We introduce a geometric entropy for quantum preparations, defined as the logarithm of the Hilbert-space volume of pure states compatible with a given set of constraints. This construction extends Boltzmann's counting perspective to the quantum setting, where compatible states need not be orthogonal and the relevant notion of"number of states"is naturally replaced by a volume in state space. We analyze three classes of constraints: restriction to a subspace, fixed expectation values, and coarse-grained subsystem descriptions. For representative examples, including subspace projection, spin expectation values, partial trace, and an imperfect detector map, we obtain explicit scaling laws and closed-form expressions for the associated volumes. The resulting framework provides a geometric measure of quantum ignorance at the level of the preparation and complements entropy notions based on density matrices and coarse graining.
Textbook quantum superposition refers to the feature that certain linear combinations of Hilbert space rays, each representing a valid quantum state, are themselves valid states. This notion is not operational, and it relies on the underlying Hilbert space formalism. Recent proposals for experimental tests of indefinite causal order, as well as tests probing the non-classicality of gravity, pivot on superposition, thereby calling for a theory-independent, operational formalisation of the concept. Here, we define superposition within the framework of Generalised Probabilistic Theories, based on observed statistics in prepare-and-measure experiments. Using this, we formulate three superposition principles to investigate which structural features of quantum theory carry over to other theories. We study conditions under which these principles carry over from subsystems to their compositions; to this end, we show that the quantum tensor product emerges as the largest composition rule for quantum systems respecting all three principles. Furthermore, we show how non-classical features such as entanglement and preparational uncertainty can be viewed as special forms of superposition.
Recently, we introduced a geometric object analogous to an orthonormal frame in the Cartan formalism to study the parameter space of quantum systems; we called it N-bein, with N being the number of parameters that characterize the quantum system. Acting as the ‘square root’ of the quantum geometric tensor (QGT), the N-bein allows us to define new tensors to improve our understanding of the structure beneath the parameter space of quantum mechanics. In this work, we extend this mathematical framework surrounding the N-bein to analyze the parameter space of quantum systems with degenerate spectra. As in the non-degenerate case, we define a non-Abelian two-state QGT to identify possible transitions between degenerate states after two consecutive parameter variations. Additionally, using the Wilczek–Zee connection, we introduce a torsion-like tensor as the covariant derivative of the N-bein. This torsion captures the noncommutativity of successive parameter variations and coincides with the antisymmetric part of the two-state QGT. We also present a geometrical formulation using differential forms and discuss the physical implications of the newly defined tensors. Furthermore, we construct several gauge-invariant observables from the N-bein and its derivatives to highlight the utility of the new tensors. Finally, to illustrate the convenience and applications of this formalism, we apply the theoretical framework to a system of coupled harmonic oscillators immersed in an electric field. The coupling between the oscillators results in a degenerate system. Thus, using the new formalism, we found correlations among the quantum states quantified by the new invariants.
Jorge Romero, Carlos A. Velasquez, J. David Vergara· Journal of Physics A: Mathem...· 0 citations
We propose that quantum mechanics emerges as the effective description of a deterministic physical reality from the perspective of an internal observer that constitutes a subsystem of a finite closed world. Because such an observer cannot access the complete ontological state of the world to which it belongs, its description is intrinsically incomplete. We refer to this fundamental limitation as subjective incompleteness. It naturally induces a factorization of the ontological state space with respect to gauge degrees of freedom that are indistinguishable to the observer. Within the proposed framework, the projective geometry of quantum state space is understood as a direct consequence of subjective incompleteness. This provides a common geometric origin for characteristic quantum phenomena, including nonlocality, contextuality, and intrinsic randomness.
A. Belinsky, Alexandr Kaminsky· Laser Physics Letters· 0 citations
We study the one-sided time evolution of thermofield double (TFD) states in random matrix theory, where the Hamiltonian is taken to be a $D\times D$ random matrix drawn from a unitarily invariant emsemble of Hermitian matrices. We argue that the Krylov basis with the maximally entangled state taken as the initial vector gives a semiclassical Hilbert space description of these TFD states in random matrix theory at any $O(1)$ temperature and time in the $D\to \infty$ limit, very analogous to the ``chord Hilbert space''construction in the double-scaled SYK (DSSYK) model. We study this semiclassical description in detail for ensembles where the spectral density in the $D\to \infty$ limit is even, compactly supported on an interval and has square root edges. With a few more conditions on the analytic structure of the spectral density, we observe that the semiclassical Hamiltonian has the same asymptotic behavior at large Krylov depth as that of DSSYK, with the corresponding parameter $\mathfrak{q}=e^{-\lambda}$ being related to the location of the nearest zero of the spectral density away from the spectral cut. Furthermore, in a large class of models corresponding to ultraviolet deformations of the DSSYK spectral density, i.e., where the spectrum in the UV is modified while leaving the near-edge behavior unchanged, we show that the semiclassical effective Hamiltonian in the Krylov basis reduces to the Liouville Hamiltonian of JT gravity in a low-energy, continuum limit. This suggests that our semiclassical Hilbert space should be interpreted as the bulk Hilbert space of a dual gravity description.
The geometry of quantum states is a fundamental research area with applications ranging from band theory in condensed matter to variational algorithms in quantum information. Due to their relative simplicity, pure states are usually studied, while mixed ones are needed in general, for instance to allow for finite temperatures. The geometry of mixed states, however, is much more challenging, with important aspects yet to be explored. Here, we provide one missing piece by identifying the structure that relates distance measure and curvature for general (mixed or pure) quantum states. This structure is shown to carry the physical meaning of fluctuation-dissipation---by directly relating distance measure with fluctuations and curvature with linear response---and it turns into the corresponding well-known structure for pure states, where it is K\"ahler, and only then. We thus obtain a general, simple fluctuation-dissipation picture, which, among its consequences, implies that response functions inherently probe the mixed-state quantum geometric tensor. We end by using this picture to give geometric characterizations of generic transport behaviors.
We argue for the universality of quantum theory using a dynamical consistency argument, within a specific Hamiltonian setting. We analyse two different types of coupling between simple quantum harmonic oscillators. Each illustrates an aspect of the free and interacting quantum fields and shows the inadequacy of semiclassical models. In particular, we establish that requiring the canonical algebra to be preserved under joint unitary dynamics rules out specific hybrid classical-quantum models. We apply our reasoning to the gravitational field in the linear regime, coupled to the quantised electromagnetic field and, separately, to quantised matter. We conclude with a comparison to DeWitt's analysis of quantum measurement, in which the apparatus, if classical, must be at least stochastic to preserve the Heisenberg Uncertainty Principle. We also note that stochastic models are inconsistent with the strict version of conservation principles, even if they comply with a probabilistic (on average) conservation.
C. Marletto, V. Vedral· 0 citations
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