Qubit State

In one sentence: A qubit is a quantum bit that can be a mix of 0 and 1, and two numbers set how likely each answer is when you check it.

Read it out loud: “Ket psi equals alpha times ket zero, plus beta times ket one. Here psi is the name of the qubit's state, alpha and beta are its two amplitudes, and ket zero and ket one are the two basis states. The bars around alpha and beta mean the size of each number. The second rule says that the squared size of alpha, plus the squared size of beta, equals one.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

A classical bit and a qubit: A classical bit, the kind stored in a phone or a laptop memory cell, is always either 0 or 1. A qubit is the quantum version: a tiny two-level system, such as the spin of an electron or the tilt of a single particle of light called a photon, that can hold a mix of the two basic states. The formula describes that mix.
Ket notation: $|\psi\rangle$ , $|0\rangle$ and $|1\rangle$: The symbol $|\,\rangle$ is called a ket. It is only a label, like a name tag on a box. $|0\rangle$ names the basis state 0, and $|1\rangle$ names the basis state 1. A basis state is one of a standard set of reference states from which every other state can be built. $|\psi\rangle$ is the name of the qubit’s state.
Complex numbers and the imaginary unit $i$: A complex number has two parts: an ordinary number, and a second part multiplied by $i$ . The imaginary unit $i$ is defined by $i^2 = -1$ , so it is not an ordinary number. For example, $3 + 4i$ has real part 3 and imaginary part 4. A complex number stores both a size and a turn, which is why the amplitudes $\alpha$ and $\beta$ are complex.
Absolute value bars $|x|$ and squared size $|\alpha|^2$: Straight bars around a number mean its size, which is its distance from zero with the sign removed. For example $|-3| = 3$ . For a complex number the size comes from its two parts, so $|3 + 4i| = 5$ . The squared size $|\alpha|^2$ means the size multiplied by itself. If $\alpha = 0.6$ , then $|\alpha|^2 = 0.36$ .
Probability: A probability is a chance written as a number from 0 to 1, where 0 means never and 1 means always. Multiply it by 100 to get a percentage, so 0.36 is 36 percent. The probabilities of all possible outcomes add up to 1, because one of them must happen. A fair coin has probability 0.5 of landing heads.
Square root, like $\sqrt{3}$: The square root of a number is the value that, multiplied by itself, gives the number. So $\sqrt{9} = 3$ , because $3 \times 3 = 9$ . The square root of 3 is about 1.732, and $\sqrt{2/3}$ is about 0.816. Squaring a square root undoes it: $(\sqrt{3})^2 = 3$ .
Superposition, the plus sign in $\alpha|0\rangle + \beta|1\rangle$: A superposition is one state made from two or more basis states at once, written as a sum. The plus sign does not mean the qubit contains two separate answers that you can read out together. It means the state is a single mix, and two amplitudes describe that mix.

What each symbol means

Symbol	Meaning
$\|\psi\rangle$	The state of the qubit: the full description of the qubit at one moment, written as a ket. It has no unit, because it is a recipe made of two amplitudes, not a length or an energy. Think of it as the complete recipe for the odds of every possible measurement result.
$\|0\rangle$	The basis state 0, one of the two standard results a measurement can give. It has no unit. It stands for a qubit that would always give the result 0 when measured in the standard way.
$\|1\rangle$	The basis state 1, the other standard result a measurement can give. It has no unit. It stands for a qubit that would always give the result 1 when measured in the standard way.
$\alpha$	The amplitude of $\|0\rangle$ , a complex number with a size and a turn. It has no unit, because it is a pure number. Its size is between 0 and 1 for every valid qubit. In the example $\alpha = 1/\sqrt{3}$ , about 0.577, and its squared size $\|\alpha\|^2$ is the chance of measuring 0.
$\beta$	The amplitude of $\|1\rangle$ , also a complex number with a size and a turn. It has no unit, because it is a pure number. Its squared size $\|\beta\|^2$ is the chance of measuring 1. In the example $\beta = i\sqrt{2/3}$ , whose size is about 0.816.
$\|\alpha\|^2 + \|\beta\|^2$	The total of the two chances. It must equal exactly 1, because a measurement always gives 0 or 1. It has no unit, because it is a sum of probabilities. In the example, $1/3 + 2/3 = 1$ , so that qubit is valid.

The formula piece by piece

$\alpha$: Start with the smallest piece, the amplitude $\alpha$ for the outcome 0. It is a complex number, so it carries a size and a turn. Only its size, after squaring, decides how likely the outcome 0 is, and the turn matters only when qubits are combined.
$|\alpha|^2$: Squaring the size of $\alpha$ turns it into a probability. Squaring removes the turn and leaves a positive number from 0 to 1, which is the chance of measuring 0.
$\alpha|0\rangle$: The amplitude multiplied by the basis state $|0\rangle$ . It is the share of the qubit that is in the state 0, with the amount $\alpha$ attached to that share.
$\alpha|0\rangle + \beta|1\rangle$: Add the two shares to get the whole qubit. The plus sign combines the two basis states into one state, which is a superposition. The qubit is not two separate things. It is one state, described by two amplitudes.
$|\alpha|^2 + |\beta|^2 = 1$: Add the two chances and check the total. The sum must equal 1, because a measurement always returns either 0 or 1. A pair of amplitudes that fails this check does not describe a real qubit.
$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle, \qquad |\alpha|^2 + |\beta|^2 = 1$: A qubit’s state is one mix of the basis states 0 and 1, and the two chances of getting 0 and 1, which are the squared sizes of the two amplitudes, must add up to 1.

What the formula is doing, and why

What the formula says. A classical bit is either 0 or 1. A qubit can hold a mix of 0 and 1 in one state, as described by $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ . The two basis states $|0\rangle$ and $|1\rangle$ are the two results a measurement can give. The amplitudes $\alpha$ and $\beta$ say how much of each basis state the qubit contains. The formula is the full description of the qubit before you measure it.

Why square the size? The amplitudes $\alpha$ and $\beta$ are not probabilities themselves. The chance of each result comes from the Born rule, named after the physicist Max Born: $P(0) = |\alpha|^2$ and $P(1) = |\beta|^2$ . Squaring the size turns an amplitude into a positive number, and experiments show that this is the rule that matches measured results. For example, $\alpha = 0.6$ gives a 36 percent chance of 0, not 60 percent.

Why must the two squares add to 1? A measurement always gives either 0 or 1, so the two chances together must cover every outcome. Nothing is left over and nothing is missing. This is why the amplitudes are said to be normalized. A pair of amplitudes that breaks the rule does not describe a real qubit.

Why the plus sign and the turn matter. The plus sign means both basis states are present in one state, which is called a superposition. It does not mean one readout shows both answers. A single measurement still gives one answer. The turn, or phase, of $\alpha$ and $\beta$ does not change the odds of one measurement in the 0 and 1 basis. It does decide how qubits combine or interfere when they are rotated or joined together.

An everyday picture

Think of one hour split between two tasks. Each task gets a share of the hour, and the shares must add up to exactly one hour. A qubit works the same way for the odds: the share for 0 is $|\alpha|^2$ , the share for 1 is $|\beta|^2$ , and together they make a whole. The analogy leaves out the turn of each amplitude, which matters only when qubits interact or are rotated.

Worked example: Measurement odds for an unequal qubit

A lab prepares many identical qubits in the state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ , with $\alpha = \frac{1}{\sqrt{3}}$ and $\beta = i\sqrt{\frac{2}{3}}$ , where $i$ is the imaginary unit with $i^2 = -1$ . Each qubit is measured once. We want the chance of 0 and the chance of 1, and the number of 0 results we expect in 3,000 runs. The numbers are chosen for this example and are not measured data.

Check that the amplitudes are valid. Before finding any chance, we check the rule that the squared sizes add up to 1. The amplitudes come first, because the chances are built from them. A wrong pair of amplitudes would give odds that do not add up to 100 percent, so this check comes first.
Find the chance of 0. The chance of 0 is the squared size of $\alpha$ . Squaring $\frac{1}{\sqrt{3}}$ means squaring the top and the bottom: $|\alpha|^2 = \left(\frac{1}{\sqrt{3}}\right)^2 = \frac{1}{3} \approx 0.333$ . The square removes the square root, because $(\sqrt{3})^2 = 3$ .
Find the chance of 1. The imaginary unit $i$ sits one unit away from zero, so multiplying by it turns $\beta$ without stretching it. That means $|\beta| = 1 \times \sqrt{\frac{2}{3}}$ . Squaring gives $|\beta|^2 = \frac{2}{3} \approx 0.667$ , which is the chance of getting 1.
Check the total. Add the two chances: $\frac{1}{3} + \frac{2}{3} = \frac{3}{3} = 1$ . The total is exactly 1, so the state is valid. If the total were 0.9 or 1.2, we would have to go back and fix the amplitudes before going on.
Turn the chances into percentages. Multiply by 100. The chance of 0 is $0.333 \times 100 \approx 33.3$ percent, and the chance of 1 is $0.667 \times 100 \approx 66.7$ percent. Percent means out of every hundred runs, so about one run in three gives 0.
Count the expected results in 3,000 runs. Multiply each chance by the number of runs. The expected number of zeros is $3000 \times \frac{1}{3} = 1000$ , and the expected number of ones is $3000 \times \frac{2}{3} = 2000$ . These are averages, so a real experiment would give numbers close to them, not exactly them.
Check the units. Chances and amplitudes have no unit. They are pure numbers, so the answers 33.3 percent and 66.7 percent carry no meter, second or joule. The number of runs is a plain count of tries, which also has no unit.

Answer: The state is valid. The chance of measuring 0 is 33.3 percent, and the chance of measuring 1 is 66.7 percent. In 3,000 runs we expect about 1,000 results of 0 and 2,000 results of 1. The factor $i$ in $\beta$ does not change either number.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Thinking a qubit is 0 and 1 at the same time, with both answers showing up in one measurement. A single measurement gives one result, 0 or 1, chosen with the chances above.
Treating $\alpha$ as a probability. With $\alpha = 0.6$ , the chance of measuring 0 is $0.6^2 = 0.36$ , which is 36 percent, not 60 percent. The chance comes from the squared size.
Forgetting the total rule. If $|\alpha|^2 + |\beta|^2$ is not 1, the state is not valid, and the chances will not add up to 100 percent.
Assuming the turn of $\beta$ changes the measured odds. For a measurement in the 0 and 1 basis, only the sizes of the amplitudes matter. The turns show up when a qubit is rotated or combined with another qubit.

Where you will meet it

Quantum computers, where each qubit is held in a state like this, and quantum gates, the operations that change the amplitudes $\alpha$ and $\beta$ .
Quantum key distribution, such as the BB84 protocol, which can encode bits in the tilt, or polarization, of single photons.
Electron spin qubits in semiconductor chips, which store 0 and 1 in the spin of a single electron.
Photon polarization experiments in optics, where horizontal and vertical polarization can serve as the two basis states.

Questions people ask

What does the ket symbol mean?

The symbol $|\psi\rangle$ is a ket. It is a label for a state, like a name tag. Physicists use it because it marks a state clearly and makes the rules easy to write. The symbol $|0\rangle$ names the basis state 0 in the same way.

Can I read out $\alpha$ and $\beta$ directly?

Not from one qubit. A single measurement gives only 0 or 1, and the measurement also disturbs the state. To estimate $|\alpha|^2$ , prepare many identical qubits and count how often you get 0. The count divided by the number of runs estimates the chance.

Why are the amplitudes complex numbers?

The turn of each amplitude matters when states are combined or rotated, and a complex number stores both a size and a turn. The size of each amplitude decides the chance of a single measurement, while the turn decides how the parts interfere. The Bloch sphere page shows how that turn is drawn.

Is a qubit the same as a random bit?

No. A random bit is either 0 or 1, with a known chance, like a coin toss. A qubit’s amplitudes carry a turn, and quantum operations use that turn to combine the parts, so the state holds more than a simple chance.

Bloch Sphere: The Bloch sphere is a round picture of a qubit: every possible state is one point on its surface, and the top and bottom points are 0 and 1.
Born Rule: The Born rule turns a quantum wave into chances: the probability of a result is the squared size of the wave for that result, so bigger waves mean more likely results.
Wave Function Normalization: Normalizing a quantum wave means scaling it so the chances of finding the particle, added over every spot, total exactly 1.