Entanglement in QFT Preview

Quantum field theory uses the same reduced-state ideas as ordinary composite quantum mechanics, but the meaning of “subsystem” becomes more delicate. The relevant split may be a set of modes, a spatial region, a local algebra of observables, or a regulated lattice approximation to a continuum field.

A regulated lattice makes partial traces explicit. Removing the short-distance cutoff requires more care: a continuum local algebra generally does not carry the ordinary density-matrix trace of a tensor factor. We will use a pair of coupled oscillators to see the origin of vacuum entanglement, then identify which entropy formulas depend on a regulator.

Helpful background. Mode decompositions and many-body spatial cuts distinguish the subsystems. Second quantization supplies field-operator language. Logarithms on this page are natural, so entropy and relative entropy are in nats; divide by $\ln2$ for bits.

From Finite Factors to Fields

In a finite bipartite system one starts with

$$\mathcal H = \mathcal H_A\otimes\mathcal H_B, \qquad \rho_A = \operatorname{Tr}_B\rho_{AB}.$$

This is the clean algebraic model behind most introductory entanglement theory. An unconstrained lattice model with a tensor factor at each site can be treated in the same way, even when an oscillator site has an infinite-dimensional Hilbert space: choose a set of lattice sites $A$, trace over the complement $\bar A$, and compute

$$S(A) = -\operatorname{Tr} \bigl( \rho_A\log\rho_A \bigr).$$

The continuum limit changes the situation. Local field operators are distributions, spatial regions have infinitely many short-distance degrees of freedom near their boundaries, and the Hilbert space associated with a continuum region need not factorize as a simple tensor product in the same way as a finite spin system. In ordinary continuum relativistic QFT, local algebras are typically type III: a regional state cannot be represented by a density operator with an intrinsic ordinary trace on that algebra. This is stronger than saying the factor is merely infinite-dimensional. A lattice cutoff or a specified separation between regions can provide a tensor-factor approximation, but smearing a few operators alone does not turn the full regional algebra into such a factor (Witten, 2018).

Mode Entanglement

Mode entanglement is entanglement across a chosen mode decomposition. A free bosonic field in a finite box with an ultraviolet cutoff can be expanded as finitely many oscillator modes:

$$H = \sum_{\mathbf k} \hbar\omega_{\mathbf k} \left( a_{\mathbf k}^\dagger a_{\mathbf k} +\frac12 \right),$$

with a Fock space organized by occupations

$$\lvert n_{\mathbf k_1},n_{\mathbf k_2},\ldots\rangle.$$

Once modes are named, one can ask whether a state factorizes across a split such as mode set $M$ versus mode set $\bar M$. This is the same logical structure as the two-mode entanglement story in quantum optics.

A state can be product in one mode split and entangled in another. However, the oscillator vacuum stays product under a passive unitary mixing only annihilation operators: every new annihilation operator still kills it. Relating normal-mode vacua to local field coordinates can instead involve different oscillator frequency scales, and relating different particle notions can involve Bogoliubov transformations that mix creation and annihilation operators. In a continuum limit, such transformations need not all be unitarily implementable in one Fock representation.

Spatial-Region Entanglement

Spatial-region entanglement asks how a state is correlated between a region $A$ and its spatial complement $\bar A$. On an unconstrained lattice with site factors,

$$\mathcal H_{\mathrm{lattice}} \cong \mathcal H_A\otimes\mathcal H_{\bar A}.$$

One computes

$$\rho_A = \operatorname{Tr}_{\bar A} \lvert\Psi\rangle\langle\Psi\rvert, \qquad S_A = -\operatorname{Tr} \bigl( \rho_A\log\rho_A \bigr).$$

In a continuum QFT, this formula is best read as a regulated starting point. One may discretize space with a specified cutoff to obtain ordinary reduced density operators, or formulate the continuum question directly using local algebras. The resulting entropy often depends on the regulator, but that does not make the question meaningless. The dependence itself reflects the accumulation of short-distance correlations across the boundary of $A$.

This is the field-theory version of the theme developed in Entanglement Depends on a Decomposition: the word “entanglement” becomes precise only after the relevant regions, modes, or observable algebras are specified.

Vacuum Entanglement

The QFT vacuum is not an empty tensor product over spatial regions. Even when it is the lowest-energy state and has no particles in a particular Fock description, it generally has nonzero correlations between local observables in separated regions.

For a regulated vacuum state $\lvert0\rangle$ and a spatial region $A$,

$$\rho_A = \operatorname{Tr}_{\bar A} \lvert0\rangle\langle0\rvert$$

is typically mixed. That mixedness is not ordinary ignorance about particles hiding in $A$. It reflects entanglement between degrees of freedom associated with $A$ and degrees of freedom associated with $\bar A$.

This distinction is easy to miss because the same word “vacuum” appears in ordinary Fock space. The vacuum state page explains the no-particle vector in nonrelativistic Fock space. In QFT, the vacuum also carries local correlation structure, symmetry constraints, and sometimes observer-dependent particle interpretations.

A two-oscillator vacuum calculation

Take $\hbar=m=1$, $\omega>0$, and $\kappa\ge0$, with

$$H=\frac12[p_1^2+p_2^2+\omega^2(q_1^2+q_2^2)+\kappa(q_1-q_2)^2].$$

The normal coordinates $q_\pm=(q_1\pm q_2)/\sqrt2$ have frequencies $\Omega_+=\omega$ and $\Omega_-=\sqrt{\omega^2+2\kappa}$. The ground state is a product of Gaussian oscillator vacua in those coordinates. In the site split, however,

$$\langle q_1^2\rangle=\frac14(\Omega_+^{-1}+\Omega_-^{-1}),\qquad \langle p_1^2\rangle=\frac14(\Omega_++\Omega_-).$$

The symmetrized $q_1,p_1$ covariance vanishes. Its reduced symplectic eigenvalue is

$$\nu_1=\frac{\Omega_++\Omega_-}{4\sqrt{\Omega_+\Omega_-}}\ge\frac12.$$

Equality holds only when $\kappa=0$. For any nonzero coupling, the one-site Gaussian reduction is mixed and the pure ground state is entangled between sites. For example, $\Omega_+=1$, $\Omega_-=3$ gives reduced purity $1/(2\nu_1)=\sqrt3/2$. A free lattice field contains many coupled oscillators of this kind; “no normal-mode particles” does not mean “no spatial entanglement.”

Area-Law Divergences

For a continuum QFT in $d$ spatial dimensions, the leading entanglement entropy of a smooth spatial region often has the schematic cutoff dependence

$$S_A \sim \alpha\, \frac{\operatorname{Area}(\partial A)}{\epsilon^{d-1}} + \cdots, \qquad d>1,$$

where $\epsilon$ is a short-distance cutoff. The coefficient $\alpha$ is generally not universal. It depends on the regulator and microscopic details. The boundary scaling is the important structural feature: most of the leading contribution comes from short-distance correlations straddling $\partial A$.

In one spatial dimension, the boundary of an interval is zero-dimensional, and conformal field theory gives a logarithmic form for the vacuum entropy of an interval of length $\ell$ on an infinite line:

$$S_A = \frac{c}{3} \log\frac{\ell}{\epsilon} + \mathrm{const}.$$

Here the state is the zero-temperature vacuum of a unitary, nonchiral $1+1$-dimensional CFT on the infinite line, $c$ is its usual central charge, and $\ell\gg\epsilon$. Boundaries, finite circumference, temperature, and chiral conventions change the formula. The coefficient of $\ln\ell$ is universal, while the additive constant depends on the cutoff convention.

The many-body bridge gives the lattice and tensor-network version of area-law thinking. The QFT lesson is sharper: spatial entanglement is often dominated by ultraviolet boundary correlations, so one must ask which pieces are universal, finite, or regulator-dependent.

Mutual Information as a Safer Quantity

The entropy $S(A)$ of one continuum region is often divergent. For two separated regions $A$ and $B$, the mutual information

$$I(A:B) = S(A)+S(B)-S(A\cup B)$$

can cancel many local boundary divergences when the regions are separated by a nonzero distance. It is not an entanglement measure for arbitrary mixed states, but it is a useful finite diagnostic of total correlation in many QFT settings.

This is the same finite-dimensional formula developed on the Mutual Information page. In QFT the value depends on geometry, state, separation, regulator assumptions, and operator-algebra choices.

Modular Hamiltonians

For a faithful reduced density operator $\rho_A$ in a regulated factor, its modular Hamiltonian is defined by

$$K_A = -\log\rho_A, \qquad \rho_A = e^{-K_A}.$$

Zero eigenvalues give infinite $-\log\rho_A$ outside the support; one can instead work on its support. Writing $\rho_A=e^{-K_A}/Z_A$ permits an additive constant in $K_A$. That constant cancels from modular flow and normalized-state expectation differences. Continuum modular theory generalizes the flow without requiring a regional density matrix.

For a generic region and interacting theory, $K_A$ is highly nonlocal and hard to compute. A major exception is the vacuum of a Lorentz-invariant QFT restricted to a half-space. In units with $\hbar=c=k_B=1$, the Bisognano–Wichmann result gives a local expression proportional to the boost generator:

$$K_R = 2\pi \int_{x^1>0} d^d x\, x^1 T_{00}(0,\mathbf x),$$

where $d$ is the number of spatial dimensions. The integration includes $x^1$ and all $d-1$ transverse coordinates. This is the right-wedge vacuum modular-flow generator, formally written as a one-sided modular Hamiltonian; in the continuum it must not be interpreted as proving the existence of an ordinary trace-normalized $\rho_R$. This formula is not a generic formula for every region. It is important precisely because generic modular Hamiltonians are much less local.

Relative Entropy Preview

For two density operators on the same regulated Hilbert-space factor, the quantum relative entropy is

$$D(\rho\Vert\sigma) = \operatorname{Tr} \bigl[ \rho(\log\rho-\log\sigma) \bigr].$$

In finite dimension this is finite when $\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma$, and is $+\infty$ otherwise. In infinite dimension even support inclusion does not guarantee finiteness. If the following entropy and expectation differences are well defined and $K_\sigma=-\log\sigma$, then

$$D(\rho\Vert\sigma) = \Delta\langle K_\sigma\rangle - \Delta S,$$

where

$$\Delta\langle K_\sigma\rangle = \operatorname{Tr}(\rho K_\sigma) - \operatorname{Tr}(\sigma K_\sigma), \qquad \Delta S = S(\rho)-S(\sigma).$$

The inequality

$$D(\rho\Vert\sigma)\ge0$$

is one reason relative entropy is so useful in QFT. It compares two states on the same region, and many ultraviolet divergences cancel between the two terms when the comparison is well posed. This makes relative entropy a more robust object than either entanglement entropy alone in many continuum arguments.

Continuum algebraic relative entropy is defined without subtracting two undefined infinite entropies. The displayed difference is useful when a common regulator and a suitable limit justify it; formal cancellation alone is not a proof of finiteness.

Holography Preview

In gauge/gravity duality, some strongly coupled quantum field theories admit a gravitational description in a higher-dimensional spacetime. In that setting, the Ryu–Takayanagi formula relates the entanglement entropy of a boundary region $A$ to the area of a bulk minimal surface $\gamma_A$ on a static time slice, anchored at $\partial A$ and homologous to $A$:

$$S_A = \frac{\operatorname{Area}(\gamma_A)}{4G_N},$$

at leading order in the static semiclassical limit of an Einstein-gravity dual, with $\hbar=1$ and compatible boundary/bulk cutoffs. The surface is codimension two in bulk spacetime. Time-dependent versions use an appropriate extremal surface; quantum and higher-curvature corrections change the expression.

This is a profound bridge between entanglement and geometry, but it should not be overgeneralized. It is not a generic formula for arbitrary quantum systems, and it does not replace ordinary reduced density operators. It is a statement within particular holographic dualities, with controlled assumptions about the field theory, state, spacetime, and gravitational limit.

Gauge-Theory and Algebra Caveats

Gauge theories add a further subtlety: the physical Hilbert space may not factorize cleanly across a spatial boundary because gauge constraints relate the two sides. Depending on the formulation, one may introduce extended Hilbert spaces, edge modes, centers of observable algebras, or algebraic definitions of entropy.

These choices are not cosmetic. They determine which observable algebra is assigned to the region and which boundary or charge-sector data are counted. This is why QFT entanglement claims should state the theory, state, region, regulator, and algebraic convention.

The warning is not that QFT entanglement is unusable. The warning is that the finite-dimensional formula

$$\rho_A = \operatorname{Tr}_{\bar A}\rho$$

must be interpreted through the physical and mathematical structure of the field theory.

References

• J. J. Bisognano and E. H. Wichmann, “On the duality condition for a Hermitian scalar field,” Journal of Mathematical Physics 16, 985–1007 (1975), doi:10.1063/1.522605.
• P. Calabrese and J. Cardy, “Entanglement entropy and quantum field theory,” Journal of Statistical Mechanics P06002 (2004), doi:10.1088/1742-5468/2004/06/P06002.
• H. Casini, “Relative entropy and the Bekenstein bound,” Classical and Quantum Gravity 25, 205021 (2008), doi:10.1088/0264-9381/25/20/205021.
• H. Casini and M. Huerta, “Entanglement entropy in free quantum field theory,” Journal of Physics A 42, 504007 (2009), doi:10.1088/1751-8113/42/50/504007.
• S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Physical Review Letters 96, 181602 (2006), doi:10.1103/PhysRevLett.96.181602.
• M. Srednicki, “Entropy and area,” Physical Review Letters 71, 666–669 (1993), doi:10.1103/PhysRevLett.71.666.
• E. Witten, “APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory,” Reviews of Modern Physics 90, 045003 (2018), doi:10.1103/RevModPhys.90.045003. The preprint is titled “Notes on Some Entanglement Properties of Quantum Field Theory.”

Exercises

1. Mode split versus spatial split. Explain why the statement “the vacuum is a product state” can be true in one mode description but misleading for spatial regions.

Solution

For a free field in a chosen normal-mode basis, the Fock vacuum is annihilated by every $a_{\mathbf k}$ and can be described as the product of oscillator ground states for those modes. That is a statement about the chosen mode decomposition.

Spatial regions define a different split. Local field correlations across the boundary of a region usually make the reduced state

$$\rho_A = \operatorname{Tr}_{\bar A} \lvert0\rangle\langle0\rvert$$

mixed. Thus productness in one mode basis does not imply productness across spatial regions.

2. Area-law scaling. In a regulated QFT with three spatial dimensions, a ball of radius $R$ has boundary area proportional to $R^2$. What is the leading cutoff scaling expected for its entanglement entropy?

Solution

For $d=3$, the schematic area-law divergence is

$$S_A \sim \alpha\, \frac{\operatorname{Area}(\partial A)}{\epsilon^2}.$$

Since $\operatorname{Area}(\partial A)\propto R^2$, the leading scaling is

$$S_A \propto \frac{R^2}{\epsilon^2},$$

up to a nonuniversal coefficient and subleading terms.

3. Relative entropy identity. In a finite-dimensional factor, let $\sigma$ be full rank and $K_\sigma=-\log\sigma$. Show that

$$D(\rho\Vert\sigma) = \Delta\langle K_\sigma\rangle - \Delta S.$$

Solution

Start from

$$D(\rho\Vert\sigma) = \operatorname{Tr}(\rho\log\rho) - \operatorname{Tr}(\rho\log\sigma).$$

Since $S(\rho)=-\operatorname{Tr}(\rho\log\rho)$ and $K_\sigma=-\log\sigma$,

$$D(\rho\Vert\sigma) = -S(\rho) + \operatorname{Tr}(\rho K_\sigma).$$

For $\sigma$ itself,

$$0 = D(\sigma\Vert\sigma) = -S(\sigma) + \operatorname{Tr}(\sigma K_\sigma).$$

Subtracting this zero gives

$$D(\rho\Vert\sigma) = \operatorname{Tr}(\rho K_\sigma) - \operatorname{Tr}(\sigma K_\sigma) - \bigl( S(\rho)-S(\sigma) \bigr),$$

which is

$$D(\rho\Vert\sigma) = \Delta\langle K_\sigma\rangle - \Delta S.$$

4. Holography caution. What assumption is missing from the claim “entanglement entropy equals area divided by $4G_N$”?

Solution

The claim needs a holographic duality and the appropriate gravitational limit. The Ryu–Takayanagi formula applies in special field theories with gravitational dual descriptions, and even there one must specify the state, boundary region, bulk surface, and whether quantum corrections are being included. It is not a generic formula for arbitrary quantum systems.

5. Local vacuum purity. In the coupled-oscillator example take $\omega=1$ and $\kappa=4$. Find the normal frequencies and local purity, and explain why passive mixing of annihilation operators is not sufficient to interpret the site vacuum.

Solution

The normal frequencies are $\Omega_+=1$ and $\Omega_-=\sqrt{1+8}=3$. The site covariance has variances $\langle q_1^2\rangle=1/3$ and $\langle p_1^2\rangle=1$, so $\nu_1=1/\sqrt3$ and purity $\sqrt3/2<1$. Normal-mode annihilation operators use their respective frequency-dependent position and momentum scales. A site annihilation operator built using a single local frequency generally mixes those normal annihilation operators with creation operators, not merely by a passive rotation.