Measuring Entanglement Entropy of a Generic Many-Body System with a Quantum Switch

Entanglement entropy has become an important theoretical concept in condensed matter physics because it provides a unique tool for characterizing quantum mechanical many-body phases and new kinds of quantum order. However, the experimental measurement of entanglement entropy in a many-body system is widely believed to be unfeasible, owing to the nonlocal character of this quantity. Here, we propose a general method to measure the entanglement entropy. The method is based on a quantum switch (a two-level system) coupled to a composite system consisting of several copies of the original many-body system. The state of the switch controls how different parts of the composite system connect to each other. We show that, by studying the dynamics of the quantum switch only, the Rényi entanglement entropy of the many-body system can be extracted. We propose a possible design of the quantum switch, which can be realized in cold atomic systems. Our work provides a route towards testing the scaling of entanglement in critical systems as well as a method for a direct experimental detection of topological order.

Figure 1

(a) The chain represents a general interacting 1D system with short-ranged hopping and interactions. We describe a method to measure EE between left ($\mathcal{L}$) and right ($\mathcal{R}$) subsystems. (b) Energy spectrum of the finite chain. The ground state is separated from other excitations by an energy gap $\Delta$, which is given by the thermodynamic gap for gapped phases, and by the finite-size gap for gapless phases, Eq. (9). (c),(d) Proposed setup for measuring $n=2$ Rényi EE. Two pairs of half-chains ${\mathcal{L}}_i$ (${\mathcal{R}}_i$), identical to the $\mathcal{L}$ ($\mathcal{R}$) half-chain in (a), are arranged in a cross geometry. The quantum switch, positioned at the center of the cross, controls the way in which the two pairs of half-chains are connected by selectively forbidding tunneling to one of the neighbors. The overlap of the ground states of the two configurations corresponding to different connections is related to $n=2$ Rényi EE and can be measured by studying Rabi oscillations of the quantum switch.

Figure 2

The dynamics of the coupled switch-chain system can be used to measure $n=2$ Rényi entropy. (a) We introduce dynamics of the quantum switch by turning on weak tunneling with amplitude $T$ between the $|\uparrow ⟩$ and $|\downarrow ⟩$ states. (b) Schematic of the energy spectrum of the switch-chains system. The ground state is doubly degenerate and is separated from the excited states by a gap. Assuming that $T\ll \Delta$, the system will perform Rabi oscillations between the two ground states, with the renormalized effective tunneling between them; see Eq. (11). By measuring the renormalized Rabi frequency, it is possible to extract the overlap of the ground states of the two configurations of the many-body system shown in Figs. 1c, 1d and, therefore, the Rényi entropy.

Figure 3

Setup proposed for measuring higher Rényi EE. The case $n=4$ is shown. $2n$ half-chains ${\mathcal{L}}_i$ and ${\mathcal{R}}_i$ are arranged in a star geometry in an alternating fashion. Initially, tunneling to both neighboring half-chains is allowed. The quantum switch blocks tunneling to one of the neighbors; thus, the two states of the switch realize two configurations, in which the half-chains are connected differently (each ${\mathcal{L}}_i$ is connected to its clockwise neighbor ${\mathcal{R}}_i$ for the $|\uparrow ⟩$ state, and it is connected to the counterclockwise neighbor ${\mathcal{R}}_{i+1}$ for the $|\downarrow ⟩$ state.)

Figure 4

Proposed design of the quantum switch in cold atomic systems. Between the end sites of the four half-chains, four quantum wells $a$, $b$, $c$, $d$ are positioned. When empty, they do not affect the tunneling between the end sites (a). Two dipolar molecules (red circles) interacting via long-range repulsive interactions with each other are put in the quantum wells (b),(c). The ground state of the molecules is doubly degenerate, corresponding to the occupation of either $a$, $c$ wells (b) or $b$, $d$ wells (c), such that the repulsive interactions between them are minimized. The dipolar molecules are assumed to be interacting with the particles constituting the many-body chains, such that depending on their state they can block the tunneling between neighboring sites (b),(d). The two states represent the $|\uparrow ⟩$ and $|\downarrow ⟩$ states of the quantum switch.