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Positive Operator-valued Measure
In functional analysis and quantum measurement theory, a positive operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum measurement described by PVMs (called projective measurements). In rough analogy, a POVM is to a PVM what a mixed state is to a pure state. Mixed states are needed to specify the state of a subsystem of a larger system (see purification of quantum state); analogously, POVMs are necessary to describe the effect on a subsystem of a projective measurement performed on a larger system. POVMs are the most general kind of measurement in quantum mechanics, and can also be used in quantum field theory. They are extensively used in the field of quantum information. Definition In the simplest case, of a POVM with a finite number of elements acting on a f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigma Algebra
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used as an operator for summation. When used at the end of a letter-case word (one that does not use all caps), the final form (ς) is used. In ' (Odysseus), for example, the two lowercase sigmas (σ) in the center of the name are distinct from the word-final sigma (ς) at the end. The Latin letter S derives from sigma while the Cyrillic letter Es derives from a lunate form of this letter. History The shape (Σς) and alphabetic position of sigma is derived from the Phoenician letter ( ''shin''). Sigma's original name may have been ''san'', but due to the complicated early history of the Greek epichoric alphabets, ''san'' came to be identified as a separate letter in the Greek alphabet, represented as Ϻ. Herodotus reports that "san" wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Operation
In quantum mechanics, a quantum operation (also known as quantum dynamical map or quantum process) is a mathematical formalism used to describe a broad class of transformations that a quantum mechanical system can undergo. This was first discussed as a general stochastic transformation for a density matrix by George Sudarshan. The quantum operation formalism describes not only unitary time evolution or symmetry transformations of isolated systems, but also the effects of measurement and transient interactions with an environment. In the context of quantum computation, a quantum operation is called a quantum channel. Note that some authors use the term "quantum operation" to refer specifically to completely positive (CP) and non-trace-increasing maps on the space of density matrices, and the term "quantum channel" to refer to the subset of those that are strictly trace-preserving. Quantum operations are formulated in terms of the density operator description of a quantum mechanica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Matrix
In 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''. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, without describing their combined state. Density matrices are thus crucial tools in areas of quantum mechanics that deal with mixed states, such as quantum statistical mechanics, open quantum systems, quantum decoherence, and quantum information. Definition and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Formulation Of Quantum Mechanics
The 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 ( ''L''2 space mainly), and operators on these spaces. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space. These formulations of quantum mechanics continue to be used today. At the heart of the description are ideas of ''quantum state'' and ''quantum observables'', which are radically different from those used in previous models of physical r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Measurement
In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis. Quantum physics has proven to be an empirical success and to have wide-ranging applicability. However, on a more philosophical level, debates continue about the meaning of the measurement concept. Mathematical formalism "Observables" as self-adjoint operators In quantum mechanics, each physical system is associated with a Hilbert space, each element of which represents a possible state of the physical system. The approach codified by John von Neumann represents a measurement upon a physical system by a self-adjoint operator on that Hilbert space termed an "observable". These observables play the role of measurab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIC-POVM
A symmetric, informationally complete, positive operator-valued measure (SIC-POVM) is a special case of a generalized measurement on a Hilbert space, used in the field of quantum mechanics. A measurement of the prescribed form satisfies certain defining qualities that makes it an interesting candidate for a "standard quantum measurement", utilized in the study of foundational quantum mechanics, most notably in QBism. Furthermore, it has been shown that applications exist in quantum state tomography and quantum cryptography, and a possible connection has been discovered with Hilbert's twelfth problem. Definition Due to the use of SIC-POVMs primarily in quantum mechanics, Dirac notation will be used throughout this article to represent elements in a Hilbert space. A POVM over a d-dimensional Hilbert space \mathcal is a set of m positive-semidefinite operators \left\_^m that sum to the identity: \sum_^m F_i = I. If a POVM consists of at least d^2 operators which span the space of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum State Discrimination
The term quantum state discrimination collectively refers to quantum-informatics techniques, with the help of which, by performing a small number of measurements on a physical system , its specific quantum state can be identified . And this is provided that the set of states in which the system can be is known in advance, and we only need to determine which one it is. This assumption distinguishes such techniques from quantum tomography, which does not impose additional requirements on the state of the system, but requires many times more measurements. If the set of states in which the investigated system can be is represented by orthogonal vectors , the situation is particularly simple. To unambiguously determine the state of the system, it is enough to perform a quantum measurement in the basis formed by these vectors. The given quantum state can then be flawlessly identified from the measured value. Moreover, it can be easily shown that if the individual states are not orthogona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Money
A quantum money scheme is a quantum cryptographic protocol that creates and verifies banknotes that are resistant to forgery. It is based on the principle that quantum states cannot be perfectly duplicated (the no-cloning theorem), making it impossible to forge quantum money by including quantum systems in its design. The concept was first proposed by Stephen Wiesner circa 1970 (though it remained unpublished until 1983), and later influenced the development of quantum key distribution protocols used in quantum cryptography. Wiesner's quantum money scheme Wiesner's quantum money scheme was first published in 1983. A formal proof of security, using techniques from semidefinite programming, was given in 2013. In addition to a unique serial number on each bank note (these notes are actually more like cheques, since a verification step with the bank is required for each transaction), there is a series of isolated two-state quantum systems.Lo, Spiller & Popescu, ''Introduction to Qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Coin Flipping
Consider two remote players, connected by a channel, that don't trust each other. The problem of them agreeing on a random bit by exchanging messages over this channel, without relying on any trusted third party, is called the coin flipping problem in cryptography. Quantum coin flipping uses the principles of quantum mechanics to encrypt messages for secure communication. It is a cryptographic primitive which can be used to construct more complex and useful cryptographic protocols, e.g. Quantum Byzantine agreement. Unlike other types of quantum cryptography (in particular, quantum key distribution), quantum coin flipping is a protocol used between two users who do not trust each other.Stuart Mason Dambort"Heads or tails: Experimental quantum coin flipping cryptography performs better than classical protocols" ''Phys.org'', March 26, 2014 Consequently, both users (or players) want to win the coin toss and will attempt to cheat in various ways. It is known that if the communication be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Cryptography
Quantum cryptography is the science of exploiting quantum mechanical properties to perform cryptographic tasks. The best known example of quantum cryptography is quantum key distribution which offers an information-theoretically secure solution to the key exchange problem. The advantage of quantum cryptography lies in the fact that it allows the completion of various cryptographic tasks that are proven or conjectured to be impossible using only classical (i.e. non-quantum) communication. For example, it is impossible to copy data encoded in a quantum state. If one attempts to read the encoded data, the quantum state will be changed due to wave function collapse (no-cloning theorem). This could be used to detect eavesdropping in quantum key distribution (QKD). History In the early 1970s, Stephen Wiesner, then at Columbia University in New York, introduced the concept of quantum conjugate coding. His seminal paper titled "Conjugate Coding" was rejected by the IEEE Information T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bloch Sphere Representation Of Optimal POVM And States For Unambiguous Quantum State Discrimination
Bloch is a surname of German origin. Notable people with this surname include: A–F * (1859-1914), French rabbi *Adele Bloch-Bauer (1881-1925), Austrian entrepreneur *Albert Bloch (1882–1961), American painter * (born 1972), German motor journalist and presenter * (1878–?), Russian lawyer, journalist, lawyer, and revolutionary *Alexandre Bloch (1857–1919), French painter *Alfred Bloch (born 1877), French footballer * (1915–1983), Swiss linguist * (1904–1979), German-British engineer * Aliza Bloch (born 1957), First female mayor of Bet Shemesh, Israel * (1768–1838), Swiss Benedictine monk *André Bloch (composer) (1873–1960), French composer and music educator *André Bloch (mathematician) (1893–1948), French mathematician * (1914–1942), French agent of the Special Operations Executive *Andreas Bloch (1860–1917), Norwegian painter, illustrator and costume designer *Andy Bloch (born 1969), American poker player *Anna Bloch (1868–1953), Danish actress *Armand Blo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
