# Measurement in the context of unitary evolution

My text describes how "projective measurements together with unitary dynamics are sufficient to implement a general measurement" and goes on (pp. 94-95) to demonstrate why this is so, but there are several aspects of this demonstration I'm not sure I fully understand.

The crux of this demonstration is to show that any general measurement of a quantum system $Q$ can expressed as the unitary evolution, $U$, of a (closed) composite system $Q\oplus M$, where $M$ has an orthonormal basis $\{\left|m\right>\}$ in one-to-one correspondence to a set of measurement operators $\{M_m\}$:

$$U\left|\psi\right>\left|0\right>\equiv\sum_{m}M_m\left|\psi\right>\left|m\right>$$

given pre-measurement states $\left|\psi\right>$ and $\left|0\right>$ for $Q$ and $M$ for respectively.

An essential step in this demonstration is to establish that the probability of an outcome $m$ matches the definition of general measurement, which is accomplished by "supposing" we preform a "projective measurement" of the form $P_m=I_Q\otimes\left|m\right>\left<m\right|$ on the joint system and calculate probabilities of its (corresponding, and thus identically labeled) outcomes:

\begin{aligned} p(m) &= \left<\psi\right|\left<0\right|U^ {\dagger}P_mU\left|\psi\right>\left|0\right>\\ &= \sum_{m',m''}\left<\psi\right|M_{m'}^{\dagger}\left<m'\right|\left(I_Q\otimes\left|m\right>\left<m\right|\right)M_{m''}\left|\psi\right>\left|m''\right> \\ &= \sum_{m',m''}\left<\psi\right|M_{m'}^{\dagger}I_QM_{m''}\left|\psi\right>\otimes\left<m'|m\right>\left<m|m''\right>\\ &=\left<\psi\right|M_{m}^{\dagger}M_{m}\left|\psi\right>\, \square\\ \end{aligned}

My questions about this all have to do with the role of measurement in the context of unitary evolution:

The role of "distinguishability"

First, I assume that while the orthonormality of $\{\left|m\right>\}$ is essential (final step) for the demonstration above, there's nothing there (or in the definition of general measurement, generally) that requires the outcomes, $\{m\}$, of $\{M_m\}$ to be distinguishable (and thus the $\{\left|\psi_m\right>\}$ need not be orthogonal). That is, $M$, could end up in distinct states with indistinguishable states of $Q$.

Can the system being measured end up in indistinguishable states even when the corresponding states of the "observer" system are orthogonal?

Probabilities for unitary evolution

The text's demonstration applies the projective measurement to unitary evolution. But isn't it more appropriate to incorporate it into the unitary evolution:

\begin{aligned} \sum_{m'}P_{m'}U\left|\psi\right>\left|0\right>&\equiv\sum_{m'}P_{m'}\sum_{m}M_m\left|\psi\right>\left|m\right>\\ I_{Q\otimes M}U\left|\psi\right>\left|0\right>&\equiv\sum_{m',m}P_{m'}M_m\left|\psi\right>\left|m\right>\\ U\left|\psi\right>\left|0\right>&\equiv\sum_{m}P_{m}M_m\left|\psi\right>\left|m\right>\\ \end{aligned}

from which, applying the general definition of measurement to the measurement operators $\{P_mM_m\}$, we arrive directly at

$$p(m) =\left<m\right|\left<\psi\right|M_m^{\dagger} P_m^2M_m\left|\psi\right>\left|m\right>=\left<\psi\right|M_m^{\dagger} M_m\left|\psi\right>\square$$

This may be a distinction without a difference; but it seems to me that the first demonstration is, strictly speaking, calculating the "probabilities" of "outcomes", $p_U(m)$, for the unitary evolution of the system, while the latter is calculating probabilities, $p_M(m)$, as experienced by elements of $M$. Only in the latter case however does it make sense to speak of "probabilities".

Is this reformulation equivalent to the one presented above from my text? Am I right to think that it is a more appropriate treatment of outcome probabilities?

Projection and “ancilla” as integral to all measurement

Even if the preceding calculation or argument aren't quite right, from a conceptual point of view it seems essential to leave the unitary evolution "undisturbed" in demonstrating the relationship between unitary evolution and measurement: in the end, the $\{P_m\}$ must be subsumed by some unitary evolution and the minimal such evolution is precisely $U$.

Of course this is all a matter of perspective: if we are, as it were, the fortunate/unfortunate cat then the unitary evolution of the cat-in-a-box system is not our concern and the text's formulation applies; but to a human outside the closed box, the projection onto the cat's state space must be a part of the unitary evolution of the cat-in-a-box system, so the alternative formulation is more appropriate. This perspective shifts when the human opens the box, and the projection of its state onto theirs ends the isolated unitary evolution of cat-in-a-box system. But again. crucially, that projection is part of a further "outer" unitary evolution of some closed system of which the box and human are a part.

So I'm a bit confused when the text says that the "ancilla system can be regarded as merely a mathematical device appearing in the construction". Is that a fair characterization?

In fact (partly for the reason given above, but even if the previous account isn't quite right or necessary), aren't projection and the ancilla — specifically projection onto an additional system whose eigenstates correspond to $\{M_m\}$ — integral to measurement (and not merely a way of doing the accounting)?

In other words, isn't every measurement of the state of a system $Q$ in fact a measurement of a composite system consisting of $Q$ and an additional system sufficient to instantiate the measurement operators $\{M_m\}$, and with corresponding eigenstates onto which the outcomes are projected? Moreover, when the additional system encompasses as much of the world is affected by those outcomes, doesn't that measurement correspond to unitary evolution?