Quantum lost property: a possible operational meaning for the Hilbert-Schmidt product

Minimum error state discrimination between two mixed states \rho and \sigma can be aided by the receipt of "classical side information" specifying which states from some convex decompositions of \rho and \sigma apply in each run. We quantify this phenomena by the average trace distance, and give lower and upper bounds on this quantity as functions of \rho and \sigma. The lower bound is simply the trace distance between \rho and \sigma, trivially seen to be tight. The upper bound is \sqrt{1 - tr(\rho\sigma)}, and we conjecture that this is also tight. We reformulate this conjecture in terms of the existence of a pair of "unbiased decompositions", which may be of independent interest, and prove it for a few special cases. Finally, we point towards a link with a notion of non-classicality known as preparation contextuality.Comment: 3 pages, 1 figure. v2: Less typos in text and less punctuation in titl

On the rough Gronwall lemma and its applications

We present a rough path analog of the classical Gronwall Lemma introduced recently by A. Deya, M. Gubinelli, M. Hofmanov\'a, S. Tindel in [arXiv:1604.00437] and discuss two of its applications. First, it is applied in the framework of rough path driven PDEs in order to establish energy estimates for weak solutions. Second, it is used in order to prove uniqueness for reflected rough differential equations

High Thermoelectric Figure of Merit by Resonant Dopant in Half-Heusler Alloys

Half-Heusler alloys have been one of the benchmark high temperature thermoelectric materials owing to their thermal stability and promising figure of merit ZT. Simonson et al. early showed that small amounts of vanadium doped in Hf0.75Zr0.25NiSn enhanced the Seebeck coefficient and correlated the change with the increased density of states near the Fermi level. We herein report a systematic study on the role of vanadium (V), niobium (Nb), and tantalum (Ta) as prospective resonant dopants in enhancing the ZT of n-type half-Heusler alloys based on Hf0.6Zr0.4NiSn0.995Sb0.005. The V doping was found to increase the Seebeck coefficient in the temperature range 300-1000 K, consistent with a resonant doping scheme. In contrast, Nb and Ta act as normal n-type dopants, as evident by the systematic decrease in electrical resistivity and Seebeck coefficient. The combination of enhanced Seebeck coefficient due to the presence of V resonant states and the reduced thermal conductivity has led to a state-of-the-art ZT of 1.3 near 850 K in n-type (Hf0.6Zr0.4)0.99V0.01NiSn0.995Sb0.005 alloys.Comment: Submitted to AIP Advance