Trace as Coarse-Graining in Markov Dynamics

Urs R. Hähner, Peter Staar, Mi Jiang +2 more

Dariusz Chruściński, Andrzej Kossakowski

Huzihiro Araki, Dénes Petz

Monotonicity under coarse-graining is a crucial property of the quantum relative entropy. The aim of this paper is to investigate the condition of equality in the monotonicity theorem and in its consequences as the strong sub-additivity of von Neumann entropy, the Golden-Thompson trace inequality and the monotonicity of the Holevo quantitity. The relation to quantum Markov states is briefly indicated.

Maxim V. Churilov

A coarse description of a dynamical system generally fails to evolve autonomously: evolving and coarse-graining in different orders gives different predictions, and the compressed propagator fails the semigroup law. We show that this path dependence is the observable boundary of missing state. For a finite continuous-time Markov chain with generator $L$, coarse channel $C$, and stochastic section $R$ satisfying $CR=I$, define $P=RC$, $Q=I-P$, and $K=CLR$. The exact coarse propagator $T_t=C\mathrm{e}^{tL}R$ obeys a Volterra equation with memory kernel $\mathcal{M}(t)=CLQ\,\mathrm{e}^{tQLQ}\,QLR$. Every memory jet $\mathcal{M}^{(k)}(0)$ is recursively recoverable from the short-time derivatives of $T_t$ alone. The rank of the block Hankel operator $\mathbb{H}_{\infty}=(\mathcal{M}^{(i+j)}(0))_{i,j\geq 0}$ is exactly the smallest number of auxiliary linear state variables required to turn the coarse dynamics into an exact autonomous, time-homogeneous system. A minimal completion is constructible by Ho--Kalman factorization and is unique up to similarity. If $d=\min\{k:\mathcal{M}^{(k)}(0)\neq 0\}$, then every smooth classical $f$-divergence generically scales as $t^{2d+4}$. Exact Markov examples realize exponents $4,6,8$ and closure ranks $1,2,3$. We also derive finite-sample singular-value certificates for hidden predictive coordinates, including a matrix-Bernstein bound with intrinsic-variance control. Finally, histories are identified when they induce the same future law under every admissible intervention, yielding the minimal counterfactual predictive state. The results turn coarse-graining into a constructive state-completion problem and separate ordinary linear realization from the harder requirement of a positive or Markov realization.

William C. Saslaw

DÉNES PETZ

Monotonicity under coarse-graining is a crucial property of the quantum relative entropy. The aim of this paper is to investigate the condition of equality in the monotonicity theorem and in its consequences as the strong sub-additivity of von Neumann entropy, the Golden–Thompson trace inequality and the monotonicity of the Holevo quantitity. The relation to quantum Markov states is briefly indicated.