In classical mechanics, a dynamical system of interacting bodies with 2N-dimensional phase space is said to be integrable if there exist N conserved functions (charges) whose Poisson brackets vanish. For an integrable system in quantum field theory (QFT) there exists an infinite set of commuting conserved charges. The existence of the conserved charges allows us to solve the physical system exactly and in this way to describe the modeled phenomena without any approximation. Although the integrability is restricted to low dimensions, the exact solution often provides general information about the physical phenomena. At present, we know precisely how to generate systematically integrable models and how to solve them, explicitly or implicitly in the form of integral equations.

Integrable models cover many domains of quantum mechanics and statistical physics:

• Non-relativistic one-dimensional (1D) continuum Fermi and Bose quantum gases with specific types of singular and short-range interactions.

• 1D lattice and continuum quantum models of condensed-matter physics, like the Heisenberg model of interacting quantum spins, the Hubbard model of hopping electrons with one-site interactions between electrons of opposite spins, the Kondo model describing the interaction of a conduction band with a localized spin impurity, microscopic models of superconductors, etc.

• Relativistic models of QFT in a (1+1)-dimensional spacetime like the sine-Gordon model and its fermionic analog, the Thirring model, and so on.

• Two-dimensional (2D) lattice and continuum classical models in thermal equilibrium like the lattice Ising model of interacting nearest-neighbor ±1 spins, the six- and eight-vertex models, the continuum Coulomb gas of ±1 charges interacting by a logarithmic potential, etc.

The goal of the covering lemma is perhaps opposite to that of the packing lemma because it applies in a setting where one party wishes to make messages indistinguishable to another party (instead of trying to make them distinguishable as in the packing lemma of the previous chapter). That is, the covering lemma is helpful when one party is trying to simulate a noisy channel to another party, rather than trying to simulate a noiseless channel. One party can accomplish this task by randomly covering the Hilbert space of the other party (this viewpoint gives the covering lemma its name).

One can certainly simulate noise by choosing a quantum state uniformly at random from a large set of quantum states and passing along the chosen quantum state to a third party without telling which state was chosen. But the problem with this approach is that it could potentially be expensive if the set from which we choose a random state is large, and we would really like to use as few resources as possible in order to simulate noise. That is, we would like the set from which we choose a quantum state uniformly at random to be as small as possible when simulating noise. The covering lemma is similar to the packing lemma in the sense that its conditions for application are general (involving bounds on projectors and an ensemble), but it gives an asymptotically efficient scheme for simulating noise when we apply it in an IID setting.