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.
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.
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 2logd 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.
Abstract The peculiar uncertainty or randomness of quantum measurements stems from quantum coherence, whose information‐theoretic characterization is currently under investigation. The resource theory of coherence investigates interpretations of coherence measures and the interplay with other quantum properties, such as quantum correlations and intrinsic randomness. Coherence can be viewed as a resource for generating intrinsic randomness by measuring a state in the computational basis. It is observed in a previous work that the coherence of formation, which measures the asymptotic coherence dilution rate, indeed quantifies the uncertainty of a correlated party (classical system) about the system measurement outcome. In this work, the result is re‐derived from a quantum point of view, and then the intrinsic randomness is connected to the relative entropy of coherence, another important coherence measure that quantifies the asymptotic distillable coherence. Even though there do not exist bound coherent states, these two coherence measures—intrinsic randomness quantified by coherence of formation and by relative entropy of coherence—are different. Interestingly, it is shown that this gap is equal to the quantum discord, a general form of quantum correlations, in the state of the system of interest and the correlated party, after a local measurement on the former system.
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.
The peculiar uncertainty or randomness of quantum measurements stems from coherence, whose information-theoretic characterization is currently under investigation. Under the resource theory of coherence, it is interesting to investigate interpretations of coherence measures and the interplay with other quantum properties, such as quantum correlations and intrinsic randomness. Coherence can be viewed as the resource for the intrinsic randomness in the measurement outcomes of a state in the computational basis. We observed in our previous work that the coherence of formation, which measures the asymptotic coherence dilution rate, indeed quantifies the uncertainty of a (classical) correlated party about the system measurement outcome. In this work, we re-derive the result from a quantum point of view and then connect the intrinsic randomness to the relative entropy of coherence, another important coherence measure that quantifies the asymptotic distillable coherence. Even though there does not exist bound coherent states, these two intrinsic randomness quantified by coherence of formation and the relative entropy of coherence are different. Interestingly, we show that this gap is equal to the quantum discord, a general form of quantum correlations, in the state of the system of interest and the correlated party, after a local measurement on the former system.
Measurements of quantum systems can be used to generate classical data that are truly unpredictable for every observer. However, this true randomness needs to be discriminated from randomness due to ignorance or lack of control of the devices. We analyze the randomness gain of a measurement-device-independent setup, consisting of a well-characterized source of quantum states and a completely uncharacterized and untrusted detector. Our framework generalizes previous schemes as arbitrary input states and arbitrary measurements can be analyzed. Our method is used to suggest simple and realistic implementations that yield high randomness generation rates of more than one random bit per qubit for detectors of sufficient quality.
Abstract In the standard Bell scenario, when making a local projective measurement on each system component, the amount of randomness generated is restricted. However, this limitation can be surpassed through the implementation of sequential measurements. Nonetheless, a rigorous definition of random numbers in the context of sequential measurements is yet to be established, except for the lower quantification in device-independent scenarios. In this paper, we define quantum intrinsic randomness in sequential measurements and quantify the randomness in the Collins–Gisin–Linden–Massar–Popescu inequality sequential scenario. Initially, we investigate the quantum intrinsic randomness of the mixed states under sequential projective measurements and the intrinsic randomness of the sequential positive-operator-valued measure (POVM) under pure states. Naturally, we rigorously define quantum intrinsic randomness under sequential POVM for arbitrary quantum states. Furthermore, we apply our method to one-Alice and two-Bobs sequential measurement scenarios, and quantify the quantum intrinsic randomness of the maximally entangled state and maximally violated state by giving an extremal decomposition. Finally, using the sequential Navascues–Pironio–Acin hierarchy in the device-independent scenario, we derive lower bounds on the quantum intrinsic randomness of the maximally entangled state and maximally violated state.