Higher Order Observations in Quantum Computing

Question

In the Quantum Gate model one generally is restricted to a set of universal Quantum gates, and an "observation" gate from which they can build circuits that carry out Quantum algorithms and are then "observed" to collapse their state.

Is there a way to do "weaker" observations, example: consider 2 qubits $q_1,q_2$ initially both at $1|0 \rangle + 0|1\rangle$ and an operator $$W(q_1, q_2)$$

which takes them to

$$ \frac{1}{2} |00 \rangle - \frac{1}{2} |01 \rangle + \frac{1}{2} |10\rangle + \frac{1}{2} |11\rangle $$

Now given that there is a way to observe an individual qubit, allowing you to effectively collapse to one of:

$$ \frac{1}{\sqrt{2}} | 00 \rangle - \frac{1}{\sqrt{2}} | 01 \rangle,\frac{1}{\sqrt{2}} | 10 \rangle + \frac{1}{\sqrt{2}} | 11 \rangle $$

I feel as though there ought to be "weaker observations", or as I call them "generalized observations", example, the observation operator $q_1 \wedge q_2$ which collapses the state to either

$$ |11\rangle , \frac{1}{\sqrt{3}}|00\rangle - \frac{1}{\sqrt{3}}|01\rangle + \frac{1}{\sqrt{3}} |10 \rangle $$

Depending on the outcome of the measurement

Is there any particular reason this wouldn't be allowed? Or is there a way to simulate an $n$ bit classical action on the qubits, using just the traditional observation and more unitary matrices?

Answer 1

In the context of a circuit, you implement fancy measurements by using ancilla qubits. For example, to measure whether the state is $|11\rangle$ or not you CCNOT onto an ancilla then measure the ancilla:

Mathematically, fancy measurements are often represented as POVMs (positive-operator valued measure). A set of non-negative Hermitian operators that sum to the identity matrix. See the wikipedia page for details. POVMs are somewhat abstract, which makes them useful when doing algebra, but you can always translate them into a circuit if you want something operationally concrete.