Why is there always a $k$-outcome experiment associated to operators such that $\sum_{i=1}^k M_i^* M_i=I$?

When we want to observe a system, we make quantum measurements, which are always described by a class of operators $$\{ M_i \}_i$$

The probability that we observe the outcome $$i$$, given that the system is in state $$|\psi\rangle$$ before we measure, is given by $$p_i = \|M_i \psi \|^2$$ and if we observe the outcome i, then the system changes to the state $$\frac{M_i\psi}{\|M_i\psi \|}$$

Moreover, as the sum of the probabilities of all possible outcomes must equal $$1$$, we have $$\sum_{i=1}^k M_i^*M_i= I$$ , when we consider a quantum experiment with at most $$k$$ possible outcomes.

In some notes we can also read :

Now let $$\{M_i : 1 \leq i \leq k\}$$ be a collection of bounded operators such that $$\sum_{i=1}^k M_i^*M_i= I$$ , then there is a $$k$$-outcome quantum experiment with these measurement operators.

Why is this the case? why is there always such an experiment?

1 Answer

Imagine you want to make a measurement on a state $$|\psi\rangle$$ (and we will make the solution work for all possible $$|\psi\rangle$$). We introduce an ancilla system (Hilbert space dimension at least equal to the number of measurement operators), prepared in the fixed state $$|0\rangle$$, and define a unitary $$U|\psi\rangle=\sum_m(M_m|\psi\rangle)|m\rangle.$$ This state is properly normalised thanks to the normalisation condition on the measurement operators, and if you check for two orthogonal states that are being measured, the outputs are orthogonal. This means that the unitary is well-defined, and really is a unitary that we can construct for any set of measurement operators.

The point now is that the desired measured can be implemented by

1. introducing the ancilla qubit.
2. applying $$U$$.
3. measuring the ancilla in the projective basis.

A few more details are given in Nielsen & Chuang section 2.2.8.

• I don't understand, I first thought that you will exhibit an experiment for a given set of measurements (operators with this property), but then what is 'ancilla' ? Just to clarify, are we talking about a physical experiment? that we can do in a lab? May 19, 2023 at 2:39
• Yes, this is an actual experiment that you can do. An ancilla is just an extra quantum system. The point is to turn the measurements into projective measurements, but to do that you need a larger Hilbert space. May 19, 2023 at 5:42