For Hermitian and non-Hermitian Hamiltonian matrices H, we present the Schr¨odinger equation for qudit (spin-j system, N-level atom) with the state vector |ψ〉 in a new form of the linear eigenvalue equation for the matrix ℋ = (H ⊗ 1_N) and the probability eigenvector |p〉 identified with quantum states in the probability representation of quantum mechanics. We discuss the possibility to experimentally detect the difference between the system states described by the solutions, corresponding to the Schrödinger equation with Hermitian and non-Hermitian Hamiltonians, by measuring the probabilities of artificial spin-1/2 projections m = ± 1/2, sets of which are identified with qudit states. We show that different symmetries of systems, including 𝒫𝒯 -symmetry and broken 𝒫𝒯 -symmetry, are determined by a set of N complex eigenvalues of the Hamiltonian matrix H.

对于Hermitian和非Hermitian哈密顿矩阵H，我们用状态向量|给出了Qudit（自旋j系统，N级原子）的Schr？odinger方程。ψ >线性特征方程的用于矩阵的新形式ℋ =（ħ ⊗1 _Ñ）和概率特征向量| p〉在量子力学的概率表示中用量子态标识。我们讨论了通过测量人工spin-1 /的概率来实验检测由解表示的系统状态之间的差异的可能性，该状态对应于具有Hermitian和非Hermitian哈密顿量的Schrödinger方程。2个投影m =±1 / 2，其中的每个集合均以qudit状态标识。我们表明，系统的不同对称性，包括𝒫𝒯-对称性和破碎𝒫𝒯-对称性，是由哈密​​顿矩阵H的一组N个复特征值确定的。