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 a given value for an observable can be computed by constructing one of these idempotent operators, which map a state onto a desired subspace.
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 Duke | Emory, Duke, Yale | Y | 2 | 10.00 | 100% | 0% | 0% |
| 2023 ARCADIA at Emory | Emory, Duke, Yale | Y | 3 | 16.67 | 100% | 33% | 33% |
| 2023 ARCADIA at Imperial | Imperial | Y | 3 | 20.00 | 100% | 67% | 33% |
| 2023 ARCADIA at Maryland | Maryland, Online | Y | 1 | 20.00 | 100% | 100% | 0% |
| 2023 ARCADIA at Ohio State | Ohio State, Texas | Y | 1 | 10.00 | 100% | 0% | 0% |
| 2023 ARCADIA Online | Maryland, Online | Y | 2 | 15.00 | 100% | 50% | 0% |
| 2023 ARCADIA at Texas | Ohio State, Texas | Y | 2 | 20.00 | 100% | 50% | 50% |
Data
| Oxford | Cambridge B | 10 | 0 | 10 | 20 |
| Edinburgh | Imperial A | 0 | 0 | 10 | 10 |
| Cambridge A | Imperial B | 10 | 10 | 10 | 30 |