Projection-valued measureIn mathematics, particularly in functional analysis, a projection-valued measure (PVM) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. Projection-valued measures are formally similar to real-valued measures, except that their values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space.
Quantum stateIn quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a quantum mechanical prediction for the system represented by the state. Knowledge of the quantum state together with the quantum mechanical rules for the system's evolution in time exhausts all that can be known about a quantum system. Quantum states may be defined in different ways for different kinds of systems or problems.
Hidden-variable theoryIn physics, hidden-variable theories are proposals to provide explanations of quantum mechanical phenomena through the introduction of (possibly unobservable) hypothetical entities. The existence of fundamental indeterminacy for some measurements is assumed as part of the mathematical formulation of quantum mechanics; moreover, bounds for indeterminacy can be expressed in a quantitative form by the Heisenberg uncertainty principle.
Measurement in quantum mechanicsIn quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. A fundamental feature of quantum theory is that the predictions it makes are probabilistic. The procedure for finding a probability involves combining a quantum state, which mathematically describes a quantum system, with a mathematical representation of the measurement to be performed on that system. The formula for this calculation is known as the Born rule.
Probability amplitudeIn quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density. Probability amplitudes provide a relationship between the quantum state vector of a system and the results of observations of that system, a link was first proposed by Max Born, in 1926. Interpretation of values of a wave function as the probability amplitude is a pillar of the Copenhagen interpretation of quantum mechanics.
Bell's theoremBell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories, given some basic assumptions about the nature of measurement. "Local" here refers to the principle of locality, the idea that a particle can only be influenced by its immediate surroundings, and that interactions mediated by physical fields cannot propagate faster than the speed of light.
Density matrixIn quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed states.
Quantum channelIn quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information is a text document transmitted over the Internet. More formally, quantum channels are completely positive (CP) trace-preserving maps between spaces of operators. In other words, a quantum channel is just a quantum operation viewed not merely as the reduced dynamics of a system but as a pipeline intended to carry quantum information.
Unitarity (physics)In quantum physics, unitarity is (or a unitary process has) the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. This is typically taken as an axiom or basic postulate of quantum mechanics, while generalizations of or departures from unitarity are part of speculations about theories that may go beyond quantum mechanics. A unitarity bound is any inequality that follows from the unitarity of the evolution operator, i.
Mathematical formulation of quantum mechanicsThe mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces (L2 space mainly), and operators on these spaces.
Interpretations of quantum mechanicsAn interpretation of quantum mechanics is an attempt to explain how the mathematical theory of quantum mechanics might correspond to experienced reality. Although quantum mechanics has held up to rigorous and extremely precise tests in an extraordinarily broad range of experiments, there exist a number of contending schools of thought over their interpretation. These views on interpretation differ on such fundamental questions as whether quantum mechanics is deterministic or stochastic, local or non-local, which elements of quantum mechanics can be considered real, and what the nature of measurement is, among other matters.
No-communication theoremIn physics, the no-communication theorem or no-signaling principle is a no-go theorem from quantum information theory which states that, during measurement of an entangled quantum state, it is not possible for one observer, by making a measurement of a subsystem of the total state, to communicate information to another observer. The theorem is important because, in quantum mechanics, quantum entanglement is an effect by which certain widely separated events can be correlated in ways that, at first glance, suggest the possibility of communication faster-than-light.
Gleason's theoremIn mathematical physics, Gleason's theorem shows that the rule one uses to calculate probabilities in quantum physics, the Born rule, can be derived from the usual mathematical representation of measurements in quantum physics together with the assumption of non-contextuality. Andrew M. Gleason first proved the theorem in 1957, answering a question posed by George W. Mackey, an accomplishment that was historically significant for the role it played in showing that wide classes of hidden-variable theories are inconsistent with quantum physics.
Max BornMax Born (ˈmaks ˈbɔʁn; 11 December 1882 – 5 January 1970) was a German-British physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 1930s. Born was awarded the 1954 Nobel Prize in Physics for his "fundamental research in quantum mechanics, especially in the statistical interpretation of the wave function".
John von NeumannJohn von Neumann (vɒn_ˈnɔɪmən ; Neumann János Lajos ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest coverage of any mathematician of his time and was said to have been "the last representative of the great mathematicians who were equally at home in both pure and applied mathematics". He integrated pure and applied sciences.
Measurement problemIn quantum mechanics, the measurement problem is the problem of how, or whether, wave function collapse occurs. The inability to observe such a collapse directly has given rise to different interpretations of quantum mechanics and poses a key set of questions that each interpretation must answer. The wave function in quantum mechanics evolves deterministically according to the Schrödinger equation as a linear superposition of different states. However, actual measurements always find the physical system in a definite state.
Quantum logicIn the mathematical study of logic and the physical analysis of quantum foundations, quantum logic is a set of rules for manipulation of propositions inspired by the structure of quantum theory. The formal system takes as its starting point an observation of Garrett Birkhoff and John von Neumann, that the structure of experimental tests in classical mechanics forms a Boolean algebra, but the structure of experimental tests in quantum mechanics forms a much more complicated structure.
ObservableIn physics, an observable is a physical property or physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum physics, it is an operator, or gauge, where the property of the quantum state can be determined by some sequence of operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value.
Wave functionIn quantum physics, a wave function (or wavefunction), represented by the Greek letter Ψ, is a mathematical description of the quantum state of an isolated quantum system. In the Copenhagen interpretation of quantum mechanics, the wave function is a complex-valued probability amplitude; the probabilities for the possible results of the measurements made on a measured system can be derived from the wave function. The most common symbols for a wave function are the Greek letters ψ and Ψ (lower-case and capital psi, respectively).
