Rescaling transformations and the Grothendieck bound formalism in a single quantum system

• URL: http://arxiv.org/abs/2409.07270v1
• Date: Wed, 11 Sep 2024 13:45:36 GMT
• Title: Rescaling transformations and the Grothendieck bound formalism in a single quantum system
• Authors: A. Vourdas,
• Abstract summary: The Grothendieck bound formalism is studied using rescaling transformations' The Grothendieck formalism considers a classical' quadratic form $cal C(theta)$ which takes values less than $1$, and the corresponding quantum' quadratic form $cal Q(theta)$ which takes values greater than $1$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: The Grothedieck bound formalism is studied using `rescaling transformations', in the context of a single quantum system. The rescaling transformations enlarge the set of unitary transformations (which apply to isolated systems), with transformations that change not only the phase but also the absolute value of the wavefunction, and can be linked to irreversible phenomena (e.g., quantum tunnelling, damping and amplification, etc). A special case of rescaling transformations are the dequantisation transformations, which map a Hilbert space formalism into a formalism of scalars. The Grothendieck formalism considers a `classical' quadratic form ${\cal C}(\theta)$ which takes values less than $1$, and the corresponding `quantum' quadratic form ${\cal Q}(\theta)$ which takes values greater than $1$, up to the complex Grothendieck constant $k_G$. It is shown that ${\cal Q}(\theta)$ can be expressed as the trace of the product of $\theta$ with two rescaling matrices, and ${\cal C}(\theta)$ can be expressed as the trace of the product of $\theta$ with two dequantisation matrices. Values of ${\cal Q}(\theta)$ in the `ultra-quantum' region $(1,k_G)$ are very important, because this region is classically forbidden (${\cal C}(\theta)$ cannot take values in it). An example with ${\cal Q}(\theta)\in (1,k_G)$ is given, which is related to phenomena where classically isolated by high potentials regions of space, communicate through quantum tunnelling. Other examples show that `ultra-quantumness' according to the Grothendieck formalism (${\cal Q}(\theta)\in (1,k_G)$), is different from quantumness according to other criteria (like quantum interference or the uncertainty principle).

Related papers

• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• Universal contributions to charge fluctuations in spin chains at finite temperature [5.174839433707792]
  We show that $gamma(theta)$ only takes non-zero values at isolated points of $theta$, which is $theta=pi$ for all our examples. In two exemplary lattice systems we show that $gamma(pi)$ takes quantized values when the U(1) symmetry exhibits a specific type of 't Hooft anomaly with other symmetries.
  arXiv  Detail & Related papers  (2024-01-17T19:05:07Z)
• Ultra-quantum coherent states in a single finite quantum system [0.0]
  A set of $n$ coherent states is introduced in a quantum system with $d$-dimensional Hilbert space $H(d)$. They resolve the identity, and also have a discrete isotropy property. A finite cyclic group acts on the set of these coherent states, and partitions it into orbits.
  arXiv  Detail & Related papers  (2023-11-17T10:05:00Z)
• Quantum Current and Holographic Categorical Symmetry [62.07387569558919]
  A quantum current is defined as symmetric operators that can transport symmetry charges over an arbitrary long distance. The condition for quantum currents to be superconducting is also specified, which corresponds to condensation of anyons in one higher dimension.
  arXiv  Detail & Related papers  (2023-05-22T11:00:25Z)
• Grothendieck bound in a single quantum system [0.0]
  Grothendieck's bound is used in the context of a single quantum system. The Grothendieck theorem is reformulated here in terms of arbitrary matrices.
  arXiv  Detail & Related papers  (2022-12-22T13:06:31Z)
• Algebraic Aspects of Boundaries in the Kitaev Quantum Double Model [77.34726150561087]
  We provide a systematic treatment of boundaries based on subgroups $Ksubseteq G$ with the Kitaev quantum double $D(G)$ model in the bulk. The boundary sites are representations of a $*$-subalgebra $Xisubseteq D(G)$ and we explicate its structure as a strong $*$-quasi-Hopf algebra. As an application of our treatment, we study patches with boundaries based on $K=G$ horizontally and $K=e$ vertically and show how these could be used in a quantum computer
  arXiv  Detail & Related papers  (2022-08-12T15:05:07Z)
• A New Look at the $C^{0}$-formulation of the Strong Cosmic Censorship Conjecture [68.8204255655161]
  We argue that for generic black hole parameters as initial conditions for Einstein equations, the metric is $C0$-extendable to a larger Lorentzian manifold. We prove it violates the "complexity=volume" conjecture for a low-temperature hyperbolic AdS$_d+1$ black hole dual to a CFT living on a ($d-1$)-dimensional hyperboloid $H_d-1$.
  arXiv  Detail & Related papers  (2022-06-17T12:14:33Z)
• The Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) Equation for Two-Dimensional Systems [62.997667081978825]
  Open quantum systems can obey the Franke-Gorini-Kossakowski-Lindblad-Sudarshan (FGKLS) equation. We exhaustively study the case of a Hilbert space dimension of $2$.
  arXiv  Detail & Related papers  (2022-04-16T07:03:54Z)
• On quantum algorithms for the Schr\"odinger equation in the semi-classical regime [27.175719898694073]
  We consider Schr"odinger's equation in the semi-classical regime. Such a Schr"odinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics.
  arXiv  Detail & Related papers  (2021-12-25T20:01:54Z)
• Anharmonic oscillator: a solution [77.34726150561087]
  The dynamics in $x$-space and in $(gx)-space corresponds to the same energy spectrum with effective coupling constant $hbar g2$. A 2-classical generalization leads to a uniform approximation of the wavefunction in $x$-space with unprecedented accuracy.
  arXiv  Detail & Related papers  (2020-11-29T22:13:08Z)
• Quantum phase transitions in nonhermitian harmonic oscillator [0.0]
  Stone theorem requires that in a physical Hilbert space $cal H$ the time-evolution of a stable quantum system is unitary. We show that in the dynamical regime of unavoided level crossings a reconstruction of $cal H$ becomes feasible.
  arXiv  Detail & Related papers  (2020-08-10T10:32:28Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.