Range Criterion

	Range Criterion In quantum mechanics, in particular quantum information Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both th ..., the Range criterion is a necessary condition that a state must satisfy in order to be separable. In other words, it is a ''separability criterion''. The result Consider a quantum mechanical system composed of ''n'' subsystems. The state space ''H'' of such a system is the tensor product of those of the subsystems, i.e. H = H_1 \otimes \cdots \otimes H_n. For simplicity we will assume throughout that all relevant state spaces are finite-dimensional. The criterion reads as follows: If ρ is a separable mixed state acting on ''H'', then the range of ρ is spanned by a set of product vectors. Proof In general, if a matrix ''M'' is of the form M = \sum_i v_i v_i^, the rang ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
picture info	Quantum Mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limits to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Quantum Information Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both the technical definition in terms of Von Neumann entropy and the general computational term. It is an interdisciplinary field that involves quantum mechanics, computer science, information theory, philosophy and cryptography among other fields. Its study is also relevant to disciplines such as cognitive science, psychology and neuroscience. Its main focus is in extracting information from matter at the microscopic scale. Observation in science is one of the most important ways of acquiring information and measurement is required in order to quantify the observation, making this crucial to the scientific method. In quantum mechanics, due to the uncertainty principle, non-commuting observables cannot be precisely measured simultaneously, as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Separable States In quantum mechanics, separable states are quantum states belonging to a composite space that can be factored into individual states belonging to separate subspaces. A state is said to be entangled if it is not separable. In general, determining if a state is separable is not straightforward and the problem is classed as NP-hard. Separability of bipartite systems Consider first composite states with two degrees of freedom, referred to as ''bipartite states''. By a postulate of quantum mechanics these can be described as vectors in the tensor product space H_1\otimes H_2. In this discussion we will focus on the case of the Hilbert spaces H_1 and H_2 being finite-dimensional. Pure states Let \_^n\subset H_1 and \_^m \subset H_2 be orthonormal bases for H_1 and H_2, respectively. A basis for H_1 \otimes H_2 is then \, or in more compact notation \. From the very definition of the tensor product, any vector of norm 1, i.e. a pure state of the composite system, can be written ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]