This paper presents a method for experimentally determining the evolution operators that describe the dynamics of a quantum black box, an arbitrary open quantum system. The authors show that it is possible to determine the quantum transfer function of such a system by using a matrix of complex numbers, χ, which fully describes the system's behavior. They illustrate this approach with examples involving one and two quantum bits (qubits), and discuss how the resulting description can be used to evaluate the performance of quantum systems. The paper begins by introducing the concept of quantum operations, which are linear maps that describe the dynamics of a quantum system. These operations can be represented using an operator-sum form, which allows for the description of various quantum processes, including unitary operations, measurements, and environmental effects. The authors then describe a general experimental procedure for determining the operators that define these quantum operations. This involves preparing a set of pure quantum states, measuring the output states, and using quantum state tomography to reconstruct the quantum operation. The procedure is further detailed for the specific case of a single qubit, where the quantum operation is described by a 12-parameter matrix χ. This matrix includes parameters that describe both unitary transformations and environmental correlations. The authors show how these parameters can be measured using a set of experiments and how the resulting matrix can be used to recover the operator-sum representation of the quantum operation. For two qubits, the authors demonstrate that the parameters describing the quantum operation can be expressed in terms of a matrix of measured density matrices. They also discuss related quantities such as entanglement fidelity and quantum channel capacity, which can be used to evaluate the performance of quantum systems. The paper concludes by noting that the proposed method can be used to determine the form of Lindblad operators used in Markovian master equations, which describe the time evolution of open quantum systems. The authors also mention that the method can be adapted to evaluate quantum operations describing measurements, such as quantum-nondemolition measurements. The paper emphasizes the importance of this method for the experimental study of quantum computation, quantum error correction, quantum cryptography, and other quantum information processing tasks.This paper presents a method for experimentally determining the evolution operators that describe the dynamics of a quantum black box, an arbitrary open quantum system. The authors show that it is possible to determine the quantum transfer function of such a system by using a matrix of complex numbers, χ, which fully describes the system's behavior. They illustrate this approach with examples involving one and two quantum bits (qubits), and discuss how the resulting description can be used to evaluate the performance of quantum systems. The paper begins by introducing the concept of quantum operations, which are linear maps that describe the dynamics of a quantum system. These operations can be represented using an operator-sum form, which allows for the description of various quantum processes, including unitary operations, measurements, and environmental effects. The authors then describe a general experimental procedure for determining the operators that define these quantum operations. This involves preparing a set of pure quantum states, measuring the output states, and using quantum state tomography to reconstruct the quantum operation. The procedure is further detailed for the specific case of a single qubit, where the quantum operation is described by a 12-parameter matrix χ. This matrix includes parameters that describe both unitary transformations and environmental correlations. The authors show how these parameters can be measured using a set of experiments and how the resulting matrix can be used to recover the operator-sum representation of the quantum operation. For two qubits, the authors demonstrate that the parameters describing the quantum operation can be expressed in terms of a matrix of measured density matrices. They also discuss related quantities such as entanglement fidelity and quantum channel capacity, which can be used to evaluate the performance of quantum systems. The paper concludes by noting that the proposed method can be used to determine the form of Lindblad operators used in Markovian master equations, which describe the time evolution of open quantum systems. The authors also mention that the method can be adapted to evaluate quantum operations describing measurements, such as quantum-nondemolition measurements. The paper emphasizes the importance of this method for the experimental study of quantum computation, quantum error correction, quantum cryptography, and other quantum information processing tasks.