph] .
  * Wiseman  _et al._ [2007] H. M. Wiseman, S. J. Jones, and A. C. Doherty, Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox, Phys. Rev. Lett. 98, 140402 (2007), arXiv:quant-ph/0612147 .
  * Anwer  _et al._ [2020] H. Anwer, S. Muhammad, W. Cherifi, N. Miklin, A. Tavakoli, and M. Bourennane, Experimental characterization of unsharp qubit observables and sequential measurement incompatibility via quantum random access codes, Phys. Rev. Lett. 125, 080403 (2020), arXiv:2001.04768 [quant-ph] .
  * Hong  _et al._ [1987] C. K. Hong, Z. Y. Ou, and L. Mandel, Measurement of subpicosecond time intervals between two photons by interference, Phys. Rev. Lett. 59, 2044 (1987).