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Concurrence (quantum Computing)
In quantum information science, the concurrence is a state invariant involving qubits. Definition The concurrence is an entanglement monotone (a way of measuring entanglement) defined for a mixed state of two qubits as: : \mathcal(\rho)\equiv\max(0,\lambda_1-\lambda_2-\lambda_3-\lambda_4) in which \lambda_1,...,\lambda_4 are the eigenvalues, in decreasing order, of the Hermitian matrix :R = \sqrt with :\tilde = (\sigma_\otimes\sigma_)\rho^(\sigma_\otimes\sigma_) the spin-flipped state of \rho and \sigma_y a Pauli spin matrix. The complex conjugation ^* is taken in the eigenbasis of the Pauli matrix \sigma_z. Also, here, for a positive semidefinite matrix A, \sqrt denotes a positive semidefinite matrix B such that B^2=A. Note that B is a unique matrix so defined. A generalized version of concurrence for multiparticle pure states in arbitrary dimensions (including the case of continuous-variables in infinite dimensions) is defined as: : \mathcal_(\rho)=\sqrt in which \r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Information Science
Quantum information science is a field that combines the principles of quantum mechanics with information theory to study the processing, analysis, and transmission of information. It covers both theoretical and experimental aspects of quantum physics, including the limits of what can be achieved with quantum information. The term quantum information theory is sometimes used, but it does not include experimental research and can be confused with a subfield of quantum information science that deals with the processing of quantum information. Scientific and engineering studies Quantum teleportation, Quantum entanglement, entanglement and the manufacturing of quantum computers depend on a comprehensive understanding of quantum physics and engineering. Google and IBM have invested significantly in quantum computer hardware research, leading to significant progress in manufacturing quantum computers since the 2010s. Currently, it is possible to create a quantum computer with over 100 qub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entanglement Monotone
In quantum information and quantum computation, an entanglement monotone or entanglement measure is a function that quantifies the amount of entanglement present in a quantum state. Any entanglement monotone is a nonnegative function whose value does not increase under local operations and classical communication. Definition Let \mathcal(\mathcal_A\otimes\mathcal_B)be the space of all states, i.e., Hermitian positive semi-definite operators with trace one, over the bipartite Hilbert space \mathcal_A\otimes\mathcal_B. An entanglement measure is a function \mu:\to \mathbb_such that: # \mu(\rho)=0 if \rho is separable; # Monotonically decreasing under LOCC, viz., for the Kraus operator E_i\otimes F_i corresponding to the LOCC \mathcal_, let p_i=\mathrm E_i\otimes F_i)\rho (E_i\otimes F_i)^/math> and \rho_i=(E_i\otimes F_i)\rho (E_i\otimes F_i)^/\mathrm E_i\otimes F_i)\rho (E_i\otimes F_i)^/math>for a given state \rho, then (i) \mu does not increase under the average over all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two spin states (left-handed and the right-handed circular polarization) can also be measured as horizontal and vertical linear polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of multiple states simultaneously, a property that is fundamental to quantum mechanics and quantum computing. Etymology The coining of the term ''qubit'' is attributed to Benjamin Schumacher. In the acknow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pauli Spin Matrix
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in connection with isospin symmetries. \begin \sigma_1 = \sigma_x &= \begin 0&1\\ 1&0 \end, \\ \sigma_2 = \sigma_y &= \begin 0& -i \\ i&0 \end, \\ \sigma_3 = \sigma_z &= \begin 1&0\\ 0&-1 \end. \\ \end These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation, which takes into account the interaction of the spin of a particle with an external electromagnetic field. They also represent the interaction states of two polarization filters for horizontal/vertical polarization, 45 degree polarization (right/left), and circular polarization (right/left). Each Pauli matrix is Hermitian, and together with the id ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Matrix
In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed states. These arise in quantum mechanics in two different situations: # when the preparation of a system can randomly produce different pure states, and thus one must deal with the statistics of possible preparations, and # when one wants to describe a physical system that is entangled with another, without describing their combined state. This case is typical for a system interacting with some environment (e.g. decoherence). In this case, the density matrix of an entangled system differs from that of an ensemble of pure states that, combined, would give the same statistical results upon measurement. Density matrices are thus crucial tools in areas of quantum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entanglement Of Formation
The entanglement of formation is a quantity that measures the entanglement of a bipartite quantum state. Definition For a pure bipartite quantum state , \psi\rangle_, using Schmidt decomposition, we see that the reduced density matrices of systems A and B, \rho_A and \rho_B, have the same spectrum. The von Neumann entropy S(\rho_A)=S(\rho_B) of the reduced density matrix can be used to measure the entanglement of the state , \psi\rangle_. We denote this kind of measure as E_(, \psi\rangle_)=S(\rho_A)=S(\rho_B) , and call it the entanglement entropy. This is also known as the entanglement of formation of a pure state. For a mixed bipartite state \rho_, a natural generalization is to consider all the ensemble realizations of the mixed state. We define the entanglement of formation for mixed states by minimizing over all these ensemble realizations, :E_f (\rho_)= \inf\left\ , where the infimum is taken over all the possible ways in which one can decompose \rho_ into pure states ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Roof Extension
Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope, a polytope with a convex set of points ** Convex metric space, a generalization of the convexity notion in abstract metric spaces * Convex function, when the line segment between any two points on the graph of the function lies above or on the graph * Convex conjugate, of a function * Convexity (algebraic geometry), a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces Economics and finance * Convexity (finance), second derivatives in financial modeling generally * Convexity in economics * Bond convexity, a measure of the sensitivity of the duration of a bond to changes in interest rates * Convex preferences, an individual's ordering of various outcomes Other uses * Convex Com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monogamy Of Entanglement
In quantum physics, monogamy is the property of quantum entanglement that restrict entanglement from being freely shared between arbitrarily many parties. In order for two qubits ''A'' and ''B'' to be maximally entangled, they must not be entangled with any third qubit ''C'' whatsoever. Even if ''A'' and ''B'' are not maximally entangled, the degree of entanglement between them constrains the degree to which either can be entangled with ''C''. In full generality, for n \geq 3 qubits A_1, \ldots, A_n, monogamy is characterized by the Coffman–Kundu–Wootters (CKW) inequality, which states that :\sum_^ \tau(\rho_) \leq \tau(\rho_) where \rho_ is the density matrix of the substate consisting of qubits A_1 and A_k and \tau is the "tangle", a quantification of bipartite entanglement equal to the square of the concurrence. Monogamy, which is closely related to the no-cloning property, is purely a feature of quantum correlations, and has no classical analogue. Supposing that tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with A Mathematical Theory of Communication, a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
