IEEE Aerospace and Electronic Systems Magazine - November 2020 - 45

Lanzagorta and Uhlmann

Figure 3.

Figure 4.

Theoretical upper bound on the detection error probability  for a
quantum radar with M ¼ 109 entangled photon states (Pq - solid
line), a coherent unentangled photon state radar (Pl - dashed line),
and a classical radar described by the Albersheim's formula (Pc dotted line) with respect to SNR in dB.

Upper bound on the detection error probability when the signalto-noise ratio per photon is snr ¼ À90 dB with respect to log 10 M
for entangled (solid line) and unentangled (dashed line) photon
states.

10À7 Pfa 10À3 . Based on these range of values, for
subsequent comparison analysis we will favorably assume
a conservative value of Pfa ¼ 10À7 that is optimistic with
respect to the true performance of a classical radar system
described by the Albersheim's equation.
Figure 3 compares the relative performance of the
three sensors based on the theoretical models that determine the upper-bound detection error probability. For the
classical case, we have extrapolated the line that corresponds to the Albersheim's equation to smaller values of
the detection probability (0:0126 Pd 0:999).
It can be observed that in the low SNR and lowbrightness regime, the quantum sensor significantly outperforms the coherent and the classical sensors, and these
results are consistent with those previously reported in the
literature [34], [49]. This advantage comes from the quantum sensor's ability to extract additional entanglementderived information from the relatively small fraction of
signal photons that are received.
As the SNR becomes very small, the detection error
probability of all three sensors rapidly approaches the same
noninformative limit of 0.5. As the SNR becomes very large,
the performance of the three sensors similarly converge but
to the idealized limit in which the detect versus no-detect status of each observation is always correctly assigned. It has
been argued, however, that in theory the performance of the
quantum sensor should dominate between these two limits
because its signature will be amplified by additional entanglement-derived information [49], [50], but at present there
is no way to determine whether this advantage can actually
be realized in practice, i.e., whether entanglement-derived
information can exceed the information cost incurred by the
practical overhead required to obtain it.

entangled (solid line) and unentangled (dashed line) photon
states when the signal-to-noise ratio per photon is
snr ¼ À90 dB. It can be seen that both sensors converge to
the noninformative limit of  ¼ 0:5 as M becomes very
small while in the case of very large M the advantage of
the quantum sensor is lost because the coherent sensor will
tend toward the same maximum information limit due to
the plentiful availability of unentangled photons. In other
words, there is decreasing benefit gained from additional
entanglement-derived information because there is no
shortage of classical information available to both sensors
from the large number of received photons.
This motivates the determination of the optimal number of signal photons to maximize the potential advantage
achievable by a quantum sensor in the high-noise regime
under consideration. This is of interest because it can be
used to identify a particular context in which a quantum
advantage is most likely to be achieved in practice. As
already mentioned, it is the tradeoff between improved
information exploitation and the practical constraints
incurred by a more complex system that will determine
whether a true quantum advantage can be realized. Let D
be the difference of the detection error upper bounds for
the quantum (entangled) and nonquantum sensors

SIZE OF THE QUANTUM PULSE
Figure 4 shows the upper-bound detection error probability
 as a function of log 10 M (log of the number of signals) for
NOVEMBER 2020

D  Pl À Pq :

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Thus, the goal is to determine the value of M that tends to
maximize D.
Figure 5 shows how D varies with respect to log10 M
(still with an assumed signal-to-noise ratio per photon
snr ¼ À90 dB) and there is a single clear and maximizing
~ can be anavalue. This value, which we will denote as M,
lytically determined from
d
1  ÀMa a ÀMa=4 
¼0
À e
D ¼
ae
dM
2
4

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IEEE Aerospace and Electronic Systems Magazine - November 2020

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