IEEE Aerospace and Electronic Systems Magazine - November 2020 - 67

Bourassa and Wilson
where G is the amplifier power gain and h^ corresponds to
^ ¼ Namp . If both the
the added amplifier noise with hh^y hi
signal and idler are amplified identically, the amplified
field quadratures all have a variance identical to
Var½I^A  ¼ G coshð2rÞ
þ ðG À 1ÞNamp þ 1=2,
whereas
2
the amplified covariance is hI^A I^B i ¼ G sinhð2rÞ=2. In
order to preserve entanglement, from the inequality (26),
it is then required that, for a given amplifier noise figure,
the amplifier gain must be limited to
G <

Namp þ 12

À2r

Namp þ e 2

:

(28)

The gain upper bound is surprisingly low with G ¼ 2 for
quantum limited amplification with Namp ¼ 1=2 and
ideal squeezing r ! 1. Similar calculations for laser
amplification of optical fields were done previously by
Agarwal and Chaturvedi in [4], with the authors arriving
at same conclusions.
Even if amplifying the quantum signal breaks entanglement, we can still ask if there may be a " practical "
advantage based on using the amplified quantum signal.
That is, it may be that amplifying the quantum signal is
just a good way to produce highly correlated noise signals.
To verify this hypothesis, we assume then that we have
some noisy classical source to which we want to compare
such that it has some detected classical power P C ¼
NS þ NC þ 1 and covariance CC ¼ NS , where NC is the
additional noise present for the classical source. Again
using symmetrical amplification of the signal and idler
and assuming that the detected noise power from the
amplified quantum source is identical P AQ ¼ P C , the
covariances in both situations are found to respect
CC ¼

CAQ þ ðG À 1ÞNamp À ðNC þ 1=2Þ:

(29)

From this, we immediately get the result that the classical
correlations exceed the amplified quantum correlations
CAQ when
ðG À 1ÞNamp > NC þ 1=2:

(30)

In other words, once the amplified noise exceeds the noise
of the classical source, there is no practical advantage.
The severe limits to amplification obtained here may be
caused by the amplified noise being added symmetrically on
both the signal and idler. Indeed, in [4], the authors extended
their calculations to asymmetric laser amplification and their
results indicate that it may provide a pathway to entanglement-preserving amplification.

ENTANGLEMENT MULTIPLEXING
A straightforward engineering solution to higher entanglement transmission powers is to use multiplexing. Both
code-division and frequency-division multiplexing are
NOVEMBER 2020

natural to noise-based QI protocols. As we will describe
in more detail briefly, this multiplexing can be done by
increasing the bandwidth of the JPA source, or by using
multiple independent amplifiers.
In code-division multiplexing, multiple signals at the
same frequency are encoded using multiple orthogonal
code words such that the covariance of the signals is zero
even when they occur in the same time and frequency
windows. Since the output noise from independent JPAs
is uncorrelated, we can consider the amplifiers to naturally produce signals that exhibit code-division multiplexing. Essentially, the output of the other sources
would just look like a small increase of the atmospheric
background noise. This should allow the output of multiple, identical JPAs to simply be combined to produce a
higher signal power.
In frequency-division multiplexing, different real or
effective sources would be designed to work in different
frequency bands. For a realistic quantum two-mode
squeezing source with an emission bandwidth of W
matched to the detection bandwidth, heterodyning with an
integration time T¼ 1=B provides a frequency resolution
of B and the ability to distinguish a number M ¼
W =B ) 1 of independent and identically distributed
entangled pairs of signal-idler fields at different frequencies. Consequently, the integrated noise power P ¼
M Â P and covariance C ¼ M Â C grow proportionally
with the number of detectable modes
pﬃﬃﬃﬃﬃ M, whereas their
measured precision increases as M for large enough M
as imposed by the Central Limit theorem.
Generally, JPAs include a narrow-band ($ 10 MHz)
resonant mode such that they can, for a brightness of
NS ¼ 1, generate M $ 106 independent entangled pairs
every second. Placing several independently controlled
JPAs operating at different frequencies on the same
chip within the same dilution refrigerator unit would
increase the number of modes while keeping the integration time constant. Another route would be to use a
more broadband entanglement source such as an open
transmission-line terminated by a SQUID, where twomode squeezing with a bandwidth of several GHz has
been demonstrated [28]. One could also use a superconducting device known as a Josephson traveling-wave
parametric amplifier, an engineered nonlinear transmission-line made from an array of thousands of Josephson
junctions [21], with a predicted two-mode squeezing
bandwidth of a few GHz and requiring only a single
control pulse [15].
As we can see, there are a number of prospects for
overcoming the limitations in amplification gain and
low source brightness. In fact, all of the multiplexing
methods just described can be combined in a noise
radar setup, which is not necessarily the case for a protocol employing joint measurement. In the near future,
it may then very well be possible to realize the

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