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arXiv:quant-ph/0412221 · Physical Review A

Entangled states of light and their robustness against photon absorption

S. J. van Enk, Osamu Hirota

Published Jun 17, 2005• 12 citations
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Abstract

We study how photon absorption losses degrade the bipartite entanglement of entangled states of light. We consider two questions: (i) What state contains the smallest average number of photons given a fixed amount of entanglement? (ii) What state is the most robust against photon absorption? We explain why the two-mode squeezed state is the answer to the first question but not quite to the second question.

DOI
10.1103/physreva.71.062322

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•Jan 1, 2015•0 citations
We study how photon absorption losses degrade the bipartite entanglement of entangled states of light. We consider two questions: (i) What state contains the smallest average number of photons given a fixed amount of entanglement? (ii) What state is the most robust against photon absorption? We explain why the two-mode squeezed state is the answer to the first question but not quite to the second question.
For entangled states of light both the amount of entanglement and the sensitivity to noise generally increase with the number of photons in the state. The entanglement-sensitivity tradeoff is investigated for a particular set of states, multidimensional entangled coherent states. Those states possess an arbitrarily large amount of entanglement EEE provided the number of photons is at least of order 22E{2}^{2E}22E. We calculate how fast that entanglement decays due to photon absorption losses and how much entanglement is left. We find that for very small losses the amount of entanglement lost is equal to $2∕\mathrm{ln}(2)\ensuremath{\approx}2.89$ ebits per absorbed photon, irrespective of the amount of pure-state entanglement EEE one started with. In contrast, for larger losses it tends to be the remaining amount of entanglement that is independent of EEE. This may provide a useful strategy for creating states with a fixed amount of entanglement.
We address the problem of the persistence of entanglement of quantum light under mode transformations, where orthogonal modes define the parties between which quantum correlations can occur. Since the representation of a fixed photonic quantum state in different optical mode bases can substantially influence the entanglement properties of the said state, we devise a constructive method to obtain families of states with the genuine feature of remaining entangled for any choice of mode decomposition. In the first step, we focus on two-photon states in a bipartite system and optimize their entanglement properties with respect to unitary mode transformations. Applying a necessary and sufficient entanglement witness criteria, we are then able to prove that the class of constructed states is entangled for arbitrary mode decompositions. Furthermore, we provide optimal bounds to the robustness of the mode-independent entanglement under general imperfections. In the second step, we demonstrate the power of our technique by showing how it can be straightforwardly extended to higher-order photon numbers in multipartite systems, together with providing a generally applicable and rigorous definition of mode-independent separability and inseparability for mixed states.
We discuss how two-photon absorption (TPA) of squeezed and coherent states of light can be detected in measurements of the transmitted light fields. Such measurements typically suffer from competing loss mechanisms such as experimental imperfections (i.e., imperfect photodetectors) and other linear scattering losses inside the sample itself, which can lead to incorrect assessments of the two-photon-absorption cross section. We evaluate the sensitivity with which TPA can be detected and find that at sufficiently large photon numbers TPA sensitivity of squeezed vacua or squeezed coherent states can become independent of linear losses that occur after the TPA event has taken place. In particular, this happens for measurements of the photon number or of the antisqueezed field quadrature, where large fluctuations counteract and exactly cancel the degradation caused by single-photon losses.
We report about the generation of entangled states obtained by applying delocalized single-photon addition between two uncorrelated light modes populated by coherent states. Such entangled states can contain an arbitrary large number of photons while preserving their non-classical correlations. Besides their importance to study quantum effects in large-scale systems, these states are found to be surprisingly robust against losses and thus well suited to travel over long distances and to be stored in atomic ensembles.
Subtracting photons from a two-mode squeezed state is a well-known method to increase entanglement. We analyze different strategies of local photon subtraction from a two-mode squeezed state in terms of entanglement gain and success probability. We develop a general framework that incorporates imperfections and losses in all stages of the process: before, during, and after subtraction. By combining all three effects into a single efficiency parameter, we provide analytical and numerical results for subtraction strategies using photon-number-resolving and threshold detectors. We compare the entanglement gain afforded by symmetric and asymmetric subtraction scenarios across the two modes. For a given amount of loss, we identify an optimized set of parameters, such as initial squeezing and subtraction beam splitter transmissivity, that maximize the entanglement gain rate. We identify regimes for which asymmetric subtraction of different Fock states on the two modes outperforms symmetric strategies. In the lossless limit, subtracting a single photon from one mode always produces the highest entanglement gain rate. In the lossy case, the optimal strategy depends strongly on the losses on each mode individually, such that there is no general optimal strategy. Rather, taking losses on each mode as the only input parameters, we can identify the optimal subtraction strategy and required beam splitter transmissivities and initial squeezing parameter. Finally, we discuss the implications of our results for the distillation of continuous-variable quantum entanglement.
In the first part of this dissertation a framework for categorizing entropic measures of nonclassical correlations in bipartite quantum states is presented. The measures are based on the difference between a quantum entropic quantity and the corresponding classical quantity obtained from measurements on the two systems. Three types of entropic quantities are used, and three different measurement strategies are applied to these quantities. Many of the resulting measures of nonclassical correlations have been proposed previously. Properties of the various measures are explored, and results of evaluating the measures for two-qubit quantum states are presented. To demonstrate how these measures differ from entanglement we move to the set of Bell-diagonal states for two qubits, which can be depicted as a tetrahedron in three dimensions. We consider the level surfaces of entanglement and of the correlation measures from our framework for Bell-diagonal states. This provides a complete picture of the structure of entanglement and discord for this simple case and, in particular, of their nonanalytic behavior under decoherence. The pictorial approach also indicates how to show that all of the proposed correlation measures are neither convex nor concave. In the second part we look at two practical interferometric setups that use nonclassical states of light to enhance their performance. First we consider an interferometer powered by laser light (a coherent state) into one input port and ask the following question: what is the best state to inject into the second input port, given a constraint on the mean number of photons this state can carry, in order to optimize the interferometer's phase sensitivity? This question is the practical question for high-sensitivity interferometry. We answer the question by considering the quantum Cram\'er-Rao bound for such a setup. The answer is squeezed vacuum. Then we analyze the ultimate bounds on the phase sensitivity of an interferometer, given the constraint that the state input to the interferometer's initial 50:50 beam splitter B is a product state of the two input modes. Requiring a product state is a natural restriction: if one were allowed to input an arbitrary, entangled two-mode state ∣Ξ⟩|\Xi \rangle∣Ξ⟩ to the beam splitter, one could generally just as easily input the state B∣Ξ⟩B|\Xi \rangleB∣Ξ⟩ directly into the two modes after the beam splitter, thus rendering the beam splitter unnecessary. We find optimal states for a fixed photon number and for a fixed mean photon number. Our results indicate that entanglement is not a crucial resource for quantum-enhanced interferometry. Initially the analysis for both of these setups is performed for the idealized case of a lossless interferometer. Then the analysis is extended to the more realistic scenario where the interferometer suffers from photon losses.