I'll be the first to admit it: qubits are hard to understand. We all need helpful visualizations to wrap our heads around complicated things.
The Bloch representation is a simple mapping from qubit states to points on a sphere.
Try moving the sliders to see how the Bloch sphere changes!
The Bloch sphere helps with two key pieces of intuition.
Intuition #1
If you measure the qubit, your probability of getting a result depends on how aligned the arrow is with that result's direction. For example, a point on the top pole has a 100% chance of measuring to |0⟩; a point on the equator gives 50/50 odds.
At the top pole
100% chance of measuring |0⟩
On the equator
50-50 chance of measuring |0⟩ or |1⟩
(but 100% chance of measuring |+⟩)
You don't have to know this, but the exact rule is P = cos²(α/2), where α is the angle between the arrow and the direction of the result. This is known as the Born rule.
Intuition #2
The same alignment rule holds for any measurement direction, not just up-and-down. Every axis through the sphere defines a pair of measurement outcomes at its two poles. Measure along the x-axis and the outcomes are |+⟩ and |−⟩. Measure along the y-axis and they're |+i⟩ and |−i⟩.
How do we calculate the Bloch representation?
We know we can write the state of a qubit like this:
α and β are complex numbers, with |α|² + |β|² = 1
From those two complex amplitudes, we can compute θ and φ:
To go from the Bloch angles to a statevector:
θ ∈ [0, π] tilts you between |0⟩ at the top and |1⟩ at the bottom. φ ∈ [0, 2π) spins you around the equator.
Gates and measurements
When you view qubits using the Bloch sphere, a lot of the math of quantum computing becomes intuitive.
Single-qubit gates are just rotations. Every single-qubit gate is a rigid 3D rotation of the sphere about some axis, by some angle. Pick a gate below and watch the state vector arc to its new direction.
Click a gate to watch the state vector rotate.
Measurements collapse the sphere to whichever pair of antipodal points you measure along. Measure along z and the state snaps to |0⟩ or |1⟩. Along x, it snaps to |+⟩ or |−⟩. The probability of each outcome is set by how far the current state's arrow already leans toward that pole.
Pick an axis. The state collapses to one of the two opposite poles along it, with probability set by where the vector currently points.
Are there Bloch spheres in real life?
Yes. For some kinds of qubits, the Bloch sphere isn't a metaphor.
The clearest example is an electron spin qubit. Electrons are tiny magnets: they have a magnetic moment that points in some direction in 3D space. That direction is the Bloch vector. North pole of the Bloch sphere = spin pointing up in the lab. South pole = spin pointing down. Anywhere else on the sphere = spin pointing somewhere else.
Some pedants would point out that, while it's useful to think of the electron's spin as pointing in a direction, you can only ever measure spin as “up” or “down” along whichever axis you choose. So strictly speaking, the spin doesn't “point” anywhere until you measure it.
When physicists write down the state of such an electron, they often switch back and forth: convert the lab direction to (θ, φ), pull the statevector out of the formula, do some algebra, then convert back.
What is relative phase?
Relative phase is the name given to the difference of the phases between α and β, when we write the statevector like this:
The relative phase is φ = φβ − φα.
It's hard to get a feel for what relative phase actually does, until you look at how the statevector formalism was invented in the first place.
The way we usually write a quantum state — |ψ⟩ = α|0⟩ + β|1⟩ — was invented by John von Neumann, often called one of the smartest minds of the 20th century. But even von Neumann himself came to doubt that it was the right way to describe quantum mechanics. In 1935 he wrote to the mathematician Garrett Birkhoff:
“I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space anymore.”
The Bloch sphere is more intuitive: it maps directly to how qubits actually behave. So the relative phase on a statevector is best thought of as the formalism's way of carrying the rest of the Bloch-sphere information.
If we only stored α and β as real numbers, they'd only tell us probabilities in the |0⟩/|1⟩ basis. But we can measure in any basis (including |+⟩/|−⟩ and |+i⟩/|−i⟩), and to predict those outcomes we also need to know φ.
Relative phase, in a statevector, is just a way of storing the rest of the 3D information from the Bloch sphere. It's embedded in the statevector in a way that extends easily to more than one qubit, which we'll get to in the next part.
If Bloch spheres are so intuitive, why use statevectors?
While statevectors are less intuitive, they let us deal with multiple qubits easily. There are simple rules for how to combine them, manipulate them, and so on. They also turn out to fit very neatly onto a computer.
I'm reminded of a line from the mathematician Michael Atiyah, on the trade between geometry and algebra:
“Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine.”
Statevectors are the same deal. Give up your visual intuition and get a way to track entanglement with just numbers. For now we'll happily make the trade for two qubits and up. For one qubit, the Bloch sphere is plenty.
For more on multi-qubit visualizations, read our whitepaper on visualizing multi-qubit states.
Side note: degrees of freedom
Why does a single-qubit state, which is described by four real numbers, collapse so neatly down to just two angles on a sphere?
Start with 4 real numbers. α has two real numbers in it, and β has two.
Minus normalization, leaves 3. Since |α|² + |β|² = 1, this removes one degree of freedom.
Minus global phase, leaves 2. As you must know already, multiplying a statevector by any global phase doesn't affect anything. Therefore, we remove one more degree of freedom.
That's it. Two real degrees of freedom, and they're exactly θ and φ on the sphere.
How does a model Bloch sphere work?
Qubi is a handheld, interactive model of a qubit. When the qubit is unentangled with other qubits, it shows you its Bloch sphere representation with a little white light!
It's a great way to build strong intuition for the topic in this guide, and plenty more.
Bloch sphere FAQ
Is the Bloch sphere a real, physical sphere?+
No. The Bloch sphere is a mathematical visualization of a single-qubit state, not a physical object. Every pure state of one qubit corresponds to exactly one point on the surface of the unit sphere. Some quantum technologies (like Qubi) build a tangible model of it, but the underlying object lives in math.
What does a point on the Bloch sphere represent?+
A direction. The arrow from the center of the sphere to a point on the surface tells you everything measurable about a pure single-qubit state: the probability of every measurement outcome, in every basis. The north pole is |0⟩, the south pole is |1⟩, and points around the equator are equal-weight superpositions with different relative phases.
How do you compute the Bloch vector from a state?+
Write the state as |ψ⟩ = cos(θ/2)|0⟩ + eiφ sin(θ/2)|1⟩. Then the Bloch vector is (sin θ cos φ, sin θ sin φ, cos θ). θ is the polar angle from the z-axis and φ is the azimuthal angle from the x-axis. Equivalently, the components of the Bloch vector are the expectation values of the three Pauli operators, ⟨X⟩, ⟨Y⟩, ⟨Z⟩.
Why is it called the Bloch sphere?+
Named after Felix Bloch, the Swiss-American physicist who introduced the representation in 1946 in the context of nuclear magnetic resonance. The same geometry was independently used by Henri Poincaré for the polarization of light, so it's sometimes called the Bloch–Poincaré sphere.
Does the Bloch sphere only work for one qubit?+
Yes — the surface picture is exact only for a single qubit. Two qubits live in a 6-dimensional state space that can't be drawn this cleanly. Useful generalizations exist (e.g., projecting one qubit at a time), but no two-qubit picture preserves all the same intuition.
Is the Bloch sphere only for pure states?+
The surface represents pure states. Mixed states — statistical mixtures of pure states — sit inside the sphere. The whole solid ball is the Bloch ball; the maximally mixed state sits exactly at the center.
Want to play with one? Open the interactive Bloch sphere visualizer.