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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.

The quantum properties of quantum measurements are indispensable resources in quantum information processing and have drawn extensive research interest. The conventional approach to revealing quantum properties relies on the reconstruction of entire measurement operators by quantum detector tomography. However, many specific properties can be determined by a part of the matrix components of the measurement operators, which makes it possible to simplify the characterization process. We propose a general framework to directly obtain individual matrix elements of the measurement operators by sequentially measuring two noncompatible observables. This method allows us to circumvent the complete tomography of the quantum measurement and extract the required information. We experimentally implement this scheme to monitor the coherent evolution of a general quantum measurement by determining the off-diagonal matrix elements. The investigation of the measurement precision indicates the good feasibility of our protocol for arbitrary quantum measurements. Our results pave the way for revealing the quantum properties of quantum measurements by selectively determining the matrix components of the measurement operators.

Many of the most fundamental observables—position, momentum, phase point, and spin direction—cannot be measured by an instrument that obeys the orthogonal projection postulate. Continuous-in-time measurements provide the missing theoretical framework to make physical sense of such observables. The elements of the time-dependent instrument define a group called the instrumental group (IG). Relative to the IG, all of the time dependence is contained in a certain function called the Kraus-operator density (KOD), which evolves according to a classical Kolmogorov equation. Unlike the Lindblad master equation, the KOD Kolmogorov equation is a direct expression of how the elements of the instrument (not just the total quantum channel) evolve. Shifting from continuous measurements to sequential measurements more generally, the structure of combining instruments in sequence is shown to correspond to the convolution of their KODs. This convolution promotes the IG to an involutive Banach algebra (a structure that goes all the way back to the origins of POVM and C*-algebra theory), which will be called the instrumental group algebra (IGA). The IGA is the true home of the KOD, similar to how the dual of a von Neumann algebra is the true home of the density operator. Operators on the IGA, which play the analogous role for KODs as superoperators play for density operators, are called ultraoperators and various important examples are discussed. Certain ultraoperator–superoperator intertwining relationships are also considered throughout, including the relationship between the KOD Kolmogorov equation and the Lindblad master equation. The IGA is also shown to have actually two distinct involutions: one respected by the convolution ultraoperators and the other by the quantum channel superoperators. Finally, the KOD Kolmogorov generators are derived for jump processes and more general diffusive processes.

The Kochen–Specker theorem, and the associated notion of quantum contextuality, can be considered as the starting point for the development of a notion of non-classical correlations for single systems. The subsequent debate around the possibility of an experimental test of Kochen–Specker-type contradiction stimulated the development of different theoretical frameworks to interpret experimental results. Starting from the approach based on sequential measurements, we will discuss a generalization of the notion of non-classical temporal correlations that goes beyond the contextuality approach and related ones based on Leggett and Garg's notion of macrorealism, and it is based on the notion of memory cost of generating correlations. Finally, we will review recent results on the memory cost for generating temporal correlations in classical and quantum systems. The present work is based on the talk given at the Purdue Winer Memorial Lectures 2018: probability and contextuality. This article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.

Tripartite nonlocality holds significant theoretical value and has numerous applications in quantum information and computation. With dichotomic measurement settings, several aspects of tripartite nonlocality are identified by violations of the Svetlichny inequality, Mermin inequality, the 46 facet inequalities of Sliwa, and the inequalities established by Bancal et al. Silva et al. explored the concept of nonlocality sharing among multiple copies of a party. This paper focuses on determining the number of observers who can simultaneously share distinct tripartite nonlocal correlations under a one-sided sequential measurement formalism. Specifically, we examine a scenario where three spatially separated parties-Alice, Bob, and multiple Charlies-share three spin- 12 particles. For multiple Charlies, we have identified constraints on θ (pure state) and p (mixed state), demonstrating the sequential sharing of various forms of tripartite nonlocality with a single Alice and Bob. Notably, under sequential measurement formalism, a non-maximally pure entangled state exhibits the same sharing capacity as a maximally entangled state.

As first shown by Popescu (1995 Phys. Rev. Lett. 74 2619), some quantum states only reveal their nonlocality when subjected to a sequence of measurements while giving rise to local correlations in standard Bell tests. Motivated by this manifestation of 'hidden nonlocality' we set out to develop a general framework for the study of nonlocality when sequences of measurements are performed. Similar to Gallego et al (2013 Phys. Rev. Lett. 109 070401) our approach is operational, i.e. the task is to identify the set of allowed operations in sequential correlation scenarios and define nonlocality as the resource that cannot be created by these operations. This leads to a characterization of sequential nonlocality that contains as particular cases standard nonlocality and hidden nonlocality.

Genuine entanglement serves as a valuable resource for multipartite quantum information processing. Recently, the ability to share the genuine entanglement among a long sequence of independent observers has gained considerable attention, especially when the state preparation faces significant limitations. Firstly, this paper focuses on investigating the sharing ability of genuine tripartite entanglement under unilateral sequential measurements. We propose a measurement strategy for generalized tripartite Greenberger-Horne-Zeilinger (GHZ) states to enable arbitrarily long sequence of independent observers to detect the genuine entanglement using entanglement witnesses. Moreover, when the sharpness parameters of each sequential observer’s measurement settings are equal, we give the correlation between the controllable angle of generalized GHZ state and the maximum number of sequential observers capable of detecting genuine entanglement. In particular, we identify the controllable angle range within which the same number of such observers can be achieved for generalized GHZ states as with the maximally entangled GHZ state. We also determine the controllable angle range for generalized tripartite W states, where the number of sequential observers can be same as that with maximally entangled W state. Furthermore, we design a measurement strategy for the trilateral sequential scenario, demonstrating that the genuine tripartite entanglement of any generalized GHZ state can be shared by arbitrarily many groups of observers.

A method for lightweight-gypsum material design using waste stone dust as the foaming agent is described. The main objective is to reach several physical properties which are inversely related in a certain way. Therefore, a linear optimization method is applied to handle this task systematically. The optimization process is based on sequential measurement of physical properties. The results are subsequently point-awarded according to a complex point criterion and new composition is proposed. After 17 trials the final mixture is obtained, having the bulk density equal to (586 ± 19) kg/m and compressive strength (1.10 ± 0.07) MPa. According to a detailed comparative analysis with reference gypsum, the newly developed material can be used as excellent thermally insulating interior plaster with the thermal conductivity of (0.082 ± 0.005) W/(m·K). In addition, its practical application can bring substantial economic and environmental benefits as the material contains 25 % of waste stone dust.

What guarantees the “peaceful coexistence” of quantum nonlocality and special relativity? The tension arises because entanglement leads to locally inexplicable correlations between distant events that have no absolute temporal order in relativistic spacetime. This paper identifies a relativistic consistency condition that is weaker than Bell locality but stronger than the no-signaling condition meant to exclude superluminal communication. While justifications for the no-signaling condition often rely on anthropocentric arguments, relativistic consistency is simply the requirement that joint outcome distributions for spacelike separated measurements (or measurement-like processes) must be independent of their temporal order. This is necessary to obtain consistent statistical predictions across different Lorentz frames. We first consider ideal quantum measurements, derive the relevant consistency condition on the level of probability distributions, and show that it implies no-signaling (but not vice versa). We then extend the results to general quantum operations and derive corresponding operator conditions. This will allow us to clarify the relationships between relativistic consistency, no-signaling, and local commutativity. We argue that relativistic consistency is the basic physical principle that ensures the compatibility of quantum statistics and relativistic spacetime structure, while no-signaling and local commutativity can be justified on this basis.

1 We modelled the growth in estimated biomass of individuals in experimental populations of Chenopodium album grown at two densities and measured sequentially nine times over 128 days. Competition is modelled by coupling individual growth equations and, within the population, the growth rate of a plant at any point in time is a function of its size to the power a, a measure of the degree of size-asymmetry of competition. 2 The growth of individuals in these crowded populations was significantly better fit by a Richards growth model than by models with one fewer parameter (e.g. logistic or Gompertz growth models). The additional parameter determines the location of the inflection point and provides great flexibility in fitting growth curves. Density had a significant effect on this parameter. 3 At the higher density, the maximum size that plants achieved was decreased and the exponential phase of growth was reduced. The estimate of the size-asymmetry parameter a was always greater than 1 and it increased significantly at the higher density. Growth was reduced and the number of very small individuals increased at the higher density, although a few plants still achieved a large size. 4 Our approach combines several recent innovations in the modelling of stand development, including (a) modelling of individual growth with biologically meaningful growth models and (b) modelling the relationship between size and growth of individuals within the population. Sequential, non-destructive data on the growth of individuals over time, in combination with modern statistical computing techniques, can lead to major advances in the modelling of plant population development, providing powerful tools for exploring variation in individual growth and for testing a wide range of hypotheses.