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In mathematics and computer science, in the field of coding theory, the Hamming bound is a limit on the parameters of an arbitrary block code: it is also known as the sphere-packing bound or the volume bound from an interpretation in terms of packing balls in the Hamming metric into the space of all possible words.
The theorem giving a lower bound on the quantum capacity of any channel is colloquially known as the LSD theorem, after the authors Lloyd, [1] Shor, [2] and Devetak [3] who proved it with increasing standards of rigor. [4]
In coding theory, the Singleton bound, named after Richard Collom Singleton, is a relatively crude upper bound on the size of an arbitrary block code with block length , size and minimum distance . It is also known as the Joshibound [ 1 ] proved by Joshi (1958) and even earlier by Komamiya (1953) .
Hamming was born in Chicago, Illinois, on February 11, 1915, [2] the son of Richard J. Hamming, a credit manager, and Mabel G. Redfield. [3] His father was Dutch, and his mother was a Mayflower descendant. [4] He grew up in Chicago, where he attended Crane Technical High School and Crane Junior College. [3]
The quantum information-theoretic interpretation of the above inequality is that the probability of obtaining outcome from a quantum measurement acting on the state is upper bounded by the probability of obtaining outcome on the state summed with the distinguishability of the two states and .
In general, a quantum code for a quantum channel is a subspace , where is the state Hilbert ... According to the quantum Hamming bound, encoding a single logical ...
One particle: N particles: One dimension ^ = ^ + = + ^ = = ^ + (,,) = = + (,,) where the position of particle n is x n. = + = = +. (,) = /.There is a further restriction — the solution must not grow at infinity, so that it has either a finite L 2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum): [1] ‖ ‖ = | |.
In quantum information theory, a set of bases in Hilbert space C d are said to be mutually unbiased if when a system is prepared in an eigenstate of one of the bases, then all outcomes of the measurement with respect to the other basis are predicted to occur with an equal probability inexorably equal to 1/d.