The Bloch Sphere

A handheld GPS locates you on Earth with two numbers: latitude and longitude. Your phone does not need to know the curve of every road you have walked, every step you have taken, every mountain in your way — it needs exactly two angles, and those two angles place you on the globe. Two numbers, one sphere, every point on Earth covered.

Every state a single qubit can be in has the same structure. The qubit is sitting in some superposition of |0\rangle and |1\rangle, with a complex amplitude attached to each. On paper the amplitudes look like four separate numbers — two real parts and two imaginary parts — and the bookkeeping gets ugly fast. But once you strip away the numbers that do not matter, you are left with exactly two angles. Those two angles locate the state on a sphere. The sphere is called the Bloch sphere, and it is the single most useful picture in all of quantum computing.

If you can draw a point on a sphere, you can draw any single-qubit state. If you can rotate a sphere, you can apply any single-qubit gate. Learn this picture once, and the next twenty chapters of this track stop being algebra and start being geometry.

Why two angles are enough

A general single-qubit state is

where \alpha and \beta are complex numbers. A complex number needs two real numbers to describe it (its real part and its imaginary part, or equivalently its magnitude and its phase). So |\psi\rangle looks like it has four real parameters.

Four is too many. The sphere picture only works because two of those four parameters carry no physical information and can be thrown away.

The first constraint is normalization. Every state must satisfy |\alpha|^2 + |\beta|^2 = 1, because the probabilities of measuring 0 and 1 have to add up to 1. That is one equation relating the four numbers, so it eliminates one degree of freedom. Four minus one is three.

The second constraint is global phase. If you multiply |\psi\rangle by an overall phase factor e^{i\gamma}, you get a new vector e^{i\gamma}|\psi\rangle that, as far as any experiment is concerned, is the same state. Every measurement probability is unchanged; every expectation value is unchanged. The phase \gamma is unobservable — it is like an arbitrary zero for a clock. So you get to pick one phase convention and throw the freedom away. Three minus one is two.

Two real parameters. A two-dimensional surface. The natural candidate is a sphere.

Why a sphere and not a square or a disc: the surface is finite (states are normalised) and has no boundary (you can rotate between any two states continuously). The only simple surface with both those properties is a sphere.

The map of the sphere — six cardinal states

Before you learn the full parameterisation, it helps to know six specific states and where they sit. These six come up constantly.

The poles — the computational basis.

• |0\rangle is at the north pole (top of the sphere).
• |1\rangle is at the south pole (bottom).

The equator at the X-axis — the equal superpositions with real amplitudes.

• |+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) is where the positive X-axis pierces the equator.
• |-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle) is where the negative X-axis pierces.

The equator at the Y-axis — the equal superpositions with imaginary amplitudes.

• |{+}i\rangle = \frac{1}{\sqrt{2}}(|0\rangle + i|1\rangle) is on the positive Y-axis.
• |{-}i\rangle = \frac{1}{\sqrt{2}}(|0\rangle - i|1\rangle) is on the negative Y-axis.

Three facts fall out of this map that every quantum programmer eventually internalises:

1. Opposite points on the sphere are orthogonal states. |0\rangle and |1\rangle are on opposite poles, and they are the two basis states — orthogonal. |+\rangle and |-\rangle are on opposite ends of the X-axis, and they too are orthogonal. Antipodal on the sphere ⟺ orthogonal in Hilbert space.

2. Every point on the equator is an equal-superposition state. Whether you land at |+\rangle or |{+}i\rangle or anywhere between them, the probability of measuring 0 or 1 is exactly 50-50. The only difference between equatorial states is phase — which determines how the state will interfere later, but not what the next measurement in the Z-basis will give.

3. Latitude tells you measurement probability. The higher up on the sphere you are (closer to |0\rangle), the more likely you are to measure 0. At the north pole, you measure 0 with probability 1. At the equator, 50-50. At the south pole, you measure 1 with probability 1.

Why the last fact is not a rule you have to memorise: it falls directly out of the formula you are about to see. The probability of measuring 0 is \cos^2(\theta/2), which is 1 at \theta=0 (north pole), \tfrac{1}{2} at \theta=\pi/2 (equator), and 0 at \theta=\pi (south pole).

The formula — two angles, one sphere

Every pure single-qubit state can be written as

where \theta \in [0, \pi] is the polar angle (measured down from |0\rangle) and \varphi \in [0, 2\pi) is the azimuthal angle (measured around the equator from the |+\rangle direction).

Reading the formula. The amplitude on |0\rangle is \cos(\theta/2). This is a real number that starts at 1 (when \theta = 0, at the north pole) and falls to 0 (when \theta = \pi, at the south pole). The amplitude on |1\rangle is \sin(\theta/2), times a complex phase e^{i\varphi}. Its magnitude is 0 at the top and 1 at the bottom. The phase \varphi rotates you around the equator — it tells you which kind of superposition you have, once \theta has told you how much of one.

Why the half-angle. A natural question: why \theta/2 and not \theta? The answer is a consequence of how quantum amplitudes relate to probabilities. The Bloch angle \theta runs from 0 (where you measure 0 with probability 1) to \pi (where you measure 1 with probability 1). But the amplitudes have to run from (1, 0) to (0, 1) over that range, which is a quarter-turn, not a half-turn. Dividing the angle by 2 makes the amplitudes turn through the right arc.

Another way to put it: antipodal points on the sphere are orthogonal states, and orthogonal states have no overlap. But \cos and \sin only give zero overlap when their arguments differ by \pi/2, not \pi. Dividing by 2 restores the match.

Worked examples — placing states on the sphere

Gates are rotations

The payoff of the Bloch sphere is not just that states become points. It is that every single-qubit gate becomes a rotation of the sphere. This is not a metaphor. Every 2 \times 2 unitary matrix — the Pauli gates, the Hadamard, the phase gates, rotation gates, anything — has a corresponding rotation axis and rotation angle on the Bloch sphere.

Some rotations you will meet almost immediately:

• The Pauli X gate rotates the sphere by 180° about the x-axis. |0\rangle (north pole) swings to |1\rangle (south pole). |+\rangle stays put (it is on the rotation axis). |{+}i\rangle swings to |{-}i\rangle.
• The Pauli Z gate rotates by 180° about the z-axis. |0\rangle and |1\rangle are on the axis, so they do not move (up to a phase). |+\rangle swings to |-\rangle.
• The Hadamard gate rotates by 180° about the axis halfway between x and z. |0\rangle swings to |+\rangle and back. This is why applying H twice returns you to the starting state.
• The rotation gates R_x(\theta), R_y(\theta), R_z(\theta) rotate by a continuous angle \theta about their named axis. These are the workhorses of variational algorithms — you will see them in Part 16.

The geometric picture replaces algebraic pattern-matching with visual reasoning. Want to know what H|0\rangle is? Imagine the Hadamard axis (halfway between x and z). Imagine rotating the sphere 180° about it. |0\rangle at the north pole swings around and lands at... |+\rangle on the positive x-axis. That is the answer, and you did not compute a single matrix product.

Every single-qubit chapter in this track (Ch 23 onwards) is a special case of this one picture.

Measurement is projection onto an axis

The measurement postulate says: if you measure in the Z-basis (the computational basis), the probability of outcome 0 is |\alpha|^2 = \cos^2(\theta/2).

Now read that geometrically. The state |\psi\rangle is a point on the Bloch sphere at polar angle \theta from the |0\rangle pole. Its projection onto the z-axis — the vertical component — is \cos(\theta). And \cos^2(\theta/2) = \tfrac{1}{2}(1 + \cos\theta). So:

Measuring along the z-axis projects the Bloch vector onto z, and the result — scaled and shifted into [0, 1] — is the probability of getting outcome 0. After the measurement, the state collapses onto whichever pole the outcome named: if the result is 0, the state becomes exactly |0\rangle (the north pole); if the result is 1, the state becomes exactly |1\rangle (the south pole).

The generalisation matters. If you measure in some other basis — say, the X-basis, whose eigenstates are |+\rangle and |-\rangle — the probability of outcome |+\rangle is the projection of the Bloch vector onto the positive x-axis, not the z-axis. Every orthonormal measurement basis corresponds to an axis through the centre of the sphere, and the probabilities are the projections.

This is why the Bloch sphere is not just a visualisation — it is an engine for reasoning about measurement.

Common confusions

• "The radius of the sphere is in units of 'quantum-ness.'" No — the sphere has radius 1 and the units are nothing. Every pure state sits on the surface; the interior of the sphere is reserved for mixed states (classical probabilistic mixtures of pure states), which you will meet in Part 13.

• "If I add two Bloch vectors, I get the Bloch vector of the sum of states." No. Adding states in Hilbert space is linear, but the map from state to Bloch vector is nonlinear (it goes through \theta/2). You cannot add Bloch vectors to get a new state; you have to translate back to amplitudes, add, renormalise, and translate back.

• "Antipodal points on the sphere differ by a global phase." No — antipodal points are orthogonal states, which is stronger than differing by a global phase. |0\rangle and |1\rangle are antipodal and have no overlap; |0\rangle and e^{i\gamma}|0\rangle are the same point on the sphere regardless of \gamma.

• "The Bloch sphere works for any number of qubits." No — the sphere only describes a single qubit's state. Two qubits live in a four-dimensional complex space, not an eight-dimensional real sphere. There is no clean analog of the Bloch sphere for multi-qubit states, and the search for one has been going on for decades. When this picture stops working, you will know you have left single-qubit land.

• "The factor of \theta/2 is a weird convention I should ignore." It is a convention, but it is not arbitrary — it is what makes antipodal points orthogonal and what aligns the rotation angles on the sphere with the physical angles of gates. Every textbook uses it; every simulator uses it; you will see it in every Qiskit function that touches single-qubit states. Learn to read it at a glance.

Going deeper

If you are just here to understand what the Bloch sphere is, you have it — a point, two angles, rotations, projections. The rest of this section is for readers who want the connections to density matrices, rotation-gate parameterisations, and the reason the sphere has exactly three independent rotation axes.

Density matrices and the Bloch vector

Write a pure state |\psi\rangle as a density matrix \rho = |\psi\rangle\langle\psi|. For a single qubit, any density matrix can be written

where \vec{r} = (r_x, r_y, r_z) is a real three-component vector called the Bloch vector and \vec{\sigma} = (X, Y, Z) is the vector of Pauli matrices.

For pure states, |\vec{r}| = 1 and the vector lies on the surface of the Bloch sphere. For mixed states, |\vec{r}| < 1 and the vector lies inside. At \vec{r} = 0, you have the maximally mixed state \rho = I/2 — the centre of the sphere, representing complete ignorance of which of |0\rangle or |1\rangle you have.

In the pure-state parameterisation you already learned, the Bloch-vector components are

These are just the Cartesian coordinates on the unit sphere — no quantum magic, just geometry.

Why three rotation axes are independent

A general single-qubit unitary (up to global phase) has three real parameters — the Euler angles, or equivalently an axis \hat{n} (two parameters) and an angle \alpha (one parameter). The rotation it induces on the Bloch sphere is

The three independent rotations — R_x, R_y, R_z — are rotations about the three coordinate axes. Any rotation in 3D can be decomposed into rotations about exactly three orthogonal axes (the ZYZ or ZXZ Euler decomposition). That is why any single-qubit unitary can be written as

for some angles (\alpha, \beta, \gamma, \delta). The Bloch sphere is not just a pretty picture — it is the reason single-qubit gate decomposition is possible.

Where the picture breaks

For two qubits, you might hope for a 6-sphere or a 15-sphere or some higher-dimensional analog. It does not happen cleanly. The state space of two qubits is the complex projective space \mathbb{CP}^3, which is a 6-dimensional real manifold but is not topologically a sphere — it has nontrivial curvature and no global coordinate chart. The Schmidt decomposition (Part 5) is the right tool for visualising two-qubit entanglement, not a higher-dimensional Bloch sphere.

When you see someone draw "two Bloch spheres side by side" to represent a 2-qubit state, that picture is only valid for product states — states that can be written as |\psi_1\rangle \otimes |\psi_2\rangle. The moment the state is entangled, the side-by-side picture stops working, because neither qubit has a pure state of its own. (Each subsystem's reduced density matrix is mixed — its Bloch vector is shorter than 1 — and the individual Bloch spheres cannot capture the correlations between the two qubits.)

Where this leads next

• Pauli X, Y, Z — the three gates that rotate the sphere about its three coordinate axes.
• The Hadamard gate — the rotation that swaps the computational basis with the X-basis.
• Rotation gates R_x, R_y, R_z — continuous-angle rotations that form the backbone of variational algorithms.
• Measurement in arbitrary bases — how to measure along any axis of the sphere by rotating first.
• A preview of density matrices — how the interior of the sphere describes mixtures and subsystems.

References

1. Nielsen and Chuang, Quantum Computation and Quantum Information (2010), §1.2 and §4.2 — Cambridge University Press.
2. John Preskill, Lecture Notes on Quantum Computation, Ch. 2 — theory.caltech.edu/~preskill/ph229.
3. Wikipedia, Bloch sphere — the canonical reference for the (θ, φ) parameterisation and rotations.
4. Wikipedia, Qubit — single-qubit states, gates, and the Bloch-vector representation.
5. Qiskit Textbook, Representing Qubit States — interactive Bloch-sphere visualisations and code.
6. IBM Quantum Learning, Single-qubit gates — gate-as-rotation picture with live simulators.