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 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][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 Wikipedia

 

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

References

1. Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. ISBN 9781107002173.
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. arXiv:2108.01311. doi:10.1109/TE.2022.3144943.
5. Kohnle, Antje (2013). "A new introductory quantum mechanics curriculum". European Journal of Physics. 35 (1). arXiv:1307.1484. doi:10.48550/arXiv.1307.1484.
6. Rybicki, George B.; Lightman, Alan P. (1979). Radiative Processes in Astrophysics. Wiley-VCH. p. 15. ISBN 9780471827597.
7. 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.
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. MIT Press. ISBN 9780262514293.
10. Cywiński, Łukasz (2008). "How to enhance dephasing time in superconducting qubits". Physical Review B. 77 (17): 174509. arXiv:0712.2225. doi:10.1103/PhysRevB.77.174509.