Postulate 1: State Space & Global Phase

Stage 1 so far has been pure mathematics: complex vector spaces, inner products, Dirac notation, operators and the spectral theorem. Quantum mechanics is the physical theory that tells us how to use that mathematics to describe a real system — a trapped ion, a superconducting circuit, a photon's polarisation. It is organised as a short list of postulates. They are not derived from anything more basic; they are the axioms whose consequences have survived a century of experiments. The first postulate answers the most basic question: what mathematical object is the state of a system?

The answer is surprisingly economical. A classical bit is one of two values; a classical probabilistic bit is a probability vector $(p_0,p_1)$ with $p_0+p_1=1$. A quantum bit is a vector of complex amplitudes $(\alpha,\beta)$ with $\lvert\alpha\rvert^2+\lvert\beta\rvert^2=1$. The change from real, non-negative numbers summing to one, to complex numbers whose squared moduli sum to one, is the whole source of interference — and, as we shall see, it also introduces a redundancy: the overall complex phase of the vector carries no physical information.

Why may we identify $\lvert\psi\rangle$ with $e^{i\gamma}\lvert\psi\rangle$? Because nothing we can ever measure distinguishes them. Every experimental statistic in quantum mechanics has one of two forms: a probability $\lVert P\psi\rVert^2=\langle\psi\vert P\vert\psi\rangle$ of an outcome whose projector is $P$ (the Born rule, Postulate 3), or an expectation value $\langle\psi\vert A\vert\psi\rangle$ of an observable $A$ (a weighted sum of such probabilities). The next theorem shows that both are invariant under $\lvert\psi\rangle\mapsto e^{i\gamma}\lvert\psi\rangle$. The key algebraic fact is that the inner product is conjugate-linear in its first slot, so the phase appears once as $e^{-i\gamma}$ and once as $e^{i\gamma}$, and $e^{-i\gamma}e^{i\gamma}=\lvert e^{i\gamma}\rvert^2=1$.

Consequences. Since probabilities, expectation values, post-measurement states and time evolution all respect the equivalence $\lvert\psi\rangle\sim e^{i\gamma}\lvert\psi\rangle$, a single-qubit state has only two real degrees of freedom, not four: writing $\alpha=\lvert\alpha\rvert e^{i a}$, $\beta=\lvert\beta\rvert e^{ib}$, normalisation removes one parameter and the global phase $e^{ia}$ removes another, leaving $\lvert\psi\rangle\sim\cos\tfrac\theta2\lvert0\rangle+e^{i\phi}\sin\tfrac\theta2\lvert1\rangle$ with $\theta\in[0,\pi]$, $\phi=b-a\in[0,2\pi)$. These are exactly the polar and azimuthal angles of the Bloch sphere. The same argument shows that gates are only defined up to a global phase: $Z$ and $-Z=e^{i\pi}Z$ are the same physical operation when applied to the whole register.

A warning. "Global" means multiplying the entire state vector. A phase that multiplies only some components is a relative phase and is physical. Likewise $e^{i\gamma}U$ is indistinguishable from $U$, but a controlled version of it is not: controlled-$(-I)$ equals $Z\otimes I$, which flips the sign of $\lvert10\rangle$ and $\lvert11\rangle$ only — the phase kickback that powers phase estimation.

Consider $\lvert+\rangle=\tfrac1{\sqrt2}(\lvert0\rangle+\lvert1\rangle)$ and $\lvert-\rangle=\tfrac1{\sqrt2}(\lvert0\rangle-\lvert1\rangle)$. Measured in the computational basis both give $P(0)=P(1)=\tfrac12$, so a $Z$-basis measurement alone cannot tell them apart. They are nonetheless different states — they are orthogonal. To reveal the difference we first rotate the basis with the Hadamard gate and then measure. The Hadamard matrix is unitary and Hermitian, $H=H^\dagger=H^{-1}$, as the unitarity check below confirms.

In the PhaseExplorer, move the $\gamma$ slider: every amplitude rotates in the complex plane by the same angle, and both the $Z$-basis and the $X$-basis histograms stay frozen. Now move $\varphi$: the $Z$-basis histogram still shows $\tfrac12,\tfrac12$, but the $X$-basis histogram (equivalently, "apply $H$ then measure") follows $\cos^2\tfrac\varphi2$ and $\sin^2\tfrac\varphi2$. This is the operational meaning of Postulate 1: information stored in relative phases is invisible in one basis and fully visible in another.

Superposition is not a mixture. The state $\lvert+\rangle$ gives $0$ or $1$ with probability $\tfrac12$ each, like a fair coin, but it is not "a qubit that is secretly $\lvert0\rangle$ or $\lvert1\rangle$ with probability $\tfrac12$". The relative-phase derivation shows the difference: after $H$, $\lvert+\rangle$ gives $0$ with certainty, while the coin-like mixture would still give $\tfrac12,\tfrac12$ (half the time $H\lvert0\rangle=\lvert+\rangle$, half the time $H\lvert1\rangle=\lvert-\rangle$, each yielding $\tfrac12,\tfrac12$). Amplitudes add before squaring — $\tfrac12(1+1)$ versus $\tfrac12(1-1)$ — and that interference is precisely what probabilities cannot do. Mixtures are handled by density operators in Module 1.4.

Counting parameters. An $n$-qubit state is a unit vector in $\mathbb C^{2^n}$: $2^{n+1}$ real numbers, minus one for normalisation, minus one for the global phase, leaves $2^{n+1}-2$ real parameters. For $n=1$ this is $2$ (the Bloch angles); for $n=10$ it is already $2046$. This is the exponential description cost that the statevector simulator pays and that quantum algorithms try to exploit.