Quantum Spread-Spectrum CDMA Communication Systems: Mathematical Foundations

TL;DR

The fundamental principles and mathematical foundations of quantum spread spectrum code division multiple access (QCDMA) communication systems are described and the principles of narrow-band filtering of quantum signals required at the receiver are detailed.

Abstract

This paper describes the fundamental principles and mathematical foundations of quantum spread spectrum code division multiple access (QCDMA) communication systems. The evolution of quantum signals through the direct-sequence spread spectrum multiple access communication system is carefully characterized by a novel approach called the decomposition of creation operators. In this methodology, the creation operator of the transmitted quantum signal is decomposed into the chip-time interval creation operators each of which is defined over the duration of a chip. These chip-time interval creation operators are the invariant building blocks of the spread spectrum quantum communication systems. With the aid of the proposed chip-time decomposition approach, we can find closed-form relations for quantum signals at the receiver of such a quantum communication system. Further, the paper details the principles of narrow-band filtering of quantum signals required at the receiver, a crucial step in designing and analyzing quantum communication systems. We show that by employing coherent states as the transmitted quantum signals, the inter-user interference appears as an additive term in the magnitude of the output coherent (Glauber) state, and the output of the quantum communication system is a pure quantum signal. On the other hand, if the transmitters utilize particle-like quantum signals (Fock states) such as single photon states, entanglement and a spread spectrum version of the Hong-Ou-Mandel effect can arise at the receivers. The important techniques developed in this paper are expected to have far-reaching implications for various applications in the exciting field of quantum communication and signal processing.

Figures (15)

• Figure 1: Effect of quantum encoder on the transmitted quantum communication signal with photon wavepacket ${\color{black}\xi}(t)$ with processing gain (code length) of $N_c= \frac{{\color{black}T_p}}{T_c}$.
• Figure 2: Temporal decomposition of photon-wavepackets for $N_c=2$. A photon-wavepackets, ${\color{black}\xi}(t)$, can be viewed as a combination of chip-time wavepackets, ${\color{black}\xi}_k(t)$.
• Figure 3: Hilbert space view of chip-time interval decomposition. A chip-time Hilbert space is associated with each chip-time interval of the quantum communication signal. The encoded and decoded quantum signals can be represented in the tensor product space of these chip-time Hilbert spaces.
• Figure 4: Effect of spreading operator on the creation operators. The spreading operator converts ${\color{black}\xi}(t)$ to ${\color{black}\xi}^e(t)$. According to Theorem \ref{['thm:spreading']}, this evolution is equivalent to applying the code sequence on the individual chip time interval creation operators.
• Figure 5: A point-to-point quantum spread spectrum communication system.

Theorems & Definitions (46)

• Definition 1
• Proposition 1
• proof
• Theorem 1: Chip-time interval creation operators
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Theorem 2