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Quantum Information Theory

Mathematical Preliminaries

0.1 Euclidean Vectors

Euclidean vectors are objects, represented by arrows, which can be used to represent physical quantities, such as velocities or forces. Any two vectors can be added to produce a third, and thus a linear combination of vectors is just another vector.

0.2 Vector spaces

As per the rules of Peano, a vector space is a structure in which the concept of linear combinations works well.

More specifically, a complex vector space, is a set VV, such that any two vectors a,ba,b, (which are elements of the space: (α,β∈V\alpha, \beta \in V)), can be combined to form the following linear combination:

aα+bβ=γ,such that  γ∈V,where  γ  is a complex vector.a\alpha +b\beta = \gamma, \text{such that}\; \gamma \in V, \text{where}\; \gamma\; \text{is a complex vector}.

A subspace of VV is any subset of VV, which is closed under addition and multiplication by complex numbers.

Here we start using the Dirac bra-ket notation to write vectors, with labels.

A basis in VV is a collection of some vectors ∣e1⟩,∣e2⟩,∣e3⟩...∣en⟩\footnotesize |e_1\rangle, |e_2\rangle, |e_3\rangle...|e_n\rangle, such that every vector in the space (∣vi⟩∈V\small |v_i\rangle \in V), can be written as a linear combination of the basis vectors: ∑ivi∣ei⟩.\sum\limits_{i}v_i|e_i\rangle.

The number of elements in some defined basis, gives us the dimension of the space.

The most common n-dimensional complex vector space, is the space of ordered n-tuples of complex numbers, often presented as column vectors, which is the space that will be used throughout most of this text:

α∣a⟩+β∣b⟩=\alpha|a\rangle + \beta|b\rangle =

[αa1+βb1 αa2+βb2 ⋮ αan+βbn]\begin{bmatrix} \alpha a_1 + \beta b_1 \ \alpha a_2 + \beta b_2 \ \vdots \ \alpha a_n + \beta b_n \end{bmatrix}

0.3 Bras and kets

An inner product on a vector space (defined over the field of complex numbers), is a function that relates a pair of vectors ∣u⟩,∣v⟩∈V|u\rangle, |v\rangle \in V with a complex number ⟨u∣v⟩\langle u|v\rangle, such that the following axioms are satisfied:

⟨u∣v⟩=⟨v∣u⟩∗\small \langle u | v \rangle = \langle v | u \rangle^*

⟨v∣v⟩⩾0,∀  ∣v⟩∈V.\small \langle v | v \rangle \geqslant 0, \forall\; |v\rangle \in V.

∀\forall is read as "For all".

Referenced in

Quantum Information Theory