Distillation of Quantum Steering

R. V. Nery, M. M. Taddei, P. Sahium, S. P. Walborn, L. Aolita, and G. H. Aguilar

Phys. Rev. Lett. 124, 120402 – Published 26 March 2020

Abstract

We show—both theoretically and experimentally—that Einstein-Podolsky-Rosen steering can be distilled. We present a distillation protocol that outputs a perfectly correlated system—the singlet assemblage—in the asymptotic infinite-copy limit, even for inputs that are arbitrarily close to being unsteerable. As figures of merit for the protocol’s performance, we introduce the assemblage fidelity and the singlet-assemblage fraction. These are potentially interesting quantities on their own beyond the current scope. Remarkably, the protocol works well also in the nonasymptotic regime of few copies, in the sense of increasing the singlet-assemblage fraction. We demonstrate the efficacy of the protocol using a hyperentangled photon pair encoding two copies of a two-qubit state. This represents to our knowledge the first observation of deterministic steering concentration. Our findings are not only fundamentally important but may also be useful for semi-device-independent protocols in noisy quantum networks.

Received 12 June 2019
Revised 31 October 2019
Accepted 13 February 2020

DOI:https://doi.org/10.1103/PhysRevLett.124.120402

Physics Subject Headings (PhySH)

Nonlocality Quantum correlations in quantum information Quantum optics Quantum protocols

General PhysicsQuantum Information, Science & Technology

Authors & Affiliations

R. V. Nery^*, M. M. Taddei, P. Sahium, S. P. Walborn, L. Aolita, and G. H. Aguilar

Instituto de Física, Universidade Federal do Rio de Janeiro, P. O. Box 68528, Rio de Janeiro, Rio de Janeiro 21941-972, Brazil

^*rnery@iip.ufrn.br

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Vol. 124, Iss. 12 — 27 March 2020

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Figure 1
(a) Bipartite semi-DI scenario. Alice can only perform uncharacterized measurements on her device, which is then effectively treated as a black box. Bob has full quantum control on his system, allowing complete knowledge of his quantum state. Together with Alice’s probability distribution for her black box, the systems compose the assemblage $Σ_{A | X}$ . (b) Depiction of a generic 1W-LOCC. The assemblage can be manipulated locally by both parties and Bob is allowed to communicate any classical parameter to Alice. Alice’s wirings, represented by gray boxes, allow creation of new random variables from previous ones by an arbitrary distribution. (c) Protocol 1 for two copies of the original assemblage. Bob applies a local filter on one of his qubits. A successful outcome results in a singlet assemblage shared between Alice and Bob, while failure produces an unsteerable assemblage. In both cases, communication of Bob’s result to Alice is used and the appropriate subsystem is then chosen by both parties.
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Figure 2
(a),(b) Production of entangled photons via SPDC. The down-converted light is spectrally filtered to ( $650 \pm 10$ ) nm and collimated by a lens (not shown in figure), which converts the photons’ momentum to spatial modes parallel to the pump beam. Only two pairs of correlated spatial modes are used in the remainder of the setup, as shown in (b), corresponding to two additional qubits besides polarization. (c) Setup for quantum steering distillation. Alice’s interferometer allows for measurements both on polarization and on the spatial DOF and comprises Alice’s two initial black boxes. On Bob’s side, the amplitude filter is implemented by the variable reflectivity mirror (VM); reflectivity is tuned so that spatial DOF amplitudes become equalized when the photon is transmitted. Polarization is ignored and tomography of spatial mode DOF ensues in the upper branch of Bob’s setup. Legend for the components: $Q$ —Quarter-wave plate; $H$ —Half-wave plate; BD—Beam displacer; PBS—Polarizing beam splitter; VM—Variable reflectivity mirror. Subindices indicate to which party ( $A$ —Alice, $B$ —Bob) and to which type of measurement ( $p$ —polarization, $s$ —spatial mode) the component pertains.
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Figure 3
Singlet-assemblage fraction (top) and LHS robustness of steering (bottom) as a function of Bob’s reduced state amplitude imbalance $α^{2} - β^{2}$ . Circles (red) correspond to the original assemblage $Σ_{A | X}^{(α)}$ shared as a base for the copies; crosses (blue) are postselected successfully distilled assemblages, obtained when Bob obtains the outcome $ω = 0$ on his local filter; triangles (orange) correspond to the resulting average assemblages obtained by applying Protocol 1 to two copies of the original assemblage. Successful distillation can be observed as the values of both measures increase after the process, even for imbalances as high as 0.81.
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