When you measure a quantum system, the outcome will be one of the eigenvalues of the measured (Hermitian) operator. If you perform this measurement over an ensemble of (normalized) pure states $|\psi\rangle$, you will readily find the expectation value $\langle \psi | A | \psi \rangle$ of the measured operator $A$, which obviously resides within its spectrum. That at least, was what most of us were taught in our first quantum mechanics course. It turns out, however, that the above depends on how exactly your perform your measurement and what is the statistical analysis you make.

In 1988 Aharonov, Albert and Vaidman (AAV) proposed a new quantum measurement scheme known as weak measurement [1]. Instead of the above strong, projective measurement, they suggested to weakly couple the measured system with the measuring pointer. One can still use the von Neumann measurement scheme, but one ought to make sure that the shift of the pointer is much smaller than its uncertainty. In that case, the amount of information one acquires with each strong measurement of the pointer is typically very small, so one will customarily need to repeat the measurement many times over similarly prepared quantum system (some exceptions do exist). But importantly, and unlike the standard paradigm, these strong measurements of the pointer barely change the states of the measured systems (this is often denoted as the non-invasiveness property of weak measurements).

That is already interesting, but AAV went one step further and combined the weak measurement protocol with a post-selection procedure to yield another intriguing concept known as the weak value. You prepare (preselect) an ensemble of systems all having the state $|\psi\rangle$, then you weakly couple some operator $A$ (not necessarily an observable) acting on the system with a measuring pointer, and finally you post-select on some $|\phi\rangle$ (which is not orthogonal to $\psi$, though here too, generalizations are known). In other words, one chooses to examine only those cases in which the system ends up in a state $|\phi\rangle$ upon a final projective measurement and discards all other instances (at least for the purpose of calculating the weak value), thereby actually using a pre- and postselected ensemble. Then, quite amazingly, one finds that the measuring pointer was shifted by an amount proportional to $\langle \phi | A | \psi\rangle /\langle \phi | \psi \rangle$ – the so-called weak value. In general, the weak value is not equal to the expectation value, and conceptually it is argued to be quite a different entity [2]. Moreover, it may reside far outside the spectrum of $A$. Such weak values are often called anomalous weak values. And indeed, AAV described a simple case where the spin value of a pre- and postselected spin-1/2 particle turns out to be 100 this way.

Although encountered with much criticism, this protocol was soon demonstrated in a laboratory experiment [3] preceding numerous others. Many researchers have been using since then the concept of weak values for foundational purposes, e.g. the study of contextuality [4], as well as for practical purposes, e.g. weak value amplification of small signals even in the vicinity of noise (see for example [5,6,7,8]). The price of large amplification, namely, a small postselection probability, should be also taken into account.

Regardless of the particular application of interest, it seemed that anomalous weak values basically result from two fundamental ingredients – weak coupling and post-selection. Recently, it turned out that the former ingredient can be relaxed to some extent [9,10,11,12], but can we relax also the latter ingredient? Can one measure an anomalous weak value even without post-selection, i.e. without discarding some of the measurement outcomes?

In [13], Abbott, Silva, Wechs, Brunner and Branciard answered this question in the affirmative (see also the previous related analysis by Diósi [14]). For doing so, they employed sequential weak measurement [15], an interesting, feasible tool on its own merits [14,16,17,18]. That is, they suggested to perform a series of weak measurements of non-commuting operators one after the other. In particular, Abbott et al. analyzed a case where a sequential weak measurement of two projectors (whose eignvalues belong to the set $\{0,1\}$) gives rise to an anomalous weak value being roughly -1/8. Unlike the current paradigm, strong post-selection was not needed, though the second weak measurement could be thought of as an “effective” or “weak” postselection. This can, and in fact, has been generalized to an arbitrary number of sequential measurements [13], with the eventual aim of distinguishing between different causal structures. Note however, that as of now, no one has apparently reported sequential weak measurements of more than 3 operators [19], with one of the problems being a constant of order $\epsilon^n$ multiplying the pointer shift when performing $n$ weak measurements of strength $\epsilon$. Another interesting limitation is the following: while weak values in pre- and postselected ensembles can be arbitrarily amplified (provided that the coupling strength is correspondingly adjusted), Abbott et al. proved that weak values are still confined in magnitude (though not in sign) to the spectrum of the measured operator. Finally, it should be noted that the same anomaly does not hold for joint weak measurements [20] performed on different substituent systems of a multipartite system. In the latter case, the measured operators commute, which takes the fundamental sting out of sequential measurements.

The analysis in [13] therefore broadens the notion of quantum measurement in general, and weak measurement in particular. It may also support the conceptual strength and physical meaning of anomalous weak values, which need not be accompanied now with any particular statistical procedure. Finally, we may gain new insights regarding uncertainty, contextuality, and causal structures.

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