Performing perfect/conclusive quantum state exclusion means to be able to discard with certainty at least one out of $n$ possible quantum state preparations by performing a measurement of the resulting state. This task of state exclusion has recently been studied at length in , and it is at the heart of the celebrated PBR thought experiment . When all the preparations correspond to pure states and there are no more of them than their common dimension, it is an open problem whether POVMs give any additional power for this task with respect to projective measurements. This is the case even for the simple case of three states in three dimensions, which is mentioned in  as unsuccessfully tackled. In this paper, we give an analytical proof that in this case considering POVMs does indeed not give any additional power with respect to projective measurements. To do so, we first make without loss of generality some assumptions about the structure of an optimal POVM. The justification of these assumptions involves arguments based on convexity, rank and symmetry properties. We show then that any pure states perfectly excluded by such a POVM meet the conditions identified in  for perfect exclusion by a projective measurement of three pure states in three dimensions. We also discuss possible generalizations of our work, including an application of Quadratically Constrained Quadratic Programming that might be of special interest.
 Chris Aholt, Sameer Agarwal, and Rekha Thomas. A QCQP approach to triangulation. In European Conference on Computer Vision, pages 654-667. Springer, 2012. 10.1007/978-3-642-33718-5_47.
 Srinivasan Arunachalam, Abel Molina, and Vincent Russo. Quantum hedging in two-round prover-verifier interactions. In LIPIcs-Leibniz International Proceedings in Informatics, volume 73. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. 10.4230/LIPIcs.TQC.2017.5.
 Koenraad MR Audenaert and Stefan Scheel. Quantum tomographic reconstruction with error bars: a Kalman filter approach. New Journal of Physics, 11 (2): 023028, 2009. 10.1088/1367-2630/11/2/023028.
 Somshubhro Bandyopadhyay, Rahul Jain, Jonathan Oppenheim, and Christopher Perry. Conclusive exclusion of quantum states. Physical Review A, 89 (2): 022336, 2014. 10.1103/physreva.89.022336.
 Jonathan Barrett, Eric G Cavalcanti, Raymond Lal, and Owen JE Maroney. No $\psi$-epistemic model can fully explain the indistinguishability of quantum states. Physical Review Letters, 112 (25): 250403, 2014. 10.1103/physrevlett.112.250403.
 Ingemar Bengtsson and Karol Zyczkowski. Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, 2007. 10.1017/9781139207010.
 Michael R Beran and Scott M Cohen. Nonoptimality of unitary encoding with quantum channels assisted by entanglement. Physical Review A, 78 (6): 062337, 2008. 10.1103/PhysRevA.78.062337.
 Subhonmesh Bose, Dennice F Gayme, K Mani Chandy, and Steven H Low. Quadratically constrained quadratic programs on acyclic graphs with application to power flow. IEEE Transactions on Control of Network Systems, 2 (3): 278-287, 2015. 10.1109/tcns.2015.2401172.
 Todd A Brun, Min-Hsiu Hsieh, and Christopher Perry. Compatibility of state assignments and pooling of information. Physical Review A, 92 (1): 012107, 2015. 10.1103/physreva.92.012107.
 Carlton M Caves, Christopher A Fuchs, and Rüdiger Schack. Conditions for compatibility of quantum-state assignments. Physical Review A, 66 (6): 062111, 2002. 10.1103/physreva.66.062111.
 Yonina C Eldar, Alexandre Megretski, and George C Verghese. Designing optimal quantum detectors via semidefinite programming. IEEE Transactions on Information Theory, 49 (4): 1007-1012, 2003. 10.1109/tit.2003.809510.
 Yonina C Eldar, Mihailo Stojnic, and Babak Hassibi. Optimal quantum detectors for unambiguous detection of mixed states. Physical Review A, 69 (6): 062318, 2004. 10.1103/physreva.69.062318.
 Youping Fan and Bernd Tibken. Optimization problems of determining the C-numerical range. IFAC Proceedings Volumes, 41 (2): 10051-10056, 2008. 10.3182/20080706-5-kr-1001.01701.
 Jaromír Fiurášek and Miroslav Ježek. Optimal discrimination of mixed quantum states involving inconclusive results. Physical Review A, 67 (1): 012321, 2003. 10.1103/physreva.67.012321.
 Erkka Haapasalo, Teiko Heinosaari, and Juha-Pekka Pellonpää. Quantum measurements on finite dimensional systems: relabeling and mixing. Quantum Information Processing, 11 (6): 1751-1763, 2012. 10.1007/s11128-011-0330-2.
 Teiko Heinosaari and Oskari Kerppo. Antidistinguishability of pure quantum states. Journal of Physics A: Mathematical and Theoretical, 51 (36): 365303, 2018. 10.1088/1751-8121/aad1fc.
 Yongwei Huang and Daniel P Palomar. Randomized algorithms for optimal solutions of double-sided QCQP with applications in signal processing. IEEE Transactions on Signal Processing, 62 (5): 1093-1108, 2014. 10.1109/tsp.2013.2297683.
 Quanzhong Li, Qi Zhang, and Jiayin Qin. A special class of fractional QCQP and its applications on cognitive collaborative beamforming. IEEE Trans. Signal Processing, 62 (8): 2151-2164, 2014. 10.1109/tsp.2014.2309072.
 Yeong-Cherng Liang and Andrew C Doherty. Bounds on quantum correlations in Bell-inequality experiments. Physical Review A, 75 (4): 042103, 2007. 10.1103/physreva.75.042103.
 Zi-Wen Liu, Christopher Perry, Yechao Zhu, Dax Enshan Koh, and Scott Aaronson. Doubly infinite separation of quantum information and communication. Physical Review A, 93 (1): 012347, 2016. 10.1103/physreva.93.012347.
 KR Parthasarathy. Extremal decision rules in quantum hypothesis testing. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2 (04): 557-568, 1999. 10.1142/s0219025799000321.
 Rudolf Ernst Peierls. More Surprises in Theoretical Physics, volume 19. Princeton University Press, 1991. ISBN 9780691025223.
 Christopher Perry. Conclusive exclusion of quantum states and aspects of thermo-majorization. PhD thesis, UCL (University College London), 2016. URL http://discovery.ucl.ac.uk/id/eprint/1473944.
 Christopher Perry, Rahul Jain, and Jonathan Oppenheim. Communication tasks with infinite quantum-classical separation. Physical Review Letters, 115 (3): 030504, 2015. 10.1103/physrevlett.115.030504.
 G Sentís, B Gendra, SD Bartlett, and AC Doherty. Decomposition of any quantum measurement into extremals. Journal of Physics A: Mathematical and Theoretical, 46 (37): 375302, 2013. 10.1088/1751-8113/46/37/375302.
 Guo Chuan Thiang. Some attempts at proving the non-existence of a full set of mutually unbiased bases in dimension 6. arXiv preprint, 2010. URL https://arxiv.org/abs/1012.3147.
 MAP Touzel, RBA Adamson, and Aephraim M Steinberg. Optimal bounded-error strategies for projective measurements in nonorthogonal-state discrimination. Physical Review A, 76 (6): 062314, 2007. 10.1103/physreva.76.062314.
 T Vértesi and E Bene. Two-qubit Bell inequality for which positive operator-valued measurements are relevant. Physical Review A, 82 (6): 062115, 2010. 10.1103/physreva.82.062115.
 Graeme Weir, Stephen M. Barnett, and Sarah Croke. Optimal discrimination of single-qubit mixed states. Physical Review A, 96: 022312, Aug 2017. 10.1103/physreva.96.022312.
 Huangjun Zhu, Yong Siah Teo, and Berthold-Georg Englert. Two-qubit symmetric informationally complete positive-operator-valued measures. Physical Review A, 82 (4): 042308, 2010. 10.1103/physreva.82.042308.
 Teiko Heinosaari and Oskari Kerppo, "Communication of partial ignorance with qubits", Journal of Physics A: Mathematical and Theoretical 52 39, 395301 (2019).
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