Error probability analysis in quantum tomography: a tool for evaluating experiments

We expand the scope of the statistical notion of error probability, i.e., how often large deviations are observed in an experiment, in order to make it directly applicable to quantum tomography. We verify that the error probability can decrease at most exponentially in the number of trials, derive the explicit rate that bounds this decrease, and show that a maximum likelihood estimator achieves this bound. We also show that the statistical notion of identifiability coincides with the tomographic notion of informational completeness. Our result implies that two quantum tomographic apparatuses that have the same risk function, (e.g. variance), can have different error probability, and we give an example in one qubit state tomography. Thus by combining these two approaches we can evaluate, in a reconstruction independent way, the performance of such experiments more discerningly.Comment: 14pages, 2 figures (an analysis of an example is added, and the proof of Lemma 2 is corrected

Mutation of the N-terminal proline 9 of BLMA from Streptomyces verticillus abolishes the binding affinity for bleomycin

AbstractA gene, blmA, from bleomycin (Bm)-producing Streptomyces verticillus, encodes a Bm-binding protein, designated BLMA. The expression of BLMA conferred resistance to Bm in the Escherichia coli host, whereas a mutant protein, designated Pro-9/Leu, with the N-terminal proline 9 residue in BLMA replaced by leucine, did not. We created a fusion protein between the maltose-binding protein (MBP) and a mutant protein Pro-9/Leu/Leu with Met-94 in Pro-9/Leu replaced by leucine. Pro-9/Leu/Leu from the fusion protein, obtained by digestion with CNBr digestion, did not inhibit DNA-cleaving and antibacterial activities of Bm. Native-polyacrylamide gel electrophoresis (PAGE) and gel filtration column chromatographic analysis showed that the molecular size of Pro-9/Leu/Leu is roughly half of that of BLMA, suggesting that the mutant protein cannot form dimeric structure. Furthermore, Far-UV circular dichroism (CD) spectrum of Pro-9/Leu/Leu was quite different from that of BLMA and similar to the spectra obtained from unordered proteins [Venyaminov, S.Y. and Vassilenko, K.S. (1994) Anal. Biochem. 222, 176–184], suggesting that the secondary structure of Pro-9/Leu/Leu is disrupted. These results indicate that the mutation abolishes not only dimer formation but also the secondary structure of BLMA, which results in the loss of its function as a Bm-resistance determinant

Effect of nonnegativity on estimation errors in one-qubit state tomography with finite data

We analyze the behavior of estimation errors evaluated by two loss functions, the Hilbert-Schmidt distance and infidelity, in one-qubit state tomography with finite data. We show numerically that there can be a large gap between the estimation errors and those predicted by an asymptotic analysis. The origin of this discrepancy is the existence of the boundary in the state space imposed by the requirement that density matrices be nonnegative (positive semidefinite). We derive an explicit form of a function reproducing the behavior of the estimation errors with high accuracy by introducing two approximations: a Gaussian approximation of the multinomial distributions of outcomes, and linearizing the boundary. This function gives us an intuition for the behavior of the expected losses for finite data sets. We show that this function can be used to determine the amount of data necessary for the estimation to be treated reliably with the asymptotic theory. We give an explicit expression for this amount, which exhibits strong sensitivity to the true quantum state as well as the choice of measurement.Comment: 9 pages, 4 figures, One figure (FIG. 1) is added to the previous version, and some typos are correcte

Adaptive experimental design for one-qubit state estimation with finite data based on a statistical update criterion

We consider 1-qubit mixed quantum state estimation by adaptively updating measurements according to previously obtained outcomes and measurement settings. Updates are determined by the average-variance-optimality (A-optimality) criterion, known in the classical theory of experimental design and applied here to quantum state estimation. In general, A-optimization is a nonlinear minimization problem; however, we find an analytic solution for 1-qubit state estimation using projective measurements, reducing computational effort. We compare numerically two adaptive and two nonadaptive schemes for finite data sets and show that the A-optimality criterion gives more precise estimates than standard quantum tomography.Comment: 15 pages, 7 figure