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• W2963794655 abstract "What is the minimal number of elements in a rank-1 positive operator-valued measure (POVM) which can uniquely determine any pure state in d-dimensional Hilbert space $$mathcal {H}_d$$ ? The known result is that the number is no less than $$3d-2$$ . We show that this lower bound is not tight except for $$d=2$$ or 4. Then we give an upper bound $$4d-3$$ . For $$d=2$$ , many rank-1 POVMs with four elements can determine any pure states in $$mathcal {H}_2$$ . For $$d=3$$ , we show eight is the minimal number by construction. For $$d=4$$ , the minimal number is in the set of $${10,11,12,13}$$ . We show that if this number is greater than 10, an unsettled open problem can be solved that three orthonormal bases cannot distinguish all pure states in $$mathcal {H}_4$$ . For any dimension d, we construct $$d+2k-2$$ adaptive rank-1 positive operators for the reconstruction of any unknown pure state in $$mathcal {H}_d$$ , where $$1le k le d$$ ." @default.
• W2963794655 created "2019-07-30" @default.
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• W2963794655 creator A5073285157 @default.
• W2963794655 date "2018-01-25" @default.
• W2963794655 modified "2023-10-18" @default.
• W2963794655 title "Pure state ‘really’ informationally complete with rank-1 POVM" @default.
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• W2963794655 doi "https://doi.org/10.1007/s11128-018-1812-2" @default.
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