Statistical mixtures of states can be more quantum than their superpositions: Comparison of nonclassicality measures for single-qubit states

Abstract

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, τ , 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 τ and NP in the whole range [0, 1] of their values, as well as the NP for a given value of τ such that τ > 0.3154. Conversely, pure states are the most nonclassical single-qubit states with respect to τ for a given D, NP versus D, and τ 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.

Maximum violations of the quantum-witness equality

“…This connection should also be useful in calculating the maximum violation possible for a given quantum evolution, as well as determining the measurements required to obtain this value. This result also ties the experimentally-testable quantum witness to more formally-defined measures of "quantumness" [13,21]. Indeed, the trace norm itself has recently been considered as a measure of quantumness [12,13], were the density matrix under scrutiny is compared with a particular set of classical states.…”

“…This result also ties the experimentally-testable quantum witness to more formally-defined measures of "quantumness" [13,21]. Indeed, the trace norm itself has recently been considered as a measure of quantumness [12,13], were the density matrix under scrutiny is compared with a particular set of classical states. The quantum witness studied here can thus be seen as an indicator of this class in which the comparator state is the very specific state that results from the witness' blind measurement.…”

“…In deriving Eq. (4), we shall show how the quantum witness is related to the trace distance [11], a measure of distance between two density matrices, recently studied itself as a measure of "quantumness" [12,13]. Equation (4) also opens up the possibility of using the quantum witness as a dimension witness [14,15] for quantummechanical systems.…”

Interplay between nonclassicality and PT symmetry in an effective two-level system with open system effects

“…Specifically, at the output of the beam-splitter a two-mode entangled state is obtained if and only if the state (other than the vacuum state) fed into the input is a single-mode nonclassical state. Thus, the nonclassicality of the single mode input state gets transferred to a two mode entangled state, and one can try to measure the nonclassicality of the input state by measuring the entanglement of the output state [32,34]. For a particular beam-splitter setting, the behavior of the nonclassical properties of the output state is entirely controlled by the input state parameters.…”

“…The nonclassical potential for the single mode input state ρ is defined to be the amount of nonclassicality of the output state ρ out . Consequently, the concurrence potential (CP) and negativity potential (NP) are defined as [32] CP (ρ) = C(ρ out ) N P (ρ) = N (ρ out ).…”