Chapter 20: Information Geometry & Quantum Information

The Fisher Information Metric

Every family of probability distributions carries a natural geometry. Given a parametric family \(\{p(x;\theta)\}\) indexed by parameters\(\theta = (\theta^1, \ldots, \theta^n)\), the Fisher information matrix defines a Riemannian metric on the parameter space:

\[ g_{ij}(\theta) = \mathbb{E}\!\left[\frac{\partial \log p}{\partial \theta^i}\frac{\partial \log p}{\partial \theta^j}\right] \]

This metric, introduced by C. R. Rao in 1945, is the unique (up to scale) Riemannian metric on the space of probability distributions that is invariant under sufficient statistics. It captures how distinguishable nearby distributions are: large\(g_{ij}\) means small changes in \(\theta\) produce large changes in the distribution, making distributions easy to tell apart.

Shun-ichi Amari systematized this into information geometry from the 1980s onward, showing that the space of probability distributions carries not just a metric but a family of connections (the \(\alpha\)-connections), a dually flat structure, and deep links to statistical inference, neural networks, and thermodynamics. The geodesics of the Fisher metric are the paths of steepest statistical change.

From Shannon to von Neumann: Entropy Goes Quantum

Shannon's entropy \(H = -\sum_i p_i \log p_i\) (1948) measures the uncertainty or information content of a classical probability distribution. Von Neumann's quantum entropy, defined for a density matrix \(\rho\), extends this to quantum mechanics:

\[ S(\rho) = -\operatorname{Tr}(\rho \log \rho) \]

Von Neumann introduced this formula in 1932, decades before Shannon. For a pure quantum state \(|\psi\rangle\), the density matrix \(\rho = |\psi\rangle\langle\psi|\)has \(S(\rho) = 0\): no classical uncertainty. For a maximally mixed state,\(S\) is maximal. The von Neumann entropy quantifies the quantum indeterminacy that is irreducibly part of the state itself.

The quantum Fisher information metric on the space of density matrices is the Bures metric, and it determines the ultimate precision limits of quantum measurement through the quantum Cramér–Rao bound. The geometry of quantum state space — known as the Bloch sphere for qubits — is intrinsically non-Euclidean and carries deep physical meaning.

Entanglement as a Geometric Resource

Quantum entanglement — the non-classical correlations between subsystems of a composite quantum state — can be quantified geometrically. For a bipartite system \(AB\) in state \(\rho_{AB}\), the entanglement entropy is the von Neumann entropy of the reduced density matrix:

\[ S_A = -\operatorname{Tr}_A(\rho_A \log \rho_A), \quad \rho_A = \operatorname{Tr}_B(\rho_{AB}) \]

For a pure state, \(S_A = 0\) if and only if the state is a product state (unentangled). Maximum entanglement corresponds to maximum entropy of the subsystem. This measure — and its generalizations, the Rényi entropies\(S_n = \frac{1}{1-n}\log\operatorname{Tr}(\rho^n)\) — has become the central quantity in quantum information theory.

Entanglement is a genuinely non-classical resource: it cannot be created by local operations and classical communication (LOCC). The geometry of entangled states within the space of all quantum states is extraordinarily complex. Separable (unentangled) states form a convex set; entangled states are everything outside it. Detecting and quantifying entanglement geometrically remains an active research area.

The Ryu–Takayanagi Formula: Geometry from Information

In 2006, Shinsei Ryu and Tadashi Takayanagi proposed a formula that stunned the theoretical physics community. In the AdS/CFT correspondence, the entanglement entropy of a region \(A\) in the boundary field theory equals the area of the minimal surface in the bulk that is homologous to \(A\):

\[ S_A = \frac{\text{Area}(\gamma_A)}{4G_N} \]

This is a direct quantum generalization of the Bekenstein–Hawking entropy formula for black holes (\(S = A/4G_N\)). It says that the geometry of the bulk spacetime — specifically the areas of minimal surfaces — is encoded in the entanglement structure of the boundary quantum state.

The implications are profound: spacetime geometry emerges from entanglement. Regions of the bulk that are geometrically connected correspond to regions of the boundary whose quantum states are highly entangled. Remove entanglement, and the geometry falls apart. Mark van Raamsdonk made this intuition vivid in 2010 by showing that decreasing entanglement between two boundary halves corresponds to the bulk spacetime pinching off into disconnected pieces.

Quantum Error Correction, Tensor Networks, and ER=EPR

The connection between holography and quantum error correction was uncovered by Almheiri, Dong, and Harlow in 2015. The bulk spacetime acts as a quantum error-correcting code: local operators in the bulk are encoded redundantly in the boundary degrees of freedom, so that even if a boundary region is removed (erased), the bulk information can be reconstructed from the remaining boundary.

Tensor networks make this precise. The MERA (Multi-scale Entanglement Renormalization Ansatz) network, introduced by Vidal, encodes a quantum state in a hierarchical structure whose geometry matches hyperbolic space — the geometry of \(\text{AdS}\). The entanglement structure of the quantum state literally constructs the bulk geometry.

Maldacena and Susskind's ER=EPR conjecture (2013) goes further: every pair of entangled particles is connected by a microscopic Einstein–Rosen bridge (wormhole). The Einstein–Podolsky–Rosen entanglement of quantum mechanics and the Einstein–Rosen wormhole of general relativity are not merely analogous — they are the same thing.