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arXiv:1811.02792 · Physics Letters B

Coarse graining as a representation change

Norbert Bodendorfer, Fabian Haneder

Published Mar 14, 2019• 30 citations
arXivHTMLPDF

Abstract

We discuss how SU(1,1) coherent states from the discrete series allow for a natural coarse graining operation. The physical application is quantum theories based on a set of three extensive observables whose Poisson algebra is isomorphic to su(1,1). In particular, we show that a Perelomov coherent state with representation label Nj0 and spinor label z encodes the physics of N independent subsystems with labels j0,z. This property is suggested by existing results for the expectation values and variances of the observables. We prove that it holds for all higher moments. Our results in particular apply to a recent quantum cosmology model that has been derived using SU(1,1) group theoretic quantisation techniques. For it, it follows that a certain notion of fiducial cell independence holds exactly at the quantum level when using the coherent states.

DOI
10.1016/j.physletb.2019.03.020

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We discuss how SU(1,1) coherent states from the discrete series allow for a natural coarse graining operation. The physical application is quantum theories based on a set of three extensive observables whose Poisson algebra is isomorphic to su(1,1). In particular, we show that a Perelomov coherent state with representation label Nj0 and spinor label z encodes the physics of N independent subsystems with labels j0,z. This property is suggested by existing results for the expectation values and variances of the observables. We prove that it holds for all higher moments. Our results in particular apply to a recent quantum cosmology model that has been derived using SU(1,1) group theoretic quantisation techniques. For it, it follows that a certain notion of fiducial cell independence holds exactly at the quantum level when using the coherent states.
We extend recent results on expectation values of coherent oscillator states and SU(2) coherent states to the case of the discrete representations of su(1,1). Systematic semiclassical expansions of products of arbitrary operators are derived. In particular, the leading order of the energy uncertainty of an arbitrary Hamiltonian is found to be given purely in terms of the time dependence of the classical variables. The coherent states considered here include the Perelomov-Gilmore (PG) coherent states. As an important application we discuss the pseudoharmonic oscillator and compare the PG states with the states introduced by Barut and Girardello. The latter ones turn out to be closer to the classical limit as their relative energy variance decays with the inverse square root of energy, while in the former case a constant is approached.
Abstract We extend recent results on expectation values of coherent oscillator states and SU(2) coherent states to the case of the discrete representations of su(1,1). Systematic semiclassical expansions of products of arbitrary operators are derived. In particular, the leading order of the energy uncertainty of an arbitrary Hamiltonian is found to be given purely in terms of the time dependence of the classical variables. The coherent states considered here include the Perelomov–Gilmore (PG) coherent states. As an important application we discuss the pseudoharmonic oscillator and compare the PG states with the states introduced by Barut and Girardello. The latter ones turn out to be closer to the classical limit as their relative energy variance decays with the inverse square root of energy, while in the former case a constant is approached.
One can think of some physical evolutions as being the emergent-effective result of a microscopic discrete model. Inspired by classical coarse graining procedures, we provide a simple procedures to coarse-grain color-blind quantum cellular automata, following Goldilock rules. The procedure consists in (i) space-time grouping the quantum cellular automaton (QCA) in cells of size NNN, (ii) projecting the cells states onto its borders, connecting them with the fine dynamics, which is reminiscent of the edge-bulk correspondence in quantum field theory; (iii) describe the overall dynamics by the border states, that we call signals; (iv) construct the coarse-grained dynamics for different sizes NNN of the cells. A byproduct of this simple toy-model is a general discrete analogous of the Stokes law. Moreover, we proved that in the spacetime limit, the automaton converge to a Dirac free Hamiltonian. The QCA we introduced here, can be implemented by present-day quantum platforms, such as Rydberg arrays, trapped ions, and superconducting qbits. We hope our study can pave the way to a richer understanding of those systems with limited resolution.
Abstract We quantify the nonclassicality of quantum states associated with the Lie group SU(1,1) by regarding states as observables and considering their variances in the SU(1,1) Perelomov coherent states. Combining the resolution of identity induced by the SU(1,1) Perelomov coherent states, we propose a quantifier for nonclassicality of a state based on the average uncertainty (variance) of the state (regarded as an observable) in the SU(1,1) Perelomov coherent states. This quantifier is easy to calculate and possesses several operational interpretations. We reveal its basic properties and illustrate it by several prototypical examples.
One can think of some physical evolutions as being the emergent-effective result of a microscopic discrete model. Inspired by classical coarse-graining procedures, we provide a simple procedure to coarse-grain color-blind quantum cellular automata that follow the so-called Goldilocks rules. The procedure consists in (i) space-time grouping the quantum cellular automaton (QCA) in cells of size NNN; (ii) projecting the states of a cell onto its borders, connecting them with the fine dynamics; (iii) describing the overall dynamics by the border states, that we call signals; and (iv) constructing the coarse-grained dynamics for different sizes NNN of the cells. A by-product of this simple toy model is a general discrete analogous of the Stokes law. Moreover we prove that in the space-time limit, the automaton converges to a Dirac free Hamiltonian. The QCA we introduce here can be implemented by present-day quantum platforms, such as Rydberg arrays, trapped ions, and superconducting qubits. We hope our study can pave the way to a richer understanding of those systems with limited resolution.
This work proposes a conceptual framework for connecting quantum closure structures, local observation, fine-grained binary records, and conventional binary coarse-graining. At the binary level, the framework introduces a structural representation distinct from ordinary ordered binary notation. The states 00, 11, and \{01,10\} are first grouped into three basic relational classes. Adding a positive/reverse distinction produces six states. Assigning one of four source labels, A, B, C, or D, to each of the two binary positions further generates 3\times2\times4\times4=96 labeled structural states. These 96 states are interpreted as a fine-grained record space rather than as 96 new conventional binary values. A separate quantum-level hypothesis is then introduced. In this candidate model, externally observed quantum values 0 and 1 are associated with complementary internal closure structures represented as 01 and 10. The labels A,B,C,D refer to four correlated positions or orientations within a single closed structure rather than to independent binary labels. Local observation of such a structure may yield source-labeled outcomes such as 0^A or 1^B. Repeated local observations generate source-labeled binary sequences, while conventional binary representation is obtained after discarding source, orientation, and positive/reverse information. The framework distinguishes microscopic structural complementarity from macroscopic statistical balance. A closed microscopic structure may be exactly complementary, while short binary observation sequences may still contain arbitrary local concentrations such as 0000 or 1111. Under symmetric and unbiased sampling conditions, long-run binary statistics may approach balance without requiring local balance at every observation step. The work also introduces a dual-projection viewpoint. The same underlying binary object can be represented through different observational rules. Conventional ordered observation yields the familiar four states 00,01,10,11, whereas relational observation yields the three classes 00, 11, and \{01,10\}, which may then be refined into the 3\rightarrow6\rightarrow96 hierarchy. These descriptions are treated as different coarse-grainings or projections of the same underlying object rather than as mutually exclusive binary systems. The 3\rightarrow6\rightarrow96 construction is presented here as a formally defined representational structure. By contrast, the proposed quantum 01/10 internal closure, the physical meaning of the A,B,C,D orientations, and any claimed quantum-to-binary correspondence remain hypotheses requiring mathematical formalization and experimental testing. Future work will focus on formal mathematical modeling, exhaustive state analysis, numerical simulation, comparison with standard quantum-mechanical predictions, falsifiable experimental protocols, and possible applications in quantum readout, feedback control, error diagnosis, and information processing. Version v0.1 is intended to establish a time-stamped record of the original hypothesis, definitions, conceptual boundaries, and research roadmap. It does not claim experimental confirmation of the proposed internal quantum structure.