Remarks on the Extended Characteristic Uncertainty Relations

In the recent papers [1, 2] the conventional uncertainty relation (UR) of Robertson [5] (which includes the Heisenberg and Schrödinger UR [4] as its particular cases) have been extended to all characteristic coefficients of the uncertainty matrix [2] and to the case of several states [1]. In this letter we present three remarks on these extended characteristic URs. The first remark refers to the form of the extended URs [1]: we note that they can be written in terms of the principal minors of the matrices involved and write the entangled Schrödinger UR [1] in a stronger form. The second remark is concerned with the extension of the characteristic inequalities to the case of mixed states and nonHermitian operators. The extension is based on the suitably constructed Gram matrix for n operators and mixed states. The characteristic inequalities for n non-Hermitian operators are in fact URs for their 2n Hermitian components. The last remark is about the domain problem of the operators involved in the URs. The proper generalization of the uncertainty matrix is suggested as the symmetric part of the corresponding Gram matrix. For pure states which are in the domain of all product of the operators involved this Gram matrix coincides with the Robertson matrix, and its symmetric part is equal to the conventional uncertainty matrix.