A note on relative entropy

Satoshi Kawakami

Lajos Molnár

Let H be a finite dimensional Hilbert space. The aim of this short note is to prove that every bijective map on the space S(H) of all density operators on H which preserves the relative entropy is of the form ϕ(ρ)=UρU* with some unitary or antiunitary operator U on H.

Romàn R. Zapatrin

Generalizing the notion of relative entropy, the difference between a priori and a posteriori relative entropy for quantum systems is drawn. The former, known as quantum relative entropy, is associated with quantum states recognition. The latter -- a posteriori relative quantum entropy is shown to be related with state reconstruction due to the following property: given a density operator $ρ$, ensembles of pure states with Gibbs distribution with respect to the defined distance are proved to represent the initial state $ρ$ up to an amount of white noise (completely mixed state) which can be made arbitrary small.

Yuya O. Nakagawa, Gábor Sárosi, Tomonori Ugajin

Mark J. Balas, Vinod P. Gehlot, Tristan D. Griffith

An essential quantity in quantum information theory is the von Neumann entropy which depends entirely on the quantum density operator. Once known, the density operator reveals the statistics of observables in a quantum process, and the corresponding von Neumann entropy yields the full information content. However, the state, or density operator, of a given system may be unknown. Quantum state observers have been proposed to infer the unknown state of a quantum system. In this note, we show (i) that the von Neumann entropy of the state estimate produced by our quantum state observer is exponentially convergent to that of the system’s true state, and (ii) the relative entropy between the system and observer’s state converges exponentially to zero as long as the system starts in a full-rank state.

Alfred Wehrl