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Draft:Bloch sphere representation in mode-counting quantum models

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  • Comment: This is WP:SYNTH and WP:OR. It combines in a random manner quantum information and standard physics to create a concept that has no general science relevance. Ldm1954 (talk) 13:15, 19 August 2025 (UTC)
  • Comment: Please fix the "Check |isbn= value: checksum" error in reference #9. GoingBatty (talk) 23:05, 16 August 2025 (UTC)

Bloch sphere representation in mode-counting quantum models
The Bloch sphere is a way of picturing the state of a single qubit, the fundamental unit of quantum computing. It represents all possible states of a qubit as points on the surface of a sphere, which makes abstract quantum concepts easier to visualize.

In quantum physics there are two main approaches: qubit systems, which use discrete states, and continuous-variable systems, which use continuous ranges such as position or momentum. The Bloch sphere is normally used for qubits, but some researchers have shown that it can also be linked to so-called "mode-counting" models in continuous-variable systems. This provides a common framework to compare different types of quantum information processing.[1][2][3]

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:

 [2]

where:

  •   — polarization factor[2]
  •   — spatial radius
  •   — time window
  •   — maximum frequency cut-off[1]
  •   — speed of light[4]

Capacity parameter

 
Illustration of the capacity parameter   defined from mode dimension  .

The capacity per mode is given by:

 [2]

Examples:

  • Qubit encoding:  , so  [1]
  • Continuous-variable encoding:  , where   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.[5][6][7][8]

Implications and limitations

 
Conceptual depiction of an energy cut-off in continuous-variable encodings.
  • Hardware constraints: Large   does not guarantee usable capacity due to limitations in control and addressing.[9]
  • Coherence time: The effective time window   may be reduced by decoherence and dephasing.[9]
  • Energy cut-offs: CV systems require finite energy cut-offs to keep mode dimension bounded.[2]


Model in Wikipedia
 
The Bloch sphere, a geometric representation of a two-level quantum system. Bloch sphere

References

  1. ^ a b c Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information (10th anniversary ed.). Cambridge University Press. ISBN 978-1107002173.
  2. ^ a b c d e f Braunstein, S. L.; van Loock, P. (2005). "Quantum information with continuous variables". Reviews of Modern Physics. 77 (2): 513–577. arXiv:quant-ph/0410100. Bibcode:2005RvMP...77..513B. doi:10.1103/RevModPhys.77.513.
  3. ^ Asfaw, Abraham (2022). "Building a quantum engineering undergraduate program". IEEE Transactions on Education. 65 (3): 220–242. arXiv:2108.01311. Bibcode:2022ITEdu..65..220A. doi:10.1109/TE.2022.3144943.
  4. ^ Kohnle, Antje (2013). "A new introductory quantum mechanics curriculum". European Journal of Physics. 35 (1): 015001. arXiv:1307.1484. doi:10.1088/0143-0807/35/1/015001.
  5. ^ Rybicki, George B.; Lightman, Alan P. (1979). Radiative Processes in Astrophysics. Wiley-VCH. p. 15. ISBN 978-0471827597.
  6. ^ Barnes, William L.; Björnshauge, B. (2020). "Classical antennas, quantum emitters, and densities of optical states". Journal of Optics. 22 (7): 073501. doi:10.1088/2040-8986/ab9d63 (inactive 18 August 2025).{{cite journal}}: CS1 maint: DOI inactive as of August 2025 (link)
  7. ^ Zmuidzinas, Jonas (2012). "Superconducting microresonators: physics and applications". Annual Review of Condensed Matter Physics. 3: 169–214. doi:10.1146/annurev-conmatphys-020911-125022.
  8. ^ Mahajan, Sanjoy (2010). Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving. MIT Press. ISBN 978-0262514293.
  9. ^ a b Cywiński, Łukasz (2008). "How to enhance dephasing time in superconducting qubits". Physical Review B. 77 (17) 174509. arXiv:0712.2225. Bibcode:2008PhRvB..77q4509C. doi:10.1103/PhysRevB.77.174509.

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