Quantum Information Dynamics

• Quantum Information Dynamics is a framework that employs state-space geometry and Bregman relative entropy over W*-algebras to capture fine-grained, nonlinear quantum evolution.
• It integrates methods such as exponential charts, fibrewise Liouvillean perturbations, and coherent-state path integrals to model local controls and entropic priors.
• The framework provides new insights into nonequilibrium quantum statistical mechanics and post-quantum theories by linking geometric, algebraic, and renormalization techniques.

The local quantum information dynamics framework posits that the geometric, algebraic, and entropic structure of spaces of normal quantum states over W*-algebras (von Neumann algebras) can be used to describe the fine-grained evolution of quantum information, including phenomena beyond conventional linear quantum theory. Central ingredients are the geometry of the state space (endowed with a quantum Bregman distance), fibrewise dynamical generators (Liouvilleans) incorporating local controls, path-integral quantization with locally entropic priors, and the renormalization group flow of information-theoretic quantities under coarse graining and control-variable elimination. This approach yields new insights for nonequilibrium quantum statistical mechanics and post-quantum information theory.

1. Geometry of the Manifold of Normal States on a W*-Algebra

Let $\mathcal{N}$ be a W*-algebra and $\mathcal{N}_*$ its predual. The manifold $\mathcal{M}$ of normal states consists of all faithful, normal, positive normalized linear functionals $\rho : \mathcal{N} \to \mathbb{C}$. At each $\rho \in \mathcal{M}$, the tangent space is

$$T_\rho \mathcal{M} \cong \left\{ \xi \in \mathcal{N}_* \mid \langle 1, \xi \rangle = 0 \right\}$$

modeled by appropriate noncommutative Orlicz spaces $L_\Psi(\mathcal{N})$.

Local Charts:

A local exponential chart at $\rho$ is

$$\varphi_\rho: \xi \in U \subset L_\Psi(\mathcal{N}) \longmapsto \rho^\xi = \exp( \log \rho + \xi - \psi_\rho(\xi) )$$

where $\psi_\rho(\xi)$ ensures normalization ($\mathcal{N}_*$0).

Bregman Relative Entropy:

A quantum Bregman divergence (e.g., the Umegaki–Araki relative entropy) is

$\mathcal{N}_*$1

This can be generalized: For a convex functional $\mathcal{N}_*$2 on $\mathcal{N}_*$3,

$\mathcal{N}_*$4

with $\mathcal{N}_*$5.

Geometry from Eguchi Construction:

Assuming $\mathcal{N}_*$6 is smooth and satisfies the strict positivity of its Hessian,

• Riemannian metric at $\mathcal{N}_*$7:

$\mathcal{N}_*$8

• Dual affine connections $\mathcal{N}_*$9 and $\mathcal{M}$0 by third derivatives:

$\mathcal{M}$1

$\mathcal{M}$2

These structures make $\mathcal{M}$3 a Norden–Sen manifold; in finite dimensions, if the connections are flat, one recovers a dually-flat Hessian geometry.

2. Fibrewise Perturbation of Liouvilleans for Local Dynamics

GNS Hilbert Bundle:

To each $\mathcal{M}$4 corresponds a GNS Hilbert space $\mathcal{M}$5. These fibres form a Banach Hilbert bundle $\mathcal{M}$6.

Standard Liouvillean:

If $\mathcal{M}$7 is a Banach–Lie–Poisson manifold under the action of a Lie algebra $\mathcal{M}$8, any real Hamiltonian function $\mathcal{M}$9 generates a Poisson flow $\rho : \mathcal{N} \to \mathbb{C}$0. The standard (unperturbed) Liouvillean $\rho : \mathcal{N} \to \mathbb{C}$1 acts as

$\rho : \mathcal{N} \to \mathbb{C}$2

with $\rho : \mathcal{N} \to \mathbb{C}$3.

Fibrewise Perturbed Liouvillean:

A local control variable (possibly nonlinear) $\rho : \mathcal{N} \to \mathbb{C}$4, with $\rho : \mathcal{N} \to \mathbb{C}$5, modifies the dynamics: $\rho : \mathcal{N} \to \mathbb{C}$6 or, for the commutator perturbation,

$\rho : \mathcal{N} \to \mathbb{C}$7

As $\rho : \mathcal{N} \to \mathbb{C}$8 is $\rho : \mathcal{N} \to \mathbb{C}$9-dependent, the global evolution becomes nonlinear.

Nonlinear Poisson Flow on $\rho \in \mathcal{M}$0:

On the base manifold, the effective Hamiltonian 1-form is

$\rho \in \mathcal{M}$1

and the total vector field is

$\rho \in \mathcal{M}$2

so the evolution is

$\rho \in \mathcal{M}$3

In (local) coordinates $\rho \in \mathcal{M}$4: $\rho \in \mathcal{M}$5

3. Path-Integral Generalisation with Coherent States

Daubechies–Klauder Coherent-State Propagator:

The propagator from $\rho \in \mathcal{M}$6 to $\rho \in \mathcal{M}$7 over time $\rho \in \mathcal{M}$8 is

$\rho \in \mathcal{M}$9

• Action functional:

$$T_\rho \mathcal{M} \cong \left\{ \xi \in \mathcal{N}_* \mid \langle 1, \xi \rangle = 0 \right\}$$0

where $$T_\rho \mathcal{M} \cong \left\{ \xi \in \mathcal{N}_* \mid \langle 1, \xi \rangle = 0 \right\}$$1 is (possibly) a Kähler connection or Berry connection.

• Entropic prior:

$$T_\rho \mathcal{M} \cong \left\{ \xi \in \mathcal{N}_* \mid \langle 1, \xi \rangle = 0 \right\}$$2

• Wiener measure:

$$T_\rho \mathcal{M} \cong \left\{ \xi \in \mathcal{N}_* \mid \langle 1, \xi \rangle = 0 \right\}$$3

with $$T_\rho \mathcal{M} \cong \left\{ \xi \in \mathcal{N}_* \mid \langle 1, \xi \rangle = 0 \right\}$$4 as in Section 1.

Local Entropic Priors:

A second-order expansion in $$T_\rho \mathcal{M} \cong \left\{ \xi \in \mathcal{N}_* \mid \langle 1, \xi \rangle = 0 \right\}$$5 yields

$$T_\rho \mathcal{M} \cong \left\{ \xi \in \mathcal{N}_* \mid \langle 1, \xi \rangle = 0 \right\}$$6

showing that the local "kinetic term" in the path-integral weighting comes from the Bregman distance's Hessian at $$T_\rho \mathcal{M} \cong \left\{ \xi \in \mathcal{N}_* \mid \langle 1, \xi \rangle = 0 \right\}$$7, making the entropic prior coincide with a metric-based Wiener measure.

Curvature Corrections:

Expanding to third order, the Christoffel symbols and thus the state-space curvature enter. These yield quantum corrections to the semiclassical action (Van Vleck determinant and higher-loop structures) that are fundamentally geometric.

4. Role of Bregman Relative Entropy in Jaynes–Mitchell–Favretti Renormalisation

Information Mass and RG Flow:

Let $$T_\rho \mathcal{M} \cong \left\{ \xi \in \mathcal{N}_* \mid \langle 1, \xi \rangle = 0 \right\}$$8 ("information mass"). Upon coarse graining or control-variable elimination (see Jaynes–Mitchell "source theory"), the induced submanifold metric transforms as

$$T_\rho \mathcal{M} \cong \left\{ \xi \in \mathcal{N}_* \mid \langle 1, \xi \rangle = 0 \right\}$$9

Implying the mass flows as

$L_\Psi(\mathcal{N})$0

where $L_\Psi(\mathcal{N})$1 is the linear transformation eliminating the "control" subspace. More generally, the renormalization group (RG) equation for $L_\Psi(\mathcal{N})$2 is

$L_\Psi(\mathcal{N})$3

with $L_\Psi(\mathcal{N})$4 the external scaling parameter and $L_\Psi(\mathcal{N})$5 the Ricci scalar of the dual connection.

Curvature and Zero-Point Energy:

When control directions are eliminated, the dual (star) connection's scalar curvature transforms as

$L_\Psi(\mathcal{N})$6

The zero-point energy ("vacuum energy") shift is

$L_\Psi(\mathcal{N})$7

Thus, zero-point energy invariance is tied to the local redefinition of information mass under entropic RG flow.

5. Implications for Nonequilibrium Quantum Statistical Mechanics

Quantum Orlicz Spaces and Extended Structure:

The most general construction equips $L_\Psi(\mathcal{N})$8 with charts modeled on noncommutative Orlicz spaces $L_\Psi(\mathcal{N})$9, admitting general convex "gauge" functionals $\rho$0. The tangent and cotangent bundles have duality

$\rho$1

Transport of states and expectation values is described by Banach–Lie group coadjoint actions, with the Poisson geometry encoding non-linear control and local statistical inference.

Synthesis of Structures:

• Geometric: Bregman-relative entropy induces informational Riemannian geometry, dual affine connections, and a rich theory of geodesics and curvature effects in state space.
• Entropic: Probabilistic state-space exploration and path integrals are governed by local quantum-relative entropic priors and their geometric expansions.
• Algebraic: The use of operator algebras and their Banach–Lie structure generalizes standard quantum mechanical dynamics and provides a rigorous setting for local post-quantum theories with nonlinear and locally gauge-invariant flows.

Altogether, this framework provides a comprehensive, flexible, and information-geometric foundation for quantum dynamics on general state spaces—including nonequilibrium regimes, renormalization, and control—built on the interplay between geometry, entropy, and the rich structure of operator algebras (Kostecki, 2016).