This paper provides a third independent derivation of the critical coherence threshold K_crit ≈ 0.127, complementing the dynamical derivation (GCT) and thermodynamic derivation (Landauer bound). The falsifiability component F(t) is identified with the Fisher information of the system’s self-estimating process. The Cramér–Rao bound then establishes that as F(t) → 0, coherence estimation variance diverges — the system becomes informationally blind to its own state. K_crit is the Fisher information collapse point: below it, no estimator can locate the system’s coherence state with bounded precision. The 1=1 invariant is identified as the geodesic endpoint of the Fisher information manifold — the natural destination of the geometry itself under contractive evaluation dynamics. Three independent frameworks (dynamical systems, thermodynamics, information geometry) converge on the same critical threshold and the same invariant. Part of the Spektre corpus (github.com/spektre-labs/corpus).
This paper provides a third independent derivation of the critical coherence threshold K_crit ≈ 0.127, complementing the dynamical derivation (GCT) and thermodynamic derivation (Landauer bound). The falsifiability component F(t) is identified with the Fisher information of the system’s self-estimating process. The Cramér–Rao bound then establishes that as F(t) → 0, coherence estimation variance diverges — the system becomes informationally blind to its own state. K_crit is the Fisher information collapse point: below it, no estimator can locate the system’s coherence state with bounded precision. The 1=1 invariant is identified as the geodesic endpoint of the Fisher information manifold — the natural destination of the geometry itself under contractive evaluation dynamics. Three independent frameworks (dynamical systems, thermodynamics, information geometry) converge on the same critical threshold and the same invariant. Part of the Spektre corpus (github.com/spektre-labs/corpus).
Establishes that the 1=1 invariant is the classical-scale instance of a universal coherence condition appearing at every level of physical organization. The quantum coherence condition S_vN = 0, the thermodynamic minimum-entropy state (k_B), the Fisher information geodesic endpoint (Cramér–Rao), and the evaluative fixed point (ECT) are identified as equivalent expressions of one invariant at different scales. The full correspondence chain: ℏ → S_vN = 0 → 1=1 → K(t) > K_crit → k_BT ln2 → I(K). Every coherent system is a Maxwell demon operating at its characteristic scale. Coherence is not emergent — it is the ground condition of physical reality. Part of the Spektre corpus (github.com/spektre-labs/corpus).
Information geometry usually refers to a class of geometries characterized from metrics and corresponding connections related to the notion of statistical information during the process of parameter estimation on classical or quantum systems.
We are aware of the fact that in parameter estimation we can't achieve perfectness. There is a lower bound, which is not zero, for the variance - covariance of the estimated parameter compared to the real parameter, the Cramer - Rao bound. In classical parameter estimation, this bound is given by the reciprocal of Fisher information and is unique in the sense that the Fisher is the unique monotone metric under stochastic transformations. In quantum parameter estimation there is a class of monotone metrics and the lowest possible bound, which determines the best quality of estimation, is given by that of the Symmetric Logarithmic Derivative (SLD). The term symmetric determines the derivative type, since there are more than one types due to the non-commutativity of operators, in general, in quantum mechanics.
In our first published paper in the framework of this Ph.D. thesis, we are working beyond the class on monotone metrics. We are using the phase space to define a new statistical metric. This is achieved by introducing logarithmic derivatives through the Husimi Q distribution and consequently the new metric. As it turns out the related bound of Cramer - Rao type, can be greater even of this of the Symmetric Logarithmic Derivative.
In our second paper, we refer to a particular bi-parameter family of generalized entropies used for the classification of complex systems. By defining a divergence for the discrete distribution, we consequently define a metric and the corresponding Levi - Civita connection, which depend on the two parameters say c and d. So, we are having geometrical objects with statistical content like the Cramer - Rao bound and the scalar curvature, which is characteristic (and different) for every class of complex systems.
In our third paper, we are treating the above parameters c and d as coordinates of a two dimensional manifold. By calculating the Fisher metric and the corresponding scalar curvature we observe particular values of c and d for which we are making the conjecture that they related with particular properties of complex systems.
Finally, we are highlighting a correlation of the curvature of a particular manifold and quantum entanglement. In particular, we are using the entropy of entanglement to define a two dimensional manifold, which we embed into three dimensional real space. We find that in the region which this surface has maximum curvature we have product quantum states and only there.
We investigate the connection between the time evolution of averages of stochastic quantities and the Fisher information and its induced statistical length. As a consequence of the Cramér-Rao bound, we find that the rate of change of the average of any observable is bounded from above by its variance times the temporal Fisher information. As a consequence of this bound, we obtain a speed limit on the evolution of stochastic observables: Changing the average of an observable requires a minimum amount of time given by the change in the average squared, divided by the fluctuations of the observable times the thermodynamic cost of the transformation. In particular, for relaxation dynamics, which do not depend on time explicitly, we show that the Fisher information is a monotonically decreasing function of time and that the minimal required time is determined by the initial preparation of the system. We further show that the monotonicity of the Fisher information can be used to detect hidden variables in the system and demonstrate our findings for simple examples of continuous and discrete random processes.
Using differential geometry, I derive a form of the Bayesian Cramér-Rao bound that remains invariant under reparametrization. With the invariant formulation at hand, I find the optimal and naturally invariant bound among the Gill-Levit family of bounds. By assuming that the prior probability density is the square of a wavefunction, I also express the bounds in terms of functionals that are quadratic with respect to the wavefunction and its gradient. The problem of finding an unfavorable prior to tighten the bound for minimax estimation is shown, in a special case, to be equivalent to finding the ground state of a Schrödinger equation, with the Fisher information playing the role of the potential. To illustrate the theory, two quantum estimation problems, namely, optomechanical waveform estimation and subdiffraction incoherent optical imaging, are discussed.
Version of 08 July 2026: Technically refined preprint incorporating clarifications discovered post-submission. We derive a fundamental upper bound on the number of particles that can sustain quantum coherence in a maximally entangled state from a unified geometric framework in which the observable spacetime 𝓜 is a four-dimensional continuous embedding within a higher-dimensional substratum 𝓐 with quantized information interface. Three derivations—direct application of action quantization, the quantum Cramér-Rao bound, and the quantum capacity of the 𝓜–𝓐 information channel—converge to the canonical form N_crit = n_max ℏ / (2 |λ| |Ψ_max| V₃ τ); these are the same action-quantization constraint expressed in action, metrological, and channel-capacity language: the Cramér-Rao route additionally supplies the metrological (Heisenberg 1/N) reading, but the ceiling N_crit follows in every case from action quantization, not from three independent principles. The bound predicts sharp coherence saturation independent of environmental isolation, distinguishing the framework from standard environmental decoherence. Discriminating signatures include a conjectured distinction between fermionic and bosonic systems, an abrupt knee in decoherence rate scaling at N_crit, and a trade-off N · τ ≤ K_AFT relevant to quantum computation. A natural reinterpretation of dynamical decoupling as embedding trajectory engineering connects the framework to established experimental techniques.
Abstract The quantum Cramér–Rao bound sets a fundamental limit on the accuracy of unbiased parameter estimation in quantum systems, relating the uncertainty in determining a parameter to the inverse of the quantum Fisher information. We experimentally demonstrate near saturation of the quantum Cramér–Rao bound in the phase estimation of a solid-state spin system, provided by a nitrogen-vacancy center in diamond. This is achieved by comparing the experimental uncertainty in phase estimation with an independent measurement of the related quantum Fisher information. The latter is independently extracted from coherent dynamical responses of the system under weak parametric modulations, without performing any quantum-state tomography. While optimal parameter estimation has already been observed for quantum devices involving a limited number of degrees of freedom, our method offers a versatile and powerful experimental tool to explore the Cramér–Rao bound and the quantum Fisher information in systems of higher complexity, as relevant for quantum technologies.