Abstract Quantum coherence serves as a defining characteristic of quantum mechanics, finding extensive applications in quantum computing and quantum communication processing. This study explores quantum block coherence in the context of projective measurements, focusing on the quantification of such coherence. Firstly, we define the correlation function between the two general projective measurements P and Q, and analyze the connection between sets of block incoherent states related to two compatible projective measurements P and Q. Secondly, we discuss the measure of quantum block coherence with respect to projective measurements. Based on a given measure of quantum block coherence, we characterize the existence of maximal block coherent states through projective measurements. This research integrates the compatibility of projective measurements with the framework of quantum block coherence, contributing to the advancement of block coherence measurement theory.
In a recent paper it was shown that all the Hilbert space formulas for quantum probabilities can be realized as functions of geometric properties of the associated projective space, but those functions were expressed using the structures of the associated Hilbert space. In this paper a direct description of all these probabilities is given as formulas involving only the geometric properties of the projective space itself without referring to the associated Hilbert space theory. In large part this depends on a projection theorem for complex projective space which is analogous to the projection theorem for Hilbert spaces. The importance of this is that this exhibits quantum probability in terms of the geometry of a Riemannian metric in a non-linear Kähler manifold without any reference to a linear Hilbert space. As such this is a part of a larger program of the geometrization of physics. This opens the possibility of generalizations of quantum theory in other similar geometric settings. The theory presented includes projective spaces of both finite and infinite dimension. Some comments explain how quantum theory based on a von Neumann algebra is compatible with this approach.
In this chapter, we study the properties of quantum entanglement, and more generally the way correlations can appear in quantum mechanics. Quantum entanglement is an important notion that we have already discussed, for instance in the context of the Von Neumann chain or of the Schrödinger cat, but here we give more details on its properties. In classical physics, the notion of correlation is well known. It hinges on the calculation of probabilities and on linear averages over a number of possibilities. A distribution gives the probability of having the first system in a some given state and the second in another state. If this distribution is not a product, the two systems are correlated. If it is a product, they are uncorrelated; measuring the properties of one system does not bring any information on the other. This is in particular the case if the state of each of the two systems is perfectly defined (which also defines the state of the whole system perfectly well). The notion of correlation between sub-systems therefore stems from the multiplicity of possible states of the whole system; fluctuations of this state are necessary to give its full meaning to the classical notion of correlation. In quantum mechanics, the situation is different: as we have seen (in particular in Chapter 4), even a physical system that is perfectly defined by a given state vector already contains fluctuations.
What happens when a quantum system undergoing unitary evolution in time is subject to repeated projective measurements to the initial state at random times? A question of general interest is: How does the survival probability
Sm, namely, the probability that an initial state survives even after
m number of measurements, behave as a function of
m? In the context of two paradigmatic quantum systems, one evolving in discrete and the other in continuous time, we present a wealth of numerical and analytical results that hint at the curious nature of quantum measurement dynamics. In particular, we unveil that when evolution after every projective measurement continues with the projected component of the instantaneous state, the average and the typical survival probability decay as an exponential in
m for large
m. By contrast, if the evolution continues with the leftover component, namely, what remains of the instantaneous state after a measurement has been performed, the survival probability decays asymptotically as a power law in
m with exponent
3/2. These results hold independently of the choice of the distribution of times between successive measurements, implying robustness and ubiquity of our derived results.
The original purpose of measurements is to provide us with information about a previously unknown physical property of the systeIn observed.In the Hilbert space formalism of quantum mechanics, this physical meaning of measurement information is not immediately apparent.In order to study the relationship between the Hilbert space coherence of the quantum state and the measurement information obtained in the laboratory, we introduce a generalized measurement postulate for finite resolution measurements.With this measurement model, correlations between non-commuting observables can be investigated.These experimentally accessible correlations reveal nonclassical features in their dependence on the operator ordering, reflecting the particular measurement context by which they are determined.