We demonstrate how the dynamical coarse-graining approach can be systematically extended to higher orders in the coupling between system and reservoir. Up to second order in the coupling constant we explicitly show that dynamical coarse-graining unconditionally preserves positivity of the density matrix -- even for bath density matrices that are not in equilibrium and also for time-dependent system Hamiltonians. By construction, the approach correctly captures the short-time dynamics, i.e., it is suitable to analyze non-Markovian effects. We compare the dynamics with the exact solution for highly non-Markovian systems and find a remarkable quality of the coarse-graining approach. The extension to higher orders is straightforward but rather tedious. The approach is especially useful for bath correlation functions of simple structure and for small system dimensions.
We analyze Lindblad-Gorini-Kossakowski-Sudarshan-type generators for selected periodically driven open quantum systems. All these generators can be obtained by temporal coarse-graining procedures, and we compare different coarse-graining schemes. Similar as for undriven systems, we find that a dynamically adapted coarse-graining time, effectively yielding non-Markovian dynamics by interpolating through a series of different but invididually Markovian solutions, gives the best results among the different coarse-graining schemes, albeit at highest computational cost.
Using the Naimark dilation theory we investigate the question under what conditions an observable which is a coarse graining of another observable is a function of it. To this end, conditions for the separability and for the Boolean structure of an observable are given.
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 $\{|\barΓ^ρ\rangle\}$ associated with density operator $ρ$ and micro-states are elements of $\{|\barΓ^ρ\rangle\}$. 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 comp…
A fuzzy observable is regarded as a smearing of a sharp observable, and the structure of covariant fuzzy observables is studied. It is shown that the covariant coarse-grainings of sharp observables are exactly the covariant fuzzy observables. A necessary and sufficient condition for a covariant fuzzy observable to be informationally equivalent to the corresponding sharp observable is given.
The connection between coarse-graining of measurement and emergence of classicality has been investigated for some time, if not well understood. Recently in (PRL $\textbf{112}$, 010402, (2014)) it was pointed out that coarse-graining measurements can lead to non-violation of Bell-type inequalities by a state which would violate it under sharp measurements. We study here the effects of coarse-grained measurements on bipartite cat states. We show that while it is true that coarse-graining does indeed lead to non-violation of a Bell-type inequality, this is not reflected at the state level. Under…
We formulate an effective-description framework for the dynamics of open quantum systems by extending the time-coarse-graining formalism to open systems. Our coarse-graining procedure efficiently removes high-frequency processes which are responsible for coherences between lower- and upper-manifold states and are irrelevant when considering only low-energy dynamics. We investigate the regime of validity of the resulting coarse-grained master equation by applying it to multi-level atoms driven by far-detuned lasers. Except for such high-frequency coherences, we find good agreement between the e…
Let H and K be Hilbert spaces and T be a coarse-graining from B(H) to B(K). Assume that density matrices D_1 and D_2 acting on H are given. In the paper the consequences of the existence of a coarse-graining S from B(K) to B(H) satisfying ST(D_1)=D_1 and ST(D_2)=D_2 are given. (This condition means the sufficiency of T for D_1 and D_2.) Sufficiency implies a particular decomposition of the density matrices. This decomposition allows to deduce the exact condition for equality in the strong subadditivity of the von Neumann entropy.
We develop a systematic coarse graining procedure for systems of $N$ qubits. We exploit the underlying geometrical structures of the associated discrete phase space to produce a coarse-grained version with reduced effective size. Our coarse-grained spaces inherit key properties of the original ones. In particular, our procedure naturally yields a subset of the original measurement operators, which can be used to construct a coarse discrete Wigner function. These operators also constitute a systematic choice of incomplete measurements for the tomographer wishing to probe an intractably large sy…
The coarse-graining approach to deriving the quantum Markovian master equation is revisited, with close attention given to the underlying approximations. It is further argued that the time interval over which the coarse-graining is performed is a free parameter that can be given a physical measurement-based interpretation. In the case of the damping of composite systems to reservoirs of different temperatures, currently of much interest in the study of quantum thermal machines with regard to the validity of `local' and `global' forms of these equations, the coupling of the subsystems leads to …
We first define the coarse-graining of probability measures in terms of stochastic kernels. We define when a probability measure is part of another probability measure and say that two probability measures coexist if they are both parts of a single probability measure. We then show that any two probability measures coexist. We extend these concepts to observables and instruments and mention that two observables need not coexist. We define the discretization of an observable as a special case of coarse-graining and show that these have \zeroone stochastic kernels. We next consider finite observ…
We present a first-principles derivation of the Markovian semi-group master equation without invoking the rotating wave approximation (RWA). Instead we use a time coarse-graining approach which leaves us with a free timescale parameter, which we can optimize. Comparing this approach to the standard RWA-based Markovian master equation, we find that significantly better agreement is possible using the coarse-graining approach, for a three-level model coupled to a bath of oscillators, whose exact dynamics we can solve for at zero temperature. The model has the important feature that the RWA has a…
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 to marginali…
Three ideas are introduced that when brought together characterize the realistic quasiclassical realms of our quantum universe as particular kinds of sets of alternative coarse-grained histories defined by quasiclassical variables: (1) Branch dependent adaptive coarse grainings that can be close to maximally refined and can simplify calculation. (2) Narrative coarse grainings that describe how features of the universe change over time and allow the construction of an environment. (3) A notion of strong decoherence that characterizes realistic mechanisms of decoherence.
Observing quantum effects such as superpositions and entanglement in macroscopic systems requires not only a system that is well protected against environmental decoherence, but also sufficient measurement precision. Motivated by recent experiments, we study the effects of coarse-graining in photon number measurements on the observability of micro-macro entanglement that is created by greatly amplifying one photon from an entangled pair. We compare the results obtained for a unitary quantum cloner, which generates micro-macro entanglement, and for a measure-and-prepare cloner, which produces a…
The dynamics of systems subjected to noise is called Markovian in the absence of memory effects, i.e. when its immediate future only depends on its present. Time correlations in the noise source may generate non-Markovian effects that, sometimes, can be erased by appropriately coarse graining the time evolution of the system. In general, the coarse graining time $t_\text{CG}$ is taken to be much larger than the correlation time $τ$ but no direct relation between them is established. Here we analytically obtain a relation between $t_\text{CG}$ and $τ$ for the dynamics of a qubit subjected to a …
Quantum state tomography often operates in the highly idealised scenario of assuming perfect measurements. The errors implied by such an approach are entwined with other imperfections relating to the information processing protocol or application of interest. We consider the problem of retrodicting the quantum state of a system, existing prior to the application of random but known phase errors, allowing those errors to be separated and removed. The continuously random nature of the errors implies that there is only one click per measurement outcome -- a feature having a drastically adverse ef…
We take up the question why the initial entropy in the universe was small, in the context of evolution of the entropy of a classical system. We note that coarse-graining is a an important aspect of entropy evaluation which can reverse the direction of the increase in entropy, i.e., the direction of thermodynamic arrow of time. Then we investigate the role of decoherence in the selection of coarse-graining and explain how to compute entropy for a decohered classical system. Finally, we argue that the requirement of low initial entropy imposes constraints on the decoherence process.
We argue that refining, coarse graining and entangling operators can be obtained from time evolution operators. This applies in particular to geometric theories, such as spin foams. We point out that this provides a construction principle for the physical vacuum in quantum gravity theories and more generally allows to construct a (cylindrically) consistent continuum limit of the theory.
Formulating the QFT's as coarse grained 'low' energy sectors of a postulated complete quantum theory of everything with the 'high' energy modes integrated out or 'clustering' into 'low' energy objects, we can evaluate the Feynman amplitudes by solving a series of natural differential equations which automatically dissolves the necessity of infinity subtraction and the associated subtleties. This new strategy has direct implications to the scheme dependence problem.
A model of a quantum measurement process is presented: a system consisting of a qubit in a superposition interacts with a measuring apparatus consisting of a N qubit state. Looking at the emerging, effective description of the apparatus given by the action of a coarse-graining channel, we have been able to recover information about the superposition coefficients of the system.We have also been able to visualize the death of quantum correlations between system and apparatus and the death of quantum coherences in the apparatus' effective state, in the limit of a strong coarse-graining action - a…
This is a derivation of the quantum-optical master equation using coarse-graining of time, which brings new insights into a decades old technique. My derivation is quite similar to derivations using quantum stochastic methods or Kraus operators, though I go through the derivation without explicitly invoking any of these concepts, so it may be easier to follow as an introduction. I also address the major pitfall of nearly all microscopic derivations of the master equation, namely that they assume the state of the system and bath factorize for all times. I show why this assumption actually holds…
We study the coarse-graining approach to derive a generator for the evolution of an open quantum system over a finite time interval. The approach does not require a secular approximation but nevertheless generally leads to a Lindblad-Gorini-Kossakowski-Sudarshan generator. By combining the formalism with Full Counting Statistics, we can demonstrate a consistent thermodynamic framework, once the switching work required for the coupling and decoupling with the reservoir is included. Particularly, we can write the second law in standard form, with the only difference that heat currents must be de…
We present the natural arguments for the rationality of a recently proposed simple approach for renormalization which is based solving differential equations. The renormalization group equation is also derived in a natural way and recognized as a decoupling theorem of the UV modes that underlie a QFT. This new strategy has direct implications to the scheme dependence problem.
The nature of coarse graining is intuitively "obvious", but it is rather difficult to find explicit and calculable models of the coarse graining process (and the resulting entropy flow) discussed in the literature. What we would like to have at hand is some explicit and calculable process that takes an arbitrary system, with specified initial entropy S, and monotonically and controllably drives the entropy to its maximum value. This does not have to be a physical process, in fact for some purposes it is better to deal with a gedanken-process, since then it is more obvious how the "hidden infor…
Stochastic modelling 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…
Starting from the basic path integral in phase space we reconsider the functional approach to the RG flow of the one particle irreducible effective average action. On employing a balanced coarse-graining procedure for the canonical variables we obtain a functional integral with a non trivial measure which leads to a modified flow equation. We first address quantum mechanics for boson and fermion degrees of freedom and we then extend the construction to quantum field theories. For this modified flow equation we discuss the reconstruction of the bare action and the implications on the computatio…
Spin foams are models of quantum gravity and therefore quantum space time. A key open issue is to determine the possible continuum phases of these models. Progress on this issue has been prohibited by the complexity of the full four--dimensional models. We consider here simplified analogue models, so called spin nets, that retain the main dynamical ingredient of spin foams, the simplicity constraints. For a certain class of these spin net models we determine the phase diagram and therefore the continuum phases via a coarse graining procedure based on tensor network renormalization. This proced…
We propose a novel coarse graining tensor renormalization group method based on the higher-order singular value decomposition. This method provides an accurate but low computational cost technique for studying both classical and quantum lattice models in two- or three-dimensions. We have demonstrated this method using the Ising model on the square and cubic lattices. By keeping up to 16 bond basis states, we obtain by far the most accurate numerical renormalization group results for the 3D Ising model. We have also applied the method to study the ground state as well as finite temperature prop…
Our everyday descriptions of the universe are highly coarse-grained, following only a tiny fraction of the variables necessary for a perfectly fine-grained description. Coarse graining in classical physics is made natural by our limited powers of observation and computation. But in the modern quantum mechanics of closed systems, some measure of coarse graining is inescapable because there are no non-trivial, probabilistic, fine-grained descriptions. This essay explores the consequences of that fact. Quantum theory allows for various coarse-grained descriptions some of which are mutually incomp…
We argue that the entanglement entropy offers us a useful coarse-grained entropy in time-dependent AdS/CFT. We show that the total von-Neumann entropy remains vanishing even when a black hole is created in a gravity dual, being consistent with the fact that its corresponding CFT is described by a time-dependent pure state. We analytically calculate the time evolution of entanglement entropy for a free Dirac fermion on a circle following a quantum quench. This is interpreted as a toy holographic dual of black hole creations and annihilations. It is manifestly free from the black hole informatio…
