Question
On general Hilbert spaces, measurements are formalized using a measure that assigns subsets to these operators. For 10 points each:
[10m] Name these operators that can be expressed as sums of outer products. The probability of measuring an observable can be computed by constructing one of these idempotent operators for the desired subspace and then taking its expectation.
ANSWER: projection operators [accept projection-valued measure]
[10h] For a generic state, this object can be written as a convex combination of projection operators. The expectation value of an observable A can be written as the trace of the following: this object times A.
ANSWER: density matrix [or density operator; prompt on rho]
[10e] By the completeness relation, the sum of the projection operators for each eigenspace is this matrix. This matrix is a diagonal matrix whose diagonal entries are all equal to one.
ANSWER: identity matrix [prompt on I]
<RA, Physics>
Summary
| 2023 ARCADIA at UC Berkeley | Premiere | Y | 2 | 15.00 | 100% | 50% | 0% |
| 2023 ARCADIA at Carleton University | Premiere | Y | 2 | 15.00 | 100% | 50% | 0% |
| 2023 ARCADIA at Claremont Colleges | Premiere | Y | 1 | 20.00 | 100% | 0% | 100% |
| 2023 ARCADIA at Indiana | Premiere | Y | 4 | 15.00 | 100% | 25% | 25% |
| 2023 ARCADIA at RIT | Premiere | Y | 2 | 20.00 | 100% | 50% | 50% |
| 2023 ARCADIA at WUSTL | Premiere | Y | 3 | 10.00 | 100% | 0% | 0% |
Data
| Chicago A | Purdue A | 10 | 0 | 10 | 20 |
| Illinois A | Notre Dame A | 0 | 10 | 10 | 20 |
| Illinois B | Indiana | 0 | 0 | 10 | 10 |
| Vanderbilt | Chicago B | 0 | 0 | 10 | 10 |