Recently, several hybrid approaches to quantum information emerged which utilize both continuous- and discrete-variable methods and resources at the same time. In this work, we investigate the bipartite hybrid entanglement between a finite-dimensional, discrete-variable quantum system and an infinite-dimensional, continuous-variable quantum system. A classification scheme is presented leading to a distinction between pure hybrid entangled states, mixed hybrid entangled states (those effectively supported by an overall finite-dimensional Hilbert space), and so-called truly hybrid entangled states (those which cannot be described in an overall finite-dimensional Hilbert space). Examples for states of each regime are given and entanglement witnessing as well as quantification are discussed. In particular, using the channel map of a thermal photon noise channel, we find that true hybrid entanglement naturally occurs in physically important settings. Finally, extensions from bipartite to multipartite hybrid entanglement are considered.
We consider the problem of optimally identifying the state of a probe qudit, prepared with given prior probability in a pure state belonging to a finite set of possible states which together span a
D-dimensional subspace of the
d-dimensional Hilbert space the qudit is defined in. It is assumed that we do not know some or all of the states in the set, but for each unknown state we are given a reference qudit into which this state is encoded. We show that from the measurement for optimal state identification with
d=D one can readily determine the optimal figure of merit for qudits with
d>D, without solving a new optimization problem. This result is applied to the minimum-error identification and to the optimal unambiguous identification of two qudit states with $d\ensuremath{\ge}2$, where either one or both of the states are unknown, and also to the optimal unambiguous identification of
N equiprobable linearly independent unknown pure qudit states with $d\ensuremath{\ge}N$. In all cases the optimal figure of merit, averaged over the unknown states, increases with growing dimensionality
d of the qudits.
We introduce a class of states of a composite quantum system, the so-called cross states, that turn out to play a major role in the theory of entanglement for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilbert spaces of the bipartition is finite-dimensional, all states are cross states, whereas, in the genuinely infinite-dimensional setting where the dimension of both Hilbert spaces is not finite, the cross states form a trace-norm dense, convex, proper subset of the set of all states. In the latter case, the cross states can be regarded as those physical states that possess a finite amount of entanglement; accordingly, all separable states are of this kind. We prove that, for any Hilbert space dimension, the separable states can be characterized as those cross states that minimize a suitable norm, i.e., the projective norm associated with the projective tensor product of two trace classes; all other cross states are density operators belonging to the projective tensor product space. This is a generalization of the classical cross norm criterion of separability. Finally, we define an extended real-valued entanglement function and study its main properties. Coherently with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the projective norm, precisely on the cross states of the system.