The allusion to vectors representing quantum states can be made precise. Using a notation introduced by Dirac, it is common to represent quantum states as abstract vectors such as |V> (vertical photon polarization) or |H> (horizontal photon polarization) in Hilbert space. These objects are known as "kets," (the right portion of the word "bracket"). For our purposes we need not be concerned with the formal properties of this vector space, and will merely concern ourselves with some of the basic results.
One of the important properties of vectors is their length, or more generally the length of the component of one vector along another, which is known as the scalar product of the two vectors. In quantum notation, it is useful to introduce the complex conjugate vectors, <V| and <H|, known as "bras" (from the left hand portion of the word bracket) to define the scalar product using the relations:
< V|V> = <H|H> = 1, <V|H> = <H|V> = 0
The length of a quantum state vector is related to the probability of an experimental outcome or measurement, which mathematically is represented by the idea of projection: the component of a vector along another vector. For example, a measurement of "V" polarization would be represented by the projection operator, P(V) = |V><V|. Thus, a measurement for "V" polarization on a |V> photon state produces the state |V>, corresponding to the 100% probability that this state has the "V" polarization on measurement. Conversely, there is no component of an |H> state along the vector |V> which corresponds to the experimental result that an "H" photon will never pass a test for "V"-ness.
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