### Quantum Fisher Information: Energy-Time Uncertainty Relations

#### (supervisor: John Gough)

Nature of project: **theory**, theory

Available to
students on full-time physics degree schemes only.

#### Project description and methodology

The project aims to understand the ideas behind Fisher information, the Cramer-Rao inequality and the generalization to quantum probabilities.

As an application we study the analogue of the energy-time uncertainty relations - in principle there is no time operator in quantum mechanics so it is not possible to derive an inequality as in the case of position and momentum. However, we can formulate as an estimation problem.

*A successful project will develop beyond the above in one/some of the following directions:*

Understanding the issues around time as an observable, and why there is no analogue of Heisenberg's uncertainty principle for energy and time in the usual formulation of quantum mechanics.

However, the estimation problem offers a way to understand this.

When considering where to take your project, please bear in mind the time available. It is preferable to do fewer things well than to try many and not get conclusive results on any of them. However, sometimes it is useful to have a couple of strands of investigation in parallel to work on in case delays occur.

*Additional scope or challenge if taken as a Year-4 project:* A full description of the technical details behind the theory.

Please speak to **John Gough** if you consider doing this project.

*Initial literature for students:*

- Neilsen & Chuang, Quantum Computation and Quantum Information,Cambridge University Press New York, NY, USA, ISBN:1107002176 9781107002173
- Vedral, V. (8 March 2002).
- Ohya, M., Petz, Denes, Quantum Entropy and Its Use, Springer-Verlag Berlin Heidelberg
- O E Barndorff-Nielsen and R D Gill, Fisher information in quantum statistics, 2000 J. Phys. A: Math. Gen. 33 4481,

#### Novelty, degree of difficulty and amount of assistance required

Fisher information is not part of the curriculum and will need to be researched. neither, of course is its quantum analogue. Understanding these concepts will be important.

The setting up of the physical problem as a estimation problem will be a key step in deriving the energy-time uncertainty relations.

#### Project milestones and deliverables (including timescale)

milestone | to be completed by |

basic theory and references | end of November |

formulation of problem - outlook | Christmas |

deriving the energy-time uncertainty relations | Easter |

writing up/ new applications | end of October |