Final answer:
To determine if a quantum channel E that transforms a state ρi into ρ′i exists, we must ensure that the wave function is continuous, smooth, and normalizable, all of which are conditions derived from the physical interpretation of quantum mechanics.
Step-by-step explanation:
To determine if there exists a Completely Positive Trace-Preserving (CPTP) map E such that E(ρi) ⇒ ρ′i for i within a set S, we must first consider certain conditions that are both necessary and sufficient for the existence of such a quantum channel. When we discuss the transformation of states via such a CPTP map, there are physical and mathematical constraints that need to be satisfied.
Necessary and Sufficient Conditions
- The wave function must avoid sudden jumps or discontinuities. This is essential to ensure physically meaningful evolution as per the first condition.
- According to the second condition, the wave function must be smooth, which means it must be differentiable, except in special cases where potential spikes are considered.
- The third condition pertains to the normalization of the wave function, which is rooted in Born's interpretation of quantum mechanics, ensuring that the probability interpretation remains valid.
These conditions hint at the broader principles that guide the construction of a valid quantum channel. Specifically, for the case of transforming two quantum states as per the problem statement, an added dimension that must be considered is the indistinguishability of particles, as well as exchange symmetry. The latter may manifest as either symmetric or antisymmetric behavior with respect to the wave function of the system. Finally, when considering transformations of quantum states, it must also hold true that the laws of physics are consistent across different frames of reference as stipulated by Einstein's relativity.