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Background

Nielsen and Chuang give a postulate for "general measurements" on quantum systems (which are provably equivalent to unitary time evolution + projective measurements on ancillas) which is that there are measurement operators $M_i$ s.t. $$\sum_i M_i^\dagger M_i = I$$ and the probability of outcome $i$ is given by $$\langle{\psi}|M_i^\dagger M_i | \psi\rangle$$ and in the case of outcome $i$, the new state will be $M_i |\psi\rangle$ (suitably normalized).

They show that this measurement can be realized by starting with $|\psi\rangle$, adding an ancilla of appropriate dimension to get the combined system $|\psi\rangle|0\rangle$, applying a well chosen unitary map to leave us in state $$\sum_i (M_i |\psi \rangle) \otimes | i \rangle,$$ then projectively measure the ancilla bit. Thus we see general measurements in their definition follow from projective measurements.

Question

We can perform an even more general measurement though in a similar manner. Note that the only projective measure on the combined system that we used was a measure of the ancilla bit. If we measure the combined system in a more "entangled" way, then we get different outcomes. E.g. let's start with a 2D state $|\psi\rangle = \alpha |0\rangle + \beta | 1 \rangle$ and add a 2D ancilla $|0\rangle$, giving us combined state $\alpha |00\rangle+ \beta | 1 0\rangle,$ then perform the projective measurement with respect to the orthonormal basis $$\frac{1}{\sqrt{2}}\left(|00\rangle + |11\rangle\right), \frac{1}{\sqrt{2}}\left(|00\rangle - |11\rangle\right), \frac{1}{\sqrt{2}}\left(|01\rangle + |10\rangle\right), \frac{1}{\sqrt{2}}\left(|01\rangle - |10\rangle\right)$$ then we end up in state $$\frac{1}{\sqrt{2}}\left(|00\rangle + |11\rangle\right)$$ with probability $|\alpha|^2$ and end up in $$\frac{1}{\sqrt{2}}\left(|00\rangle - |11\rangle\right)$$ with probability $|\beta|^2$. Tracing over the ancilla bit shows that in either case, the density operator describing the first bit we end up with after this measurement is $$\frac{1}{2} \left(|0\rangle \langle 0| + |1\rangle \langle 1|\right).$$ Thus this measurement operation we've given cannot possibly described by the formalism given by Nielsen and Chuang. For one thing, we are left with a (non-pure) density operator, even if we start with a pure state. For another, we can record distinct outcomes with different probabilities, but the system under study ends up in the same end configuration regardless of the outcome.

Have measurement operations of this type been studied? Do they have a name and/or a standard reference? Is there a reason Nielsen and Chuang don't consider these kind of measurements as part of their general measurement formalism?

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  • $\begingroup$ @ChiralAnomaly The question is why this isn't considered as part of N&C's measurement formalism, where as they specifically talk about more general measurements than projective measurements, but only consider measurements that retain the purity of the state. It seemed odd to me that they only went "halfway" to full generality in describing measurements. Perhaps it's just because things get kind of messy in the more general case? $\endgroup$ Nov 13, 2021 at 0:39
  • $\begingroup$ @ChiralAnomaly not 100% sure what you mean. Their general measurement postulate they give is pretty much the same as POVMs, except with POVMs we don't know what the state is after a specified measurement outcome. Their general measurement postulate is what I gave in the beginning of my question and takes pure states inputs to pure state outputs. $\endgroup$ Nov 13, 2021 at 1:11
  • $\begingroup$ not sure whether this is what you are asking, but the measurement you refer to can definitely be described as a POVM, and thus by the "generalized measurement" formalism of N&C (which I personally find to be better presented in terms of channels rather than measurements, but anyway). What you are saying about a projective measurement on an enlarged system is Naimark's dilation. Any POVM can be written that way, including the one you present. The trick is that it uses an isometric evolution which transforms the "entangled measurement" into a projective measurement on the ancilla. $\endgroup$
    – glS
    Nov 13, 2021 at 18:54
  • $\begingroup$ the discussion in quantumcomputing.stackexchange.com/q/12275/55 might be of interest here $\endgroup$
    – glS
    Nov 13, 2021 at 18:55

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The most general measurement which includes a post-measurement state is given by $\rho\mapsto M_i\rho M_i^\dagger$, where in addition, you are allowed to "forget" part of the measurement outcome -- that is, the general post-measurement states will be of the form $$ \rho_J = \sum_{i_J} M_i\rho M_i^\dagger\ , $$ where the $J$ denote disjoint subsets of the index set, with outcome probability $p_J = \mathrm{tr}[\rho_J]$. Your measurement is of that kind.

In some sense, however, this type of measurement is stricly "weaker" then measuring all the $M_i$ individually, in that the former can be classically recovered after performing the latter measurement. Nevertheless, it should indeed be included when describing the most general formalism.

Note that in the "bare" POVM measurement formalism, which assigns probabilities $p_i=\mathrm{tr}[\rho F_i]$ to a state $\rho$ for a measurement described by a positive operator-valued measure (POVM) $F_i$, $\sum F_i=I$, but does not talk of post-measurement states (which often are, in fact, unphysical), does not have this shortcoming, as we can define $F_J = \sum_{i\in J} M_i^\dagger M_i$.

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