Talk:Fidelity of quantum states

self-promotion
It seems that the edit by IP address 86.73.72.40 : fidelity in measurements and the reference T. Amri, Quantum behavior of measurement apparatus, arXiv:1001.3032 (2010). is not relevant to the encyclopedic nature of wikipedia. It may even be self promotion. The IP address 86.73.72.40 has gone about adding this reference and information related to this reference at several Physics pages. Kanwarpreet Grewal 09:45, 2 April 2010 (UTC) —Preceding unsigned comment added by Kp grewal (talk • contribs)

I don't know whether that "fidelity in quantum measurements" paragraph is self-promotion, but it is certainly not a useful part of the article. I'm deleting it. Aram.harrow (talk) 03:30, 23 May 2014 (UTC)

Common mistaken definition
The correct definition of the fidelity is $$Tr[\sqrt{\sqrt{\rho}\sigma \sqrt{\rho}}]$$, which is equal to $$||\sqrt{\rho}\sqrt{\sigma}||_1$$, However, the formula $$Tr[\sqrt{\rho}\sqrt{\sigma}]$$ is not correct but is commonly written as a mistake by beginners (presumably because it's simple and easy to understand). In particular, editors have replaced the correct formula with the incorrect one. This is a subtle error that can persist for a while without being caught, and will probably cause people grief. Jess_Riedel (talk) 18:10, 21 September 2017 (UTC)

Different definitions of fidelity
Two different definitions of “fidelity” exist in the literature, namely $$F=\langle\psi|\rho|\psi\rangle$$ and $$F'=\sqrt{\langle\psi|\rho|\psi\rangle}$$ with $$F'=\sqrt{F}$$. They both have different advantages and disadvantages. In particular, F has the interpretation of a probability. The definition F' is mainly propagated by the famous Quantum information book by Nielsen and Chuang. However, the majority of books use the definition F, and so does the vast majority of scientific papers in the quantum physics community. Therefore I adapted the article to reflect this and use F as the main definition. The existence of two different definitions is better highlighted now. Here are a few examples of books that use the now-adopted definition of F: Geometry of Quantum States: An Introduction to Quantum Entanglement (Bengtsson), Quantum Information: An Overview (Jaeger), Exploring the Quantum (Haroche), Quantum Information (Barnett), A Guide to Experiments in Quantum Optics (Bachor), Quantum Optics (Walls & Milburn), Fundamentals of Quantum Optics and Quantum Information (Lambropoulos & Petrosyan), Quantum State estimation (Paris). --Geek3 (talk) 22:38, 6 April 2018 (UTC)

Equivalent expression via characteristic polynomials
I have added a new subsection "Equivalent expression via characteristic polynomials", but since this is sourced to one of my papers in Physical Review A I obviously have a massive conflict of interest here. I would be grateful for some eyes on this in case my addition is inappropriate, and in any case to esnure that it is clear. There's an extended discussion of this question at Stack Exchange Quantum Computing which people might find useful and which includes links to alternative proofs in case you prefer these to mine. Jonathan A Jones (talk) 13:54, 6 December 2023 (UTC)


 * I noticed the new subsection "Equivalent expression via characteristic polynomials" and wanted to clarify that the argument seems to reflect my unpublished work found here: https://arxiv.org/abs/2309.10565, rather than the cited paper in Physical Review A. This discrepancy could mislead readers and affect academic crediting. To avoid confusion and maintain accurate citation practices, could we consider removing this section until a published source that explicitly uses this argument is available? Noxafy (talk) 22:04, 24 January 2024 (UTC)
 * As you know neither of our proofs are original: this result is regularly rediscovered from time to time. Indeed it's almost embarrassing how frequently it is rediscovered, and it's for that precise reason that I think it important that it be included here: something has to be done to reduce the chance of somebody else rediscovering it and erroneously thinking the result is novel. The proofs fall into a number of broad categories, with your proof and my proof being essentially equivalent: I prove that the characteristic polynomial is invariant under cyclic permutation for a specific matrix product which arises in the fidelity calculation, while you start from the general invariance of the characteristic polynomial under cyclic permutation. The argument I used on the page is closer to yours than to mine, though fundametally it is based on an email I got from Sarah Plosker, who proved the result back in 2015, well before either of us, though of course she wasn't the first either. The exact form was chosen to benefit from existing lemmas with articles that coud be linked to, so that the proof flows nicely on the page here.
 * So I think this result unquestionably belongs on the page. What we source it to is a different argument, and one that should really be decided by somebody less conflicted than we are. I went for my paper becauuse it (1) is in the peer reviewed literature, and (2) it places the result clearly in the context of quantum fidelity, rather than as a side issue to something else. I don't think there's any other paper that clearly meets those criteria, but I may have missed something, or my whole argument might just be special pleading. Should your manuscript be publlished then obviously (at least to me) it should be added here, and there's some argument for doing that with the ArXiv paper, but that seems to me to be a matter better decided by others. Jonathan A Jones (talk) 21:59, 25 January 2024 (UTC)