entanglement

Prerequisites: Canonical Quantum Mechanics. Full path (17 pages)

Intuitive
Concrete
Abstract

Intuitive

When a particle with spin $0$ decays into two particles with spin $\frac{1}{2}$ , the total spin of the pair must still be zero. Before a measurement neither particle has a definite spin direction. But if we measure the spin of one particle along any axis, we know with certainty that the spin of the other particle along the same axis is opposite, no matter how far apart they are. This holds for every choice of axis, and Bell's theorem shows that no definite spin values fixed at the moment of the decay can reproduce all of these correlations. In quantum physics we then say that the spin state of the two particles is entangled.

Concrete

Composite systems. If system $A$ has Hilbert space $\mathcal H_A$ and system $B$ has $\mathcal H_B$ , the joint system has the tensor product $\mathcal H_A\otimes\mathcal H_B$ . A pure state is a product state if it can be written as $|\psi\rangle = |a\rangle\otimes|b\rangle$ . Then each subsystem has a definite state of its own. A pure state that cannot be written in this form is entangled.

For two spin- $\frac12$ particles (qubits) with basis $|0\rangle, |1\rangle$ , a standard example is the Bell state

$|\Phi^+\rangle = \frac{1}{\sqrt 2}\big(|0\rangle|0\rangle + |1\rangle|1\rangle\big).$

Suppose it were a product, $(\alpha|0\rangle + \beta|1\rangle)(\gamma|0\rangle + \delta|1\rangle)$ . Comparing coefficients requires $\alpha\gamma = \beta\delta = 1/\sqrt2$ and $\alpha\delta = \beta\gamma = 0$ , which is impossible. So $|\Phi^+\rangle$ is entangled.

Schmidt decomposition. Every pure state of $\mathcal H_A\otimes\mathcal H_B$ can be written as

$|\psi\rangle = \sum_i \sqrt{\lambda_i}\, |a_i\rangle\otimes|b_i\rangle, \qquad \lambda_i \geq 0, \quad \sum_i \lambda_i = 1,$

with orthonormal sets $\{|a_i\rangle\}$ and $\{|b_i\rangle\}$ . This follows from the singular value decomposition of the coefficient matrix $c_{jk}$ in $|\psi\rangle = \sum_{jk} c_{jk}|j\rangle|k\rangle$ . The state is entangled exactly when more than one $\lambda_i$ is nonzero.

Reduced states and entanglement entropy. The state of subsystem $A$ alone is the reduced density matrix $\rho_A = \mathrm{Tr}_B\,|\psi\rangle\langle\psi| = \sum_i \lambda_i |a_i\rangle\langle a_i|$ . For a product state $\rho_A$ is pure. For an entangled state it is mixed, even though the total state is pure. The amount of entanglement of a pure state is measured by the entanglement entropy

$S_A = -\mathrm{Tr}\,\rho_A\log\rho_A = -\sum_i \lambda_i\log\lambda_i ,$

which is the same for $A$ and $B$ . For $|\Phi^+\rangle$ we get $\rho_A = \frac12\mathbb 1$ and $S_A = \log 2$ , the maximum for a qubit.

Correlations beyond classical ones. In the spin singlet $|\Psi^-\rangle = \frac{1}{\sqrt2}(|{\uparrow}\downarrow\rangle - |{\downarrow}\uparrow\rangle)$ , the two spins are anticorrelated along every axis, and for measurement directions $\vec a$ and $\vec b$

$\langle(\vec a\cdot\vec\sigma)\otimes(\vec b\cdot\vec\sigma)\rangle = -\vec a\cdot\vec b .$

A classical correlation, like two balls of opposite colour put in separate boxes, can be explained by values fixed at the source. Bell showed that every model of this kind, a local hidden variable model, obeys inequalities that these quantum correlations violate. In the CHSH form, with settings $a, a'$ for $A$ and $b, b'$ for $B$ and outcomes $\pm1$ ,

$|E(a,b) + E(a,b') + E(a',b) - E(a',b')| \leq 2$

for local hidden variable models, while the singlet reaches $2\sqrt2$ for suitably chosen directions. Experiments agree with quantum mechanics. Entanglement does not allow signalling: no operation on $B$ changes $\rho_A$ , so the measurement statistics at $A$ alone are unaffected.

Entanglement is a resource in quantum information: quantum teleportation and superdense coding both use a shared Bell pair.

Abstract

Mixed states. For a density operator $\rho$ on $\mathcal H_A\otimes\mathcal H_B$ , the definition is: $\rho$ is separable if it is a convex combination of product states,

$\rho = \sum_k p_k\, \rho_A^{(k)}\otimes\rho_B^{(k)}, \qquad p_k\geq 0,\ \sum_k p_k = 1,$

(or a limit of such), and entangled otherwise. Separable states are exactly those that can be prepared from product states by local operations and classical communication (LOCC). Deciding separability is hard in general. A necessary condition is the Peres criterion: if $\rho$ is separable, its partial transpose $\rho^{T_B}$ is positive semidefinite. By the Horodecki theorem the condition is also sufficient for $\mathbb C^2\otimes\mathbb C^2$ and $\mathbb C^2\otimes\mathbb C^3$ , but not in higher dimensions.

Measures. Entanglement measures are functions that do not increase on average under LOCC (entanglement monotones). For pure bipartite states, the entanglement entropy $S(\rho_A)$ is singled out: asymptotically, $n$ copies of $|\psi\rangle$ can be converted by LOCC into approximately $nS(\rho_A)$ Bell pairs (with $\log_2$ ) and back, with vanishing error as $n\to\infty$ . For pure states, the full LOCC convertibility of single copies is governed by majorization of the Schmidt coefficients (Nielsen's theorem). For mixed states several inequivalent measures exist, such as entanglement of formation, distillable entanglement and negativity.

Entanglement in quantum field theory. In quantum field theory the Hilbert space does not factorize as a tensor product over spatial regions. The algebraic formulation assigns to each spacetime region $\mathcal O$ an algebra of observables $\mathfrak A(\mathcal O)$ , and under standard assumptions the algebras of bounded regions in relativistic QFT are von Neumann algebras of type $\mathrm{III}_1$ . Such algebras have no normal pure states and no density matrices, so the entanglement entropy of a region is not finite without a cutoff. The Reeh-Schlieder theorem states that the vacuum is cyclic for every local algebra: acting with operators localized in an arbitrarily small region on the vacuum produces a dense set of states. This reflects the strong entanglement of the vacuum between a region and its complement. With a short-distance cutoff $\epsilon$ , the entanglement entropy of a region in $3+1$ dimensions is dominated by a term proportional to the area of its boundary divided by $\epsilon^2$ (area law), with a coefficient that depends on the regularization.

Path to this page

Leads to: Bell's Theorem.