# Linear Algebra Dictionary

## Ray

A ray is a general name for an equivalence class of vectors . Quantum states are rays as global phases of this type are indistinguishable under measurement. Rays are denoted using the 'ket' , and their dual-vector is denoted by the 'bra' , such that the dual vector maps us from a state in the vector space to the space of complex numbers.

Explicitly, the dual-vector corresponds to the complex-conjugate .

## Hilbert Space

A Hilbert Space is a complex [vector space][#vector-space] on the complex numbers , replete with an inner product such that it is 1. Positive: 2. Linear: 3. Skew symmetric:

It is also complete under a norm .

In quantum mechanics, the inner product is given by the 'braket' of the states .

## Vector Space

## Mutually Unbiased Basis

The notion of mutually unbiased bases arose in the context of defining the minimum set of observables required to full determine a quantum state (given a bunch of copies, obviously). The intuitve intuition is that we see a set of measurements tht are 'as different' as possible, such that their probabilities are relatively independent and we gain maximal information from each measurement.

Given a pair of bases on a Hilbert space , then the bases are said to be mutually unbiased if and only if

A dimensional quantum system has amplitudes, of which we can find independent ampltudes from measurements. As a result, we need to make mutually unbiased measurements (measurements in a mutually unbiased basis) to fully characterise the state. In fact, a quantum system has at most independent mutually unbiased bases when . If the system has dimension , then we have operators which fall in to maximal noncomuting systems. Each basis is made up of orthogonal unitary matrices such that , where the basis vectors described above are the +1 eigenstates of these obvservables.