Framed Hilbert space: hanging the quasi-probability pictures of quantum theory

@article{Ferrie2009FramedHS,
  title={Framed Hilbert space: hanging the quasi-probability pictures of quantum theory},
  author={Christopher Ferrie and Joseph Emerson},
  journal={New Journal of Physics},
  year={2009},
  volume={11},
  pages={063040}
}
Building on earlier work, we further develop a formalism based on the mathematical theory of frames that defines a set of possible phase-space or quasi-probability representations of finite-dimensional quantum systems. We prove that an alternate approach to defining a set of quasi-probability representations, based on a more natural generalization of a classical representation, is equivalent to our earlier approach based on frames, and therefore is also subject to our no-go theorem for a non… 
View on IOP Publishing
doi.org
73 Citations
Highly Influential Citations
4
Background Citations
30
Methods Citations
6

Tables from this paper

73 Citations

From the Attempt of Certain Classical Reformulations of Quantum Mechanics to Quasi-Probability Representations
The concept of an injective affine embedding of the quantum states into a set of classical states, i.e., into the set of the probability measures on some measurable space, as well as its relation to…
Quantum Phase Space Representations and Their Negativities
A classical simulation scheme of quantum computation given a restricted set of states and measurements may be—occasionally, but only occasionally—interpreted naturally as a statistical simulation of…
Necessity of negativity in quantum theory
A unification of the set of quasiprobability representations using the mathematical theory of frames was recently developed for quantum systems with finite-dimensional Hilbert spaces, in which it was…
A Classical Formulation of Quantum Theory?
TLDR
The main challenge in this paper is to find a simple way of characterizing the allowed sets of classical pictures, and one promising approach is presented and shown how it works out for the case of a single qubit.
Quasi-probability representations of quantum theory with applications to quantum information science
This paper comprises a review of both the quasi-probability representations of infinite-dimensional quantum theory (including the Wigner function) and the more recently defined quasi-probability…
Quasiprobability representation of quantum coherence
We introduce a general method for the construction of quasiprobability representations for arbitrary notions of quantum coherence. Our technique yields a nonnegative probability distribution for the…
Non-classicality as a computational resource
TLDR
This thesis complete Spekkens' toy theory with measurement update rules and a mathematical framework that generalises it to systems of any finite dimensions (prime and non-prime) and extends the operational equivalence between the toy theory and stabilizer quantum mechanics to all odd dimensions via Gross' Wigner functions.
Minimal informationally complete measurements for probability representation of quantum dynamics
In the present work, we suggest an approach for describing dynamics of finite-dimensional quantum systems in terms of pseudostochastic maps acting on probability distributions, which are obtained via…
Geometry of Quantum States from Symmetric Informationally Complete Probabilities
It is usually taken for granted that the natural mathematical framework for quantum mechanics is the theory of Hilbert spaces, where pure states of a quantum system correspond to complex vectors of…
A Quantum-Bayesian Route to Quantum-State Space
In the quantum-Bayesian approach to quantum foundations, a quantum state is viewed as an expression of an agent’s personalist Bayesian degrees of belief, or probabilities, concerning the results of…
...
...

References

SHOWING 1-10 OF 59 REFERENCES
FAST TRACK COMMUNICATION: Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations
Several finite-dimensional quasi-probability representations of quantum states have been proposed to study various problems in quantum information theory and quantum foundations. These…
Wigner-Weyl correspondence in quantum mechanics for continuous and discrete systems-a Dirac-inspired view
Drawing inspiration from Dirac's work on functions of non-commuting observables, we develop an approach to phase-space descriptions of operators and the Wigner-Weyl correspondence in quantum…
Gleason-Type Derivations of the Quantum Probability Rule for Generalized Measurements
We prove a Gleason-type theorem for the quantum probability rule using frame functions defined on positive-operator-valued measures (POVMs), as opposed to the restricted class of orthogonal…
Contextuality for preparations, transformations, and unsharp measurements
The Bell-Kochen-Specker theorem establishes the impossibility of a noncontextual hidden variable model of quantum theory, or equivalently, that quantum theory is contextual. In this paper, an…
Quantum systems with finite Hilbert space
Quantum systems with finite Hilbert space are considered, and phase-space methods like the Heisenberg–Weyl group, symplectic transformations and Wigner and Weyl functions are discussed. A…
Quantum Theory From Five Reasonable Axioms
The usual formulation of quantum theory is based on rather obscure axioms (employing complex Hilbert spaces, Hermitean operators, and the trace formula for calculating probabilities). In this paper…
Hudson's theorem for finite-dimensional quantum systems
We show that, on a Hilbert space of odd dimension, the only pure states to possess a non-negative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary…
A Wigner-function formulation of finite-state quantum mechanics
Quantum states and generalized observables: a simple proof of Gleason's theorem.
  • P. Busch
  • Physics
    Physical review letters
  • 2003
TLDR
A simple proof of the result, analogous to Gleason's theorem, that any quantum state is given by a density operator, and a von Neumann-type argument against noncontextual hidden variables is obtained.
Quantum mechanics as a statistical theory
An attempt is made to interpret quantum mechanics as a statistical theory, or more exactly as a form of non-deterministic statistical dynamics. The paper falls into three parts. In the first, the…
...
...