Phase Retrieval in $\mathbb C^4$ Requires Exactly Eleven Measurements

Abstract:Determining the minimal number of intensity measurements required for phase retrieval in $\mathbb{C}^4$ has been a long-standing open problem. Prior to this work, the best-known results implied that this minimum was either $10$ or $11$. In this paper, we leverage characteristic classes and cohomology groups from differential topology to prove that no family of $10$ vectors in $\mathbb{C}^4$ possesses the phase retrieval property. Combining our lower bound with Vinzant's explicit eleven-vector construction establishes that the exact minimum is $11$. Our result yields a significant consequence for pure state quantum tomography, namely, a rank-one POVM on $\mathbb{C}^4$ requires exactly $11$ elements to be informationally complete for pure states. This further implies that three orthonormal bases are insufficient to uniquely distinguish all pure states in $\mathbb{C}^4$. Because four orthonormal bases are already known to be sufficient, we conclude that exactly four bases are required, thereby completely resolving the problem left in [C. Carmeli, T. Heinosaari, J. Schultz, A. Toigo, Eur. Phys. J. D].

Quantum Physics (quant-ph); Information Theory (cs.IT); Differential Geometry (math.DG)

arXiv:2607.27719 [quant-ph]

(or arXiv:2607.27719v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2607.27719 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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