On PSI-complete and PSIR-complete measurements

Minimal Informationally Complete Measurements for Pure States

We consider measurements, described by a positive-operator-valued measure (POVM), whose outcome probabilities determine an arbitrary pure state of a D-dimensional quantum system. We call such a measurement a pure-state informationally complete (PSI-complete) POVM. We show that a measurement with 2D-1 outcomes cannot be PSI-complete, and then we construct a POVM with 2D outcomes that suffices, thus showing that a minimal PSI-complete POVM has 2D outcomes. We also consider PSI-complete POVMs that have only rank-one POVM elements and construct an example with 3D-2 outcomes, which is a generalizat…

The asymmetry properties of pure quantum states

The asymmetry properties of a state relative to some symmetry group specify how and to what extent the given symmetry is broken by the state. Characterizing these is found to be surprisingly useful for addressing a very common problem: to determine what follows from a system's dynamics (possibly open) having that symmetry. We demonstrate and exploit the fact that the asymmetry properties of a state can be understood in terms of information-theoretic concepts. We show that for a pure state psi and a symmetry group G, they are completely specified by the characteristic function of the state, def…

使用最小秩一POVM的纯态量子层析

Pure-State Quantum Tomography with Minimal Rank-One POVMs

本文研究纯态量子层析中最小秩一POVM的vital性，证明n维空间中其大小在实数域和复数域的严格上界并给出构造，还揭示实数情形下与区组设计的联系及显式构造。

The theory of manipulations of pure state asymmetry: basic tools and equivalence classes of states under symmetric operations

If a system undergoes symmetric dynamics, then the final state of the system can only break the symmetry in ways in which it was broken by the initial state, and its measure of asymmetry can be no greater than that of the initial state. It follows that for the purpose of understanding the consequences of symmetries of dynamics, in particular, complicated and open-system dynamics, it is useful to introduce the notion of a state's asymmetry properties, which includes the type and measure of its asymmetry. We demonstrate and exploit the fact that the asymmetry properties of a state can also be un…

Root Estimator of Quantum States

The root estimator of quantum states based on the expansion of the psi function in terms of system eigenfunctions followed by estimating the expansion coefficients by the maximum likelihood method is considered. In order to provide statistical completeness of the analysis, it is necessary to perform measurements in mutually complementing experiments (according to the Bohr terminology). Estimation of quantum states by the results of coordinate, momentum, and polarization (spin) measurements is considered.