Proposal for Implementing Device-Independent Quantum Key Distribution Based on a Heralded Qubit Amplifier

In device-independent quantum key distribution (DIQKD), the violation of a Bell inequality is exploited to establish a shared key that is secure independently of the internal workings of the QKD devices. An experimental implementation of DIQKD, however, is still awaited, since hitherto all optical Bell tests are subject to the detection loophole, making the protocol unsecured. In particular, photon losses in the quantum channel represent a fundamental limitation for DIQKD. Here we introduce a heralded qubit amplifier based on single-photon sources and linear optics that provides a realistic solution to overcome the problem of channel losses in Bell tests.

Figure 1

Principle of DIQKD. Alice and Bob repeatedly choose for their QKD devices inputs $x$ and $y$ (the measurements on entangled particles) and obtain outputs $a$ and $b$ (the measurement outcomes). They then use an authenticated public channel to compare a sample of their data in order to estimate the conditional probability distribution $P(a,b\mid x,y)$. If $P(a,b\mid x,y)$ violates the CHSH inequality by a sufficient amount, then Alice and Bob can use standard error correction and privacy amplification to distill a secret key out of the remaining data. To establish security, nothing has to be known or assumed about Alice’s and Bob’s black boxes, except that they can be described by quantum physics. Note, however, that it is assumed that Alice and Bob are each located in a secure place and control the information going in and out of their locations (dotted lines). In particular, the value of the inputs $x$ and $y$ and of the outputs $a$ and $b$ should not leak out unwillingly of Alice’s and Bob’s secure place. This is the only part of the protocol that cannot be untrusted: Alice and Bob should either enforce these conditions (e.g., by closing a “door”) or test it (e.g., by monitoring the output signals of the boxes).

Figure 2

(a) Heralded amplifier for single photons as proposed in Ref. 16. A beam splitter with transmission coefficient $t$ turns an incoming photon into the entanglement of modes $c$ and out which can be used to teleport an arbitrary state $\alpha |0⟩+\beta {\mathrm{in}}^{†}|0⟩$ with the help of a partial Bell state analyzer. If $t=\frac{1}{2}$, this is standard quantum teleportation; i.e., the outcoming state $\alpha |0⟩\pm \beta {\mathrm{out}}^{†}|0⟩$ is similar to the incoming one, up to a possible unitary transformation depending on which detector clicked. But if $t>\frac{1}{2}$, a successful Bell state measurement projects the outcoming state in the incoming one but shifted towards the single-photon state $|1⟩$: $\sqrt{1-t}\alpha |0⟩\pm \sqrt{t}\beta {\mathrm{in}}^{†}|0⟩$. (b) Setup for amplifying polarization qubits in a heralded way. This scheme is similar to the single-photon amplifier except that a product state of two photons with orthogonal polarization are sent through the partial beam splitter. The probabilistic Bell measurement is based on a 50–50 beam splitter followed by polarization measurements in the $h/v$ basis (which require a polarization beam splitter and two photodetectors).

Figure 3

Proposed setup for the implementation of DIQKD based on a heralded qubit amplifier. The entangled-photon source is located close to Alice’s location. Each of Alice’s and Bob’s black boxes includes a measurement apparatus. Furthermore, Bob’s box contains the qubit amplifier which gives an heralding signal each time an entangled pair has been successfully distributed. Since Bob performs a measurement or, in other words, inputs a $y$, only when he gets the heralding signal, Alice and Bob can safely discard all events where a photon got lost in the quantum channel. Note that the detectors can be either out or in the boxes depending on whether they can be trusted or not. In the figure, they are outside the black boxes.

Figure 4

Key rate vs distance for DIQKD with imperfect devices (log-log scale). (Red) curves labeled (a) correspond to untrusted detectors of efficiency ${\eta }_d=0.95$ (seen as part of the QKD black boxes); (blue) curves labeled (b) correspond to trusted detectors of efficiency ${\eta }_d=0.8$ (moved out of the QKD black boxes). The dotted vertical line represents the maximal distance above which no secret key can be extracted in the absence of an amplification process that counterbalances transmission losses. The two lower curves give the key rate (in $\mathrm{bit}/\min $) as a function of the distance for an amplifier based on heralded single-photon sources; the two upper curves represent the key rate (in $\mathrm{bit}/s$) for an amplifier with on-demand single-photon sources.