Superposition is the core feature that sets quantum theory apart from classical physics. Here, we investigate whether sets of quantum measurements can be modelled by using devices limited to performing jointly diagonalisable measurements. This leads us to introduce classical measurement models, where many such devices are connected together through classical randomness and post-processing. Such classical measurement models are intermediate between the notion of commutative measurements and joint measurability. Towards understanding these models we (i) identify exact noise and loss rates at which all projective measurements admit a classical model, (ii) propose numerical methods to construct classical models for finite sets of measurements, and (iii) show how to construct witnesses for ruling out the existence of classical models. In addition, we show that classical measurement models also have operational implications in non-disturbance tasks where sequential quantum measurements are implemented with classical side-information. Our work provides a new approach to the role of superposition in quantum measurements.
Superposition is the core feature that sets quantum theory apart from classical physics. Here, we investigate whether sets of quantum measurements can be modelled by using devices limited to performing jointly diagonalisable measurements. This leads us to introduce classical measurement models, where many such devices are connected together through classical randomness and post-processing. Such classical measurement models are intermediate between the notion of commutative measurements and joint measurability. Towards understanding these models we (i) identify exact noise and loss rates at which all projective measurements admit a classical model, (ii) propose numerical methods to construct classical models for finite sets of measurements, and (iii) show how to construct witnesses for ruling out the existence of classical models. In addition, we show that classical measurement models also have operational implications in non-disturbance tasks where sequential quantum measurements are implemented with classical side-information. Our work provides a new approach to the role of superposition in quantum measurements.
A classical state-preparation device cannot generate states in relative superposition. We introduce classical models in which devices that are individually unable to generate states with relative superposition can be stochastically coordinated to simulate sets of quantum states. These models have natural operational interpretation in prepare-and-measure scenarios and they can account for many non-commuting quantum state sets. We develop systematic methods both for classically simulating quantum sets and for showing that no such simulation exists, thereby certifying quantum coherence. In particular, we determine the exact noise rates required to classically simulate the entire state space of quantum theory. We also reveal connections between the operational classicality of sets and the well-known fundamental concepts of joint measurability and Einstein-Podolsky-Rosen steering. Here, we present an avenue to understand how and to what extent quantum states defy generic models based on classical devices, which also has relevant implications for quantum information applications.
Quantum instruments describe both the classical outcome and the updated state associated with a quantum measurement. We ask whether these processes can be simulated using only a natural subset of resources, namely projective measurements on the system and quantum processing of the postmeasurement states. We show that the simulability of instruments can be connected to an entanglement classification problem. This leads to a computationally efficient necessary condition for simulation of generic instruments and to a complete characterisation for qubits. We use this to address relevant quantum information tasks, namely (i) the noise tolerance of standard qubit unsharp measurements, (ii) nonprojective advantages in information-disturbance trade-offs, and (iii) increased sequential Bell inequality violations under projective measurements. Moreover, we consider also d-dimensional Lüders instruments that correspond to weak versions of standard basis measurements and show that for large d these can permit scalable noise advantages over projective implementations.
We investigate the compression of quantum information with respect to a given set M of high-dimensional measurements. This leads to a notion of simulability, where we demand that the statistics obtained from M and an arbitrary quantum state ρ are recovered exactly by first compressing ρ into a lower-dimensional space, followed by some quantum measurements. A full quantum compression is possible, i.e., leaving only classical information, if and only if the set M is jointly measurable. Our notion of simulability can thus be seen as a quantification of measurement incompatibility in terms of dimension. After defining these concepts, we provide an illustrative example involving mutually unbiased bases, and develop a method based on semidefinite programming for constructing simulation models. In turn we analytically construct optimal simulation models for all projective measurements subjected to white noise or losses. Finally, we discuss how our approach connects with other concepts introduced in the context of quantum channels and quantum correlations.
Quantum measurements can be incompatible, i.e., they can fail to be jointly measurable. Recently, a weaker notion of joint-measurability, called partial joint-measurability, was proposed by Masini et al. in [Quantum 8, 1574 (2024)]. In this work, we further generalize this notion to the setting where only a subset of the outcomes of each measurement is required to be jointly determined by classical variables. We provide two mathematical formulations of partial joint-measurability and show that, like full joint-measurability, it can be decided by solving a single semidefinite program. We prove that in the case of an untrusted measurement device, an adversary Eve, limited to classical side information, can perfectly guess the outcomes of the measurement device if and only if the set of measurements is partially jointly measurable. We derive analytical thresholds on the detection efficiency below which generic measurements become partially jointly measurable. Such bounds directly yield limits on the robustness of device-independent and semi-device-independent quantum cryptographic protocols against detection inefficiency. In particular, our results highlight the importance of a careful treatment of postselection in security analyses.
Generalized contextuality is a possible indicator of non-classical behaviour in quantum information theory. In finite-dimensional systems, this is justified by the fact that noncontextual theories can be embedded into some simplex, i.e. into a classical theory. We show that a direct application of the standard definition of generalized contextuality to continuous variable systems does not envelope the statistics of some basic measurements, such as the position observable. In other words, we construct families of fully classical, i.e. commuting, measurements that nevertheless can be used to show contextuality of quantum theory. To overcome the apparent disagreement between the two notions of classicality, that is commutativity and noncontextuality, we propose a modified definition of generalised contextuality for continuous-variable systems. The modified definition is based on a physically-motivated approximation procedure, that uses only finite sets of measurement effects. We prove that in the limiting case this definition corresponds exactly to an extension of noncontextual models that benefits from non-constructive response functions. In the process, we discuss the extension of a known connection between contextuality and no-broadcasting to the continuous-variable scenario, and prove structural results regarding fixed points of infinite-dimensional entanglement breaking channels.
In this thesis, we consider the properties of measurements in quantum theory and other operational theories. After having introduced the framework of operational theories, we consider a communication scheme based on an experimental prepare-and-measure scenario and demonstrate this with different communication tasks. This gives us context for how the different communication tasks can be implemented in different theories, in doing so establishing quantum theory intuitively as an operational theory among other theories. The main property of measurements we focus on in this work is the simulation of measurements, which consists of manipulating the inputs and outputs of the measurement devices. We study how using this process on existing measurement devices can be used to operationally imitate new devices, and what kind of structure the simulation process induces on measurements. We also consider applications of simulability. Firstly, we consider operational restrictions imposed upon measurements. We argue that the restricted set of physical measurements must be closed with respect to the simulation process since the simulation of physical devices must lead to other physically feasible devices. We demonstrate different types of restrictions by classifying them. As a second application we see how the simulation of measurements relates to compatibility of measurements and how it can be viewed as a generalisation of it. This allows us to present an operational principle previously known to quantum theory, the no-free-information principle, according to which any measurement that is compatible with all other measurement must not provide any useful, and therefore free, information about the system. Whilst this principle holds in quantum theory, there are non-classical theories for which it is violated, and so enforcing this principle may be considered a way to exclude some unphysical theories.