Monotonicity of Quantum Relative Entropy

Quantum finance is freshly developing, for example, to model interest rates and price options, see, for example. In order to explore quantum portfolio and analyze its risk, this work will develop quantum relative entropy in theory of information for quantitative analysis in banks and finance firms, and for practitioners for fixed income securities and foreign exchange in decision making. One quantum relative entropy is introduced without Radon-Nikodym derivative, therefore much simpler to compute than Araki-Umegaki type, Belavkin-Staszewski type, and Hammersley-Belavkin type, in particular, its monotonicity is obtained.