lecturenotes2 - Dirac Notation and Basic Linear Algebra for...

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Dirac Notation and Basic Linear Algebra for Quantum Computing Dave Bacon “Mathematicians tend to despise Dirac notation, because it can prevent them from making important distinctions, but physicists love it, because they are always forgetting such distinctions exist and the notation liberates them from having to remember.” - David Mermin In quantum theory, the basic mathematical structure we deal with is a complex Hilbert space. A complex Hilbert space is a complex vector space with an inner product and which is also complete with respect to the norm defined by the inner product (complete here means that every Cauchy sequence of vectors converges to a vector where convergence is measured by the norm.) In quantum computing we will be dealing almost exclusively with the case where this Hilbert space is the vector space of complex N dimensional vectors, C N and the inner product between the vectors v = [ v 0 v 1 ··· v N - 1 ] T and w = [ w 0 w 1 ··· w N - 1 ] T (here T denotes transpose so we are writing the vectors as “column” vectors) is given by h w,v i = N - 1 X i =0 w * i v i (1) Here I’d like to introduce you to the Dirac bra-ket notation. Generally this notation can be used form any complex Hilbert space (and really in even more general settings, but we certainly won’t need to worry about this) but since we will be dealing with the vector space C N and the above inner product, it is useful to introduce this notation and show what it explicitly corresponds to in this complex Hilbert space. Kets : Vectors in a complex Hilbert space are denoted in the bra-ket notation by kets. Let’s call our Hilbert space H . Then we denote a vector in this space as a ket | v i ∈ H . When the vector space we are dealing with is C N , then | v i is nothing more than an ordered n-tuple of complex numbers. In particular we will find it useful to think about | v i as a column vector of N complex numbers: | v i ↔ v 0 v 1 . . . v N - 1 (2) v i C . We can do everything with kets that we can do with vectors: we can add them | v i + | w i , multiply them by a scalar α | v i , etc. Now there is a special vector in a vector space, the zero vector. For the zero vector we will never write it as | 0 i (you’ll see why soon.) Instead we will always just write it as 0, so | v i + 0 = | v i . Bras : Recall that for a vector space V we can define it a dual vector space, V * . This is the space of linear functionals on V : i.e. scalar-valued linear transformations on V . What does this mean? Well it means that an element of the dual space takes a vector and turns it into a complex number (functional on V ). Further this transform is linear, meaning we can add these transforms and multiply them by a scalar. Elements of the dual vector space for a Hilbert space H are written as “bra”s: h w | ∈ H * . When we are dealing with C N and the above inner product, then bras are
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lecturenotes2 - Dirac Notation and Basic Linear Algebra for...

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