Draft:Bloch sphere representation in mode-counting quantum models

Representation of quantum states using the Bloch sphere and mode-counting models

Review waiting, please be patient. This may take 6 weeks or more, since drafts are reviewed in no specific order. There are 2,058 pending submissions waiting for review. If the submission is accepted, then this page will be moved into the article space. If the submission is declined, then the reason will be posted here. In the meantime, you can continue to improve this submission by editing normally. Where to get help If you need help editing or submitting your draft, please ask us a question at the AfC Help Desk or get live help from experienced editors. These venues are only for help with editing and the submission process, not to get reviews. If you need feedback on your draft, or if the review is taking a lot of time, you can try asking for help on the talk page of a relevant WikiProject. Some WikiProjects are more active than others so a speedy reply is not guaranteed. How to improve a draft Wikipedia:Contributing to Wikipedia – a basic overview on how to edit Wikipedia. Help:Wikitext – how to use the markup Help:Referencing for beginners – how to include references Wikipedia:Article development – how to develop your article Wikipedia:Writing better articles – how to improve your article Wikipedia:Verifiability – make sure your article includes reliable third-party sources You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article. Improving your odds of a speedy review To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags. Add tags to your draft Editor resources Easy tools: Citation bot (help) | Advanced: Fix bare URLs Reviewer tools Instructions · What links here · Bloch sphere representation in mode-counting quantum models (talk: + · bio) · (log) · Copyvios report · reFill · Citation Bot · (Search: Google, Wikipedia) · Submitted 13 days ago by Harold Foppele (talk: D · +) · Last edited 11 days ago by Harold Foppele

Submission declined on 16 August 2025 by Stuartyeates (talk). The lead needs to explain this in a manner that a layperson can understand. IT also needs to link to related terms. If you would like to continue working on the submission, click on the "Edit" tab at the top of the window. If you have not resolved the issues listed above, your draft will be declined again and potentially deleted. If you need extra help, please ask us a question at the AfC Help Desk or get live help from experienced editors. Please do not remove reviewer comments or this notice until the submission is accepted. Where to get help If you need help editing or submitting your draft, please ask us a question at the AfC Help Desk or get live help from experienced editors. These venues are only for help with editing and the submission process, not to get reviews. If you need feedback on your draft, or if the review is taking a lot of time, you can try asking for help on the talk page of a relevant WikiProject. Some WikiProjects are more active than others so a speedy reply is not guaranteed. How to improve a draft Wikipedia:Contributing to Wikipedia – a basic overview on how to edit Wikipedia. Help:Wikitext – how to use the markup Help:Referencing for beginners – how to include references Wikipedia:Article development – how to develop your article Wikipedia:Writing better articles – how to improve your article Wikipedia:Verifiability – make sure your article includes reliable third-party sources You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article. Improving your odds of a speedy review To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags. Add tags to your draft Editor resources Easy tools: Citation bot (help) | Advanced: Fix bare URLs Declined by Stuartyeates 13 days ago. Last edited by Harold Foppele 11 days ago. Reviewer: Inform author. This draft has been resubmitted and is currently awaiting re-review.

• Comment: Please fix the "Check |isbn= value: checksum" error in reference #9. GoingBatty (talk) 23:05, 16 August 2025 (UTC)

• Comment: See advice previously given at Wikipedia:Teahouse#help for sandbox. Andy Mabbett (Pigsonthewing); Talk to Andy; Andy's edits 12:36, 16 August 2025 (UTC)

---

This is a draft article. It is a work in progress open to editing by anyone. Please ensure core content policies are met before publishing it as a live Wikipedia article. Find sources: Google (books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL Easy tools: Citation bot (help) | Advanced: Fix bare URLs · Article logs · Draft logs. Last edited by Harold Foppele (talk | contribs) 11 days ago. (Update) This draft has been submitted and is currently awaiting review.

Bloch sphere representation in mode-counting quantum models
The Bloch sphere is a geometric representation of the states of a single qubit, the basic unit of quantum information. Each point on the surface of the sphere corresponds to a possible quantum state, making it a valuable visual tool in quantum mechanics.^[1]

In some approaches to continuous-variable quantum information (CV), such as "mode-counting" models, connections can be drawn to the Bloch sphere picture. This allows a direct comparison between CV techniques—which use observables with continuous spectra—and qubit-based methods, which use finite-dimensional Hilbert spaces.^[2][3]

Some mode-counting approaches in CV quantum systems can be mapped onto this framework, allowing direct comparison with qubit-based methods.^[4]

Mode-counting formula

 

Overview schematic of a mode-counting CV model and its comparison point with the Bloch sphere.

A general expression for the number of effective quantum modes is:

${\displaystyle N_{\mathrm {eff} }=\alpha \cdot {\frac {4}{3}}\pi R^{3}T\cdot {\frac {f_{\mathrm {max} }^{4}}{c^{3}}}}$ ^[2]

where:

• ${\displaystyle \alpha \approx 2}$  — polarization factor^[2]
• ${\displaystyle R}$  — spatial radius
• ${\displaystyle T}$  — time window
• ${\displaystyle f_{\mathrm {max} }}$  — maximum frequency cut-off^[1]
• ${\displaystyle c}$  — speed of light^[5]

Capacity parameter

 

Illustration of the capacity parameter ${\displaystyle C}$  defined from mode dimension ${\displaystyle D_{\mathrm {mode} }}$ .

The capacity per mode is given by:

${\displaystyle C^{2}=\log _{2}(D_{\mathrm {mode} })}$ ^[2]

Examples:

• Qubit encoding: ${\displaystyle D_{\mathrm {mode} }=2}$ , so ${\displaystyle C^{2}=1}$ ^[1]
• Continuous-variable encoding: ${\displaystyle D_{\mathrm {mode} }=m_{\mathrm {max} }+1}$ , where ${\displaystyle m_{\mathrm {max} }}$  is the maximum photon number.^[2]

Illustrative scenarios

 

Illustrative scenarios comparing orders of magnitude for effective mode counts across platforms.

Mode counting, i.e. evaluating the density of states, is a standard method in statistical and quantum physics, used for example in derivations of black-body radiation, in optical local density of states, and in superconducting microwave resonators; these contexts are often introduced using order-of-magnitude estimates.^[6][7][8][9]

Implications and limitations

 

Conceptual depiction of an energy cut-off in continuous-variable encodings.

• Hardware constraints: Large ${\displaystyle N_{\mathrm {eff} }}$  does not guarantee usable capacity due to limitations in control and addressing.^[10]
• Coherence time: The effective time window ${\displaystyle T}$  may be reduced by decoherence and dephasing.^[10]
• Energy cut-offs: CV systems require finite energy cut-offs to keep mode dimension bounded.^[2]

Model in Wilipedia

 

The Bloch sphere, a geometric representation of a two-level quantum system. Bloch_sphere

References

1. Nielsen, M. A.; Chuang, I. L. (2010). Quantum Computation and Quantum Information (10th anniversary ed.). Cambridge University Press. ISBN 978-1107002173.
2. Braunstein, S. L.; van Loock, P. (2005). "Quantum information with continuous variables". Reviews of Modern Physics. 77 (2): 513–577. doi:10.1103/RevModPhys.77.513.
3. "Continuous-variable quantum information". Wikipedia. Retrieved 17 August 2025.
4. Asfaw, Abraham (2022). "Building a quantum engineering undergraduate program". IEEE Transactions on Education. 65 (3): 220–242. doi:10.1109/TE.2022.3144943. arXiv:2108.01311.
5. Kohnle, Antje (2013). "A new introductory quantum mechanics curriculum". European Journal of Physics. 35 (1). doi:10.48550/arXiv.1307.1484.
6. Rybicki, George B.; Lightman, Alan P. (1979). Radiative Processes in Astrophysics. Wiley-VCH. ISBN 978-0-471-82759-7. p. 15.
7. Barnes, William L.; Björnshauge, B. et al. (2020). "Classical antennas, quantum emitters, and densities of optical states". Journal of Optics. 22 (7): 073501. doi:10.1088/2040-8986/ab9d63.
8. Zmuidzinas, Jonas (2012). "Superconducting microresonators: physics and applications". Annual Review of Condensed Matter Physics. 3: 169–214. doi:10.1146/annurev-conmatphys-020911-125022.
9. Mahajan, Sanjoy (2010). Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving. Cambridge University Press. ISBN 978-0-262-51429-3.
10. Cywiński, Łukasz (2008). "How to enhance dephasing time in superconducting qubits". Physical Review B. 77 (17): 174509. doi:10.1103/PhysRevB.77.174509. arXiv:0712.2225.