Quantum control theory and applications: A survey <s> Quantum states <s> An inexpensive faucet aerator is provided by three molded parts and a thin metal cup that holds two of the molded parts in assembled relationship. The first molded part is an elongated annulus provided with upstream and downstream recesses separated by an inner ring that helps to break up the liquid flow and serves as an abutment to support a second molded, jet-forming, part in the upstream recess and is arranged to be engaged by the third, molded, part located in the downstream recess and aiding in defining air intake means to the aerator. <s> BIB001 </s> Quantum control theory and applications: A survey <s> Quantum states <s> The key approaches for machine learning, particularly learning in unknown probabilistic environments, are new representations and computation mechanisms. In this paper, a novel quantum reinforcement learning (QRL) method is proposed by combining quantum theory and reinforcement learning (RL). Inspired by the state superposition principle and quantum parallelism, a framework of a value-updating algorithm is introduced. The state (action) in traditional RL is identified as the eigen state (eigen action) in QRL. The state (action) set can be represented with a quantum superposition state, and the eigen state (eigen action) can be obtained by randomly observing the simulated quantum state according to the collapse postulate of quantum measurement. The probability of the eigen action is determined by the probability amplitude, which is updated in parallel according to rewards. Some related characteristics of QRL such as convergence, optimality, and balancing between exploration and exploitation are also analyzed, which shows that this approach makes a good tradeoff between exploration and exploitation using the probability amplitude and can speedup learning through the quantum parallelism. To evaluate the performance and practicability of QRL, several simulated experiments are given, and the results demonstrate the effectiveness and superiority of the QRL algorithm for some complex problems. This paper is also an effective exploration on the application of quantum computation to artificial intelligence. <s> BIB002
In quantum mechanics, the state of a closed quantum system can be represented by a unit vector |ψ (this notation is known as the Dirac representation  and the state |ψ is also called wavefunction) in a complex Hilbert space H. It is worth mentioning that the "state" in quantum mechanics is different from that in classical systems. For a classical system, the state usually describes some real physical properties such as position and momentum, which are generally observable. However, a quantum state cannot be directly observed and also does not directly correspond to physical quantities of the quantum system. Since the global phase of a quantum state has no observable physical effect, we often say that the vectors |ψ and e iα |ψ (where i = √ −1 and α ∈ R) describe the same physical state. A quantum state which can be represented with a unit vector |ψ is called a pure state. For example, in quantum information theory, information is coded using two-state (two-level) quantum systems (called qubits) and the state |ψ of a qubit can be written as where θ ∈ [0, π], ϕ ∈ [0, 2π]. Then |0 and |1 correspond to the states zero and one for a classical bit BIB001 , BIB002 . In practical applications, the quantum systems to be controlled are usually not simple closed systems. They may be quantum ensembles or open quantum systems and their states cannot be written in the form of unit vectors. In this case, it is necessary to introduce the density operator or density matrix ρ : H → H to describe quantum states of quantum ensembles or open quantum systems. A density operator ρ is positive and has trace equal to one. Suppose that a quantum system is in an ensemble {p j , |ψ j } of pure states; i.e., in a mixture of a number of pure states |ψ j with respective probabilities p j . The density matrix for the system is defined as follows BIB001 : where ψ j | = (|ψ j ) † and j p j = 1. Here, the operation (·) † refers to the adjoint. For a pure state |ψ , ρ = |ψ ψ| and tr(ρ 2 ) = 1. If the state ρ of a quantum system satisfies tr(ρ 2 ) < 1, we call the quantum state a mixed state. A composite quantum system (assumed to be made up of two subsystems A and B) is defined on a Hilbert space H = H A ⊗ H B , which is the tensor product of the Hilbert spaces H A and H B on which the subsystems A and B are defined. For the composite quantum system, its state ρ AB can be described by the tensor product of the states of its subsystems; i.e., ρ AB = ρ A ⊗ ρ B (e.g., see BIB001 ). Considering any bipartite pure state |ψ AB , if it can be written as a tensor product of pure states |ϕ A ∈ H A and |χ B ∈ H B , we call it a separable state; otherwise, we call it an entangled state  . Quantum entanglement is a uniquely quantum mechanical phenomenon which plays a key role in many interesting applications of quantum communication and quantum computation, and a detailed discussion can be found in the references BIB001 and  .