Streetonomics: Quantifying Culture Using Street Names

Melanie Bancilhon1, Marios Constantinides2*, Edyta Paulina Bogucka4, Luca
Maria Aiello2,5, Daniele Quercia2,3

1 Washington University in Saint Louis, Saint Louis, USA

2 Nokia Bell Labs, Cambridge, UK

3 CUSP, King’s College London, London, UK

4 Technical University of Munich, Munich, Germany

5 IT University, Copenhagen, DK

*<EMAIL_ADDRESS>
## Abstract

Quantifying a society’s value system is important because it suggests what
people deeply care about—it reflects who they actually are and, more
importantly, who they will like to be. This cultural quantification has been
typically done by studying literary production. However, a society’s value
system might well be implicitly quantified based on the decisions that people
took in the past and that were mediated by what they care about. It turns out
that one class of these decisions is visible in ordinary settings: it is
visible _in street names_. We studied the names of 4,932 honorific streets in
the cities of Paris, Vienna, London and New York. We chose these four cities
because they were important centers of cultural influence for the Western
world in the 20th century. We found that street names greatly reflect the
extent to which a society is gender biased, which professions are considered
elite ones, and the extent to which a city is influenced by the rest of the
world. This way of quantifying a society’s value system promises to inform new
methodologies in Digital Humanities; makes it possible for municipalities to
reflect on their past to inform their future; and informs the design of
everyday’s educational tools that promote historical awareness in a playful
way.

## Introduction

Culture is a complex multi-level construct [1]. Anthropologists claim that it
is intangible and imperceptible due to our inability to quantify it [2]. In
this study, we define a society’s culture as its set of collective behaviors,
norms and values.

To study geographical variations of cultural aspects, traditional approaches
have resorted to surveys, census, or literary production. More recently,
however, the availability of online open-data sources has made it possible to
study and quantify cultural aspects on a larger scale. Michel et al. [3]
introduced ‘culturomics’: a large-scale study of millions of digitized books
that uncovered historical trends. Montalto et al. [4] quantified a city’s
culture by investigating indicators such as cultural vibrancy (e.g., presence
and attractiveness of cultural venues and facilities), creative economy (e.g.,
the capacity of culture to generate jobs and innovation), and creative
potential (e.g., conditions that enable creative processes to thrive), and
they did so for multiple European cities.

We connect these two veins of research by proposing _‘streetonomics’_
(http://social-dynamics.net/streetonomics), an alternative way of quantifying
cultural indicators using a type of urban features that is visible yet often
overlooked: street names. “The main merit of commemorative street names is
that they introduce an authorized version of history into ordinary settings of
everyday life”, as Azaryahu [5] put it. Street names are more than spatial
indicators, and, since the beginning of time, rulers have used spatial
engineering as a form of social engineering. As a result, street names mirror
a city’s social, cultural, political, and even religious values [6].

For the first time, we combined heterogeneous open-data sources to offer a
unique and cheap way to study the footprint of a city’s cultural dimension
through time and space. This approach could offer tools that would not only
create dwellers’ awareness of the ‘historical memory’ of a city, but would
also be used by public authorities to reflect on past street-naming decisions
and to inform future ones. In doing so, we made three main contributions:

  * •

We formulated four main research questions (§RESEARCH QUESTIONS3) that are
meant to quantify: _i)_ gender biases in naming streets; _ii)_ whether streets
speak to a distant past _vs._ a closer present; _iii)_ which professions have
been historically celebrated and which eventually died off; and _iv)_ whether
culture historically had a local _vs._ global focus.

  * •

We curated and made publicly available a dataset containing street names and
information about the figures—‘honorees’—these streets were named after
(§METHODS4). The dataset contains 4,932 streets in the cities of Paris,
Vienna, London, and New York.

  * •

As we hypothesized, we found that street names reflect the value system of a
society (§RESULTS5). Street names are gender-biased against women in two
cities: the highest female proportion is found in Vienna (54%), followed by
London (40%), then Paris (32%), and then New York (26%). Streets speak to a
distant past, all the more so for those in London, Vienna and in Paris rather
than those in New York. Certain professions have been consistently celebrated
over centuries (e.g., artists, scientists, and writers), while others went out
of fashion (e.g., military). Finally, based on the number of streets named
after foreigners, among the four cities, Vienna was the city that celebrated
nations other than its own.

## Related work

### Urban Informatics

By 2025, we will see another 1.2 billion people living in cities, and that
will require new ways of managing the increased complexity that comes with
larger cities. Since culture is one of the most important factors for urban
success [7], we need new ways of ‘quantifying’ it. Recently, Hristova et al.
[8] quantified the cultural capital of neighborhoods in London and New York.
They did so by operationalizing sociologist Pierre Bourdieu’s theory of
cultural capital upon geo-referenced pictures. They found that neighborhood’s
cultural capital rather than its economic capital is more predictive of future
house prices and, alas, of gentrification as well. By going beyond culture,
researchers have recently quantified important sociological aspects of city
life: from Jane Jacobs’s urban vitality [9] to Stanley Milgram’s urban
ambiance [10].

In line with this research, we propose the use of open data to quantify a
city’s culture, contributing to the fields of social computing and
computational social science, in that, we answered questions typical of
studies concerning culture or urbanism, and did so in a computational way.

### Streets and Place Naming

Naming streets and places is linked to power. Cohen and Kliot’s [11] analyzed
how place naming functioned as a strategy for governments to promote specific
historical events over others and, in so doing, shape national identity. Guyot
et al. [12] described place naming as a territorialization process that shapes
place identify. In a similar way, Azaryahu [5] described street naming as an
administrative control process. To see this type of processes in practice,
consider South Africa during the apartheid period. Place names originated by
the fusion of European and Afrikaans names and served colonial powers in their
attempt to erase indigenous groups’ identities. To partly counter that, the
new territorial demarcations have sought to address the inequalities stemming
from the apartheid and shape a new national identity [12, 13]. In a similar
way, in the United States, studies have shown that streets named after Martin
Luther King were placed on minor streets rather than major roads [14], thus
reinforcing traditional racial boundaries [15], and recent street (re)naming
has tried to change the country’s course. Oto-Peralias [6] analyzed the
religiosity expressed by street names in Spain, and found that it was
correlated with cultural and economic factors. Contrary to this previous work,
our study differs in two ways: _i)_ we analyzed a large corpus of streets
comprised of four cities as opposed to a single city; and _ii)_ we explored
honorees’ ‘meta’ information such as their genders, professions, countries of
origin as opposed to studying naming patterns.

Overall, previous studies were limited in scale, or were restricted to the use
of street names only. This study, instead, collated a large dataset of street
names covering four large cities across two centuries, and relied on open-
source knowledge bases to enriched these names with information about the
historical figures after whom the streets were named, including age, gender,
occupation, and country of origin.

### Historical background of Paris, Vienna, London, and New York

Paris The early changes in rational planning in Paris occurred in the
Enlightenment period, where cemeteries were banned from the city due to the
potential health hazard caused by the large number of corpses buried in
churches during the French Revolution. Henri Lefebrve once remarked that “new
social relationships call for new space, and vice verca”. This is reflected in
the decisions that the revolutionaries of 1789 made to establish the Conseil
des Batiments Civils and built courts and prisons to reflect the tenets of the
Age of Reason [16]. Later in 1860, Napoleon III decided to transform Paris
into a modern empire by expanding the walls of the city and redesigning the
urban scene. He hired urban planner Georges-Eugene Haussmann and together they
started rebuilding and expanding the city. Between 1852 and 1870, the center
was rebuilt with wide new avenues, parks and squares, and the city went from
12 to its current 20 “arrondissements”. They built the luxurious Rue de Rivoli
[16], one of the most iconic streets in the city. As new streets were being
built, a street naming convention was established to assign names to new
streets and rename the existing ones.

Vienna In 1860, liberals in Austria took over the Habsburg Empire and
transformed its institutions in accordance to the liberal identity and the
cultural values of the middle class. In the process, they gained power over
the city of Vienna. The city became a cultural and economic hub, and the
center of intellectual life. From the moment of their accession to power,
liberals began to reshape the city in their own image and transformed the face
of Vienna. This era is commonly called the Ringstrasse era (“Ring Street
Era”). This was the name of a new major road that was built to replace the old
city walls and that separated the old city from its suburbs [17]. Since then,
the city experienced a rapid expansion and the practice of naming new streets
after historical figures started.

London A defining moment for the British capital was 1666. During that time
the Great Fire broke out, and the city of London was reborn from its ashes.
The fire destroyed about 60% of the city and, within a few days of the fire,
plans for the rebuilding of the city commenced; these included restoration of
churches, development of new streets and roads, and expansion [18]. In the
following centuries, there was a period of rapid growth. The early signs of
the Industrial Revolution were evident, and London became the centre of the
evolving British Empire. Until 1920, it was the largest city in the world for
a century.

New York When the city of New York was still occupied by the Dutch, street
names were primarily used to commemorate the city’s European heritage. These
names were later on Anglicized by the British: streets were renamed first
after landowners to mark property ownership, and later after well-known
figures, including merchants, war heroes and politicians. Despite the
diffusion of this conventional naming practice, the municipality encountered
issues establishing a complete system of street signs. As the city expanded,
the Commissioners’ Plan of 1811 was adopted, a grid urban planning was
introduced, and streets were numbered rather than named. Only very few streets
would be named after certain privileged people. Soon after, street co-naming
was adopted in ‘honorific streets’ [19], where people’s name were added to
street numbers.

## RESEARCH QUESTIONS

As discussed in the previous section, we considered the four cities largely
because, by looking at them, one looks at some of the Western world’s most
important cultural centers in the past century: in the first part of the 20th
century, Vienna was the cultural capital of the world with key figures such as
Klimt and Freud; then, around the 1930s, Paris became the ‘go-to place’ for
American artists, and Hemingway and Gershwin were two notable examples; then,
for a century until 1920s, London had become the largest city in the world;
finally, in the 1950s, New York took center stage by redefining how art
(Jackson Pollock) and literature (Jack Kerouac) were meant to look like in the
modern world, and that influence still persists today. For those cities, we
set out to answer four main research questions and, in so doing, we
investigate whether explicit policies and public debates left any traces in
our dataset.

Q1 Are streets names in Paris, Vienna, London, and New York gender-biased?
Throughout history, western societies have often been masculinity-oriented,
and women’s achievements have been severely overlooked. For centuries, women
were relegated to the private sphere, while men dominated the public sphere
[20]. Urban planning was no exception; cities were built by men and for
men—and that made street naming gender biased [21]. In Harlem, through the
1970s and 1980s, the names of African American women were removed from places
of memory, which shows how commemorative strategies can reflect a combination
of racial and gender discrimination [22].

Gender equality in street names has been demanded by several feminist
organizations around the world, including by the group called
“OsezLeFeminisme” in Paris who protested that “it is time to reinstate the
place of women in history” (https://www.citylab.com/design/2015/08/the-
streets-of-paris-renamed-for-women/402526). As a result, in 2015, Paris Mayor
Anne Hildalgo has renamed several streets after iconic women such as Nina
Simone, Assia Djabar, Simone de Beauvoir, and Rosa Parks. We set out to
explore whether these policies left any traces in our dataset.

Q2 Are the historical times in which the honorees lived close or distant from
present times?
The long/short term orientation captures the balance between the tendency to
keep and honor past traditions, and the drive for pragmatic, future-oriented
problem solving. This cultural dimension might be partly reflected by the
historical times the honorees represent—whether they represent a distant or a
closer past. The history of the four cities has been rich of major societal
and cultural revolutions that spanned the whole two past centuries, and that
have influenced contemporary culture to different extents. We looked at how
the protagonists of these past events are celebrated through street names.

Q3 What are the most celebrated professions?
The way in which the honorees’ professions are celebrated might give an
indication about the values that are associated to personal status. Societies
characterized by higher levels of power distance are more inclined to build
highly hierarchical social structures and to celebrate high-status
individuals—for example, those who have elite professions. The occupation
might also contain information about the types of achievements that are most
celebrated, which ties back to the notion of individualism: individualistic
societies value more personal achievements (e.g., a sportsman breaking a world
record) rather than collective achievements (e.g., a civil right activist).
Avenues of Washington DC are named after the member states of the Union [5],
while streets in Bucharest celebrate the working class [23]. Similarly, we
explored the professions of the individuals commemorated in the streets of the
four cities as a way to determine what these societies valued over the years.

Q4 Are foreigners celebrated?
Collectivist societies, as opposed to individualistic ones, tend to be more
patriotic [24, 25]: street naming is often an expression of power and
patriotism, and the majority of streets are named after local and national
leaders. But, at times, streets are named after foreigners as well. The extent
to which this happens is largely determined by historical events (alliances,
colonization, or immigration), and by the position of society in the spectrum
of patriotism. Therefore, we looked at the extent to which street names
celebrate foreign cultures.

## METHODS

Table 1: Dataset Statistics. Each city is listed with its streets’ denomination periods and the historical periods in which the corresponding honorees lived. These are shown as (min, max) ranges. City | Honorific Streets | Denomination Period | Historical Period Honorees
---|---|---|---
Paris | 1428 | 1202 - 2011 | 60 B.C. - 1940
Vienna | 1662 | 1778 - 2018 | 700 - 1982
London | 770 | 1030 - 2013 | 18 B.C. - 1961
New York | 1072 | 1998 - 2013 | 1474 - 2003

### Datasets

We gathered street name information about the four cities from open data
sources. For each street, we collected eight types of information: the
district that the street belongs to, the renaming date (denomination) of the
street, the person name (honoree) that the street was named after, the
honoree’s gender, the honoree’s occupation, the honoree’s country of origin,
the honoree’s date of birth (dob), and date of death (dod). These curated
datasets include only information about honorific streets, namely streets that
are dedicated to historical figures — this is a subset of the total set of
streets. In this study, we focus on honorific streets and we do not consider
streets that are not named after people (e.g., places, events, or natural
elements). Next, we describe how we obtained, cleaned, and structured this
data (Table 1), which we made publicly available (https://social-
dynamics.net/streetonomics/data) to support future research.

_Paris._ We used both Wikidata and Wikipedia. We queried the structured
database Wikidata through its SPARQL endpoint query service, and obtained all
the instances of the “street class” located in Paris, which amounted to 3,413
streets. Each returned street object had a “named after” field, which was a
link to the corresponding eponym. Streets in Paris are named mostly after
regions, events, or people. Since we were interested in streets named after
people, we took all eponyms instances of the “person class” (leaving us with
1,808 streets). We then crawled these persons’ Wikipedia pages, and that
resulted in a dataset in which each row contained a street name; its borough;
and its honoree’s name, gender, occupation, date of birth, date of death, and
country of origin. After further data cleaning, we ended up with the final
Paris dataset containing information for 1,428 honorific streets.

_Vienna._ We started from the ‘Vienna History Wiki’ platform
(https://www.geschichtewiki.wien.gv.at), which brings together historical
knowledge about the city, and it does so in structured webpages. Each webpage
contains the history of a given street and an infoBox (similar to Wikipedia)
containing street creation date, denomination (if the street was named after a
person), and district. We initially resolved data about 2,481 streets. We
translated German names to English in two steps: first, we used an off-the-
shelf python package for language translation using Google Translate API
(https://pypi.org/project/googletrans). Second, to resolve any disambiguation
from the automatic translation, one of the authors manually corrected 5% of
them (e.g., disambiguation due to special characters in the German language).
After data cleaning (i.e., streets whose honoree full information could not be
found), we ended up with the final Vienna dataset containing 1,662 streets.

_London._ We crawled Wikipedia entries that referred to London streets,
narrowing down the search on entries that contained mentions that typically
characterize pages about streets such as ‘named after’, ‘honor’, and
‘celebrate’. This process resulted into 2500 streets, which were then
annotated using the Amazon Mechanical Turk crowd-sourcing platform: each entry
was annotated by at least 2 workers, and conflicts were manually resolved. The
resulting dataset consisted of 770 streets.

_New York._ From urban planner Gilbert Tauber (http://nycstreets.info), we
obtained a curated dataset that contains 1,459 honorific streets, and each
street is associated with its honoree name and gender, and the street’s
denomination and borough. After data cleaning, we ended up with the final New
York dataset containing 1,072 streets.

We then mapped all this information into each city’s neighborhoods defined by
the official polygons.

(a) Paris

(b) Vienna

(c) London

(d) New York

Fig 1: The fraction of streets existing today that were (re)named over the
decades. Results are shown
by city.

(a) Paris

(b) Vienna

(c) London

(d) New York

Fig 2: Number of honorees (after whom all the city’s streets were named)
who were alive at each historical decade.

### Data Processing and Formulae

Having obtained these four datasets, we processed them as follows.

_Temporal and spatial mapping._ We mapped our data spatially in terms of
district _D_ (neighborhood); and temporally in terms of two types of decade:
denomination decade _DD_ (the decade in which a street was denominated), and
focus on historical decade _FHD_ (the extent to which an historical decade is
represented across a city’s streets, which is based on the decades during
which all the honorees lived).

_Focus on Historical Decade (FHD)._ We used an honoree’s dob and dod to
reflect his/her lifespan. We then counted the number of times the honoree’s
lifespan coincides with each historical decade. We repeated that for all the
honorees (i.e., for all the city’s streets in our data). The resulting
measure—the (city) focus on historical decade—reflects the extent to which
each decade is represented across the city’s streets.

_Gender._ We computed a female proportion (_f_prop_) per denomination decade
DD and per district D. The female proportion per D is computed by first
grouping streets associated with females into their districts, and by then
counting the proportion of these streets in each district D.

$f\\_prop@DD=\frac{\\#f\\_streets@DD}{\\#all\\_streets@DD}$ (1)
$f\\_prop@D=\frac{\\#f\\_streets@D}{\\#all\\_streets}$ (2)

_Foreigners._ After marking as _foreigners_ the honorees whose countries of
origin were different than the city’s country, we computed the proportion of
foreigners (_for_prop_) per denomination decade DD and per district D:

$for\\_prop@DD=\frac{\\#for\\_streets}{\\#all\\_streets@DD}$ (3)
$for\\_prop@D=\frac{\\#for\\_streets@D}{\\#all\\_streets}$ (4)

_Occupation._ Based on the International Standard Classification of
Occupations, we mapped each occupation to any of the following 12 occupational
groups: creative and performing artists, authors, journalists and linguists,
science and engineering professionals, legal, social and cultural
professionals, craft and related trades workers, business and administration
professionals, legislators, commissioned armed forces officers, religious,
health associate professionals, and teaching professionals. This category-
based classification schema allowed us to map historical professions and
modern professions under a unified set of categories. For example, bishops,
archibishops, or clergy were mapped into the ‘religous’ category; in a similar
way, we mapped the rest of the professions’ categories.

#### Deciding the starting date.

We set the starting decade for Paris to 1860, which is when the city began its
expansion and became what it is nowadays. The number of streets (re)named
peaked in the 1920-30s (Figure 1a). That period is the post-World War I era
during which writers and artists from all over the world were attracted to
Paris. For Vienna, our analysis started in 1860, the time when the modern city
was re-purposed. After the World War II (Figure 1b), the city underwent a
large-scale urban development, and streets were named after key figures of the
war. In London, we started in 1666, when the Great Fire broke out and the city
was reborn from its ashes (Figure 1c). In New York, we started our analysis in
1998, when the street co-naming was introduced. We observed that there was a
drastic increase in the number of streets being named after the 9/11 terrorist
attack (Figure 1d), where hundreds of streets were named in memory of victims,
emergency responders, and people in the community who provided help.

## RESULTS

(a) streets named after females

(b) streets named after foreigners

Fig 3: Over decades, the proportion of streets named after: (a) female
figures using Formula (1), and (b) foreigners using Formula (3). Historical
decades refer to the date of (re)naming of a street.

(a)

(b)

Fig 4: Fraction of streets named after: _(top)_ (a) women computed
with formula (2); and _(bottom)_ (b) foreigners computed with formula (4).
The more intense the color, the higher the fraction.

### Gender

Q1 Are streets names in Paris, Vienna, London, and New York gender-biased?
Despite Paris and New York being centers of progressive culture, they still
fail to commemorate men and women in equal measure. Sadly, what is an
historically legacy will stay with us in the future, if these two cities do
not change their course. As men’s and women’s aspiration converge, street
naming should reflect that. More specifically, in Paris, 4% of today’s streets
are named after women, and these streets are clustered in the 1st
arrondissement, the very center of Paris (Figure 4a), and that proportion used
to be only 1% prior to 1980 (Figure 3a). In the 1880s, in Vienna (Figure 3a),
the proportion of streets named after females was below 5%, reaching a peak of
10% in 1920-1940s. In 1980s, more and more streets started being named and
renamed after female figures, resulting into a proportion of female streets as
high as 41% in 2000s and 54% in 2010s. These streets are predominantly
clustered in the 22nd district (Donaustadt), in the east side of the city
(Figure 4a). In London, the proportion of female steets reached its highest
proportion of 40% around the 1980s (Figure 3a), and are predominantly
clustered in the South West boroughs. Finally, in New York, the proportion of
female streets rose steadily over the last three decades, up to 26% in 2010s
(Figure 3a), and are clustered in Manhattan (Figure 4a).

### Historical decades

Q2 Are the historical times in which the honorees lived close or distant from
present times?
Streets in Paris are named mostly after people who lived in the 1860s, as the
peaks in the two distributions (Figure 1a and 2a) suggest approximately the
same time period. This reflects the era of the Second Empire during which
urban planner Haussmann worked with Napoleon III to transform Paris into an
empire city. Form comes with function, and Haussmann’s vision transformed the
city both spatially and socially. His urban changes entailed better
infrastructure, including the reconstruction of the entire Paris’s sewage
system [26], and left a mark that is still visible today [27].

In Vienna, most of the historical figures were living through the 1900s
(Figure 2b). Around this time, the city had expanded. The historical period
most represented in the city’s streets appear to be that between World War I
and the Nazi invasion [28]. World War I destroyed the city and, around 1940s,
Austria was invaded by Nazi Germany.

London streets are named mostly after people who lived through the 1700s and
1800s. Within that temporal window, the city experienced the Great Fire, grew
because of the large-scale interventions promoted by King George III, and
benefited from the vibrant atmosphere created by the introduction of the
printing press.

Finally, in New York, streets celebrate people who lived through 1950s to
2000s (Figure 2c), and a considerable part of them (36%) are named after 9/11
victims and emergency responders.

Fig 5: The rise and fall of occupations. Each occupation is ranked by
its overall frequency among streets at each decade.

### Occupation

Q3 What are the most celebrated professions?
We analyzed the professions of the individuals commemorated in the
streetscapes of Paris, Vienna, London, and New York to determine what these
societies valued over the decades.

As Paris’s value system changed, the importance of professions changed as
well. After World War II, most streets commemorated people in the military
(ranked 1st during 1940-1950s). In the decades that followed, Paris was reborn
from its ashes and honored more artists, writers and scientists, leading the
military profession to be ranked at the 5th position nowadays. Religious
occupations did not received much attention, except the period between 1950s
and 1980s, during which they ranked at the 4th position.

Vienna’s streets celebrate artists (Figure 5a). Yet its streets still reflect
historical changes in society as it has been in the case of Paris. People in
the military ranked at the 3rd position in 1860s (after the Austria-Hungary
war), increasingly lost importance over time, and finally ended up at the 9th
position today. Compared to Paris, scientists and writers have not been valued
as much, while those in legal and social occupations have (they are at the 3th
position today).

London’s streets celebrate the British royal family, politicians, and military
professionals. Artists, on the other hand, were neglected for centuries and
have only recently been represented, reflecting the more contemporary image of
the city.

In New York, the number of honorific street names peaked in 2001 after the
9/11 attack, and these streets today represent 36% of the total (Figure 1c).
This has resulted in the celebration of people who worked in the ‘business and
community’ area (e.g., civil servants). The profession that has remained
constant is that of artists, which have constantly ranked at the 4th position.

To study the long-term naming trends, we grouped all the decades into the four
half-centuries of 1800-1850, 1850-1900, 1900-1950, and 1950-2000. For each of
the half-centuries, we computed the average of its five correlation
coefficients; each coefficient is the Kendall Tau coefficient between the
occupational ranking in a given decade (e.g., 1850) and that of the next
(e.g., 1860). We did so for all the cities but New York. That is because its
street renaming patterns have been comparatively recent and, as such, they can
be studied one decade over another but are not amenable to be studied over the
two centuries.

Figure 6 shows the average correlations for all the four half-centuries. For a
given half-century, the lower the correlation, the more changes the city
experienced in terms of celebrated occupations. We see that Paris experienced
incremental and constant changes throughout the centuries, Vienna mostly in
the $20^{th}$ century (which followed the biggest expansion started with the
Ringstrasse era), and London during the first half of that century (during
which the city experienced its largest expansion).

Fig 6: The stability of the types of occupations celebrated by a city’s
streets. The higher a city’s value, the more stable the occupations celebrated
by its streets remained in a given half-century.

### Foreigners

Q4 Are foreigners celebrated?
The overall proportion of streets named after foreigners (Figure 3b) is
highest in Vienna (44.6%), followed by London (14.6%), Paris (10.9%), and then
New York (3.2%).

In Paris, the great majority of streets named after foreigners are clustered
in the 8th arrondissment (Figure 4b), in the northwest part of the city. At
the beginning of great expansion (1860-1880), about 18% of the streets were
named after foreigners (Figure 3b). That was after the French revolution
(1785) and, at that point, Belgium was part of France. After the Belgian
revolution (1830-1839), the two nations split, and the number of streets named
after foreigners decreased to 10%, a proportion that lasted for a century.
Quite recently (1980s), the proportion rose again to reach 29% in 2000s.
Interestingly, in the last decade, there was not any new street (re)named
after a foreigner.

As for Vienna, 44.6% of its streets are named after foreigners (Figure 3b),
reflecting the city’s historical ties with other Central and Eastern European
countries. The proportion of foreigners in street names peaked in the 1870s
(70%), right after the formation of the Austrian-Hungary republic. After
1900s, the proportion started to decrease, down to 45% in 1910s, and to 18%
during 2000s. As opposed to Paris, in the last decade, the proportion rose
again and reached 30% in present times. As one expects, most of these streets
are located in the center of the city (Figure 4b), that is, in the inner old
town (now Innere Stadt, 1st district), and in the northern part (Brigittenau,
the 20th district).

London streets celebrate foreigners less than what Vienna does, but more than
what the two other cities do (Figure 3b). A decline in the number of streets
named after foreigners was experienced shortly after the two world wars,
suggesting a more inward-looking perspective. There was then a reversal of
fortune, with a peak being registered in the 1970s, period in which the city
financially and culturally took center stage.

The case of New York is different. Only 5% of its streets are named after
foreigners (Figure 3b), clustered mostly in the Bronx (Figure 4b). The
proportion was only marginally higher (7%) two decades back. New York is the
very model of a global city, but it is more so financially than culturally.
That might be partly because the city is quite young compared to Vienna or
Paris, and partly because it tends to focus on itself. New York is the world’s
financial capital. Yet art, music and fashion have contributed to the city’s
economy for a long time, but their contribution has been overlooked. This
cultural production “relies more than ever on the social encounters that come
from living shoulder to shoulder with your peers”
(https://www.economist.com/books-and-arts/2007/09/06/art-and-the-city). Since
a considerable number of these peers are foreigners, we expect that New York
will change its course and start to celebrate some of these individuals, in
line with its truly global nature.

Table 2: Validity check. City |  Total streets (from OSM) |  People (%) (our dataset) |  People (%) (200 streets) |  Female (%) (our dataset) |  Female (%) (sample)
---|---|---|---|---|---
Paris | 6953 | 21 | 46 | 6 | 4
Vienna | 7498 | 22 | 56 | 12 | 11
London | 55921 | 1 | 30 | 10 | 24
New York | 10438 | 10 | 14 | 20 | 14

### Validation

To ensure the robustness of our findings, we conducted an additional validity
check, with the main goal of estimating two quantities: i) the coverage of our
curated datasets over the total set of honorific streets existing in the four
cities, and ii) the actual male/female ratio in the full set of honorific
streets, as a simple way to appraise the error of the measures we used in our
study.

Tracing back the name of a street to its honoree is a demanding task, and it
requires the investigation of multiple historical records that are often not
available in digital format. Attempting to do so without relying on such
historical sources faces several challenges including that of name expansion
(i.e., inferring the honoree’s first name from the last name) and
disambiguation (i.e., when a street name could refer to multiple entities,
such as a place or a person). With these challenges in mind, we adopted a
best-effort approach to obtain a rough estimate of the two target quantities
on a random set of streets.

We used a two-step methodology. First, we obtained road segments within
administrative borders of each city from OpenStreetMap
(http://download.geofabrik.de). The dataset includes road segments of
different types such as residential roads or pedestrian tracks
(https://wiki.openstreetmap.org/wiki/Key:highway). We excluded motorways,
trunks, cycleways and paths in addition to streets that included numbers or no
name was attached to them. This yielded a total number of 6,953 in Paris,
7,498 in Vienna, 55,921 in London, and 10,438 in New York. Second, for each
city, we randomly sampled 200 streets uniformly distributed across the city’s
boroughs/districts and searched those street names on Wikipedia. We marked a
street as a honorific if, among all the search results, there was an explicit
mention about its history. For all honorific streets obtained as such, we
obtained the gender of the honoree (Table 2). In the city of Paris, we found a
total of 92 streets linked with people, whereas the rest 108 are not
honorific. Out of the 46% of honorific streets, we found a total of 6% to be
named after a female figure. Similarly, in Vienna, we found 56% streets in the
random sample to be honorific, and out of these 11% were name after women. In
London, we found 30% out of the 200 streets to be honorific, and out of these
a total of 10% were named after women. In New York, we found 14% out of the
200 streets to be honorific, and out of these a total of 14% were named after
women. This additional validity check illustrates that, to a great extent, our
findings are aligned with a smaller random sample manually checked.

## Discussion and conclusion

### Main Results

By analyzing honorific streets (i.e., streets that were named after a person),
we revealed social-cultural characteristics of the four cities under study.
Our findings, however, refer to a subset of streets (i.e., honorific streets)
in those cities, and they should be interpreted as such. We found that streets
in Paris, Vienna, London and New York are gender-biased. Such a bias has been
mitigated by a recent trend of naming new streets after outstanding female
figures. Yet, Paris, for example, despite being at the center of gender
equality movements and public debates, still remains the most gender-biased
city among the three.

As one expects, street names reflect a country’s culture and can be used as
proxies for the country’s historical and cultural characteristics (e.g., as an
indicator of religiosity [6]). New York’s street naming is more present-
oriented than Vienna’s or Paris’s or London’s. The streets of the three
European capitals are mostly named after people who lived in the 19th century,
while those in New York, even when not considering the 9/11 victims as part of
the picture, are mostly named after people who had spent significant part of
their lives in the second half of the 20th century.

The representation of occupations over time shows an interesting evolution.
Military professions were celebrated after major conflicts, and then declined
as the western world entered in its longest period of peace after World War
II. The streetscape in Paris (and all the more so in Vienna) have always been
dominated by artists or writers, and that perfectly reflects the role of these
two cities as the world’s avant-garde for arts, fashion, and beauty in the
20th century. Interestingly, streets named after high-status professions such
as lawyers and jurists declined over time in Austria, while they increased in
France. As one expects, London tend to celebrate royals. In New York, most of
the streets are named after people who worked in the ‘business and community’
area (e.g., civil servants).

Finally, Vienna has more than 44.6% of streets named after foreigners. This
mainly reflects the city’s historical ties with Central and Eastern European
countries other than Austria (e.g., with Hungary through the creation of the
Austrian-Hungary Empire of dual monarchy in 1870s, with Germany through the
invasion of Austrian-born Adolf Hitler in 1938). London ranked second among
the four cities with a fraction of 14.6% of its streets named after
foreigners. Paris has a small fraction of its streets named after foreigners
(10.9%), and the Belgian’s influence on those streets is easily explained:
Belgium and France had been one country for a short while. Finally, despite
being considered a global city, New York tends to be self-focused by almost
exclusively celebrating Americans (only 3.2% of its streets are named after
foreigners).

### Implications

Our methodology offers an alternative way to study urban culture through open
data. Cultural studies have been typically conducted through the manual
analysis of literary production. More recently, they have been conducted at
scale through the analysis of millions of digitized books [3]. In line with
previous work, we have shown that also street names are indicators of culture
because they encapsulate a society’s beliefs and, consequently, their analysis
could be used to study hypotheses that have been left hitherto untested, and
these hypotheses could be taken from different fields, from Urban Studies to
Digital Humanities.

From a practical standpoint, our work might benefit two main stakeholders.
First, public authorities and municipalities could reflect on their past to
inform their future. To see how, consider the city of Madrid. To mark its
distance from the dictatorship period, it has recently started to change the
street names that were assigned during that period. Our tools can stimulate
this type of public reflection by enabling policy makers to take well-informed
decisions, and for citizens to participate in discussions. Ultimately, the
idea behind these tools is to increase historical awareness, promote cultural
heritage, and reshape the collective memory positively. Second, as people tend
to forget their historical past [29, 30], our methodology could help raise
dwellers’ cultural and historical awareness. Educational tools in the form of
simple games or visual analytics might well improve people’s awareness of
their city’s history and enable new forms of grass-root engagement [31]. As a
case in point, consider a mobile app that monitors its user’s mobility,
quantify the historical bits the user is exposed to on a daily basis, and
returns summaries of key historical events back on a weekly basis.

### Limitations and future work

This study has four limitations that call for further work. First, our
methodology is not fully automatic. The procedure for obtaining and creating
structured datasets from heterogeneous data sources is automatized, but cases
in which data was not present or badly formatted were manually resolved. As
such, this study cannot be easily scaled to the entire world. Language
translation, data collection, and data integration are just some of the
technical challenges to be solved. This leads us to our second limitation: our
analysis is restricted to three cities located in Europe, and one in the
United States. This set contains world-class cities that are excellent
candidates to study the evolution of culture in the Western world through
centuries, but it does not represent other blocks of cultural or religious
identities (e.g., Islamic world, Latin-American cultures). The third
limitation concerns potential selection biases in the open data sources used
in the study. A case in point is women. They are represented in Wikipedia
pages less than men [32]. Future work could explore the extent to which the
street naming biases we reported are further aggravated by the biases of the
external data sources we relied on. Finally, the fourth limitation is that our
analysis is limited to an important yet tiny portion of history: the last 200
years, and three decades in the case of New York (during that time honorific
streets were introduced in New York) Yet the very same methodology could be
applied in different historical periods, which could lead to additional cross-
cultural comparisons.

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# Quantum Inception Score

Akira Sone<EMAIL_ADDRESS>Department of Physics, University of
Massachusetts, Boston, Massachusetts 02125, USA  Naoki Yamamoto
<EMAIL_ADDRESS>Quantum Computing Center, Keio University, Hiyoshi
3-14-1, Kohoku, Yokohama 223-8522, Japan  Department of Applied Physics and
Physico-Informatics, Keio University, Hiyoshi 3-14-1, Kohoku, Yokohama
223-8522, Japan

###### Abstract

Motivated by the great success of classical generative models in machine
learning, enthusiastic exploration of their quantum version has recently
started. To depart on this journey, it is important to develop a relevant
metric to evaluate the quality of quantum generative models; in the classical
case, one such examples is the inception score. In this paper, we propose the
quantum inception score, which relates the quality to the classical capacity
of the quantum channel that classifies a given dataset. We prove that, under
this proposed measure, the quantum generative models provide better quality
than their classical counterparts because of the presence of quantum coherence
and entanglement. Finally, we harness the quantum fluctuation theorem to
characterize the physical limitation of the quality of quantum generative
models.

##  I Introduction

A burgeoning advancement of the artificial intelligence (AI) is a hallmark of
the contemporary Information Age. The principal objective of AI research is
the creation of machines which exhibit human-like capabilities of performing
complicated tasks including learning, analyzing and reasoning. Presently, the
prevailing landscape of AI technologies is predominantly underpinned by the
machine learning algorithms, such as convolutional neural network and support
vector machine, which have been embracing various applications [1, 2, 3, 4,
5]. However, there has been a recent warning regarding machine learning and
its adherence to a Moore’s law-like trend concerning the size of data sets
[6], namely the curse of the dimensionality. Considering that people seek to
encode data into larger feature spaces to facilitate pattern discovery, there
is a pressing need for an alternative approach to reduce complexity in the
context of big data analysis.

Quantum computers hold significant potential to attack such challenge, owing
to their intrinsic greater information storage and information processing
capacities compared to the classical devices [7]. Actually, we find
enthusiastic exploration for the quantum-enhanced machine learning [8, 9, 10,
11, 12]. Especially our primary focus is on unsupervised learning [13, 14],
which is a field dedicated to uncovering hidden patterns within unlabeled data
sets and further classifying them. One of the most successful framework in
classical unsupervised learning is the generative model [15, 16], where the
generator is trained to produce data, such as images, with high accuracy and
substantial diversity from a vast amount of unlabeled data. Prominent examples
of the generative models include variational autoencoders (VAEs) [17] and
generative adversarial networks (GANs) [18]. It is then important to have a
relevant metric to evaluate the accuracy and diversity of the generator, and
the so-called inception score (IS) [19, 20] is an example of such metric. In
practice, achieving a higher IS equates to better quality of a generator that
enjoys both greater accuracy and diversity. Quantum generative models [21, 22,
23] serve as the quantum counterparts to classical generative models. Various
quantum generative models have been proposed, which include variational
quantum autoencoders (VQAEs) [24, 25, 26, 27], the quantum generative
adversarial networks (qGANs) [28, 29, 30] and quantum Boltzmann machines [31,
32, 33]. Furthermore, quantum generative models have recently found
applications in various domains, such as quantum many-body physics [34, 35,
36].

To assess the quality of quantum generative models and highlight the factors
that distinguish their performance compared to classical models, we need a
meaningful and physically attainable metric. In this paper, we propose the
quantum inception score (qIS), which can be defined through the use of Holevo
information [37, 38]. This allows us to connect the physical interpretation of
the quality of a generative model, with the classical capacity of the quantum
channel playing a role as a classifier. The proposed metric leads us to
demonstrate that the quantum advantage in this metric is attributed to the
presence of quantum coherence, and further that the entanglement and the
entangled measurement play a significant role in asymptotically achieving the
best qIS. Our results not only serve as another illustrative example
highlighting the advantage of quantum generative models arising from
entanglement and coherence, but also support the claim by Gao et al. [22],
demonstrating the enhancement of expressivity of the quantum generative models
due to the Greenberger–Horne–Zeilinger (GHZ) state and cluster state, from a
different perspective; moreover, our results support the claim by Huang et al.
[39], demonstrating the quantum advantage in learning achieved by the
entangled measurement. Finally, leveraging the qIS, we employ the information-
theoretic fluctuation theorem [40, 41, 42] to characterize the physical
limitation of the quality of quantum generative models based on the concept of
quantum efficacy [42, 43, 44]. Figure 1 presents four main results summarized
in Def. 1 and Theorem 1, 2 and 3. These results corroborate the significance
of exploring the quantum foundation and communication approach to study the
quantum machine learning protocols.

Figure 1: Summary of the main results. The quality of the quantum generative
model, which is quantified by the qIS $\xi_{q}$, is better than that of its
classical counterpart, which is quantified by the classical IS $\xi_{c}$,
because of the presence of the quantum coherence. The qualtiy is degraded
mainly due to pure dephasing. The ratio between the quantum and classical IS
is lower bounded by using the quantum efficacy $\gamma\in(0,1]$ as
$\xi_{q}/\xi_{c}\geqslant\gamma^{-1}$, which implies the physical limitation
of the quality of quantum generative models. We also demonstrate that the best
quality, quantified by the maximum of the qIS, $\Xi_{q}$ called the
entanglement-enhanced qIS, can be asymptotically achievable.

This paper is organized as follows. In Sec. II, we briefly review the concept
of IS in the classical generative models and introduce the qIS in Sec. III.
Then, we compare the quantum and classical IS in Sec. IV, and further discuss
the role of quantum coherence in Sec. V. Finally, we utilize the quantum
fluctuation theorem to characterize the degradation of the quality in Sec. VI,
followed by the conclusion in Sec. VII.

##  II Inception score

Let us first review the concept of IS based on Refs. [19, 20]. Let
$\mathcal{X}\equiv\\{x_{1},x_{2},\cdots,x_{r}\\}$ be a dataset produced from
an unknown probability distribution. The aim of generative modeling is to
construct a model $p(x)$ that approximates the unknown probability
distribution producing $\mathcal{X}$. Given a data $x_{i}$, we also aim to
construct a classifier that produces a relevant label
$y_{j}\in\mathcal{Y}=\\{y_{1},y_{2},\cdots,y_{\ell}\\}$, where $\ell$ denotes
the number of the labels; we model the classifier via the conditional
probability $p(y|x)$, which is usually constructed via a neural network. Note
here that, in the generative model, we always have $r\geqslant\ell$. The
empirical marginal probability for the label data, $p(y_{j})$, is given by
$p(y_{j})=\sum_{i=1}^{r}p(y_{j}|x_{i})p(x_{i})$. Therefore, the input and
output of the network are $p(x_{i})$ and $p(y_{j})$, respectively. Then, the
IS of $p(x)$ relative to $p(y|x)$ is defined by

$\displaystyle\xi\equiv\exp\left(\sum_{i=1}^{r}p(x_{i})D(p(y|x_{i})\|p(y))\right),$
(1)

where

$\displaystyle
D(p(y|x_{i})\|p(y))\equiv\sum_{j=1}^{\ell}p(y_{j}|x_{i})\ln\frac{p(y_{j}|x_{i})}{p(y_{j})}$
(2)

is the empirical Kullback–Leibler (KL) divergence of the conditional label
probability $p(y_{j}|x_{i})$ with respect to $p(y_{j})$. Note that $\ln\xi$
can be expressed as $\ln\xi=\sum_{i}p(x_{i})\sum_{j}p(y_{j}|x_{i})\ln
p(y_{j}|x_{i})-\sum_{j}p(y_{j})\ln p(y_{j})$, that is, the sum of the expected
negative entropy of $p(y|x)$ and the entropy of $p(y)$. The former quantifies
the accuracy of the label assigned to the data and the latter does the
diversity of the data, implying that a generative model with bigger IS may
cast as a high-quality data generator. Therefore, in this scenario, the
primary objective is to construct a generative model, with the help of
relevant design of the classifier, that achieves the higher IS; for this
reason, IS has been often used for GAN.

##  III Quantum inception score

We consider a general setup of quantum generative model with a system
described by $d$-dimensional Hilbert space $\mathcal{H}_{S}$. Let
$\mathcal{B}(\mathcal{H}_{S})$ denotes the set of density operators acting on
$\mathcal{H}_{S}$. Suppose that the generator encodes a classical data
$x_{i}\in\mathcal{X}$ to an input state
$\rho_{\text{in}}(x_{i})\in\mathcal{B}(\mathcal{H}_{S})$, for $i=1,\ldots,r$.
As in the usual machine learning scenario that tries to solve problems by
encoding data into a larger dimensional space, we assume $d\geqslant r$. The
encoded quantum states are then processed by a completely positive and trace-
preserving (CPTP) map
$\Phi:\mathcal{B}(\mathcal{H}_{S})\to\mathcal{B}(\mathcal{H}_{S}^{\prime})$,
where $\mathcal{H}_{S}^{\prime}$ denotes the Hilbert space of the output
states. Thus the output state
$\rho_{\text{out}}\in\mathcal{B}(\mathcal{H}_{S}^{\prime})$ is related to the
input as $\rho_{\text{out}}=\Phi(\rho_{\text{in}})$.

Now, let us write the input state $\rho_{\text{in}}$ as an ensemble of
$\rho_{\text{in}}(x_{i})\in\mathcal{B}(\mathcal{H}_{S})$ encoding the
classical input data $x_{i}\in\mathcal{X}$ sampled from a probability
distribution $p(x_{i})$, i.e.,

$\displaystyle\rho_{\text{in}}=\sum_{i=1}^{r}p(x_{i})\rho_{\text{in}}(x_{i}).$
(3)

Note that in general $\rho_{\text{in}}(x_{i})$ does not necessarily commute
with $\rho_{\text{in}}(x_{i^{\prime}})$ when $i\neq i^{\prime}$. Then, the
output state becomes

$\displaystyle\rho_{\text{out}}=\sum_{i=1}^{r}p(x_{i})\rho_{\text{out}}^{(i)}\,,$
(4)

where

$\displaystyle\rho_{\text{out}}^{(i)}\equiv\Phi(\rho_{\text{in}}(x_{i})).$ (5)

Let us write the ensemble of the output state as
$\mathcal{E}\equiv\\{p(x_{i}),\rho_{\text{out}}^{(i)}\\}_{i=1}^{r}$. Because
the supports of $\rho_{\text{out}}^{(i)}$ and $\rho_{\text{out}}$ satisfy
$\text{supp}(\rho_{\text{out}}^{(i)})\subseteq\text{supp}(\rho_{\text{out}})$,
the quantum relative entropy
$S(\rho_{\text{out}}^{(i)}\|\rho_{\text{out}})\equiv{\rm
Tr}[\rho_{\text{out}}^{(i)}\ln\rho_{\text{out}}^{(i)}]-{\rm
Tr}[\rho_{\text{out}}^{(i)}\ln\rho_{\text{out}}]$ takes a finite value. Then
the Holevo information $\chi(\mathcal{E};\rho_{\text{out}})$ of the output
ensemble is defined as [37, 38]

$\displaystyle\chi(\mathcal{E};\rho_{\text{out}})$
$\displaystyle\equiv\sum_{i=1}^{r}p(x_{i})S(\rho_{\text{out}}^{(i)}\|\rho_{\text{out}})\,.$
(6)

The Holevo-Schumacher-Westmoreland (HSW) capacity (or Holevo capacity) [45,
46, 47, 48, 49, 50, 51] is defined as the maximization of the Holevo
information over all the possible $p(x_{i})$ and $\rho_{\text{in}}(x_{i})$:

$\displaystyle\chi_{\max}(\Phi)\equiv\max_{\\{p(x_{i}),\rho_{\text{in}}(x_{i})\\}}\chi(\mathcal{E};\rho_{\text{out}})\,.$
(7)

In the most generic scenario, the maximal transmittable amount of classical
information through the quantum channel $\Phi$ is measured by the regularized
classical capacity $C(\Phi)$ [49, 50, 51, 52, 53]

$\displaystyle
C(\Phi)\equiv\lim_{n\to\infty}\frac{1}{n}\chi_{\max}(\Phi^{\otimes n})\,,$ (8)

which is asymptotically achievable with infinite number of $\Phi$, and we
always have [50, 51, 52, 53]

$\displaystyle C(\Phi)\geqslant\chi_{\max}(\Phi)\,.$ (9)

From these facts, as our first main result, we define the qIS as follows.

######  Definition 1 (Quantum Inception Score).

The quantum inception score (qIS) is defined as

$\displaystyle\xi_{q}\equiv\exp\left(\chi(\mathcal{E};\rho_{\text{out}})\right)\,.$
(10)

Suppose now that the task of quantum generator is to generate the optimal set
of $\\{p(x_{i}),\rho_{\text{in}}(x_{i})\\}$ to maximize $\xi_{q}$. Then,
because the asymptotically achievable maximal qIS is given by $\exp(C(\Phi))$
from Eqs. (8) and (9), $\exp(C(\Phi))$ is regarded as an indicator of the best
achievable quality by the generator. Adopting an example of the image
recognition, these results are consistent to the intuition that having a
greater volume of information enables the generator to produce images that are
not only more accurate but also exhibit a broader range of diversity.

Particularly, in the asymptotic setup, when the entangled measurement on
$\Phi^{\otimes n}(\bar{\rho}_{\rm in})$ with entanglement input
$\bar{\rho}_{\rm in}\in\mathcal{B}(\mathcal{H}_{S}^{\otimes n})$ is allowed,
we always have $C(\Phi)>\chi_{\max}(\Phi)$ [53, 52]. With this special case,
we define the maximally achievable qIS as follows.

######  Definition 2.

With the entangled measurement and entanglement input, the maximally
achievable qIS is given by

$\displaystyle\Xi_{q}\equiv\exp\left(C(\Phi)\right)\,,$ (11)

which we call the entanglement-enhanced qIS.

Note that Eq. (8) still holds; therefore, the best quality is still
asymptotically achievable. This result implies that the entanglement and
entangled measurement play a significant role as a resource in achieving the
best quality of the quantum generative model. This result can be regarded as
an example supporting both the claim by Gao et al. [22], which demonstrated
the quantum advantage of the quantum generative models thanks to the
entanglement, and the claim by Huang et al. [39], which demonstrated the
quantum advantage in learning achieved by the entangled measurement.

##  IV Quantum vs classical inception score

Here, we compare the qIS with its classical counterpart. By introducing
projection measurements, we can recover the classical IS as follow. To
classify the data into $\ell$ labels, we assume
$\ell=\dim(\mathcal{H}_{S}^{\prime})$ and consider the complete set of
orthogonal projectors $\\{\Pi_{j}\\}_{j=1}^{\ell}$ on
$\mathcal{H}_{S}^{\prime}$ satisfying $\sum_{j=1}^{\ell}\Pi_{j}=\openone$ and
$\Pi_{j}\Pi_{k}=\Pi_{j}\delta_{jk}$ with $\delta_{jk}$ the Kronecker’s delta.
Regarding the dimensions of the Hilbert spaces, hence, the meaningful setup is
$d\geqslant r\geqslant\ell$. Therefore, $\Phi$ belongs to the class of tree-
like quantum classifiers composed of the hierarchical quantum circuits [54,
55, 56], where the compressed states are expected to carry the features of the
encoded data $\mathcal{X}$.

For a state $\rho\in\mathcal{B}(\mathcal{H}_{S}^{\prime})$, let us write the
post-projection-measurement state as

$\displaystyle\mathcal{P}(\rho)\equiv\sum_{j=1}^{\ell}\Pi_{j}\rho\Pi_{j}\,,$
(12)

where $\Pi_{j}\equiv|y_{j}\rangle\\!\langle y_{j}|$ is the projection
measurement onto the orthogonal state $|y_{j}\rangle$. Then, the probability
that the output state takes $|y_{j}\rangle$ is given by $p(y_{j})\equiv\langle
y_{j}\hskip 1.0pt|\rho_{\text{out}}|\hskip 1.0pty_{j}\rangle$. From Eqs. (3),
(4), and (5), we have $\langle y_{j}\hskip
1.0pt|\rho_{\text{out}}^{(i)}|\hskip 1.0pty_{j}\rangle=p(y_{j}|x_{i})$.
Because $\mathcal{P}(\rho_{\text{out}}^{(i)})$ and
$\mathcal{P}(\rho_{\text{out}})$ have the same eigenspaces
$\\{|y_{j}\rangle\\}_{j=1}^{\ell}$, we have
$S(\mathcal{P}(\rho_{\text{out}}^{(i)})\|\mathcal{P}(\rho_{\text{out}}))=D(p(y|x_{i})\|p(y))$,
meaning that the classical IS in Eq. (1) can be recovered from Eq. (10) by
projection measurements. Therefore, writing the Holevo information of the
projected output ensemble due to $\mathcal{P}$ as

$\displaystyle\chi_{\mathcal{P}}(\mathcal{E};\rho_{\text{out}})\equiv\sum_{i=1}^{r}p(x_{i})S(\mathcal{P}(\rho_{\text{out}}^{(i)})\|\mathcal{P}(\rho_{\text{out}}))\,,$
(13)

we can define the classical IS dependent on the choice of $\mathcal{P}$ as

$\displaystyle\xi_{c}(\mathcal{P})\equiv\exp\left(\chi_{\mathcal{P}}(\mathcal{E};\rho_{\text{out}})\right)\,.$
(14)

For each
$S(\mathcal{P}(\rho_{\text{out}}^{(i)})\|\mathcal{P}(\rho_{\text{out}}))$, due
to
$\text{supp}(\rho_{\text{out}}^{(i)})\subseteq\text{supp}(\rho_{\text{out}})$,
we can define the so-called projectively measured relative entropy, which was
introduced by Berta et al. in Ref. [57], as

$\displaystyle
S^{\mathbb{P}}(\rho_{\text{out}}^{(i)}\|\rho_{\text{out}})\equiv\sup_{\mathcal{P}}S(\mathcal{P}(\rho_{\text{out}}^{(i)})\|\mathcal{P}(\rho_{\text{out}}))\,,$
(15)

where the supremum is achieved when $\rho_{\text{out}}^{(i)}>0$ and
$\rho_{\text{out}}>0$. Furthermore, the following relation holds [57]

$\displaystyle\\!\\!\\!\\!S(\rho_{\text{out}}^{(i)}\|\rho_{\text{out}})=S^{\mathbb{P}}(\rho_{\text{out}}^{(i)}\|\rho_{\text{out}})\Longleftrightarrow[\rho_{\text{out}}^{(i)},\rho_{\text{out}}]=0\,,$
(16)

which plays the central role in deriving our third main result (Theorem 2).
Given these quantities, we define the optimal classical IS as

$\displaystyle\xi_{c}^{\mathbb{P}}\equiv\exp\left(\sum_{i=1}^{r}p(x_{i})S^{\mathbb{P}}(\rho_{\text{out}}^{(i)}\|\rho_{\text{out}})\right)\,.$
(17)

Note that the optimal classical IS can also be defined by using the positive
operator-valued measures (POVMs). In Refs. [58, 59], the measured relative
entropy is defined as
$S^{\mathbb{M}}(\rho_{\text{out}}^{(i)}\|\rho_{\text{out}})\equiv\sup_{\mathcal{M}}S(\mathcal{M}(\rho_{\text{out}}^{(i)})\|\mathcal{M}(\rho_{\text{out}}))$
over all the POVMs $\mathcal{M}$. Berta et al. proved the relation
$S^{\mathbb{M}}(\rho_{\text{out}}^{(i)}\|\rho_{\text{out}})=S^{\mathbb{P}}(\rho_{\text{out}}^{(i)}\|\rho_{\text{out}})$
[57]. Therefore, although we cannot necessarily recover the classical IS from
the POVMs, the optimal classical IS can be written in terms of the measured
relative entropy,
$\xi_{c}^{\mathbb{P}}=\xi_{c}^{\mathbb{M}}\equiv\exp\left(\sum_{i=1}^{r}p(x_{i})S^{\mathbb{M}}(\rho_{\text{out}}^{(i)}\|\rho_{\text{out}})\right)$,
which implies that the following theorems regarding the optimal classical IS
are generally valid for all the POVMs.

Given the above discussion, we can obtain our second main result as follows.

######  Theorem 1.

The inception scores, $\xi_{q}$, $\xi_{c}(\mathcal{P})$, and
$\xi_{c}^{\mathbb{P}}$, satisfy

$\displaystyle\xi_{q}\geqslant\xi_{c}^{\mathbb{P}}\geqslant\xi_{c}(\mathcal{P})~{}(\forall\mathcal{P})\,.$
(18)

###### Proof.

From
$S^{\mathbb{P}}(\rho_{\text{out}}^{(i)}\|\rho_{\text{out}})\equiv\sup_{\mathcal{P}}S(\mathcal{P}(\rho_{\text{out}}^{(i)})\|\mathcal{P}(\rho_{\text{out}}))$,
by definition of the projectively measured relative entropy, we have
$S^{\mathbb{P}}(\rho_{\text{out}}^{(i)}\|\rho_{\text{out}})\geqslant
S(\mathcal{P}(\rho_{\text{out}}^{(i)})\|\mathcal{P}(\rho_{\text{out}}))$.
Also, due to the monotonicity of the quantum relative entropy under the
projection measurements [7], we have
$S(\rho_{\text{out}}^{(i)}\|\rho_{\text{out}})\geqslant
S^{\mathbb{P}}(\rho_{\text{out}}^{(i)}\|\rho_{\text{out}})$. Therefore, we can
obtain
$\xi_{q}\geqslant\xi_{c}^{\mathbb{P}}\geqslant\xi_{c}(\mathcal{P})~{}(\forall\mathcal{P})$.
∎

Theorem 1 implies that the quantum generative model has better quality than
its classical counterpart.

##  V Quantum coherence and quality

From Theorem 1, it is natural to ask what is the quantum resource for this
advantage. The answer to this question is quantum coherence. Let us write
$G(\rho,\mathcal{P})$ as the relative entropy of coherence with fixed basis
given by $\mathcal{P}$, which is defined as [60]

$\displaystyle G(\rho,\mathcal{P})\equiv S(\rho\|\mathcal{P}(\rho))\,.$ (19)

Therefore, $G(\rho,\mathcal{P})=0$ if and only if $\mathcal{P}(\rho)=\rho$,
meaning that the projectors $\Pi_{j}$ is the eigenbasis of $\rho$.

Let us define $\mathcal{P}_{k}$ as the projection onto the eigenbases of the
$k$th output $\rho_{\text{out}}^{(k)}$, i.e.
$\mathcal{P}_{k}(\rho_{\text{out}}^{(k)})=\rho_{\text{out}}^{(k)}$, which
means that when $\Pi_{j}^{(k)}$ denotes the projector onto the $j$th
eigenbasis of $\rho_{\text{out}}^{(k)}$, for any quantum state $\rho$, we have
$\mathcal{P}_{k}(\rho)=\sum_{j=1}^{\ell}\Pi_{j}^{(k)}\rho\Pi_{j}^{(k)}$. When
$\rho_{\text{out}}^{(k)}$ and $\rho_{\text{out}}$ are nondegenerate, we can
obtain our third main result as follows.

######  Theorem 2.

Let $\mathcal{P}_{k}$ be the projection onto the eigenbases of the $k$th
output $\rho_{\rm{out}}^{(i)}$, i.e.
$\mathcal{P}_{k}(\rho_{\rm{out}}^{(k)})=\rho_{\rm{out}}^{(k)}$. Then, if
$\rho_{\rm{out}}^{(k)}>0~{}{(\forall k)}$ and $\rho_{\rm{out}}>0$ are
nondegenerate density matrices, we have

$\displaystyle\xi_{q}>\xi_{c}^{\mathbb{P}}\Longleftrightarrow
G(\rho_{\rm{out}},\mathcal{P}_{k})\neq 0~{}(\exists k)\,.$ (20)

###### Proof.

The whole proof is based on the assumption that $\rho_{\text{out}}^{(k)}>0$
and $\rho_{\text{out}}>0$ are nondegenerate. We prove the statement

$\displaystyle\xi_{q}=\xi_{c}^{\mathbb{P}}\Longleftrightarrow
G(\rho_{\text{out}},\mathcal{P}_{k})=0~{}(\forall k)\,,$ (21)

which is equivalent to Eq. (20) by contraposition.

For the sufficiency, because $p(x_{k})\neq 0~{}(\forall k)$ and
$S(\rho_{\text{out}}^{(k)}\|\rho_{\text{out}})\geqslant
S^{\mathbb{P}}(\rho_{\text{out}}^{(k)}\|\rho_{\text{out}})$ always hold, when
$\xi_{q}=\xi_{c}^{\mathbb{P}}$, i.e.
$\sum_{k=1}^{r}p(x_{k})S(\rho_{\text{out}}^{(k)}||\rho_{\text{out}})=\sum_{k=1}^{r}p(x_{k})S^{\mathbb{P}}(\rho_{\text{out}}^{(k)}||\rho_{\text{out}})$,
we must have
$S(\rho_{\text{out}}^{(k)}\|\rho_{\text{out}})=S^{\mathbb{P}}(\rho_{\text{out}}^{(k)}\|\rho_{\text{out}})~{}(\forall
k)$. From
$S(\rho_{\text{out}}^{(k)}\|\rho_{\text{out}})=S^{\mathbb{P}}(\rho_{\text{out}}^{(k)}\|\rho_{\text{out}})\Longleftrightarrow[\rho_{\text{out}}^{(k)},\rho_{\text{out}}]=0$,
we must have $[\rho_{\text{out}}^{(k)},\rho_{\text{out}}]=0~{}(\forall k)$.
Because we suppose that $\rho_{\text{out}}^{(k)}$ and $\rho_{\text{out}}$ are
nondegenerate, all $\rho_{\text{out}}^{(k)}$ share common eigenspace with
$\rho_{\text{out}}$; therefore, we have
$\mathcal{P}_{k}(\rho_{\text{out}})=\rho_{\text{out}}~{}(\forall k)$. From the
definition of the relative entropy of coherence, we can conclude

$\displaystyle\xi_{q}=\xi_{c}^{\mathbb{P}}\Longrightarrow
G(\rho_{\text{out}},\mathcal{P}_{k})=0~{}(\forall k)\,.$ (22)

For necessity, when $G(\rho_{\text{out}},\mathcal{P}_{k})=0~{}(\forall k)$, we
have $\mathcal{P}_{k}(\rho_{\text{out}})=\rho_{\text{out}}~{}(\forall k)$,
which leads to $[\rho_{\text{out}}^{(k)},\rho_{\text{out}}]=0~{}(\forall k)$.
From
$S(\rho_{\text{out}}^{(k)}\|\rho_{\text{out}})=S^{\mathbb{P}}(\rho_{\text{out}}^{(k)}\|\rho_{\text{out}})\Longleftrightarrow[\rho_{\text{out}}^{(k)},\rho_{\text{out}}]=0$,
we have
$S(\rho_{\text{out}}^{(k)}\|\rho_{\text{out}})=S^{\mathbb{P}}(\rho_{\text{out}}^{(k)}\|\rho_{\text{out}})~{}(\forall
k)$. Therefore, we can conclude

$\displaystyle G(\rho_{\text{out}},\mathcal{P}_{k})=0~{}(\forall
k)\Longrightarrow\xi_{q}=\xi_{c}^{\mathbb{P}}\,.$ (23)

From Eqs. (22) and (23), Eq. (21) holds. Given
$\xi_{q}\geqslant\xi_{c}^{\mathbb{P}}$, by contraposition, we can finally
obtain $\xi_{q}>\xi_{c}^{\mathbb{P}}\Longleftrightarrow
G(\rho_{\text{out}},\mathcal{P}_{k})\neq 0~{}(\exists k)$. ∎

Theorem 2 demonstrates that the qIS is larger than its classical counterpart
if and only if the quantum coherence of $\rho_{\text{out}}$ is observed. This
implies that the quantum coherence is the resource for the quantum generative
model to have better quality than its classical counterpart.

##  VI Physical limitation

Finally, we discuss the physical limitation of the quality of quantum
generative models. From Theorem 2, we can also expect that the pure dephasing
(or decoherence) contributes to degrading the quality into the classical one.
Here, we analyze this observation in a quantitative way by employing the
information-theoretic fluctuation theorems [40, 41, 42] based on the concept
of the quantum efficacy [42, 43, 44]. Following Ref. [42], for
$\rho_{\text{out}}>0$ and $\rho_{\text{out}}^{(i)}>0$, when $\Delta a$ is a
random variable whose average is $\langle\Delta
a\rangle=\chi(\mathcal{E};\rho_{\text{out}})-\chi_{\mathcal{P}}(\mathcal{E};\rho_{\text{out}})$,
the corresponding quantum fluctuation theorem is given as $\langle\exp(-\Delta
a)\rangle=\gamma$, where $\gamma$ is called quantum efficacy 111For the origin
of $\gamma$ regarding the Holevo information, refer to Eqs. (11) – (14) in
Ref. [42]. and satisfies $0<\gamma\leqslant 1$.

From the Jensen’s inequality, we have
$\chi(\mathcal{E};\rho_{\text{out}})-\chi_{\mathcal{P}}(\mathcal{E};\rho_{\text{out}})\geqslant-\ln\gamma$.
Here, note that $\gamma$ is strictly dependent on $\Phi$ and the choice of the
projection measurement $\mathcal{P}$. Then, we can obtain our fourth main
result as follows.

######  Theorem 3.

The quantum inception score can be lower bounded by using quantum efficacy
$\gamma$ as

$\displaystyle\xi_{q}\geqslant\frac{\xi_{c}(\mathcal{P})}{\gamma}~{}~{}(0<\gamma\leqslant
1)\,.$ (24)

###### Proof.

By definitions, we have
$\xi_{q}\equiv\exp\left(\chi(\mathcal{E};\rho_{\text{out}})\right)$ and
$\xi_{c}(\mathcal{P})\equiv\exp\left(\chi_{\mathcal{P}}(\mathcal{E};\rho_{\text{out}})\right)$.
From $\exp(-\ln\gamma)=\gamma^{-1}$, we can obtain
$\xi_{q}\geqslant\gamma^{-1}\xi_{c}(\mathcal{P})$ with $0<\gamma\leqslant 1$.
∎

From Refs. [42, 62], when $\gamma=1$, the equality of Eq. (24) holds if and
only if $\mathcal{P}$ is the projection measurement onto the common eigenspace
of all $\rho_{\text{out}}^{(i)}$, i.e.
$\mathcal{P}(\rho_{\text{out}}^{(i)})=\rho_{\text{out}}^{(i)}~{}~{}(\forall
i)$. Therefore, from Eq. (4), in this case, we must have
$\rho_{\text{out}}=\mathcal{P}(\rho_{\text{out}})$. Suppose that initially
$\rho_{\text{out}}$ satisfies
$\chi(\mathcal{E};\rho_{\text{out}})>\chi_{\mathcal{P}}(\mathcal{E};\rho_{\text{out}})$.
This means that $\rho_{\text{out}}$ has non-zero off-diagonal elements in the
bases of $\mathcal{P}$, which implies the presence of the quantum coherence.
After a noisy process, if $\xi_{q}$ coincides with $\xi_{c}(\mathcal{P})$, the
final state becomes $\mathcal{P}(\rho_{\text{out}})$. This state preserves the
diagonal elements of $\rho_{\text{out}}$ in the bases of $\mathcal{P}$, but
has zero off-diagonal elements. This implies that the quality degradation of
the quantum generative model is primarily due to the pure dephasing process.

From this analysis, we can also provide a physical interpretation to the
quantum efficacy $\gamma$. From Eq. (24), $\gamma$ can be regarded as a
quantity characterizing the physical limitation of the quality due to the
information loss. In particular, the negative logarithm $(-\ln\gamma)$
accounts for the information contents contained by the quantum coherence,
which is preserved during the entire quantum process.

##  VII Conclusion

We have proposed the quantum inception score as a quality measure of quantum
generative models, and obtained the following three main claims by connecting
to the Holevo information, the maximally transmittable amount of classical
information through the quantum classifier channel. First, the quantum
coherence plays a central role in enhancing the quality of quantum generative
models. Second, the best quality can be further achieved with the entanglement
input and entangled measurement. Third, the quality degradation of quantum
generative models is due to the decoherence, which can be characterized by
utilizing the quantum fluctuation theorem.

Finally we remark that IS is not currently a widely used measure for assessing
generative models because of its potential constraints in facilitating useful
model comparison as pointed out in Ref. [19]. Therefore, these constraints
could extend to the qIS in the similar manner. Nonetheless, the results
obtained in this paper based on qIS enable us to emphasize the significance of
further investigation of characterizing the quantum machine learning protocols
from the fundamental perspectives, such as quantum information transmission
and information thermodynamics.

## Acknowledgement

We are grateful to Haidong Yuan, Paola Cappellaro, Mingda Li, Sebastian
Deffner, Marco Cerezo, Quntao Zhuang and Tharon Holdsworth for insightful and
helpful discussions. A.S. is supported by the NSF under Grant No. MPS-2328774.
N.Y. is supported by MEXT Quantum Leap Flagship Program Grants No.
JPMXS0118067285 and No. JPMXS0120319794 and JST Grant No. JPMJPF2221.

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