# Fundamental Spacetime Representations of Quantum Antenna Systems
## 6 Quantum Antenna Radiation Patterns: Basic Constructions
### The Probability Law of Receiving Radiated Quantum States: Source-Receiver Coupling Gain Estimation  
Consider the configuration shown in Figure 5. The quantum antenna communication system is characterized by the _gain functional_  
\[\mathcal{G}:\mathcal{X}\times\mathcal{X}\to\mathbb{C},\ \ \mathcal{G}= \mathcal{G}[J_{\mathrm{s}},J_{\mathrm{r}}], \tag{74}\]  
which is a bilinear functional in the source and receiver currents \(J_{\mathrm{s}}(x^{\prime})\in\mathcal{X},x^{\prime}\in D_{\mathrm{s}}\), and \(J_{\mathrm{r}}(x)\in\mathcal{X},x\in D_{\mathrm{r}}\), respectively. Here, \(\mathcal{X}\) is the space of real-valued functions on the four-dimensional domains \(D_{\mathrm{s},\mathrm{r}}\subset\mathbb{M}^{4}\). The complex number \(\mathcal{G}\) gives the probability amplitude of information transmission (coupling amplitude) between source and receiver. This gain plays the role of \"transmission coefficient\" or the \"coupling coefficient\" often used in conventional electromagnetic communication systems, e.g., see [33,88]. From the fundamental theory of quantum antennas developed above, we may easily deduce the general expression of this functional by simply treating the receiver as a source with the \"quantum reverse\" process of that of the transmitter, i.e., the dual of the transmitter problem, e.g., by converting kets to bras, taking adjoints, complex conjugates, etc. Since for _real_ sources and receiver currents \(J_{\mathrm{r}s}(x^{\prime})\), the problem is fully reciprocal,24 we can immediately write down the q-antenna system gain expression as follows:  
\[\mathcal{G}=\langle 0|\int_{x\in D_{\mathrm{r}}}\mathrm{d}^{4}x\int_{x^{ \prime}\in D_{\mathrm{s}}}\mathrm{d}^{4}x^{\prime}\,J_{\mathrm{r}}(x) \mathcal{T}\phi(x)\phi^{\dagger}(x^{\prime})J_{\mathrm{s}}(x^{\prime})|0\rangle, \tag{75}\]  
where (43) was used to realize the transmitter, and a similar form was adapted for the receiver. **Example 6 (Construction of practical angular gain pattern in q-antenna systems). ** We will use the general relativistic expression (75) in order to construct an angular expression characterizing quantum radiation for possible use in practical settings. To do that, we first need to break relativistic covariance in order to simplify the problem and facilitate calculations. The most natural choice for a preferred frame is to use a coordinate system at rest with either the lab frame of the transmitter or the receiver. We choose the source  
Figure 5: Angular gain measurement configuration scenario for a generic quantum antenna source system \(J_{\mathrm{s}}(\mathbf{\mathrm{x}},t),\mathbf{\mathrm{x}}\in S_{\mathrm{s}},t\in T_{ \mathrm{s}}\). frame below. Consider a standard test receiver to be used for performing quantum measurement on the radiation emitted by the source \(J_{\rm s}(\mathbf{x}^{\prime},t^{\prime})\), where \(\mathbf{x}^{\prime}\in S_{s}\), and \(S_{s}\) is the spatial support of the source region. The receiver spatial region is \(S_{\rm r}\), and its strength is given by \(J_{\rm r}(\mathbf{x},t)\). Both \(t^{\prime}\) and \(t^{\prime}\) will be measured with a single clock at rest in the source lab frame. Let the test (receiver) current be reduced to a concentrated delta source  
\[J_{\rm r}(x)=J_{\rm r}(\mathbf{x},t)=\delta(\mathbf{x}-\mathbf{x}_{\rm r})J_{ \rm r}(t),\ \ \ \ \forall\,t\in T_{\rm r}. \tag{76}\]  
That is, we measure the response at a sharp spatial location \(\mathbf{x}_{\rm r}\), while we impose the receiver function restriction \(J_{\rm r}(t)\) on the time measurement interval \(T_{\rm r}\).