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Lieb Conjecture
In quantum information theory, the Lieb conjecture is a theorem concerning the Wehrl entropy of quantum systems for which the classical phase space is a sphere. It states that no state of such a system has a lower Wehrl entropy than the SU(2) coherent states. The analogous property for quantum systems for which the classical phase space is a ''plane'' was conjectured by Alfred Wehrl in 1978 and proven soon afterwards by Elliott H. Lieb, who at the same time extended it to the SU(2) case. The conjecture was only proven in 2012, by Lieb and Jan Philip Solovej Jan Philip Solovej (born 14 June 1961) is a Danish mathematician and mathematical physicist working on the mathematical theory of quantum mechanics. He is a professor at University of Copenhagen. Biography Solovej obtained his Ph.D. in 198 .... References External links Video of a lecture by Lieb discussing the conjecture and outlining its proof. Quantum mechanical entropy Conjectures that have been proved ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Wehrl Entropy
In quantum information theory, the Wehrl entropy, named after Alfred Wehrl, is a classical entropy of a quantum-mechanical density matrix. It is a type of quasi-entropy defined for the Husimi Q representation of the phase-space quasiprobability distribution. See for a comprehensive review of basic properties of classical, quantum and Wehrl entropies, and their implications in statistical mechanics. Definitions The Husimi function is a " classical phase-space" function of position and momentum , and in one dimension is defined for any quantum-mechanical density matrix by :Q_\rho(x,p)=\int \phi(x,p , y)^* \rho (y, y')\phi (x,p, y')dy dy', where is a " (Glauber) coherent state", given by :\phi(x,p, y)=\pi^\exp(-, y-x, ^2/2)+i\, px). (It can be understood as the Weierstrass transform of the Wigner quasi-probability distribution.) The Wehrl entropy is then defined as : S_W(\rho) = -\int Q_\rho(x,p) \log Q_\rho(x,p) \, dx \, dp ~. The definition can be easily generalized to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. It is the outer product of direct space and reciprocal space. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs. Introduction In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane. For every possible state of the system or allowed combination of values of the system's parameters, a point is included in the multidimensional space. The system's evolving state over time traces a path (a phase-space trajectory for the system) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent States In Mathematical Physics
Coherent states have been introduced in a physical context, first as quasi-classical states in quantum mechanics, then as the backbone of quantum optics and they are described in that spirit in the article Coherent states (see alsoJ-P. Gazeau,''Coherent States in Quantum Physics'', Wiley-VCH, Berlin, 2009.). However, they have generated a huge variety of generalizations, which have led to a tremendous amount of literature in mathematical physics. In this article, we sketch the main directions of research on this line. For further details, we refer to several existing surveys.S.T. Ali, J-P. Antoine, J-P. Gazeau, and U.A. Mueller, Coherent states and their generalizations: A mathematical overview, ''Reviews in Mathematical Physics'' 7 (1995) 1013-1104.S.T. Ali, J-P. Antoine, and J-P. Gazeau, ''Coherent States, Wavelets and Their Generalizations'', Springer-Verlag, New York, Berlin, Heidelberg, 2000. A general definition Let \mathfrak H\, be a complex, separable Hilbert space, X a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jan Philip Solovej
Jan Philip Solovej (born 14 June 1961) is a Danish mathematician and mathematical physicist working on the mathematical theory of quantum mechanics. He is a professor at University of Copenhagen. Biography Solovej obtained his Ph.D. in 1989 from Princeton University with the thesis on "Universality in the Thomas-Fermi-von Weizsäcker Model of Atoms and Molecules" supervised by Elliott H. Lieb. As a post-doctoral researcher, he went to the University of Michigan in 1989/90 and to the University of Toronto in 1990. In 1991 (and 2003/04) he was a member at the Institute for Advanced Study. From 1991 to 1995, he was Assistant Professor in the Department of Mathematics at Princeton University. From 1995 to 1997, he was a research professor at the University of Aarhus. Since 1997, he has been a professor in the Department of Mathematics at the University of Copenhagen. Since 2016, he has been the Centre leader of VILLUM Centre of Excellence for the Mathematics of Quantum Theory ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanical Entropy
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantization". This means that the magnitude of the physical property can take on only discrete values consisting of integer multiples of one quantum. For example, a photon is a single quantum of light (or of any other form of electromagnetic radiation). Similarly, the energy of an electron bound within an atom is quantized and can exist only in certain discrete values. (Atoms and matter in general are stable because electrons can exist only at discrete energy levels within an atom.) Quantization is one of the foundations of the much broader physics of quantum mechanics. Quantization of energy and its influence on how energy and matter interact (quantum electrodynamics) is part of the fundamental framework for understanding and describing nature. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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