Quantum Entanglement
Bell states, the EPR paradox, Bell inequalities, and the experimental violation of local realism
Quantum entanglement is arguably the most striking departure from classical physics: two particles can share a quantum state such that measurements on them are correlated in ways that no classical local theory can explain. Einstein called it spukhafte Fernwirkung — “spooky action at a distance” — and spent decades trying to explain it away. Bell’s theorem and subsequent experiments show he was wrong.
Composite systems and the tensor product
In quantum mechanics, the Hilbert space of a composite system is the tensor product of the subsystem spaces:
\begin{equation} \mathcal{H}_{AB} = \mathcal{H}_A \otimes \mathcal{H}_B. \end{equation}
| A state $$ | \psi\rangle_{AB} \in \mathcal{H}_{AB}\(is **separable** if it can be written as\) | \psi\rangle = | \phi\rangle_A \otimes | \chi\rangle_B$$. Otherwise, it is entangled — it cannot be factored into a product of states for each subsystem. |
Bell states
The four maximally entangled two-qubit states (Bell basis):
\begin{equation} |\Phi^\pm\rangle = \frac{1}{\sqrt{2}}\left(|00\rangle \pm |11\rangle\right), \qquad |\Psi^\pm\rangle = \frac{1}{\sqrt{2}}\left(|01\rangle \pm |10\rangle\right). \end{equation}
| Consider $$ | \Psi^-\rangle = \frac{1}{\sqrt{2}}( | 01\rangle - | 10\rangle)$$: if Alice measures spin-up along any axis, Bob’s particle is immediately in the spin-down state along that same axis — regardless of the spatial separation between them. |
The EPR paradox
| Einstein, Podolsky, and Rosen (1935) argued that quantum mechanics is incomplete. Their argument: for $$ | \Psi^-\rangle\(, Alice can measure along\)\hat{x}\(or\)\hat{z}\(and predict Bob's outcome with certainty. Since Alice's measurement cannot physically disturb Bob's distant particle, Bob's particle must have had a definite value along both axes *before* measurement. But quantum mechanics forbids simultaneous definite values for non-commuting observables (\)[\hat{\sigma}_x, \hat{\sigma}_z] \neq 0$$). Therefore, quantum mechanics must be incomplete — there must be hidden variables completing the description. |
Bell’s theorem
John Bell (1964) showed that any local hidden variable (LHV) theory — one in which measurement outcomes are determined by pre-existing hidden variables, and in which no signal travels faster than light — satisfies an inequality that quantum mechanics violates.
| Consider spin measurements along directions \(\hat{a}, \hat{b}, \hat{c}\) on the singlet $$ | \Psi^-\rangle$$. The CHSH inequality (Clauser, Horne, Shimony, Holt, 1969) states that for any LHV theory: |
\begin{equation}\label{eq.CHSH} |S| \leq 2, \quad S = E(a, b) - E(a, b’) + E(a’, b) + E(a’, b’), \end{equation}
where \(E(\hat{n}, \hat{m}) = \langle \hat{\sigma}_A\cdot\hat{n}\;\hat{\sigma}_B\cdot\hat{m}\rangle\) is the correlation between the two outcomes.
Quantum prediction: for the singlet, \(E(\hat{n}, \hat{m}) = -\cos\theta_{nm}\). Choosing \(\hat{a}\) at \(0°\), \(\hat{a}'\) at \(90°\), \(\hat{b}\) at \(45°\), \(\hat{b}'\) at \(-45°\):
\begin{equation} S_{\rm QM} = -\cos45° - \cos(-45°) - \cos45° - \cos135° = 2\sqrt{2} \approx 2.828. \end{equation}
Quantum mechanics violates \eqref{eq.CHSH} by a factor \(\sqrt{2}\). This is the Tsirelson bound — the maximum quantum violation.
Experimental tests
A succession of increasingly rigorous experiments confirmed the quantum prediction:
| Experiment | Year | \(\lvert S\rvert\) measured | Loopholes closed |
|---|---|---|---|
| Freedman–Clauser | 1972 | \(2.00 \pm 0.05\) | First test |
| Aspect et al. | 1982 | \(2.697 \pm 0.015\) | Locality |
| Weihs et al. | 1998 | \(2.731 \pm 0.026\) | Fast switching |
| Hensen et al. | 2015 | \(2.42 \pm 0.20\) | Both (loophole-free) |
| Giustina et al. | 2015 | \(2.37 \pm 0.09\) | Both |
The 2015 loophole-free experiments closed simultaneously the locality loophole (settings chosen after particles are separated) and the detection loophole (high-efficiency detectors). Local hidden variable theories are experimentally ruled out. The 2022 Nobel Prize in Physics was awarded to Aspect, Clauser, and Zeilinger for this work.
Entanglement entropy
| For a pure state $$ | \psi\rangle_{AB}$$, the degree of entanglement is measured by the von Neumann entropy of the reduced density matrix: |
\begin{equation} S_A = -\mathrm{Tr}(\rho_A \ln \rho_A), \quad \rho_A = \mathrm{Tr}_B(|\psi\rangle\langle\psi|). \end{equation}
For a Bell state, \(\rho_A = \frac{1}{2}\mathbb{1}\) and \(S_A = \ln 2\) (one ebit of entanglement). For a product state, \(S_A = 0\).
| The Schmidt decomposition guarantees that any bipartite pure state can be written as $$ | \psi\rangle = \sum_k \lambda_k | \alpha_k\rangle_A | \beta_k\rangle_B\(with\)\lambda_k \geq 0\(,\)\sum_k\lambda_k^2 = 1$$. Entanglement iff more than one Schmidt coefficient is non-zero. |
Applications
| Quantum teleportation (Bennett et al., 1993): transmit an arbitrary qubit state $$ | \phi\rangle$$ using one Bell pair and two classical bits, without physically transmitting the qubit. The state is reconstructed at the receiver via local unitaries conditioned on the classical message. |
Superdense coding: transmit two classical bits by sending one qubit, using pre-shared entanglement — doubling the classical capacity of a quantum channel.
Quantum key distribution (E91 protocol, Ekert 1991): the Bell inequality violation itself certifies that an eavesdropper has not tampered with the shared key — device-independent security.
Entanglement as a resource underlies all of quantum computing’s potential speedups: Shor’s factoring algorithm, Grover’s search, quantum simulation of many-body systems. The study of entanglement structure — area laws, topological entanglement entropy, tensor network states — is now a cornerstone of condensed matter and quantum gravity.