3 years ago

# Pure state ‘really’ informationally complete with rank-1 POVM

Yu Wang, Yun Shang

### 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 $$1\le k \le d$$ .

DOI: 10.1007/s11128-018-1812-2

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