Experimentally Certified Transmission of a Quantum Message through an Untrusted and Lossy Quantum Channel via Bell's Theorem

TL;DR

A protocol in a device-independent framework, which allows for the certification of practical quantum transmission links in scenarios in which minimal assumptions are made about the functioning of the certification setup, and takes unavoidable transmission losses into account.

Abstract

Quantum transmission links are central elements in essentially all protocols involving the exchange of quantum messages. Emerging progress in quantum technologies involving such links needs to be accompanied by appropriate certification tools. In adversarial scenarios, a certification method can be vulnerable to attacks if too much trust is placed on the underlying system. Here, we propose a protocol in a device independent framework, which allows for the certification of practical quantum transmission links in scenarios where minimal assumptions are made about the functioning of the certification setup. In particular, we take unavoidable transmission losses into account by modeling the link as a completely-positive trace-decreasing map. We also, crucially, remove the assumption of independent and identically distributed samples, which is known to be incompatible with adversarial settings. Finally, in view of the use of the certified transmitted states for follow-up applications, our protocol moves beyond certification of the channel to allow us to estimate the quality of the transmitted quantum message itself. To illustrate the practical relevance and the feasibility of our protocol with currently available technology we provide an experimental implementation based on a state-of-the-art polarization entangled photon pair source in a Sagnac configuration and analyze its robustness for realistic losses and errors.

Figures (14)

• Figure 1: Sketch of the problem. Alice's goal is to send a qubit, potentially part of a larger system, in state $\rho_i$, through an untrusted quantum channel $\mathcal{E}$ (green path). To do so, she sometimes tests the channel by sending half an entangled state (blue path). Alice and Bob can then measure the output state $\Phi_o$, to assess how close the action of the physical channel $\mathcal{E}$ is to an ideal reference channel $\mathcal{E}_0$ on the transmitted state $\rho_i$.
• Figure 2: Protocol sketch in a one-sided device independent scenario: Alice prepares $N$ copies of the probe state $\Phi_i$, and sends them through the untrusted channel $\mathcal{E}$ that varies with time, as well as $\rho_i$ at a random secret position $r$. Some states are lost such that Bob only receives a fraction of them. Alice tells Bob the value of $r$. If $\rho_i$ was lost, then the protocol aborts. Otherwise, Bob stores $\rho_i$ and, together with Alice, tests the violation of the steering inequality with the output probe states. They deduce the average channel's quality over the protocol, which informs on the probability that the message $\rho_i$ was accurately transmitted to Bob, up to isometries.
• Figure 3: Experimental setup for photonic certified quantum communication through an unstrusted channel. Photon pairs are generated via type-II SPDC, in a ppKTP crystal ($30mm$-long, $46.2µm$ poling period), and entangled in polarization in a Sagnac interferometer. The source is pumped with a $770nm$ continuous laser. Signal and idler photons are emitted around $1540nm$, separated from the pump by a dichroic mirror, and from each other by the polarizing beam splitter (PBS) of the interferometer. They are then coupled into single-mode fibers, and sent to the different players. The idler photon is both used as Alice's part of the maximally-entangled pair and to herald the probe state. The signal photon is sent to Bob through the untrusted lossy channel. A variable optical attenuator (VOA) allows to simulate an honest channel with a tunable amount of loss. The biphoton state is measured with polarization analyzers, each made of two waveplates (WPs), a fibered PBS, and $>80\%$-efficiency Superconducting Nanowire Single-Photon Detectors (SNSPDs). The WPs are mounted on motorized stages, allowing to both regularly randomize the measurement basis and implement dishonest channels. Detection events are then sent to a fast coincidence counter which gathers all the data required in order to evaluate the quantum correlations and channel's transmissivity.
• Figure 4: Schematic decomposition of the untrusted channel $\mathcal{E}$, into an equivalent channel $\mathcal{E}'$ that the protocol effectively certifies, and a trusted channel, corresponding to the characterized and homogeneous losses $\lambda_c$ trusted by Alice.
• Figure 5: Minimum fidelity $F(\rho_i,\rho_o)$ certified via our protocol as a function of the measured heralding efficiency, tuned with a VOA, and for different trusted losses $\lambda_c$ (colored curves). The curves are plotted by taking the average fidelity of the probe state to a Bell state $\overline{F_i}$, and the average of the deviation from maximum violation $\epsilon$, over all protocol attempts. Experimental results deviate from these curves, as $F^i$ and $\epsilon$ vary between experiments. Errors induced by the finite statistics are directly subtracted from the certified fidelity, as detailed in Methods (see Eqs. (\ref{['eq:RelativeDifferencePassProbaMain']}) and (\ref{['ineq:FinalResultMain']}) in particular). Error bars include effects induced by the unbalance in detectors' efficiency and the propagation of errors on $F^i$. We also display the fidelity $F(\rho_i,\rho_o)$ measured via quantum state tomography, for $\rho_i = \Phi_i$.

Theorems & Definitions (13)

• Definition 1: Self-testing of a CPTD map
• Lemma 1: Extended Processing Inequality
• Lemma 2: Channel's Metrics Equivalence
• Theorem 1: Kraus' Theorem
• Lemma 3
• Lemma 1: Extended Processing Inequality
• Lemma 2: Channel's Metrics Equivalence
• Corollary 2
• Lemma 4: Bound on the transmissivity
• Lemma 5: Probabilistic Channel Certification