|
Classical Capacity
In quantum information theory, the classical capacity of a quantum channel is the maximum rate at which classical data can be sent over it error-free in the limit of many uses of the channel. Holevo, Schumacher, and Westmoreland proved the following least upper bound on the classical capacity of any quantum channel \mathcal: : \chi(\mathcal) = \max_ I(X;B)_ where \rho^ is a classical-quantum state of the following form: : \rho^ = \sum_x p_X(x) \vert x \rangle \langle x \vert^X \otimes \rho_x^A , p_X(x) is a probability distribution, and each \rho_x^A is a density operator that can be input to the channel \mathcal. Achievability using sequential decoding We briefly review the HSW coding theorem (the statement of the achievability of the Holevo information rate I(X;B) for communicating classical data over a quantum channel). We first review the minimal amount of quantum mechanics needed for the theorem. We then cover quantum typicality, and finally we prove the theorem using a ... [...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] |
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] |
Alexander Holevo
Alexander Semenovich Holevo(russian: Алекса́ндр Семéнович Хóлево, also spelled as Kholevo and Cholewo) is a Soviet and Russian mathematician, one of the pioneers of quantum information science. Biography Steklov Mathematical Institute, Moscow, since 1969. He graduated from Moscow Institute of Physics and Technology in 1966, defended a PhD Thesis in 1969 and a Doctor of Science Thesis in 1975. Since 1986 A.S. Holevo is a Professor (Moscow State University and Moscow Institute of Physics and Technology). Research A.S. Holevo made substantial contributions in the mathematical foundations of quantum theory, quantum statistics and quantum information theory. In 1973 he obtained an upper bound for the amount of classical information that can be extracted from an ensemble of quantum states by quantum measurements (this result is known as '' Holevo's theorem''). A.S. Holevo developed the mathematical theory of quantum communication channels, the noncommutat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holevo Information
Holevo's theorem is an important limitative theorem in quantum computing, an interdisciplinary field of physics and computer science. It is sometimes called Holevo's bound, since it establishes an upper bound to the amount of information that can be known about a quantum state (accessible information). It was published by Alexander Holevo in 1973. Accessible information As for several concepts in quantum information theory, accessible information is best understood in terms of a 2-party communication. So we introduce two parties, Alice and Bob. Alice has a ''classical'' random variable ''X'', which can take the values with corresponding probabilities . Alice then prepares a quantum state, represented by the density matrix ''ρX'' chosen from a set , and gives this state to Bob. Bob's goal is to find the value of ''X'', and in order to do that, he performs a measurement on the state ''ρ''''X'', obtaining a classical outcome, which we denote with ''Y''. In this context, the amo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Quantum State
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces. Pure states are also known as state vectors or wave functions, the latter term applying particularly when they are represented as functions of position or momentum. For example, when dealing with the energy spectrum of the electron in a hydrogen at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Operator
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 m ... [...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] |
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] |
POVM
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] |
Typical Subspace
In quantum information theory, the idea of a typical subspace plays an important role in the proofs of many coding theorems (the most prominent example being Schumacher compression). Its role is analogous to that of the typical set in classical information theory. Unconditional quantum typicality Consider a density operator \rho with the following spectral decomposition: : \rho=\sum_p_( x) \vert x\rangle \langle x\vert . The weakly typical subspace is defined as the span of all vectors such that the sample entropy \overline( x^) of their classical label is close to the true entropy H( X) of the distribution p_( x) : : T_^\equiv\text\left\ , where : \overline( x^) \equiv-\frac\log( p_( x^) ) , :H( X) \equiv-\sum_p_( x) \log p_( x) . The projector \Pi_^ onto the typical subspace of \rho is defined as : \Pi_^\equiv\sum_\vert x^\rangle \langle x^\vert , where we have "overloaded" the symbol T_^ to refer also to the set of \delta-typical sequences: : T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entanglement-assisted Classical Capacity
In the theory of quantum communication, the entanglement-assisted classical capacity of a quantum channel is the highest rate at which classical information can be transmitted from a sender to receiver when they share an unlimited amount of noiseless quantum entanglement, entanglement. It is given by the quantum mutual information of the channel, which is the input-output quantum mutual information maximized over all pure bipartite quantum states with one system transmitted through the quantum channel, channel. This formula is the natural generalization of Shannon's noisy channel coding theorem, in the sense that this formula is equal to the capacity, and there is no need to regularize it. An additional feature that it shares with Shannon's formula is that a noiseless classical or quantum feedback channel cannot increase the entanglement-assisted classical capacity. The entanglement-assisted classical capacity theorem is proved in two parts: the direct coding theorem and the conver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.jpg "Click for more on -> Quantum Information Theory")
