In the space of mixed states the Schrödinger-Robertson uncertainty relation holds though it can never be saturated.
Two tight extensions of this relation in the space of mixed states exist; one proposed by Dodonov
and Man'ko, where the lower limit on the uncertainty depends on the purity of the state, and another where
the uncertainty is bounded by the von Neumann entropy of the state proposed by Bastiaans. Driven by the
needs that have emerged in the field of quantum information, in a recent work we have extended the puritybounded
uncertainty relation by adding an additional parameter characterizing the state, namely its degree of
non-Gaussianity. In this work we alternatively present a extension of the entropy-bounded uncertainty relation.
The common points and differences between the two extensions of the uncertainty relation help us to draw more
general conclusions concerning the bounds on the non-Gaussianity of mixed states.
We propose a general method for introducing extensive characteristics of quantum entanglement. The method relies on polynomials of nilpotent raising operators, that create entangled states acting on a reference vacuum state. By introducing the notion of tanglemeter (the logarithm of the state vector represented in a special canonical form and expressed via polynomials of nilpotent variables), we show how this description provides a simple criterion for entanglement as well as a universal method for constructing the invariants characterizing entanglement.
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