  When the von Neumann entropy of a quantum system is increased by
measurements, part of its information is lost. To faithfully reflect such a
gain-loss relation, we propose the Information Retrievability (IR) and
Information Loss (IL), which depend only on the density matrix of the system
before and after measurements. We explain that, after a pure quantum state
collapses to a maximal mixed state, it gets the maximal entropy together with
the maximal info loss, and its discrete uniform distribution contains only
classical info. Then we compute the entropy, IR and IL for systems of
single-qubit, entangled qubits (like Bell, GHZ, W state) and the 2-qubit Werner
(mixed) state with their dependence on various parameters. We notice that,
since the data-exchange between the two observers in Bell tests can recover
certain critical quantum info, the related quantum entropy should be removable
(a possible dilemma linked to quantum non-locality). We show that, measuring
the Bell, GHZ and the marginally entangled Werner state will produce the same
minimal entropy gain, accompanied by equal minimal info loss.
