APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory^*

These are notes on some entanglement properties of quantum field theory, aiming to make accessible a variety of ideas that are known in the literature. The main goal is to explain how to deal with entanglement when—as in quantum field theory—it is a property of the algebra of observables and not just of the states.

Figure 1

(a) A function $g(u)$ holomorphic in the upper half $u$ plane can be computed by a Cauchy integral formula: any contour $\gamma$ in the upper half plane can be used to compute $g(u)$ for $u$ in the interior of $\gamma$. (b) If $g(u)$ is continuous on the boundary of the upper half plane, one can take $\gamma$ to run partly along the boundary. If in addition $g(u)=0$ along part of the boundary—indicated here by dashed lines—then that part of the contour can be dropped. In this case, the Cauchy integral formula remains holomorphic as $u$ is moved through the gap and into the lower half plane, implying that $g(u)$ is holomorphic on that part of the real axis and is identically zero.

Figure 2

(a) An open set $\mathcal{U}$ in Minkowski spacetime, and its domain of dependence $\widehat{\mathcal{U}}$ (the union of $\mathcal{U}$ with the regions labeled as $\widehat{\mathcal{U}}$ in the figure), which in this case is a causal diamond and coincides with the causal completion ${\mathcal{U}}^{\prime \prime }$ of $\mathcal{U}$. (b) The two open sets $\mathcal{U}$ and ${\mathcal{U}}^{\prime }$ are causal complements; each is the largest open set that is spacelike separated from the other. (c) A quite different open set $\mathcal{U}$ whose causal completion ${\mathcal{U}}^{\prime \prime }$ (the union of $\mathcal{U}$ and the regions labeled ${\mathcal{U}}^{\prime \prime }$) is the same causal diamond as in (a).

Figure 3

The right wedge ${\mathcal{U}}_r$ and the left wedge ${\mathcal{U}}_{\ell }$ in Minkowski spacetime. They are the domains of dependence of the right half and left half of the initial value surface $t=0$, which are labeled as ${\mathcal{V}}_r$ and ${\mathcal{V}}_{\ell }$.

Figure 4

(a) The path integral on the half-space $\tau <0$ as a function of boundary values of the fields gives a way to compute the vacuum wave function $\mathrm{\Omega }$. (b) To compute the reduced density matrix of the vacuum for the right half of the surface $\tau =0$ by a Euclidean path integral, we use the path integral on the lower half-space $\tau <0$ to compute a vacuum bra $⟨\mathrm{\Omega }|$, and the path integral on the upper half-space $\tau >0$ to compute a vacuum ket $|\mathrm{\Omega }⟩$. Then we glue together the left halves of the boundaries of the $\tau <0$ and $\tau >0$ half-spaces, identifying the field variables on those boundaries in the bra and the ket. The net effect—a path integral on the upper half-space and the lower half-space together with an integral over field variables on half of the $\tau =0$ hypersurface—produces a path integral on the space depicted here. It can be obtained from Euclidean space ${\mathbb{R}}^D$ by making a “cut” along the half-hyperplane $\tau =0$, $x\ge 0$. (c) Sketched here is a Euclidean wedge of opening angle $\theta$.

Figure 5

(a) The state $\mathsf{a}|\mathrm{\Omega }⟩$ can be obtained by a path integral in the lower half-plane, with $\mathsf{a}$ inserted on the right half of the boundary. (b) Acting with $\exp (2\pi \alpha K_{\ell })\exp (-2\pi \alpha K_r)\mathsf{a}|\mathrm{\Omega }⟩$ adds a wedge of opening angle $2\pi \alpha$ to the right boundary and removes one from the left boundary. If we rotate the picture so that the boundary is again horizontal, it looks like this; the operator $\mathsf{a}$ is now inserted on a ray that is at an angle $2\pi \alpha$ from the horizontal. (c) By the time we get to $\alpha =1/2$, $\mathsf{a}$ is inserted on the left boundary of the lower half-plane. We cannot extend this process further.

Figure 6

An accelerating trajectory $\mathsf{x}(\tau )$ in the right quadrant of the $xt$ plane. The point $\tau =0$ is marked. Shown in dotted lines, on the left, is the mirror trajectory $-\mathsf{x}(\tau )$, which can be obtained from the first by a shift in imaginary time. The two trajectories are spacelike separated.

Figure 7

Two spacelike separated open sets $\mathcal{U}$ and ${\mathcal{U}}^{\prime }$ in Minkowski spacetime, with a gap between them.

Figure 8

(a) If a function $F(z)$ is holomorphic in the strip $0>\mathrm{Im}z>-1/2$ and continuous at the lower boundary of the strip, we can write a Cauchy integral formula with a contour that runs partly on the lower boundary. (b) If $F(z)$ is holomorphic for $-1/2>\mathrm{Im}z>-1$ and continuous on the upper boundary of that strip, we can write a Cauchy integral formula with a contour that runs partly on the upper boundary. (c) If $F(z)$ satisfies both conditions, we can combine the contours from (a) and (b), choosing them so that the part that runs on the line $\mathrm{Re}z=-1/2$ cancels. The resulting Cauchy integral formula shows that $F(z)$ is holomorphic on that line. The argument sketched in Fig. 1 of Sec. 2b is actually the special case of this in which $F(z)$ vanishes in the lower strip.