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arXiv:2605.18291 · Quantum Science and Technology

Quantum randomness beyond projective measurements

Fionnuala Curran

quant-phSubmitted May 18, 2026• Announced May 19, 2026• 0 citations
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Abstract

The unpredictability of quantum physics gives rise to intrinsic randomness. In an adversarial scenario, any additional degrees of freedom must be attributed to an eavesdropper with correlations to the measurement set-up. The true randomness is then quantified by the probability that she correctly guesses the measurement outcomes, optimised over all possible strategies. Extremal measurements are appealing here, since they do not allow information to leak to such an eavesdropper. Beyond projective measurements, however, a simple question remains open: how much intrinsic randomness can be generated by a given extremal measurement? In a step towards solving it, we characterise the randomness generated by any unbiased extremal rank-one measurement acting on any state, solving the problem explicitly in dimension two. Four-outcome qubit measurements of this type are tomographic, so these results hold for fully source-device-dependent randomness too. The tetrahedral symmetric informationally complete (SIC) measurement, we find, has the least intrinsic randomness within this class. We also present the skewed SIC family of measurements, and use them to partially solve an open problem: we prove that $2 \log d$ bits of randomness, the maximal amount, can be generated device-dependently (or source-device-independently) in any dimension in which there exists a SIC measurement.

Journal reference
Quantum Sci. Technol. 11 045095 (2026)
DOI
10.1088/2058-9565/aeb3a9
Comments
12 + 36 pages, 4 figures

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The unpredictability of quantum physics gives rise to intrinsic randomness. In an adversarial scenario, any additional degrees of freedom must be attributed to an eavesdropper with correlations to the measurement set-up. The true randomness is then quantified by the probability that she correctly guesses the measurement outcomes, optimised over all possible strategies. Extremal measurements are appealing here, since they do not allow information to leak to such an eavesdropper. Beyond projective measurements, however, a simple question remains open: how much intrinsic randomness can be generated by a given extremal measurement? In a step towards solving it, we characterise the randomness generated by any unbiased extremal rank-one measurement acting on any state, solving the problem explicitly in dimension two. Four-outcome qubit measurements of this type are tomographic, so these results hold for fully source-device-dependent randomness too. The tetrahedral symmetric informationally complete (SIC) measurement, we find, has the least intrinsic randomness within this class. We also present the skewed SIC family of measurements, and use them to partially solve an open problem: we prove that 2log⁡d2 \log d2logd bits of randomness, the maximal amount, can be generated device-dependently (or source-device-independently) in any dimension in which there exists a SIC measurement.
Quantum physics exhibits an intrinsic and private form of randomness with no classical counterpart. Any setup for quantum randomness generation involves measurements acting on quantum states. In this work, we consider the following question: Given a quantum measurement, how much randomness can be generated from it? In real life, measurements are noisy and thus contain an additional, extrinsic form of randomness due to ignorance. This extrinsic randomness is not private since, in an adversarial model, it takes the form of quantum side information held by an eavesdropper who can use it to predict the measurement outcomes. Randomness of measurements is then quantified by the guessing probability of this eavesdropper, when minimized over all possible input states. This optimization is in general hard to compute, but we solve it here for any two-outcome qubit measurement and for projective measurements in arbitrary dimension mixed with white noise. We also construct, for a given measured probability distribution, different realizations with (i) a noisy state and noiseless measurement (ii) a noiseless state and noisy measurement and (iii) a noisy state and measurement, and we show that the latter gives an eavesdropper significantly higher guessing power.
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.
Quantum measurements can produce unpredictable randomness arising from the uncertainty principle. When measuring a state with von Neumann measurements, the intrinsic randomness can be quantified by the quantum coherence of the state on the measurement basis. Unlike projection measurements, there are additional and possibly hidden degrees of freedom in an apparatus for generic measurements. We propose an adversary scenario to characterize the intrinsic randomness of general measurements with arbitrary input states. Interestingly, we discover that certain measurements, including symmetric and information-complete ones, generate nonzero randomness for all states, which suggests a new approach for designing source-independent random number generators without state characterization. Furthermore, our results reveal that intrinsic randomness can quantify coherence under general measurements, which generalizes the result in the standard resource theory of state coherence.
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.
We investigate how much randomness can be extracted from a generic partially entangled pure state of two qubits in a device-independent setting, where a Bell test is used to certify the correct functioning of the apparatus.For any such state, we first show that two bits of randomness are always attainable both if projective measurements are used to generate the randomness globally or if a nonprojective measurement is used to generate the randomness locally.We then prove that the maximum amount of randomness that can be generated using nonprojective measurements globally is restricted to between approximately 3.58 and 3.96 bits.The upper limit rules out that a bound of four bits potentially obtainable with extremal qubit measurements can be attained.We point out this is a consequence of the fact that nonprojective qubit measurements with four outcomes can only be self-tested to a limited degree in a Bell experiment.
Device-independent random number generation (DIQRNG) is the most secure form of generating private randomness using quantum physical processes. Its strength lies in producing numbers that are impossible to predict by any eavesdropper restricted by the laws of quantum theory. Moreover, security is proven solely from observed measurement statistics, without the need to characterise or trust the devices used in random number generation. Implementing DIQRNG is, however, costly, as it requires high-quality entangled systems. It is therefore important to make the best use of available resources. In this work, we show that using projective measurements -- which are most readily implementable experimentally -- one can certify 2log⁡(d)2\log(d)2log(d) bits of device-independent randomness from a bipartite system of local dimension ddd for every d≥2d \ge 2d≥2, thus reaching the theoretically maximum possible rate of DIQRNG. We provide explicit protocols reaching 2log⁡(d)2\log(d)2log(d) bits based on mutually unbiased bases. Furthermore, we compute numerical bounds on the rate for the case of imperfect implementations, showing that our protocols are robust to experimental noise.