## Friday, 6 September 2013

### Postulates of Quantum Mechanics: State Space

A quantum state $\left|\psi\right>$ is an element of \emph{Hilbert space} $\mathcal{H}$ with unit norm. A Hilbert space is a complete vector space over the complex numbers $\mathbb{C}$ that has an inner product. This inner product allows the norms and relative angles between different states in the Hilbert space to be defined. A set of basis states can be associated to the Hilbert space $\mathcal{H}$. If $\mathcal{H}$ is finite dimensional with dimension $N$, then there exists a set of $N$ \emph{orthonormal} basis vectors $\left|1\right>, \left|2\right>, . . . , \left|N\right>$. Orthonormaility means that the inner product between any two basis states $\left|r\right>$ and $\left|s\right>$ is $0$ if $r\not=s$, and is $1$ if $r=s$. Thus, every basis vector in an orthonormal basis has unit norm. In this way, an arbitrary state $\left|\psi\right>\in\mathcal{H}$ can be expressed as a linear combination, or superposition, of basis states
$\left|\psi\right>= \alpha_1\left|1\right>+ \alpha_2 \left|2\right>+ . . . + \alpha_N \left|N\right>,$
where $\alpha_i \in \mathbb{C}$ for all $i\in\{1,2, . . . , N\}$ and $\displaystyle\sum\limits_{i=1}^N |\alpha_i|^2 = 1$. This latter constraint is called the normalization constraint and implies that $\left|\psi\right>$ is of unit norm. Such a state is said to be normalized.

## Combining States

This postulate about combining states can be thought of as an extension of the first. It tells us how to construct a new state representing the joint system of multiple states using the tensor product $\otimes$. First, the properties of the tensor products will be made precise.

Let $\mathcal{V}$ denote the set of Hilbert spaces. Then the tensor product is a map $\otimes: \mathcal{V}\times\mathcal{V}\rightarrow \mathcal{V}$. If $V$ and $W$ are $m$ and $n$ dimensional Hilbert spaces, respectively, then the tensor product of $V$ with $W$ is denoted by $V\otimes W$ and is a Hilbert space of dimension $mn$. An element of $V\otimes W$ can be written in the form $\left|v\right>\otimes \left|w\right>$, where $\left|v\right>\in\mathcal{V}$ and $\left|w\right>\in\mathcal{W}$. In particular, if $\left|i\right>\in\mathcal{V}$ and $\left|j\right>\in\mathcal{W}$ are orthonormal basis vectors of $\mathcal{V}$ and $\mathcal{W}$, respectively, then the tensor state $\left|i\right>\otimes \left|j\right>$ will be an orthonormal basis vector of the space $V\otimes W$. Hence, a general state of $V\otimes W$ will be given as a linear combination of basis states of the form $\left|i\right>\otimes \left|j\right>$ where $i\in\{1,2,...,n\}$ and $j\in\{1,2,...,m\}$. The tensor product also satisfies the following three properties.

(1) For all $\lambda\in \mathbb{C}$, $\left|v\right>\in\mathcal{V}$ and $\left|w\right>\in\mathcal{W}$,
$\lambda(\left|v\right>\otimes\left|w\right>)=(\lambda\left|v\right>)\otimes\left|w\right>=\left|v\right>\otimes(\lambda\left|w\right>)$

(2) For all $\left|v_1\right>, \left|v_2\right>\in\mathcal{V}$, and $\left|w\right>\in\mathcal{W}$,

$(\left|v_1\right>+\left|v_2\right>)\otimes\left|w\right>=\left|v_1\right>\otimes\left|w\right>+\left|v_2\right>\otimes\left|w\right>$

(3) For all $\left|v\right>\in\mathcal{V}$ and $\left|w_1\right>, \left|w_2\right>\in\mathcal{W}$,

$\left|v\right>\otimes(\left|w_1\right>\otimes\left|w_2\right>)=\left|v\right>\otimes\left|w_1\right>+\left|v\right>\otimes\left|w_2\right>$

As a notational convenience let the following equivalently represent the tensor product of two states $\left|v\right>$ and $\left|w\right>$:

$\left|v\right>\otimes\left|w\right>\equiv\left|v\right>\left|w\right>\equiv\left|vw\right>.$

We are now finally in position to state the second axiom. Given two states
$\left|\psi_1\right>\in\mathcal{H}_1$ and $\left|\psi_2\right>\in\mathcal{H}_2,$ a new state representing the joint system is given by
$\left|\psi\right>=\left|\psi_1\right>\otimes \left|\psi_2\right> \in\mathcal{H}.$
Here, $\mathcal{H}=\mathcal{H}_1\otimes \mathcal{H}_2$ and $dim(\mathcal{H})=dim(\mathcal{H}_1) \cdot dim(\mathcal{H}_2)$, where $dim(\mathcal{H})$ denotes the dimension of the Hilbert space $\mathcal{H}$. Continuing in this way, the tensor product of arbitrarily many states can be constructed.