A simplified BB84 protocol

The 3-state BB84 protocol

Another simplification consists of using the 3-state BB84 protocol which was proposed in 2006 by Fung and Lo [31]. In this protocol, Alice sends two states in theZbasis and only one in theXbasis. Its security has been proven in 2014 by Tamakiet al. [32]. In addition, they have shown that not all four BB84 states are necessary and that equal SKR can be achieved with three or four states.

Using the 3-state BB84 protocol with the encoding of equation (2.4), we obtain the following states:

|^ψi ∈ {|⁰i^,|¹i^,|+i}^{. (2.7)}

This enables a simplification of the preparation setup. Since|−iis not used anymore, phase modulation is not required and the three states can be generated using only an IM.

A BB84 with less states detected by Bob

In the work presented in appendix A.6, we propose further simplifications to the 3-state BB84 which allow to simplify its experimental implementation and in particular Bob’s setup. In the 3-state BB84 protocol (as in the standard BB84 protocol), Bob has the ability to measure the states in the two bases,ZandX, that we denote asdataandmonitoring lines, respectively. The corresponding states detected are |⁰i^,|¹i ^and |+i^,|−i^{. It is} however possible to use a scheme where Bob measures only three states. We consider the case where he has access to |⁰i ^and |¹i ^{in the Z} basis but only to |−i ^{in the X} basis. This requires a new expression for the estimation of the phase error rate. We derive an expression with terms which do not depend on events where Bob detects|+i^. Note, however, that this expression takes into account detections in both the data and the monitoring lines. This has a small drawback, namely that it is necessary to know precisely the probability of choosing each basis as well as the detection efficiency of each detector. It should be noted that the BB84 protocol with less states detected by Bob is not specific to the time-bin encoding.

Implementing theXbasis measurement with the encoding of equation (2.7) requires an unbalanced interferometer (as shown in figure 2.1b). Bob can use different setups which are sketched in figure 2.4. When the full detection scheme with four states is considered, in order to measure |+i^and|−i, it is necessary to monitor both outputs of the interferometer (each output corresponding to one state). This can be achieved either by using a Mach-Zehnder type interferometer which has two outputs (a) or by using a Michelson type interferometer preceded by a circulator (b). The scheme with

2.2. Simplifications to the BB84 protocol

SPDs

(a)

FM FM SPDs

CIR

(b)

FM SPD FM

(c)

Figure 2.4: Possible implementations of Bob’s measurement scheme in the X basis.

Setups (a) and (b) enable the measurement of both|+i^and|−i, while (c) can measure only|−i. (a) with a Mach-Zehnder interferometer, one detector can be placed at each output. (b) in a Michelson interferometer, the second output coincides with the input.

A circulator is therefore required to detect the second state. (c) to detect only one state, it is sufficient to place a detector in one arm of a Michelson interferometer. SPD:

single-photon detector; CIR: circulator; FM: Faraday mirror.

Alice Bob det

(a)

t0 t1 t2 t0 t1 t2 t0 t1 t2

Alice Bob det

(b)

+

t0 t1 t2 t0 t1 t2 t1 t0+t2 t1

t0+t2 t0+t2

Figure 2.5: States prepared by Alice and detected by Bob in the monitoring line. Since the interferometer can delay the pulses by one time-bin, there are three detection times for each qubit. (a)t₀andt₂depend only on the time-bins early and late, respectively.

The interference between the time-bins early and late takes place att₁. Since Bob projects onto the state|−i, when Alice sends the state|+ithe interference is destructive and consequently no detection should occur int₁. (b) When the qubit duration corresponds to the double of the time-bin duration, there is an overlap of the detection timest0and t₂of consecutive qubits.

only three states detected by Bob enables to implement the measurement in theXbasis using a Michelson type interferometer and only one detector (c). It is clear that this latter solution is favourable. First, it allows to use only one detector instead of two (in theXbasis). Moreover, using a Mach-Zehnder requires polarisation management which implies a more complex setup and consequently reduces the phase stability of the interferometer. Finally, option (b) requires to characterise (and trust) the circulator loss as well as the detectors’ efficiencies.

Application to the time-bin encoding

In the case of time-bin encoding, the detection in the monitoring line, after the inter-ferometer, can occur at three different times denoted as t0,t₁,t2(see figure 2.5a). The interference between the time-bins early and late occurs att₁which is therefore an actual

basis, bit state µ1 µ2

Z, 0 |0i Z, 1 |1i X, 0 |+i

Figure 2.6: Encoding of the states sent by Alice in the time-bin 1-decoy simplified BB84 protocol.

measurement in theX basis. By contrast,t0 (t2) depends only on the time-bin early (late). This means that detections at timest₀andt₂can be seen as projections in theZ basis. This feature enables the derivation of an expression for the phase error rate which depends only on detections events in the monitoring line.

In general, if the qubit duration (the inverse of the repetition rate) is long enough compared to the time-bin duration (the time delay between|⁰i^and|¹i), Bob has access to the three detection times after the interferometer². To increase the repetition rate, the most efficient way of implementing the time-bin encoding is when the duration of a qubit is two times the duration of a time-bin (see figure 2.5b). However, when such a scheme is employed, the time-bint₀ of one qubit overlaps with the time-bint₂of the previous qubit. We therefore adapt once again the phase error rate calculation to match this situation. Finally, we show how to combine the simplified time-bin BB84 presented above with the 1-decoy encoding.

Summary: a simplified time-bin BB84 protocol

To sum up, we have developed a simplified BB84 protocol where Alice prepares only three states ({|⁰i^,|¹i^,|+i}) and Bob projects onto only three states ({|⁰i^,|¹i^,|−i}^). Fig-ure 2.6 shows the states prepared by Alice when this protocol is implemented with the time-bin encoding and the 1-decoy state method. This enables an efficient implementa-tion where Alice employs only one IM and Bob has a stable and passive measurement apparatus with two detectors in total. This experimental setup is described in detail in the next section.

2This is the case when the qubit duration is at least three times the time-bin duration.

3 Time-bin QKD - implementation

Time-bin QKD has been used in numerous experimental demonstrations [21, 14]. In particular, most of the experiments realised in our group are based on time-bin encod-ing [33, 34, 35, 13]. This encodencod-ing has the advantage that it requires no stabilisation of the quantum channel. It is therefore also used in commercial systems [36, 37]. In this chapter we describe our main setup, a time-bin QKD platform which has a repetition rate of 2.5 GHz. The goal is to highlight the requirements of each component, describe and motivate our technical choices as well as to provide a characterisation of the system building blocks.

The general overview of the experimental platform is depicted in figure 3.1. We describe first the lower dashed box which contains the state generation (Alice’s optics and electronics), the state transmission (quantum channel) and the state acquisition (Bob’s optics and electronics). Then, we describe the upper dotted box, that consists of the signal generation and the experiment control by the FPGAs and the PCs. The optical setup is sketched in figure 3.2.

3.1 A source of BB84 states clocked at 2.5 GHz