Luong et al.
Figure 2.
Figure 1. Solid orange curve is a sinusoidal waveform as a function of time (or, equivalently, of phase). If classical physics were true, a measurement of the waveform under theoretically ideal conditions would give the solid curve exactly. Quantum physics predicts that there will always be noise in the measurement result. The blue dots depict what the measurement result would look like according to quantum theory, again under ideal conditions. Units are arbitrary.
suggests that one regime in which the quantum radar can have an immense impact is in the low-power regime—a suggestion supported by experiment, as we will see later. The key takeaway from all of this is that we must regard I and Q voltages as random variables with nonzero variances. (As a matter of fact, this follows from the Heisenberg uncertainty principle, which sets a lower bound on the product of their variances.) The clean and noiseless waveforms of classical physics specify merely the average behavior of an electromagnetic signal; they do not quite tell the whole story, especially at low signal powers.
ENTANGLEMENT AS CORRELATED QUANTUM NOISE Suppose now that we have two electromagnetic signals. As explained above, each one must have a certain amount of quantum noise. If the quantum noise in each signal is independent of each other, it follows that the two signals can never be 100% correlated. This seems to put a hard limit on how strongly correlated the two signals can be, but that is not so. Quantum physics dictates that the two signals must be noisy; it does not say that the noise in each signal must be independent. Therefore, the “hard limit” does not exist. There does exist a level of correlation, however, beyond which the quantum noise in the two signals cannot remain independent. If the correlation is higher than this level, the two signals are entangled. This is depicted in Figure 2. Note that entanglement is not merely a statement that the quantum noise is correlated. Two signals whose quantum noise components are correlated, but exhibit low levels of “classical” correlation, are nevertheless not entangled. Only when the overall correlation is so high that the quantum noise is forced to be correlated can we say that the signals are entangled. In summary, then, if two electromagnetic signals are so highly correlated that their quantum noise cannot be independent, the two signals are entangled. Although this description is very far from a comprehensive description of entanglement, it is all we need. APRIL 2020
This number line depicts the Pearson correlation coefficient r between two electromagnetic signals. Because of the presence of quantum noise, the coefficient cannot go to 1 unless the quantum noise in the two signals are correlated. There is a certain “maximum classical correlation” beyond which the quantum noise must be correlated. In this regime, the two signals are entangled.
Note that the point labeled “Max classical correlation” in Figure 2 is a hard value: quantum states are either entangled or not entangled. The location of this point depends on the frequency and power of the two correlated signals. The quantum literature does not give results in terms of the correlation coefficient; however, we have done so here because it is easier to understand. For more on the topic of entanglement testing and entanglement in general, the reader is referred to the large body of scientific literature on the topic, e.g., [10] and [11]. The discussion so far has focused on bipartite entanglement, in which two random variables are highly correlated. The idea can be extended to multipartite entanglement, in which greater numbers of random variables are correlated. Multipartite entanglement finds the use in quantum computers, for example, but is not necessary for the quantum radar. As we will see, only bipartite entanglement is required for the latter.
CHALLENGES IN GENERATING ENTANGLED SIGNALS The generation of entangled signals is a very deep topic, and it would be outside the scope of this article to enter into an extensive discussion of the details. Here, we cannot do more than touch on a few basic facts and show that entanglement generation is a nontrivial task. For further details, the reader may consult a book on quantum optics, such as that by Scully and Zubairy [12]. Naively, we might think that generating a pair of highly correlated signals should be straightforward. For example, we could use two arbitrary waveform generators to generate exactly the same signal, which would then be 100% correlated. This cannot work because arbitrary waveform generators cannot control the quantum noise in the signals. As a consequence, the generators cannot cause the quantum noise in the two signals to be correlated. Therefore, by definition, the two signals are not entangled. Another idea is to generate one signal, then use a beamsplitter to split it in two. Perhaps in this way we could have 100% correlation? Unfortunately, beamsplitters cannot create entanglement out of classical signals such as sinusoidal waves. The reason is somewhat subtle. It turns out that, according to quantum optics, every beamsplitter must have
IEEE A&E SYSTEMS MAGAZINE
25
Authorized licensed use limited to: Khwaja Fareed University of Eng & IT. Downloaded on July 04,2020 at 01:05:04 UTC from IEEE Xplore. Restrictions apply.
