Back to Hilbert space

Kam Wai Clifford Chan, Juan P. Torres, Joseph H. Eberly

We show that, although the amount of mutual entanglement of photons propagating in free space is fixed, the type of correlations between the photons that determine the entanglement can dramatically change during propagation. We show that this amounts to a migration of entanglement in Hilbert space, rather than real space. For the case of spontaneous parametric down-conversion, the migration of entanglement in transverse coordinates takes place from modulus to phase of the biphoton state and back again. We propose an experiment to observe this migration in Hilbert space and to determine the full entanglement.

Andrea Beggi, Ilaria Siloi, Claudia Benedetti +4 more

Abstract Quantum states of systems made of many identical particles, e.g. those described by Fermi–Hubbard and Bose–Hubbard models, are conveniently depicted in the Fock space. However, in order to evaluate some specific observables or to study system dynamics, it is often more effective to employ the Hilbert space description. Moving effectively from one description to the other is thus a desirable feature, especially when a numerical approach is needed. Here we recall the construction of the Fock space for systems of indistinguishable particles, and then present a set of recipes and advice for students and researchers with the need to commute back and forth from one description to the other. The two-particle case is discussed in some detail, and a few guidelines for numerical implementations are given.

Aaron David Fairbanks, Peter Selinger

Abstract Traced monoidal categories model processes that can feed their outputs back to their own inputs, abstracting iteration. The category of finite-dimensional Hilbert spaces with the direct sum tensor is not traced. But surprisingly, in 2014, Bartha showed that the monoidal subcategory of isometries is traced. The same holds for coisometries, unitary maps, and contractions. This suggests the possibility of feeding outputs of quantum processes back to their own inputs, analogous to iteration. In this paper, we show that Bartha’s result is not specifically tied to Hilbert spaces, but works in any dagger additive category with Moore–Penrose pseudoinverses (a natural dagger-categorical generalization of inverses).