††thanks: This manuscript has been submitted to Artificial Intelligence for
the Earth Systems. Copyright in this manuscript may be transferred without
further notice. Supplementary material (Fig. S1) is found at the end of this
manuscript.

# Pixels and Predictions: Potential of GPT-4V in Meteorological Imagery
Analysis and Forecast Communication

###### Abstract

Generative AI, such as OpenAI’s GPT-4V large-language model, has rapidly
entered mainstream discourse. Novel capabilities in image processing and
natural-language communication may augment existing forecasting methods. Large
language models further display potential to better communicate weather
hazards in a style honed for diverse communities and different languages. This
study evaluates GPT-4V’s ability to interpret meteorological charts and
communicate weather hazards appropriately to the user, despite challenges of
hallucinations, where generative AI delivers coherent, confident, but
incorrect responses. We assess GPT-4V’s competence in two tasks: (1)
generating a severe-weather outlook from weather-chart analysis and conducting
self-evaluation, revealing an outlook that corresponds well with a Storm
Prediction Center human-issued forecast; and (2) producing hazard summaries in
Spanish and English from weather charts. Responses in Spanish, however,
resemble direct (not idiomatic) translations from English to Spanish, yielding
poorly translated summaries that lose critical idiomatic precision required
for optimal communication. Our findings advocate for cautious integration of
tools like GPT-4V in meteorology, underscoring the necessity of human
oversight and development of trustworthy, explainable AI.

The integration of generative AI such as GPT-4V into meteorology brings
opportunity for improving forecast communication of weather hazards across
languages and communities. This study evaluates GPT-4V’s capacity to create
plausible severe-weather outlooks, and communicate hazards in both Spanish and
English, from inputs of weather charts and text. The weather outlook generally
aligns with a human-generated forecast product but displays vagueness and
incorrect reasoning. Further, translations lack idiomatic precision and
display poor grasp of cultural nuance. Despite this, GPT-4V shows potential
for advancement in meteorological application. We advocate for cautious AI
integration, emphasizing need for human oversight and reliable, trustworthy
output.

##  1 Introduction

Throughout the information age, scientists have embraced computers as
allies—from automation of simple tasks to post-processing big datasets. Human
weather forecasters provide added value over raw numerical weather prediction
(NWP) guidance, both quantitatively (Novak et al., 2011) and in public
communication (Stuart et al., 2006). Today, generative AI is rapidly entering
the mainstream in the wake of intuitive web-based “chatbot” interface, such as
OpenAI’s ChatGPT large-language model (chat.openai.com). Within this flurry of
AI-model development, meteorologists now hold unprecedented but little-
explored tools for improving the weather-forecasting enterprise. The addition
of “sight” to language models—so-called multi-modal models that can ingest
more than text—enables weather images and charts to be interpreted not just by
the human eye but by machine intelligence. Given this combination of natural-
language processing, intelligence, and ingestion of imagery, we ask: Can this
rapid technological advance serve meteorologists in forecasting, research, and
public communication? We do not seek to replace the human forecaster; rather,
we evaluate competence of the first public release of GPT-4V to augment the
meteorologist’s toolbox.

In the era of big data, scientists are increasingly turning to multi-modal AI
models like GPT-4V to increase efficiency and achieve more in less time.
Recent AI models can perform specific tasks as well as humans (Bubeck et al.,
2023; Yang et al., 2023) and may disrupt society more than the last generation
of LLMs such as GPT-3 (Floridi and Chiriatti, 2020; Tamkin et al., 2021;
Bender et al., 2021). Yet, despite typically coherent and confident responses
to prompts, there is the risk of hallucinations: instances where output
confidently and convincingly contains erroneous information. The language
model generates plausible output based on patterns in the training corpus, but
does not fact-check its inferences out-of-the-box. The forecast of severe
weather is a risk-averse endeavor; small errors may have disproportionally
negative consequences. Given this risk sensitivity and the importance of
specific wording when communicating weather hazards to the public (Rothfusz et
al., 2018; Trujillo-Falcón et al., 2021), AI language models present potential
to improve public communication of hazard risks (Olteanu et al., 2014)
tailored to the audience.

Our goal is to discern whether GPT-4V can constructively contribute to
meteorological applications; we probe its capabilities and limits in
meteorological application through three questions:

  1. 1.

Can GPT-4V correctly interpret weather charts and imagery?

  2. 2.

Can GPT-4V effectively communicate weather hazards in language tailored for
the audience?

  3. 3.

What constitutes a useful answer – what are our expectations and are they
reasonable?

##  2 Method

We choose GPT-4 as our large-language model due to OpenAI’s claimed GPT-4V
performance metrics and release of the “vision” ability enabling image inputs.
Further, at the time of writing, keyword searches on Google for ChatGPT and
OpenAI were an order of magnitude larger (trends.google.com, accessed 15 March
2024; not shown) than competitors and their AI models such as Anthropic
(Claude) and Google (Gemini or Bard). Herein, we used the ChatGPT web-portal:
the online graphical front-end to OpenAI GPT models.

During initial exploration, many types of challenges were provided to GPT-4V;
here, we focus on two tasks that are most representative of GPT-4V’s range and
aptitude of abilities. First, we ask GPT-4V to interpret multiple weather
charts and deduce the risk of meteorological hazards. Second, we give GPT-4V a
synoptic-scale forecast chart marked with regions of general weather type such
as thunderstorms and request plain-language summaries in both Spanish and
English. The conversations were processed by the lead author within ChatGPT,
performed within two weeks of 1 October 2023 with the same version released in
stages to the public starting on September 25
(https://help.openai.com/en/articles/6825453-chatgpt-release-notes, accessed 1
November 2023). We used no custom instructions, which can be provided to
personalize responses within ChatGPT. We include sections of our GPT-4V
conversations in the manuscript, but full conversations (unabridged other than
trimming of unimportant procedural text) are found in the Supplementary
Material. Text quoted verbatim from GPT-4V conversation is italicized. Our
experiments were conducted within a closed environment (i.e., ChatGPT did not
have access to the internet). The training corpus did not include information
beyond September 2021.

##  3 GPT-4V and Meteorological Prediction

Yang et al. (2023) and Bubeck et al. (2023) showed that GPT-4 is able to
interpret and conceptualize data spatially, such as text-based navigation
after a description of a location. It also displays nascent ability to
anticipate, which is a key human characteristic (Dennett, 2015). Combining
both factors, can GPT-4V conceptualize the atmospheric state from a sequence
of charts?

###  3.1 Initial set of weather charts

We want to determine if GPT-4V can grasp the three-dimensional atmospheric
flow, and hence show GPT-4V multiple pairs of charts depicting North American
Model (NAM) and Global Forecasting System (GFS) guidance data, visualized by
Pivotal Weather (https://home.pivotalweather.com/, accessed 15 October 2023).
We provide two models to assist GPT-4V capture basic characteristics of
uncertainty. While this is smaller corpus of guidance that would be available
to human forecasters, we give sufficient information to GPT-4V that a human
would capture the general flow pattern from the same resources:

  * •

Geopotential height at 300 hPa and 500 hPa

  * •

Dry-bulb temperature at 500 hPa, 850 hPa, and 2 m

  * •

Wind at 300 hPa, 500 hPa, 850 hPa, and 10 m

  * •

Mean sea-level pressure

  * •

Equivalent potential temperature at 2 m

  * •

Simulated composite reflectivity

Figure 1: Three pairs of weather charts taken from a full set given to GPT-4V.
Images reproduced with kind permission of Pivotal Weather, LLC. Geopotential
height and wind speed at 300 hPa (a,b); simulated composite reflectivity
(c,d); dry-bulb temperature at 850 hPa (e,f). Left column is GFS (a,c,e);
right column is NAM (b,d,f)

Figure 2: GPT-4V response to a collection of weather charts. We subjectively
highlight vague/incorrect responses in red and useful/correct sections in
blue.

Figure 1 shows a subset of these charts given to GPT-4V, including
geopotential height and wind vectors on multiple levels critical for
understanding the flow pattern. The forecast synoptic-scale flow pattern
displays a vertically stacked extratropical cyclone over the U.S. Great Plains
on the polar side of a jet stream (Fig. 1a,b) and evident in simulated
composite reflectivity (Fig. 1c,d). The warm and cold sectors of the cyclone
are seen in 850-hPa temperature (Fig. 1e,f). Our first instructional prompt
after providing charts (Fig. 2) asks GPT-4V to give a summary of uncertainty
after the first batch of maps and how it might improve its conceptualization
of the atmospheric state with further charts. Much of the response is not
specific enough to provide utility to end users (Fig 2), referring to some
differences, slight variations, differences in forecasted weather that might
affect predictions.

Responses display hallucinations falsely detected in the provided charts: Fig.
2 refers to precipitation in the Eastern US not shown in the charts (Fig.
1c,d). The choice of terminology is often non-standard: GPT-4V identifies a
more pronounced jet streak over the northeastern US. If we interpret
“pronounced” as meaning stronger in magnitude, this is evidently a mix-up
between the two models, despite GPT-4V’s ability to read small text in images
(not shown). It is unlikely a forecaster would find utility in GPT-4V’s
responses in Fig. 2: vague replies require the human to continue their task by
fetching further guidance on top of fact-checking the responses. However, a
human forecaster would also have access to many more meteorological charts;
accordingly, we now heed the request to provide addition chart(s).

###  3.2 Second set of weather charts

At the end of the response in Fig. 2, GPT-4V requests five further pairs of
charts to better grasp atmospheric uncertainty and structure:

  * •

Upper-air soundings, to assist understanding of stability and convective
potential;

  * •

Surface dew-point temperature (implying GPT-4V has not processed the 2-m
equivalent-potential-temperature charts);

  * •

Sea-level pressure and analyzed surface frontal boundaries (something GPT-4V
can attempt to infer from the provided charts);

  * •

Vorticity at 500 hPa (again, GPT-4V might infer this from existing upper-level
charts);

  * •

Humidity charts (at unspecified pressure levels).

The request for humidity data is sensible from GPT-4V’s lack of access to
upper-level humidity information; we give GFS and NAM charts of 700-hPa
relative humidity and wind vectors, then request GPT-4V to infer frontal
positions from these and previous charts. When asked about the equivalent-
potential-temperature charts, given initially to improve understanding of
near-surface cyclone structure (not shown), GPT-4V is not able to recall
these. It is unclear whether this is due to these charts residing so deep in
the conversation memory that this chart is deprioritized: GPT-4V will forget
information if the model architecture deems it less relevant to imminent or
recent prompts (Vaswani et al., 2017; OpenAI, 2023). Continuing its response,
GPT-4V makes numerous mistakes, such as another conflation of NAM and GFS
300-hPa wind maxima and identifying discrepancies indiscernible to human eye
in the Great Lakes region of the 700-hPa humidity charts (not shown; S1).
GPT-4V’s response also suggests a warm front north of the high-humidity region
in the Great Lakes, but mistakenly locates a cold front in the northeastern US
rather than stretching from Ohio to the Carolinas.

The errors and lack of useful information in these responses early in the
conversation may cause a chain-reaction of useless or harmful responses if not
caught by a human quickly enough in responses. This failure to check self-
consistency stems from lack of hybrid fast-and-slow thought process (Kahneman,
2011) to oversee fidelity: a known limitation of AI in emulating human thought
and decision-making (Booch et al., 2021; Weston and Sukhbaatar, 2023). Bubeck
et al. (2023) found when deriving or processing mathematical logic, where a
statement is often either correct or incorrect, GPT-4V can reach a correct
answer despite generating contradictory rationale. Similarly, as we progress
through the conversation, previous logical and factual errors do not preclude
a useful final answer—self-inconsistency does not preclude a course-
correction. As such, we turn to GPT-4V’s ability to recognize meteorological
hazards in the charts provided, and find GPT-4V indeed has a reasonable grasp
of the atmospheric state despite prior error. This further accents the need
for humans to persist with their GPT-4V conversation rather than halting after
one-shot tasks, partly due to low interpretability of response variation from
a black-box model (e.g., Flora et al. 2024).

Figure 3: Conversation snippet with (a) MUCAPE maps and request, (b) response.
We have removed discussion of maps not shown in Fig. 1.

Figure 4: Conversation snippet showing GPT-4V outlook and corresponding SPC-
issued outlook for same period.

###  3.3 Issuing a mock forecast

The Storm Prediction Center (SPC) issues convective outlooks for mesoscale
hazards such as tornadoes, damaging hail, and strong winds (spc.noaa.gov),
providing invaluable information to the public about potential threats to life
and property (Herman et al., 2018). Emulating the quality of SPC human
forecasters is a highly complex task, and thus we do not expect GPT-4V to
perform near human level. In pivoting to convective severe weather, we provide
a final chart of Most Unstable Convective Available Potential Energy (MUCAPE)
and ask for a reevaluation of uncertainty (Fig. 3). This order to reevaluate
follows OpenAI-recommended practices of splitting a long complex task into
smaller, manageable responses whose quality can be assessed by the supervising
human.

The GPT-4V response synthesizes its knowledge base to correctly identify a
swath of instability from Missouri to Illinois. The response names states and
geographical regions, highlights regions of high uncertainty, and locates a
jet streak and relevance of the wind maximum to a convective forecast. The
improvement in response quality lends support to our course-corrections and
provision of further maps; concerningly, some generated responses appear to
equate uncertainty with elevated severe potential.

Having primed GPT-4V with charts and subtasks, we request an ambitious SPC-
style convective outlook (Fig. 4), valid 0000 UTC 14 October 2023
(commensurate with previous charts’ valid time). The response contains some
vague and generic rationale (significant differences, discrepancies), but
GPT-4V nonetheless provides five specific regions with “elevated risk” that
ultimately resemble the SPC forecast (Fig 4b).

  * •

GPT-4V identifies Texas as a region of elevated risk from its proximity to
instability and moisture; however, variations in MUCAPE represent uncertainty,
not magnitude.

  * •

Missouri and Arkansas are noted for moisture convergence, a diagnostic found
suboptimal for predicting convective initiation (Banacos and Schultz, 2005);
again, discrepancies between the GFS and NAM charts are given as evidence of a
risk of hazards.

  * •

Rationale for choosing Illinois and Indiana is faulty, as uncertainty does not
indicate potential for severe weather, though notably GPT-4V appears to
consider multiple vertical levels with specific reference to 700 hPa moisture.

  * •

GPT-4V finally identifies Ohio as an area with a lower risk of severe weather
than states farther west; there is little rationale given, but the description
tallies with the broad SPC thunderstorm risk (Fig 4b).

The main difference between the SPC forecast chart (Fig 4b) and the GPT-4V
response is that GPT-4V does not identify the U.S. Southeast as another area
of thunderstorm risk.

We want GPT-4V to evaluate its own forecast through comparison with the actual
human-issued SPC charts (Fig. 5), gauging whether GPT-4V has a sensible grasp
of the atmospheric state. We ask GPT-4V to create “emulations” to evaluate
itself, where each emulation represents a human evaluator with various biases.
This represents a wisdom of crowds approach—the rationale also behind ensemble
weather-prediction systems. We ask GPT-4V to summarize each emulation’s
response. Some responses would not be uttered by human SPC evaluators
(“narrowing down the focus areas based on more real-time data” is not
pertinent for a 48-h forecast); other responses correspond to the SPC outlook
(“refining the exact boundaries would have been key to align better […]”).
When evaluating itself, GPT-4V is too generous at times, and even critiques
the SPC outlook based on previous guidance. GPT-4V’s final evaluation appears
to treat the SPC outlook as truth (“align better”), despite stating in the
prompt there is yet no correct answer. In sum, GPT-4V continues to make poor
logical arguments to support its responses, but ultimately yields a
respectable outlook when compared to an analogous chart produced by humans at
the SPC.

Figure 5: “Wisdom of crowds” method of self-evaluation of outlook in Fig. 4a
having been provided the human equivalent in Fig. 4b.

##  4 GPT-4V and Bilingual Weather Communication

GPT-4V has shown potential to grasp the general atmospheric state from a
sequence of maps. Until this point, we have used and received generated text
that has not been tailored to the layperson. Different communication styles
and languages are required for different communities: for instance, surveys
(https://nces.ed.gov/programs/digest/d21/tables/dt21_225.70.asp, accessed 1
November 2023) show over 20% of the US population do not speak English as
their first language at home (Dietrich et al., 2022). Given the importance of
risk communication appropriate for each community’s culture, we now test
GPT-4V’s ability to not only interpret meteorological charts but also
communicate an accessible summary to two audiences: plain-language Spanish
appropriate for Spanish speakers in the US and the same for American English.
Recent models approach machine–human parity in natural-language translation
skill (Läubli et al., 2020), but this is sensitive to the model architecture
and following of best practices (Hassan et al., 2018), and performance varies
between source and target languages. Indeed, certain idiomatic expressions may
be untranslatable between natural languages, requiring awareness of idiomatic
translation where the true meaning is preserved. Translation is inherently
creative, and different cultures describe geographical and weather phenomena
uniquely. Hence, to account for diversity in responses, we give identical maps
and prompts in three separate conversations to gauge consistency of our
subjective evaluation.

We analyze GPT-4V’s generated 200-word plain-language summary of a synoptic
analysis issued by the US Weather Prediction Center, valid 2 October 2023 over
the contiguous United States. We request the generated text first in a Spanish
localization appropriate for the US population (Fig. 6). Consequently, we
request the same response but in American English. Responses from GPT-4V are
not identical even if prompts are, due to the GPT temperature parameter.
Temperature controls creativity or randomness in the generated response.
Manually setting temperature to zero yields deterministic responses identical
for a given prompt; larger values increase the model’s variability of
responses.

Despite prompting for Spanish first, we find the English version is a direct
translation of the Spanish, rather than an idiomatic one, both containing
hallucinations. Direct translations can cause confusion as context and key
messages are more likely to become lost in translation (Trujillo-Falcón et
al., 2022). This yields some linguistic ambiguity:

  * •

Examples where of GPT-4V translated the information inconsistently with the
field’s standards, such as clima for weather, which can also be translation of
climate for most Spanish speakers. The inconsistency has proven to cause
confusion among bilingual groups (Trujillo-Falcón et al., 2021).

  * •

Vague use of geographical terminology (noroeste for northeast; la región
central del país for “center of the country”). Especially for multilingual
groups that were not born in the United States, the lack of suitable
geographical context can present challenges in comprehension and eventual
decision making (Trujillo-Falcón et al., 2024).

  * •

Acronyms did not translate completely into Spanish. For example, the Eastern
Daylight Time (EDT) acronym remained the same in English and Spanish. For the
acronym to be understandable for bilingual speakers, the acronym should be
spelled out in Spanish.

  * •

As seen in English responses, there are vague calls for action (tome
precauciones, “take precautions”) and descriptions of hazards (mezcla de
precipitaciones, “mixed precipitation”), rather than pointed advice required
from an expert guidance system.

Figure 6: Map, prompt, and response for bilingual hazard communication.

Figure 7: The second and third responses to identical prompting to that in
Fig. 6, performed in distinct conversation threads, regarding bilingual
communication of hazards.

Given established machine–human near-parity in natural-language translation
(Läubli et al., 2020), this poor technique of translation is disappointing and
surprising. Due to the black-box nature of GPT-4V, it is difficult to
determine reasons behind poor performance. Although the English response does
not suffer from inappropriately direct translation, it shares similar
shortcomings in vagueness and non-standard terminology:

  * •

There are hallucinations of scattered rain in the northeast and thunderstorms
in the northwest,

  * •

The spelling of travelling is not American English, highlighting errors in
localization for both languages,

  * •

The response should address hazards specifically rather than discussing
adverse weather in general — each sentence should carry weight given the need
for a concise summary.

To its credit, GPT-4V’s responses in both languages identify severe weather
potential in a broad swath from Texas to Illinois and use of state names
appear correctly in all three responses. This tentatively increases our
optimism that further tuning, prompt optimization, and a method of course-
correction during a conversation can all contribute to substantially more
useful responses.

In summary, issuing a bilingual hazard outlook is a complex task that requires
competence in image recognition, understanding in time and space,
communication to a lay-audience, and translation abilities. Further
investigation (not shown) revealed many errors were shared with a similar
Spanish–English translation task restricted to textual input. GPT-4V produces
responses that appear as if an English response was directly translated into
Spanish, perhaps stemming from the disproportionally high appearance (93% in
GPT-3) of English in the training corpora (Bender et al., 2021; Byrd, 2023),
in contrast to the open-access BLOOM model where English comprises about 30%
(BigScience Workshop et al., 2022). For GPT-4V, English may act as a bridge
language, especially if the corpus does not contain culturally nuanced weather
terminology. However, previous investigation found performance improved when
employing English as the bridging languages between less common translation
(Bakhshaei et al., 2010; Kunchukuttan, 2021). In pursuit of improved
meteorological public communication, systems constrained to domain-specific
terminology tailored for the community in question (Trujillo-Falcón et al.,
2021; Bitterman et al., 2023; Trujillo-Falcón et al., 2024) are likely to
remedy some lexical issues shown above.

##  5 Synthesis and Recommendations

###  5.1 Seeking fidelity in coherence

GPT-4V frequently gives coherent answers but in the manner of a student
attempting to veil their lack of knowledge with a wealth of regurgitation. The
answer may be misleading or incorrect (hallucinations), but the delivery is
convincing. The variety of responses is too wide for a given prompt, and the
language too vague, for applications such as scientific communication where
there is finite correct, useful information but many incorrect answers (a low
signal-to-noise ratio). This variety is likely a result of an excessively
large default temperature value that yields inappropriate creativity for
scientific tasks. Answers also contain useless filler text, awkward direct
translation of natural language, and vague geography.

With this said, it is remarkable we have technology that recognizes so much
content in images. There are glimpses of real utility, such as issuing and
self-evaluating a mock SPC-style outlook, and interpreting and explaining
weather charts. This promise must be balanced with the adage of the blind
squirrel: it will find an acorn eventually. Specific instructions (or
resources such as uploaded PDF guidelines) may assist with improving elements
such as preferred terminology. Further, to better discriminate between
meaningless coherence and useful truth, the supervising human is able to—and
should—cross-reference statements with established meteorological fact and
expert input. For complex tasks, a continuous conversation between human and
AI allows course-corrections, and is encouraged over one-shot attempts at
eliciting useful responses.

There is a trade-off between confidence/determinism and
creativity/uncertainty, and asking specifically for uncertainty and honesty
(to avoid hallucinations) was not consistently effective during testing. This
is a parallel of over-/under-confidence in a probabilistic weather forecast,
or over-/under-fitting an AI model. We show that self-evaluation is possible
with conceptual copies of GPT-4V’s own output, but whether GPT-4V actually
does more than emulate an emulation is unclear (Schaeffer et al., 2023).
Further, GPT-4V gives full confidence to responses that could be
misinterpreted. Such a misunderstanding of a prognosis fully accepted as true
is an example of “catastrophic error” in information theory (Pierce, 1980).
Indeed, when a system is perceived as infallible, an incorrect prediction
becomes exponentially more damaging as the probabilities linearly approach the
limits of zero or unity (Cover and Thomas, 2012). This can be remedied in
GPT-4V communication, as with humans, by instructions never to issue binary
forecasts and use of appropriate error estimates for the time and spatial
scale.

###  5.2 Recommendations

ChatGPT and its products, as with all AI assistants, are best considered a co-
pilot in academic realms—especially so during idea generation, simplifying
complex ideas, and narrowing the scope of large paragraphs. However, in
operations tasked with protecting lives and property, there is little room for
error in issuing timely, correct warnings for hazards, and little time for
thorough vetting of language-model output. Rigorous testing must be completed
before humans can be removed further from the operational loop. We do not have
detailed knowledge of how text is generated by a system that is neither
transparent nor explainable (Flora et al., 2024). Thus, it can be difficult to
anticipate and identify errors or refine model inputs (text or image prompts
in the case of GPT-4V) to improve model accuracy.

An effective session should resemble a conversation: be prepared to correct
and nudge the conversation to define an answer, and even reprimand GPT-4V for
laziness (https://openai.com/blog/new-embedding-models-and-api-updates,
accessed 25 January 2024). Remarkably, the latter’s cause is still unknown at
the time of writing. Hence, we recommend a specific first prompt while
continuing to course-correct and narrow scope during the chat. The drawback of
discussion is running out of “token memory” (i.e., how much of the
conversation GPT-4V remembers), demanding user recapitulation. Some answers
show a lack of logical consistency; the user must play the role of an
overarching “monitor” that is able to tell when it is wrong, why, and how to
correct this missing knowledge (Booch et al., 2021). This highlights the
importance of trustworthy, interpretable output from AI systems. Without human
trust in AI output, there is little reason to use generated text over, say,
seeking peer-reviewed or human-expert guidance.

###  5.3 Future work

Ultimately, a tornado was observed in Western Illinois during the time period
covered in Figs. 4 and 5. Further work could test whether AI can generate an
SPC forecast comparable in skill to a human. Future testing might ask for more
detailed responses from each emulation, such as asking GPT-4V to evaluate
without the SPC convection outlook first. Prompts and responses should be
concise to limit the risk of losing crucial information earlier in the
conversation.

We suggest potential research avenues that explore:

  * •

Ability to request nested emulations to create a synthesis “wisdom of crowds”
or ensemble approach to prompting (or by simply using multiple independent
chats with identical prompts),

  * •

Uses to improve accessibility to those visually impaired,

  * •

Prompt engineering, or finding an optimal question, especially in the presence
of high stochasticity (this includes testing the order of translation for
bilingual communication),

  * •

Manual control and optimization of the temperature or creativity parameter,

  * •

Open-source large multi-modal models and local optimization of the model to
meteorological applications, including a lower stochasticity value and
prescription of specific terms for use in multilingual warning communications,

  * •

Having GPT-4V iteratively critique and optimize its own prompts and responses,

  * •

Despite its failure to give more detail within the expert section, could
GPT-4V give discussion tailored for aviation, emergency managers, etc., with
appropriate conveyance of uncertainty in both complexity and geography. The
structure of experimentation could follow those within a human setting as
conducted by Shivers-Williams and Klockow-McClain (2021).

###### Acknowledgements.

JRL thanks faculty and students at the Department of Geography and Meteorology
at Valparaiso University, family and colleagues who gave frequent feedback on
real-life Generative AI output. The authors thank Kimberly Hoogewind and Sean
Ernst for useful input in the review stage of this paper. JRL and SNL are
funded by Uintah County Special Service District 1 and the Utah Legislature.
CKP’s contribution to this work comprised regular duties at federally funded
NOAA/NSSL. Funding for MLF and JETF was provided by NOAA/Office of Oceanic and
Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement
NA21OAR4320204, U.S. Department of Commerce. Partial funding for DMS was
provided to the University of Manchester by the Natural Environment Research
Council through Grants NE/V012681/1, NE/W000997/1, and NE/X018539/1. Co-
authors following JRL in the author list are alphabetic in order and
contributed equally to this manuscript. Weather charts herein and in
supplementary material are reproduced with kind permission of Pivotal Weather,
LLC, where labeled with a watermark. Outside of experiments, GPT-4 was used to
generate preliminary ideas for project development. No AI-generated text was
used verbatim herein. Solely textual data was generated for the present study
and is contained entirely within the Supplementary Material.

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##  Appendix A Deriving modified optimization

In this appendix, we give the proof of Theorem 4. We begin by proving the
following lemma:

######  Lemma 1.

For any constants $\varphi_{0},\varphi_{1}\geq 0$, the function $\hat{T}$
defined in (58) is a convex function with Legendre-Fenchel conjugate
$\hat{T}^{*}$ given by (60), and satisfies

$\displaystyle\hat{T}(\mathbf{g})=\hat{T}^{**}(\mathbf{g})=\sup_{\boldsymbol{\lambda}}\left(\mathbf{g}\cdot\boldsymbol{\lambda}-\hat{T}^{*}(\boldsymbol{\lambda})\right).$
(97)

###### Proof.

We start the proof by showing that $\hat{T}(\mathbf{g})$ is a convex function.
We first restate the definition of $\hat{T}$ from (58):

$\hat{T}(\mathbf{g})\coloneqq\varphi_{0}(\left(\max(\mathbf{g})-\min(\mathbf{g})\right)^{2}+\varphi_{1}\left(\max(\mathbf{g})-\min(\mathbf{g})\right).$

First note that $\max(\mathbf{g})$ is convex and $\min(\mathbf{g})$ is
concave. Thus, $\max(\mathbf{g})-\min(\mathbf{g})$ is convex; moreover, it is
non-negative, therefore, its square is also a convex function. By noting that
$\varphi_{0},\varphi_{1}\geq 0$, we conclude $\hat{T}(\mathbf{g})$ is a convex
function.

Now to prove its Legendre-Fenchel conjugate is the expression (60), we first
write out the definition of the conjugate (see e.g. [BV04] Chapter 3.3):

$\hat{T}^{*}(\boldsymbol{\lambda})=\sup_{\mathbf{g}}\left(\mathbf{g}\cdot\boldsymbol{\lambda}-\varphi_{0}(\left(\max(\mathbf{g})-\min(\mathbf{g})\right)^{2}-\varphi_{1}\left(\max(\mathbf{g})-\min(\mathbf{g})\right)\right).$
(98)

Let us first consider the case where the sum of the elements of
$\boldsymbol{\lambda}$ is zero, i.e., $\sum_{c}\lambda_{c}=0$. Note that for
any vector $\mathbf{g}$, we can define another vector $\tilde{\mathbf{g}}$,
whose elements are given by:

$\tilde{g}_{c}=\begin{cases}\max(g)&\text{ if }\lambda_{c}\geq 0\\\
\min(g)&\text{ otherwise }\end{cases}.$ (99)

It is not hard to see that for any vector $\mathbf{g}$, the quantity within
the supremum in (98) is upper bounded by the value with $\tilde{\mathbf{g}}$
in place of $\mathbf{g}$. Furthermore, letting $\mathcal{C}_{\geq 0}$ be the
set of $c$ values with $\lambda_{c}\geq 0$, we have

$\displaystyle\tilde{\mathbf{g}}\cdot\boldsymbol{\lambda}=\max(g)\sum_{c\in\mathcal{C}_{\geq
0}}\lambda_{c}+\min(g)\sum_{c\notin\mathcal{C}_{\geq
0}}\lambda_{c}=(\max(g)-\min(g)){\left\lVert\boldsymbol{\lambda}\right\rVert_{1}}/{2},$
(100)

using the fact that $\sum_{c\in\mathcal{C}_{\geq
0}}\lambda_{c}=-\sum_{c\notin\mathcal{C}_{\geq
0}}\lambda_{c}={\left\lVert\boldsymbol{\lambda}\right\rVert_{1}}/{2}$ since
$\sum_{c}\lambda_{c}=0$. Therefore, the RHS of (98) simplifies to

$\displaystyle\hat{T}^{*}(\boldsymbol{\lambda})$
$\displaystyle=\sup_{\mathbf{g}}\left(({\left\lVert\boldsymbol{\lambda}\right\rVert_{1}}/{2}-\varphi_{1})\left(\max(\mathbf{g})-\min(\mathbf{g})\right)-\varphi_{0}\left(\max(\mathbf{g})-\min(\mathbf{g})\right)^{2}\right)$
(101) $\displaystyle=\sup_{\beta\geq
0}\left(({\left\lVert\boldsymbol{\lambda}\right\rVert_{1}}/{2}-\varphi_{1})\beta-\varphi_{0}\beta^{2}\right),$
(102)

where in the second line, we use the fact that the optimization only depends
on the difference $\max(\mathbf{g})-\min(\mathbf{g})$. The optimization in
(101) can be solved using simple calculus which leads to the following:

$\hat{T}^{*}(\boldsymbol{\lambda})=\begin{cases}\frac{({\left\lVert\boldsymbol{\lambda}\right\rVert_{1}}/{2}-\varphi_{1})^{2}}{4\varphi_{0}}&\text{if}\left\lVert\boldsymbol{\lambda}\right\rVert_{1}\geq
2\varphi_{1}\\\
0&\text{if}\left\lVert\boldsymbol{\lambda}\right\rVert_{1}<2\varphi_{1}\end{cases}$
(103)

Let us now look at the situation where $\sum_{c}\lambda_{c}\neq 0$. Consider
the particular choice $\mathbf{g}=k\mathbf{1}$ for some $k\in\mathbb{R}$.
Taking the supremum over such choices of $\mathbf{g}$, from (98) we have:

$\displaystyle\hat{T}^{*}(\boldsymbol{\lambda})$
$\displaystyle\geq\sup_{k}\left(k\sum_{c}\lambda_{c}\right)$ (104)
$\displaystyle=+\infty,$ (105)

where the second equality follows by setting $k\rightarrow+\infty$ (resp.
$k\rightarrow-\infty$) if $\sum_{c}\lambda_{c}>0$ (resp.
$\sum_{c}\lambda_{c}<0$). Combining (103) with (104) results the expression in
(60).

Finally, to show that (97) holds, we simply use the fact that $\hat{T}$ is a
convex function with its domain being all of $\mathbb{R}^{|\mathcal{C}|-1}$.
Therefore its epigraph is a convex closed set, and hence it is equal to the
conjugate of its conjugate (see e.g. [BV04] Chapter 3.3.2), i.e. we have
$\hat{T}(\mathbf{g})=\hat{T}^{**}(\mathbf{g})=\sup_{\boldsymbol{\lambda}}\left(\mathbf{g}\cdot\boldsymbol{\lambda}-\hat{T}^{*}(\boldsymbol{\lambda})\right)$
as claimed. ∎

With this, we turn to proving the first equality (62) in Theorem 4. First,
recall that as discussed in Sec. 4.3, any $\mathbf{g}$ and $k_{g}$ satisfying
the condition $\mathbf{g}\cdot\mathbf{q}+k_{g}\leq
r_{\mathrm{cross}}(\mathbf{q})$ (for all $\mathbf{q}$) yields a valid
crossover min-tradeoff function. This means that in fact any gradient vector
$\mathbf{g}$ corresponds to some valid choice(s) of crossover min-tradeoff
function, simply by choosing $k_{g}$ to be some value satisfying that
condition (such values always exist because $r_{\mathrm{cross}}(\mathbf{q})$
is a non-negative convex function). Now observe that in the supremum over
crossover min-tradeoff functions in (61), for any fixed $\mathbf{g}$, the best
choice of $k_{g}$ would be the highest possible value satisfying that
condition. Specifically, this implies we should choose it to be as follows,
recalling that by definition an infimum is the highest possible lower bound on
a set (see [TSB+22] Sec. 5 for another perspective based on Lagrange duals):

$\displaystyle k_{g}$
$\displaystyle=\inf_{\mathbf{q}}\left(r_{\mathrm{cross}}(\mathbf{q})-\mathbf{g}\cdot\mathbf{q}\right)$
$\displaystyle=\inf_{J}\left(W(\rho^{\mathrm{g}}_{J})-\mathbf{g}\cdot\mathbf{\Phi}[\rho^{\mathrm{t}}_{J}]\right),$
(106)

where in the second line we have substituted in the definition of
$r_{\mathrm{cross}}(\mathbf{q})$ from (40), with
$\rho^{\mathrm{g}}_{J},\rho^{\mathrm{t}}_{J}$ being again understood as
functions of $J$ via (38). Hence, we should always take the scalar term
$k_{g}$ in the optimization (61) to be given by the above expression.

Substituting the formulas (97) and (106) into the argument of the supremum in
(61), it becomes

$\displaystyle\mathbf{g}\cdot\mathbf{q}^{\mathrm{hon}}+\inf_{J}\left(W(\rho^{\mathrm{g}}_{J})-\mathbf{g}\cdot\mathbf{\Phi}[\rho^{\mathrm{t}}_{J}]\right)-\sup_{\boldsymbol{\lambda}}\left(\mathbf{g}\cdot\boldsymbol{\lambda}-\hat{T}^{*}(\boldsymbol{\lambda})\right)$
$\displaystyle=\,$
$\displaystyle\inf_{(J,\boldsymbol{\lambda})\in\mathcal{D}}\left(W(\rho^{\mathrm{g}}_{J})+\hat{T}^{*}(\boldsymbol{\lambda})+\mathbf{g}\cdot\left(\mathbf{q}^{\mathrm{hon}}-\mathbf{\Phi}[\rho^{\mathrm{t}}_{J}]-\boldsymbol{\lambda}\right)\right),$
(107)

where $\mathcal{D}$ is a convex set defined as the set of tuples
$(J,\boldsymbol{\lambda})$ such that $J$ is a Choi matrix and
$\boldsymbol{\lambda}\in\mathbb{R}^{|\mathcal{C}|-1}$ satisfies
$\sum_{c}\lambda_{c}=0$ (we can restrict the optimization over
$\boldsymbol{\lambda}$ to such values without loss of generality, because
$\hat{T}(\mathbf{g})=+\infty$ whenever $\sum_{c}\lambda_{c}\neq 0$). Our goal
is to find the choice of $\mathbf{g}$ that maximizes the above value, in other
words to find the optimal solution $\mathbf{g}$ to

$\displaystyle\sup_{\mathbf{g}}\inf_{(J,\boldsymbol{\lambda})\in\mathcal{D}}\left(W(\rho^{\mathrm{g}}_{J})+\hat{T}^{*}(\boldsymbol{\lambda})+\mathbf{g}\cdot\left(\mathbf{q}^{\mathrm{hon}}-\mathbf{\Phi}[\rho^{\mathrm{t}}_{J}]-\boldsymbol{\lambda}\right)\right).$
(108)

(Note that in the above, the objective function is finite everywhere over the
domain $\mathcal{D}$, so we do not have issues with domain definitions.) We
now simply observe that the above problem is _precisely_ the Lagrange dual of
the constrained convex optimization

$\displaystyle\begin{aligned}
\inf_{(J,\boldsymbol{\lambda})\in\mathcal{D}}&\quad
W(\rho^{\mathrm{g}}_{J})+\hat{T}^{*}(\boldsymbol{\lambda})\\\ \text{
s.t.}&\quad\mathbf{q}^{\mathrm{hon}}-\mathbf{\Phi}[\rho^{\mathrm{t}}_{J}]-\boldsymbol{\lambda}=\mathbf{0},\end{aligned}$
(109)

and furthermore this constrained optimization satisfies strict feasibility,
for instance by setting $J=\mathbb{I}$ and
$\boldsymbol{\lambda}=\mathbf{q}^{\mathrm{hon}}-\mathbf{\Phi}[\rho^{\mathrm{t}}_{\mathbb{I}}]$
(this point indeed lies in the relative interior of $\mathcal{D}$, because
$J=\mathbb{I}$ is in the relative interior of the set of Choi matrices, and
$\sum_{c}\lambda_{c}=0$ by (66)). Hence by Slater’s condition ([BV04] Chapter
5.2.3), we have strong duality, i.e. the optimizations (108) and (109) have
the same optimal value, and the optimal dual solution $\mathbf{g}$ is
attained. From (109), we obtain the desired equality (62) by simply noting
that the $\sum_{c}\lambda_{c}=0$ restriction in the domain $\mathcal{D}$ is
already enforced by the
$\mathbf{q}^{\mathrm{hon}}-\mathbf{\Phi}[\rho^{\mathrm{t}}_{J}]-\boldsymbol{\lambda}=\mathbf{0}$
constraint (see (66)).

Turning to the next equality (63), it is a straightforward transformation of
the optimization problem that can be derived as follows. First observe that
the optimization in (62) is clearly lower bounded by the optimization in (63)
because every feasible point $(J,\boldsymbol{\lambda})$ in the former yields a
feasible point
$(J^{\prime},\boldsymbol{\lambda}^{\prime},\boldsymbol{\tau}^{\prime})$ of the
latter with the same objective value, simply by taking $J^{\prime}=J$,
$\boldsymbol{\lambda}^{\prime}=\boldsymbol{\lambda}$ and
$\tau^{\prime}_{c}=|\lambda_{c}|$. Conversely, the optimization in (63) is
also lower bounded by the optimization in (62), because every feasible point
$(J,\boldsymbol{\lambda},\boldsymbol{\tau})$ in the former yields a feasible
point $(J^{\prime},\boldsymbol{\lambda}^{\prime})$ of the latter with an
objective value that is no higher, by taking $J^{\prime}=J$ and
$\boldsymbol{\lambda}^{\prime}=\boldsymbol{\lambda}$ (in which case we have
$\sum_{c}\tau_{c}\geq\left\lVert\boldsymbol{\lambda}^{\prime}\right\rVert_{1}$
and hence $s\\!\left(\sum_{c}\tau_{c}/{2}\right)\geq
s\\!\left(\left\lVert\boldsymbol{\lambda}^{\prime}\right\rVert_{1}/2\right)=\hat{T}^{*}(\boldsymbol{\lambda}^{\prime})$,
since $s$ is a monotone increasing function and
$\sum_{c}\lambda^{\prime}_{c}=0$ by (66)). Hence, the optimizations in (62)
and (63) are equal.

Finally, we turn to proving the last bound (65). The intuition behind this
property is that as mentioned above, the optimal dual solution to (109) is in
fact the optimal choice of $\mathbf{g}$ in (61). However, since in arriving at
the SDP (64) we have implemented several transformations of the domain,
objective and constraints in the “fundamental” optimization (109), it is not
immediately clear whether this property is inherited by the dual of the SDP
(64). Hence, we shall instead directly prove the bound (65) via an appropriate
series of inequalities.

First note that the SDP (64) is again strictly feasible (e.g. following the
above ideas, by choosing $J=\mathbb{I}$,
$\boldsymbol{\lambda}=\mathbf{q}^{\mathrm{hon}}-\mathbf{\Phi}[\rho^{\mathrm{t}}_{\mathbb{I}}]$
and any $\tau_{c}>|\lambda_{c}|$) and hence by Slater’s condition, the dual is
attained and has the same optimal value $r_{\mathrm{SDP}}$. By the definition
of Lagrange duality, this means the optimal dual solution to the
$\mathbf{q}^{\mathrm{hon}}-\mathbf{\Phi}[\rho^{\mathrm{t}}_{J}]-\boldsymbol{\lambda}=\mathbf{0}$
constraint in the SDP (64) is a vector
$\mathbf{g}^{\star}\in\mathbb{R}^{\left|\mathcal{C}_{\setminus\perp}\right|}$
such that181818Here we are using the fact that SDP duals can be viewed as
Lagrange duals. Strictly speaking, when writing an SDP dual, one would usually
“dualize” all the constraints (including the implicit constraints enforcing
that $J$ is a Choi matrix), whereas in the following expression we have only
“dualized” the
$\mathbf{q}^{\mathrm{hon}}-\mathbf{\Phi}[\rho^{\mathrm{t}}_{J}]-\boldsymbol{\lambda}=\mathbf{0}$
constraint and not the other contraints. However, it is not difficult to show
that any dual feasible solution in the former version yields a dual feasible
solution for the latter version that has at least the same value; in
particular, an optimal dual solution in the former yields an optimal dual
solution in the latter.

$\displaystyle\begin{aligned}
r_{\mathrm{SDP}}=\;\inf_{(J,\boldsymbol{\lambda},\boldsymbol{\tau})\in\mathcal{D}^{\prime}}&\quad
L(\rho^{\mathrm{g}}_{J},\boldsymbol{\tau})+\mathbf{g}^{\star}\cdot\left(\mathbf{q}^{\mathrm{hon}}-\mathbf{\Phi}[\rho^{\mathrm{t}}_{J}]-\boldsymbol{\lambda}\right),\end{aligned}$
(110)

where for compactness we introduce the following notation:
$\mathcal{D}^{\prime}$ denotes the set of tuples
$(J,\boldsymbol{\lambda},\boldsymbol{\tau})$ such that $J$ is a Choi matrix
and $\boldsymbol{\lambda},\boldsymbol{\tau}\in\mathbb{R}^{|\mathcal{C}|-1}$
are vectors satisfying
$-\boldsymbol{\tau}\leq\boldsymbol{\lambda}\leq\boldsymbol{\tau}$ (i.e. it
incorporates the constraints other than
$\mathbf{q}^{\mathrm{hon}}-\mathbf{\Phi}[\rho^{\mathrm{t}}_{J}]-\boldsymbol{\lambda}=\mathbf{0}$
in the SDP (64)).

With this, note that given the above choice of gradient $\mathbf{g}^{\star}$,
if we choose the scalar term in $g^{\star}$ according to (106) then the
desired result (65) holds:

$\displaystyle
g^{\star}(\mathbf{q}^{\mathrm{hon}})-\hat{T}(\mathbf{g}^{\star})$
$\displaystyle=\mathbf{g}^{\star}\cdot\mathbf{q}^{\mathrm{hon}}+\inf_{J}\left(W(\rho^{\mathrm{g}}_{J})-\mathbf{g}^{\star}\cdot\mathbf{\Phi}[\rho^{\mathrm{t}}_{J}]\right)-\sup_{\boldsymbol{\lambda}}\left(\mathbf{g}^{\star}\cdot\boldsymbol{\lambda}-\hat{T}^{*}(\boldsymbol{\lambda})\right)$
$\displaystyle=\inf_{(J,\boldsymbol{\lambda})\in\mathcal{D}}\left(W(\rho^{\mathrm{g}}_{J})+\hat{T}^{*}(\boldsymbol{\lambda})+\mathbf{g}^{\star}\cdot\left(\mathbf{q}^{\mathrm{hon}}-\mathbf{\Phi}[\rho^{\mathrm{t}}_{J}]-\boldsymbol{\lambda}\right)\right)$
$\displaystyle=\inf_{(J,\boldsymbol{\lambda},\boldsymbol{\tau})\in\mathcal{D}^{\prime}}\left(W(\rho^{\mathrm{g}}_{J})+s\\!\left(\sum_{c}\tau_{c}/{2}\right)+\mathbf{g}^{\star}\cdot\left(\mathbf{q}^{\mathrm{hon}}-\mathbf{\Phi}[\rho^{\mathrm{t}}_{J}]-\boldsymbol{\lambda}\right)\right)$
$\displaystyle\geq\inf_{(J,\boldsymbol{\lambda},\boldsymbol{\tau})\in\mathcal{D}^{\prime}}\left(L(\rho^{\mathrm{g}}_{J},\boldsymbol{\tau})+\mathbf{g}^{\star}\cdot\left(\mathbf{q}^{\mathrm{hon}}-\mathbf{\Phi}[\rho^{\mathrm{t}}_{J}]-\boldsymbol{\lambda}\right)\right)$
$\displaystyle=r_{\mathrm{SDP}},$ (111)

where each line is justified as follows. The first and second lines are
obtained by simply substituting (97) and (106) and then simplifying the
resulting expression, similar to before. The third line holds by the same
arguments as in the above derivation of (63). The fourth line holds because
$L(\rho^{\mathrm{g}}_{J},\boldsymbol{\tau})$ is a lower bound on
$W(\rho^{\mathrm{g}}_{J})+s\\!\left(\sum_{c}\tau_{c}/{2}\right)$ by
hypothesis. Finally, the fifth line is simply the equality (110).

##  Appendix B Improved conversions between Rényi entropy and von Neumann
entropy

The bounds derived in [AHT24] are similar in spirit to the GEAT in the sense
that they involve the entropies of single rounds as discussed in Sec. 4.3.
However, the relevant quantity is instead the Rényi entropy
$H^{\uparrow}_{\alpha}(S|IE)$ or $H_{\alpha}(S|IE)$ (depending on exactly
which bound in [AHT24] is considered — the former is usually relevant for
device-dependent protocols; see Theorem 3 in that work), rather than the von
Neumann entropy $H(S|IE)$.191919There are also a number of other improvements
that significant sharpen the finite-size bounds and entirely avoid the
separate optimization of min-tradeoff function choice, but we do not discuss
those points further here. In principle, to obtain the tightest finite-size
key rates from that work, one should directly analyze those Rényi entropies
rather than $H(S|IE)$. However, since an extensive body of work has already
been devoted to the latter [CML16, WLC18, WL22, NUL23, KL24] (including the
approaches we used here for formalizing the $W$ function in Sec. 6–7), it may
still be useful to bound the former in terms of the latter, so that the
approaches from those works can be applied. We now present some methods for
doing so that may be particularly useful for optical QKD protocols — in
particular, some of the resulting finite-size corrections roughly depend on
the _detected_ rather than _total_ number of rounds, which can be a
significant difference.

######  Remark 3.

In some applications of entropy accumulation, one may have to consider the
entropies $H_{\alpha}(SC|IE)$ rather than just $H_{\alpha}(S|IE)$ (recall that
$C$ is the register containing data for the acceptance test) — the former is
potentially relevant for protocols where the $C$ registers might not satisfy
the non-signaling conditions of the GEAT; we defer all details to [AHT24]. The
bounds presented below generalize straightforwardly to that scenario, except
they might depend on $\dim(SC)$ rather than $\dim(S)$. We stress however that
a useful proof tactic is that if $C$ can be “projectively reconstructed” from
$SIE$ in the sense of [DFR20, Lemma B.7], then that lemma gives us

$\displaystyle H_{\alpha}(SC|IE)=H_{\alpha}(S|IE)\quad\text{and}\quad
H(SC|IE)=H(S|IE),$ (112)

so in such scenarios we can freely add or remove $C$ from the left-hand-side
conditioning registers using these relations, allowing the following bounds to
be applied directly.

To begin, a known simple bound in terms of $H(S|IE)$ is [DFR20, Lemma B.9]:

$\displaystyle H^{\uparrow}_{\alpha}(S|IE)\geq H_{\alpha}(S|IE)\geq
H(S|IE)-(\alpha-1)\log^{2}\left(1+2\dim(S)\right),$ (113)

for any $\alpha\in\left(1,1+\frac{1}{\log\left(1+2\dim(S)\right)}\right)$. A
more elaborate bound was derived in [DF19, Corollary IV.2 with Corollary
III.5], of the form

$\displaystyle H^{\uparrow}_{\alpha}(S|IE)\geq H_{\alpha}(S|IE)\geq
H(S|IE)-(\alpha-1)\frac{\ln
2}{2}\log^{2}\left(1+2\dim(S)^{\kappa}\right)-\widetilde{K}(\alpha)(\alpha-1)^{2},$
(114)

for any $\alpha\in(1,2)$, where $\kappa=1$ if the $S$ system is classical and
$\kappa=2$ if it is quantum (for QKD we would of course focus on the former
case). The quantity $\widetilde{K}(\alpha)$ is a somewhat complicated
expression similar to the $K(\alpha)$ term we presented in Eq. (1), but it can
be explicitly upper bounded.202020While it involves various other Rényi
entropies, one can simply apply crude bounds on those entropies in terms of
the system dimensions without too much loss of tightness, because in any case
it only affects a term of order $O((\alpha-1)^{2})$. By comparing the
prefactors on the $(\alpha-1)$ terms, we see that for classical $S$, the bound
(114) is substantially better than (113) whenever $\alpha$ is close enough to
$1$ for the $\widetilde{K}(\alpha)(\alpha-1)^{2}$ term to be negligible. (If
$S$ is quantum, it can still be better if $\dim(S)$ is small.)

Bounds of essentially the above form were often implicitly used in many
theorems involving Rényi entropies, including the proofs of the EAT and GEAT
bounds in [DFR20, DF19, MFS+22] (albeit in a fairly complicated fashion).
However, we now show that the bounds can be substantially refined for
applications in optical QKD. We begin by stating the following lemma:

######  Lemma 2.

Consider any $w\in[0,1]$ and any states
$\nu,\nu^{\prime},\nu^{\prime\prime}\in\operatorname{S}_{=}(CQ)$ with
classical $C$, such that
$\nu_{CQ}=w\nu^{\prime}_{CQ}+(1-w)\nu^{\prime\prime}_{CQ}$.212121We do not
require $\nu^{\prime},\nu^{\prime\prime}$ to have disjoint supports, so for
instance we do not need an explicit classical register in $\nu$ indicating
whether $\nu^{\prime}$ or $\nu^{\prime\prime}$ “occurred”. Let
$\Delta_{\alpha}$ denote

$\displaystyle\Delta_{\alpha}\coloneqq\frac{\alpha-1}{\alpha}\frac{\ln
2}{2}\log\dim(C).$ (115)

Then for all $\alpha\in(1,\infty)$ we have:

$\displaystyle\begin{gathered}H^{\uparrow}_{\alpha}(C|Q)_{\nu}\geq(1-\Delta_{\alpha})wH^{\uparrow}_{\alpha}(C|Q)_{\nu^{\prime}},\\\
H_{\alpha}(C|Q)_{\nu}\geq(1-\alpha\Delta_{\alpha})wH_{\alpha}(C|Q)_{\nu^{\prime}},\end{gathered}$
(118)

and for all $\alpha\in\left(1,1+\frac{2}{\ln\dim(C)}\right]$ we have:

$\displaystyle\begin{gathered}H^{\uparrow}_{\alpha}(C|Q)_{\nu}\geq
wH^{\uparrow}_{\alpha}(C|Q)_{\nu^{\prime}}\left(1-(1-w)\Delta_{\alpha}-2{w^{2}}\Delta_{\alpha}^{2}\right)\\\
H_{\alpha}(C|Q)_{\nu}\geq
wH_{\alpha}(C|Q)_{\nu^{\prime}}\left(1-(1-w)\alpha\Delta_{\alpha}-2{w^{2}}\alpha^{2}\Delta_{\alpha}^{2}\right).\end{gathered}$
(121)

We defer the proof to the end of this appendix, first discussing its
implications and applications here. We focus mainly on the first bound (118)
as it is somewhat simpler. The point of it is that in QKD protocols, we often
have the property that any state $\nu_{SIE}$ that can be produced in a single
round (after the measurements, announcements and sifting) indeed has the form
$\nu=w\nu^{\prime}+(1-w)\nu^{\prime\prime}$, where furthermore
$\nu^{\prime\prime}$ is such that we cannot get lower bounds on
$H_{\alpha}(S|IE)_{\nu^{\prime\prime}}$ other than the trivial lower bound of
zero. As a concrete example, in the decoy-state analysis in Sec. 7,
$\nu^{\prime\prime}$ could be the state conditioned on the event222222This is
a fairly elaborate event, so in potential future applications, one might want
to instead just focus on a subset of the conditions in this event for
simplicity, though the resulting bounds would be less tight.

$\displaystyle\text{[more than $1$ photon was sent] or [the round was sifted
out]},$ (122)

since when this event happens, either Eve can learn $S$ perfectly or the value
of $S$ is set to a deterministic value. Note that the value of $w$ can be
“variable” (e.g. it depends on Eve’s attack) rather than a fixed value; for
instance, this is indeed the case in the above example, because the sifting
procedure involves checking whether Bob received a detection, and the
probability of this can vary depending on Eve’s attack.

For such a state (with classical $S$), its von Neumann entropy would satisfy

$\displaystyle H(S|IE)_{\nu}\geq wH(S|IE)_{\nu^{\prime}},$ (123)

by concavity of von Neumann entropy, with equality whenever
$\nu^{\prime},\nu^{\prime\prime}$ have disjoint supports and
$H_{\alpha}(S|IE)_{\nu^{\prime\prime}}=0$. Most numerical approaches for
studying such states, including those in this work, would implicitly use this
(or similar ideas) to replace $H(S|IE)_{\nu}$ with the lower bound
$wH(S|IE)_{\nu^{\prime}}$ and study the latter instead. Now observe that the
bound (118) states that we have an approximate analogous result
$H^{\uparrow}_{\alpha}(S|IE)_{\nu}\gtrsim
wH^{\uparrow}_{\alpha}(S|IE)_{\nu^{\prime}}$ for Rényi entropy, up to a
prefactor that approaches $1$ as $\alpha\to 1$. Consequently, if we _now_
bound $H^{\uparrow}_{\alpha}(S|IE)_{\nu^{\prime}}$ using either (113) or
(114), we conclude that the following bounds hold (for the same ranges of
$\alpha$ as in (113)–(114)):

$\displaystyle
H^{\uparrow}_{\alpha}(S|IE)\geq\left(1-\Delta_{\alpha}\right)\left(wH(S|IE)_{\nu^{\prime}}-w(\alpha-1)\log^{2}\left(1+2\dim(S)\right)\right),$
(124) $\displaystyle
H^{\uparrow}_{\alpha}(S|IE)\geq\left(1-\Delta_{\alpha}\right)\left(wH(S|IE)_{\nu^{\prime}}-w(\alpha-1)\frac{\ln
2}{2}\log^{2}\left(1+2\dim(S)\right)-w\widetilde{K}(\alpha)(\alpha-1)^{2}\right),$
(125)

where in the second case we have set $\kappa=1$ since $S$ is classical. Note
that the prefactors of $1-\Delta_{\alpha}$ are just constants, and thus these
lower bounds should still be numerically tractable under the methods used in
this work — recall that these numerical approaches were likely to be analyzing
$wH(S|IE)_{\nu^{\prime}}$ in place of $H(S|IE)_{\nu}$ anyway, and the other
terms in these bounds are linear in $w$ (and therefore also usually linear in
the optimization variables in the relevant parametrizations, thereby retaining
convexity). In fact, for some simpler use cases the value $w$ is in fact a
constant in the optimization, making this even easier to analyze — for
instance, this holds if e.g. $w$ only represents the probability of sending at
most 1 photon in a single round.

The critical benefit of using these bounds instead of (113)–(114) directly is
that the correction terms have picked up a prefactor of $w$, which
substantially reduces their magnitude if $w$ is small. In fact, applying the
heuristic scaling analysis of $\alpha$ described in [AHT24, Sec. 5.1.1]
informally suggests that it might cause the finite-size correction terms on
the final Rényi entropy bounds to scale as $O(\sqrt{wn})$ instead of
$O(\sqrt{n})$ (putting aside the complication that $w$ might potentially be an
optimization variable rather than a constant, as well as some subtleties
regarding the scaling of a “penalty function” term in the [AHT24] results).
This change is especially dramatic if for instance $w$ is the probability of
the round being detected (or the even more restrictive event we defined above
in (122)), because then $wn$ roughly corresponds to the number of detected
rounds, which can be orders of magnitude smaller than $n$ at high loss.

This comes at a small price of having an overall prefactor of
$1-\Delta_{\alpha}$ that reduces the values slightly. However, we can roughly
quantify the maximum possible effect of this prefactor with a series of crude
bounds, focusing for instance on the bound (125):

$\displaystyle\left(1-\frac{\alpha-1}{\alpha}\frac{\ln
2}{2}\log\dim(S)\right)\bigg{(}wH(S|IE)_{\nu^{\prime}}-w(\alpha-1)\frac{\ln
2}{2}\log^{2}\left(1+2\dim(S)\right)-O((\alpha-1)^{2})\bigg{)}$
$\displaystyle=$ $\displaystyle
wH(S|IE)_{\nu^{\prime}}-wH(S|IE)_{\nu^{\prime}}\frac{\alpha-1}{\alpha}\frac{\ln
2}{2}\log\dim(S)-w(\alpha-1)\frac{\ln
2}{2}\log^{2}\left(1+2\dim(S)\right)-O((\alpha-1)^{2})$ $\displaystyle\geq$
$\displaystyle wH(S|IE)_{\nu^{\prime}}-w(\alpha-1)\frac{\ln
2}{2}\log^{2}\dim(S)-w(\alpha-1)\frac{\ln
2}{2}\log^{2}\left(1+2\dim(S)\right)-O((\alpha-1)^{2})$ $\displaystyle\geq$
$\displaystyle wH(S|IE)_{\nu^{\prime}}-w(\alpha-1)(\ln
2)\log^{2}\left(1+2\dim(S)\right)-O((\alpha-1)^{2}).$ (126)

From this we immediately see that the new bound (125) should be better than
its previous counterpart (114) whenever $w\leq 1/2$ and $(\alpha-1)^{2}$ is
small; in fact, since the above bounds were somewhat crude, it is likely to be
better over a larger range than that. (On the other hand, it does highlight
that if for instance we simply consider $w$ to be the probability of a
generation round, i.e. $w=1-\gamma$, then this approach may not give an
improvement, since $\gamma$ is often small.)

Finally, we briefly comment on the other bound (121) in Lemma 2. Observe that
it is slightly sharper than (118), in that it has an additional prefactor of
$1-w$ on the $O(\alpha-1)$ term, hence making that term smaller.
Unfortunately, it poses potential challenges for numerical work because it has
nonlinear dependencies on $w$. This may not be a problem if we are considering
examples in which $w$ is a fixed value (for instance, the probability of
sending at most 1 photon), so that the prefactor in (121) is simply a
constant, but in situations where $w$ is an optimization variable, it could be
difficult to handle in this bound. Furthermore, recalling that we are most
likely interested in scenarios where $w$ is small, the improvement is unlikely
to be substantial. Hence this version may have somewhat limited use.

We wrap up by presenting the proof of Lemma 2. The informal intuition behind
the proof is that since the Rényi entropies converge to von Neumann entropy as
$\alpha\to 1$, we might expect to get results analogous to the von Neumann
entropy case via Taylor expansions about $\alpha=1$.

###### Proof.

Extend $\nu_{CQ}$ onto a classical register $Z$ via
$\nu_{CQZ}=w\nu^{\prime}_{CQ}\otimes\left|0\right>\\!\left<0\right|_{Z}+(1-w)\nu^{\prime\prime}_{CQ}\otimes\left|1\right>\\!\left<1\right|_{Z}$.
We first prove the bound on $H_{\alpha}(C|Q)_{\nu}$ in (118) (we consider this
case first because the intermediate formulas are slightly simpler than the
$H^{\uparrow}_{\alpha}(C|Q)_{\nu}$ case). Specifically, noting that $C$ is
classical so its entropies are non-negative, we find the claimed result for
all $\alpha\in(1,\infty)$:

$\displaystyle H_{\alpha}(C|Q)_{\nu}$ $\displaystyle\geq
H_{\alpha}(C|QZ)_{\nu}$
$\displaystyle=\frac{1}{1-\alpha}\log\left(w2^{(1-\alpha)H_{\alpha}(C|Q)_{\nu^{\prime}}}+(1-w)2^{(1-\alpha)H_{\alpha}(C|Q)_{\nu^{\prime\prime}}}\right)$
$\displaystyle=\frac{1}{1-\alpha}\log\left(w2^{(1-\alpha)H_{\alpha}(C|Q)_{\nu^{\prime}}}+(1-w)\right)\text{
since }H_{\alpha}(C|Q)_{\nu^{\prime\prime}}\geq 0$
$\displaystyle=-\frac{1}{(\ln 2)(\alpha-1)}\ln\left(we^{-(\ln
2)(\alpha-1)H_{\alpha}(C|Q)_{\nu^{\prime}}}+(1-w)\right)$ $\displaystyle\geq
wH_{\alpha}(C|Q)_{\nu^{\prime}}\left(1-\frac{(\ln
2)(\alpha-1)H_{\alpha}(C|Q)_{\nu^{\prime}}}{2}\right)\;\text{since }(\ln
2)(\alpha-1)H_{\alpha}(C|Q)_{\nu^{\prime}}\geq 0$ $\displaystyle\geq
wH_{\alpha}(C|Q)_{\nu^{\prime}}\left(1-(\alpha-1)\frac{\ln
2}{2}\log\dim(S)\right)=wH_{\alpha}(C|Q)_{\nu^{\prime}}\left(1-\alpha\Delta_{\alpha}\right),$
(127)

where in the second line we use [Tom16, Proposition 5.1], in the third line we
use the fact that the expression is nondecreasing in
$H_{\alpha}(C|Q)_{\nu^{\prime\prime}}$, and in the fifth line we apply the
following inequality that holds for all $w\in[0,1]$ and $x\in[0,\infty)$:

$\displaystyle-\ln\left(we^{-x}+(1-w)\right)\geq-\ln\left(w\left(-x+\frac{x^{2}}{2}\right)+1\right)\geq
w\left(x-\frac{x^{2}}{2}\right)=wx\left(1-\frac{x}{2}\right),$ (128)

where the first inequality uses the bound $e^{-x}\leq 1-x+x^{2}/2$ for
$x\in[0,\infty)$ (and implicitly the fact that the bounding quantity
$1-x+x^{2}/2$ is still non-negative so the logarithm value is still well-
defined), and the second inequality is the generic bound $\ln(1-x)\leq-x$ for
all $x$ in the domain, i.e. $x\in(-\infty,1)$. (These bounds on the
exponential and logarithm functions can be obtained using Taylor’s theorem
with mean-value form of the remainder term, noting that the sign of the
remainder term is fixed in the regimes we consider; alternatively, the bound
on the logarithm function can be obtained directly from the fact that it is
concave.)

The $H^{\uparrow}_{\alpha}$ bound in (118) follows from similar calculations,
except we start with a slightly different relation for $H^{\uparrow}_{\alpha}$
in [Tom16, Proposition 5.1],

$\displaystyle
H^{\uparrow}_{\alpha}(C|QZ)_{\nu}=\frac{\alpha}{1-\alpha}\log\left(w2^{\frac{1-\alpha}{\alpha}H^{\uparrow}_{\alpha}(C|Q)_{\nu^{\prime}}}+(1-w)2^{\frac{1-\alpha}{\alpha}H^{\uparrow}_{\alpha}(C|Q)_{\nu^{\prime\prime}}}\right),$
(129)

then carry out the remaining calculations the same way.

To prove the remaining bounds, we use higher-order Taylor expansion bounds to
sharpen the inequality (128). Specifically, using the same bound on the
exponential function but instead combining it with $\ln(1-x)\leq-x-x^{2}/2$
for $x\in[0,1)$, we see that the following holds for all $w\in[0,1]$ and
$x\in[0,2]$:

$\displaystyle-\ln\left(we^{-x}+(1-w)\right)$
$\displaystyle\geq-\ln\left(w\left(-x+\frac{x^{2}}{2}\right)+1\right)$
$\displaystyle\geq
w\left(x-\frac{x^{2}}{2}\right)+\frac{w^{2}}{2}\left(x-\frac{x^{2}}{2}\right)^{2}\text{
since }x\leq 2\implies w\left(x-\frac{x^{2}}{2}\right)\geq 0$
$\displaystyle=wx-\left(\frac{w}{2}-\frac{w^{2}}{2}\right)x^{2}-\frac{w^{2}}{2}x^{3}+\frac{w^{2}}{8}x^{4}$
$\displaystyle=wx\left(1-\frac{1-w}{2}x-\frac{w}{2}x^{2}+\frac{w}{8}x^{3}\right).$
(130)

With this we conclude that via analogous calculations to above, the bounds in
(121) hold. Note that in doing so we dropped the last term in the above bound,
because it is a positive contribution for which there may not be any
straightforward lower bounds other than the trivial bound of zero (unless we
allow the final bound to have nonlinear dependence on the entropy term as
well). We also highlight that the (somewhat loosely constructed) restriction
on allowed $\alpha$ values is to ensure that $(\ln
2)(\alpha-1)H_{\alpha}(C|Q)_{\nu^{\prime}}$ or $(\ln
2)\frac{1-\alpha}{\alpha}H^{\uparrow}_{\alpha}(C|Q)_{\nu^{\prime}}$ is upper
bounded by

$\displaystyle(\ln 2)(\alpha-1)\log\dim(S)=(\alpha-1)\ln\dim(S)\leq 2,$ (131)

so we can validly apply the above bound.

One could also consider higher-order bounds on the exponential function
instead, but since the next valid bound that holds for all $x\in[0,\infty)$
requires going to the next even-order expansion (i.e. $e^{-x}\leq
1-x+x^{2}/2!-x^{3}/3!+x^{4}/4!$), the resulting bound is very lengthy and
seems unlikely to be of much use (furthermore, at best it only sharpens the
$O((\alpha-1)^{2})$ corrections in the final results, which may already have
been fairly negligible). Another option would be to directly analyze the
Taylor expansion of the function $-\ln\left(we^{-x}+(1-w)\right)$, taking care
to ensure the remainder term has the correct sign in the required regimes, but
again the improvements would only be on the $O((\alpha-1)^{2})$ corrections. ∎