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Choi–Jamiołkowski Isomorphism
In quantum information theory and operator theory, the Choi–Jamiołkowski isomorphism refers to the correspondence between quantum channels (described by completely positive maps) and quantum states (described by density matrices), this is introduced by Man-Duen Choi and Andrzej Jamiołkowski. It is also called channel-state duality by some authors in the quantum information area, but mathematically, this is a more general correspondence between positive operators and the complete positive superoperators. Definition To study a quantum channel \mathcal from system S to S', which is a trace-preserving completely positive map from operator spaces \mathcal(\mathcal_S) to \mathcal(\mathcal_), we introduce an auxiliary system A with the same dimension as system S. Consider the maximally entangled state Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each part ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Information Theory
Quantum information is the information of the quantum state, 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 Observable, observables cannot be precisely mea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Channel
In 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. (Some authors use the term "quantum operation" to also include trace-decreasing maps while reserving "quantum channel" for strictly trace-preserving maps.) Memoryless quantum channel We will assume for the moment that all state spaces of the systems considered, classical or quantum, are finite-dimensional. The memoryless in the section title carries the same meaning as in classical information theory: the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completely Positive Map
In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition. Definition Let A and B be C*-algebras. A linear map \phi: A\to B is called positive map if \phi maps positive elements to positive elements: a\geq 0 \implies \phi(a)\geq 0. Any linear map \phi:A\to B induces another map :\textrm \otimes \phi : \mathbb^ \otimes A \to \mathbb^ \otimes B in a natural way. If \mathbb^\otimes A is identified with the C*-algebra A^ of k\times k-matrices with entries in A, then \textrm\otimes\phi acts as : \begin a_ & \cdots & a_ \\ \vdots & \ddots & \vdots \\ a_ & \cdots & a_ \end \mapsto \begin \phi(a_) & \cdots & \phi(a_) \\ \vdots & \ddots & \vdots \\ \phi(a_) & \cdots & \phi(a_) \end. We say that \phi is k-positive if \textrm_ \otimes \phi is a positive map, and \phi is called completely positive if \phi is k-positive for all k. Properties * Positi ... [...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] |
Andrzej Jamiołkowski
Andrzej Edmund Jamiołkowski (born March 22, 1946 in Toruń) is a Polish theoretical physicist, rector of the Nicolaus Copernicus University in Toruń (in the years 1993–1999 and 2005–2008). He is known for the Choi–Jamiołkowski isomorphism in quantum information theory. Jamiołkowski graduated from the Faculty of Physics at the University of Warsaw in 1969. After graduation, he began research work at the Nicolaus Copernicus University in Toruń, where in 1973 he obtained a doctoral degree, and in 1982 a postdoctoral degree. In 1990, he was appointed Professor. In the 1960s, he spent over four years on foreign scholarships, including in Munich, Oldenburg and Marburg. Jamiołkowski held many important functions at the Nicolaus Copernicus University: in the years 1985–1986 he was the deputy director of the Institute of Physics, and then in the years 1986–1993 the Vice-Rector of the Nicolaus Copernicus University for science and cooperation with foreign countries. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Channel-state Duality
In quantum information theory, the channel-state duality refers to the correspondence between quantum channels and quantum states (described by density matrices). Phrased differently, the duality is the isomorphism between completely positive maps (channels) from ''A'' to C''n''×''n'', where ''A'' is a C*-algebra and C''n''×''n'' denotes the ''n''×''n'' complex entries, and positive linear functionals (states) on the tensor product :\mathbb^ \otimes A. Details Let ''H''1 and ''H''2 be (finite-dimensional) Hilbert spaces. The family of linear operators acting on ''Hi'' will be denoted by ''L''(''Hi''). Consider two quantum systems, indexed by 1 and 2, whose states are density matrices in ''L''(''Hi'') respectively. A quantum channel, in the Schrödinger picture, is a completely positive (CP for short), trace-preserving linear map :\Phi : L(H_1) \rightarrow L(H_2) that takes a state of system 1 to a state of system 2. Next, we describe the dual state correspond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximally Entangled State
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics. Measurements of physical properties such as position, momentum, spin, and polarization performed on entangled particles can, in some cases, be found to be perfectly correlated. For example, if a pair of entangled particles is generated such that their total spin is known to be zero, and one particle is found to have clockwise spin on a first axis, then the spin of the other particle, measured on the same axis, is found to be anticlockwise. However, this behavior gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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