The common idea behind complexity reduction in physical systems is separation of information into ``physically meaningful'' and ``safely ignorable.'' Here we consider a generic notion of such separation---implemented by coarse-graining the state space---and address the question of what information is indeed safely ignorable if we want to reduce the complexity of dynamics. The general condition for reducibility of dynamics under coarse graining will be presented for stochastic and quantum systems. In the process we develop the quantum notion of state-space coarse graining that allows us to marginalize selected degrees of freedom. One of our main findings is that there is a broader class of symmetries, beyond those that are considered by Noether's theorem, that can play a role in the reduction of dynamics. Some examples of quantum coarse grainings and the reduction of dynamics with symmetries will be discussed.
The common idea behind complexity reduction in physical systems is separation of information into physically meaningful'' and safely ignorable.'' Here we consider a generic notion of such separation---implemented by coarse-graining the state space---and address the question of what information is indeed safely ignorable if we want to reduce the complexity of dynamics. The general condition for reducibility of dynamics under coarse graining will be presented for stochastic and quantum systems. In the process we develop the quantum notion of state-space coarse graining that allows us to marginalize selected degrees of freedom. One of our main findings is that there is a broader class of symmetries, beyond those that are considered by Noether's theorem, that can play a role in the reduction of dynamics. Some examples of quantum coarse grainings and the reduction of dynamics with symmetries will be discussed.
The idea that symmetries simplify or reduce the complexity of a system has been remarkably fruitful in physics, and especially in quantum mechanics. On a mathematical level, symmetry groups single out a certain structure in the Hilbert space that leads to a reduction. This structure is given by the irreducible representations of the group, and in general it can be identified with an operator algebra (a.k.a. C*-algebra or von Neumann algebra). The primary focus of this thesis is the extension of the framework of reductions from symmetries to operator algebras, and its applications in finite-dimensional quantum mechanics. Finding the irreducible representations structure is the principal problem when working with operator algebras. We will therefore review the representation theory of finite-dimensional operator algebras and elucidate this problem with the help of two novel concepts: minimal isometries and bipartition tables. One of the main technical results that we present is the Scattering Algorithm for analytical derivations of the irreducible representations structure of operator algebras. For applications, we will introduce a symmetry-agnostic approach to the reduction of dynamics where we circumvent the non-trivial task of identifying symmetries, and directly reduce the dynamics generated by a Hamiltonian. We will also consider quantum state reductions that arise from operational constraints, such as the partial trace or the twirl map, and study how operational constraints lead to decoherence. Apart from that, we will extend the idea of reduction beyond operator algebras to operator systems, and formulate a quantum notion of coarse-graining that so far only existed in classical probability theory. We will also characterize how the uncertainty principle transitions to the classical regime under coarse-grained measurements and discuss the implications in a finite-dimensional setting.
Stochastic modeling of complex systems plays an essential, yet often computationally intensive, role across the quantitative sciences.Recent advances in quantum information processing have elucidated the potential for quantum simulators to exhibit memory advantages for such tasks.Heretofore, the focus has been on lossless memory compression, wherein the advantage is typically in terms of lessening the amount of information tracked by the model, while-arguably more practical-reductions in memory dimension are not always possible.Here, we address the case of lossy compression for quantum stochastic modeling of continuous-time processes, introducing a method for coarse graining in quantum state space that drastically reduces the requisite memory dimension for modeling temporal dynamics while retaining near-exact statistics.In contrast to classical coarse graining, this compression is not based on sacrificing temporal resolution and brings memory-efficient high-fidelity stochastic modeling within reach of present quantum technologies.
In this paper we introduce a canonical quantum coarse-graining and use negentropy to connect ignorance as measured by quantum information entropy and ignorance related to quantum coarse-graining. For our procedure, macro-states are the set of purifications {∣Γˉρ⟩} associated with density operator ρ and micro-states are elements of {∣Γˉρ⟩}. Unlike other quantum coarse-graining procedures, ours always gives a well-defined unique coarse-graining of phase space. Our coarse-graining is also unique in that the volumes of phase space associated with macro-states are computed from differential manifolds whose metric components are constructed from the Lie group symmetries that generate {∣Γˉρ⟩}. We call these manifolds surfaces of ignorance, and their volumes quantify the lack of information in ρ as measured by quantum information entropies. To show that these volumes behave like information entropies, we compare them to the von Neumann and linear entropies for states whose symmetries are given by SO(3), SU(2), and SO(N). We also show that our procedure reproduces features of Boltzmann's original coarse-graining by showing that the majority of phase space consists of states near or at equilibrium. As a consequence of this coarse-graining, it is shown that an inherent flag variety structure underlies composite Hilbert spaces.
The universality of classical thermodynamics rests on the central limit theorem, due to which, measurements of thermal fluctuations are unable to reveal detailed information regarding the microscopic structure of a macroscopic body. When small systems are considered and fluctuations become important, thermodynamic quantities can be understood in the context of classical stochastic mechanics. A fundamental assumption behind thermodynamics is therefore that of coarse graining, which stems from a substantial lack of control over all degrees of freedom. However, when quantum systems are concerned, one claims a high level of control. As a consequence, information theory plays a major role in the identification of thermodynamic functions. Here, drawing from the concept of gauge symmetry-essential in all modern physical theories-we put forward a new possible intermediate route. Working within the realm of quantum thermodynamics, we explicitly construct physically motivated gauge transformations which encode a gentle variant of coarse graining behind thermodynamics. As a first application of this new framework, we reinterpret quantum work and heat, as well as the role of quantum coherence.
Reduced density matrices are a powerful tool in the analysis of entanglement structure, approximate or coarse-grained dynamics, decoherence, and the emergence of classicality. It is straightforward to produce a reduced density matrix with the partial-trace map by tracing out part of the quantum state, but in many natural situations this reduction may not be achievable. We investigate the general problem of identifying how the quantum state is reduced given a restriction on the observables. For example, in an experimental setting, the set of observables that can actually be measured is usually modest (compared to the set of all possible observables) and their resolution is limited. In such situations, the appropriate state-reduction map can be defined via a generalized bipartition, which is associated with the structure of irreducible representations of the algebra generated by the restricted set of observables. One of our main technical results is a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. We demonstrate the viability of this approach with two examples of limited-resolution observables. The definition of quantum state reductions can also be extended beyond algebras of observables. To accomplish this task we introduce a more flexible notion of bipartition, the partial bipartition, which describes coarse grainings preserving information about a limited set (not necessarily algebra) of observables. We describe a variational method to choose the coarse grainings most compatible with a specified Hamiltonian, which exhibit emergent classicality in the reduced state space. We apply this construction to the concrete example of the one-dimensional Ising model. Our results have relevance for quantum information, bulk reconstruction in holography, and quantum gravity.
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.