In quantum mechanics, a quantity is said to be "invariant under unitary transformations in phase-space" if its value remains unchanged even if you apply a specific type of transformation to the system's state. Here's a breakdown of the key terms:

• Unitary transformation: This refers to a specific mathematical operation applied to a quantum state (represented by a wavefunction) that preserves the overall probability of finding the system in any particular state. It essentially rotates or stretches the wavefunction in a way that doesn't change the underlying probabilities.
• Phase-space: In quantum mechanics, phase-space is a mathematical construct that combines position (q) and momentum (p) of a quantum particle. It's a way to visualize the possible states the particle can be in.

So, if a quantity is invariant under unitary transformations in phase-space, it means that regardless of how you rotate or stretch the system's wavefunction in phase-space (as long as you maintain the probabilities), the value of that quantity will stay the same.

Here are some examples of quantities that might be invariant under unitary transformations in phase-space:

• The Heisenberg uncertainty principle: This principle states a fundamental limit on how precisely you can know both the position and momentum of a particle simultaneously. It relates the product of the uncertainties in position and momentum (Δq * Δp) to a constant value (h-bar). This product remains constant even under unitary transformations.
• The total energy of a closed system: In a closed system (one that doesn't exchange energy with the surroundings), the total energy remains constant regardless of the transformations applied to the system's state.

Understanding invariance under unitary transformations is important because it helps us identify properties of a quantum system that are independent of the specific basis we choose to represent its state. It reflects fundamental aspects of the system that are independent of how we choose to describe it mathematically.

IYH Dear Jian Zhou

W/o using too much jargon: In quantum key distribution, we want to make sure that the method we use to share secure keys remains secure even when certain transformations happen. This property is called "invariant under unitary transformations in phase-space."

Think of phase-space as a special way of representing quantum systems using both position and momentum information. Unitary transformations involve changing the states and operations we use in a particular way. To satisfy unitary operation invariance, we need to design the quantum key distribution protocol in such a way that it remains secure even if we apply these transformations to the states and operations. It's like making sure that, even if we change how we encode and measure quantum information, our method of keeping the shared key secret still works reliably.