Necessary condition for existence of conditional SIC-POVM (Mathematics for Uncertainty and Fuzziness)

Abstract

This paper is an announcement of our result and the detailed version will be submitted to somewhere.(See [2].)POVMs (positive operator valued measure) on quantum systems are considered as a measurement in quantum physics.If POVMs have a good condition, then we can determine a state by results of a measurement.In this case, the POVM is called informationally complete and the process is called quantum state tomography.First, we introduce SIC-POVMs (symmetric informationally complete POVM).SIC-POVMs is generated by vectors in $\mathbb{C}^{n}$ whose the absolute values of inner products of each vectors are same.Zauner conjectured that there exist such $n^{2}$ vectors in $\mathbb{C}^{n}$ for any $n$ .But the existence is only proved when $n\leq 15$ and $n=19$ , 24, 35,48[5, 8].Next, we introduce conditional SIC-POVMs.A state of a quantum system is a density matrix which has several parameters.When a few parameters are known, then SIC-POVM is not the best measurement to determine the state.Hence we need another POVM and it is a conditional SIC-POVM.Conditional SIC-POVMs are also generated by vectors in $\mathbb{C}^{n}$ .But the existence of conditional SIC-POVMs depends on the system.We will discuss the details in Sect. 4. 2 Preliminaries Definition 2.1 $\rho\in M_{n}(\mathbb{C})$ is called a density matrix (or state) if $\rho\geq 0$ and $\prime b(\rho)=1.$For any density matrix $\rho$ , we can define a state $\hat{\rho}$ by $\hat{\rho}(X)=h(\rho X)$ .Conversely, any state is written by the above form.Therefore, there exists a one-t $(\succ$ one correspondence between density matrices and states.Definition 2.2 A set of positive operators $\{P_{i}\}_{i=1}^{k}\subset M_{n}(\mathbb{C})$ is called a positive operator valued measure (POVM) if $P_{i}\geq 0(1\leq i\leq k)$ and $\sum_{i=1}^{k}P=I.$If $P_{1}$ is a projection, then $\{P_{i}\}_{i=1}^{k}$ is called a projection valued measure $(PVM)$ .