Distortion operator and Entanglement Information Rate Distortion of Quantum Gaussian Source

Quantum random variable, distortion operator are introduced based on canonical operators. As the lower bound of rate distortion, the entanglement information rate distortion is achieved by Gaussian map for Gaussian source. General Gaussian maps are further reduced to unitary transformations and additive noises from the physical meaning of distortion. The entanglement information rate distortion function then are calculated for one mode Gaussian source. The rate distortion is accessible at zero distortion point. For pure state, the rate distortion function is always zero. In contrast to the distortion defined via fidelity, our definition of the distortion makes it possible to calculate the entanglement information rate distortion function for Gaussian source. Distortion operator: Two major parts in classical information theory are channel capacity and rate distortion theory. They concern respectively with the reliability and effectiveness of information transmission. In quantum information theory, channel capacity has been widely investigated, but little effort has been put into developing quantum rate-distortion theory[1][2]. It was proven [1] that the quantum rate-distortion function R(D) is lower bounded by entanglement information rate-distortion function R(D). For a given source R(D) is defined by R(D) = min E|d(E)≤D Ic(ρ, E). (1) where d is some distortion function, and E is the channel. The result is proven under the assumption of distortion function defined by transmission fidelity. It is linear among different modes. We can extend the distortion function to a more general form. The result will also be true if it is linear among different modes. One of the useful distortion function is mean square function as used in classical information theory for Gaussian source. The mean square distortion in classical theory is d = E(d(Y, Y )) = ∫ p(y, y)d(y, y)dydy, with d(y, y) = (y − y). Where Y is the input random variable and Y ′ is the output, p(y, y) is the joint density distribution function. The same idea should be extended to quantum information theory. What is the quantum corresponding of random variable? We prefer the canonical operatorsX and P . Then the distortion operator will be introduced as