# The State Postulate > Quantum mechanics is built on four postulates. The first answers the most basic question: *what > is the state of a system?* The answer — a unit vector in a Hilbert space, identified up to a global > phase — is deceptively simple, but it already encodes superposition, the irrelevance of global > phase, and the physical reality of *relative* phase. Everything else in the theory acts on this > object. ## Learning Objectives After this lesson you will be able to: 1. State **Postulate 1** precisely and explain why states are unit vectors (rays), not arbitrary vectors. 2. Apply the **normalization** condition and **normalize** an arbitrary vector to a valid state. 3. Distinguish **global phase** (unobservable) from **relative phase** (observable) and justify the difference. 4. Use the **superposition principle** and interpret amplitudes via the (forthcoming) Born rule. 5. Represent and normalize states in NumPy and reason about phase equivalence. --- ## Intuition A classical bit is in state $0$ or $1$. A quantum system is described instead by a **vector of complex amplitudes** — one amplitude per classically-distinguishable configuration. For a qubit there are two configurations, $\lvert 0\rangle$ and $\lvert 1\rangle$, so the state is $$ \lvert\psi\rangle = \alpha\lvert 0\rangle + \beta\lvert 1\rangle, \qquad \alpha,\beta\in\mathbb{C}. $$ The amplitudes are not probabilities — they're complex numbers whose **squared magnitudes** are probabilities, and whose **relative phase** governs interference. Two requirements pin down which vectors are physical: the probabilities must sum to $1$ (**normalization**), and multiplying the whole vector by a phase $e^{i\gamma}$ changes nothing measurable (**global phase freedom**). A physical state is therefore not quite a vector — it's a vector modulo these redundancies, a **ray**. --- ## Theory ### Postulate 1 (State space) > **Postulate 1.** Associated to any isolated physical system is a complex Hilbert space > $\mathcal{H}$, the **state space**. The state of the system is completely described by a **unit > vector** $\lvert\psi\rangle\in\mathcal{H}$, i.e. $\langle\psi|\psi\rangle = 1$. Two vectors that > differ only by a global phase, $\lvert\psi\rangle$ and $e^{i\gamma}\lvert\psi\rangle$ ($\gamma\in\mathbb{R}$), > represent the **same** physical state. For a single qubit, $\mathcal{H} = \mathbb{C}^2$ with computational basis $\{\lvert0\rangle,\lvert1\rangle\}$ (from [0.1.1](../../term-0-foundations/c01-linear-algebra/01-complex-vector-spaces.md)). For $n$ qubits, $\mathcal{H} = (\mathbb{C}^2)^{\otimes n} = \mathbb{C}^{2^n}$ (Postulate 4, [1.1.4](04-composite-systems-postulate.md)). ### Normalization Writing $\lvert\psi\rangle = \sum_k c_k\lvert k\rangle$ in an orthonormal basis, normalization is $$ \langle\psi|\psi\rangle = \sum_k |c_k|^2 = 1 . $$ By the (upcoming) **Born rule**, $|c_k|^2$ is the probability of obtaining outcome $k$ on measurement in that basis; normalization is exactly "the probabilities sum to one." Any nonzero vector $\lvert v\rangle$ becomes a state by **normalizing**: $\lvert\psi\rangle = \lvert v\rangle/\sqrt{\langle v|v\rangle}$ (Parseval, [0.1.2](../../term-0-foundations/c01-linear-algebra/02-inner-products-norms.md)). ### The superposition principle Because $\mathcal{H}$ is a vector space, any linear combination of states is (after normalization) a state: if $\lvert\psi_1\rangle,\lvert\psi_2\rangle$ are valid states, so is $\alpha\lvert\psi_1\rangle + \beta\lvert\psi_2\rangle$ for appropriate $\alpha,\beta$. This **superposition principle** is the mathematical heart of quantum computing — a single register can hold a superposition over all $2^n$ bit-strings — and is a direct consequence of states living in a *linear* space. ### Global phase is unobservable; relative phase is not These two facts look similar but are opposite in consequence. **Global phase.** The states $\lvert\psi\rangle$ and $e^{i\gamma}\lvert\psi\rangle$ are physically identical. *Why:* every physical prediction is an expectation value $\langle\psi|A|\psi\rangle$ or a probability $|\langle\phi|\psi\rangle|^2$ (Postulate 2). Under $\lvert\psi\rangle\mapsto e^{i\gamma}\lvert\psi\rangle$, $$ \langle\psi|A|\psi\rangle \mapsto e^{-i\gamma}\langle\psi|A|\psi\rangle e^{i\gamma} = \langle\psi|A|\psi\rangle, \qquad |\langle\phi|\psi\rangle|^2 \mapsto |e^{i\gamma}|^2\,|\langle\phi|\psi\rangle|^2 = |\langle\phi|\psi\rangle|^2 . $$ The phase cancels in every observable quantity. Hence a physical state is really an equivalence class $\{e^{i\gamma}\lvert\psi\rangle\}$ — a **ray** in $\mathcal{H}$ (a point of *projective* Hilbert space). **Relative phase.** Inside a superposition, the *relative* phase between components **is** physical. Compare $$ \lvert +\rangle = \tfrac1{\sqrt2}(\lvert0\rangle + \lvert1\rangle), \qquad \lvert -\rangle = \tfrac1{\sqrt2}(\lvert0\rangle - \lvert1\rangle). $$ These differ only by a relative sign ($e^{i\pi}$) on the $\lvert1\rangle$ component, yet they are **orthogonal** ($\langle +|-\rangle = 0$) — perfectly distinguishable states. A global phase rescales *everything*; a relative phase reweights components against each other, and that survives into measurable interference (e.g. measuring $\lvert\pm\rangle$ in the $X$ basis gives opposite certain outcomes). This distinction is *the* reason quantum mechanics needs $\mathbb{C}$, foreshadowed in [0.1.1](../../term-0-foundations/c01-linear-algebra/01-complex-vector-spaces.md). An interferometer makes the relative phase visible. A Mach–Zehnder is $H$, then $\operatorname{diag}(1, e^{i\varphi})$ on one arm, then $H$ again. On its own, each arm still holds the photon with probability $\tfrac12$ for every $\varphi$, yet detector $D_0$ fires with probability $\cos^2(\varphi/2)$. Slide $\varphi$ from $0$ to $\pi$ and the state between the splitters turns from $\lvert+\rangle$ into $\lvert-\rangle$, sending every photon to the other detector: ```qm-demo mach-zehnder ``` > **Caution.** "Global phase doesn't matter" is true for a state *in isolation*. A relative phase > between two branches of a superposition — including one introduced by a *controlled* operation — > absolutely matters. Phase-kickback (Term 2.4) weaponizes exactly this. A preview of that trick, on the same interferometer. The lower arm no longer carries a phase shifter; instead the photon's path *controls* a phase gate $P(\varphi)$ on a second qubit. With the target in the eigenstate $\lvert1\rangle$, the eigenvalue $e^{i\varphi}$ lands on the path's $\lvert1\rangle$ branch and the fringes are the shifter's. With $\lvert+\rangle$, which is not an eigenstate, the path entangles with the target and the fringes fade: ```qm-demo mach-zehnder { "kickback": true, "phase": 180 } ``` --- ## Worked Examples ### Example 1 — Normalizing a state Let $\lvert v\rangle = 2\lvert0\rangle + (1-i)\lvert1\rangle$. Then $\langle v|v\rangle = |2|^2 + |1-i|^2 = 4 + 2 = 6$, so the normalized state is $$ \lvert\psi\rangle = \tfrac1{\sqrt6}\big(2\lvert0\rangle + (1-i)\lvert1\rangle\big). $$ Measurement-in-the-computational-basis probabilities (Born rule): $p_0 = 4/6 = 2/3$, $p_1 = 2/6 = 1/3$, summing to $1$. ✓ ### Example 2 — Global vs relative phase Consider $\lvert\psi\rangle = \tfrac1{\sqrt2}(\lvert0\rangle + \lvert1\rangle) = \lvert+\rangle$. - Apply a **global** phase $e^{i\pi/3}$: $e^{i\pi/3}\lvert+\rangle$. Overlap with any $\lvert\phi\rangle$ has the same modulus, so it is physically $\lvert+\rangle$ — same state. - Apply a **relative** phase $e^{i\pi}=-1$ to the $\lvert1\rangle$ component: $\tfrac1{\sqrt2}(\lvert0\rangle - \lvert1\rangle) = \lvert-\rangle$. Now $\langle+|-\rangle = 0$ — a *different, orthogonal* state. Same-looking operation, opposite physical consequence. --- ## Hands-on (Python) ```python import numpy as np ket0 = np.array([1, 0], dtype=complex) ket1 = np.array([0, 1], dtype=complex) def normalize(v): return v / np.linalg.norm(v) # Example 1: v = 2 * ket0 + (1 - 1j) * ket1 psi = normalize(v) print(np.round(psi, 4)) # [0.8165+0.j 0.4082-0.4082j] print(np.abs(psi) ** 2) # [0.6667 0.3333] = Born probabilities, sum 1 ``` ```python def same_physical_state(a, b, tol=1e-9): """Equal up to GLOBAL phase iff || = 1 for normalized a, b.""" a, b = normalize(a), normalize(b) return np.isclose(abs(np.vdot(a, b)), 1.0, atol=tol) plus = normalize(ket0 + ket1) minus = normalize(ket0 - ket1) # Global phase e^{iπ/3} on |+> -> same physical state: print(same_physical_state(plus, np.exp(1j * np.pi / 3) * plus)) # True # Relative phase (sign on |1>) turns |+> into |-> -> DIFFERENT, orthogonal state: print(same_physical_state(plus, minus)) # False print(np.vdot(plus, minus)) # 0j (orthogonal) ``` > **Looking ahead to Braket.** A simulator stores the amplitude vector you just built. Because global > phase is unobservable, two simulators may report state vectors differing by an overall phase yet be > physically identical — always compare via $|\langle\phi|\psi\rangle|$, never raw components > ([Appendix B §8](../../appendix/B-python-numpy-refresher.md)). --- ## Exercises **E1 (easy).** Normalize $\lvert v\rangle = 3\lvert0\rangle + 4i\lvert1\rangle$ and give the computational-basis measurement probabilities.

Solution

$\langle v|v\rangle = 9 + 16 = 25$, so $\lvert\psi\rangle = \tfrac15(3\lvert0\rangle + 4i\lvert1\rangle)$. Probabilities $p_0 = 9/25 = 0.36$, $p_1 = 16/25 = 0.64$ (sum $1$).

**E2 (easy).** Are $\lvert\psi\rangle = \tfrac1{\sqrt2}(\lvert0\rangle + i\lvert1\rangle)$ and $\lvert\phi\rangle = \tfrac{i}{\sqrt2}(\lvert0\rangle + i\lvert1\rangle)$ the same physical state?

Solution

$\lvert\phi\rangle = i\lvert\psi\rangle$, a global phase $e^{i\pi/2}$. Same physical state. Check: $|\langle\psi|\phi\rangle| = |i|\,|\langle\psi|\psi\rangle| = 1$.

**E3 (medium).** Show that for normalized states, $|\langle\phi|\psi\rangle|^2$ is invariant if *either* state is multiplied by a global phase, and explain why this means rays (not vectors) are physical.

Solution

Under $\lvert\psi\rangle\mapsto e^{i\gamma}\lvert\psi\rangle$, $\langle\phi|\psi\rangle\mapsto e^{i\gamma}\langle\phi|\psi\rangle$, whose modulus squared is unchanged ($|e^{i\gamma}|=1$); same for a phase on $\lvert\phi\rangle$. Since *all* predictions are such overlaps/expectations, the global phase is operationally inaccessible — the physical object is the equivalence class $\{e^{i\gamma}\lvert\psi\rangle\}$, a ray.

**E4 (medium).** A student claims $\lvert0\rangle + \lvert1\rangle$ and $\lvert0\rangle - \lvert1\rangle$ "differ by a phase, so they're the same state." Where is the error?

Solution

The $-1$ is a **relative** phase on the $\lvert1\rangle$ component only, not a global phase on the whole vector. After normalization they are $\lvert+\rangle$ and $\lvert-\rangle$, which are orthogonal and perfectly distinguishable. A global phase would multiply *both* components equally; here only one flips.

**E5 (hard).** A qubit state up to global phase has how many *real* parameters? Derive the count and connect it to the Bloch sphere (Term 1.2).

Solution

A vector in $\mathbb{C}^2$ has $4$ real parameters (two complex amplitudes). Normalization $|\alpha|^2+|\beta|^2=1$ removes $1$; global-phase freedom removes another $1$. That leaves $4-2 = 2$ real parameters — exactly the two angles $(\theta,\varphi)$ of a point on the **Bloch sphere** $\lvert\psi\rangle = \cos\tfrac\theta2\lvert0\rangle + e^{i\varphi}\sin\tfrac\theta2\lvert1\rangle$ (developed in [1.2.2](../c02-qubits-bloch/02-bloch-sphere.md)). This matches the $\dim_\mathbb{R}\mathbb{C}^2 = 4$ count from [0.1.1 E5](../../term-0-foundations/c01-linear-algebra/01-complex-vector-spaces.md).

--- ## Checkpoint 1. State Postulate 1. Why a *unit* vector, and why "up to global phase"? 2. What does normalization mean physically? 3. Give an operation that is unobservable and one that is observable, both "just a phase." 4. State the superposition principle and why it follows from $\mathcal{H}$ being a vector space. 5. How many real parameters describe a qubit state, and why?

Answers

1. The state is a unit vector $\lvert\psi\rangle\in\mathcal H$ with $\langle\psi|\psi\rangle=1$, defined up to a global phase. Unit norm makes Born probabilities sum to $1$; global phase cancels in all predictions, so it carries no information. 2. The measurement-outcome probabilities $\sum_k|c_k|^2 = 1$ — total probability is one. 3. Unobservable: multiplying the whole state by $e^{i\gamma}$ (global phase). Observable: a relative phase between components, e.g. $\lvert0\rangle+\lvert1\rangle \to \lvert0\rangle-\lvert1\rangle$. 4. Any linear combination of states is (after normalization) a state, because $\mathcal H$ is closed under addition and scalar multiplication. 5. Two: $4$ real parameters of $\mathbb C^2$, minus one for normalization and one for global phase.

--- ## Further Reading - **[NC]** Nielsen & Chuang, §2.2.1 (Postulate 1) — the state-space postulate. - **[Sak]** Sakurai & Napolitano, §1.1–1.3 — kets, states, and the physical setup. - **[Pre]** Preskill, *Ph219*, Ch. 2 — axioms of quantum theory. --- ← Prev: [Term 0 · Reversible Computation](../../term-0-foundations/c04-computation-complexity/03-reversible-computation.md) · Up: [Term 1](../README.md) · Next: [Observables & the Measurement Postulate](02-observables-measurement-postulate.md) →