Hilbert SpaceDate: 02/04/98 at 00:24:59 From: Renee Anderson Subject: Hilbert Space I am an eleventh grade high school student, and I'm doing reseach on quantum computations. Could you please tell me what Hilbert Space is? Date: 02/04/98 at 16:21:48 From: Doctor Rob Subject: Re: Hilbert Space See the following URL on the World Wide Web: http://www.astro.virginia.edu/~eww6n/math/HilbertSpace.html It is a vector space over a field with an inner product, such that the inner product yields a norm turning the vector space into a metric space. That is a terribly technical definition, which you may not understand. Over the field of real numbers, the set of real n-tuples with componentwise addition and the usual dot-product forms a Hilbert space. The norm is the square root of the sum of the squares of the components of the n-tuple. The norm of the difference between two n- tuples is then the distance between them, and it satisfies the usual properties of a distance: it is never negative, it is zero if and only if the two n-tuples are the same, and it satisfies the triangle inequality. (In fact, it is the usual Euclidean distance.) This distance function makes the set a metric space. Over the complex numbers, the set of complex n-tuples with componentwise addition and the inner product <u,v> = u . conjugate(v) forms a Hilbert space. The norm is ||v|| = Sqrt[<v,v>] = square root of the sum of the squares of the absolute values of the components of v. The norm of the difference between two n-tuples is then the distance between them, and it satisfies the usual properties of a distance: it is never negative, it is zero if and only if the two n-tuples are the same, and it satisfies the triangle inequality. This distance function makes the set a metric space. The above Hilbert spaces have dimension n. Some Hilbert spaces can be infinite-dimensional. In quantum computation, the metric spaces are of the second type described above, and the dimension is usually 2^k, when there are k quantum bits. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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