Orthogonality and non-sensical mathematics

John Harding, Remi Salinas Schmeis

Leo Radzihovsky, Stephen J. Nettel

In 1946 Cameron and Martin concluded a paper [‘‘The orthogonal development of non-linear functionals in series of Fourier–Hermite functionals,’’ Ann. Math. 48, 385–392 (1947)] by saying: ‘‘This paper opens up the whole set of techniques of Orthogonal Developments...for use...in the study of non-linear functionals... .’’ Here, the Cameron–Martin–Wiener expansion is developed with mathematics familiar to physicists, and its use is illustrated and applied to a quantum theory of electron transport.

Yebo Gu, Tao Shen, Jian Song +1 more

Contemporary research has highlighted that the design possibilities for Orthogonal Artificial Noise (AN) are circumscribed by the inherent limitations in channel degrees of freedom, yielding a scant assortment of design alternatives. In this letter, Non-Orthogonal Artificial Noise (NORAN) emerges as a viable substitute. Its applications have been recognized in relay communication systems such as those involving Unmanned Aerial Vehicles (UAVs) and Intelligent Reflecting Surfaces (IRS). Despite this, the mathematical exploration of the Secrecy Capacity (SC) function within wireless communication systems incorporating NORAN has been scant. This letter pioneers the examination of the concave-convex nature of the SC function in such systems. The findings reveal that, subject to certain transmit power conditions, the SC function for NORAN is inherently convex, implying a decrement in SC upon NORAN’s deployment, which negates its intended effect. In response to this conundrum, this discourse introduces Pilot-based Codebook Non-Orthogonal Artificial Noise (PCAN), wherein the codebook capitalizes on pilot signals exchanged between the receiver and sender as an encryption key. The receiver, with the aid of the pilot signals, retrieves the forthcoming PCAN from the codebook, thereby neutralizing the PCAN’s influence during the reception of information. This innovation equates the SC functions of PCAN and AN within the same channel framework. Additionally, this study delves into the optimization conundrum of PCAN, considering scenarios of both perfect and imperfect Channel State Information (CSI). Through theoretical analysis, it is established that with an ideal CSI and an signal-to-noise ratio (SNR) surpassing zero, the quintessential power allocation for PCAN equates to half of the total transmission power. Contrariwise, under imperfect CSI parameters, PCAN fails to deliver efficacy.

Thomas Durt

Antonio Acín

Dallin Durfee, James L. Archibald

Using concepts of geometric orthogonality and linear independence, we logically deduce the form of the Pauli spin matrices and the relationships between the three spatially orthogonal basis sets of the spin-1/2 system. Rather than a mathematically rigorous derivation, the relationships are found by forcing expectation values of the different basis states to have the properties we expect of a classical, geometric coordinate system. The process highlights the correspondence of quantum angular momentum with classical notions of geometric orthogonality, even for the inherently non-classical spin-1/2 system. In the process, differences in and connections between geometrical space and Hilbert space are illustrated.