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 properties 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 ˆ ρ 2 and 37% if the ‘superpurity’ limr→0[Tr( ˆ ρ 1+1/r )] r is chosen as a measure of deviation).