### Bits and States

A **bit** is a question that only has exactly one of two possible answers. We often consider both classical and quantum bits. In fact, all bits are **qubits** (quantum bits) but their quantum nature are usually too small to be noticed.

A **classical bit** is usually represented by the set $\{0,1\}$. We can easily map real experiments to this set - for example, the flipping of a coin results in an element of the set $\{\mathrm{Heads}, \mathrm{Tails}\}$. A **state** or **configuration** is just an element of this set.

We introduce **Dirac notation**, $\ket{0}, \ket{1}$, to denote states - these are actually called **ket** vectors. Later we shall see **bra** vectors $\bra{0}, \bra{1}$. Note that $\ket{0}, \ket{1}$ are just labels - the zero and one could be replaced with any other symbols - e.g. $\ket{h}, \ket{t}$.

We can describe multi-bit sytems representing a string of bits (questions that have exactly one of two possible answers). For example, a 3-bit sytems has states in the set $$\{\:\ket{000}, \ket{001}, \ket{010}, \ket{011}, \ket{100}, \ket{101}, \ket{110}, \ket{111} \:\}$$

We note that the number of states, $N_s=2^n$ where $n$ is the number of bits.

### Rules for updating

Technically, a physical system is a pair of objects,

- a collection of states that can be represented by bits
- a set of rules for updating - that is, a set of rules that transform states into other states over a discrete time step.

Nb. For continuous systems, this entails knowing both the position and momentum of each object in the system. See the lectures on Classical Mechanics for details

### Examples

#### 1. Real Numbers.

We can use binary numbers to represent integers, $$\begin{align*}0 &\rightarrow 0\\1 &\rightarrow 1\\2 &\rightarrow 10\\3 &\rightarrow 11\\4 &\rightarrow 100\\&\cdots\end{align*}$$

We can approximate real numbers to any precision by just including the fractional parts and scaling the bit accordingly. For example, if we consider real numbers to one decimal place, then our bit-unit would be $0.1$. Thus, $$\begin{align*}0.0 &\rightarrow 0\\0.1 &\rightarrow 1\\0.2 &\rightarrow 10\\0.3 &\rightarrow 11\\0.4 &\rightarrow 100\\&\cdots\end{align*}$$

#### 2. Fields in space.

Suppose we wanted to measure the temperature in a room. We could chop up an area (2D for simplicity) into grid cells small enough so that the temperature does not vary much across the cell. The temperature in each cell can be written as a multi-bit real number.

0100 | 1001 | 0011 | 0001 |

0100 | 0001 | 0111 | 0111 |

1000 | 1000 | 1110 | 1011 |

0100 | 1101 | 0110 | 1111 |

The final state would be something like $\ket{01001001001100010100 \dots 1111}$ - just the tabular numbers spliced together.

#### 3. Motion of particles

Again, we could chop up a volume into a lattice of cells, but in this case we would make the cells small enough so that only one particle could fit at a given time. Either the cell contains a particle (1), or not (0).

0 | 1 | 0 | 1 |

1 | 1 | 0 | 0 |

0 | 0 | 1 | 0 |

0 | 0 | 0 | 1 |

Splicing theses together leads to the state $\ket{0101110000100001}$.

In principle, any physical system can be represented by bits - a fact that computers rely on, of course. We may need to refine our description, by including more lattices for example.

### A single-bit system

Take, for example, the single-bit system of a coin, where the states H(eads) and T(ails) can be represented simply by points in space, as shown in figure 1.1 (a).

Nb. We've dropped the Dirac notation here.

In (b), we show the pair of rules: $\left\{ \begin{matrix} H \rightarrow H \\ T \rightarrow T \end{matrix} \right.$

Notice that we need two rules - one for each state. There is only one other possible pair of rules for a single-bit system, and that's shown in (c), where $\left\{ \begin{matrix} H \rightarrow T \\ T \rightarrow H \end{matrix} \right.$

Each pair of rules in (b) and (c) is called a **Law of physics** for this set of states. In general, laws of physics are of course much more complicated than this, but this would just mean more bits and rules to describe it.

Nb. Not all sets of rules are laws of physics - we shall see that only rules that result in permutations of the whole collection are valid.

### A two-bit system

The collection of possible states for a two-bit system is $\{HH,TH,HT,TT\}$ (using coins again). Here are some possible rules:

(a) $\left\{ \begin{matrix} HH \rightarrow TH \\ TH \rightarrow TT \\ TT \rightarrow HT \\ HT \rightarrow HH \end{matrix} \right.$

This can be written as a 4-**cycle**: $(HH,TH,TT,HT)$ where all the arrows are implied.

(b) $\left\{ \begin{matrix} HH \rightarrow TH \\ TH \rightarrow HH \\ HT \rightarrow TT \\ TT \rightarrow HT \end{matrix} \right.$

As a pair of 2-**cycles**, this is: $\left\{ \begin{matrix}(HH,TH) \\ (HT,TT) \end{matrix} \right.$.

(c) $\left\{ \begin{matrix} HH \rightarrow TH \\ TH \rightarrow HT \\ HT \rightarrow TT \\ TT \rightarrow HH \end{matrix} \right.$

This can be written as a 4-**cycle**: $(HH,TH,HT,TT)$.

(d) $\left\{ \begin{matrix} HH \rightarrow TH \\ TH \rightarrow TT \\ TT \rightarrow HH \\ HT \rightarrow HH \end{matrix} \right.$

This cannot be written as a cycle, since $HH$ is the result of more than one rule. This is an example of a set of rules that isn't a law of physics. Technically, it would be OK going forward (in time steps), but the rule for $HH$ would be ambiguous going backward in time. That is, this set of rules is not **reversible**.

Nb. In all, there are $24=4!$ permutations, hence 24 sets of rules.

In general, all classical systems are reversible. This is a principle of classical physics. It is based on the assumption that we can, in principle, measure any variable perfectly. Reversibility implies that a valid set of rules results in a set of cycles that *fill *the state space. These cycles are then called **trajectories**, and over time, each state belongs to a unique trajectory.

Nb. This property is referred to as **unitarity** in quantum physics.