What Is a Quantum State? Definition, Types & Examples

TL;DR

Quantum computing prepares qubit states, transforms amplitudes and relative phase with gates, creates entanglement when needed, and measures the final state to obtain classical results.
A quantum state is the mathematical description used to predict measurement outcomes and the future evolution of a quantum system.
For a pure qubit, the state can be written as $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle, \quad \text{where } |\alpha|^2 + |\beta|^2 = 1$ . The complex amplitudes contain both probability and phase information.
A pure state can use a statevector. A density matrix can describe both pure and mixed quantum states, including noisy systems and subsystems of entangled systems.
Measurement depends on the chosen basis. The Born rule gives outcome probabilities, and an ideal projective measurement updates the post-measurement state.
A qubit has two computational basis states, $|0\rangle \text{ and } |1\rangle$ , but infinitely many possible pure states formed from normalized superpositions.

What is a quantum state?

A quantum state is a mathematical description of a quantum system that contains the information to calculate probabilities for possible measurement outcomes and predict how the system evolves.

Quantum states infographic showing pure, mixed, eigen, product, and entangled states, plus the Bloch sphere, measurement, and state evolution. — Quantum states used in quantum computing.

In classical mechanics, an object can have a definite position and speed at a particular moment. Quantum mechanics works differently. Before measurement, a quantum state can contain several possible outcomes. It tells us how likely those outcomes are and how they can interact through a quantum effect called interference.

A quantum system is simply the physical object, or part of an object, that we describe using quantum mechanics. Examples include an electron, a photon, an atom, or a qubit.

A qubit is the simplest example. It is a two state quantum system based on the states $|0\rangle \text{ and } |1\rangle$ . A general pure qubit state is:

|\psi\rangle = \alpha|0\rangle + \beta|1\rangle

Here, α and β are called probability amplitudes. Their squared sizes must add up to 1:

|α|² + |β|² = 1.

If we measure the qubit using the $|0\rangle \text{ and } |1\rangle$ states:

$|\alpha|^2$ is the probability of measuring 0.
$|\beta|^2$ is the probability of measuring 1.

For example, if both probabilities are 0.5, the measurement has a 50% chance of producing 0 and a 50% chance of producing 1.

The term quantum state can refer to several related concepts, including pure states, mixed states, eigenstates, and entangled states. These terms describe different features of a quantum system, so they are not necessarily separate or mutually exclusive categories.

It also helps to separate four important ideas:

State: Describes the quantum system and its possible behavior.
Basis: A set of reference states used to write the state or perform a measurement.
Observable: A property we can measure, such as energy, position, or spin.
Outcome: The actual classical result produced by the measurement.

How is a quantum state represented mathematically?

A pure quantum state is commonly represented by a normalized vector in a complex Hilbert space so we can calculate measurement probabilities and predict how the system will change. For a qubit, we can represent the same state in several ways, including ket notation, a column vector, a point on the Bloch sphere, or a density matrix.

Statevector and ket notation

A single qubit can be written as:

|\psi\rangle = \alpha|0\rangle + \beta|1\rangle

Here:

$|\psi\rangle$ represents the qubit’s quantum state.
$|0\rangle$ and $|1\rangle$ are the two basic qubit states.
α and β are numbers called probability amplitudes.

The amplitudes determine the probability of each measurement result:

$|\alpha|^2$ is the probability of measuring 0.
$|\beta|^2$ is the probability of measuring 1.

Because the measurement must produce either 0 or 1, these probabilities must add up to one:

|\alpha|^2 + |\beta|^2 = 1

We can also represent the basic states as column vectors:

|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \qquad |1\rangle = \begin{bmatrix} 0 \\ 1 \end{bmatrix}

The general qubit state then becomes:

|\psi\rangle = \begin{bmatrix} \alpha \\ \beta \end{bmatrix}

Ket notation and column-vector notation describe the same state. They simply write it in different forms.

Wavefunction

When describing a particle’s position, we often represent its state using a wavefunction:

\psi(x)

The wavefunction assigns an amplitude to every possible position x. Its squared size,

|\psi(x)|^2

gives the probability density for finding the particle near that position.

A wavefunction is therefore one way to represent a quantum state. However, the terms are not always interchangeable because not every quantum state is most conveniently described as a position-based wavefunction.

Quantum phase

A probability amplitude contains both a size and a quantum phase. The size helps determine measurement probabilities, while the phase affects how different quantum possibilities interfere.

There are two important types of phase:

Global phase: Applies equally to the entire state and does not change measurement predictions.
Relative phase: Describes the phase difference between parts of a state and can change interference results.

Consider these two states:

|+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}, \quad |-\rangle = \frac{|0\rangle – |1\rangle}{\sqrt{2}}

If we measure either state using the $|0\rangle \text{ and } |1\rangle$ basis, both have:

A 50% probability of producing 0
A 50% probability of producing 1

However, they are not the same state. The plus and minus signs create different relative phases. If we measure them using another basis called the X basis, ∣+⟩ always produces the + result, while ∣−⟩ always produces the − result.

Density matrix

A density matrix provides a more general way to represent a quantum state.

For a pure state ∣ψ⟩, its density matrix is:

\rho = |\psi\rangle\langle\psi|

A density matrix can also represent a mixture in which the system has a probability $p_i$ of being prepared in each state $\rho = \sum_i p_i |\psi_i\rangle\langle\psi_i|$

This makes density matrices useful for describing:

Uncertainty about how a system was prepared
Noise and loss of quantum coherence
Quantum systems interacting with their environment
One part of a larger entangled system
Realistic quantum-computing hardware

A statevector describes a pure state, while a density matrix can describe both pure and mixed states.

One qubit, four equivalent views

Consider the state:

|+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}

We can represent this same state in four ways:

Ket notation:

|+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}

Column vector:

\frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ 1 \end{bmatrix}

Bloch sphere: A point on the sphere pointing in the positive x-direction.
Density matrix:

\rho = \frac{1}{2} \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}

These are not four different quantum states but four mathematical ways to describe the same physical state. The representation may change, but the underlying state does not.

What are the main types of quantum states?

The main distinction is between pure and mixed states. Other terms, such as eigenstate, product state, entangled state, stationary state, bound state, and vacuum state, describe different properties of a quantum state.

These categories can overlap. For example, one quantum state can be pure, entangled, and an eigenstate at the same time.

Pure state

A pure state provides the most complete description of a quantum system allowed by quantum mechanics. We can represent it using a single normalized statevector.

For example, a qubit may exist in a superposition of the basis states ∣0⟩ and ∣1⟩: ∣ψ⟩=α∣0⟩+β∣1⟩

A pure state does not mean that the measurement result is already known. It means that the uncertainty comes from the quantum nature of the system rather than from our lack of knowledge about how the system was prepared.

For instance, the state ∣+⟩ is a pure state even though measuring it in the computational basis gives equal probabilities of 0 and 1.

Mixed state

A mixed state describes a system when we do not know which of several possible pure states was prepared. It can also describe one part of a larger entangled system.

Unlike a pure state, a mixed state generally requires a density matrix. For example, an equal mixture of ∣0⟩ and ∣1⟩ is: ρ=21[1001]

This mixed state produces a 50% probability of measuring 0 and a 50% probability of measuring 1.

The pure state ∣+⟩ produces the same probabilities when measured in the computational basis. However, the two states behave differently in other measurement bases because the pure superposition preserves phase and interference, while the mixture does not.

Eigenstate

An eigenstate is a iuiuiu with a definite value for a particular measurable property, such as energy or spin along a chosen direction.

If ∣a⟩ is an eigenstate of an observable represented by A, then: A∣a⟩=a∣a⟩

Here, a is the value obtained when that observable is measured.

The term eigenstate always refers to a specific observable. A state can be an eigenstate of one observable without being an eigenstate of another. For example, a qubit may have a definite spin value along the z-direction but not along the x-direction.

Product state

A product state describes a combined system whose individual parts have separate states. The complete state can be divided into the states of its subsystems.

For two systems A and B, a product state has the form: ∣ψ⟩AB=∣ψ⟩A⊗∣ϕ⟩B

In this case, each subsystem has its own independent state description. Measuring one subsystem may reveal information about that subsystem, but the combined state does not contain quantum entanglement between them.

Entangled state

An entangled state describes two or more quantum systems whose states cannot be described independently. The complete system has a definite quantum state, but its individual parts do not have separate pure-state descriptions.

A standard example is the Bell state: ∣Φ+⟩=2∣00⟩+∣11⟩

If both qubits are measured in the computational basis, the results are always correlated. The measurement produces either 00 or 11, each with a 50% probability.

Before measurement, however, the state is not simply one unknown classical option. It is a coherent superposition of both possibilities. This distinction allows entangled states to produce correlations that product states cannot reproduce.

Bound state

A bound state describes a particle that remains confined to a particular region because it does not have enough energy to escape its potential.

Its energy usually belongs to a set of discrete allowed values: H∣ψn⟩=En∣ψn⟩

Electrons bound to an atomic nucleus are a common example. They can occupy only particular energy states rather than any possible energy.

Stationary state

A stationary state is an energy eigenstate whose measurable probability distribution does not change over time when the Hamiltonian is constant.

Its time evolution is: ∣ψ(t)⟩=e−iEt/ℏ∣ψ(0)⟩

The state gains a time-dependent phase, but this overall phase does not change its observable probability distribution. The word “stationary” therefore does not mean that the state stops evolving mathematically. It means that its measurable distribution remains unchanged.

Vacuum state

A vacuum state is the lowest-energy quantum state of a field or physical mode. It does not mean complete emptiness.

The vacuum state is commonly written as: ∣0⟩

Even when no particles are present, a quantum field can still have fluctuations and a nonzero ground-state energy. The vacuum is therefore a physical quantum state, not the absence of a state.

How does measurement change a quantum state?

A quantum measurement does not simply reveal a value that was already hidden inside the system. It asks the system a specific question, produces one of the allowed outcomes, and changes the quantum state based on the result.

The measurement we choose determines the possible outcomes. The state determines the probability of each outcome. After we record a result, the state changes to match that result.

In the ideal projective model, measurement projects the state into the state or subspace connected to the observed outcome. Real measurements can be more complex, but this model explains the basic theory clearly.

The Born rule

The Born rule connects a quantum state to the probabilities of its measurement outcomes.

Consider a qubit:

|\psi\rangle = \alpha|0\rangle + \beta|1\rangle

If we measure it in the computational, or Z, basis, the probability of obtaining 0 is $|\alpha|^2$ , while the probability of obtaining 1 is $|\beta|^2$ .

The amplitudes do not tell us which result a single measurement will produce. They let us predict the pattern that will appear when we repeat the same preparation and measurement many times.

The measurement basis matters

A measurement basis defines the set of possible results. The same quantum state can produce different results when measured in different bases.

Consider the state:

|+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}

If we measure it in the Z basis, the outcomes 0 and 1 each have a 50% probability. If we measure it in the X basis, the + outcome occurs with certainty.

The state has not changed before the measurement. We have changed the question asked of the system. This is why one measurement cannot provide a complete description of an unknown quantum state.

The post-measurement state

Measurement changes what we can predict about future measurements.

Suppose we prepare $|+\rangle$ and measure it in the Z basis. If the outcome is 0, the state becomes:

|\psi_{\text{after}}\rangle = |0\rangle

If the outcome is 1, it becomes:

|\psi_{\text{after}}\rangle = |1\rangle

If we immediately repeat the same Z-basis measurement, we obtain the same result with certainty, provided the system does not evolve and no noise affects it between measurements.

Repeated measurement and eigenstates

An eigenstate already has a definite value for the observable being measured. An ideal measurement of that observable returns the corresponding value with certainty.

This relationship is written as:

A|a\rangle = a|a\rangle

Here, A represents the observable, ∣a⟩ is one of its eigenstates, and a is the measured value.

The measurement leaves the state inside the same eigenspace. However, the system may still have uncertain outcomes when measured using another, incompatible observable.

How do quantum states work in quantum computing?

A quantum computer stores and processes information using qubits. A quantum circuit prepares an initial quantum state, applies gates to change it, and measures the final state to produce a classical result.

Quantum algorithms do not usually calculate the answer in one direct step. Instead, they control amplitudes and phases so that incorrect possibilities cancel while useful outcomes become more likely.

A qubit is a two-state system

A classical bit can hold either 0 or 1. A qubit uses the basis states $|0\rangle \quad \text{and} \quad |1\rangle$ , but it can also exist in a superposition of both:

|\psi\rangle = \alpha|0\rangle + \beta|1\rangle

The amplitudes α and β determine the probabilities of measuring 0 or 1.

A single qubit therefore has two computational basis states but infinitely many possible pure states. For n qubits, there are $2^n$ computational basis states, along with a continuous range of possible superpositions over them.

The Bloch sphere represents one qubit

The Bloch sphere provides a visual representation of a single-qubit pure state. Every point on its surface represents a different pure state, apart from an unobservable global phase.

A general state on the Bloch sphere is:

|\psi\rangle = \cos\left(\frac{\theta}{2}\right)|0\rangle + e^{i\phi}\sin\left(\frac{\theta}{2}\right)|1\rangle

The angles $\theta \quad \text{and} \quad \phi$ locate the state on the sphere. Pure states appear on the surface, while mixed states appear inside the sphere, often called the Bloch ball.

The Bloch sphere is useful for understanding how single-qubit gates rotate a state. However, it cannot directly represent systems containing several qubits.

Quantum gates change the state

Quantum gates transform a quantum state in a controlled and reversible way. In the ideal circuit model, each gate acts as a unitary transformation.

The Hadamard gate, for example, changes $|0\rangle$ into an equal superposition:

H|0\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}

Different gates perform different operations:

Hadamard gates create or remove superpositions.
Phase gates change the relative phase between amplitudes.
Rotation gates move a qubit around the Bloch sphere.
Controlled gates connect the behavior of multiple qubits and can create entanglement.

Relative phase is important because quantum algorithms depend on interference. Some possible computational paths reinforce one another, while others cancel. The algorithm arranges this interference so that useful measurement results become more probable.

Multi-qubit states and entanglement

A system containing n qubits requires $2^n$ complex amplitudes in its general statevector. Each amplitude belongs to one computational basis state, ranging from $|00\ldots0\rangle \quad \text{to} \quad |11\ldots1\rangle$ .

Entanglement occurs when the state of the complete system cannot be separated into independent states for its individual qubits. A common two-qubit example is the Bell state:

|\Phi^+\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}

When measured in the computational basis, this state produces either 00 or 11, each with a 50% probability. The two qubits always produce matching results.

The number of amplitudes doubles with every additional qubit. This exponential growth is why classical computers struggle to store and simulate arbitrary large quantum systems.

Measurement produces classical output

A quantum circuit ends by measuring some or all of its qubits. Measurement converts the final amplitudes into probabilities for classical bit strings.

If the final one-qubit state is:

|\psi\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}

a computational-basis measurement produces 0 or 1, each with a 50% probability.

One run provides only one result but to analyze the distribution of outputs we usually need to execute a circuit many times.

Short Qiskit example

The following Qiskit example creates one qubit, applies a Hadamard gate, and calculates the resulting statevector:

import sys
!{sys.executable} -m pip install qiskit
from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

qc = QuantumCircuit(1)
qc.h(0)

state = Statevector.from_instruction(qc)
print(state)
print(state.probabilities_dict())

import sys
!{sys.executable} -m pip install qiskit
from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

qc = QuantumCircuit(1)
qc.h(0)

state = Statevector.from_instruction(qc)
print(state)
print(state.probabilities_dict())

The qubit begins in $|0\rangle$ . The Hadamard gate prepares the $|+\rangle$ state, with an amplitude of $\frac{1}{\sqrt{2}}$ for both $|0\rangle \quad \text{and} \quad |1\rangle$ . The probability output therefore assigns 0.5 to each result.

Developers can prepare, transform, simulate, and measure these states using different quantum programming languages and frameworks.

How are quantum states prepared, evolved, and verified?

Working with a quantum state involves three main stages: preparing the intended state, controlling how it changes, and checking whether the final state matches the target.

Real quantum hardware is affected by noise and interactions with its surroundings. So researchers must verify the state instead of assuming that every operation worked perfectly.

State preparation

State preparation creates the starting state required for an experiment or algorithm.

In circuit-based quantum computing, qubits usually begin in the state:

|0\ldots0\rangle

Quantum gates then transform this initial state into the desired state. For example, applying a Hadamard gate to a qubit in $|0\rangle$ prepares the $|+\rangle$ state.

The physical preparation method depends on the hardware. Quantum systems may use laser or microwave pulses, cooling, optical pumping, or photon detection. Regardless of the method, the goal is the same: repeated preparations should produce consistent measurement statistics.

State evolution

After preparation, the quantum state changes over time or in response to quantum gates.

For an isolated system, the Schrödinger equation describes this evolution:

i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle = H|\psi(t)\rangle

Here, H represents the system’s energy and determines how the state evolves.

In an ideal quantum circuit, gates perform controlled transformations that preserve the state’s total probability. However, real systems interact with their surroundings. Heat, electromagnetic interference, imperfect controls, and unwanted coupling can damage the state.

This process is known as decoherence. It can weaken superposition and entanglement, causing the actual state to move away from the intended one. Density matrices and quantum channels provide the tools needed to describe these noisy systems.

Quantum state tomography

A single measurement cannot reveal an unknown quantum state completely. Measurement provides only one result and usually changes the state in the process.

Quantum state tomography solves this problem by preparing many copies of the same state and measuring them in different bases. Researchers combine the resulting statistics to estimate the original state.

A single-qubit density matrix can be reconstructed from measurements along the $x,\ y,\ \text{and}\ z$ directions:

\rho = \frac{1}{2}\left(I + r_x X + r_y Y + r_z Z\right)

Here, $r_x,\ r_y,\ \text{and}\ r_z$ come from the average measurement results in the three directions.

Full tomography becomes expensive as the number of qubits increases because the number of values required to describe the system grows rapidly. Large experiments therefore often measure only the properties needed for the task or use more scalable methods such as classical shadows and randomized benchmarking.

Quantum state fidelity

Fidelity measures how closely a prepared state matches a target state.

For two pure states, one common definition is:

F = |\langle\psi|\phi\rangle|^2

Here, $|\psi\rangle$ represents the target state and $|\phi\rangle$ represents the prepared state.

The fidelity ranges from 0 to 1:

F=1 means the two states are equivalent.
F=0 means the states are completely distinguishable.
A value between 0 and 1 shows partial similarity.

Some sources define fidelity without squaring the overlap. Technical work should therefore state which convention it uses. Qiskit’s state_fidelity function uses the squared convention.

Researchers use fidelity to evaluate state preparation, compare simulations with hardware results, and check the quality of a state reconstructed through tomography.

A practical verification process

A small quantum experiment can follow this process:

Define the target quantum state.
Prepare many copies using the same gates or physical controls.
Measure the copies in the required bases.
Use the results to estimate the properties of the prepared state.
Perform full tomography only when a complete reconstruction is necessary.
Compare the estimate with the target using fidelity or another suitable metric.
Investigate noise and calibration errors when the result moves away from the target.

This process connects the mathematical description of a quantum state with real experimental work. Preparation creates the state, controlled evolution transforms it, and verification determines whether the hardware produced the intended result.

Key takeaways

A quantum state describes a quantum system and helps predict how it will evolve and what measurement results it may produce. Pure states use statevectors or wavefunctions, while density matrices describe both pure and mixed states. In quantum computing, gates transform quantum states by changing their amplitudes and phases, entanglement connects multiple qubits, and measurement converts quantum information into classical results.

Frequently asked questions about quantum states

What is a quantum state in simple terms?

A quantum state is the mathematical description that quantum mechanics uses to predict what can happen when a system is measured and how the system changes over time. This is the simplest quantum state meaning: the state stores predictive information, while a measurement produces one specific classical outcome.

Is a quantum state the same as a wavefunction?

No. A wavefunction is one representation of a quantum state, usually in a position or momentum basis. Pure states can also be written as kets or statevectors, and mixed states require density matrices. So the wavefunction is important, but the broader quantum state definition is not limited to wave mechanics.

What is a pure state in quantum mechanics?

Pure state in quantum mechanics is a state that can be represented by one normalized statevector $|\psi\rangle$ . A pure state’s density matrix is $\rho = |\psi\rangle\langle\psi|$ and satisfies $\operatorname{Tr}(\rho^2) = 1$ . A coherent superposition can be pure; a classical probabilistic mixture is generally mixed.

What is the difference between a pure state and a mixed state?

A pure state has a single statevector description. A mixed state uses a density matrix to represent classical uncertainty over preparations or a reduced subsystem of a larger correlated system. Two states can produce the same statistics in one basis while still differ in coherence and therefore in other measurement bases.

How many quantum states can a qubit have?

A qubit has two computational basis states, $|0\rangle \quad \text{and} \quad |1\rangle$ , but infinitely many possible pure quantum states formed by normalized superpositions $\alpha|0\rangle + \beta|1\rangle$ . This resolves the common how many quantum states are there question: two basis labels do not mean only two possible states.

How does observation change a quantum state?

How does observation change quantum state? In the standard projective model, measurement produces an allowed outcome with a probability set by the pre-measurement state, then updates the state to the corresponding outcome subspace. The result depends on the measurement basis, so measurement is not just reading a basis-independent hidden value.

What is an eigenstate?

To define eigenstate, choose an operator $A$ . A state $|a\rangle$ is an eigenstate when $A|a\rangle = a|a\rangle$ . The value a is the eigenvalue. This eigenstate definition is always relative to an operator. A state can be an eigenstate of one observable and a superposition of eigenstates of another.

Can two particles have the same quantum state?

Yes in many contexts, but particle statistics matter. Distinct systems can be prepared in the same single-particle state. Identical bosons can occupy the same one-particle state. Identical fermions in the same antisymmetrized system cannot occupy the same complete single-particle state because of the Pauli exclusion principle.

What is the difference between a quantum state and a quantum number?

A quantum state is the full mathematical object used to make predictions. A quantum number is a label, often an eigenvalue or index, associated with a particular observable or basis. Several quantum numbers may label a basis state, but those labels do not always capture an arbitrary superposition or mixed state.

How is a quantum state measured or reconstructed?

You do not learn an arbitrary unknown quantum state from one measurement. A measurement gives one outcome. To reconstruct a state, quantum state tomography repeats the preparation many times, measures in multiple bases, and estimates a density matrix or statevector consistent with the data. Large systems often use more targeted verification methods.