Draft:Bloch sphere representation in mode-counting quantum models

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

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• 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)

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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 of Bloch sphere]

 

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

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.