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

Abstract

What is the minimal number of elements in a rank-1 positiveoperator-valued measure (POVM) which can uniquely determine any pure state in d-dimensional Hilbert space 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 of 4d−3. For d = 2, many rank-1 POVMs with four elements can determine any pure states in H2. 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 can not distinguish all pure states in H4. For any dimension d, we construct d + 2k − 2 adaptive rank-1 positive operators for the reconstruction of any unknown pure state in H d , where 1 ≤ k ≤ d.

Set of 4d–3 observables to determine any pure qudit state

“…While this amount of projectors represents approximately d more measurement outcomes than reported in Ref. [21] for an adaptive scheme, our method allows us to certify the assumption of purity. Conversely, our method requires in the order of d measurements less than the method reported in Ref.…”

“…In Ref. [21], Wang et al have proven that a set of 4d − 3 fixed projectors are enough to reconstruct any pure qudit. Additionally, they showed that this number can be reduced to at most 3d − 2 in an adaptive scheme, where the results of a first measurement of the particular unknown state in the canonical base (d projections) are used to choose the remaining projectors.…”

Pure State Tomography with Fourier Transformation

“…[ 30,40 ] With the adaptive strategy, a measure zero set from all the pure states can be neglected, and thus, only $$3d-2$$ outcomes are required. [ 25,41,42 ] When a qudit is measured using an observable, d outcomes may appear if there are no ancilla qubits, with each outcome corresponding to a projected eigenstate. The different eigenstates of an observable should be orthogonal.…”