A property called entanglement entropy helps to describe the quantum states of interacting particles, and it has at last been measured. The findings open the door to a deeper understanding of quantum systems. See Article p.77
A puzzling aspect of quantum mechanics is entanglement: the idea that the combined state of two particles can be completely specified, but that the state of each entangled particle is completely random when measured alone. Entanglement entropy marries the concept of entanglement with that of entropy — the degree of randomness of a system — and has become a useful theoretical tool with which to characterize many-body states in condensed-matter physics. On page 77 of this issue, Islam et al.1 report the first experimental measurement of entanglement entropy in a small system of atoms trapped in a lattice of light, a model of a solid-state system.
Quantum mechanics, the theory of the microscopic world, has many features that run counter to our everyday experiences in a classical world. The possibility of entanglement in a quantum system of two or more particles has been a challenging and stimulating idea for many years. Einstein and his colleagues were famously bothered by the idea that measuring one particle of an entangled pair seemingly instantaneously determined the state of its partner — “spooky action at a distance”, as they put it2. But the existence of entanglement was made concrete through the theoretical work of the physicist John Bell3, and experimental tests of Bell's inequalities (constraints derived from Bell's work) have unambiguously verified the quantum-mechanical description of the microscopic world (see ref. 4, for example).
Although an understanding of two-particle entanglement is quite well in hand, there is no specific measure of the amount of entanglement in three or more particles. Yet entanglement has become an important tool for understanding the states of many-body systems. When many particles interact with one another, even through simple interactions, the low-energy quantum states can be surprisingly complicated, with lots of entanglement. Entanglement entropy has become a favoured theoretical measure for categorizing such complex states.
To understand what many-particle entanglement means, let's start by considering a non-entangled system. If I create a system that has N particles, each in an identical state independent of their N – 1 neighbours, then its many-body description is simple, and measuring one particle or partitioning the sample has little impact on the overall system. Not that such states are uninteresting — this is a good description of a state of matter called a Bose–Einstein condensate, for example. Similarly, if each particle is in its own different state, with no relationship to its neighbours, then measurement or partitioning has no global effect.
But if the particles are entangled with one another, either pairwise or in a more complex fashion, then measurement of one particle affects the state of other particles. Entanglement entropy measures the increase in entropy (which can be thought of as increased randomness) that occurs if we partition such a system5. Identifying emergent, complex, lowest-energy states of seemingly simple systems of interacting particles is a particularly challenging task, for which entanglement entropy can be used to understand the nature of the state and to probe its 'quantumness'.
Until now, entanglement entropy has been a purely theoretical construct in condensed-matter physics, because it is difficult to partition a solid-state system and measure its constituents. Islam et al. have performed the first such measurements using two identical copies of a small system of four atoms trapped in an optical lattice (an array of interfering laser beams). If the potential-energy 'landscape' of the optical lattice is not too deep, the particles can tunnel from one site to the next and feel the presence of their neighbours. This leads to a many-body state that exhibits entanglement. But if the lattice is deep, the particles act as individuals, and are free of entanglement.
The authors performed their experiment in a quantum gas microscope6, in which a single layer of an optical lattice is generated just below a high-resolution optical microscope. When Islam et al. relaxed some of the optical confining fields, the two copies of the four-atom systems could tunnel into one another and, through quantum interference (the Hong–Ou–Mandel effect7), leave a signature of their state in the number of atoms in each lattice site (Fig. 1). The authors simply counted the atoms using the microscope and extracted the entanglement entropy (the second-order Rényi entanglement entropy5, for those in the know) from the number of atoms. In this way, they show that their four-atom system can have less entropy as a whole than when it is partitioned, something that is not possible without entanglement, nor in any classical system.
As the first measurement of its kind, this is a milestone. But as with any first, the experimental techniques involved have been pushed to their limits. The 'many-body' systems therefore consist of only four particles, primarily limited by how well the two copies interfere. With improvements, it should be possible to study larger numbers of atoms and more-interesting interacting systems. An intriguing possibility would be to measure higher-order entanglement — the current experiment measures second-order entanglement, whereas nth-order entanglement would require n interfering copies. This would give further access to the entanglement spectrum, which yields complete knowledge of the quantum state of a system.
Understanding the way in which complex many-body states appear and evolve in systems out of equilibrium is a hot topic in condensed-matter physics, because much of our world is not in equilibrium. This is an especially interesting question in closed systems for which there is no means of driving the system to a thermal equilibrium. Entanglement entropy will be a crucial tool for understanding non-equilibrium systems, and Islam and colleagues' experimental approach is easily adaptable to such studies. The authors' proof-of-principle experiment also opens the door to a greater understanding of the role of entanglement in complex many-body systems through direct experimental observations. Given that both entanglement and entropy are sometimes perplexing concepts, the ability to acquire tangible information about them in the laboratory will certainly benefit their study.