We analyze the entropy behaviour of a discrete quantum channel defined by the composition of a unitary automorphism and a projective conditional expectation onto a finite-dimensional subalgebra. Let [Formula: see text], where [Formula: see text] is unitary and [Formula: see text] is the conditional expectation associated with an orthogonal projector decomposition of the identity. We note that the von Neumann entropy is monotone nondecreasing under repeated application of [Formula: see text]. Equality is characterized explicitly in terms of the invariant algebra of the channel. We further identify the fixed-point space as the intersection of the commutant of [Formula: see text] with the block-diagonal von Neumann algebra defined by the coarse-graining projectors. Finally, we describe the asymptotic structure under repeated application of the channel via spectral properties of completely positive trace-preserving maps. The results demonstrate that entropy growth in this setting arises solely from algebraic noncommutativity between unitary action and conditional expectation, without invoking thermodynamic limits, statistical ensembles, or environmental assumptions.
We analyze the entropy behaviour of a discrete quantum channel defined by the composition of a unitary automorphism and a projective conditional expectation onto a finite-dimensional subalgebra. Let [Formula: see text], where [Formula: see text] is unitary and [Formula: see text] is the conditional expectation associated with an orthogonal projector decomposition of the identity. We note that the von Neumann entropy is monotone nondecreasing under repeated application of [Formula: see text]. Equality is characterized explicitly in terms of the invariant algebra of the channel. We further identify the fixed-point space as the intersection of the commutant of [Formula: see text] with the block-diagonal von Neumann algebra defined by the coarse-graining projectors. Finally, we describe the asymptotic structure under repeated application of the channel via spectral properties of completely positive trace-preserving maps. The results demonstrate that entropy growth in this setting arises solely from algebraic noncommutativity between unitary action and conditional expectation, without invoking thermodynamic limits, statistical ensembles, or environmental assumptions.
We build a framework for the thermodynamics of macroscopic quantum systems. In contrast with approaches requiring access to the full density matrix, our framework relies on a coarse-grained description, based on measurement statistics of a few observables. When these observables commute, the outcomes define classical macrostates whose entropy is quantified by observational entropy, accounting for uncertainty about both the macrostate and the microstate within it. We extend this notion to non-commuting observables forming a subalgebra of the operator space, and use Jaynes' principle to define an algebra-dependent entropy interpolating between von Neumann and observational entropies. Given initial and final measurement sets, connected by internal and/or environment-induced dynamics, we derive a second law for the coarse-grained dynamics. Unlike formulations based on von Neumann entropy, our inequality captures irreversibility from both non-unitary environment-induced dynamics and internal equilibration. It takes the usual form of a positive entropy production when the system is initially at internal equilibrium, while correction terms capture nonequilibrium resources ignored by the coarse-graining. We also derive fluctuation theorems for coarse-grained thermodynamic quantities. Along a quasi-static path of measurement schemes, we identify quantum macroscopic notions of work and heat fulfilling the first and second laws, including an additional work contribution from manipulating the algebra to which the system is confined, through external constraints or quantum measurement backaction. Finally, we apply our framework to examples illustrating the impact of varying the coarse-graining scheme. Our approach unifies macroscopic and stochastic thermodynamics in a genuinely quantum framework, laying the basis for a versatile, experimentally friendly toolbox to analyze complex quantum dynamics.
Abstract In this paper, we study the multiplicative behaviour of quantum channels, mathematically described by trace preserving, completely positive maps on matrix algebras. It turns out that the multiplicative domain of a unital quantum channel has a close connection to its spectral properties. A structure theorem (theorem 2.5), which reveals the automorphic property of an arbitrary unital quantum channel on a subalgebra, is presented. Various classes of quantum channels (irreducible, primitive, etc) are then analysed in terms of this stabilising subalgebra. The notion of the multiplicative index of a unital quantum channel is introduced, which measures the number of times a unital channel needs to be composed with itself for the multiplicative algebra to stabilise. We show that the maps that have trivial multiplicative domains are dense in completely bounded norm topology in the set of all unital completely positive maps. Some applications in quantum information theory are discussed.
If a set of von Neumann subalgebras has a trivial intersection in finite dimension, then the sum of relative entropies of a given density to its projection in each such algebra is larger than a multiple of its relative entropy to its projection in the trivial intersection. This results in a subadditivity of relative entropy with a dimension and algebra-dependent, multiplicative constant. As a primary application, this inequality lets us derive relative entropy decay estimates in the form of modified logarithmic-Sobolev inequalities for complicated quantum Markov semigroups from those of simpler constituents.
We develop a submodular framework for the von Neumann entropy of discrete quantum convolutions, providing a noncommutative counterpart to the direct side of entropic additive combinatorics. We first introduce globally weighted quantum convolutions, which form compatible families indexed by admissible subsets of a fixed collection of inputs. Our main theorem reveals a polymatroidal geometry underlying their entropy growth: relative to any fixed admissible input block, the entropy gains admit a normalized, monotone, submodular extension to all subsets of the remaining inputs. The theorem yields convolutional strong subadditivity, quantum Ruzsa triangle inequality, and quantum entropic Plünnecke--Ruzsa inequalities for arbitrary input states. For repeated inputs, it gives sharp comparisons of entropy growth across admissible scales; in particular, the quantum doubling constant controls all higher admissible convolution entropies with optimal exponents. Together, these results bring submodular methods from additive combinatorics into the quantum setting and provide a systematic route to broad families of convolutional entropy inequalities.
We establish a sharp quantitative stability theorem for the von Neumann entropy under a natural class of convex constraints induced by orthogonal block decompositions of finite-dimensional Hilbert spaces. Given a block–convex set of block-diagonal quantum states, we prove that any state whose entropy is within ε of the minimal value must lie within O(ε)O(\sqrt{\varepsilon})O(ε) in trace norm of the manifold of entropy minimizers. The stability exponent is shown to be optimal, and purely quadratic stability is characterized precisely in the confining (fixed-support) regime. The proof relies on a canonical decomposition of entropy into classical (Shannon) and conditional components, combined with sharp relative entropy estimates and geometric properties of the marginal polytope. As an application, we obtain quantitative non-concentration bounds for spectral measures associated with finite spectral decompositions of self-adjoint operators. The framework is entirely finite-dimensional, independent of arithmetic or dynamical assumptions, and provides a general geometric stability principle for entropy minimization under structured convex constraints.
We prove recovery theorems for α-z Rényi divergences under normal unital positive maps between von Neumann algebras. Previous results in this setting required 2-positivity, while recent finite dimensional work showed that this assumption can be relaxed to mere positivity. Our proof uses sufficiency for JW*-subalgebras and Lp-spaces over them to characterize equality in the data processing inequality. As a further application of our methods, we solve a problem discussed by Haagerup and Stormer: Every conditional expectation of a von Neumann algebra onto a JW*-subalgebra factors through a conditional expectation onto the generated von Neumann subalgebra.