84
6 Beyond Point-to-Point Quantum Key Distribution
[1, 2] on the secret key rate of any point-to-point QKD protocol, which only depend
on the channel transmittance η. In particular, the secret key rate of any QKD protocol
performed over a lossy channel of transmittance η is upper bounded by the PirandolaLaurenza-Ottaviani-Banchi (PLOB) bound [2]:
r PLOB = − log(1 − η),
(6.1)
where the logarithm is intended in base 2, as usual. In the high-loss regime (η 1),
we can expand the logarithm in (6.1):
r PLOB ≈ 1.44 η,
(6.2)
and observe that the key rate cannot scale better than linearly with the transmittance
of the channel, thus decreasing exponentially with the channel length.
The only way to overcome such severe limitations on the achievable key rate is
to employ one or more intermediate nodes in the quantum channel connecting the
users. However, this fact alone is not sufficient in general to yield key rates with an
improved scaling compared to the PLOB bound.
Consider for instance the MDI-QKD protocol presented in the Chap. 5. Despite
featuring an intermediate measuring station that splits the channel between Alice
and Bob of transmittance η in two channels of transmittance
√ η each,
1 the key rate
does not scale better than the PLOB bound. Indeed, in order to have a successful
detection, both photons sent by Alice and Bob need to arrive at the central relay,
which occurs with probability
√
η ·
√
η = η. Thus, the gain and hence the key rate
cannot scale better than linearly with the transmittance η of the whole channel.
A possible solution is instead represented by quantum repeaters [3, 4], which
guarantee a polynomial scaling of the communication efficiency with the distance.
However, such devices are still very challenging to implement as they require either
quantum memories [4–6] or quantum error correction [7, 8].
Other viable options are evolutions of the original MDI-QKD scheme, like
memory-assisted MDI-QKD featuring quantum memories [9, 10] or adaptive MDIQKD with quantum non-demolition measurements [11]. In both cases, the protocol
adapts to the photon losses ensuring that the Bell-state measurement is performed
between pulses from Alice and Bob that actually arrived, even combining pulses
sent in different rounds. In this way, only one photon per round is required to arrive,
thus yielding a key rate proportional to
√
η. Despite the square-root improvement in
the key rate scaling, the evolved MDI schemes still rely on two-photon interference
events and their implementation is far from being practical.
In the next Section we introduce a novel QKD protocol which represents the
simplest solution, found so far, to improve the key rate scaling of QKD beyond the
PLOB bound thus reaching further distances.
1 Each channel has length L/2, if L is the total channel length between Alice and Bob. The transmittance of each channel is thus e −γ L/(2·10) =
√ η.
6 Beyond Point-to-Point Quantum Key Distribution
[1, 2] on the secret key rate of any point-to-point QKD protocol, which only depend
on the channel transmittance η. In particular, the secret key rate of any QKD protocol
performed over a lossy channel of transmittance η is upper bounded by the PirandolaLaurenza-Ottaviani-Banchi (PLOB) bound [2]:
r PLOB = − log(1 − η),
(6.1)
where the logarithm is intended in base 2, as usual. In the high-loss regime (η 1),
we can expand the logarithm in (6.1):
r PLOB ≈ 1.44 η,
(6.2)
and observe that the key rate cannot scale better than linearly with the transmittance
of the channel, thus decreasing exponentially with the channel length.
The only way to overcome such severe limitations on the achievable key rate is
to employ one or more intermediate nodes in the quantum channel connecting the
users. However, this fact alone is not sufficient in general to yield key rates with an
improved scaling compared to the PLOB bound.
Consider for instance the MDI-QKD protocol presented in the Chap. 5. Despite
featuring an intermediate measuring station that splits the channel between Alice
and Bob of transmittance η in two channels of transmittance
√ η each,
1 the key rate
does not scale better than the PLOB bound. Indeed, in order to have a successful
detection, both photons sent by Alice and Bob need to arrive at the central relay,
which occurs with probability
√
η ·
√
η = η. Thus, the gain and hence the key rate
cannot scale better than linearly with the transmittance η of the whole channel.
A possible solution is instead represented by quantum repeaters [3, 4], which
guarantee a polynomial scaling of the communication efficiency with the distance.
However, such devices are still very challenging to implement as they require either
quantum memories [4–6] or quantum error correction [7, 8].
Other viable options are evolutions of the original MDI-QKD scheme, like
memory-assisted MDI-QKD featuring quantum memories [9, 10] or adaptive MDIQKD with quantum non-demolition measurements [11]. In both cases, the protocol
adapts to the photon losses ensuring that the Bell-state measurement is performed
between pulses from Alice and Bob that actually arrived, even combining pulses
sent in different rounds. In this way, only one photon per round is required to arrive,
thus yielding a key rate proportional to
√
η. Despite the square-root improvement in
the key rate scaling, the evolved MDI schemes still rely on two-photon interference
events and their implementation is far from being practical.
In the next Section we introduce a novel QKD protocol which represents the
simplest solution, found so far, to improve the key rate scaling of QKD beyond the
PLOB bound thus reaching further distances.
1 Each channel has length L/2, if L is the total channel length between Alice and Bob. The transmittance of each channel is thus e −γ L/(2·10) =
√ η.