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arXiv:1507.08587 · Physical Review A

Increasing relative nonclassicality quantified by standard entanglement potentials by dissipation and unbalanced beam splitting

Adam Miranowicz, Karol Bartkiewicz, Neill Lambert, Yueh-Nan Chen, Franco Nori

Published Dec 8, 2015• 40 citations
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Abstract

If a single-mode nonclassical light is combined with the vacuum on a beam splitter, then the output state is entangled. As proposed in [Phys. Rev. Lett. 94, 173602 (2005)], by measuring this output-state entanglement for a balanced lossless beam splitter, one can quantify the input-state nonclassicality. These measures of nonclassicality (referred to as entanglement potentials) can be based, in principle, on various entanglement measures, leading to the negativity (NP) and concurrence (CP) potentials, and the potential for the relative entropy of entanglement (REEP). We search for the maximal relative nonclassicality, which can be achieved by comparing two entanglement measures for (i) arbitrary two-qubit states and (ii) those which can be generated from a photon-number qubit via a balanced lossless beam splitter, where the qubit basis states are the vacuum and single-photon states. Surprisingly, we find that the maximal relative nonclassicality, measured by the REEP for a given value of the NP, can be increased (if $\mathrm{NP}<0.527$) by using either a tunable beam splitter or by amplitude damping of the output state of the balanced beam splitter. We also show that the maximal relative nonclassicality, measured by the NP for a given value of the REEP, can be increased by phase damping (dephasing). Note that the entanglement itself is not increased by these losses (since they act locally), but the possible ratios of different measures are affected. Moreover, we show that partially dephased states can be more nonclassical than both pure states and completely dephased states, by comparing the NP for a given value of the REEP. Thus, one can conclude that not all standard entanglement measures can be used as entanglement potentials. Alternatively, one can infer that a single balanced lossless beam splitter is not always transferring the whole nonclassicality of its input state into the entanglement of its output modes. The application of a lossy beam splitter can solve this problem, at least for the cases analyzed in this paper.

DOI
10.1103/physreva.92.062314

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If a single-mode nonclassical light is combined with the vacuum on a beam splitter, then the output state is entangled. As proposed in [Phys. Rev. Lett. 94, 173602 (2005)], by measuring this output-state entanglement for a balanced lossless beam splitter, one can quantify the input-state nonclassicality. These measures of nonclassicality (referred to as entanglement potentials) can be based, in principle, on various entanglement measures, leading to the negativity (NP) and concurrence (CP) potentials, and the potential for the relative entropy of entanglement (REEP). We search for the maximal relative nonclassicality, which can be achieved by comparing two entanglement measures for (i) arbitrary two-qubit states and (ii) those which can be generated from a photon-number qubit via a balanced lossless beam splitter, where the qubit basis states are the vacuum and single-photon states. Surprisingly, we find that the maximal relative nonclassicality, measured by the REEP for a given value of the NP, can be increased (if NP<0.527\mathrm{NP}<0.527NP<0.527) by using either a tunable beam splitter or by amplitude damping of the output state of the balanced beam splitter. We also show that the maximal relative nonclassicality, measured by the NP for a given value of the REEP, can be increased by phase damping (dephasing). Note that the entanglement itself is not increased by these losses (since they act locally), but the possible ratios of different measures are affected. Moreover, we show that partially dephased states can be more nonclassical than both pure states and completely dephased states, by comparing the NP for a given value of the REEP. Thus, one can conclude that not all standard entanglement measures can be used as entanglement potentials. Alternatively, one can infer that a single balanced lossless beam splitter is not always transferring the whole nonclassicality of its input state into the entanglement of its output modes. The application of a lossy beam splitter can solve this problem, at least for the cases analyzed in this paper.
If a single-mode nonclassical light is combined with the vacuum on a beam splitter, then the output state is entangled. As proposed in [Phys. Rev. Lett. 94, 173602 (2005)], by measuring the output-state entanglement for a balanced lossless beam splitter, one can quantify the input-state nonclassicality. These measures of nonclassicality (referred to as entanglement potentials) can be based, in principle, on various entanglement measures, leading to the negativity (NP) and concurrence (CP) potentials, and the potential for the relative entropy of entanglement (REEP). We search for the maximal nonclassicality, which can be achieved by comparing two entanglement measures for arbitrary two-qubit states and those which can be generated from a photon-number qubit via a balanced lossless beam-splitter, where the qubit basis states are the vacuum and single-photon states. Surprisingly, we find that the maximal nonclassicality, measured by the REEP for a given value of the NP, can be increased (if NP<0.527) by using either a tunable beam splitter or by amplitude damping of the output state of the balanced beam splitter. We also show that the maximal nonclassicality, measured by the NP for a given value of the REEP, can be increased by dephasing. The entanglement itself is not increased by these local losses, but the possible ratios of different measures are affected. Moreover, we show that partially-mixed states can be more nonclassical than both pure states and completely-mixed states, by comparing the NP for a given value of the REEP. This implies that not all entanglement measures can be used as entanglement potentials. Alternatively, a single balanced lossless beam splitter is not always transferring the whole nonclassicality of its input state into the entanglement of its output modes. Applying a lossy beam splitter can solve this problem at least for the cases analyzed in this paper.
Entanglement potentials (EPs) enable the characterization and quantification of the nonclassicality of single-mode optical fields by measuring the entanglement generated through beam splitting. We experimentally generated single-photon states and tomographically reconstructed the corresponding two-qubit states to determine EPs defined via popular two-qubit measures of entanglement. These include the potentials for the relative entropy of entanglement (REEP), concurrence, and negativity. Among our experimental states, we found those that are very close (at least for some ranges of parameters) to the theoretical upper and lower bounds on relative EPs (or relative nonclassicality), i.e., when one EP is maximized or minimized for a given value of another EP. We experimentally confirmed the counterintuitive theoretical result of Miranowicz et al. [Phys. Rev. A 92, 062314 (2015)] that the relative nonclassicality (specifically, the negativity potential for given values of the REEP) of single-photon states can be increased by dissipation.
Entanglement potentials are popular measures of the nonclassicality of single-mode optical fields. These potentials are defined by the amount of entanglement (measured by, e.g., the negativity or concurrence) of the two-mode field generated by mixing a given single-mode field with the vacuum on a balanced beam splitter. We generalize this concept to define the potentials for Bell nonlocality and quantum steering in specific measurement scenarios, in order to quantify single-mode nonclassicality in a more refined way. Thus, we can study the hierarchy of three types of potentials in close analogy to the well-known hierarchy of the corresponding two-mode quantum correlations. For clarity of our presentation, we focus on the analysis of the nonclassicality potentials for arbitrary vacuum-one-photon superpositions (VOPSs), corresponding to a photon-number qubit. We discuss experimentally feasible implementations for the generation of single-mode VOPS states, their mixing with the vacuum on a balanced beam splitter, and their two-mode Wigner-function reconstruction using homodyne tomography to determine the potentials. We analyze the effects of imperfections, including phase damping and unbalanced beam splitting on the quality of the reconstructed two-mode states and nonclassicality potentials. Although we focus on the analysis of VOPS states, single-mode potentials can also be applied to study the nonclassicality of qudits or continuous-variable systems.
We study the relation between single-mode nonclassicality and two-mode entanglement in a beam splitter. We show that single-mode nonclassicality (the entanglement potential) of incident light cannot be transformed into two-mode entanglement completely after a single beam splitter. Some of the entanglement potential remains as single-mode nonclassicality in the two entangled output modes. Two-mode entanglement generated in the process can be equivalently quantified as an increase in the minimum uncertainty widths (or decrease in the squeezing) of the output states compared to the input states. We use the nonclassical depth and logarithmic negativity as single-mode nonclassicality and entanglement measures, respectively. We realize that a conservation relation between the two quantities can be adopted for Gaussian states, if one works in terms of uncertainty width. This conservation relation is extended to many sets of beam splitters.
A bosonic state is commonly considered nonclassical (or quantum) if its Glauber-Sudarshan PPP 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 PPP function into a positive one; (2) the nonclassical distance DDD, 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 DDD for a fixed value of both the depth $\ensuremath{\tau}$ and NP in the whole range [0,1][0,1][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 DDD, NP versus DDD, 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.
Abstract Nonclassicality of a single-mode state can be quantified by the entanglement (potential) the state creates at the output of a beam splitter. Thus, beam splitter transformation can be utilized for obtaining single-mode nonclassicality conditions (quantifications) from two-mode entanglement criteria (measures). Here, (i) we employ this relation to quantify the single-mode nonclassicality (SMNc) of a nonlinear crystal field generating second harmonic signals. (ii) We also manage to witness the SMNc of superradiant emission, a non-Gaussian state, in a Dicke phase transition. Entanglement potential, at the beam splitter output, is obtained in terms of measurable single-mode quantities. Thus, presence of a beam splitter in an experiment is not necessary. Employing the beam splitter method, one can also utilize hybrid entanglement criteria as SMNc conditions working well in different classes of single-mode states. (iii) We also work on essentials of the entanglement-SMNc relation: one obtains the generalized squeezing condition ⟨ â † â ⟩ < |⟨ â 2 ⟩| as the SMNc condition when an optimized form of Duan–Giedke–Cirac–Zoller criterion [PRL 84, 2722] is used at the output of the beam splitter.