The preparation of quantum many-body systems faces the difficulty that in a realistic scenario only few control parameters of the system may be accessible. In this context, an interesting connection between the energy fluctuations during state preparation and its geometric length as measured by the Fubini-Study metric was discussed in [5]. An inspiring conjecture lower bounding the energy fluctuations by the minimal geometric length of all accessible state preparation protocols was put forward together with supporting examples and numerical evidence. However, we here show that the conjecture does not hold but can be violated if the accessible parameter space has extrinsic curvature, when embedded into the space of all dynamically accessible states. We illustrate this by a number of generic qubit, qutrit and harmonic oscillator systems.

The fermionic von Neumann entropy, the fermionic entanglement entropy and the fermionic relative entropy are defined for causal fermion systems. Our definition makes use of entropy formulas for quasi-free fermionic states in terms of the reduced one-particle density operator. Our definitions are illustrated in various examples for Dirac spinors in two- and four-dimensional Minkowski space, in the Schwarzschild black hole geometry and for fermionic lattices. We review area laws for the two-dimensional diamond and a three-dimensional spatial region in Minkowski space. The connection is made to the computation of the relative entropy using modular theory.

The numerical analysis of causal fermion systems is advanced by employing differentiable programming methods. The causal action principle for weighted counting measures is introduced for general values of the integer parameters f (the particle number), n (the spin dimension) and m (the number of spacetime points). In the case n= 1, the causal relations are clarified geometrically in terms of causal cones. Discrete Dirac spheres are introduced as candidates for minimizers for large m in the cases n = 1,f = 2 and n = 2,f = 4. We provide a thorough numerical analysis of the causal action principle for weighted counting measures for large m in the cases n = 1,2 and f = 2,3,4. Our numerical findings corroborate that all minimizers for large m are good approximations of the discrete Dirac spheres. In the example n = 1,f = 3 it is explained how numerical minimizers can be visualized by projected spacetime plots. Methods and prospects are discussed to numerically investigate settings in which hitherto no analytic candidates for minimizers are known.