Quantum Tomography under Prior Information

Abstract

Abstract. We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify an unknown quantum state which is constrained by prior information? We show that if the prior information restricts the possible states to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the set of all states. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that the minimal number of measurement outcomes (POVM elements) necessary to identify all pure states in a d-dimensional Hilbert space is 4d − 3 − c(d)α(d) for some c(d) ∈ [1, 2] and α(d) being the number of ones appearing in the binary expansion of (d − 1).

Pure-state tomography with the expectation value of Pauli operators

“…The same argument can be used to prove that the 8-by-2 submatrices formed by columns (1,8), (2,7) or (3, 6) also have rank at most 1.…”

“…It is known that there exists a family of 4d − 4 observables such that any d-dimensional pure state is UDP [7], and 5d − 6 observables such that any d-dimensional pure state is UDA [6]. Many other techniques for pure-state tomography have been developed, and experiments have been performed to demonstrate the reduction of the number of measurements needed [8][9][10][11][12][13].…”

“…If there is a zero element in the submatrix formed by rows (1,2,7,8) Now, the only case we left is the following:…”

“…Therefore, under our assumption that H has only 1 positive eigenvalue, H also has only 1 negative eigenvalue. However, let us consider the 3-by-3 submatrix of H formed by (5,7,8)-th rows and (1, 2, 3)-th columns . Its determinant is (−i + 2b 1 )(i + 2b 1 ) 2 (1+2ib 2 )(i+2b 2 ) 2 (i−2b 3 ) 2 (i+2b 3 ) 2 r((1+…”

Phase retrieval versus phaseless reconstruction

“…In the complex case, they showed that a generic set of (4m-2)-vectors does phaseless reconstruction. Heinossaari, Mazzarella and Wolf [6] show that n-vectors doing phaseless reconstruction in C m requires n ≥ 4m − 4 − 2α, where α is the number of 1 ′ s in the binary expansion of (m-1). Bodmann [3] showed that phaseless reconstruction in C m can be done with (4m − 4)-vectors.…”