This paper presents the first experimental realization of a toolbox for simulating open quantum systems with up to five qubits using trapped ions. The researchers demonstrate coherent and dissipative operations to control the dynamics of open quantum systems, including the dissipative preparation of entangled states, simulation of coherent many-body spin interactions, and quantum non-demolition measurement of multi-qubit observables. The system is implemented using a quantum computing architecture with trapped ions, where multi-qubit gates are combined with optical pumping to achieve both coherent and dissipative processes. The study shows that by adding controlled dissipation to coherent operations, novel prospects for open-system quantum simulation and computation can be achieved. The paper discusses the fundamental challenges of controlling open quantum systems, which are inherently coupled to their environment. While progress has been made in isolating systems from their environment and coherently controlling qubit dynamics, engineering the dynamics of many particles through controlled coupling to an environment remains largely unexplored. The researchers demonstrate that by engineering the system-environment coupling, it is possible to simulate open quantum systems and control their dynamics. This includes the dissipative preparation of entangled states, the simulation of coherent many-body spin interactions, and the quantum non-demolition measurement of multi-qubit observables. The study also shows that by combining suitable coherent and dissipative time steps, it is possible to realize the most general nonunitary open-system evolution of a many-particle system. This engineering of the system-environment coupling generalizes the concept of Hamiltonian quantum simulation to open quantum systems. The researchers demonstrate all essential coherent and dissipative elements for controlling general open-system dynamics, including the dissipative preparation and manipulation of many-body states and quantum phases, as well as quantum computation based on dissipation. The paper presents an experimental demonstration of a complete toolbox for controlling the dynamics of open systems through coherent and dissipative manipulations of a multi-qubit system. In a string of trapped ions, each ion encoding a qubit, the qubits are subdivided into "system" and "environment". The system-environment coupling is then engineered through the universal set of quantum operations available in ion-trap quantum computers and a dissipative mechanism based on optical pumping. The researchers illustrate this engineering by dissipatively preparing a Bell state in a 2+1 ion system and a 4-qubit GHZ-state in a 4+1 ion system. They also demonstrate coherent n-body interactions by implementing the fundamental building block for 4-spin interactions and a readout of n-particle observables in a non-destructive way with a quantum-nondemolition (QND) measurement of a 4-qubit stabilizer operator. The paper also discusses the experimental implementation of Bell-state cooling, where the dynamics of an open quantum system S coupled to an environment E can be described by the unitary transformation ρ_SE → U ρ_SE U†, with ρ_SE the joint density matrix of the composite system SThis paper presents the first experimental realization of a toolbox for simulating open quantum systems with up to five qubits using trapped ions. The researchers demonstrate coherent and dissipative operations to control the dynamics of open quantum systems, including the dissipative preparation of entangled states, simulation of coherent many-body spin interactions, and quantum non-demolition measurement of multi-qubit observables. The system is implemented using a quantum computing architecture with trapped ions, where multi-qubit gates are combined with optical pumping to achieve both coherent and dissipative processes. The study shows that by adding controlled dissipation to coherent operations, novel prospects for open-system quantum simulation and computation can be achieved. The paper discusses the fundamental challenges of controlling open quantum systems, which are inherently coupled to their environment. While progress has been made in isolating systems from their environment and coherently controlling qubit dynamics, engineering the dynamics of many particles through controlled coupling to an environment remains largely unexplored. The researchers demonstrate that by engineering the system-environment coupling, it is possible to simulate open quantum systems and control their dynamics. This includes the dissipative preparation of entangled states, the simulation of coherent many-body spin interactions, and the quantum non-demolition measurement of multi-qubit observables. The study also shows that by combining suitable coherent and dissipative time steps, it is possible to realize the most general nonunitary open-system evolution of a many-particle system. This engineering of the system-environment coupling generalizes the concept of Hamiltonian quantum simulation to open quantum systems. The researchers demonstrate all essential coherent and dissipative elements for controlling general open-system dynamics, including the dissipative preparation and manipulation of many-body states and quantum phases, as well as quantum computation based on dissipation. The paper presents an experimental demonstration of a complete toolbox for controlling the dynamics of open systems through coherent and dissipative manipulations of a multi-qubit system. In a string of trapped ions, each ion encoding a qubit, the qubits are subdivided into "system" and "environment". The system-environment coupling is then engineered through the universal set of quantum operations available in ion-trap quantum computers and a dissipative mechanism based on optical pumping. The researchers illustrate this engineering by dissipatively preparing a Bell state in a 2+1 ion system and a 4-qubit GHZ-state in a 4+1 ion system. They also demonstrate coherent n-body interactions by implementing the fundamental building block for 4-spin interactions and a readout of n-particle observables in a non-destructive way with a quantum-nondemolition (QND) measurement of a 4-qubit stabilizer operator. The paper also discusses the experimental implementation of Bell-state cooling, where the dynamics of an open quantum system S coupled to an environment E can be described by the unitary transformation ρ_SE → U ρ_SE U†, with ρ_SE the joint density matrix of the composite system S