Entanglement is a very fundamental and amazing feature of quantum theory. It measures non-classical non-local correlations between different parts of a quantum system. If the quantum system is in a highly entangled state, measuring an observable on a part of the system can strongly and instantaneously affect the state of another part of the system, even very far away from the point of measurement. Many applications in the field of quantum information and computation, such as quantum teleportation or quantum cryptography, exploit these powerful correlations. They allow to do tasks that are impossible classically. As another example of application, Hawking’s radiation is closely related to entanglement between the region inside a black hole and the region outside (the only accessible one for an observer).
When a bipartite quantum system (product Hilbert space HA*HB) is in a pure state, a well-known measure of entanglement is the Von Neumann entropy (or more generally the Renyi entropy) of either subsystem, which is the quantum version of the classical Shannon entropy. The entropy is zero for unentangled (« separable ») states and positive for entangled states.
An interesting problem is to study entanglement of random states. In a large system where the Hamiltonian is not known precisely, the wavefunction can be modelled as a random superposition of the basis states, in the same spirit that Wigner introduced random matrices to study large nuclei.
For quantum computation, it is desirable to construct highly entangled states to exploit quantum correlations as best as possible. Random states are suitable candidates as their average entanglement entropy is known to be almost maximal.
We have recently computed the full distribution of entanglement entropy for a random pure state in a bipartite quantum system. For a random pure state in a bipartite system, the eigenvalues of the reduced density matrix of either subsystem are distributed exactly as the eigenvalues of a specific random matrix ensemble (« Wishart ») in presence of an additional constraint (sum of the eigenvalues is one). This allows us to use techniques from random matrix theory such as the Coulomb gas method. The constraint is crucial and leads to unexpected phase transitions in the entropy distribution. We indeed find two critical points at which the entropy distribution changes shape and has a nonanalyticity. These changes are the direct consequence of two phase transitions in the associated Coulomb gas. In particular, at the second transition point, the maximal eigenvalue becomes suddenly much larger than the other eigenvalues. This transition is reminiscent of the Bose Einstein condensation in cold atoms.
With the full distribution of the entropy, we can also get the extreme tails of the distribution. In particular, we show that the common idea that a random pure state is maximally entangled is not quite correct. We indeed find that, although random pure states are highly entangled on average, the probability of an almost maximally entangled state is actually very small.