12. Composite Quantum Systems
12.1Two-qubit states and the four-dimensional state space
The fourth postulate says composite systems combine by tensor product: a two-qubit state lives in the 4-dimensional space ℂ² ⊗ ℂ², with computational basis |00⟩, |01⟩, |10⟩, |11⟩ — that is, "first qubit, second qubit", not a binary number you can interpret freely. A general state is α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩ with four complex amplitudes, Σ|·|² = 1. In numpy, a state vector for qubits ordered q0, q1 is np.kron(psi_q0, psi_q1). Note the bookkeeping shock: one extra qubit doubles the amplitudes you store, while the classical description of two bits still needs two bits. The four amplitudes are not four bits of information — measurement still returns two bits total — but they are four numbers your simulator must carry and evolve.
12.2n-qubit state space and exponential growth
Iterating: n qubits span a 2ⁿ-dimensional complex space with 2ⁿ amplitudes. The growth is brutal and worth feeling numerically: 10 qubits → 1,024 amplitudes (16 KB as complex128); 20 → 1 M (16 MB); 26 → 67 M (1 GB); 30 → 16 GB; 40 → 16 TB; 50 → beyond any laptop. This exponential is not a "trick" quantum computers do — it is simply the size of the description, and it cuts both ways: it is why classical simulation is a supercomputer-class task past ~45 qubits (yet state-of-the-art tensor-network methods push specific structured circuits far further, Part XIV), and it is the headroom quantum algorithms spend. You have already felt the wall in Part I's memory experiment. From this point on, every algorithm is judged by how it maneuvers inside this space without ever writing it down.
n qubits -> 2^n complex amplitudes (complex128 = 16 bytes each)
10 -> 1,024 16 KB trivial
20 -> 1,048,576 16 MB fine
26 -> 67,108,864 1 GB laptop limit
30 -> 1,073,741,824 16 GB desktop limit
40 -> ~1.1e12 16 TB cluster
50 -> ~1.1e15 16 PB simulation frontier12.3Product states
The clean case: a product state is one whose state vector factors as |ψ⟩₁ ⊗ |ψ⟩₂ — each qubit has a state of its own. |00⟩, |+−⟩ = |+⟩⊗|−⟩, and (α|0⟩+β|1⟩)⊗(γ|0⟩+δ|1⟩) are all product states. Everything you learned for one qubit applies per-qubit: probabilities multiply, gates apply blockwise, entanglement is zero. Product states are also exactly what a classical computer handles cheaply: instead of 2ⁿ amplitudes, you store n two-component vectors — polynomial cost. This is why structured circuits (those that keep the state near a product or mildly correlated form) remain simulable at scales far beyond naive limits. The interesting question, then, is precise: which states are not of this form, and what can they do? That question is entanglement, next.
12.4Entangled states
An entangled state is a multi-qubit state that cannot be written as a tensor product, even after trying every split and every basis: |ψ⟩ ≠ |φ⟩₁ ⊗ |χ⟩₂ for any choices. The canonical example: (|00⟩ + |11⟩)/√2. Neither qubit individually has a state vector at all — each is maximally random alone (50/50 in every basis), yet outcomes are perfectly correlated. This is a genuinely new object, not classical correlation with extra steps: no probabilistic recipe of "pre-agreed instructions" reproduces all of its statistics (13.3 makes that a theorem). Engineering consequence: entanglement is a resource that consumes qubits irreversibly during preparation but then links them with correlations no classical channel provides. Almost everything valuable in this book — teleportation, error-correcting codes, quantum chemistry states — is built from non-product states.
12.5Bell states and Bell-state preparation
The four Bell states are the maximally entangled two-qubit states, named after John Bell: Φ⁺ = (|00⟩+|11⟩)/√2, Φ⁻ = (|00⟩−|11⟩)/√2, Ψ⁺ = (|01⟩+|10⟩)/√2, Ψ⁻ = (|01⟩−|10⟩)/√2. They form an orthonormal basis of the 4-dimensional space — a complete alternative "entangled basis". Preparing Φ⁺ is a two-gate recipe you will write a thousand times: apply H to qubit 0, then CNOT from qubit 0 to qubit 1. Read it mechanically: H creates qubit 0's superposition; CNOT copies the correlation, not the amplitude (no-cloning survives). The three other Bell states follow by a Z or X (or both) on one qubit. In circuit diagrams this recipe is the standard warm-up, and in hardware it is the first benchmark any two-qubit link must pass.
Bell-state preparation circuit:
q0: ──H────●──
│
q1: ───────X──
|00> -> (|00> + |11>)/sqrt(2) = |Phi+>12.6Bell-state measurement
A Bell-state measurement asks "which of the four Bell states am I in?" — projecting two qubits onto the Bell basis. It cannot be done by measuring each qubit separately (that reads the computational basis and destroys the entanglement), but it is achievable with gates: apply CNOT(q0→q1) then H on q0, then measure both in the computational basis. That circuit is the inverse of the preparation circuit, which is no accident — basis changes are unitary, and this one maps the Bell basis onto the computational basis. Two applications to hold onto: teleportation (13.5) is a Bell measurement whose classical outcome steers a correction; and Bell-state measurements between photons are the workhorse of quantum networking (entanglement swapping, 13.7). On today's hardware it succeeds probabilistically with linear optics; deterministic versions need matter qubits.
12.7Classical versus quantum correlations
Sharpen the distinction with numbers, because it is the conceptual heart of Part V. Classical correlated bits: a source outputs 00 half the time, 11 half the time. Outcomes are correlated, yet each bit has a definite value before you look — the randomness is ignorance about a pre-existing fact. Bell's Φ⁺ looks identical for Z-measurements: 00 and 11 with equal probability. The difference appears when you rotate the measurement basis: measuring both qubits in the X basis also gives perfect correlation, and correlations across mixed bases (one X, one Z) violate the bounds any pre-agreed-values model must obey (13.4). A handy numerical signature you can compute tonight: for Φ⁺, the correlation matrix ⟨σᵢ ⊗ σⱼ⟩ over axes i, j ∈ {x, y, z} is diag(1, −1, 1) — a pattern no classical mixture of definite spin directions matches.