Quantum Limit of Measurement and the Projection Postulate

We study an electrostatic qubit monitored by a point-contact detector. Projecting an entire qubit-detector wave function on the detector eigenstates we determine the precision limit for the qubit measurements, allowed by quantum mechanics. We found that this quantity is determined by qubit dynamics as well as decoherence, generated by the measurement. Our results show how the quantum precision limit can be improved by a proper design of a measurement procedure. Rapid experimental progress in monitoring of single quantum systems renewed the interest in the quantum mechanical limitation of measurement accuracy 1. This limitation is originated from the uncertainty relation between observables of a measured microscopic system, the so-called standard quantum limit of measurement 2. Yet, a single quantum system is not observed directly, but through the interaction with a measurement device (detector). This implies that the quantum limit of measurement is not related to a measured system only, but rather to an entire system, including detector 2. A measurement in the quantum mechanical formalism corresponds to projection of the wave function of an entire system on eigenstates of the detector, accessible by an observer. This is the so-called projection postulate 3 , analogues to the Bayes principle in any probabilistic description. Since the detector and the measured system are interacting, the above projection on the detector states affects the measured microscopic system. If the latter is projected on one of its own states, then the microscopic system can be measured with any accuracy. However, if this system is projected on the superposition of its states, the measurement cannot be precise. Its accuracy is given by a size of the corresponding wave packet. In this Letter we apply the above described procedure for a determination of the precision limit of quantum measurements. As an example we take an electron trapped inside a double-dot (electrostatic qubit) and continuous monitored by a point-contact detector 4. The total system can be treated entirely quantum mechanically , although the detector represents a macroscopic (mesoscopic) device 5. This allows us to obtain the detector eigenstates and then to perform projections of the total wave function on these eigenstates without any additional assumptions.