Independent Events and Tensor Products

Ray LaPierre

Nicolas Loizeau, Flaviano Morone, Dries Sels

Quantum many-body systems are typically endowed with a tensor product structure. A structure they inherited from probability theory, where the probability of two independent events is the product of the probabilities. The tensor product structure of a Hamiltonian thus gives a natural decomposition of the system into independent smaller subsystems. It is interesting to understand whether a given Hamiltonian is compatible with some particular tensor product structure. In particular, we ask, is there a basis in which an arbitrary Hamiltonian has a 2-local form, i.e., it contains only pairwise interactions? Here we show, using analytical and numerical calculations, that a generic Hamiltonian (e.g., a large random matrix) can be approximately written as a linear combination of two-body interaction terms with high precision; that is, the Hamiltonian is 2-local in a carefully chosen basis. Moreover, we show that these Hamiltonians are not fine-tuned, meaning that the spectrum is robust against perturbations of the coupling constants. Finally, by analyzing the adjacency structure of the couplings [Formula: see text], we suggest a possible mechanism for the emergence of geometric locality from quantum chaos.

Kalyanapuram Rangachari Parthasarathy

Suppose (Ω i , F i ), 1 ≤ i ≤ n are sample spaces describing the elementary outcomes and events concerning n different statistical systems in classical probability. To integrate them into a unified picture under the umbrella of a single sample space one takes their cartesian product (Ω, F) where Ω = Ω1 x … x Ω n , F = F1 x … x F n , the smallest σ-algebra containing all rectangles of the form F1 x F1 x … F n , F j ∈ F j for each j. Now we wish to search for an analogue of this description in quantum probability when we have n systems where the events concerning the j-th system are described by the set P(ℋ j ) of all projections in a Hubert space ℋ j j = 1, 2,…, n. Such an attempt leads us to consider tensor products of Hilbert spaces. We shall present a somewhat statistically oriented approach to the definition of tensor products which is at the same time coordinate free in character. To this end we introduce the notion of a positive definite kernel.

Ray LaPierre

T. R. Klevtsov, Dmitry Anatolevich Kronberg

Abstract The problem of determining the linear independence for a vector system arises in quantum mechanics in the context of the unambiguous state discrimination and other postselective transformations. For several copies of the same set of quantum states, the conditions of linear independence are known, whereas in this paper we consider tensor product for two general sets of vectors. We obtain a necessary and sufficient condition for the linear independence of the tensor product of two vector systems.

Matthias P. Kl�y, Charles H. Randall, David J. Foulis