Entanglement Across Fields

Entanglement connects subjects that otherwise use quite different language. A communication protocol asks what two laboratories can accomplish with a shared state. A spin-chain calculation asks how much information is shared across a spatial cut. A chemistry calculation asks which orbitals must be treated together. In each case, the subsystem choice and the available measurements determine the meaning of the result. Continuum field theory adds a further complication: local regions need not admit the simple Hilbert-space factorization familiar from finite systems.

Helpful background. Reduced states connect a joint state to local measurements. Bipartite entanglement distinguishes pure-state entropy from mixed-state entanglement tests.

Choose an application

What you want to understand	Continue with
Shared states as resources for communication	Quantum information
Spatial correlations, phases, and efficient state representations	Many-body physics
Orbital choice and electronic correlation	Quantum chemistry
Polarization, path, and quadrature measurements	Quantum optics
Entanglement, steering, and Bell tests	Foundations
Vacuum correlations and spatial regions	Quantum field theory

Communication uses entanglement under local constraints

A Bell pair supplies one ebit of pure bipartite entanglement: either qubit has entropy $1$ when the logarithm is base two, or $\ln2$ with natural logarithms. Local operations and classical communication (LOCC) cannot increase an entanglement monotone on average. This restriction gives the shared state a resource meaning: for example, teleportation consumes a Bell pair and also requires a classical message and suitable local operations.

The qualification “on average” matters. A successful branch of an entanglement-concentration experiment can contain more entanglement than the input pair; its probability and the unsuccessful branches must be included. Noisy entangled states also need not be directly useful for every communication task. See LOCC and the quantum-information application for the distinctions between entanglement, distillation, and protocol performance.

Spatial cuts reveal many-body structure

For a pure lattice state, tracing out the complement of a region $A$ gives an entropy $S(A)=-\operatorname{Tr}\rho_A\ln\rho_A$ that measures entanglement across that cut. In many local gapped ground states the leading term scales with the size of the boundary:

$$S(A)\sim\alpha|\partial A|+\cdots.$$

This is an area law, with a generally nonuniversal coefficient. It is not a statement about every local Hamiltonian or every state. For a critical one-dimensional ground state whose long-distance limit is a conformal field theory, an interval of length $\ell$ in an infinite system instead has

$$S(\ell)=\frac c3\ln\frac{\ell}{a}+s_0$$

at zero temperature and at scales $\ell\gg a$. Here $c$ is the central charge and $a$ a short-distance cutoff. Physical boundaries, finite temperature, and finite system size change the expression. Calabrese and Cardy derive these geometry-dependent results.

Entropy is a useful summary, but the full Schmidt spectrum contains more information. In one dimension, rapidly decaying Schmidt coefficients make accurate truncation and matrix-product descriptions possible. An area-law slogan alone does not establish an efficient approximation for an arbitrary state, especially in higher dimensions. The many-body application leads to the detailed entropy discussion.

One can also study an entanglement Hamiltonian $H_E=-\ln\rho_A$ on the support of $\rho_A$. Its eigenvalues are $-\ln p_j$ for positive reduced-state eigenvalues $p_j$. Zero eigenvalues correspond formally to infinite entanglement energies. Writing $\rho_A=e^{-H_E}/Z_E$ allows an arbitrary additive constant in $H_E$; this is a way to describe the reduced spectrum, not necessarily the physical Hamiltonian of region $A$.

Chemistry distinguishes exchange from correlation

Antisymmetry is required for every electronic state. A single Slater determinant already has that exchange structure; it does not automatically contain the electronic correlation that requires a superposition of determinants beyond Hartree–Fock. Orbital entropies and mutual information can help identify orbitals that should be treated together in an active-space calculation, but their values depend on the orbital partition.

A unitary rotation among occupied orbitals of a determinant changes it only by an overall phase. A rotation mixing the modes assigned to different subsystems, however, changes the subsystem question. Thus “correlation beyond a determinant” and “entanglement across these particular orbitals” are distinct claims. The chemistry application develops concrete examples and links to electronic-structure methods.

Optics measures correlations between accessible modes

Two spatially separated photons may carry the polarization state

$$|\Psi^-\rangle =\frac{|H\rangle_A|V\rangle_B-|V\rangle_A|H\rangle_B}{\sqrt2}.$$

The subscripts identify the arms or detected modes; polarization provides the local two-level systems. Other experiments correlate paths, frequencies, or oscillator quadratures. Continuous variables and modes explains why a two-mode squeezed state is pure globally and thermal in either individual mode.

A preparation device does not by itself specify the state actually measured. Loss, unresolved modes, multiple emitted pairs, and conditioning on detector outcomes can change it. The optics application relates state descriptions to these measurements.

Foundations distinguishes correlation from signalling

Entanglement, steering, and Bell nonlocality answer different questions. Entanglement rules out a mixture of product states. Steering rules out a suitable local-hidden-state account of conditional states, in a specified direction. A Bell test rules out a local-hidden-variable account of the tested correlations. Bell nonlocality implies entanglement, but a failed Bell test does not establish separability. Measurement choices and the allowed class of measurements matter; see the foundations application.

Entanglement does not permit controllable faster-than-light signalling. In an ordinary tensor-product model, a trace-preserving local channel $\mathcal E_B$ satisfies

$$\operatorname{Tr}_B[(\mathrm{id}_A\otimes\mathcal E_B)(\rho_{AB})] =\rho_A.$$

Here $\mathrm{id}_A$ is the identity map on operators. Conditioning on a reported measurement outcome can change the state assigned to $A$, but the unconditioned local statistics stay the same. Exercise 2 proves this directly.

Field theory requires care with the continuum limit

A finite lattice or finite collection of bosonic modes supplies ordinary tensor factors. In continuum field theory, local operator algebras can fail to admit a density-matrix description on a separate Hilbert-space factor for each sharply bounded region. Gauge constraints introduce further subtleties. A regulator can make a calculation possible, but the chosen local observables and the limiting procedure remain part of its interpretation.

Vacuum entropies of sharply bounded regions typically diverge as the short-distance cutoff is removed. One should not interpret that divergence as a finite count of transferable Bell pairs. Quantities such as mutual information between separated regions can avoid the leading local boundary divergences in suitable theories; differences of divergent entropies must still be defined consistently with a common regulator. The QFT introduction explains where finite-system intuition remains useful and where the algebraic description is needed.

Exercises

What would an area law predict?

A square region of side $L$ on a two-dimensional lattice has leading entropy $S(L)=4\alpha L/a$, where $a$ is the lattice spacing. Compare doubling $L$ with a volume-law prediction $S(L)=\beta(L/a)^2$.

Solution

Doubling $L$ doubles the leading area-law entropy but multiplies the volume-law expression by four. This comparison concerns leading scaling at fixed cutoff. It does not determine subleading terms or identify a phase without additional information about the state and Hamiltonian.

Why a local channel cannot signal

For a finite-dimensional system, write $\mathcal E_B(X)=\sum_\mu K_\mu X K_\mu^\dagger$ with $\sum_\mu K_\mu^\dagger K_\mu=I_B$. Prove that every local expectation $\langle O_A\rangle$ is unchanged when outcomes on $B$ are ignored.

Solution

Cyclicity of the full trace gives

$$\begin{aligned} \langle O_A\rangle' &=\sum_\mu\operatorname{Tr}\!\left[ \rho_{AB}(O_A\otimes K_\mu^\dagger K_\mu)\right]\\ &=\operatorname{Tr}[\rho_{AB}(O_A\otimes I_B)]. \end{aligned}$$

Since this holds for every $O_A$, the reduced density matrix is unchanged. A selected outcome does not satisfy the summed completeness relation, which is why a conditional state can differ.

A determinant under an occupied-orbital rotation

Let $b_i^\dagger=\sum_{j=1}^N U_{ij}c_j^\dagger$ for an $N\times N$ unitary matrix mixing only the occupied orbitals. Relate $b_1^\dagger\cdots b_N^\dagger|0\rangle$ to $c_1^\dagger\cdots c_N^\dagger|0\rangle$.

Solution

Anticommutation removes repeated-orbital terms; reordering the remaining terms gives the signed permutation sum $\det U$. The two determinants therefore differ by $\det U$, a phase because $U$ is unitary. The occupied subspace and physical state have not changed.

References

• N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Reviews of Modern Physics 86, 419–478 (2014). DOI.
• P. Calabrese and J. Cardy, “Entanglement entropy and quantum field theory,” Journal of Statistical Mechanics: Theory and Experiment (2004), P06002. Article and preprint.
• J. Eisert, M. Cramer, and M. B. Plenio, “Area laws for the entanglement entropy,” Reviews of Modern Physics 82, 277–306 (2010). DOI.
• R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Reviews of Modern Physics 81, 865–942 (2009). DOI.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010.
• A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover, 1996.
• D. F. Walls and G. J. Milburn, Quantum Optics, 2nd ed., Springer, 2008.