Quantum Tomography and Schwinger's Picture of Quantum Mechanics
- URL: http://arxiv.org/abs/2205.00170v1
- Date: Sat, 30 Apr 2022 06:10:14 GMT
- Title: Quantum Tomography and Schwinger's Picture of Quantum Mechanics
- Authors: Florio M. Ciaglia, Fabio Di Cosmo, Alberto Ibort and Giuseppe Marmo
- Abstract summary: The problem of tomographic reconstruction of states is investigated within the so-called Schwinger's picture of Quantum Mechanics.
The main goal of the paper consists in providing a reconstruction formula for states on the groupoid-algebra associated with the observables of the system.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: In this paper the problem of tomographic reconstruction of states is
investigated within the so-called Schwinger's picture of Quantum Mechanics in
which a groupoid is associated with every quantum system. The attention is
focused on spin tomography: In this context the groupoid of interest is the
groupoid of pairs over a finite set. In a nutshell, this groupoid is made up of
transitions between all possible pairs of outcomes belonging to a finite set.
In addition, these transitions possess a partial composition rule, generalizing
the notion of groups. The main goal of the paper consists in providing a
reconstruction formula for states on the groupoid-algebra associated with the
observables of the system. Using the group of bisections of this groupoid,
which are special subsets in one-to-one correspondence with the outcomes, a
frame is defined and it is used to prove the validity of the tomographic
reconstruction. The special case of the set of outcomes being the set of
integers modulo n, with n odd prime, is considered in detail. In this case the
subgroup of discrete affine linear transformations, whose graphs are linear
subspaces of the groupoid, provides a \textit{quorum} in close analogy with the
continuos case.
Related papers
- Equivalence of dynamics of disordered quantum ensembles and semi-infinite lattices [44.99833362998488]
We develop a formalism for mapping the exact dynamics of an ensemble of disordered quantum systems onto the dynamics of a single particle propagating along a semi-infinite lattice.
This mapping provides a geometric interpretation on the loss of coherence when averaging over the ensemble and allows computation of the exact dynamics of the entire disordered ensemble in a single simulation.
arXiv Detail & Related papers (2024-06-25T18:13:38Z) - Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space [45.9982965995401]
We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators.
We find conditions for their strong continuity and establish properties of their generators.
arXiv Detail & Related papers (2024-06-15T17:39:32Z) - A Lagrangian path integral approach to the qubit [0.0]
In this formalism a Feynman-like computation of its probability amplitudes is done.
The Lagrangian is interpreted as a function on the groupoid describing the quantum system.
arXiv Detail & Related papers (2024-01-24T19:25:06Z) - Infinite Permutation Groups and the Origin of Quantum Mechanics [0.0]
When the lattice is atomistic, it is isomorphic to the lattice of definably closed sets of a finitary relational structure in First Order Logic.
We show that the automorphism group must belong to a family of permutation groups known as geometric Jordan groups.
We then use the classification theorem for Jordan groups to argue that the combined requirements of probability and atomicism leave uncountably infinite Steiner 2-systems.
arXiv Detail & Related papers (2023-07-24T18:00:16Z) - Spectrum of localized states in fermionic chains with defect and
adiabatic charge pumping [68.8204255655161]
We study the localized states of a generic quadratic fermionic chain with finite-range couplings.
We analyze the robustness of the connection between bands against perturbations of the Hamiltonian.
arXiv Detail & Related papers (2021-07-20T18:44:06Z) - Schwinger's picture of quantum mechanics: 2-groupoids and symmetries [0.0]
It is shown that, given a groupoid $Grightarrows Omega$ associated with a (quantum) system, there are two possible descriptions of its symmetries.
On the other hand, taking advantage of the notion of group of bisections of a given groupoid, the global perspective allows to construct a group of symmetries out of a 2-groupoid.
The latter notion allows to introduce an analog of the Wigner's theorem for quantum symmetries in the groupoid approach to Quantum Mechanics.
arXiv Detail & Related papers (2021-04-28T16:53:56Z) - Quantum particle across Grushin singularity [77.34726150561087]
We study the phenomenon of transmission across the singularity that separates the two half-cylinders.
All the local realisations of the free (Laplace-Beltrami) quantum Hamiltonian are examined as non-equivalent protocols of transmission/reflection.
This allows to comprehend the distinguished status of the so-called bridging' transmission protocol previously identified in the literature.
arXiv Detail & Related papers (2020-11-27T12:53:23Z) - Quantum mechanics of bipartite ribbon graphs: Integrality, Lattices and
Kronecker coefficients [0.0]
We define solvable quantum mechanical systems on a Hilbert space spanned by bipartite ribbon graphs with a fixed number of edges.
The square of the Kronecker coefficient for a triple of Young diagrams is shown to be equal to the dimension of a sub-lattice in the lattice of ribbon graphs.
As an avenue to explore quantum supremacy and its implications for computational complexity theory, we outline experiments to detect non-vanishing Kronecker coefficients for hypothetical quantum realizations/simulations of these quantum systems.
arXiv Detail & Related papers (2020-10-08T15:18:46Z) - Radiative topological biphoton states in modulated qubit arrays [105.54048699217668]
We study topological properties of bound pairs of photons in spatially-modulated qubit arrays coupled to a waveguide.
For open boundary condition, we find exotic topological bound-pair edge states with radiative losses.
By joining two structures with different spatial modulations, we find long-lived interface states which may have applications in storage and quantum information processing.
arXiv Detail & Related papers (2020-02-24T04:44:12Z) - Schwinger's picture of Quantum Mechanics [0.0]
We will present what can be called Schwinger's foundational approach to Quantum Mechanics.
The basic ingredients of this formulation are the textitselective measurements, whose algebra rules define a mathematical structure called groupoid.
arXiv Detail & Related papers (2020-02-21T14:28:31Z) - Operator-algebraic renormalization and wavelets [62.997667081978825]
We construct the continuum free field as the scaling limit of Hamiltonian lattice systems using wavelet theory.
A renormalization group step is determined by the scaling equation identifying lattice observables with the continuum field smeared by compactly supported wavelets.
arXiv Detail & Related papers (2020-02-04T18:04:51Z)
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.