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

• URL: http://arxiv.org/abs/2607.27719v1
• Date: Thu, 30 Jul 2026 05:56:44 GMT
• Title: Phase Retrieval in $\mathbb C^4$ Requires Exactly Eleven Measurements
• Abstract summary: We prove that no family of $10$ vectors in $mathbbC4$ possesses the phase retrieval property.<n>A rank-one POVM on $mathbbC4$ requires exactly $11$ elements to be informationally complete for pure states.
• Score: 0.661626618020738
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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].

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