Certifying the inherent randomness in quantum state discrimination

Abstract

Quantum mechanics forbids perfect discrimination of non-orthogonal states, a limitation that can be exploited to certify intrinsic randomness. Building on this principle, this thesis investigates how limits on state discrimination can be used to generate certified randomness in prepare-and-measure protocols. In particular, we study how the performance of minimum-error quantum state discrimination constrains an adversary’s ability to predict measurement outcomes, under the sole assumption of an average energy constraint on the prepared states.

We show how observed discrimination performance bounds an adversary’s optimal guessing probability and yields certified min-entropy. Exact results are obtained for fixed-dimension qubit ensembles, including closed-form expressions for two-qubit preparations and extensions to general qubit ensembles. We then introduce dimension-unbounded semidefinite programming relaxations based on Gram-matrix techniques, providing device-independent bounds that rely only on the energy constraint.

A central result is the identification of critical discrimination thresholds beyond which deterministic strategies are excluded and genuine quantum randomness is certified. These results establish quantum state discrimination as an operational witness for randomness generation under natural physical assumptions.

Popular Abstract

Have you ever flipped a coin on a windy day and wondered whether the breeze subtly pushed it toward heads or tails? Many processes that appear random are not truly unpredictable. Coins can be biased by tiny imperfections, dice can be weighted, and computer programs only simulate randomness as long as their internal seed remains hidden. This thesis begins with a simple but deep question: how can we generate numbers that are fundamentally unpredictable, even to someone who fully understands the device producing them?

The starting point is a core principle of quantum physics. At the microscopic level, nature does not always permit sharp distinctions. Two quantum states can be different yet partially overlap, like musical tones so close in pitch that even a trained ear may confuse them. Quantum theory tells us that such states cannot be perfectly distinguished, regardless of the measurement. This intrinsic uncertainty is not a limitation of our knowledge but a fundamental feature that can be used as a resource. My work focuses on quantum state discrimination, a guessing task involving single quantum systems. Because the states overlap, there is always a nonzero probability of error, and that irreducible error becomes a source of genuine unpredictability.

A key practical constraint is that realistic devices cannot use unlimited energy. I study what happens when the average energy of the prepared states is restricted while we attempt to distinguish between them. Lower energy makes the states harder to tell apart and limits the achievable success rate. Crucially, this apparent drawback becomes an advantage, since the energy bound itself helps certify randomness that even a perfectly informed adversary cannot predict.

To make this idea concrete, imagine a hat containing two nearly identical marbles. You are allowed only a brief look under dim lighting before declaring which marble was chosen. From the observed error rate, one can calculate the amount of true randomness present, known as min-entropy, the standard measure used in secure random number generation.

The broader motivation is clear. Modern society relies on random numbers that attackers cannot foresee, from secure communication to scientific simulations. By linking energy constraints, discrimination limits, and certified entropy in a single framework, this work aims to turn quantum uncertainty into trustworthy randomness. The goal is not just to design a better coin, but to ensure that it cannot be rigged, even by its maker.

• author: Karlsson, Oliver
• supervisor: Carles Roch i Carceller; Armin Tavakoli
• organization: Mathematical Physics; Department of Physics
• course: FYSK04 20252
• year: 2026
• type: M2 - Bachelor Degree
• subject: Physics and Astronomy
• keywords: Quantum state discrimination, Prepare-and-measure, Semi-device-independent, Quantum random number generation, QRNG, Energy-constrained quantum systems, Semidefinite programming
• language: English
• id: 9221698
• date added to LUP: 2026-02-02 08:12:54
• date last changed: 2026-02-02 08:12:54