Complex Numbers & Dirac Notation
Why quantum amplitudes are complex, and how to read ⟨bra|ket⟩ notation
By the end of this topic you'll be able to
Quantum mechanics stands alone among the sciences in how it calculates probabilities: it always evaluates a probability p as the mod-square of a complex number A, so p = |A|². Binney and Skinner call A the probability amplitude for p, and note that nobody fully understands why nature insists on this extra complex-valued layer beneath ordinary probability.
A complex number z = a + bi, with real part a, imaginary part b, and i = √-1, can equally well be written as a length r and an angle θ: z = re^{iθ}. That angle θ is the phase, and it turns out to be one of the most consequential ideas in the entire subject.
Dragging θ alone (r fixed) rotates z around the circle without changing |z|² — exactly the kind of change a single measurement can't see directly, but that shows up as interference once two amplitudes combine.
The rule that makes probability amplitudes strange is how they combine. If an outcome can happen by two mutually exclusive routes S or T, the amplitude for 'S or T' is the sum A(S)+A(T) — not the probabilities p(S)+p(T) as ordinary probability theory would demand.
Mod-squaring that sum of amplitudes produces an extra cross term, 2Re(A(S)A(T)), with no counterpart in classical probability. Binney and Skinner call any departure from simple probability addition caused by this term quantum interference* — and it is this cross term, entirely dependent on the relative phase of A(S) and A(T), that is absent whenever a route is somehow monitored or distinguished.
A qubit's state α|0⟩ + β|1⟩ carries exactly this structure: α and β are complex probability amplitudes, and the probability of measuring 0 is |α|², the modulus-squared of α.
Two states can share identical measurement probabilities while differing only in phase — invisible to a single measurement in the way just described, yet exactly the ingredient that makes interference, and with it every quantum speedup in this curriculum, possible.
Dirac introduced the symbol |ψ⟩, a ket, to denote a complete set of amplitudes for a system, independent of which particular measurement's amplitudes you happen to be using to describe it — in the same way a position vector describes a geometrical point independently of which coordinate system locates it. Its adjoint, the bra ⟨ψ|, is the corresponding linear function that extracts amplitudes from kets.
Put a bra and a ket together, ⟨φ|ψ⟩, and the result is the inner product: a single complex number measuring the overlap between the two states, exactly the ⟨v,w⟩ of ordinary linear algebra wearing quantum notation. The same notation composes further — |0⟩⟨0| is a matrix, and ⟨0|H|1⟩ a single number — which is exactly why, once it clicks, it becomes so much easier to read the primary literature of the field.
Try It Yourself
Let |ψ⟩ = (1/√2)|0⟩ + (i/√2)|1⟩. Compute the inner product ⟨0|ψ⟩, then find the probability of measuring 0.
- 1⟨0|ψ⟩ picks out the coefficient of |0⟩ in the expansion of |ψ⟩, since ⟨0|0⟩ = 1 and ⟨0|1⟩ = 0. So ⟨0|ψ⟩ = 1/√2.
- 2The Born-rule probability is the modulus-squared of this amplitude: |⟨0|ψ⟩|² = |1/√2|² = 1/2.
- 3Now check normalization as a sanity test: |1/√2|² + |i/√2|² = 1/2 + 1/2 = 1, using |i|² = 1 since i = e^{iπ/2} has modulus 1 — confirming |ψ⟩ is a valid state.
⟨0|ψ⟩ = 1/√2, and the probability of measuring 0 is 1/2 — exactly like a fair coin, even though the |1⟩ amplitude carries a phase of i that |0⟩'s doesn't.
Reference
| Complex number | a,b real; i²=-1; r = |z| (modulus), θ = phase | |
| Modulus squared | Gives the Born-rule probability of an amplitude | |
| Ket | A column vector — a quantum state | |
| Bra | The conjugate transpose of a ket — a row vector | |
| Inner product | Overlap between states φ and ψ, a complex number |
Quick Check
If a qubit's amplitude for |1⟩ is β = (1/√2)e^{iπ/2}, what's the probability of measuring 1?
What does ⟨φ|ψ⟩ represent?