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Friday, 6 September 2013

Notational Remarks: Bra-ket Notation

The main mathematical axioms of quantum mechanics which will be necessary to define quantum computation are reviewed in the following sections. These axioms and definitions will make clear what is meant by quantum states, how they are transformed into other states, and what happens to them upon measurement. Before we get to it, it is worthwhile to make some remarks regarding the notation that that is conventionally used in quantum computation.

To understand the motivation and utility of the notation commonly used in quantum mechanics and quantum computation. At the core of the theories' mathematical formulation is linear algebra, which is essentially concerned with vector spaces and the appropriate transformations between them. A quantum state of some vector space \(\mathcal{H}\) representing the state space of the quantum system is symbolically represented as \(\left|\psi\right>\), where the state is labelled with some symbol \(\psi\), and the \(\left|\cdot\right>\) is included to imply that an element of the state space \(\mathcal{H}\) is being considered. A state \(\left|\psi\right>\) can be synonymously thought of as a column vector in \(\mathcal{H}\). This notation becomes more illuminating if thought about from a linear algebra perspective. In this sense, by making a suitable choice of basis vectors of \(\mathcal{H}\), a state \(\left|\psi\right>\) can be represented as a column vector of entries where the different entries in the column give the vector's components in the appropriate places. For example, if \(\left|\psi\right>\) is a state in a \(3\)-dimensional vector space, then the state \(\left|\psi\right>\) can be written as
\[\left|\psi\right>=
\begin{pmatrix}
  x\\y\\ z
 \end{pmatrix}\]
for some coefficients \(x,y\), and \(z\) which are generally complex numbers.

This notation is convenient because in quantum mechanics one often encounters operators which transform vectors (states). These operators can be conveniently expressed as matrices. Let's say \(U\) is such an operator, on a \(3\)-dimensional vector space for example, with a particular matrix representation given as follows
\[U=
\begin{pmatrix}
  a_{11}&a_{12}&a_{13}\\
  a_{21}&a_{22}&a_{23}\\
  a_{31}&a_{32}&a_{33}
 \end{pmatrix}\]
for some coefficients \(a_{ij}\) which are also generally complex numbers. When \(U\) acts on the state \(\left|\psi\right>\) what will result is some new column vector
\[\left|\psi'\right>=
\begin{pmatrix}
  x'\\
y'\\
 z'
 \end{pmatrix},\]
which is computed explicitly using the standard matrix multiplication
\[\left|\psi'\right>=U\left|\psi\right>=\begin{pmatrix}
  a_{11}&a_{12}&a_{13}\\
  a_{21}&a_{22}&a_{23}\\
  a_{31}&a_{32}&a_{33}
 \end{pmatrix}
\begin{pmatrix}
  x\\
y\\
 z
 \end{pmatrix}=
 \begin{pmatrix}
  a_{11}\cdot x+a_{12}\cdot y+a_{13}\cdot z\\
  a_{21}\cdot x+a_{22}\cdot y+a_{23}\cdot z\\
  a_{31}\cdot x+a_{32}\cdot y+a_{33}\cdot z
 \end{pmatrix}=
\begin{pmatrix}
  x'\\
y'\\
 z'
 \end{pmatrix}.\]

By using this short hand notation for states and operators in quantum mechanics we don't have to always express these things with the tedious, and sometimes excessive, notation of matrices and vectors with all of their components written out in full detail. Doing this may not always be necessary and in some circumstances inappropriate, because doing so requires making a choice of an arbitrary basis for the vector space in order to specify the coefficients.  We may not really care what  \(\left|\psi'\right>\) is in terms of its vector components, but are satisfied knowing that its just symbolically equal to \(U\left|\psi\right>\).   Actually, it should be a general guiding principle to never choose a basis and keep results basis-free unless required to!

  The notation deployed here is commonly referred to as Dirac notation or bra-ket notation. A state \(\left|\psi\right>\) represented as a column vector is called a ket. For some motivation for the actual symbolic form of the bra-ket notation we need to mention what a bra is. For every ket, \(\left|\psi\right>\), there exists an associated \emph{bra} notated \(\left<\psi\right|\) as this sort of  backwards-looking ket. Recall that previously kets were introduced specifically as column vectors. In this way, a bra can be thought of as a row vector
 \[\left<\psi\right|=
\begin{pmatrix}
  \overline x &\overline y&\overline z
 \end{pmatrix},\]
which is shown here as a \(3\)-dimensional vector merely for illustrative purposes. Every ket \(\left|\psi\right>\) is associated to a bra \(\left<\psi\right|\) by simply turning either the column vector to a row vector or vice-versa, and taking the complex conjugate of each of the coefficients, which is the operation defined as turning some complex number \(z=x+iy\) to the complex number \(\overline z=x-iy\) where \(x\) and \(y\) are real numbers. This operation will be notated as the conjugate transpose: \(\left|\psi\right>^\dagger=\left<\psi\right|\) and \(\left<\psi\right|^\dagger=\left|\psi\right>\). Therefore, \(\left|\psi\right>^{\dagger\dagger}=\left|\psi\right>\) and \(\left<\psi\right|^{\dagger\dagger}=\left<\psi\right|\).

The vector spaces used to model quantum systems will be equipped with an inner product, which is an operation on the states of the system that provides a quantitative measure of how "close" two states are in the space. Formally, an inner product is a map \(\mathcal{H}\times\mathcal{H}\rightarrow \mathbb{C}\) that assigns two states in \(\mathcal{H}\) some complex number in \(\mathbb{C}\). The "bra-ket" notation is further motivated by its utility in calculating the inner product of two states
\[\left|\psi_1\right>=
\begin{pmatrix}
  x_1\\
y_1\\
 z_1
 \end{pmatrix}\]
 and
 \[\left|\psi_2\right>=
\begin{pmatrix}
  x_2\\
y_2\\
 z_2
 \end{pmatrix}\]
 by transforming one of the bras into a ket and performing the matrix multiplication, or dot product, given by \(\left|\psi_1\right>^\dagger\left|\psi_2\right>=\left<\psi_1\right|\left|\psi_2\right>\):
\[\left<\psi_1\right|\left|\psi_2\right>=
\begin{pmatrix}
  \overline x_1&
\overline y_1&
\overline z_1
 \end{pmatrix}
\begin{pmatrix}
  x_2\
y_2\
 z_2
 \end{pmatrix}=
   \overline x_1\cdot x_2 +
\overline y_1\cdot y_2 +
\overline z_1\cdot z_2
, \]
 which is just some complex number.

As a convention, the inner product \(\left<\psi_1\right|\left|\psi_2\right>\) will be shortened to \(\left<\psi_1|\psi_2\right>\), which looks like the two states grouped in a bracket. Hence the term "bra-ket" notation! Notice that \(\left<\psi_2|\psi_1\right>=\overline{\left<\psi_1|\psi_2\right>}\). So in general the order in which the inner product of two states is taken does matter, but the complex numbers representing the values of the two inner products are just complex conjugates of one another.

Define the norm of a complex number \(z\in\mathbb{C}\) as \(|z|=\sqrt{z\overline z}\), which measures the length or distance a certain complex number is away from the origin in the complex plane. This concept of a norm can be extended to define the length or norm of some quantum state \(\left|\psi\right>\) in terms of the the inner product of the state \(\left|\psi\right>\) with itself as \(||\left|\psi\right>||=\sqrt{\left<\psi|\psi\right>}\), which consequently is always some real number. A state \(\left|\psi\right>\) is said to have unit norm if \(||\left|\psi\right>||=1\).

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