Generalized conditional entropy in bipartite quantum systems

N. GIGENA; R. ROSSIGNOLI

Taller; IV Quantum Information School; 2013

We analyze, for a general concave entropic form, the associated conditionalentropy of a quantum system A+B, obtained as a result of a local measurement onone of the systems (B). This quantity is a measure of the average mixedness ofA after such measurement, and its minimum over all local measurements is shownto be the associated entanglement of formation between A and a purifying thirdsystem C. In the case of the von Neumann entropy, this minimum determines alsothe quantum discord. For classically correlated states and mixtures of a purestate with the maximally mixed state, we show that the minimizing measurementcan be determined analytically and is universal, i.e., the same for all concaveforms. While these properties no longer hold for general states, we also showthat in the special case of the linear entropy, an explicit expression for theassociated conditional entropy can be obtained, whose minimum among projectivemeasurements in a general qudit-qubit state can be determined analytically, interms of the largest eigenvalue of a simple 3x3 correlation matrix. Suchminimum determines the maximum conditional purity of A, and the associatedminimizing measurement is shown to be also universal in the vicinity of maximalmixedness. Results for X states, including typical reduced states of spin pairsin XY chains at weak and strong transverse fields, are also provided andindicate that the measurements minimizing the von Neumann and linearconditional entropies are typically coincident in these states, beingdetermined essentially by the main correlation. They can differ, however,substantially from that minimizing the geometric discord.