Sample-Optimal Tomography of Quantum States

© 1963-2012 IEEE. It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error \epsilon in trace distance required O(dr^{2}/\epsilon ^{2}) copies for a d -dimensional density matrix of rank r. Here, we give a theoretical measurement scheme (POVM) that requires O (dr/ \delta) \ln ~(d/\delta) copies to estimate \rho to error \delta in infidelity, and a matching lower bound up to logarithmic factors. This implies O((dr / \epsilon ^{2}) \ln ~(d/\epsilon)) copies suffice to achieve error \epsilon in trace distance. We also prove that for independent (product) measurements, \Omega (dr^{2}/\delta ^{2}) / \ln (1/\delta) copies are necessary in order to achieve error \delta in infidelity. For fixed d , our measurement can be implemented on a quantum computer in time polynomial in n

Measuring processes and the Heisenberg picture

In this paper, we attempt to establish quantum measurement theory in the Heisenberg picture. First, we review foundations of quantum measurement theory, that is usually based on the Schr\"{o}dinger picture. The concept of instrument is introduced there. Next, we define the concept of system of measurement correlations and that of measuring process. The former is the exact counterpart of instrument in the (generalized) Heisenberg picture. In quantum mechanical systems, we then show a one-to-one correspondence between systems of measurement correlations and measuring processes up to complete equivalence. This is nothing but a unitary dilation theorem of systems of measurement correlations. Furthermore, from the viewpoint of the statistical approach to quantum measurement theory, we focus on the extendability of instruments to systems of measurement correlations. It is shown that all completely positive (CP) instruments are extended into systems of measurement correlations. Lastly, we study the approximate realizability of CP instruments by measuring processes within arbitrarily given error limits.Comment: v

Menunggu detik FFPVM6

Tinggal dua hari lagi Festival Filem dan Video Pelajar Malaysia Ke-6 (FFPVM6) bakal bermula. Sebagai panduan, berikut disenaraikan aktiviti-aktiviti menarik sepanjang festival tersebut