# 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…

## Tables from this paper

## 73 Citations

From the Attempt of Certain Classical Reformulations of Quantum Mechanics to Quasi-Probability Representations

- Physics
- 2013

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

- Physics
- 2018

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

- Mathematics
- 2010

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?

- PhilosophyEntropy
- 2022

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

- Physics
- 2011

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

- PhysicsPhysical Review A
- 2018

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…

Algebraic probability-theoretic characterization of quantum correlations

- Physics
- 2017

Quantum entanglement and nonlocality are inequivalent notions: There exist entangled states that nevertheless admit local-realistic interpretations. This paper studies a special class of…

Non-classicality as a computational resource

- Computer Science
- 2018

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.

Notes on qubit phase space and discrete symplectic structures

- Mathematics, Physics
- 2010

We start from Wootter's construction of discrete phase spaces and Wigner functions for qubits and more generally for finite-dimensional Hilbert spaces. We look at this framework from a…

Minimal informationally complete measurements for probability representation of quantum dynamics

- PhysicsNew Journal of Physics
- 2020

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…

## References

SHOWING 1-10 OF 59 REFERENCES

FAST TRACK COMMUNICATION: Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations

- Physics
- 2008

Several finite-dimensional quasi-probability representations of quantum states have been proposed to study various problems in quantum information theory and quantum foundations. These…

Formulation of Quantum Mechanics Based on the Quasi-Probability Distribution Induced on Phase Space

- Physics
- 1958

A formulation of quantum mechanics is postulated which is based solely on a quasi-probability function on the classical phase space. It is then shown that this formulation is equivalent to the…

Wigner-Weyl correspondence in quantum mechanics for continuous and discrete systems-a Dirac-inspired view

- Physics
- 2006

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

- Mathematics
- 2004

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…

Discrete Moyal-type representations for a spin

- Mathematics
- 2000

In Moyal's formulation of quantum mechanics, a quantum spin s is described in terms of continuous symbols, i.e., by smooth functions on a two-dimensional sphere. Such prescriptions to associate…

Contextuality for preparations, transformations, and unsharp measurements

- Philosophy
- 2005

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

- Physics, Mathematics
- 2004

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

- Physics
- 2001

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

- Mathematics
- 2006

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…