We introduce a quantum-optical notion of nonclassicality that we call the process output nonclassicality for multimode quantum channels. The motivation comes from an information-theoretic point of view and the emphasis is on the output states of a channel. We deem a channel to be ``classical'' if its outputs are always classical irrespective of the input, i.e., if the channel is nonclassicality breaking, and nonclassical otherwise. Our condition is stronger than the one considered by Rahimi-Keshari et al., [Phys. Rev. Lett. 110, 160401 (2013)] and we compare the two approaches. Using our framework we then quantify the nonclassicality of a quantum process by introducing a noise-robustness type of measure that we call the nonclassicality depth of a channel. It characterizes a certain threshold noise beyond which a given channel outputs only classical states. We achieve this by generalizing a prescription by Lee [Phys. Rev. A 44, R2775 (1991)] to multimode states and then by extension to multimode channels.
Nonclassicality and entanglement are notions fundamental to quantum-information processes involving continuous variable systems. That these two notions are intimately related has been intuitively appreciated for quite some time. An aspect of considerable interest is the behavior of these attributes of a state under the action of a noisy channel. Inspired by the notion of entanglement-breaking channels, we define the concept of nonclassicality-breaking channels in a natural manner. We show that the notion of nonclassicality breaking is essentially equivalent---in a clearly defined sense of the term ``essentially''---to the notion of entanglement breaking, as far as bosonic Gaussian channels are concerned. This is notwithstanding the fact that the very notion of entanglement breaking requires reference to a bipartite system, whereas the definition of nonclassicality breaking makes no such reference. Our analysis rests on our classification of channels into nonclassicality-based, as against entanglement-based, types of canonical forms. Our result takes ones intuitive understanding of the close relationship between nonclassicality and entanglement a step closer.
In the context of bipartite bosonic systems, two notions of classicality of correlations can be defined:
P classicality, based on the properties of the Glauber-Sudarshan
P function; and
C classicality, based on the entropic quantum discord. It has been shown that these two notions are maximally inequivalent in a static (metric) sense, as they coincide only on a set of states of zero measure. We extend and reinforce quantitatively this inequivalence by addressing the dynamical relation between these types of nonclassicality in a paradigmatic quantum-optical setting: the linear mixing at a beam splitter of a single-mode Gaussian state with a thermal reference state. Specifically, we show that almost all
P-classical input states generate outputs that are not
C classical. Indeed, for the case of zero thermal reference photons, the more
P-classical resources at the input the less
C classicality at the output. In addition, we show that the
P classicality at the input---as quantified by the nonclassical depth---does instead determine quantitatively the potential of generating output entanglement. This endows the nonclassical depth with a new operational interpretation: it gives the maximum number of thermal reference photons that can be mixed at a beam splitter without destroying the output entanglement.
A complete classification of single-mode bosonic Gaussian channels on the basis of being quantum-limited or entanglement breaking under
n-fold composition is obtained. Parametric forms for all single-mode bosonic Gaussian channels that remain quantum-limited under
n-fold composition is obtained. It is shown that the obtained parametric forms of quantum-limited amplifier, attenuator, and singular noise channels are entanglement saving, i.e., they do not break entanglement under finite
n-fold composition. All other single-mode bosonic Gaussian channels, quantum limited or not, are shown to be eventually entanglement breaking. Nonclassicality breaking under multiple composition of a single-mode bosonic Gaussian channel is also studied. We outline a family of single-mode bosonic Gaussian channels that are eventually entanglement breaking, but asymptotically nonclassicality saving. We illustrate examples of channels that are eventually nonclassicality breaking, asymptotically nonclassicality breaking, and asymptotically nonclassicality saving.
A bosonic state is commonly considered nonclassical (or quantum) if its Glauber-Sudarshan
P function is not a classical probability density, which implies that only coherent states and their statistical mixtures are classical. We quantify the nonclassicality of a single qubit, defined by the vacuum and single-photon states, by applying the following four well-known measures of nonclassicality: (1) the nonclassical depth, $\ensuremath{\tau}$, related to the minimal amount of Gaussian noise which changes a nonpositive
P function into a positive one; (2) the nonclassical distance
D, defined as the Bures distance of a given state to the closest classical state, which is the vacuum for the single-qubit Hilbert space; together with (3) the negativity potential (NP), and (4) concurrence potential, which are the nonclassicality measures corresponding to the entanglement measures (i.e., the negativity and concurrence, respectively) for the state generated by mixing a single-qubit state with the vacuum on a balanced beam splitter. We show that complete statistical mixtures of the vacuum and single-photon states are the most nonclassical single-qubit states regarding the distance
D for a fixed value of both the depth $\ensuremath{\tau}$ and NP in the whole range
[0,1] of their values, as well as the NP for a given value of $\ensuremath{\tau}$ such that $\ensuremath{\tau}>0.315\phantom{\rule{0.16em}{0ex}}4$. Conversely, pure states are the most nonclassical single-qubit states with respect to $\ensuremath{\tau}$ for a given
D, NP versus
D, and $\ensuremath{\tau}$ versus NP. We also show the ``relativity'' of these nonclassicality measures by comparing pairs of single-qubit states: if a state is less nonclassical than another state according to some measure then it might be more nonclassical according to another measure. Moreover, we find that the concurrence potential is equal to the nonclassical distance for single-qubit states. This implies an operational interpretation of the nonclassical distance as the potential for the entanglement of formation.
This thesis explores ways in which quantum channels and correlations (of both classical and quantum types) manifest themselves, and also studies the interplay between these two aspects in various physical settings. Quantum channels represent all possible evolutions of states allowed by quantum mechanics while correlations are intrinsic (nonlocal) properties of composite systems. There are four main problems that are addressed (i) the role played by initial system-bath correlations on the subsystem dynamics of an open quantum system, (ii) computation of correlations for two-qubit X-states, (iii) the robustness of non-Gaussian entanglement under symmetric local noisy attenuator and amplifier channels, and finally, (iv) the connection between nonclassicality breaking and entanglement breaking for single-mode bosonic Gaussian channels.