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arXiv:2506.14526 · PRX Quantum

Certified Randomness From Quantum Speed Limits

Caroline L. Jones, Albert Aloy, Gerard T. Higgins, Markus P. Müller

Published Feb 6, 2026• 0 citations
arXivHTMLPDF

Abstract

Quantum speed limits are usually regarded as fundamental restrictions, constraining the amount of computation that can be achieved within some given time and energy. Complementary to this intuition, here we show that these limitations are also of operational value: they enable the secure generation of certified randomness. We consider a prepare-and-measure scenario with some (experimentally determined or promised) upper bound on the energy uncertainty Δ E of the average prepared quantum state, but without any further assumptions on the devices, Hilbert space or Hamiltonian. Given that we can freely choose the time at which to apply the untrusted preparation procedure, we show that this scenario admits the generation of randomness that is secure against adversaries with additional classical information. We show how to determine the amount of certified randomness given the observed correlations, discuss how interactions with the environment are taken into account, and sketch a conceivable experimental implementation. In particular, we show that single-mode coherent states admit this kind of certification of nonzero randomness in some parameter regimes, reinforcing existing demonstrations of nonclassicality in the simple harmonic oscillator. Our results extend earlier efforts to devise semi-device-independent protocols grounded in reasonable physical assumptions, and they contribute to the understanding of time-energy uncertainty relations via their operational consequences.

DOI
10.1103/bnrq-sp5w

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•Jan 1, 2026•0 citations
Quantum speed limits are usually regarded as fundamental restrictions, constraining the amount of computation that can be achieved within some given time and energy. Complementary to this intuition, here we show that these limitations are also of operational value: they enable the secure generation of certified randomness. We consider a prepare-and-measure scenario with some (experimentally determined or promised) upper bound on the energy uncertainty Δ E of the average prepared quantum state, but without any further assumptions on the devices, Hilbert space or Hamiltonian. Given that we can freely choose the time at which to apply the untrusted preparation procedure, we show that this scenario admits the generation of randomness that is secure against adversaries with additional classical information. We show how to determine the amount of certified randomness given the observed correlations, discuss how interactions with the environment are taken into account, and sketch a conceivable experimental implementation. In particular, we show that single-mode coherent states admit this kind of certification of nonzero randomness in some parameter regimes, reinforcing existing demonstrations of nonclassicality in the simple harmonic oscillator. Our results extend earlier efforts to devise semi-device-independent protocols grounded in reasonable physical assumptions, and they contribute to the understanding of time-energy uncertainty relations via their operational consequences.
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.
One of the striking properties of quantum mechanics is the occurrence of the Bell-type non-locality. They are a fundamental feature of the theory that allows two parties that share an entangled quantum system to observe correlations stronger than possible in classical physics. In addition to their theoretical significance, non-local correlations have practical applications, such as device-independent randomness generation, providing private unpredictable numbers even when obtained using devices delivered by an untrusted vendor. Thus, determining the quantity of certifiable randomness that can be produced using a specific set of non-local correlations is of significant interest. First, we present an experimental realization of recent Bell-type operators designed to provide private random numbers that are secure against adversaries with quantum resources. We use semi-definite programming to give lower bounds on the generated randomness in terms of both min-entropy and von Neumann entropy in a device-independent scenario. Our results demonstrate the first experiment that certifies nearly two bits of randomness from binary measurements of two parties [1], [2]. Apart from single-round certification, we provide an analysis of finite-key protocol for quantum randomness expansion using the Entropy Accumulation Theorem. Second, we present an efficient and practical method for certifying quantum randomness using generalized measurements. Indeed, we have derived a method for self-testing the presence of POVMs. We present the certification of more than one bit of min-entropy from a single POVM on one of the qubits of an entangled state using a variant of the elegant Bell inequality. We also obtain more than one bit of randomness in a prepare and measure scenario of a similar structure with a POVM on a single qubit [3]. We provide numerical simulations to demonstrate the effectiveness of our randomness certification and expansion.
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.
Abstract Although quantum computers can perform a wide range of practically important tasks beyond the abilities of classical computers 1,2 , realizing this potential remains a challenge. An example is to use an untrusted remote device to generate random bits that can be certified to contain a certain amount of entropy 3 . Certified randomness has many applications but is impossible to achieve solely by classical computation. Here we demonstrate the generation of certifiably random bits using the 56-qubit Quantinuum H2-1 trapped-ion quantum computer accessed over the Internet. Our protocol leverages the classical hardness of recent random circuit sampling demonstrations 4,5 : a client generates quantum ‘challenge’ circuits using a small randomness seed, sends them to an untrusted quantum server to execute and verifies the results of the server. We analyse the security of our protocol against a restricted class of realistic near-term adversaries. Using classical verification with measured combined sustained performance of 1.1 × 10 18 floating-point operations per second across multiple supercomputers, we certify 71,313 bits of entropy under this restricted adversary and additional assumptions. Our results demonstrate a step towards the practical applicability of present-day quantum computers.
Complex solid-state systems are promising platforms for scalable, high-bandwidth quantum random-number generation, yet certifying the quantum origin of their fluctuations remains difficult because the underlying microscopic dynamics are hard to model and validate. Here we demonstrate semi-device-independent quantum randomness certification from semiconductor spin noise, to our knowledge the first such certificate on any spin-noise source, without relying on a microscopic model of the spin system. The untrusted optical source is constrained by an experimentally tested mean-photon-number bound together with a declared analogue-range and per-sample clipping ceiling, while the trusted receiver is described as a calibrated, noisy, coarse-grained homodyne measurement. Using a semidefinite programme with rigorously controlled Fock-space truncation, we certify randomness that remains private against an adversary holding arbitrary quantum side information. Offline analysis yields certified entropy rates of 3.23.23.2--3.43.43.4\,Gbit/s from a singly charged (In,Ga)As quantum-dot ensemble and 333333\,Mbit/s from nnn-GaAs in a spin-noise-matched detection mode. This exceeds the certified entropy rate of earlier spin-noise generators by more than two orders of magnitude, and the certificate tolerates a resolved per-symbol energy contribution from the solid-state emitter itself rather than requiring a near-vacuum input.
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.