Error Regions for Properties of The Quantum State
Abstract
This thesis mainly studies the method for constructing error intervals for properties of the quantum state. As a complement to point estimators for the quantum state estimation, region (interval for one dimension) estimators are proposed to supplement the error regions to the point estimator. These proposals, however, are ad hoc because they usually rely on having a lot of data, or consider all the possible data that haven't been observed. In [1], a method is provided for systematically constructing optimal error regions for quantum state estimation from the data actually observed. After identifying the prior probability as the size of a region, two types of optimal error regions--maximum-likelihood regions and smallest credible regions--are reported which are the bounded-likelihood regions that comprise all states with likelihood exceeding a threshold value. As a generalization of the above scenario for reporting optimal error regions for quantum state estimation, we propose a systematic method for constructing error intervals for a property of state directly from the experimental data. Usually, we are not interested in the full details of the quantum state, but rather care about some parameters or a few properties of the state. Moreover, it is much more difficult to estimate a high-dimensional quantum state. Therefore, a direct estimate of the properties of interest is more practical than the estimate of the whole quantum state. Analogous to error regions for quantum state estimation, the optimal error intervals are characterized by finding the constant likelihood values conditional on the property of state. For illustration, we identify the optimal error intervals for fidelity (with respect to certain target states) and purity of single-qubit states, as well as the CHSH quantity for two-qubit states. [1] J. Shang, H. K. Ng, A. Sehrawat, X. Li, and B.-G. Englert. Optimal error regions for quantum state estimation. New. J. Phys., 15:123026, 2013.
- Publication:
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Ph.D. Thesis
- Pub Date:
- 2016
- Bibcode:
- 2016PhDT........17X
- Keywords:
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- Quantum physics