Discussion Overview
The discussion revolves around the problem of identifying a unitary operator U given pairs of quantum states \left|\Psi\right> and U\left|\Psi\right>. Participants explore the minimum number of such pairs required to ascertain U with a certain level of confidence, and whether this number is less than what would be needed to determine the states independently.
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- One participant questions whether it is possible to identify U without fully determining the states \left|\Psi\right> and U\left|\Psi\right> independently.
- Another participant expresses skepticism about the sufficiency of information available to determine a general U from the proposed pairs.
- A later reply asserts that knowing \left|\Psi\right> and U\left|\Psi\right> only reveals one column of the matrix representing U.
- One participant inquires if the situation changes when \left|\Psi\right> is a tensor state of n qubits.
- Another participant suggests that if the tensor product consists of different qubit states and U is a tensor product of identical operators on each qubit, then it may be possible to learn everything about the individual operators.
Areas of Agreement / Disagreement
Participants generally disagree on the sufficiency of the information provided by the pairs of states to determine U. Some assert that it is insufficient, while others propose specific cases where more information could be gleaned.
Contextual Notes
The discussion does not resolve the mathematical steps or assumptions regarding the nature of the states or the unitary operators involved, leaving open questions about the generalizability of the claims made.