IEEE Aerospace and Electronic Systems Magazine - November 2020 - 16

Quantum Radar Cost and Practical Issues
and transmitted waveform polarization have not been analyzed thoroughly in papers on QR performance.
Seventh, we should develop a new theory of QR that
works for more than a few photons per mode per quantum
transmit device. Eighth, we should invent new QR theory
that provides more than 6 dB improvement relative to CR
for the same transmit power and bandwidth for low photon
flux per mode. The theory in [16], which predicts a 6 dB
quantum advantage, assumes Gaussian states. Nobody
knows what might happen in the non-Gaussian case.
Nobody knows what the optimal QR design looks like.
Rumor has it that physicists are working intensely on
these two topics at research labs around the world now.
For example, we could use Bayesian quantum mechanics
as explained below. In particular, we could solve the
Belavkin-Zakai equation (and the corresponding PDE
for the density matrix) using the exponential family of
multivariate densities, which is much more general than
Gaussian densities, using the theory developed in [11].
The Belavkin-Zakai equation is the Bayesian generalization of the Schr€odinger equation using macroscopic noisy
measurements [48].
Ninth, all QR so far are based on the usual textbook version of quantum mechanics, which takes the
Schr€odinger equation and the Born rule as given. But there
is another approach to quantum mechanics that derives
the Born rule from Bayes' rule, and which modifies the
Schr€odinger equation to explicitly use Bayes' rule to
model macroscopic measurements of quantum effects.
There are many distinct versions of this alternative quantum mechanics, but the most popular version uses the
Lindblad equation to replace the Schr€odinger equation.
For example, Steve Weinberg derives the Born rule as
an asymptotic solution of the Lindblad equation [10].
Weinberg cannot derive the Born rule without assuming
that the von Neumann entropy is nondecreasing with
time. At first this sounds very reasonable, corresponding
roughly to the second law of thermodynamics, but in fact
this is the exact opposite of what one would expect for a
properly posed estimation problem, in which a measurement results in the increase of information and, hence, the
reduction of entropy. The entropy might increase for the
entire physical system but not for the subsystem that corresponds to the observer. The correct Bayesian version of
the second law of thermodynamics for quantum measurements is given in [51], but it is not used today to design
QR because very few working physicists know about it.
Moreover, no real physical macroscopic measurement
happens instantaneously, and hence, the dynamics of the
time evolution of the density matrix could be exploited to
our advantage in the design of QR as well as quantum
computers and quantum communication systems. For
example, the best measurement protocols might not correspond to Born's rule. It could be that for optimal QR
designs no steady-state solution exists, and hence, the
16

Born rule is simply wrong. This is exactly what happens
in optimal estimation theory (and practice); for example,
the Kalman filter might never reach the steady state; only
the crudest approximations to optimal estimation are computed assuming steady-state solutions. Furthermore, ultrawideband QR cares about extremely short time scales
during which the steady state might never be reached, and
hence, the Born rule is irrelevant. Similar issues arise in
quantum chemistry, where researchers are trying to compute the transient solution of the Schr€
odinger equation on
very short time scales in order to understand real chemical
reactions. Hence, one can imagine fruitful cross fertilization between quantum chemistry and QR, as well as the
possibility of new physics, beyond the Born rule and
beyond the Schr€
odinger equation.
Weinberg's paper [10] is a very skillful solution of the
Lindblad equation using the best physical intuition (to
invent relevant hypotheses) with no mathematical details
suppressed; however, it does not benefit from knowing
about Kalman's beautiful theory for the stability of the
Kalman filter, using observability and controllability as
sufficient conditions for stability of the filter [24]. Moreover, we now know that controllability and observability
are not close to being necessary conditions for stability,
but rather much weaker conditions are sufficient [25]. Furthermore, for nonlinear estimation, we now know that
observability is needed but controllability is not [26]. Controllability is closely related to hypoellipticity for PDEs,
which is a condition to guarantee the existence of a solution to the Fokker-Planck equation (and many other
PDEs), and in the simplest case, it is like assuming that
the process noise covariance matrix is positive definite,
which is obviously not necessary for filter stability. Controllability is also closely related to the rigorous thermodynamic results of Caratheodory, which influenced Kalman
directly. The mathematical conditions for nonlinear filter
stability is an important open problem in mathematics,
and the analysis of the solutions to the Lindblad equation
(and more general PDEs for Bayesian quantum mechanics) would benefit from more research. For example, the
work of Mitter and Newton is a big step in the right direction to understand the information flow in Kalman filters
[27], with another significant advance reported in [28] for
nonlinear problems. Apparently none of this useful estimation theory is known to physicists working on QC or
QCOM or QR. The current state of the art in physics is to
analyze static linear problems in the steady state with no
explicit measurement models. But the real physical problems for QC and QCOM and QR require very different
methods. Nevertheless there are a few papers and books
that describe the correct Bayesian measurement theory for
quantum mechanical systems, including [46]-[50], but
essentially no working physicists actually use this theory
for QR because they do not know about it. Gisin's essay
[49] is an excellent exposition of the Bayesian methods

IEEE A&E SYSTEMS MAGAZINE

NOVEMBER 2020

IEEE Aerospace and Electronic Systems Magazine - November 2020

Table of Contents for the Digital Edition of IEEE Aerospace and Electronic Systems Magazine - November 2020