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.
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.
The prepare-and-measure framework based on energy constraints offers a practical middle ground between fully device-dependent and device-independent quantum cryptography. The only assumption on an otherwise uncharacterized prepare-and-measure device is that the energy of the prepared states is bounded. Existing security analyses of this framework assume that the preparation and measurement devices share at most classical correlations, and under this assumption certified lower bounds on the extractable randomness have been established. Recent work has shown that an adversary who pre-distributes entanglement between the devices can mount attacks that are strictly more powerful than those available when the devices share only classical correlations, reducing the extractable randomness below the previously certified rates. This leaves open the fundamental question of whether randomness can be certified at all in this scenario.
We address this open question by constructing semidefinite programming relaxations of the guessing probability by adapting the Navascués-Pironio-Acín hierarchy to the energy-constrained setting where shared entanglement between the devices is permitted. Our relaxations yield certified lower bounds on the extractable randomness without enforcing any restrictions on the dimensions of the quantum state shared between the preparation and measurement devices. We show that these certified lower bounds are strictly positive for a range of energy values, thereby answering the open question affirmatively: certified randomness generation is theoretically possible in the energy-constrained SDI framework even in the presence of a fully quantum adversary.
For quantum random number generators (QRNGs), strong security guarantees can be obtained under various assumptions. The most well-known assumptions are the non-signalling assumption (needed for device-independent (DI) QRNG) and post-quantum computational assumptions (needed for computationally bounded single-prover QRNG protocols). In this work, we study a third model: a simple prepare-and-measure QRNG whose security relies on physically motivated energy constraints. This physically motivated set-up allows for easier implementations of QRNGs and can supply better rates.
So far, previous works that studied the energy-constrained model either assumed that the adversary is classical in their security proof or supplied relatively weak bounds on the certified randomness, by bounding the conditional min-entropy. In this work, we prove a new, essentially tight, lower bound on the single-round von Neumann entropy against adversaries holding both classical and quantum side-information. Our work reveals that working with the von Neumann entropy, rather than the min-entropy, in the quantum case can lead to a major improvement, nearly matching the classical bound in many cases. We further study the robustness of our entropy bound. Since the single-round von Neumann entropy is the main quantity of interest when characterizing the key rate of QRNGs, our bound establishes the main ingredient for rigorous security proofs against general quantum attacks in this framework.
Quantum information tasks are often analyzed under varying trust assumptions about the devices involved. The semi-device-independent (SDI) framework offers a balance between needed assumptions and experimental feasibility. In this work, we study the energy-constrained SDI scenario, where the only assumption in a prepare-and-measure setup is an upper bound on the energy of the prepared quantum states. In contrast to previous studies that restricted the preparation and measurement devices to be classically correlated, we show that allowing entanglement strictly enlarges the set of achievable correlations. We identify two operational consequences of this result. The first concerns randomness certification, where we show that allowing the adversary to employ entangled strategies may significantly reduce the amount of certifiable randomness. This includes situations where the amount of randomness drops to zero in the presence of entanglement, while it remains positive when entanglement is excluded. Second, for the task of distinguishing an arbitrary quantum channel from the identity, we show that the known dimension-independent bound on the advantage conferred by entanglement is violated under an energy constraint.
Intrinsic randomness is generated when a quantum state is measured in any basis in which it is not diagonal. In an adversarial scenario, we quantify this randomness by the probability that a correlated eavesdropper could correctly guess the measurement outcomes. What if the eavesdropper is never wrong, but can sometimes return an inconclusive outcome? Inspired by analogous concepts in quantum state discrimination, we introduce the unambiguous randomness of a quantum state and measurement, and, relaxing the assumption of perfect accuracy, randomness with a fixed rate of inconclusive outcomes. We solve the maximal unambiguous randomness of any quantum state, optimised over all projective measurements, and find that it's proportional to the smallest eigenvalue of the state. We also solve both problems for any state and projective measurement in dimension two, and for an isotropically noisy state measured in an unbiased basis of any dimension. In the former case, when the setup is used to choose random bases for a prepare-and-measure quantum key distribution protocol, the unambiguous randomness quantifies the knowledge gained by an eavesdropper about the secret key, without causing any disturbance. In the latter, we find that, while experimental noise is typically ascribed to a single device, an eavesdropper correlated only to a noisy state is outperformed by an eavesdropper with joint correlations to both a noisy state and a noisy measurement. In fact, we identify a critical noise parameter beyond which the joint-noise eavesdropper achieves a perfect guessing probability, ending all hopes of private randomness.
In quantum mechanics, a measurement applied to a state in general produces some amount of intrinsic randomness. This is not only a fundamental feature of the theory, but is also at the basis of any quantum process to generate random numbers. The simplest of such processes consists of a single, fully charaterized, measurement acting on a single, fully characterized, state. Unfortunately, no general method to estimate the intrinsic randomness produced in such setups is known. In this work, we address this issue by presenting a semidefinite programming formulation of the maximum probability with which an adversary, Eve, can guess the outcomes of characterized but untrusted prepare-and-measure setups. We then present several applications of this construction. First, we apply our method to a variety of specific setups, allowing us both to benchmark the approach and, more importantly, to determine the exact amount of certifiable randomness in scenarios where only upper bounds were previously available. Then, we show that the presence of entanglement between the device preparing the state and the measurement strictly increases Eve's predictive power, already in the most elementary setup of a binary measurement acting on a qubit state.
This paper investigates semi-device-independent quantum randomness generation with a trusted binary pure-state source and an untrusted binary detector whose side information is classical. We derive a closed-form Shannon-rate expression for this setting, depending only on the trusted Gram overlap of the two source states and the observed symmetric error probability. The key point is that the full binary-qubit POVM optimisation must include the two deterministic extreme points omitted by the projective-only treatment; including them gives a substantially lower, and correct, certified rate. The closed form is an unconditional upper bound on the certified asymptotic i.i.d.\ Shannon rate, and becomes tight on a numerically verified dual-feasibility region containing all operating points used in the paper. Outside this region the same expression remains an upper bound. We then apply the result to squeezed-coherent BPSK sources, showing how squeezing changes the trade-off between state distinguishability and certified randomness in the lossless and lossy regimes. Finally, we clarify the adversary model if the adversary is allowed to hold a detector-purification register that tags the outcome.