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 \(1\le k \le d\) .

Publisher URL: https://link.springer.com/article/10.1007/s11128-018-1812-2

DOI: 10.1007/s11128-018-1812-2

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