# Higher Order Observations in Quantum Computing

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?

• That's just a plain projective measurement of an operator which has a degenerate eigenspace. E.g., if you measure $\vec S_1\cdot\vec S_2$ on a pair of spin-1/2 you get sth. with only two outcomes. – Norbert Schuch Mar 20 '17 at 9:02

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: