Purity- and entropy-bounded uncertainty relations for mixed quantum states

We give a review of different forms of uncertainty relations for mixed quantum states obtained over the last two decades and present many new results. The nonclassical proper-ties of mixed states minimizing the purity-bounded uncertainty relations (a possibility of sub-Poissonian statistics, squeezing etc) are considered. The normalized Hilbert-Schmidt distance between the minimizing states and the 'most classical' thermal states is used as a 'measure of nonclassicality' together with the Mandel parameter. For highly mixed minimizing states (whose 'purities' are very small), the normalized Hilbert-Schmidt distance tends to a finite limit, which depends on the nature of the state (15% of the maximal possible distance if the deviation from pure states is characterized by the 'standard purity' Tr (rho) over cap (2) and 37% if the 'superpurity' lim(r-->0)[Tr((rho) over cap (1+1/r))](r) is chosen as a measure of deviation).