[1]Javier Coronado-Blázquez

[1]Telefónica Tech IoT & Big Data, Ronda de la Comunicación Oeste 1, Madrid,
28050, Spain

# A NLP Approach to “Review Bombing” in Metacritic PC Videogames User Ratings

0000-0001-9643-0125<EMAIL_ADDRESS>*

###### Abstract

Many videogames suffer “review bombing” –a large volume of unusually low
scores that in may cases do not reflect the real quality of the product– when
rated by users. By taking Metacritic’s 50,000+ user score aggregations for PC
games in English language, we use a Natural Language Processing (NLP) approach
to try to understand the main words and concepts appearing in such cases,
reaching a 0.88 accuracy on a validation set when distinguishing between just
bad ratings and review bombings. By uncovering and analyzing the patterns
driving this phenomenon, these results could be used to further mitigate these
situations.

###### keywords:

Machine Learning, Natural Language Processing

##  1 Introduction

Videogames are currently the leading entertainment industry product, exceeding
$193 billion global revenue in 2021, with an annual growth of about 15% [1].
From small indie to multi-million triple-A titles, from first-person shooters
to racing simulators and even almost unclassifiable experiences, the current
gaming ecosystem is rich and diverse.

As with streaming platforms, the amount of offer can be overwhelming for a
casual player. It is only natural that they carry out a little research on the
Internet to decide whether a game is worth playing to [2], especially when
most of triple-A titles are released at an average price of $70 [3].

Similarly to the Internet Movie Data Base (IMDb) and Rotten Tomatoes for
movies and TV shows, or GoodReads for books, Metacritic was born in 1999 as a
score aggregator for different entertainment products, including videogames.
For a given title, Metacritic offers an average score –the Metascore–
pondering professional critics from specialized media. Furthermore, any
registered user can vote, in a scale of 0 to 10, and leave an optional comment
as review [4, 5]111Although the Metascore ranges from 0 to 100, in the
remaining of the paper we will always refer to the Metascore divided by 10 to
equate it to the user score..

In recent years, Metacritic has been in the news for the so-called “review
bombing”, this is, titles where the average user score is anomalously low,
especially when compared to the Metascore [6], present a large volume of
those, and the user reviews usually lead towards derivative aspects of the
game instead of the title quality itself, such as studio marketing campaigns
or commercial strategy, change of story focus in sequels, or political and
cultural controversies such as feminist perspectives or LGTBIQ+ character
representation [7, 8].

Contrary to the expected user reviews, the review bombing tends to be a
coordinated action by many individuals in forums such as Reddit or 4chan [9],
often with multiple accounts, with the intention of doing as much harm as
possible to the game studio, trying to hurt its sales and commercial
performance [10, 11].

Indeed, empirical evidence proves that online ratings are not Gaussian-
distributed (as naively expected), but rather “J-shaped”, with a raise at very
low scores, indicating that upset users are more prone to leave a review, and
this is usually in the first $\mathrm{10^{th}}$ score percentile [12, 13, 14].

Many authors have utilized automated sentiment analysis to study product
reviews and extract insights from the customers’ patterns (see, e.g., [15, 16,
17, 18]). Additionally, there are several works devoted to the trustworthiness
of reviews, and the impact they may have on customers’ decisions [19, 20, 21,
22].

Nevertheless, videogame reviews have only been considered in marketplace
reviews (and therefore subject to artificial negative reviews due to problems
with the selling platform and/or the delivery process) [23, 24, 25, 26, 27],
but have not been studied in the context of score aggregators.

While Metacritic and other platforms have recognized the problem and started
to draw a plan to limit them (from a 48-hour moratorium since the game release
date to a weighted user score), deciding whether a user review is “valid” or
can be labeled as “review bombing” is currently a highly-subjective and manual
task [28]. For example, in many cases there are day-one bugs that makes a game
almost unplayable in certain platforms, although in most cases the developers
fix those problems with free software updates, leading to a user rating
increase222The most notorious case is CD Projekt RED’s Cyberpunk 2077, a 2020
highly-anticipated triple-A title, which was released with severe problems to
guarantee a stable gaming experience, leading to extremely poor user scores
[29]. Three years later, subsequent updates have improved the game and, at the
time of writing, it holds a 7.1 average user rating..

In this paper, we propose a Natural Language Processing (NLP) approach to
study this problem, by defining some criteria to create a review-bombing
dataset, and analyze the main concepts and words driving the users towards
these ratings. This is a novel outlook, as to our knowledge there is no record
in literature of such automated, artificial intelligence-based analysis,
whereas there are hundreds of articles and web posts discussing this
problematic from a sociological and/or economic point of view. In this sense,
this work can complement those studies from an analytical perspective,
providing a technical framework. This research gap can hopefully ignite an
interest on the application of automated artificial intelligence tools to
address and properly handle those situations, improving the user experience
with unbiased reviews that reflect the actual quality of the product.

The paper is structured as follows: in Section 2 we define the criteria used
to determine whether a title has suffered review-bombing, and create a dataset
of such ratings. Section 3 details the training and evaluation process of a
NLP algorithm. In Section 4, we explore the main concepts and words present in
such reviews, as well as further insights. Finally, discussion and conclusions
are outlined in Section 5.

##  2 Defining a Metacritic “review bombing” dataset

Our starting point is a Metacritic PC games review dataset from Kaggle [30],
comprising a total of 513,250 ratings, for titles spanning from 1995 up to
2023. These include entries for yet-to-be-released games, indicated as ’TBD’
either in the date, score o review.

We decide not to translate into English reviews written in other languages, as
the translation could not be as precise as desired given the context (not only
related to gaming context, which include neologisms, but prone to orthographic
errors, or deliberately misleading language)333This is an inherent limitation
of the analysis for some games such as Blizzard’s multi-player shooter
Overwatch 2, where two thirds of the reviews are written in Chinese –with 97%
of them negative– due to the widespread shutdown of Blizzard games caused by
the ending of NetEase-Blizzard agreement in China, causing players to lose
access to their accounts and the ability to play on a national server [31]..
English reviews are detected with the langdetect library [32].

Additionally, being them our main interest, only user reviews are taken into
account. Although professional critics are also included in the data,
typically a volume of x10–300 times more user ratings are found per title when
compared to the specialized media.

Metacritic establishes a minimum 4 ratings to assign a score (either user
score or Metascore). Lots of little, modest games are usually not reaching
this threshold for users, and therefore will not be taken into account in our
sample. Likewise, lots of indie titles lack a minimum of 4 professional
reviews. Nevertheless, review bombing is known to happen mainly for triple-A
games developed by big studios, which are well known in the community and have
a good volume of both professional and user reviews.

We define review bombing candidates when a title meets the following
requirements:

  1. 1.

has both a Metascore and user score reported;

  2. 2.

has a minimum of 5 user reviews in English;

  3. 3.

has a discrepancy between the Metascore and average user rating larger or
equal than 4.0 points, in favour of the professional critics.

For example, if the Metascore for a title meeting the two first requirements
is 7.0, we will mark it as potentially review-bombing candidate if the average
user score is below 3.0. This points towards a very large discrepancy between
professional and general reviews.

Despite the 4.0 point threshold being an ad hoc choice, we believe it also
possesses two additional advantages; firstly, it removes critics’ low-scored
games – if we chose a 3.0 as threshold, a game with a Metascore of 4.0 and
average user score of 1.0 would meet the criteria, although this case would
not be as clearly review bombing as the 7.0 vs. 3.0 example we presented
before. In second place, as presented in Figure 1, there are no games
exceeding 4 points of difference between average Metascores and user scores in
favour of the latter (pointing to “cult titles”). The fact that this large gap
exists only in the opposite case is a symptom of review bombing.

Figure 1: Metascore – user rating scores for PC games (1995–2023). Shaded
areas show user-preferred games (average user score $>$ Metascore), critic-
preferred titles (average user score $<$ Metascore), and potential games which
have suffered review bombing (Metascore minus average user score $>$ 4.0).
Colour code traces the Metascore.

From Figure 1 some conclusions can already be extracted. When analyzing the
release year, we observe how review bombing –as defined with our conservative
criteria– was almost non-existing before $\sim$2010, while it dramatically
increases from 2015–2016 onwards. As we pointed before, no title exceeds a
4-point difference in the case of unfavourable professional critics, while in
the case of review bombing candidates many titles with more than 5 points of
difference are found, with a maximum of a 7-point discrepancy444The record is
held by Tom Clancy’s The Division 2: Warlords of New York, with a Metascore of
7.9 and an average user score of 0.9 at the time of writing..

With these criteria, our final sample of potential games which have suffered
review bombing consists of 52,863 individual reviews for a total of 68 PC
games –an average 777 reviews per game, while in the case of non-potential
review bombing titles this value drops to 80, across 5,152 titles–. As we
mentioned before, review bombing tends to be a coordinated action across
individual users, which substantially increases the average number of reviews
per game.

In Table 1 we show examples of review bombing and examples of just negative
reviews, for user scores below 3 points.

Review bombing |  No Review bombing
---|---
I’m just here to help that score get lower huehuehuehuehuehuehuehhuehuehuehue |  literally unplayable. Enemy console players with cheating aim assist will always be better than you. You cannot compete with the trash aim assist system that pull the bullets to the head
Boycott Blizzard for squirting out this garbage. Until everyone wakes up and STOPS giving this company money, the games they make will NEVER get any better. Warcraft 3 Reforged is proof of this. |  It’s fun game at the first, but it’s so boring when you reach high level. Plain story spesific at 2.1 version, bad character kit design, lack of content and anniversary reward is suck. This devloper become so greddy at this time. I not recomend this game
Thanks for ruining my childhood you poop faces [x6] |  Awfuly boring gameplay, primitive graphics, many connections errors…
I’m just scoring low so hopefully Blizzard will wake up and realize that no one likes DRM, and no one wants 10 expansions until they make the next one…Buy to win ******** lame man…Just lame. |  “We continue to investigate reports of ongoing connection issues. Thank you for your patience!” The gaming experience of this game is their server crashes daily and all my characters progress got rolled back
Table 1: Examples of negative Metacritic user reviews, for review bombing
(left column) and no review bombing (right column). See text for more details.

With these reviews, we perform a tokenization of the texts with spacy [33],
and create a vector matrix with a TfidfVectorizer from scikit-learn [34] using
1000 max features555We also tried using 500 and 2000 features and the results
were very similar.. With this matrix, we will be able to train different NLP
models to address the importance of different words and concepts in these
potential review bombings.

##  3 NLP models training and evaluation

Once our dataset of potential review bombings is built, we perform a machine-
learning (ML) approach with NLP algorithms, aiming to understand the
importance of words (lemmas) in those reviews, and to distinguish them from
simply bad reviews with no harming intention.

We define two class labels, depending on whether an individual user review is
equal or less than 1.0. This seems an excessively aggressive approach, as
reviews with a score of 2.0 may be review bombing. Nevertheless, when taking a
look at the user score distribution for our review bombing sample, as shown in
Figure 2, there is a clear peak below 1.0, as most of the review bombing users
aim to assign scores as low as possible. In the same figure, we show the user
score distribution for non-bombing reviews, where just $\sim$1.7% of scores
are below 3.0.

Figure 2: User ratings for Metacritic PC game reviews in the sample of
potential review bombing titles (left panel) and the sample of non-potential
review bombing titles (right panel). See text for details.

The resulting data is divided in a train/test split with 80/20 proportions and
stratification on the label frequency. Four models are considered for the
classification task: Logistic Regression (LR), Random Forest (RF), Gradient
Boosting (GB) and Multinomial Naive Bayes (MNB), all loaded from scikit-learn
[34]. To ensure the maximum performance of each of them while guaranteeing an
unbiased score estimation, a grid search hyperparameter optimization with
5-fold cross-validation is executed.

The best model is the MNB ($\alpha_{opt}=0.01$), with a 0.88 accuracy on the
validation set. Recall, precision and F1 macro averages are 0.84, 0.88 and
0.86 respectively. The rest of models present accuracies between 0.81 and
0.85, indicating that they are all competitive for this dataset. The confusion
matrix for the MNB model is shown in Figure 3.

Figure 3: Normalized confusion matrix of the MNB best NLP model, for Review
Bombing (RB) and Non-Review Bombing (Non-RB).

There are almost 7x more false negatives than false positives, i.e., the model
assigns 27% of actual review bombing ratings to the “normal” class, yet only
4% of the opposite. This suggests that review bombing language is more complex
than standard reviews, and that, while some user reviews are assigned scores
below 1.0 in critic’s acclaimed titles, review bombing only exceptionally
assigns higher scores.

##  4 Insights on review bombing main drivers

With a trained ML algorithm with satisfactory metrics on the validation set,
we can rank the importance of words and concepts appearing in those ratings
flagged as review bombing.

To do so, we extract the conditional probability from the vector matrix for
each of the feature words. By sorting these, a ranking of the most relevant
concepts is obtained. In order to enhance the visualization of concepts for
the human eye, we present those results as a wordcloud in Figure 4, where the
size of each word is proportional to its importance.

Figure 4: Wordcloud for the most relevant concepts in review bombing user
ratings. See text for more details.

As seen in the figure, most non-obvious words (ignoring “game”, ”bad”, etc.)
point towards different groups of concepts:

  1. 1.

Individual companies and studios: words such as “blizzard”,“activision” or
“rockstar” indicate that many users target titles developed by these
companies. Indeed, many games in the sample are developed by these studios,
such as Diablo III, World of Warcraft (and expansions), Grand Theft Auto
Remastered or Tom Clancy’s The Division 2. Indeed, one of the most important
features is “company”, pointing towards dislike of these individual studios.

  2. 2.

Originality and content: concepts like “original”, “classic”, “new”, “old”,
“remaster” or “childhood” point out to upsetting of users’ expectations with
the originality of newly produced games, which in many cases are remasters or
remakes of mid-90s or early 2000s titles with widespread critical acclaim. New
approaches in artistic or gameplay direction, updated scripts or bad ports are
usually vectors of review bombing. Many of these users may experiment “gaming
nostalgia” [35], having played the original titles when originally released,
and finding the new title unable to meet the expectations of their childhood
memories.

  3. 3.

Economic aspects: words such as “money”, “refund”, “spend” or “cash” suggests
financial motives driving review bombing. This is especially the case for
sequels and paid downloadable content (DLC) expanding the base game, where
users consider the additions are not worth the retail price. On the other
hand, micro-transactions and so-called “pay-for-win” are a common monetization
strategy –especially in online multiplayer titles– which is usually not well
received by users [36].

  4. 4.

Sentiment analysis: vocabulary including “garbage”, “terrible”, “disgusting”,
“trash” points towards very negative, extreme feelings towards the product.
Within the field of evaluation theory, this is known as “value-based” ratings,
depending on individual opinions which are influenced by sociocultural
factors, in opposition to the fact-based ones which are founded on verifiable
facts [37]. This indicates that most of these reviews do not criticize
objective aspects, but are rather based upon personal feelings towards the
game content.

  5. 5.

Frustration: many of the review bombings present words like “false”,
“promise”, “shame”, “hope”, “lie”, etc. Complementary to the sentiment
analysis, this suggests many users perceive some titles as a “betrayal” to
their expectations. The NLP model finds that reviews containing these words
are those most prone to be review bombing, as user groups are genuinely
deceived and hope that flooding the scores with near-zero ratings the industry
will try to avoid similar situations in the future to prevent sales drops.

It is very interesting to note no political or LGTBIQ+ keywords are found as
drivers in this PC dataset. This is actually expected when taking a look to
the games and genres in which review bombing is found in PC, being most of
them first-person shooters (FPS), massively multiplayer online role-playing
game (MMORPG) or action titles, with no such content. Although some recent
titles such as Dead Space Remake or Resident Evil 4 Remake have been subject
to some review bombing for this reason [38, 39], it has been a very little
share of ratings, which keeps the user score and Metascore discrepancy below 1
point, therefore not having any relevance at all in the context of score-
driven sales trends.

Games with relevant review bombing for political and/or LGTBIQ+ content most
notably include Horizon Forbidden West: Burning Shores and The Last of Us Part
II [40], Sony Playstation exclusives. In such cases, an overwhelming majority
of the low-score user reviews were centered on so-called “woke” content [41],
such as feminine main characters, gender-inclusive language, and multi-racial,
minorities and/or LGTBIQ+ representation. Multiple zero- or one-point reviews
of such titles were just the phrase “go woke, go broke”, usually employed as
slogan calling for boycott of entertainment products featuring such content
[42]. Future work foresees extending this analysis to console-exclusive titles
to understand the impact of these reviews in the NLP model.

##  5 Discussion and conclusions

In this work, we have studied online score aggregator Metacritic reviews to
study the so-called “review bombing” phenomenon, in which a large group of
users assign unusally low scores to a product with critical acclaim. This has
been an increasing concern in the videogame industry, with multiple episodes
in the last years.

By using a Kaggle Metacritic dataset comprising a total of 513250 individual
ratings for PC games, we limited to user scores written in English, and
defined criteria to limit the results to those which potentially have suffered
review bombing, establishing a threshold of at least 4.0 point difference
between the Metascore and average user score. The final dataset consisted on
52,863 reviews for 68 PC titles.

After cleaning and tokenizing the texts, we trained a Natural Language
Processing (NLP) machine learning model, in order to understand the main words
and concepts driving review bombing user comments. We trained four different
algorithms, namely Logistic Regression, Random Forest, Gradient Boosting and
Multinomial Naive Bayes. The latter turned out to be the better performing
when trained with a train/test split of 80/20% with a grid search
hyperparameter optimization and 5-fold cross-validation, reaching a global
accuracy of 0.88.

By ranking the conditional probabilities of the tokenized text, we were able
to gain insights on the main drivers of review bombing. We found 5 groups of
concepts: individual companies, originality and content, economic aspects,
sentiment analysis and frustration. By taking into account both these families
and the individual words the model found to be most relevant, online
aggregators can take actions against such situations.

What actions are those is a matter of discussion. Suppression as a definitive
strategy to fight review bombing can be a double-edged sword: if a review
bombing phenomenon has transcended the gaming community and reached widespread
media, there has already been damage to the platform’s reputation and
prestige. In that situation, an immediate and drastic intervention can be
perceived as unwarranted meddling and a signal that the evaluative tool lacks
impartiality in its assessment of the subject [43].

Additionally, understanding the underlying drivers of these reviews
–especially when focusing on technical issues or lack of and/or repetitive
content– can be used by studios to improve user experience and as feedback for
developers [44]. Indeed, studies of review bombs can be used as an opportunity
for further theoretical advancements in the field of information management
[45]. While it is a common occurrence in social media, the concept of meta-
opinion is a relatively novel concept in the realm of online reviews,
deviating from the standard model where users provide reviews solely about the
product itself. Instead, meta-opinions offer a distinctive perspective by
connecting and building upon prior, value-based opinions related to the
product. Their primary function is to either bolster or challenge previous
messages, thereby contributing to the evaluation of the overall credibility of
product reviews [46].

The analysis we have conducted can be very easily extended to other game score
aggregators like Steam and even others product user reviews platforms such as
IMDb or Goodreads. This way, not only can content moderators benefit from the
understanding of underlying drivers, but a transversal analysis can undercover
further sociocultural and demographic patterns depending on the platform,
product type, etc.

This is especially relevant for political and/or LGTBIQ+ content-based review
bombing (“go woke, go broke”), which are not present in PC games due to the
most notorious cases happening in console-exclusive titles such as Horizon
Forbidden West: Burning Shores or The Last of Us Part II. The impact of such
content can significantly alter the conclusions of this analysis, most likely
adding a sixth category on the insights groups of concepts.

In this sense, a further consideration for future extensions of this work is
the influence of the review date in relation to the game’s release date. As an
example of such effect, at the time of writing The Last of Us Part II holds a
9.3 Metascore and 5.8 user score. Yet, during the first days after release,
the gap widened to a 9.5 Metascore and 3.4 user score [47]. This suggests very
early, negative reviews are a coordinated action targeted towards damaging the
game reputation, while players with a more fact-based approach tend to spend
real time playing the title before delivering a verdict [40]. Such analysis
presents a problem with Metacritic criteria changes precisely aiming to combat
review bombing, in which many of these reviews are now removed and a minimum
48h period is left before accepting user reviews.

Additionally, the demographic of the user, namely sex and age, can be very
relevant. In the case of Metacritic this information is not reported, yet in
other platforms suchs as IMDb it is. As an example of such importance, HBO’s
The Last of Us –adaptation of the title of the same name– received review
bombing in its third episode for a LGTBIQ+ plot not explicitly showed (yet
implicit) in the original game. When looking at the score by demographics,
there is a big difference between the ratings by male or female users, as well
as a smaller difference by age [48, 49]. Such discrepancies may also be useful
when understanding the different drivers of this phenomenon, and different
user clusters can be profiled.

Finally, the rise of Large Language Models (LLMs) is changing the paradigm for
better and worse. LLMs can be used to enhance and automate fake review
detection [50, 51]. But on the other hand, LLMs are also becoming
exceptionally good as quick, human-like content generators which can flood
platforms such as these with review bombing and an elaborated text which can
be challenging to label as machine-generated [52, 53]. In this sense, LLMs are
a new player which can be both part of the problem and the solution.

Data Availability
The dataset analyzed in this paper is publicly available at Kaggle.

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#  Beating one bit of Communication with Quantum Correlations in Smaller
Dimensions

Peter Sidajaya<EMAIL_ADDRESS>Centre for Quantum Technologies,
National University of Singapore, 3 Science Drive 2, Singapore 117543  Valerio
Scarani  Centre for Quantum Technologies, National University of Singapore, 3
Science Drive 2, Singapore 117543  Department of Physics, National University
of Singapore, 2 Science Drive 3, Singapore 117542

###### Abstract

As a consequence of Bell’s theorem, the statistics of measurements on some
entangled states cannot be simulated with Local Hidden Variables (LHV) alone.
The amount of communication that must be supplied is an intuitive quantifier
of non-classicality. While it is obvious that this amount can be very large in
general, it has been surprisingly difficult to find simple examples of quantum
correlations, whose simulation requires more than one bit of communication. In
this paper, we report the simplest example to date, which lives in the
$(5,2,5,5)$ Bell scenario [the previously known smallest case living in the
$(7,3,16,16)$ scenario]. The proof is built on the observation that the
maximisation of the 1-bit score is equivalent to finding the best partition in
which the sum of the local scores in the two sub-games is maximal.

##  I Introduction

Since Bell’s seminal theorem, we have known that the correlations made by some
quantum systems cannot be reproduced solely with Local Hidden Variables (LHV)
[1, 2]. Consequently, some have started asking how powerful are these quantum
correlations in contrast to LHV, and compared to one another. One way to
phrase this question is to ask how much additional resources are needed to
supplement LHV in order to simulate these quantum correlations. Some have
investigated the case for nonlocal boxes [3, 4, 5], but one resource that is
most intuitive is classical communication, which we will be focusing on in
this work.

Some works have been done on the average amount of communication needed to
simulate the entanglement of a system of a generic dimension [6, 7, 8, 9]. On
the other hand, going to a small, finite dimension, a substantial body of
literature has been written on what could and could not be simulated with just
a single bit of classical communication (1-bit). For two qubits, 1-bit is
provably enough to simulate all the statistics of the maximally entangled
state [6, 10, 11, 12, 13] and of a whole range of weakly-entangled states
[14]; there is numerical evidence supporting the conjecture that all two-qubit
states can be simulated with just 1-bit [15].

Conversely, some have tried to search for a correlation that could not be
simulated by 1-bit. These correlations, or behaviors, can be sought for in
Bell scenarios with $m_{A/B}$ inputs and $o_{A/B}$ outputs for Alice/Bob,
denoted $(m_{A},m_{B},o_{A},o_{B})$. The set of 1-bit correlations forms a
polytope, akin to the local polytope, but obviously includes signalling
behaviours and is thus embedded in the full probability space. The real
dimension of this space, called probability dimension, is given by
$m_{A}m_{B}o_{A}o_{B}$ 111Taking normalisation into account, the effective
probability dimension is $m_{A}m_{B}(o_{A}o_{B}-1)$. We keep the formula in
the main text to facilitate direct comparison with Ref. [21].. Some have
characterised these polytopes in the simplest scenarios, i.e. up to
$(3,3,2,2)$, and found no violation by quantum correlations [17, 18, 19]. For
larger scenarios, the task becomes computationally unfeasible. Using a
different technique, Vértesi and Bene found a quantum correlation that is not
simulatable with 1-bit, although it requires an infinite number of settings
[20]. Eventually, just recently, Márton et al. found examples of quantum
correlations that are not simulatable with 1-bit in finite scenarios [21].
Their constructions are based on parallel repetition. The smallest scenario in
which they found examples are $(7,3,16,16)$ (probability dimension: 5376) for
fixed-directional communication, and $(7,7,16,16)$ (probability dimension:
12544) or $(63,63,2,2)$ (probability dimension 15876) for bidirectional
communication. In this work, we will present an example of a quantum
correlation that is not simulatable with 1-bit for fixed directional
communication, in the $(5,2,5,5)$ scenario (probability dimension: 250). The
statistics of interest can be generated by suitable measurements on the
maximally entangled state of two five-dimensional systems. This comes within
the range of challenging but feasible experimental demonstrations [22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38].

The paper is structured as follows. In Section II we will first present a
method to find the 1-bit bound that is faster than a brute force approach. In
Section III we will present our Bell inequality and the corresponding
correlation that lies outside the 1-bit polytope.

##  II Bell Inequalities for 1-bit of Communication

###  II.1 Defining the 1-bit polytope

Let us call the local set $\mathcal{L}$, the quantum set $\mathcal{Q}$, and
the set of 1-bit communication strategies $\mathcal{L}_{+1}$. Like
$\mathcal{L}$, $\mathcal{L}_{+1}$ is also a polytope, but one that spans
outside the no-signalling space. Since it is a polytope, the maximum value of
a linear inequality achieved by the set must be given by at least one of its
extremum points. This simplifies the problem of finding the 1-bit bound into
finding the extremum point which maximises the inequality.

Now let us call the set of inputs of Alice (Bob) as
$\mathcal{X}\;(\mathcal{Y})$ with size $m_{A}$ ($m_{B}$) and her (his) outputs
$\mathcal{A}\;(\mathcal{B})$ with size $o_{A}$ ($o_{B}$). The specific
scenario of a game or inequality is denoted by the $(m_{A},m_{B},o_{A},o_{B})$
notation. In the usual studies of Bell nonlocality, the non-signalling
constraints on $\mathcal{L}$ effectively reduce the dimension of the
probability space under study to
$m_{A}(o_{A}-1)m_{B}(o_{B}-1)+m_{A}(o_{A}-1)+m_{B}(o_{B}-1)$ [39]. However,
the polytope $\mathcal{L}_{+1}$, however, is not bound by this constraint: we
must work in the full probability space of dimension $m_{A}m_{B}o_{A}o_{B}$.

After receiving her input, Alice has $o_{A}^{m_{A}}$ possible deterministic
strategies. She then has $\left(2^{m_{A}-1}-1\right)$ deterministic
communication strategies. This number is given by Stirling’s number of the
second kind, which gives the number of ways to partition a set into a fixed
number of subsets. Finally, Bob has $o_{B}^{2m_{B}}$ possible deterministic
strategies, where the two come from Alice’s bit of communication. This number
can be reduced as there are duplicates, with the final number being

$o_{A}^{m_{A}}\left(o_{B}^{m_{B}}+(2^{m_{A}-1}-1)(o_{B}^{2m_{B}}-o_{B}^{m_{B}})\right).$

The number of extremum points grows very quickly with dimensions and direct
facet enumeration is practically impossible once we go to (3,3,2,2), where
some ingenious techniques are needed to enumerate the facets [19]. However, if
we are given a Bell inequality, we could still find its 1-bit bound by running
a brute force search on all the extremum points. Once we go above 20 million
points or so, however, which is reached in (4,3,4,4), this also starts
becoming unwieldy. This problem can be circumvented by noting that in order to
find the maximum 1-bit bound of a Bell inequality, we do not need to find the
Bell value of every single extremum point of $\mathcal{L}_{+1}$.

###  II.2 A faster computation for the 1-bit bound

A behaviour in a given scenario is given by a set of probabilities
$p(a,b|x,y)$ for all $a,b,x,y$, which can be written as a vector. Similarly, a
Bell inequality is given by a set of coefficients $V(a,b|x,y)$ for all
$a,b,x,y$, which can also be written as a vector. As mentioned, the maximum
score achieved by a polytope is obtained by at least one of its extremum
points. Thus, the task of finding the 1-bit bound of a given Bell inequality
$\vec{B}$ can be written as

$\displaystyle\max_{\vec{p}\in\\{\vec{p}_{\mathcal{L}_{+1},D}\\}}{\vec{B}\cdot\vec{p}},$
(1)

where $\\{\vec{p}_{\mathcal{L}_{+1},D}\\}$ is the set of extremum points of
$\mathcal{L}_{+1}$. Each of the extremum points corresponds to a deterministic
strategy. For each point, the output functions of Alice and Bob are given by
$A(x)=\delta_{f(x),a}$ and $B(y,c)=\delta_{g(y,c),b}$, while the communication
function is given by $C(x)=\delta_{h(x),c}$, for some assignment functions
$f,g,h$ that assigns one value $a,b,c$ to every input combinations.

In a deterministic 1-bit communication strategy, Alice deterministically sends
a 0 or 1 to Bob depending on her input. This means that Alice and Bob have
agreed on a partitioning Alice’s inputs in two sets

$\displaystyle\mathcal{J}=\\{x\in\mathcal{X}|C(x)=0\\}$ (2)

and its complement $\mathcal{X}/\mathcal{J}$. When we consider the partitioned
sub-strategies, they are local, as we have used the bit of communication.
Thus, all deterministic 1-bit communication strategies can be decomposed into
two local parts

$\displaystyle\vec{p}_{\mathcal{L}_{+1},D}=\vec{p}_{\mathcal{L}^{\prime},D}^{\;\mathcal{J}}\oplus\vec{p}_{\mathcal{L}^{\prime\prime},D}^{\;\mathcal{X}/\mathcal{J}},$
(3)

where $\oplus$ is the direct addition, $\mathcal{L}^{\prime}$ is the local
polytope in the smaller scenario of $(|\mathcal{J}|,m_{B},o_{A},o_{B})$ and
similarly for $\mathcal{L}^{\prime\prime}$. The two sub-strategies are local
in their respective partitions and thus are extremum points of the local
polytope in the smaller scenarios. Similarly, we can also partition $\vec{B}$
into two:

$\displaystyle\vec{B}=\vec{B}^{\;\mathcal{J}}\oplus\vec{B}^{\;\mathcal{X}/\mathcal{J}}.$
(4)

With these in mind, we can rewrite equation (1) in the following way:

$\displaystyle\max_{\vec{p}\in\\{\vec{p}_{\mathcal{L}_{+1},D}\\}}{\vec{B}\cdot\vec{p}}$
$\displaystyle=\max_{A(x),B(y,c),C(x)}{\vec{B}\cdot\vec{p}_{\mathcal{L}_{+1},D}}$
$\displaystyle=\max_{A(x),B(y,c),C(x)}{(\vec{B}^{\;\mathcal{J}}\oplus\vec{B}^{\;\mathcal{X}/\mathcal{J}})\cdot(\vec{p}_{\mathcal{L}^{\prime},D}^{\;\mathcal{J}}\oplus\vec{p}_{\mathcal{L}^{\prime\prime},D}^{\;\mathcal{X}/\mathcal{J}})}$
$\displaystyle=\max_{A(x),B(y,c),C(x)}\left(({\vec{B}^{\;\mathcal{J}}\cdot\vec{p}_{\mathcal{L}^{\prime},D}^{\;\mathcal{J}})+(\vec{B}^{\;\mathcal{X}/\mathcal{J}}\cdot\vec{p}_{\mathcal{L}^{\prime\prime},D}^{\;\mathcal{X}/\mathcal{J}})}\right)$
$\displaystyle=\max_{C(x)}\left(\max_{A^{\prime}(x),B(y,0)}({\vec{B}^{\;\mathcal{J}}\cdot\vec{p}_{\mathcal{L}^{\prime},D}^{\;\mathcal{J}})+\max_{A^{\prime\prime}(x),B(y,1)}(\vec{B}^{\;\mathcal{X}/\mathcal{J}}\cdot\vec{p}_{\mathcal{L}^{\prime\prime},D}^{\;\mathcal{X}/\mathcal{J}})}\right).$
(5)

In the first equality, we rewrite the maximisation variables to be the
strategies of the parties. In the second equality, we use equation (3) and
(4). In the third equality, we partition the Bell vector along $\mathcal{J}$
and use the distributive property of the dot product and the direct sum. In
the final equality, we note that the two terms are independent of each other,
i.e., the local strategy of the first sub-inequality does not affect the score
of the second sub-inequality. Thus, for a given partition $\mathcal{J}$, we
can split the maximisation into two terms. Here, we write $A^{\prime}$ and
$A^{\prime\prime}$ the strategies of Alice to denote that they are defined on
the domain of the partitioned input sets. We can now also assign a value to
the $c$ in the input arguments of $B$ since $c$ has a fixed value in each of
the partitions as defined by equation (2).

For a more intuitive description, consider that for a given Bell game, a
deterministic bit of communication allows the parties to effectively partition
the game into two. One where her input falls in $\mathcal{J}$, and one where
it does not. When she sends the bit to Bob, Bob now knows which partition of
the game her input is in. In each of these partitions, they now play a normal
Bell game. These two local strategies of the sub-games do not interact with
each other, since the strategy in the first sub-game does not affect how they
play and how they score in the second sub-game. Thus, a 1-bit strategy is
optimum in the total Bell game if and only if its sub-strategies are also
optimum in the sub-games. Thus, in order to find the maximum 1-bit strategy,
one needs only to maximise the two sub-games using local deterministic
strategies, for all the possible ways to partition the Bell game into two sub-
games.

This rewritten maximisation can be run faster than the original form since
finding the local bound of a game is a relatively fast task compared to
enumerating over all 1-bit deterministic strategies. This speedup comes from
the fact that when we rewrite this optimisation, we skip over the extremum
points which are not optimum in the partitioned sub-games defined by the
communication strategies. However, note that the result of the optimisation is
still analytically exact, and this allows us to find the exact 1-bit bound in
higher dimensions, which will be useful for the next sections.

##  III Finding quantum correlations that violate $\mathcal{L}_{+1}$

###  III.1 Five (and beyond) dimensional Bell-like inequalities for
$\mathcal{L}_{+1}$

| $y=1$ | $y=2$
---|---|---
$x=1$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
$\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$
$\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$
$\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$
$\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1
$x=2$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$
$\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$
$\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$
$\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1
$\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
$x=3$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$
$\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$
$\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1
$\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
$\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$
$x=4$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$
$\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1
$\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
$\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$
$\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$
$x=5$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1
$\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
$\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$
$\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$
$\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$ | $\cdot$ | $\cdot$ | 1 | $\cdot$
Table 1: The coefficients of the Bell inequality in $(5,2,5,5)$. The table is
written in full probability notation with the dots denoting zeros. The
polytope bounds are $S_{\mathcal{L}}=6$ and $S_{\mathcal{L}_{+1}}=7$. The
maximally entangled bipartite 6-dimensional states violate this with at least
a score of $S_{\mathcal{Q}}=7.1777$.

Our first example comes as a generalisation of a previous correlation found in
[15] to the $(5,2,5,5)$ scenario, shown in Table 1. This game is a truncated
version of the XOR-d games for $d=5$ [40, 41, 42]. Up to some relabelling, the
XOR-d games can be written as

$\displaystyle\mathcal{I}_{d}(\mathcal{P})=\sum_{a,b,x,y}^{d-1}\left[a-b=xy\;\text{mod}\;d\right]P(a,b|x,y),$
(6)

where $[\cdot]$ is the Iverson bracket, which evaluates to 1 when the
expression inside is true and 0 otherwise. While $a,b,x,y$ usually runs from
$\\{0,1,...,d-1\\}$, here we truncate $b$ to $\\{0,1\\}$.

This Bell inequality has local bound $S_{\mathcal{L}}=6$ and no-signalling
bound $S_{\mathcal{NS}}=10$. More importantly, the 1-bit bound is
$S_{\mathcal{L}_{+1}}$ = 7 while the quantum bound is in the range of
$7.1777\leq S_{\mathcal{Q}}\leq 7.1788$, where the lower bound is obtained by
an explicit example and the upper bound by the level 1+AB NPA hierarchy [43,
44]. The total dimension of the probability space of this scenario is 250,
signifying an improvement from the previously reported smallest example for
fixed directional communication: the (truncated, asymmetrised) two Magic
squares found in [21] by method of parallel repetition feature the total
dimension of 5376.

This can be further generalised to higher $d$. For example, the $d=6$ case in
$(6,2,6,6)$ yields $S_{\mathcal{L}}=7$ , $S_{\mathcal{L}_{+1}}=8$, and
$S_{\mathcal{NS}}=12$. The maximally entangled bipartite 6-dimensional states
violate this with a score of $8.3173<S_{\mathcal{Q}}<8.3693$ (level 1+AB NPA).
In general, for a $(d,2,d,d)$ scenario, $S_{\mathcal{L}}=d+1$,
$S_{\mathcal{L}_{+1}}=d+2$, and $S_{\mathcal{NS}}=2d$.

###  III.2 Violating the inequality with quantum mechanics

Figure 1: The plot shows the robustness of the violation against noise in the
state. For each data point, we add Gaussian noise to the entries of
$\rho=\ket{\Psi}\bra{\Psi}$ to get a noisy state and optimise the measurements
numerically to obtain the maximum quantum score for that state. Since the
optimisation is done heuristically, the points represent only a lower bound
and sometimes the optimisation fails to converge on the global maximum, which
gives the outliers. The fidelity of the noisy state ($F$) is the x-axis, while
the score ($S$) is the y-axis. The dotted line is the local bound
$S_{\mathcal{L}}$.

The lower bounds were obtained by numerical optimisation using MATLAB. All
were obtained by fixing the state to the maximally entangled state

$\displaystyle\ket{\Psi}=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\ket{ii},$ (7)

where $d$ is the dimension of the output. The measurements are obtained
numerically (link attached at the end), but we can still infer some patterns.
Note that due to the nonlinear and heuristic nature of the optimisation, the
precision and optimality of the resulting measurements are significantly
limited. In the following inferences we will use the symbol $\simeq$ to denote
a rough equality. In most cases, this rough equality is satisfied within a 10%
margin, but in some cases, especially when the values are close to zero, this
margin is not satisfied.

Let $\ket{a_{j}^{i}}\bra{a_{j}^{i}}$ $\Pi_{j}^{i}$ be Alice’s projector that
corresponds to the $i$-th measurement and $j$-th output. The first observation
is that, $\forall i\in\mathcal{X}$ and $\forall j,k\in\mathcal{A}$,

$\displaystyle\left|\innerproduct{a_{j}^{i}}{a_{k}^{(i+1)\text{mod}d}}\right|^{2}\simeq\left|\innerproduct{a_{j}^{(i+1)\text{mod}d}}{a_{k}^{(i+2)\text{mod}d}}\right|^{2}.$
(8)

I.e., the inner product between the $j$-th and $k$-th basis vectors from two
’neighbouring’ answers are always the same for every pair of ’neighbouring’
answers. Second, Bob’s two measurements form a mutually unbiased bases (MUB)
with each other:

$\displaystyle\left|\innerproduct{b_{i}^{1}}{b^{2}_{j}}\right|^{2}\simeq\frac{1}{d}.\quad\forall
i,j\in\mathcal{B}.$ (9)

Finally, the resulting behaviour is a convex combination of the nonlocal
behaviour that maximally violates the inequality and white noise.

$\displaystyle P(a,b|x,y)\simeq
wP_{\mathcal{NS}}(a,b|x,y)+(1-w)\frac{1}{d^{2}},$ (10)

where $P_{\mathcal{NS}}(a,b|x,y)=V(a,b|x,y)/d$, i.e., Table 1 reinterpreted as
a non-signalling behaviour. For $d=5$, $w\simeq 0.64$ while for $d=6$,
$w\simeq 0.63$.

Fig. 1 shows the heuristic quantum scores obtained by numerical optimisation
for noisy states in $d=5$. A fidelity of around $\sim 0.97$ is required to
observe a violation. This presents a possibility for a challenging, but
doable, implementation using a high dimensional two-qudit state, such as those
created in photonics, either using orbital angular momentum (OAM) [23, 24, 25,
26, 27], path [28, 29, 30, 31], time-bin [32, 33], frequency-bin [34, 35], or
time-energy encoding [36, 37]. Recently, higher dimensional entanglement has
even been demonstrated outside of photonics using trapped ions [38].

##  IV Conclusions

Here we have presented a family of quantum correlations for $d\geq 5$ which
can not be simulated with a fixed directional communication from Alice to Bob
with just a single bit of information. The $d=5$ case is, so far, the smallest
example, both in the dimension of the quantum system and the total probability
dimension, of a quantum correlation that cannot be simulated with a single bit
of fixed directional communication from Bob. Indeed, even if Ref. [21] has
examples with smaller output alphabets, the quantum realisations require
higher-dimensional states. The small Hilbert space of this correlation
presents a practical opportunity for an experimental implementation.

Our example can be extended for bidirectional communication by doing two of
the games in parallel with reversed rules of Alice and Bob in the second game,
resulting in a $(10,10,25,25)$ game. However, this does not improve on the two
magic squares $(7,7,16,16)$ presented in [21].

##  V Code Availability

The measurements to obtain the lower bounds of $S_{\mathcal{Q}}$ can be found
in https://github.com/PeterSidajaya/1-bit-part-deux.

##  VI Acknowledgements

This research is supported by the National Research Foundation, Singapore and
A*STAR under its CQT Bridging Grant.

P.S. thanks Yeong-Cherng Liang, Swati Pandey, Kai-Siang Chen, and Gelo Noel M.
Tabia for discussions and running some NPA hierarchy.

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