Question

If two quantum states can be Schmidt decomposed using the same sets of joint basis at all times, no matter the evolution they go through, are the quantum states equal?

Answer 1

The fact that two quantum states can be Schmidt decomposed using the same sets of joint basis at all times does not necessarily mean that the quantum states are equal.

Step-by-step explanation:

The fact that two quantum states can be Schmidt decomposed using the same sets of joint basis at all times does not necessarily mean that the quantum states are equal. Quantum states can have the same sets of joint basis but still differ in other properties, such as their wave functions or energy values. The Schmidt decomposition refers to a specific way of representing a quantum state using a set of orthogonal basis states. However, the overall state and its evolution can still be different.

For example, consider two electrons in an atom. Both electrons can have the same sets of quantum numbers, satisfying the Pauli exclusion principle. However, they can still have different energies and occupy different states within the atom.