The Quantum Bit

The fundamental resource and basic unit of quantum information is the quantum bit (qubit), which behaves like a classical bit enhanced by the superposition principle (see below). From a physical point of view, a qubit is represented by an ideal two-state quantum system. Examples of such systems include photons (vertical and horizontal polarization), electrons and other spin-${1\over 2}$ systems (spin up and down), and systems defined by two energy levels of atoms or ions. From the beginning the two-state system played a central role in studies of quantum mechanics. It is the most simple quantum system, and in principle all other quantum systems can be modeled in the state space of collections of qubits.

From the information processing point of view, a qubit's state space contains the two ``logical'', or ``computational'', states $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ and $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{1}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$. The so-called ``ket'' notation for these states was introduced by P. Dirac, and its variations are widely used in quantum physics. One can think of the pair of symbols `` $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}$'' and `` $\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$'' as representing the qubit system. Their content specifies a state for the system. In this context $\mathfrak{0}$ and $\mathfrak{1}$ are system-independent state labels. When, say, $\mathfrak{0}$ is placed within the ket, the resulting expression $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ represents the corresponding state of a specific qubit.

The initial state of a qubit is always one of the logical states. Using operations to be introduced later, we can obtain states which are ``superpositions'' of the logical states. Superpositions can be expressed as sums $\alpha\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$... ...t$}{\mathfrak{1}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ over the logical states with complex coefficients. The complex numbers $\alpha$ and $\beta$ are called the ``amplitudes'' of the superposition. The existence of such superpositions of distinguishable states of quantum systems is one of the basic tenets of quantum theory called the ``superposition principle''. Another way of writing a general superposition is as a vector

$$\alpha\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\... ...row\left(\!\begin{array}{c}\alpha\\ \beta\end{array}\!\right),$$ (4)

where the two-sided arrow `` $\leftrightarrow$'' is used to denote the correspondence between expressions that mean the same thing.

The qubit states that are superpositions of the logical states are called ``pure'' states: A superposition $\alpha\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfrak{0}}\mbox{$... ...t$}{\mathfrak{1}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ is a pure state if the corresponding vector has length $1$, that is $\vert\alpha\vert^2+\vert\beta\vert^2=1$. Such a superposition or vector is said to be ``normalized''. (For a complex number given by $\gamma=x+ iy$, one can evaluate $\vert\gamma\vert^2=x^2+y^2$. Here, $x$ and $y$ are the real and imaginary part of $\gamma$, and the symbol $i$ is a square root of $-1$, that is, $i^2=-1$. The conjugate of $\gamma$ is $\overline\gamma=x-iy$. Thus $\vert\gamma\vert^2=\overline\gamma \gamma$.) Here are a few examples of states given in both the ket and the vector notation:

$$\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi_1}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$} = \Big(\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\mathfr... ...ightarrow \left(\!\begin{array}{c}1/\sqrt{2}\\ 1/\sqrt{2}\end{array}\!\right),$$ (5)

$$\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi_2}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$} = {3\over 5}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\m... ...ngle$} \leftrightarrow \left(\!\begin{array}{c}3/5\\ -4/5\end{array}\!\right),$$ (6)

$$\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi_3}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$} = {i3\over 5}\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\... ...le$} \leftrightarrow \left(\!\begin{array}{c}i3/5\\ -i4/5\end{array}\!\right).$$ (7)

The state $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi_3}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ is obtained from $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi_2}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ by multiplication with $i$. It turns out that two states cannot be distinguished if one of them is obtained by multiplying the other by a ``phase'' $e^{i\theta}$. Note how we have generalized the ket notation by introducing expressions such as $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ for arbitrary states.

The superposition principle for quantum information means that we can have states that are sums of logical states with complex coefficients. There is another, more familiar type of information whose states are combinations of logical states. The basic unit of this type of information is the probabilistic bit (pbit). Intuitively, a pbit can be thought of as representing the as-yet-undetermined outcome of a coin flip. Since we need the idea of probability to understand how quantum information converts to classical information, we briefly introduce pbits.

A pbit's state space is a probability distribution over the states of a bit. One very explicit way to symbolize such a state is by using the expression $\{ p{:}\mathfrak{0},(1-p){:}\mathfrak{1}\}$, which means that the pbit has probability $p$ of being $\mathfrak{0}$ and $1-p$ of being $\mathfrak{1}$. Thus a state of a pbit is a ``probabilistic'' combination of the two logical states, where the coefficients are nonnegative real numbers summing to $1$. A typical example is the unbiased coin in the process of being flipped. If ``tail'' and ``head'' represent $\mathfrak{0}$ and $\mathfrak{1}$, respectively, the coin's state is $\{{1\over 2}{:}\mathfrak{0},{1\over 2}{:}\mathfrak{1}\}$. After the outcome of the flip is known, the state ``collapses'' to one of the logical states $\mathfrak{0}$ and $\mathfrak{1}$. In this way, a pbit is converted to a classical bit. If the pbit is probabilistically correlated with other pbits, the collapse associated with learning the pbit's logical state changes the overall probability distribution by a process called ``conditioning on the outcome''.

A consequence of the conditioning process is that we never actually ``see'' a probability distribution. We only see classical deterministic bit states. According to the frequency interpretation of probabilities, the original probability distribution can only be inferred after one looks at many independent pbits in the same state $\{p{:}\mathfrak{0},(1-p){:}\mathfrak{1}\}$: In the limit of infinitely many pbits, $p$ is given by the fraction of pbits seen to be in the state $\mathfrak{0}$. As we will explain, we can never ``see'' a general qubit state either. For qubits there is a process analogous to conditioning. This process is called ``measurement'' and converts qubit states to classical information.

Information processing with pbits has many advantages over deterministic information processing with bits. One advantage is that algorithms are often much easier to design and analyze if they are probabilistic. Examples include many optimization and physics simulation algorithms. In some cases, the best available probabilistic algorithm is more efficient than any known deterministic algorithm. An example is an algorithm for determining whether a number is prime or not. It is not known whether every probabilistic algorithm can be ``derandomized'' efficiently. There are important communication problems that can be solved probabilistically but not deterministically. For a survey, see [10].

What is the difference between bits, pbits and qubits? One way to visualize the difference and see the enrichment provided by pbits and qubits is shown in Fig. 1.

FIG. 1: Visual comparison of the state spaces of different information units. The states of a bit correspond to two points. The states of a pbit can be thought of as ``convex'' combinations of the states of a bit and therefore can be visualized as lying on the line connecting the two bit states. A qubit's pure states correspond to the surface of the unit sphere in three dimensions, where the logical states correspond to the poles. This representation of qubit states is called the ``Bloch sphere''. The explicit correspondence is discussed at the end of Sect. 2.7. See also the definition and use of the Bloch sphere in [11]. The correspondence between the pure states and the sphere is physically motivated and comes from a way of viewing a spin-${1\over 2}$ system as a small quantum magnet. Intuitively, a state is determined by the direction of the north pole of the magnet.