Draft:Bloch sphere representation in mode-counting quantum models

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)

Bloch sphere representation in mode-counting quantum models describes how certain continuous-variable (CV) quantum information frameworks can be related to the standard qubit Bloch sphere picture. The Bloch sphere is a widely used geometric representation of the pure state space of a single qubit, where each point corresponds to a possible quantum state with a given superposition and relative phase.^[1]^[2]

Some mode-counting approaches in CV quantum systems can be mapped onto this framework, allowing direct comparison with qubit-based methods.^[3]

Mode-counting formula

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

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

N_{\mathrm {eff} }=\alpha \cdot {\frac {4}{3}}\pi R^{3}T\cdot {\frac {f_{\mathrm {max} }^{4}}{c^{3}}}

^[2]

where:

$\alpha \approx 2$ — polarization factor^[2]
$R$ — spatial radius
$T$ — time window
$f_{\mathrm {max} }$ — maximum frequency cut-off^[1]
$c$ — speed of light^[4]

Capacity parameter

The capacity per mode is given by:

Illustration of the capacity parameter

C

defined from mode dimension

D_{\mathrm {mode} }

.

C^{2}=\log _{2}(D_{\mathrm {mode} })

^[2]

Examples:

Qubit encoding: $D_{\mathrm {mode} }=2$ , so $C^{2}=1$ ^[1]
Continuous-variable encoding: $D_{\mathrm {mode} }=m_{\mathrm {max} }+1$ , where $m_{\mathrm {max} }$ 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.^[5]^[6]^[7]^[8]

Implications and limitations

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

Hardware constraints: Large $N_{\mathrm {eff} }$ does not guarantee usable capacity due to limitations in control and addressing.^[9]
Coherence time: The effective time window $T$ may be reduced by decoherence and dephasing.^[9]
Energy cut-offs: CV systems require finite energy cut-offs to keep mode dimension bounded.^[2]

Edit by Harold Foppele aug 16 2025. 23:12

References

^ ^a ^b ^c Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. ISBN 9781107002173.
^ ^a ^b ^c ^d ^e ^f Braunstein, Samuel L.; van Loock, Peter (2005). "Quantum information with continuous variables". Reviews of Modern Physics. 77: 513–577. doi:10.1103/RevModPhys.77.513.
^ Asfaw, Abraham (2022). "Building a quantum engineering undergraduate program". IEEE Transactions on Education. 65 (3): 220–242. doi:10.1109/TE.2022.3144943.
^ Kohnle, Antje (2013). "A new introductory quantum mechanics curriculum". European Journal of Physics. 35 (1). doi:10.48550/arXiv.1307.1484.
^ Rybicki, George B.; Lightman, Alan P. (1979). Radiative Processes in Astrophysics. Wiley-VCH. p. 15. ISBN 978-0-471-82759-7.
^ 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. {{cite journal}}: Explicit use of et al. in: |author2= (help)
^ Zmuidzinas, Jonas (2012). "Superconducting microresonators: physics and applications". Annual Review of Condensed Matter Physics. 3: 169–214. doi:10.1146/annurev-conmatphys-020911-125022.
^ Mahajan, Sanjeev Arora (2010). Order-of-Magnitude Physics. Cambridge University Press. ISBN 978-0-521-18858-8. {{cite book}}: Check |isbn= value: checksum (help)
^ ^a ^b Cywiński, Łukasz (2008). "How to enhance dephasing time in superconducting qubits". Physical Review B. 77: 174509. doi:10.1103/PhysRevB.77.174509.

[NielsenChuang-1] Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. ISBN 9781107002173.

[BraunsteinLoock-2] ^ ^a ^b ^c ^d ^e ^f Braunstein, Samuel L.; van Loock, Peter (2005). "Quantum information with continuous variables". Reviews of Modern Physics. 77: 513–577. doi:10.1103/RevModPhys.77.513.

[Asfaw2022-3] Asfaw, Abraham (2022). "Building a quantum engineering undergraduate program". IEEE Transactions on Education. 65 (3): 220–242. doi:10.1109/TE.2022.3144943.

[4] Kohnle, Antje (2013). "A new introductory quantum mechanics curriculum". European Journal of Physics. 35 (1). doi:10.48550/arXiv.1307.1484.

[5] Rybicki, George B.; Lightman, Alan P. (1979). Radiative Processes in Astrophysics. Wiley-VCH. p. 15. ISBN 978-0-471-82759-7.

[6] 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. {{cite journal}}: Explicit use of et al. in: |author2= (help)

[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, Sanjeev Arora (2010). Order-of-Magnitude Physics. Cambridge University Press. ISBN 978-0-521-18858-8. {{cite book}}: Check |isbn= value: checksum (help)

[Cywinski2008-9] Cywiński, Łukasz (2008). "How to enhance dephasing time in superconducting qubits". Physical Review B. 77: 174509. doi:10.1103/PhysRevB.77.174509.

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