Projective Measurement In Quantum Information Theory

Quantum information science is the study of the information processing tasks that can be accomplished using quantum mechanical systems. It combines two of the great sciences of the twentieth century:the quantum theory and information theory and has been developed quickly in the past two decades. The quantum Shannon theory is one the most important reserch directions in quantum information science. It studies the fundmental question:what is possible and what is impossible in quantum information processing. Therefore, the quantum Shannon theory playes an essential role in quantum inforamtion science.As we all know, in order to acquire information from a quantum system, we must perform measurement on it. The projective measurement is the most important case of the quantum measurements. The main difference between classical and quantum measurements lies that the quantum measurement might disturb the initial state of the system. In other word, the quantum measurement may destroy the information. This is due to the non-commutation of the quantum mechanics. However, by coupling the measuring device to the system weakly, it is possible to read out certain information while limiting the disturbance to the system. This lead to two interesting questions. Firstly, in some cases it is necessary to perform a sequence of weak measurements to acquire the desired information. Although a single measurement does not disturb the system very much, the disturbace could potentially accumulate gradually when the measurements are performed in sequential fasion. So a natural question for sequential measurments is that:how the disturbance accumulates, or say, how many measurements can be performed until the final state is no longer close to the initial state. Moreover, performing a large number of measurements results in a variety of possible sequence. Then how can we estimate the probability of occurrence of the resulting sequences? Secondly, the projective measurement onto the typical subspace is an important special case of the weak measurement which has been widely used in quantum Shannon theory. However, it is not clear that how to understand the typical subspace theory from the view point of geometric. So if one can provide a geometric interpresion for the typical subspace, it would be very helpful for understanding the quantum Shannon theory. In this dissertation, we mainly investigates the two problems and the main results are summarized as follows1. The trigonometric representation of the projective measurment is presented. It is shown that any state can be trigonometric decomposed by a binary projective measurement. The distance between the states and the probability of the occurrence of a special result can be represented by the trigonometric function. This implies that the Bloch sphere accturally corresponds to a binary projective measurement.2. Using the trigonometric representation of the projective measurment, a specifical analysis of the sequential projective measurements is proposed. An upper bound for the distance between the initial state and the post-measured state after a sequence of measurements is obtained. The lower bound for the probability of the occurrence of a special result is also presented. The two bounds answer the first question proposed above and would could be very powerful tools for quantum Shannon theory.3. Based on the bounds of the sequential projective measurements, a new proof of how to achieve the Holevo bound is presented. It is shown that, the receiver can perform a sequence of measurement to indentify the codeword step by step. The error probability of this decoding strategy would approach 0 when the rate is lower than Holevo quantity.4. A generalization from sequetial projective measurement to sequential general measurement is presented.5. A geometric interpretation of quantum typical subspace is proposed. It is shown that if two states can be associated by projective measurements, then their subspaces have an approximate inclusive relation. This follows that the typical subspace corresponds to a plane in the Bloch sphere.6. Based on this geometric interpretation, a method of estimating of the eigenbasis of unknown density operator is proposed. It is shown that, the eigenbasis can be determined by two planes in the Bloch sphere and the planes can be found via a sequence of projective measurements.7. The entropy-typical subspace theory is specified. It is shown that the subspace with a smaller entropy can be approximate included by the subspace with a bigger entropy. This implies an universal data compression code.