Draft:Bloch sphere representation in mode-counting quantum models

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

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]

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

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:

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

Capacity parameter

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]

 

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

Illustrative scenarios

The implications of the mode-counting expression can be visualised with simple, order-of-magnitude scenarios (e.g., superconducting, optical, and microscale settings). These are illustrative only and depend on platform-specific constraints.

 

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

Implications and limitations

• Hardware constraints: Large ${\displaystyle N_{\mathrm {eff} }}$  does not guarantee usable capacity due to limitations in control and addressing.^[5]
• Coherence time: The effective time window ${\displaystyle T}$  may be reduced by decoherence and dephasing.^[5]
• Energy cut-offs: CV systems require finite energy cut-offs to keep mode dimension bounded.^[2]

 

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

References

1. Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. ISBN 9781107002173.
2. 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.
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. Cywiński, Łukasz (2008). "How to enhance dephasing time in superconducting qubits". Physical Review B. 77: 174509. doi:10.1103/PhysRevB.77.174509.