In this post I’d like to briefly mention some connections between log-Sobolev inequalities, and more specifically hypercontractivity, with quantum mechanics and quantum information. To anyone interested I recommend the video from Christopher King’s excellent overview talk at the BIRS workshop. The talk includes a nice high-level discussion of the origins of QFT (roughly 25 minutes in) that I won’t get into here.
Existence and uniqueness of ground states in quantum field theories. The very introduction of (classical!) hypercontractivity, before the name was even coined, was motivated by stability questions in quantum field theory. This approach apparently dates back to a paper by Nelson from 1966, “A quartic interaction in two dimensions” (not available online! If anyone has a copy…my source on this are these notes by Barry Simon; see section 4). Nelson’s motivation was to establish existence and uniqueness of the ground state of an interacting boson field theory that arose as the natural quantization of a certain popular classical field theory. Let be the resulting Hamiltonian. acts on the Hilbert space associated with the state space of a bosonic field with degrees of freedom, the symmetric Fock space .
The main technical reason that make bosonic systems much easier to study than fermionic ones is that bosonic creation and annihilation operators commute. In the present context this commutativity manifests itself through the existence of an isomorphism, first pointed out by Bargmann, between and the space , where is the Gaussian measure on . The connection arises by mapping states in to functions on local observables that lie in the algebra generated by the bosonic elementary creation and annihilation operators. Since these operators commute (when they act on distinct particles), the space of such functions can be identified with the space of functions on the joint spectra of the operators; doing this properly (as Bagmann did) leads to , with the creation and annihilation operators and mapped to and its conjugate respectively.
which under Bargmann’s isomorphism is mapped to . Up to some mild scaling I’m not sure I understand, this is precisely the Liouvillian associated with the Ornstein-Uhlenbeck noise operator that we saw in the previous post!
Here is how hypercontractivity comes in. is Hermitian, and using Bargmann’s isomorphism we can identify with a positive operator on . Performing this first step can already provide interesting results as, in the more general case of a Hamiltonian acting on a field with infinitely any degrees of freedom, it lets one argue for the existence of a ground state of by applying the (infinite-dimensional extension of the) Perron-Frobenius theorem to the positive operator . But now suppose we already know e.g. , so that does have a ground state (there is a lower bound on its spectrum), and we are interested in the existence of a ground state for , where is a small perturbation. This scenario arises for instance when building an interacting field theory as the perturbation of a non-interacting theory, the situation that Nelson was originally interested in. The question of existence of a ground state for reduces to showing that the operator is bounded in norm, for some . Using it suffices to bound, for any ,
where the inequality follows from Hölder. Assuming is bounded (let’s say this is how we measure “small perturbation” — in the particular case of the quantum field Nelson was interested in this is precisely the natural normalization on the perturbation), proving a bound on reduces to showing that there is a constant such that
for any , i.e. is hypercontractive (or at least hyperbounded) in the operator norm. This is precisely what Nelson did, for equal to the number operator (1), i.e. he proved an estimate of the form~(2) for the Ornstein-Uhlenbeck process, later obtaining optimal bounds: hypercontractivity was born! (As Barry Simons recalls in his notes the term “hypercontractive” was only coined a little later, in a paper of his and Hoeg-Krohn.)
Fermionic Hamiltonians and non-commutative spaces. This gives us motivation for studying hypercontractivity of operators on : to establish stability estimates for the existence of ground states of bosonic Hamiltonians. But things start to get even more interesting when one considers fermionic Hamiltonians. Fermions are represented on the antisymmetric Fock space, and for that space there is no isomorphism similar to Bargmann’s. His isomorphism is made possible by thinking of states as functions acting on observables; if the observables commute we obtain a well-defined space of functions acting on the joint spectrum of the observables, leading to the identification with a space of functions on , where is the number of degrees of freedom of the system.
But the fermionic creation and annihilation operators anti-commute: they can’t be simultaneously diagonalized. So states are naturally functions on the non-commutative algebra generated by these operators. Apparently Segal was the first to explore this path explicitly, and he used it as motivation to introduce non-commutative integration spaces . (As an aside, I find it interesting how the physics suggested the creation of such a beautiful mathematical theory. I used to believe that the opposite happened more often — first the mathematicians develop structures, then the physicists find uses for them — but I’m increasingly realizing that historically it’s quite the opposite that tends to happen, and this is especially true in all things quantum mechanical!) For the case of fermions the canonical anti-commutation relations make the algebra generated by the creation and annihilation operators into a Clifford algebra , and the question of existence and uniqueness of ground states now suggests us to explore the hypercontractivity properties of the associated semigroup (third example in the previous post) in . This approach was carried out in beautiful work of Gross; the hypercontractivity estimate used by Gross was generalized, with optimal bounds, by Carlen and Lieb. (I recommend the introduction to their paper for more details on how hypercontractivity comes in, and Gross’ paper for more details on the correspondance between the original fermionic Hamiltonian and the semigroup for which hyperocntractivity is needed.)
Quantum information theory. As Christopher King discussed in his talk at BIRS, aside from their use in quantum field theory non-commutative spaces and hypercontractivity are playing an increasingly important role in quantum information theory. This direction was pioneered in work of Kastoryano and Temme, who showed that, just as classical hypercontractivity and its connection with log-Sobolev inequalities has proven extremely beneficial for the fine study of convergence properties of Markov semi-groups in the “classical” commutative setting (cf. my previous post), hypercontractivity estimates and log-Sobolev inequalities for quantum channels could lead to much improved bounds (compared with estimates based on the spectral gap) on the mixing time of the associated semi-groups. In particular Kastoryano and Temme analyzed the depolarizing channel and obtained the exact value for the associated log-Sobolev constant, extending the classical results on the Bonami-Beckener noise operator that I described in the previous post. The depolarizing channel is already very important for applications, including the analysis of mixing time of dissipative systems or certain quantum algorithms such as the quantum metropolis algorithm.
For more applications and recent developments (including a few outliers…), see the workshop videos!