K. Li

The requirement for Hermiticity in quantum mechanics ensures the reality of energies, while the parity-time symmetry offers an alternative route to achieve this goal. Interestingly, in a three-level system, the parity-time symmetry-breaking can lead to a third-order exceptional point with distinctive topological properties and enhanced sensitivity. To experimentally implement this in open quantum systems, it is essential to introduce two well-controlled loss channels. However, the requirement for these two loss channels presents a challenge in experimental implementation due to the lack of methods to realize the dynamics governed by an effective non-Hermitian Hamiltonian. Here we address the challenge by employing two approaches to eliminate the effects of quantum jump terms so that the dynamics is governed by an effective non-Hermitian Hamiltonian in a dissipative trapped ion with two loss channels. Based on this, we experimentally observe the parity-time symmetry-breaking-induced third-order exceptional point through non-Hermitian absorption spectroscopy. In particular, we perform quantum state tomography to directly demonstrate the coalescence of three eigenstates into a single eigenstate at the exceptional point. Finally, we identify an intrinsic third order Liouvillian exceptional point associated with a parity-time symmetry breaking via quench dynamics. Our experiments can be extended to observe other non-Hermitian phenomena involving more than two levels and potentially find applications in quantum information technology.

Due to the exponential complexity of the resources required by quantum state tomography (QST), people are interested in approaches towards identifying quantum states which require less effort and time. In this paper, we provide a tailored and efficient method for reconstructing mixed quantum states up to $12$ (or even more) qubits from an incomplete set of observables subject to noises. Our method is applicable to any pure or nearly pure state $ρ$, and can be extended to many states of interest in quantum information processing, such as multi-particle entangled $W$ state, GHZ state and cluster states that are matrix product operators of low dimensions. The method applies the quantum density matrix constraints to a quantum compressive sensing optimization problem, and exploits a modified Quantum Alternating Direction Multiplier Method (Quantum-ADMM) to accelerate the convergence. Our algorithm takes $8,35$ and $226$ seconds respectively to reconstruct superposition state density matrices of $10,11,12$ qubits with acceptable fidelity, using less than $1 \%$ of measurements of expectation. To our knowledge it is the fastest realization that people can achieve using a normal desktop. We further discuss applications of this method using experimental data of mixed states obtained in an ion trap experiment of up to $8$ qubits.