Entropy Gain and Information Loss by Measurements

When the von Neumann entropy (VNE) of a system increases due to measurements, certain information is lost, some of which may be recoverable. We define information retrievability (IR) and information loss (IL) as functions of the density matrix through VNE to illustrate the relationship between gain and loss. We demonstrate that when a pure, unbiased m-qubit state collapses into a maximally mixed state, it experiences the maximal loss of information and the highest gain in entropy, equivalent to the m-bit classical Shannon entropy. We analyze the VNE, IR, and IL of single qubits, entangled photon pairs in Bell tests, three-qubit systems in quantum teleportation, multiple-qubit systems of GHZ and W states, and two-qubit Werner mixed states, emphasizing their IL dependence on parameters such as polarization bias and qubit count. Data exchange between two observers in Bell tests can recover some of the lost quantum information and eliminate the associated quantum entropy, even years later. The need to recover knowledge explains why no spooky action occurs at a distance. We show that measuring the Bell, GHZ, and marginally entangled Werner states yields the same minimum entropy gain (ln2) and equal minimal information loss (50 percent).