A calculation-first introduction

Quantum Information, Explained

What can quantum states be used for, and what can they not do? Fifteen chapters answer that one question at a time: why two sources with identical statistics can differ, how two gates entangle a pair, what must be sent to teleport a qubit, how flipping a sign finds a marked item, and how an error is undone without reading the state it damaged.

The chapters assume complex numbers, linear algebra and elementary probability, and introduce quantum notation as it is needed. Each opens with a question, answers it with a worked calculation and a figure, keeps longer derivations in optional sections, and ends by saying what was settled and why the next chapter follows. Read in order; each chapter names and links the earlier sections it depends on.

If quantum mechanics itself is new, the companion guide, Quantum Mechanics, Explained, covers amplitudes, spin, measurement and the formalism more slowly. It is a recommended starting point, not a requirement, and chapters here link to its specific essays where they help.

Start with qubits and relative phase →

Each chapter ends with two problems with expandable worked answers. Try them before opening the answers; being able to reproduce a calculation is a better checkpoint than recognising its final formula.

States and measurements

What a measurement reveals, what it leaves behind, and what one matrix can hold when information is incomplete.

1. 01 Qubits, amplitudes, and relative phaseHow can two preparations give identical measurement statistics and still be different states? A second measurement reveals the relative phase between a qubit's amplitudes.
2. 02 Measurement: probabilities and state changesA qubit reads 0 with probability ¾. What state is it in afterwards, and do the probabilities decide that? Two apparatuses with identical statistics leave different states.
3. 03 Density matrices and the state of a subsystemCan any measurement tell a coin flip between |0⟩ and |1⟩ from a coin flip between |+⟩ and |−⟩? One matrix holds every prediction, for mixtures and for half of an entangled pair.

Gates and control

What a circuit does to amplitudes, and how a physical pulse produces a gate.

4. 04 Gates, circuits, and universalityWhat does each gate do to the amplitudes, and what can a controlled gate change that one-qubit gates cannot? Two gates followed from |00⟩ to a Bell pair.
5. 05 Rotations and physical control of a qubitFor how long, and about which axis, must a qubit be driven to reach a chosen measurement probability? From a Hamiltonian to a rotation of the Bloch vector.

Entanglement and communication

What cannot be copied, what entanglement predicts that no local model can, and what it takes to move a state.

6. 06 Why an unknown state cannot be copiedA machine copies |0⟩ and |1⟩. What does it do to |+⟩, and could any machine copy every state? Linearity forces entanglement instead of copies.
7. 07 Entanglement and the Schmidt decompositionGiven a two-qubit state’s amplitudes, is it a product, and if not, how far from one? The coefficient matrix and its singular values answer both.
8. 08 Bell correlations and the CHSH inequalityFixed instructions from the source cap a combination of correlations at 2; a Bell pair gives 2√2. What that rules out, and what it leaves standing.
9. 09 Teleportation and dense coding, step by stepWhat must Alice send so Bob recovers an unknown qubit, and why can he not have it sooner? Every register tracked, plus the reverse protocol.

Quantum algorithms

How phases and a final change of basis answer questions with fewer queries, and what those query counts do and do not measure.

10. 10 Phase kickback and the Deutsch–Jozsa algorithmHow can one query decide whether f(0) = f(1) when neither value is read? Function values become signs, and a final change of basis reads their pattern.
11. 11 Fourier sampling, phase estimation, and periodsOne Fourier sample returns one integer. What it reveals about a hidden period, why peaks spread, and the classical work that finishes the job.
12. 12 Grover search as two reflectionsHow can a sign flip, followed by a second operation, make an answer more likely, and why must we stop repeating that pair? Two reflections make a rotation that approaches the marked item and then passes it.

Noise and information

What noise does, how encoded amplitudes are protected without being read, and how many bits a qubit can carry.

13. 13 Quantum channels: discarded environments and noiseA qubit decays while no one watches. What is left of its state, what a recorded emission changes, and why noise need not make a state more mixed.
14. 14 Error correction without reading the logical stateHow can an error be found and undone without learning the encoded state? Parity checks reveal which qubit flipped and nothing about the amplitudes.
15. 15 Entropy, accessible information, and the Holevo boundHow many bits can one transmitted qubit carry, and what does a shared entangled pair add? Entropy, information, and the resources each protocol uses.