Simultaneous Decoherence and Mode Filtering in Quantum Channels: Theory and Experiment

Gabriele Riccardi, Cristian Antonelli, Daniel E. Jones, and Michael Brodsky

Phys. Rev. Applied 15, 014060 – Published 29 January 2021

Abstract

Quantum networks require robust quantum channels for fast and reliable entanglement distribution over long distances. As quantum communication technology matures, it moves towards utilizing actual fibers and free-space optical channels; hence, there is a growing need for physical models describing decoherence. The primary challenge is to concisely account for numerous elements distributed along a lengthy optical path. We approach this by starting with an analytical model of a channel with just two lumped elements, one representing decoherence and the other representing mode filtering. Interestingly, we find that, while the order and relative orientation of the two elements produce a wealth of different biphoton states, the amount of entanglement in all those states is exactly the same. Then, we conduct experiments that implement this channel and verify our analytical findings. Finally, we expand our analysis to the most general fiber polarization channel, comprising a statistically significant number of arbitrarily oriented elements. We show that, over an ample range of parameters, our two-element analytical model is quite accurate in describing the fiber channel, which makes it an effective tool for gaining insights into channel decoherence.

Received 9 October 2020
Revised 15 December 2020
Accepted 7 January 2021

DOI:https://doi.org/10.1103/PhysRevApplied.15.014060

Physics Subject Headings (PhySH)

Optical quantum information processing Polarization of light Quantum channels Quantum communication Quantum correlations in quantum information Quantum entanglement

Communication networks Optical fibers

Quantum Information, Science & Technology

Authors & Affiliations

Gabriele Riccardi^1,*, Cristian Antonelli^1,†, Daniel E. Jones ^2,‡, and Michael Brodsky ^2,§

¹Department of Physical and Chemical Sciences, University of L’Aquila, 67100 L’Aquila, Italy
²U.S. Army Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland 20783-1197, USA

^*gabriele.riccardi@graduate.univaq.it
^†cristian.antonelli@univaq.it; Also at National Inter-University Consortium for Telecommunications - CNIT, Italy.
^‡daniel.e.jones161.civ@mail.mil
^§michael.brodsky4.civ@mail.mil; Also at U.S. Military Academy, West Point, New York 10996, USA.

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Vol. 15, Iss. 1 — January 2021

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Images

Figure 1
Schematics of the experimental apparatus. Here $τ$ is the decohering element, $γ$ is the filtering element, $γ_{C}$ is the compensating filtering element, EPS is the entangled-photon source, DSF is the dispersion-shifted fiber, PDL is the PDL emulator, PMD is the PMD emulator that applies a differential group delay $τ = 6.6$ ps, PC is the polarization controller, DS is the detector station, PA is the polarization analyzer consisting of several waveplates (red) and a polarizer (blue), and SPD is the single-photon detector. The order of the decohering and filtering elements in channel $A$ can be changed to investigate the decoherence-first and filtering-first cases.
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Figure 2
Theoretical (left) and experimental (right) representations of Eqs. (11a) and (12a) for the case where the eigenvectors of the filtering and decoherence elements are aligned [(a)–(e) and (c)–(g)] and orthogonal [(b)–(f) and (d)–(h)] in Stokes space. The matrices in the top two rows [(a)–(b)–(e)–(f)] refer to the filtering-first scenario and are expressed in the basis $| p_{A} h_{B} ⟩$ , $| p_{A} h_{B}^{'} ⟩$ , $| p_{A}^{'} h_{B} ⟩$ , $| p_{A}^{'} h_{B}^{'} ⟩$ . The matrices in the bottom two rows [(c)–(g)–(d)–(h)] refer to the decoherence-first scenario and are expressed in the basis $| h_{A} p_{B} ⟩$ , $| h_{A} p_{B}^{'} ⟩$ , $| h_{A}^{'} p_{B} ⟩$ , $| h_{A}^{'} p_{B}^{'} ⟩$ . For all matrices, $γ = 0.46$ .
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Figure 3
Concurrence as a function of the amount of filtering $γ$ in the channel. The dashed line represents the theoretical result Eq. (13). All of the markers are experimental points: squares refer to the filtering-first scenario, while circles refer to the decoherence-first scenario. In both cases, the open markers correspond to the case in which $γ$ is aligned with $τ$ , and the filled markers to the case in which they are orthogonal.
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Figure 4
Experimental results of the setup reproducing Eq. (14) for the decoherence-first (green) and filtering-first (cyan) cases when $τ$ and $γ$ are orthogonal in Stokes space. Left panel: experimental density matrices expressed in the basis $| h_{A} h_{B} ⟩$ , $| h_{A} h_{B}^{'} ⟩$ , $| h_{A}^{'} h_{B} ⟩$ , $| h_{A}^{'} h_{B}^{'} ⟩$ for $γ = 0.41$ (a),(d), 0.66 (b),(e), and 0.77 (c),(f). Right panel: coincidence probabilities, measured via the diagonal elements of the density matrix, as a function of $γ$ . Comparison of $ρ_{i, 11}$ , $ρ_{i, 22}$ , $ρ_{i, 33}$ , and $ρ_{i, 44}$ , where $i = 1, 2$ for the filtering-first and decoherence-first cases, respectively. Here $ρ_{i, 11}$ and $ρ_{i, 44}$ are independent of the ordering of the two effects; however, $ρ_{1, 22}$ is equivalent to $ρ_{2, 33}$ , and vice versa.
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Figure 5
Concurrence versus the filtering magnitude $γ$ in the filtering-first scenario. Open and filled markers correspond to the cases in which the vectors $γ$ and $τ$ are aligned and orthogonal, respectively. The upper set of data points refer to the case in which nonlocal compensation of modal filtering is implemented by passing photon $B$ through and additional filtering element. The dashed curve is the analytical result Eq. (13) and the solid curve shows the restored concurrence level, also given by the same equation with $γ = 0$ .
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Figure 6
Concurrence as a function of the angle formed by the real and imaginary parts of the complex-valued vector $\tilde{τ} = \tilde{τ_{R}} + i \tilde{τ_{I}}$ . The larger black dots refer to the decoherence-first configuration, where the vector $\tilde{τ}$ is given by Eqs. (20)–(22), while the smaller dots are obtained by randomly varying the orientation of the real and imaginary parts of the same vector.
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