# Linear Algebra Dictionary

## Ray

A ray is a general name for an equivalence class of vectors $e^{i\alpha}\ket{\psi}:\alpha\in [0,2\pi]}$. Quantum states are rays as global phases of this type are indistinguishable under measurement. Rays are denoted using the 'ket' $\ket{\psi}$, and their dual-vector is denoted by the 'bra' $\bra{\psi} : \braket{\psi}{\phi}\in\mathbb{C}$, 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 $\ket{\psi}^{\dagger}$.

## Hilbert Space

A Hilbert Space $\mathcal{H}$ is a complex [vector space][#vector-space] on the complex numbers $\mathbb{C}$, replete with an inner product $(x,y) : x, y \in \mathcal{H} \rightarrow \mathbb{C}$ such that it is 1. Positive: $(x,y)>0\forall x,y \neq 0$ 2. Linear: $(x, ay + bz) = a(x,y) + b(x,z)$ 3. Skew symmetric: $(x,y) = (y,x)^{*}$

It is also complete under a norm $\norm{x}=\sqrt{(x,x)}$.

In quantum mechanics, the inner product is given by the 'braket' of the states $(x,y) = \braket{x}{y}$.

## 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 $B_{1}=\{\ket{\phi_{i}}\}, B_{2}=\ket{\psi_{j}}\}$ on a Hilbert space $\mathcal{H}:\norm{\mathcal{H}}=d$, then the bases are said to be mutually unbiased if and only if

A $d$ dimensional quantum system has $d$ amplitudes, of which we can find $d-1$ independent ampltudes from $d$ measurements. As a result, we need to make$d+1$ mutually unbiased measurements (measurements in a mutually unbiased basis) to fully characterise the state. In fact, a quantum system has at most $d+1$ independent mutually unbiased bases when $d$. If the system has dimension $d=p^{m}$, then we have $(p+1)^{m}$ operators which fall in to $p^{m}+1$ maximal noncomuting systems. Each basis is made up of orthogonal unitary matrices such that $Tr(X^{\dagger}Y)=0\forall X,Y\in B_{i}$, where the basis vectors described above are the +1 eigenstates of these obvservables.