Composition of Privacy Guarantees: Classical and Quantum

I try to always consider the classical alternative to any quantum computation or quantum information-theoretic primitive. This is a deliberate choice. I am not a pure quantum theorist in the sense of studying quantum models in isolation, nor am I interested in quantum advantage as an article of faith. Rather, my goal is to delineate (as precisely as possible) the boundary between what classical and quantum theories can guarantee, especially when privacy guarantees are composed over time, across mechanisms, or between interacting systems.

In the context of privacy, composition is where theory meets reality: real systems are never single-shot. They involve repeated interactions, adaptive adversaries, and layered mechanisms. Quantum information introduces new phenomena (entanglement, non-commutativity, and measurement disturbance) that complicate classical intuitions about composition. At the same time, classical privacy theory has developed remarkably robust tools that often remain surprisingly competitive, even when quantum resources are allowed.

The guiding question of this post is therefore not “What can quantum systems do that classical ones cannot?” but rather:

When privacy guarantees are composed, what genuinely changes in the transition from classical to quantum. And what does not?

By keeping classical alternatives explicitly in view, we can better understand which privacy phenomena are inherently quantum, which are artifacts of modeling choices, and which reflect deeper structural principles that transcend the classical vs. quantum divide.

Classical Composition of Differential Privacy

Recall the definition of differential privacy:

Approximate Differential Privacy
Let $\mathcal{X}$ denote the data universe and let $\mathcal{D} \subseteq \mathcal{X}^n$ be the set of datasets.
Two datasets $D,D'\in\mathcal{D}$ are called neighbors, denoted $D\sim D'$ , if they differ in the data of exactly one individual.

A (possibly randomized) algorithm $\mathcal{M} : \mathcal{D} \to (\mathcal{Y},\mathcal{F})$ is said to be
$(\varepsilon,\delta)$ -differentially private if for all neighboring datasets $D\sim D'$ and all measurable events
$S \in \mathcal{F}$ ,
$\Pr[\mathcal{M}(D)\in S] \;\le\; e^{\varepsilon}\Pr[\mathcal{M}(D')\in S] + \delta$ .

It has been shown in a few references/textbooks that basic composition holds for differential privacy. We recall the statement:

Theorem (Basic sequential composition for approximate differential privacy)
Fix $k\in\mathbb{N}$ . For each $i\in\{1,\ldots,k\}$ let $\mathcal{M}_i$ be a (possibly randomized) algorithm that, on input a dataset $D$ , outputs a random variable in some measurable output space $(\mathcal{Y}_i,\mathcal{F}_i)$ .
Assume that for every $i$ , $\mathcal{M}_i$ is $(\varepsilon_i,\delta_i)$ -differentially private.

Define the $k$ -round interactive (sequential) mechanism $\mathcal{M}$ as follows: on input $D$ , for $i=1,\ldots,k$ , it outputs $Y_i \leftarrow \mathcal{M}_i (D; Y_1,\ldots,Y_{i-1}),$
where $\mathcal{M}_i(\cdot; y_{<i})$ denotes the $i$ th mechanism possibly chosen adaptively as a (measurable) function of the past transcript $y_{<i}=(y_1,\ldots,y_{i-1})$ .
Let $Y=(Y_1,\ldots,Y_k)$ denote the full transcript in the product space
$(\mathcal{Y},\mathcal{F}) := \prod_{i=1}^k (\mathcal{Y}_i,\mathcal{F}_i)$ .

Then $\mathcal{M}$ is $\left(\sum_{i=1}^k \varepsilon_i,\ \sum_{i=1}^k \delta_i\right)$ -differentially private.

In particular, if $\varepsilon_i=\varepsilon$ and $\delta_i=\delta$ for all $i$ , then $\mathcal{M}$ is $(k\varepsilon, k\delta)$ -differentially private.

What happens in the quantum setting?

Composition of Quantum Differential Privacy

A central “classical DP intuition” we have already set up is: once you have per-step privacy bounds, you can stack them, and in the simplest form the parameters add. e.g., $(\varepsilon, \delta)$ adds across rounds. In the quantum world, however, DP is commonly defined operationally against arbitrary measurements; and this makes the usual classical composition proofs, which rely on a scalar privacy-loss random variable, no longer directly applicable.

In a recent work, Theshani Nuradha and I show two complementary points, one negative (a barrier) and one positive:

Composition can fail in full generality for approximate QDP (POVM-based).
We show that if you allow correlated joint implementations when combining mechanisms/channels, then “classical-style” composition need not hold: even channels that are “individually perfectly private” can lose privacy drastically when composed in this fully general way.
Composition can be restored under explicit structural assumptions.
Then we identify a regime where you can recover clean composition statements: tensor-product channels acting on product neighboring inputs. In that regime, we propose a quantum moments accountant built from an operator-valued notion of privacy loss and a matrix moment-generating function (MGF).
How we get operational guarantees (despite a key obstacle).
A subtlety we highlight: the Rényi-type divergence we consider for the moments accountant does not satisfy a data-processing inequality. Nevertheless, we prove that controlling appropriate moments is still enough to upper bound measured Rényi divergence, which does correspond to operational privacy against arbitrary measurements.
End result: advanced-composition-style behavior (in the right setting).
Under those structural assumptions, the paper obtains advanced-composition-style bounds with the same leading-order behavior as in classical DP. i.e., you can once again reason modularly about long pipelines, but only after carefully stating what “composition” means (i.e., joint, tensor-product, factorized) physically/operationally in the quantum setting.

Check out the paper. Feedback/comments are welcome!

The Usefulness of “Useless” Knowledge (and Why AI Makes Flexner Even More Right)

I just finished reading The Usefulness of Useless Knowledge again, this time with the perspective of living through a period of rapid technological acceleration driven by AI. On an earlier reading, Flexner’s defense of curiosity-driven inquiry felt aspirational and almost moral in tone, a principled argument for intellectual freedom. On rereading, it feels more diagnostic. Many of the tensions he identified (i.e., between short-term utility and long-term understanding, between institutional incentives and genuine discovery) now play out daily in how we fund, evaluate, and deploy AI research. What has changed is not the structure of his argument, but its urgency: in a world increasingly optimized for immediate outputs, Flexner’s insistence that transformative advances often arise from questions with no obvious application reads less like an idealistic manifesto and more like a practical warning.

In 1939, on the eve of a world war, Abraham Flexner published a slim, stubbornly optimistic essay with a mischievous title: The Usefulness of Useless Knowledge. His claim is not that practical work is bad. It’s that the deep engine of civilization is often curiosity that doesn’t start with an application in mind, and that trying to force every idea to justify itself immediately is a reliable way to stop the next revolution before it begins.

Robbert Dijkgraaf’s companion essay (and related pieces written from his vantage point at the Institute for Advanced Study) updates Flexner’s argument for a world that is now built out of microelectronics, networks, and software; this is exactly the substrate on which modern AI sits. Reading them together today feels like watching two people describe the same phenomenon across two eras: breakthroughs are usually the delayed interest on “useless” questions.

Below is a guided tour of their core ideas, with a detour through the current AI moment, where “useless” knowledge is quietly doing most of the work.

Flexner’s central paradox: curiosity first, usefulness later

Flexner’s essay is a defense of a particular kind of intellectual freedom: the right to pursue questions without writing an ROI memo first.

Dijkgraaf highlights one of Flexner’s most quoted lines (and the one that best captures the whole stance): “Curiosity… is probably the outstanding characteristic of modern thinking… and it must be absolutely unhampered.”

That “must” is doing a lot of work. Flexner isn’t saying that applications are optional. He’s saying the route to them is often non-linear and hard to predict. He even makes the institutional point: a research institute shouldn’t justify itself by promising inventions on a timeline. Instead: “We make ourselves no promises… [but] cherish the hope that the unobstructed pursuit of useless knowledge” will matter later.

Notice the subtlety: he hopes it will matter, but he refuses to make that the official rationale. Why? Because if you only fund what looks useful today, you’ll underproduce the ideas that define tomorrow.

The “Mississippi” model of discovery (and why it matters for AI)

Flexner is unusually modern in how he describes the innovation pipeline: not as single geniuses striking gold, but as a long chain of partial insights that only later “click.”

He writes: “Almost every discovery has a long and precarious history… Science… begins in a tiny rivulet… [and] is formed from countless sources.”

This is basically an antidote to the myth that research can be managed like a factory. You can optimize a pipeline once you know what the pipeline is. But when you’re still discovering what questions are even coherent, “efficiency” often means “premature narrowing.”

AI is a perfect example of the Mississippi model. Modern machine learning is not one idea; it’s a confluence:

mathematical statistics + linear algebra,
optimization + numerical computing,
information theory + coding,
neuroscience metaphors + cognitive science,
hardware advances + systems engineering,
and now massive-scale data and infrastructure.

Much of that was, at some point, “not obviously useful” until it suddenly was.

Flexner’s warning: the real enemy is forced conformity

Flexner’s defense of “useless knowledge” is not only about technology; it’s about human freedom. He’s writing in a period where universities were being pushed into ideological service, and he argues that the gravest threat is not wrong ideas, but the attempt to prevent minds from ranging freely.

One of his sharpest lines: “The real enemy… is the man who tries to mold the human spirit so that it will not dare to spread its wings.”

If you read that in 2025, it lands uncomfortably close to modern pressures on research:

“Only fund what’s immediately commercial.”
“Only publish what’s trendy.”
“Only study what aligns with the current institutional incentive gradient.”
“Only build what can be shipped next quarter.”

And in AI specifically:

“Only do work that scales.”
“Only do benchmarks.”
“Only do applied product wins.”

Flexner isn’t anti-application; he’s anti-premature closure.

Dijkgraaf’s update: society runs on knowledge it can’t fully see anymore

Dijkgraaf’s companion essay takes Flexner’s stance and says, essentially: look around, Flexner won. The modern world is built out of the long tail of basic research.

He gives a crisp late-20th-century example: the World Wide Web began as a collaboration tool for particle physicists at CERN (introduced in 1989, made public in 1993). He ties that to the evolution of grid and cloud computing developed to handle scientific data, technology that now undergirds everyday internet services. Then he makes a claim that matters a lot for AI policy debates: fundamental advances are public goods (i.e., they diffuse beyond any single lab or nation).That’s an especially relevant lens for AI, where:

open ideas (architectures, optimization tricks, safety methods) propagate fast,
but compute, data, and deployment concentrate power.

If knowledge is a public good, then a society that starves basic research is quietly selling off its future, even if it still “uses” plenty of science in the present.

AI as a case study in “useful uselessness”

Here’s a helpful way to read Flexner in the age of AI:

A) “Useless” questions that became AI infrastructure

Many of the questions that shaped AI looked abstract or niche before they became inevitable:

How do high-dimensional models generalize?
When does overparameterization help rather than hurt?
What is the geometry of optimization landscapes?
How can representation learning capture structure without labels?
What are the limits of compression, prediction, and inference?

These don’t sound like product requirements. They sound like “useless” theory, until you realize they govern whether your model trains at all, whether it’s robust, whether it leaks private data, whether it can be aligned, and whether it fails safely.

Flexner’s point isn’t that every abstract question pays off. It’s that you can’t pre-identify the ones that will, and trying to do so narrows the search too early.

B) “Tool-making” is often the hidden payoff

Dijkgraaf emphasizes that pathbreaking research yields tools and techniques in indirect ways. (ias.edu)
AI progress has been exactly this: tool-making (optimizers, architectures, pretraining recipes, eval frameworks, interpretability methods, privacy-preserving techniques) that later becomes the platform everyone builds on.

C) The scary twist: usefulness for good and bad

Flexner also notes that discoveries can become instruments of destruction when repurposed. He uses chemical and aviation examples to make the point.

AI has the same dual-use character:

The same generative model family can draft medical summaries or automate phishing.
The same computer vision advances can improve accessibility or expand surveillance.
The same inference tools can find scientific patterns or extract sensitive attributes.

Flexner’s framework doesn’t solve dual-use, but it forces honesty: the ethical challenge isn’t a reason to stop curiosity; it’s a reason to pair curiosity with governance, norms, and safeguards.

A Flexnerian reading of the current AI funding wave

We’re currently living through a paradox that Flexner would recognize instantly:

AI is showered with investment because it’s visibly useful now.
That investment creates pressure to define “research” as whatever improves next quarter’s metrics.
But the next conceptual leap in AI may come from areas that look “useless” relative to today’s dominant paradigm.

If you want better long-horizon AI outcomes (i.e., robustness, interpretability, privacy, security, alignment, and scientific discovery) Flexner would argue you need institutions that protect inquiry that isn’t instantly legible as profitable.

Or in his words, you need “spiritual and intellectual freedom.”

What to do with this (three practical takeaways)

1) Keep a portfolio: fast product work + slow foundational work

Treat research like an ecosystem. If everything must justify itself immediately, you get brittle progress. Flexner’s “no promises” stance is a feature, not a bug.

2) Reward questions, not only answers

Benchmarks matter, but they can also overfit the field’s imagination. Some of the most important AI work right now is about re-framing the question (e.g., what counts as “understanding,” what counts as “alignment,” what counts as “privacy,” what counts as “truthfulness”).

3) Build institutions that protect intellectual risk

Flexner designed the Institute for Advanced Study around the idea that scholars “accomplish most when enabled” to pursue deep work with minimal distraction.
AI needs its own versions of that: spaces where the incentive is insight, not velocity.

AI is not an argument against Flexner (it’s his exhibit A)

If you hold a smartphone, use a search engine, or interact with modern AI systems, you’re touching the compounded returns of yesterday’s “useless” knowledge.

Flexner’s defense isn’t sentimental. It’s strategic: a society that wants transformative technology must also want the conditions that produce it: freedom, patience, and room for ideas that don’t yet know what they’re for. Or, as Dijkgraaf puts it in summarizing Flexner’s view: fundamental inquiry goes to the “headwaters,” and applications follow, slowly, steadily, and often surprisingly.

Main Source: https://www.ias.edu/ideas/2017/dijkgraaf-usefulness

Tradeoffs Matter: On Developing Lower Bounds

As I write this blog post, I just received news that the U.S. is designating Nigeria as a ‘country of particular concern’ over Christian persecutions. I cannot think of any other country with such a large population but (almost) equal representation of Christians and Muslims. Since before I was born, this has caused a lot of friction but somehow the country has survived all that friction (thus far!). But the friction stems from tradeoffs of having such religious heterogeneity, a topic of its own discussion. But in this post, I’ll focus on the high levels of mathematical tradeffs.

There’s something deeply human about lower bounds. They’re not just mathematical artifacts; they’re reflections of life itself. To me, a lower bound represents the minimum cost of achieving something meaningful. And in both life and research, there’s no escaping those costs.

The Philosophy: Tradeoffs Everywhere

Growing up, I was lucky enough to have certain people in my family spend endless hours guiding me: helping with schoolwork, teaching patience, pushing me toward growth. Looking back, I realize these people could have done a hundred other things with that time. But they (especially my mum) chose to invest it in me. That investment wasn’t free. It came with tradeoffs, the time she could never get back. But without that investment, I wouldn’t be who I am today.

That’s the thing about life: everything has a cost. In 2022/2023, I could have focused entirely on my research. But instead, I poured my energy into founding NaijaCoder, a technical education nonprofit for Nigerian students. It was rewarding, but also consuming. I missed out on months of uninterrupted research momentum. And yet, I have no regrets! Because that, too, was a lower bound. The minimum “cost” of building something (hopefully) lasting and impactful.

Every meaningful pursuit (i.e., love, growth, service, research) demands something in return. There are always tradeoffs. Anyone who claims otherwise is ignoring the basic laws of nature. You can’t have everything, and that’s okay. The beauty lies in understanding what must be given up for what truly matters.

Lower Bounds in Technical Domains

Mathematicians talk about lower bounds as the limits of efficiency. When we prove a lower bound, we’re saying: “No matter how clever you are, you can’t go below this.” It’s not a statement of despair: it’s a statement of truth.

Lower bounds define the terrain of possibility. They tell us what’s fundamentally required to solve a problem, whether it’s time, space, or communication. In a strange way, they remind me of the constraints in life. You can’t do everything at once. There’s a cost to progress. To prove a good lower bound is to acknowledge that reality has structure. There’s an underlying balance between effort and outcome. It’s an act of intellectual honesty: a refusal to pretend that perfection is free.

Nowhere do these tradeoffs feel more personal than in privacy. In the quest to protect individual data, we face the same universal truth: privacy has a price. If we want stronger privacy guarantees, we must give up some accuracy, some utility, some convenience. In differential privacy, lower bounds quantify that tension. They tell us that no matter how sophisticated our algorithms are, we can’t perfectly protect everyone’s data and keep every detail of the dataset intact. We must choose what to value more — precision of statistical estimates or protection.

These aren’t technical inconveniences; they’re moral lessons. They remind us that every act of preservation requires some loss. Just as my mother’s care required time, or NaijaCoder required research sacrifices, protecting privacy requires accepting imperfection.

Acceptance

The pursuit of lower bounds (in research or in life) is about humility. It’s about recognizing that limits aren’t barriers to doing good work; they’re the context in which doing good becomes possible.

Understanding lower bounds helps us stop chasing the illusion of “free perfection.” It helps us embrace the world as it is: a world where tradeoffs are natural, where effort matters, and where meaning is found not in escaping limits but in working within them gracefully.

So, whether in mathematics, privacy, or life, the lesson is the same: there are always tradeoffs. And that’s not a tragedy; it’s the very structure that gives our choices value.

I hope these ideas shape how I live, teach, and do research going forward. In my work on privacy, I’m constantly reminded that (almost) every theorem comes with a cost and that understanding those costs makes systems more honest and human. In education, through NaijaCoder, I see the same principle: every bit of growth for a student comes from someone’s investment of time and care.

Developing lower bounds isn’t just a mathematical pursuit. It’s a philosophy of life, one that teaches patience, realism, and gratitude. The world is full of limits, but within those limits, we can still create beauty, meaning, and progress: one bounded step at a time.

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