Quantum Information Dimension and Geometric Entropy

TL;DR

The paper develops geometric quantum mechanics (GQM) by treating quantum states as probability measures on the complex projective state space, introducing geometric quantum states (GQS) to capture ensemble structure. It defines the quantum information dimension $\mathfrak{D}$ and the dimensional geometric entropy $H_{\mathfrak{D}}$, adapting Renyi-style notions to quantum geometry and establishing their operational meaning via an analog of the asymptotic equipartition property. Through four case studies—finite and infinite environments, a particle in a 2D box, chaotic quantum maps, and the thermodynamic limit—the authors demonstrate that $\mathfrak{D}$ detects fractal-like support and $H_{\mathfrak{D}}$ quantifies information content, with interesting results such as $\mathfrak{D}_{\infty} \approx 0.83$ and $h_{\infty} \approx 0.66$ in the thermodynamic limit. The work provides a geometric bridge between classical information theory and quantum information, offering new tools to analyze open quantum systems and suggesting rich future directions in quantum fractals and ensemble-based quantum information. It highlights the potential for broader connections to entropy and information measures in the quantum domain and motivates further exploration of ensembles of pure states in quantum technologies.

Abstract

Geometric quantum mechanics, through its differential-geometric underpinning, provides additional tools of analysis and interpretation that bring quantum mechanics closer to classical mechanics: state spaces in both are equipped with symplectic geometry. This opens the door to revisiting foundational questions and issues, such as the nature of quantum entropy, from a geometric perspective. Central to this is the concept of geometric quantum state -- the probability measure on a system's space of pure states. This space's continuity leads us to introduce two analysis tools, inspired by Renyi's information theory, to characterize and quantify fundamental properties of geometric quantum states: the quantum information dimension that is the rate of geometric quantum state compression and the dimensional geometric entropy that monitors information stored in quantum states. We recount their classical definitions, information-theoretic meanings, and physical interpretations, and adapt them to quantum systems via the geometric approach. We then explicitly compute them in various examples and classes of quantum system. We conclude commenting on future directions for information in geometric quantum mechanics.

Figures (13)

• Figure 1: Geometric quantum state $q(Z) = \sum_{k=1}^{2^9} p_k\delta[Z-Z_k]$ on $\mathbb{C}P^1$, with coordinates $(\theta,\phi)$. It is a finite sum of $2^9$ Dirac measures. Each point is a possible pure state $\ket{Z_k} = \cos \theta_k/2 \ket{0} + \sin \theta_k/2 e^{i\phi_k}\ket{1}$ in which the system can be, with probability $p_k$. The specific value of each $p_k$ is encoded in a point's color; see legend.
• Figure 2: Quantum state space of a qutrit: (Left) A finite-dimensional quantum system with $D=3$ represented in $2D$. Section \ref{['sec:QID']} noted that canonically conjugated coordinates allow considering the full quantum state space as a classical $2-$simplex $\Delta_2$, which represents the space of classical probability distributions $(1-p_1-p_2,p_1,p_2)$. (Right) A two-torus $\mathbb{T}^2$ that accounts for the nontrivial phases $\left(\phi_1,\phi_2\right)$.
• Figure 3: Geometric quantum states visited along a single trajectory generated by the Extended Baker's Map with parameters $\lambda_a=\lambda_b=0.2$, $\beta=4\pi/10$, and initial condition $(p_0,\phi_0)=(0.32865,0.98886)$. $N=10^7$ time-steps plotted on the Bloch square $(p,\phi) \in [0,1] \times [0,2\pi]$. Over time, due to the map's chaotic nature, even a single trajectory covers a (strange) attractor, with self-similar (fractal) structure. More specifically, vertically, the attractor has a uniform structure. Horizontally, it has self-similar, fractal structure, equivalent to a generalized Cantor set. This is demonstrated, going from the left panel to the right, via successively magnifying small subsets of states.
• Figure 4: Extended Baker's Map information dimension $d_I$: The estimation incrementally decreases the coarse-graining scale $\epsilon$ and, at each step, calculates $H(Z^\epsilon)$. Then, excluding initial points to avoid saturation, it performs a least-square fit to extract the $H(Z^\epsilon)$'s growth rate as a function of $\log1/\epsilon$. We estimate $d_I = 1.31 \pm 0.01$. This is fully consistent with the analytical prediction of $d_I = 1.31$, plotted in red. See Eq. (\ref{['eq:ID_BM']}) and Ref. Farm83 for the analytical estimate.
• Figure 5: Standard Map on the Bloch Square: Quantum states $(p,\phi) \in [0,1] \times [0,2\pi]$ iterated at $K=1.15$ over a uniform grid of initial conditions. (Left) Initial distribution $n=0$: Homogeneous distribution with $30^2$ points $(p_0^{(j)},\phi_0^{(k)}) = (j/30,2\pi k/30)$, with $j,k=0,\ldots,29$ running over all initial conditions, each distinctly colored. (Middle) After only $n=25$ iterations the points begin to mix, according to whether they lie in regions of periodic, quasiperiodic, or chaotic behavior. (Right) The long-term distribution, after $n=1000$ iterations. The full range of behaviors is evident.

Theorems & Definitions (7)

• Definition 1: Quantum Information Dimension
• Definition 2: Dimensional quantum entropy
• Theorem 1: G-AEP for i.i.d. quantum processes
• Theorem 2
• Theorem 3
• Definition 1: Extended Baker's Map
• Definition 2: Standard Map