Question

In Continuous Variable -Quantum Key Distribution (CVQKD), usually Gaussian modulated coherent states are sent. This means both quadratures of a coherent state are chosen from two normal distribution. So do we can prepare a state with well-known quadratures on the phase space? My main question is, how someone can successfully create a coherent state with exactly well-known phase and amplitude quadratures. I think even if experimental devices are perfect, he/she can not make a quantum state with exact known quadratures. I think making quantum coherent states is limited by the Heisenberg uncertainty. Am I right? In other words, do these uncertainties happened not only during measurement but also during state preparation?

Answer 1

You are correct in thinking that the Heisenberg uncertainty principle limits the preparation of quantum states with exactly well-known quadratures. This principle applies not just during measurement, but also during state preparation, meaning that there is always some inherent uncertainty in the amplitude and phase quadratures of a coherent state.

Step-by-step explanation:

In Continuous Variable Quantum Key Distribution (CVQKD), Gaussian-modulated coherent states are utilized because they are well-described by quantum mechanics. The quadratures (phase and amplitude) of these coherent states are indeed drawn from two normal distributions, or Gaussian distributions. However, these states cannot be prepared with precisely known quadratures due to the intrinsic limitations imposed by the Heisenberg uncertainty principle. This principle is not only a constraint during measurement but also during state preparation, signifying that both the particle's position and momentum (which correspond to the quadratures in the context of coherent states) cannot be simultaneously known to arbitrary precision.

Even with ideal experimental devices, there would always be a minimal amount of uncertainty, as a consequence of quantum fluctuations. These fluctuations are fundamental to the quantum description of a state, represented by a wave function. This wave function encodes probabilities rather than precise values, which is essential for the probabilistic nature of quantum mechanics. As such, creating a coherent state with exactly well-known phase and amplitude quadratures is not feasible, aligning with the principles of quantum mechanics illustrated by phenomena like quantum superposition and the probabilistic distribution of a particle's location in space.