13. Entanglement

13.1What entanglement actually means

State it without mysticism: entanglement means the joint state contains information about correlations that is not contained in any individual subsystem. For Φ⁺, each qubit's reduced state (trace out the other — 13.9) is I/2, maximally random, yet the pair is in a pure, fully determined state. The information lives only in the joint description. Three clarifications matter. Entanglement is basis-independent — if a state factors in one basis it factors in all. Entanglement is between subsystems, not particles — the same math applies to modes, ions, or halves of a chip. And it does not allow signaling (13.3): the local statistics of each side are I/2 no matter what the other side does. What it enables is correlations-with-structure that classical channels cannot supply without shared pre-existing randomness.

13.2Detecting entanglement

Given a two-qubit density matrix, how do you know it is entangled? Tools, in increasing power. Purity of reduced states: for a pure two-qubit state, entangled exactly when either reduced state is mixed (Tr(ρ₁²) < 1). Schmidt decomposition: reshape the 4-vector into a 2×2 matrix and take singular values — product states have exactly one nonzero singular value (used in 12.4's experiment). PPT criterion: transpose one subsystem's part of ρ; for two qubits, negative eigenvalues prove entanglement — necessary and sufficient in this dimension. Bell-inequality violation: a sufficient (not necessary) certificate with the strongest physical meaning. In the wild, hardware tomography (Part XI) reconstructs ρ from measurements, and the first thing experimentalists compute is one of these tests. Note the honest caveat: above two qubits, full detection is computationally hard — entanglement is cheap to create, expensive to certify.

13.3Bell's theorem

In 1964 John Bell asked a question with a yes/no answer: can the correlations of entangled states be explained by local hidden variables — pre-agreed values carried by each particle, set at emission, with no faster-than-light influence? He proved mathematically that any such model satisfies quantitative bounds (Bell inequalities) that quantum mechanics predicts can be violated. Nature's verdict, across experiments from Aspect (1982) to the loophole-free tests of 2015 and the 2022 Nobel work: quantum mechanics wins; local hidden variables are dead. The engineering reading matters more than the metaphysics: Bell's theorem turns "entanglement is weird" into "entanglement's correlations are a *certified, testable, non-classical resource*" — the resource behind device-independent cryptography and certified randomness. No signaling theorem stands alongside it: violating an inequality requires comparing outcomes classically; neither party alone can send a message.

13.4Bell inequalities

The workhorse is the CHSH inequality: for two parties each choosing between two measurement settings (A₀, A₁ and B₀, B₁), any local hidden-variable model obeys S = |⟨A₀B₀⟩ + ⟨A₀B₁⟩ + ⟨A₁B₀⟩ − ⟨A₁B₁⟩| ≤ 2. Quantum mechanics allows up to 2√2 (the Tsirelson bound), reached by Φ⁺ with measurement angles 45° apart. The settings for a maximally entangled pair: A₀ = Z, A₁ = X, B₀ = (Z+X)/√2, B₁ = (Z−X)/√2 — each expectation computed by measuring in the rotated basis (11.4). Real experiments must close the detection loophole (high-efficiency detectors) and the locality loophole (space-like separation of the parties); the 2015 Delft, NIST, and Vienna experiments did both. Expect S ≈ 2.6–2.8 on noisy lab hardware rather than 2.83 — the deficit is a two-qubit fidelity meter, used exactly that way in benchmarks.

13.5Quantum teleportation

Teleportation moves an unknown state |ψ⟩ = α|0⟩ + β|1⟩ from Alice to Bob using one shared Bell pair plus two classical bits — and it transfers the state without either party learning it. Protocol: (1) Alice Bell-measures |ψ⟩ together with her half of Φ⁺; (2) she gets one of four outcomes, each equally likely, and sends the 2-bit label to Bob; (3) Bob's qubit is now |ψ⟩ up to one Pauli correction — I, X, Z, or XZ — determined by Alice's bits; (4) Bob applies the inverse Pauli. Read the resource accounting: no matter or energy transported, no state cloned (the original is destroyed by the Bell measurement — no-cloning enforced), no faster-than-light signaling (the correction needs the classical bits). Teleportation is not a curiosity; it is the routing primitive: quantum networks, module-to-module links in modular hardware, and gate teleportation in fault tolerance all run this protocol.

Teleportation:

  psi:    ──●─────────M────────── 2 classical bits ──┐
           │         │                               │  X or Z
  alice:  ──H───●────M──────────                       ▼
                │                                  bob: ──apply correction──> |psi>
  bob:   ───────X──────────────────────────

13.6Superdense coding

The mirror protocol: superdense coding sends two classical bits using one qubit of transmission plus one shared Bell pair. Alice applies I, Z, X, or ZX to her half of Φ⁺ — four options, one per 2-bit message — then sends the qubit to Bob, who performs a Bell-state measurement (12.6) and reads both bits. The accounting is exact: two classical bits of capacity from one transmitted qubit, because the pre-shared entanglement was established earlier and counted separately. This is the cleanest demonstration that entanglement is a communication resource with quantifiable value. Both directions are real: teleportation trades entanglement + classical bits to move quantum states; dense coding trades entanglement + a quantum channel to double classical capacity. Hardware demos are routine with photons and ions; the capacity advantage requires a noiseless qubit channel, so classical engineering keeps dense coding mostly a benchmark.

13.7Entanglement swapping and monogamy

Two structural facts govern how entanglement behaves at scale. Entanglement swapping: Alice and Bob share no entanglement, but Alice and Carol share a Bell pair, as do Carol and David; Carol Bell-measures her two qubits and — conditioned on her classical outcome — Alice and David are entangled, never having interacted. This is how quantum repeaters will chain short links into long-distance networks: entanglement is built up segment by segment, swapped at nodes, purified against noise. Monogamy of entanglement: if qubits A and B are maximally entangled, B cannot be entangled with C at all — maximal correlation with A excludes any correlation-with-structure toward C. Monogamy is why entanglement cannot be freely shared or copied, why eavesdroppers measurably damage QKD links, and why error-correcting codes can separate logical information from the environment. Structure: share it pairwise, spend it deliberately.

13.8Entanglement entropy

To measure entanglement in a mixed or multi-partite world you need a number. Split a pure state into subsystems A and B; the reduced density matrix ρ_A = Tr_B(|ψ⟩⟨ψ|) (trace out B) carries A's statistics alone. The entanglement entropy is the von Neumann entropy S(ρ_A) = −Tr(ρ_A log₂ ρ_A) = −Σ λᵢ log₂ λᵢ over eigenvalues λᵢ. Zero means product state; 1 bit means a qubit of A is maximally entangled with B (as for all Bell states). Computationally this is a two-liner via SVD: singular values of the reshaped amplitude matrix are √λᵢ — the Schmidt decomposition of 13.2. Beyond two qubits, entanglement entropy measures bipartite entanglement only; the full multi-partite structure is richer and partly unclassified (research frontier below). In Part XIV, entanglement entropy becomes the diagnostic for whether a quantum state is classically simulable.

13.9Entanglement as a computational resource

Is entanglement necessary for quantum speedup? Mostly yes, with a famous exception. Jozsa–Linden proved that pure-state circuits with vanishing multi-qubit entanglement are efficiently classically simulable — so exponential speedups require entanglement. The exception: the Gottesman–Knill theorem (Part X) shows a class of heavily entangling Clifford circuits that is simulable, so entanglement alone does not guarantee advantage — the entanglement must be of a classically-hard structure (volume-law scaling, high complexity). The engineering synthesis: entanglement is a necessary fuel with a quality requirement, and "how entangled is my state, in a simulability-relevant sense?" is a live measurement problem — answered numerically by the entropy of 13.8 and by tensor-network bond dimensions. Error-corrected computing is, from this angle, the discipline of maintaining a very specific, very robust entangled state for the duration of a computation.