Optimal Discrimination of Qubit States - Methods, Solutions, and Properties

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The only possibility to choose a positive operator σ x \sigma_{\mathrm{x}} such that tr ​ [ σ x ​ M x ] = 0 \mbox{tr}[\sigma_{\mathrm{x}}M_{\mathrm{x}}]=0 for a positive operator M x M_{\mathrm{x}} of rank-two is that σ x = 0 \sigma_{\mathrm{x}}=0 . This is taken into account in the the case when r x = 0 r_{\mathrm{x}}=0 in the KKT conditions, and therefore excluded. Then, POVM elements are either rank-one or the null operator

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