Performing a quantum measurement yields two different results: a classical outcome drawn from a probability distribution, according to Born's rule, and a quantum outcome corresponding to the post-measurement state. Quantum devices that provide both outcomes can be described through quantum instruments. In a realistic scenario, one can expect that the observer's obtained classical and quantum outcomes are non-ideal: this can be due to experimental limitations, but could also be explained by adversarial interference, that is, a second party that disturbs the device through a concealed measurement to obtain information. The second scenario can be interpreted through quantum compatibility, as it implies that both the observer's instrument and the adversary's measurement can be performed simultaneously. In this work, we show how the noise of the observer's device relates to the amount of information that the adversary can obtain. We study scenarios in which the adversary aims to acquire information on the same basis as the observer's measurement, or on a mutually unbiased basis with respect to the observer's basis. In both cases, we derive necessary and sufficient conditions for the compatibility of a single qubit non-ideal quantum instrument and a noisy meter, from which we obtain the maximum amount of information that the adversary can extract in terms of the noise parameters of the observer's instrument. Finally, we provide the device implementation from the adversary's point of view for the same basis scenario.
Performing a quantum measurement yields two different results: a classical outcome drawn from a probability distribution, according to Born's rule, and a quantum outcome corresponding to the post-measurement state. Quantum devices that provide both outcomes can be described through quantum instruments. In a realistic scenario, one can expect that the observer's obtained classical and quantum outcomes are non-ideal: this can be due to experimental limitations, but could also be explained by adversarial interference, that is, a second party that disturbs the device through a concealed measurement to obtain information. The second scenario can be interpreted through quantum compatibility, as it implies that both the observer's instrument and the adversary's measurement can be performed simultaneously. In this work, we show how the noise of the observer's device relates to the amount of information that the adversary can obtain. We study scenarios in which the adversary aims to acquire information on the same basis as the observer's measurement, or on a mutually unbiased basis with respect to the observer's basis. In both cases, we derive necessary and sufficient conditions for the compatibility of a single qubit non-ideal quantum instrument and a noisy meter, from which we obtain the maximum amount of information that the adversary can extract in terms of the noise parameters of the observer's instrument. Finally, we provide the device implementation from the adversary's point of view for the same basis scenario.
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.
The goal of randomness extraction is to distill (almost) perfect randomness from a weak source of randomness. When the source yields a classical string X, many extractor constructions are known. Yet, when considering a physical randomness source, X is itself ultimately the result of a measurement on an underlying quantum system. When characterizing the power of a source to supply randomness, it is hence natural to ask how much classical randomness we can extract from a quantum system. To tackle this question, we here take on the study of quantum-to-classical randomness extractors (QC-extractors). We provide constructions of QC-extractors based on measurements in a full set of mutually unbiased bases (MUBs), and certain single qubit measurements. The latter are particularly appealing since they are not only easy to implement, but also appear throughout quantum cryptography. We proceed to prove an upper bound on the maximum amount of randomness that we could hope to extract from any quantum state. Some of our QC-extractors almost match this bound. We show two applications of our results. First, we show that any QC-extractor gives rise to entropic uncertainty relations with respect to quantum side information. Such relations were previously only known for two measurements. In particular, we obtain strong relations in terms of the von Neumann (Shannon) entropy as well as the min-entropy for measurements in (almost) unitary two-designs, a full set of MUBs, and single qubit measurements in three MUBs each. Second, we resolve the central open question in the noisy-storage model by linking security to the quantum capacity of the adversary's storage device. More precisely, we show that any two party cryptographic primitives can be implemented securely as long as the adversary's storage device has sufficiently low quantum capacity. Our protocol does not need any quantum storage to implement, and is technologically feasible using present-day technology.
In the general theory of quantum measurement, one associates a positive semidefinite operator on a d-dimensional Hilbert space to each of the n possible outcomes of an arbitrary measurement. In the special case of a projective measurement, these operators are pairwise Hilbert--Schmidt orthogonal, but when n>d, orthogonality is restricted by positivity. This restriction has consequences which are not present from a classical perspective; in particular, we find it allows us to more precisely state the old quantum adage that it is impossible to extract information from a system without disturbance. Restricting attention throughout to the Luders state updating rule, we consider three properties of a measurement: its disturbance, a measure of how the expected post-measurement state deviates from the input state, its purity gain, a measure of the expected reduction of uncertainty resulting from measurement, and its orthogonality, a measure of the degree to which the measurement operators differ from an orthonormal set. We show that these quantities satisfy a simple algebraic relation amounting to an expression of an information-disturbance trade-off. Finally, we assess several classes of measurements on these grounds and identify symmetric informationally complete quantum measurements as the unique quantum analogs of a perfectly informative and nondisturbing classical ideal measurement.
A fundamental prediction of quantum mechanics is that observables can be measured through ideal measurements, that is, measurements that give the same outcome when repeated and do not disturb any observable that is jointly measurable. In terms of quantum instruments, such an ideal measurement is described by projective effects together with the associated Luders state-update rule. Here, we show that quantum mechanics allows for self-testing any ideal measurement. Using solely the output statistics of bipartite Bell tests in which a non-demolition measurement is sometimes added, it is possible to certify in a device-independent manner whether the added measurement is ideal.
In quantum information the fundamental information-containing system is the qubit. A measurement of a single qubit can at most yield one classical bit. However, a dichotomous measurement of an unknown qubit will yield much less information about the qubit state. We use Bayesian inference to compute how much information one progressively gets by making sucessive, individual measurements on an ensemble of identically prepared qubits. Perhaps surprisingly, even if the measurements are arranged so that each measurement yields one classical bit, that is, the two possible measurement outcomes are a priori equiprobable, it takes almost a handful of measurements before one has gained one bit of information about the gradually concentrated qubit probability density. We also show that by following a strategy that reaps the maximum information per measurement, we are led to a mutually unbiased basis as our measurement bases. This is a pleasing, although not entirely surprising, result.
Incompatibility of quantum devices is one of the cornerstones of quantum theory, and the incompatibility of quantum measurements and channels has been linked to quantum advantage in certain information-theoretic tasks. In this work, we focus on the less well explored question of the incompatibility of quantum instruments, that is, devices that describe the measurement process in its entirety, accounting for both the classical measurement outcome and the quantum postmeasurement state. In particular, we focus on the recently introduced notion of parallel compatibility of instruments, which has been argued to be a natural notion of instrument compatibility. We introduce, in a manner similar to the case of measurements and channels, the incompatibility robustness of quantum instruments and derive universal bounds on it. We then prove that postprocessing of quantum instruments is a free operation for parallel compatibility. Last, we provide families of instruments for which our bounds are tight and families of compatible indecomposable instruments.