  The semi-device-independent approach provides a framework for
prepare-and-measure quantum protocols using devices whose behavior must not be
characterized nor trusted, except for a single assumption on the dimension of
the Hilbert space characterizing the quantum carriers. Here, we propose instead
to constrain the quantum carriers through a bound on the mean value of a
well-chosen observable. This modified assumption is physically better motivated
than a dimension bound and closer to the description of actual experiments. In
particular, we consider quantum optical schemes where the source emits quantum
states described in an infinite-dimensional Fock space and model our assumption
as an upper bound on the average photon number in the emitted states. We
characterize the set of correlations that may be exhibited in the simplest
possible scenario compatible with our new framework, based on two
energy-constrained state preparations and a two-outcome measurement.
Interestingly, we uncover the existence of quantum correlations exceeding the
set of classical correlations that can be produced by devices behaving in a
purely pre-determined fashion (possibly including shared randomness). This
feature suggests immediate applications to certified randomness generation.
Along this line, we analyze the achievable correlations in several
prepare-and-measure optical schemes with a mean photon number constraint and
demonstrate that they allow for the generation of certified randomness. Our
simplest optical scheme works by the on-off keying of an attenuated laser
source followed by photocounting. It opens the path to more sophisticated
energy-constrained semi-device-independent quantum cryptography protocols, such
as quantum key distribution.
