# On the negative-result experiments
in quantum mechanics

Kenichi Konishi(1,2)
(1)INFN, Sezione di Pisa, Largo Pontecorvo, 3, Ed. C, 56127 Pisa, Italy
(2)Department of Physics “E. Fermi”, University of Pisa,
Largo Pontecorvo, 3, Ed. C, 56127 Pisa, Italy

<EMAIL_ADDRESS>orcid.org/0000-0002-4944-5444

###### Abstract

We comment on the so-called negative-result experiments (also known as null
measurements, interaction-free measurements, and so on) in quantum mechanics
(QM), in the light of the new general understanding of the quantum-measurement
processes, proposed recently. All experiments of this kind (null-measurements)
can be understood as improper measurements with an intentionally biased
detector set up, which introduces exclusion or selection of certain events.
The prediction on the state of a microscopic system under study based on a
null measurement, is sometimes dramatically described as “wave-function
collapse without any microsystem-detector interactions”. Though certainly
correct, such a prediction is just a consequence of the standard QM laws, not
different from the situation in the so-called state-preparation procedure.
Another closely related concept is the (first-class or) repeatable
measurements. The verification of the prediction made by a null-measurement
requires eventually a standard unbiased measurement involving the microsystem-
macroscopic detector interactions, which are nonadiabatic, irreversible
processes of signal amplification.

Keywords: Particles, Quantum measurement, Null measurement, Wave-function
collapse

###### Contents

  1. 1 Introduction
  2. 2 Solution of the quantum-measurement problem in a nutshell
    1. 2.1 A secret key
    2. 2.2 A remark
  3. 3 Negative-result experiments
    1. 3.1 Renninger
      1. 3.1.1 State preparation
      2. 3.1.2 $\alpha$ particle tracks in a cloud chamber
    2. 3.2 Elitzur-Vaidman bomb tester
    3. 3.3 Modified Stern-Gerlach set-up
  4. 4 Reflections

##  1 Introduction

A typical quantum-mechanical measurement process involves an interaction
between a microscopic quantum system with a macroscopic experimental device,
which is capable of faithfully capturing the quantum state of the microscopic
system - object of the “measurement” - and of recording the result in the form
of a classical state of matter. The process typically involves a non-
adiabatic, irreversible process of signal amplification (such as a chain
ionization and fixture of images on a photographic film). Critical discussions
on earlier attempts for constructing a theory of measurements can be found in
[1]; many original papers are collected in [2]. See also [3, 4]. Quantum
measurement processes have been analyzed recently, with a few new key
observations [5, 6].

Such a physical characterization of a quantum measurement was challenged by a
series of gedanken (as well as real) experiments [7, 8] of particular kind, in
which some negative-result (or null) measurement, hence without any
microsystem-macroscopic-device interactions typical of the standard quantum
measurements, allows one to acquire a nontrivial information on the quantum
state of the system under study. The predicted state, $|\psi^{\prime}\rangle$,
is necessarily a restriction 111Namely, a projection of the original vector in
the Hilbert space onto a vector in a space of a smaller dimension. of the
original wave function $|\psi\rangle$. This fact was somewhat dramaticaly
expressed as “an interaction-free measurement leading to a wave-function
collapse”. These negative-result-experiment arguments have been presented, and
believed by some, as counter-examples to the physical characterization of the
quantum measurement processes, i.e., non-adiabatic, irreversible processes
involving micro system - macroscopic device interactions.

The aim of this short note is twofold. The first is to review the
characteristics of a typical quantum-measurement processes, based on a few
novel observations made recently [5, 6, 9]. The second is to give a simple,
more sober interpretation of these null measurements: they all correspond to
specific, biased detector set-ups which select or exclude certain possible
events. The prediction made on the quantum state after a null-measurement,
certainly correct, is just a consequence of the standard quantum mechanical
laws: they do not in any way disprove the physical picture of a typical
quantum measurement process. Indeed, verification of the predictions made by a
null-measurement on the system being studied, eventually requires a proper,
unbiased quantum measurement, involving microsystem-macroscopic-device
interactions.

Even though the main subject of this note [7, 8] is rather an old one, and has
been discussed extensively [1], it touched some of the most subtle aspects of
the interpretations of the quantum mechanical predictions. It is thus not
entirely pointless today, perhaps, to revisit the essential aspects of these
negative-result experiments, to ensure that our understanding of the QM laws
be crystal clear.

##  2 Solution of the quantum-measurement problem in a nutshell

A measurement of a quantity $F$, done on a quantum state $\psi$

$|\psi\rangle=\sum_{n}c_{n}|n\rangle\;,\qquad\sum_{n}|c_{n}|^{2}=1\;,$ (2.1)

where $|n\rangle$ is the eigenstate of $F$ with eigenvalue, $f_{n}$, used to
be schematized as

$\displaystyle|\psi\rangle\otimes|\Phi_{0}\rangle\otimes|Env_{0}\rangle$
$\displaystyle=$
$\displaystyle\left(\sum_{n}c_{n}|n\rangle\right)\otimes|\Phi_{0}\rangle\otimes|Env_{0}\rangle$
(2.2) $\displaystyle\longrightarrow$
$\displaystyle\sum_{n}c_{n}|n\rangle\otimes|\Phi_{n}\rangle\otimes|Env_{0}\rangle$
(2.3) $\displaystyle\longrightarrow$
$\displaystyle\sum_{n}c_{n}|n\rangle\otimes|\Phi_{n}\rangle\otimes|Env_{n}\rangle\;,$
(2.4)

where $|\Phi_{n}\rangle$ represents the detector state, with has recorded the
result $F=f_{n}$ and $|Env\rangle$ stands for everything else, the state of
the experimentalist and the rest of the world. The index $0$ indicates a
neutral state, whereas the index $n$ stands for the measurement result,
$F=f_{n}$. Such a formula is found in many textbooks of quantum mechanics
(QM), and (basically) in all past discussions [1, 2].

Actually, the formulae (2.2)-(2.4) are incorrect in several accounts [5, 6].

(i) First, the factorized form for the wave function expressed by the symbol
$\otimes$ in the state after the measurement 222As for the state before the
measurement, factorization of $|\psi\rangle$ is correct: it must be so in any
ideal experiment., is not valid. In general, in the aftermath of a measurement
the microscopic system gets entangled with the device and with the
environment, typically in an uncontrolled manner 333In an exceptional class of
the so-called repeatable (or the first-class) experiments, the microscopic
system under study remains factorized and intact. These processes are indeed
closely related to the “negative-result experiments” as well as with the state
preparation procedure, as discussed below, Sec. 3. , even though the
information about its original state is faithfully recorded by the detector.

(ii) Secondly, the experimental device, $|\Phi\rangle$, typically a
macroscopic body at finite temperatures, and entangled with the rest of the
world (environment), is itself in a decohered, mixed state [10]-[16],[9]. This
is so even before the measurement, let alone after the measurement. This means
that the expression (2.3) should not be considered as a pure state (a coherent
superposition): it is a mixture 444This observation is sufficient to explain
away a “puzzle” recently discussed in [17]. .

Taking these points into account, it was proposed [5] that the detector-
environment entangled state be denoted as $|\Phi;Env\rangle$. The typical
measurement process thus looks more like

$\displaystyle|\psi\rangle\otimes|\Phi_{0};Env_{0}\rangle$ $\displaystyle=$
$\displaystyle\left(\sum_{n}c_{n}|n\rangle\right)\otimes|\Phi_{0};Env_{0}\rangle\,,\qquad{\rm(before)}$
(2.5) $\displaystyle\longrightarrow$
$\displaystyle\sum_{n}c_{n}\,|{\tilde{\Phi}}_{n};Env_{0}\rangle\,,\qquad{\rm(after)}$
(2.6) $\displaystyle\longrightarrow$
$\displaystyle\sum_{n}c_{n}\,|{\tilde{\Phi}}_{n};Env_{n}\rangle\,,\qquad{\rm(later)}\;,$
(2.7)

where $|{\tilde{\Phi}}_{n};Env_{0}\rangle$ with the symbol
${\tilde{\Phi}}_{n}$ stands for the microsystem-detector entangled state (see
[5] for more discussion) with a macroscopic marking of the recording,
$F=f_{n}$.

The essential features of the process, (2.5) $\to$ (2.6), are the following
555The second stage of the process, (2.7), in which the experimentalist sees
the result of the measurement on her computer screen, the others read about it
in Physical Review, etc., is totally irrelevant, in spite of many
sophisticated and sometimes philosophical discussions made in the past (see
[2] for some)..

(a)

Each term in (2.6) containing $|{\tilde{\Phi}}_{n};Env_{0}\rangle$ is a
complicated mixed state (point (ii) above), representing the microsystem-
detector-environment entangled state (point (i) above), with a well-defined
macroscopic marker of the measurement result, $F=f_{n}$. It is an eigenstate
of the operator $F$. Namely,

$\displaystyle F\,|n\rangle$ $\displaystyle=$ $\displaystyle f_{n}\,|n\rangle$
(2.8) $\displaystyle\to\qquad F\,|{\tilde{\Phi}}_{n};Env_{0}\rangle$
$\displaystyle=$ $\displaystyle f_{n}|{\tilde{\Phi}}_{n};Env_{0}\rangle\,.$
(2.9)

The relation between (2.8) and (2.9) defines a good, faithful measurement.

(b)

A key observation [5] is that, reflecting the pointlike nature of the
fundamental entities of our world, each measurement process is a spacetime
pointlike event (or triggered by one). This entails that the wave functions
corresponding to the different terms in (2.6) have no overlapping spacetime
supports. Thus, not only the orthonormality

$\langle{\tilde{\Phi}}_{m};Env_{0}|{\tilde{\Phi}}_{n};Env_{0}\rangle=\delta_{mn}\,,$
(2.10)

holds, but also a dynamical diagonalization

$\langle{\tilde{\Phi}}_{m};Env_{0}|G|{\tilde{\Phi}}_{n};Env_{0}\rangle=G_{n}\,\delta_{mn}\,.$
(2.11)

occurs for any local operator $G$. Note that $G_{n}$ in (2.11) is defined by
(2.11) itself: it is unrelated to the eigenvalues of the operator $G$ in the
isolated microscopic system before the experiment.

The diagonalization (2.11) is of utmost importance.

Before the measurement, in the state (2.5), the expectation value of a generic
quantity $G$ is given by

$\\{\langle\Phi_{0};Env_{0}|\otimes\langle\psi|\\}G\\{|\psi\rangle\otimes|\Phi_{0};Env_{0}\rangle\\}=\langle\psi|G|\psi\rangle=\sum_{m,n}G_{mn}c_{m}^{*}c_{n}\equiv\mathop{Tr}\nolimits\rho^{(0)}G\;,$
(2.12)

($G_{mn}\equiv\langle m|G|n\rangle$) meaning that the system is described by a
density matrix

$\rho^{(0)}_{nm}=c_{n}c_{m}^{*}\;,$ (2.13)

i.e., by the pure state, $|\psi\rangle=\sum_{n}c_{n}|n\rangle.$ For $G=F$, the
variable whose eigenstates are taken as the basis $\\{|n\rangle\\}$, one finds
of course the standard formula

$\langle\psi|F|\psi\rangle=\sum_{n}f_{n}|c_{n}|^{2}\;.$ (2.14)

After the measurement,  according to (2.11) the expectation value of a generic
variable $G$ taken in the “state” (2.6), is given (by using (2.11)) by

$\left(\sum_{n}c_{m}^{*}\,\langle{\tilde{\Phi}}_{m};Env_{0}|\right)\,G\,\left(\sum_{n}c_{n}\,|{\tilde{\Phi}}_{n};Env_{0}\rangle\right)=\sum_{n}|c_{n}|^{2}G_{n}\;.$
(2.15)

That this holds for any operator $G$ means that the density matrix of the
system has been effectively reduced to a diagonal form

$\rho^{(0)}\stackrel{{\scriptstyle(measurement)}}{{\Longrightarrow}}\rho^{(1)}=\left(\begin{array}[]{ccccc}|c_{1}|^{2}&&&&\\\
&|c_{2}|^{2}&&&\\\ &&\ddots&&\\\ &&&|c_{n}|^{2}&\\\
&&&&\ddots\end{array}\right)\;.$ (2.16)

By paraphrasing the “environment-induced superselection rule” [1], we may call
(2.16) the measurement-induced superselection rule.

(c)

The fact that the wave functions of the different terms in (2.6) have no
overlapping spacetime support means that the aftermath of each measurement
event is a single term in (2.6) 666A related fact is that the detector-
environment “state” $|\Phi_{0};Env_{0}\rangle$, even if it might look
identical macroscopically, it can never be the same quantum state at two
different measurement instants. The time evolution of the macroscopic number
of molecules and atoms in the detector and environment, means that the “state”
just before each experiment $|\Phi_{0};Env_{0}\rangle$ is a unique quantum
state, carrying a hidden index “(n)” of each measurement, with outcome, $n$. ,
that is, the time evolution in each single measurement is,

$\left(\sum_{n}c_{n}|n\rangle\right)\otimes|\Phi_{0};Env\rangle\Longrightarrow|{\tilde{\Phi}}_{m};Env\rangle\;:$
(2.17)

i.e., with a single term present the instant after the measurement (e.g., with
$F=f_{m}$). (2.17) is often (improperly) described as a “wave-function
collapse”. 777 The words evoke in our mind an image of some distribution
suddenly contracting, which does not correspond to any real physical process.
The wave function is itself not an observable.

(d)

A second crucial consequence of our description of the measurement process,
(2.5), (2.6), concerns the repeated measurements. For the measurement of the
quantity $F$, it follows from (2.9), (2.10) and (2.15) that the expectation
value is given by

$\left(\sum_{n}c_{m}^{*}\,\langle{\tilde{\Phi}}_{m};Env_{0}|\right)\,F\,\left(\sum_{n}c_{n}\,|{\tilde{\Phi}}_{n};Env_{0}\rangle\right)=\sum_{n}|c_{n}|^{2}f_{n}\;,$
(2.18)

where $f_{n}$ are the eigenvalues of $F$. This means that the relative
frequency for finding the result, $F=f_{n}$, has been found to be given by

${\cal P}_{n}=|c_{n}|^{2}\;.$ (2.19)

The derivation of the “wave-function collapse” (2.17) and of the formula for
the relative frequency (2.19), amount to the solution of the quantum-
measurement problem. 888One might object that we have just reproduced the
standard Born rule. This is not quite so. Unlike the latter, our description
explains why and how the “wave-function collapse” occurs, and yields the rule
(2.19) as the result of physical measurement processes involving the
microsystem-detector-environment interactions. Even though they look similar,
the difference in the meaning of (2.14) and (2.18) is crucial. Last, by
eliminating the fundamentally obscure concept of “probability” inherent to
Born’s rule, and by replacing it by the (normalized) relative frequency for
various outcomes in repeated experiments, it leads to a more natural
interpretation of the QM laws [5, 6]. For instance, the concept of the “wave
function of the universe” makes perfect sense now, whereas in the traditional
interpretation of QM based on Born’s rule one cannot avoid falling into a
conundrum of having nobody outside the universe, observing it and making
repeated experiments on it, etc. In the author’s understanding, the
cosmologists today are adopting this new interpretation of QM laws, when they
discuss the structure formation in the early universe, through the density
quantum fluctuations at certain stage of the inflation.

###  2.1 A secret key

An alert reader must have noticed the following subtlety. We affirmed that
each of the microsystem-detector-environment entangled state
$|{\tilde{\Phi}}_{n};Env_{0}\rangle$ is a complicated mixed state, involving
the myriad of microscopic processes, such as the scattering of air molecules
against the $\sim O(10^{25})$ atoms and molecules composing the detector, the
emission of the infrared photons from the latter, and so on. Nevertheless, we
used it as an ordinary wave function (i.e., a pure state), to evaluate the
expectation values, (2.15), (2.18). Is it consistent?

Actually here lies a secret key in the whole discussion. What might not be
widely appreciated is the fact that there are no differences of principle
between the concepts of the pure and mixed states. Consider any (pure) quantum
state $\Psi(\\{{\bf\xi}_{i}\\},\,\\{{\bf\chi}_{k}\\})$ where
${\bf\xi}_{i}\equiv({\bf r}_{i},s_{i})$ and ${\bf\chi}_{k}\equiv({\bf
r}^{\prime}_{k},s^{\prime}_{k})$ are the position and spin component of the
particles composing the whole system. We (the physicists) are assumed to have
access only to the subsystem ($A$) containing the degrees of freedom
$\\{{\bf\xi}_{i}\\}$. The rest of the world ($B$) described by
$\\{{\bf\chi}_{k}\\}$ is off-limit. By introducing an orthonormal (ON) set of
states describing the system $A$, $\\{|n\rangle^{(A)}\\}$, and expanding the
coefficient in ON states of $B$, $\\{|N\rangle^{(B)}\\}$, a generic state has
the form,

$|\Psi\rangle=\sum_{n,N}c_{n,N}\,|n\rangle^{(A)}|N\rangle^{(B)}\;.$ (2.20)

The expectation value of any variable $G$ pertinent to $A$, is then

$\langle\Psi|G|\Psi\rangle=\mathop{Tr}\nolimits{\bf G}{\rho}\;,\qquad{\bf
G}_{mn}=\langle m|G|n\rangle\;,\quad\rho_{nm}=\sum_{N}c_{n,N}c_{m,N}^{*}$
(2.21)

where $\rho$ is the density matrix. It is perfectly correct however to use the
wave function (2.20), or (2.6), e.g., in (2.15) or in (2.18), when the sum
over the system $B$ is indeed implied in the calculation 999A similar idea was
used in [9] to explain why the Ehrenfest theorem can be used to derive
Newton’s equations for the center of mass of a macroscopic body at finite body
temperatures, which is in a decohered, mixed state. .

To sum up, the concept of a mixed state (versus a pure state) is a relative
one, depending on which part of the world ($A$) is accessible to us. The
ignorance about the rest of the universe ($B$) is parametrized by the density
matrix. More interestingly, two (or more) groups of physicists studying a
common event such as a multiparticle decay, located in spacetime regions which
are spacelikely separated, might find apparently paradoxical outcomes
involving various quantum correlations. These phenomena of “quantum
nonlocality”, however as fascinating and mysterious they might continue to
appeal to human mind, are today fully understood. The peculiar “subjectivity”
of the quantum-mechanical laws also hinges upon these circumstances.

###  2.2 A remark

One of the characteristics (or requirements) of a proper quantum measurement,
which every experimentalists know well, is that the device must not have any
bias, i.e., it should be equally effective to register all possible outcomes,
$f_{n}$. For, if it were not so, the experimental average for the frequency
times various possible results would not match the theoretical prediction,
(2.19), even after many repeated measurements.

An exception occurs when there are only two possible outcomes, either
$F=f_{1}$ or $F=f_{2}$. In this case, the device which is capable of measuring
only one of the possible results, e.g., $f_{1}$ (the so-called yes-no
experiment) is sufficient to give unbiased measurement results. Event by
event, the detection of $f_{1}$ (yes) means the wave function collapse (2.17)
with $F=f_{1}$; the non detection of $f_{1}$ (no) implies that the state is
$|\psi\rangle=|2\rangle$, even if the measurement $F=f_{2}$ has not yet been
actually done. The negative-result experiments to be reviewed below rely on an
analogous logic.

##  3 Negative-result experiments

In this section a few well-known examples of the negative-result experiments
are reviewed and their essential features critically analyzed.

###  3.1 Renninger

A pair of collections of particle detectors, each covering one of the
hemispheres, surround a radioactive nucleus 101010In the original article by
Renninger [7], an excited atom is used and what is emitted is a photon.
Nothing essential changes by replacing the atom in an excited level by an
unstable nucleus, however. in the center, which emits an $\alpha$ particle,
e.g., in S-wave. If one of the shells does not observe $\alpha$, its wave
function has been “collapsed to the other hemisphere”, without any
interactions between the $\alpha$ particle and the experimental device!

To make the puzzle look sharper, the detectors in the second hemisphere (call
$\Phi^{(lower)}$) may be set at a radius much larger than the first. If the
half-life-time of the nucleus is $\tau$, the nucleus has most likely decayed
($\alpha$ has been emitted) by the time $t=30\tau$ 111111Only in an
exceptional one out of $e^{30}$ repeated experiments, on the average, the
nucleus will be found still undecayed, without $\alpha$ emission. . If the
second detector is set at the distance

$R>30c\tau\;,$ (3.1)

then by the time $t=30\tau$, if the first detector has not detected $\alpha$,
then most likely it is still travelling towards the lower-hemisphere. So it
might appear as if “the wave function had collapsed”, without any particle-
detector interactions having taken place. The “paradox” is only apparent, and
its origin can be traced to the misconception that the wave-function collapse
is a sort of real physical process in itself, as noted already.

The metastable nucleus which $\alpha$-decays in an $S$ state, can be expressed
by

$|\Psi\rangle=|\Psi^{(0)}(t)\rangle+|\Psi^{\prime}(t)\rangle|\alpha\rangle\;,$
(3.2)

At the time $t=30\tau$ the nucleus has most certainly decayed, so let us
concentrate on the second term of (3.2). The $\alpha$ particle is described by
an S-wave function 121212This is really the reduced one-body description of
the $\alpha$ particle - nucleus. ,

$|\Psi\rangle=\sum_{i}c_{i}|i\rangle\;,$ (3.3)

where $i=1,2,\ldots N$, $N\gg 1$, represent the uniformly discretized cells of
the $4\pi$ solid angle. The S-wave nature of the wave function means that

$\forall i\qquad c_{i}=\frac{1}{\sqrt{N}}\;.$ (3.4)

This is simply a discretized version of the $S$-wave wave function,

$\Psi({\bf r})=f(r)\sim\frac{e^{ikr}}{r}\;,$ (3.5)

independent of the angular variables. The standard measurement of the angular
distribution is done with

$|\psi\rangle\otimes|\Phi_{0}\rangle=\sum_{i}c_{i}|i\rangle\otimes|\Phi_{0}\rangle\;.$
(3.6)

where the detector $\Phi_{0}$ is uniformly sensitive over the $4\pi$ solid
angle. It should not have any bias as to which angular direction the particle
is eventually measured.

The experimental set-up by Renninger, instead, has a nonuniform $\Phi_{0}$.
Namely,

$|\psi\rangle|\Phi_{0}\rangle=\sum_{i=1}^{N/2}c_{i}|i\rangle\otimes|\Phi_{0}^{(upper)}\rangle+\sum_{i=N/2+1}^{N}c_{i}|i\rangle\otimes|\Phi_{0}^{(lower)}\rangle\;.$
(3.7)

where $\Phi_{0}^{(upper)}$ and $\Phi_{0}^{(lower)}$ are the first and second
groups of detectors at the upper and lower hemispheres, respectively. The
detector $\Phi_{0}^{(upper)}$ ($\Phi_{0}^{(lower)}$) is insensitive to the
$\alpha$ particle flying towards the lower (upper) hemisphere, thus the cross
terms such as
$\sum_{i=1}^{N/2}c_{i}|i\rangle\otimes|\Phi_{0}^{(lower)}\rangle$ are absent.

In the case (3.1) the two detectors are such that up to $t=30\tau$, the lower
detector is insensitive to the $\alpha$ particle, i.e., is not able to detect
it. Only the first is sensitive, and only to the $\alpha$ particle travelling
towards the upper hemisphere.

If the experiment is repeated, half of the times the detector
$\Phi_{0}^{(upper)}$ will record $\alpha$ at time $t\leq 30\tau$, and the
other half of the times it will not. This is what this particularly biased
detector will produce, in accordance with the QM prediction.

In a second type of events, where the first detector $\Phi_{0}^{(upper)}$ does
not detect the $\alpha$ particle (“the negative-result measurement”), it is
correct for us to infer that  the wave function is reduced to the scond term
of (3.7),

$|\psi\rangle|\Phi_{0}\rangle\to\sum_{i=N/2+1}^{N}c_{i}|i\rangle\otimes|\Phi_{0}^{(lower)}\rangle\;.$
(3.8)

In these events the $\alpha$ particle will be detected by $\Phi_{0}^{(lower)}$
at a later time,

$t\sim\frac{R}{c}\gg 30\tau\;,$ (3.9)

as will be eventually verified by the standard $\alpha$ \-
$\Phi_{0}^{(lower)}$ detector interactions.

The negative-result experiment such as this could look somewhat paradoxical. A
phrase such as “for us to infer that …” might indeed appear to indicate that
the “wave function collapse,” $|\psi\rangle\to\sum_{i=N/2+1}^{N}|i\rangle\,,$
has been caused by human mind - the realization of the non-detection fact. The
wave-function collapse, so it might seem, does not need any microsystem-
macroscopic device interactions as those assumed in [18, 19]. These questions
were at the center of ardent debates, partially reignited by Renninger’s work
(see for example, [20], [21]).

What happens actually in the Renninger experiment is that, in each decay
event, the $\alpha$ particle is emitted either towards the upper hemisphere or
towards the lower hemisphere. With the same relative frequencies, if the
experiment is repeated. That is all.

The aim of revisiting these old, and after all simple, issues nonetheless, was
to illustrate how the wrong wordings and the misconception about the “wave-
function-collapse” have led to nonexistent, and hence unsolvable, problems in
the past discussions on QM.

####  3.1.1 State preparation

The deduction (3.8) in the case of Renninger’s negative-result experiment, is
not essentially different from the preparation of a collimated atomic beam, by
using two successive slits, such that the particles which have passed both
slits have a more or less well-defined momentum direction, so that one can
predict that the particle, left freely propagating, will be detected in the
direction of a straight line connecting the two successive slits, within some
error (taking into account the diffraction effects).

The analogy may be made even closer, by making the upper detector
$\Phi^{(upper)}$ of Renninger cover 99 % of the $4\pi$ solid angle, leaving a
small hole in the south-pole direction. Then in the very rare event (one in
$100$) in which $\Phi^{(upper)}$ has not recorded $\alpha$, it can be
predicted that the $\alpha$ will be detected in the direction of the south-
pole direction by $\Phi^{(lower)}$, later. Instead of the beam preparation by
using two slits, here one uses just one slit and the selection of the (non-
observation, null-measurement) event.

####  3.1.2 $\alpha$ particle tracks in a cloud chamber

Renninger’s process brings us back to one of the oldest “puzzles” in QM: why
does an $\alpha$ particle emitted by a metastable nucleus, described by a
spherically symmetric wave function, (3.5), produce each an (almost) straight-
line track, instead of a sequence of ionization blots distributed all over
$4\pi$ angular directions? The answer has been given by a standard
perturbation analysis made by Mott [22], and we are not here to discuss it
anew.

The reason why we brought up this historical issue here, in spite of little
direct logical connection with the negative-experiment problems, is this. The
observation of an $\alpha$ particle track in a Wilson chamber is a quantum
measurement of the angular distribution of its momentum. It is a measurement
of intermediate type, between the general one ((2.6) and (2.17)), and the
repeatable experiments ((3.15) and (3.16) below). In the latter, special type
of experiments, the state of the microsystem under study remains factorized
and intact as a pure state: the measurement process can be used as a state
preparation.

In the former, more general type of measurements, the microscopic system gets
entangled with the device and with the environment in the process, and the
information about its quantum state gets lost completely, in general, after
the measurement event (see (2.17)).

An $\alpha$ particle track in a cloud chamber starts when $\alpha$ hits the
first atom, ionizing it. The $\alpha$\- electron scattering process is a
spacetime pointlike event, triggering the measurement event. The state of the
$\alpha$ particle is only slightly affected by the $\alpha$-electron
scattering. The large mass ratio, $m_{\alpha}/m_{e}\sim 8000$, means that the
momentum of the $\alpha$ particle is almost unaffected by the
$\alpha$-electron scattering, as is obvious from kinematics and as explicitly
verified by a concrete QM calculations [22], as it proceeds along an almost
straight path, hitting and ionizing a sequence of atoms on its way.

In conclusion, the explanation of the (apparent) wave function collapse,
(2.17), i.e., that a measurement process is effectively a spacetime local
event, is valid also in the intermediate type of measurements, such as the
Mott process, momentum measurements by using the magnetic fields, particle
tracks in the vertex detectors, and so on.

###  3.2 Elitzur-Vaidman bomb tester

A more sophisticated, amusing set-up is the so-called Elitzur-Vaidman bomb
tester experiment [8]. A single photon is sent to Mach-Zehnder interferometry.
See Fig. 1.

A bomb which is either real or fake, is introduced in the lower path. After
the passage through the first half mirror (the lower left corner in Fig. 1)
the original right moving photon $|1\rangle$ is converted to the superposition

$|1\rangle\to|\gamma\rangle=\frac{1}{\sqrt{2}}(|1\rangle+i|2\rangle)\;,$
(3.10)

where $|2\rangle$ is the wave packet of the photon (reflected and) moving
upwards 131313We follow the notation and convention of [8]. We recall that the
photon, upon reflection, acquires a phase shift of $\tfrac{\pi}{2}$. .

In the case the bomb is a fake 141414It is assumed [8] that in that case the
photon passes the region unaffected. , the photon (3.10) goes through the two
fully-silvered mirrors (at the upper-left and lower-right corners of Fig. 1),
and is in the linear superposition,

$|\gamma^{\prime}\rangle=\frac{1}{\sqrt{2}}(i|2\rangle-|1\rangle)\;,$ (3.11)

before entering the second half-silvered, beam splitter (in the upper right
corner in Fig. 1). Going through it, the wave packet $|1\rangle$ gets
transformed as in (3.10), whereas $|2\rangle$ goes into the state,

$|2\rangle\to|\gamma^{\prime\prime}\rangle=\frac{1}{\sqrt{2}}(i|1\rangle+|2\rangle)\;.$
(3.12)

Substituting (3.10) and (3.12) into (3.11), we see that the state of the
photon after the final beam splitter is

$|\gamma^{\prime\prime\prime}\rangle=-|1\rangle\;,$ (3.13)

which is purely right-moving. Due to the interference effects the $|2\rangle$
component coming from the two terms of the original split-photon state (3.10),
has canceled 151515This is one of the beautiful features of the Eitzur-Vaidman
experiment. The interference effect, characteristic of wave aspect of quantum
particles, is here seen in a single photon event. Typically the interference
effects in QM, instead, manifest itself after many identical experiments are
repeated, such as in [23]. . Only the detector $D_{1}$ is triggered (the
photon detected) by $|\gamma^{\prime\prime\prime}\rangle$, accordingly.

In the case it is a real one, the bomb is a detector inserted in the lower,
horizontal section. It is a quantum measurement device to measure the state of
the photon in $|\gamma\rangle$, (3.10). It is however a biased detector,
capable of registering only the photon travelling in the lower path,
$|1\rangle$ 161616It is thus completely analogous to the upper-half detectors
in the Renninger set-up, (3.7), in which the second, lower-hemisphere
detectors are set at a large distance. . There are two possible outcomes for
each incident photon: either detection (explosion), or non detection (no
explosion). In the first case, the photon simply gets lost, and neither the
detector $D_{1}$ nor $D_{2}$ will register the photon.

In the second case - a sort of null measurement - the wave function (3.10) is
reduced as

$|\gamma\rangle\to|2\rangle\;,$ (3.14)

in each such event. The photon gets then reflected by the mirror at the upper-
left corner ($|2\rangle\to i|1\rangle$), and arriving at the final beam
splitter, gets transformed again as in (3.10). It is detected by detector
$D_{1}$ half of the times, and by $D_{2}$ the other half of times, if the
experiment is repeated.

In conclusion, detection of the photon by detector $D_{2}$ implies that the
bomb is real. The interesting point is that we know that this is so, but we
know also that it has not exploded, i.e., the photon has not interacted with
the bomb.

Even though the phenomenon might look quite remarkable, and is certainly not
expected in classical physics, everything follows from the standard QM laws.
If any, as emphasized by Elitzur and Vaidman themselves [8], this process is
interesting as a particular, peculiar manifestation of quantum nonlocality.
The situation here might look rather different from the more familiar examples
of quantum nonlocality associated with entangled pairs of photons, electrons,
etc. Actually, quantum nonlocality manifests itself whenever a microscopic
system is in a pure quantum state with wave function having spatial support of
a macroscopic extension ((3.10) here). As noted in [5], quantum nonlocality is
due to the absence in QM of any fundamental constant with the dimension of a
length.

Figure 1: Elitsur-Vaidman bomb-tester experiment. The photon enters from the
lower left corner to a Mach-Zehnder interferometry. The detection of the
photon at the detector $D_{2}$ implies that the bomb is real, but that the
photon has not interacted with the bomb.

###  3.3 Modified Stern-Gerlach set-up

The process (3.14) - the negative-result event - can also be regarded as a
particular realization of the so-called repeatable experiment. A repeatable
measurement is an exceptional class of experiments in which the microscopic
system under study remains factorized (and intact) after the measurement,
i.e., as

$\left(\sum_{n}c_{n}|n\rangle\right)\otimes|\Phi_{0}\rangle\longrightarrow\sum_{n}c_{n}|n\rangle\otimes|\Phi_{n}\rangle\;,$
(3.15)

or focusing on a single experiment with the result, $F=f_{m}$,

$\left(\sum_{n}c_{n}|n\rangle\right)\to|m\rangle\;.$ (3.16)

A simple example of the repeatable measurement is a variation of the Stern-
Gerlach (SG) experiment. In the standard SG set-up (Fig. 2), an incoming beam
of spin $\frac{1}{2}$ atom (e.g., $A_{g}$), travelling in the $\hat{x}$
direction, is sent into a region of inhomogeneous magnetic field, ${\bf
B}=(0,0,f(z))$ with a strong gradient, $\partial f(z)/dz\neq 0$. The incident
wave packet, is divided into two,

$|\psi\rangle=c_{1}|\\!\uparrow\rangle+c_{2}|\\!\downarrow\rangle\;,\qquad|c_{1}|^{2}+|c_{2}|^{2}=1\;,$
(3.17)

with the spin-up wave packet $|\\!\\!\uparrow\rangle$ is deflected upwards and
the spin down component $|\\!\\!\downarrow\rangle$ downwards, as the atom
proceeds towards $\hat{x}$ direction. On the screen they leave the two groups
of blots whose intensities (the numbers of atoms) are proportional to
$|c_{1}|^{2}:|c_{2}|^{2}$ after many atoms have been registered.

In a possible variation of the Stern-Gerlach set-up (Fig. 3), a detector $D$
is inserted in the region where the lower wave packet passes [21]. The screen
behind the region of the inhomogeneous magnetic field is eliminated. $D$ is
analogous to the set of detectors in the upper-hemisphere in Renninger’s set-
up: it is a biased detector, capable of capturing and recording only the atoms
in the spin-down state. For each single incident atom, $D$ either registers it
(yes) or does not (no). In the first case, the spin has been measured to be in
the state $s_{z}=-\tfrac{1}{2}$, but the atom itself gets lost in the
complicated atom-detector interactions.

In the negative answer case (null measurement), no atom-detector interactions
have taken place; nonetheless its spin state is determined to be
$s_{z}=+\tfrac{1}{2}$. The atom is in the pure $|\\!\\!\uparrow\rangle$ state,
and it can be used as the initial condition for subsequent analyses, for
instance, with another SG set-ups with the magnets oriented in another
direction, etc.

The whole discussion can be readily generalized to the case of atoms with spin
$1$, $\tfrac{3}{2}$, etc. by appropriately enlarging the set of detectors, so
as to extract and prepare the state of any chosen spin state
$|s_{z}=m\rangle$, through interaction-free, null measurements.

The modified SG set-up we considered in this section can thus be seen as a
simple, prototype version of Renninger’s negative-result experiment [21], as
an example of the repeatable measurement, or as a typical “state preparation”
process, illustrating well the fact that these three concepts are closely
related to each other 171717We have already seen a similar connection also in
Sec. 3.1.1. .

Figure 2: The standard SG set-up

Figure 3: The modified SG set-up

##  4 Reflections

It is essential, in all negative-result experiments discussed in Sec. 3, that
a very weak flux of the incident particles is used, such that processes with a
single incident $\alpha$ particle, a photon, or an atom, are studied. Also,
the experimental control must be good enough so that the expected time of
arrival of each particle at the (biased) detector, is known with a reasonable
precision. The reason is that, if it were not so, the non observation of
certain event would not lead to any useful conclusion, as e.g., the particle
may not yet have arrived, or has already passed, or is unknown when it will
arrive, and so on.

In other words, an ideal null-measurement is a spacetime local event, albeit a
virtual (i.e., missed) one 181818In the Renninger experiment, even though the
spontaneous $\alpha$ emission is a spacetime local event [5], the exact
instant $\alpha$ is emitted cannot be predicted, being a manifestation of
quantum fluctuations. This is the reason why one must construct the argument
by considering a lap of time (e.g., $t\leq 30\,\tau$), to make sure that the
nucleus has decayed and the $\alpha$ emission has taken place - with
certainty, $1-O(e^{-30})$. . It represents the other side of the same medal of
the standard quantum measurement processes each of which is a local spacetime
event at its core [5]. This latter fact is the origin of the apparent “wave-
function collapse”, as reviewed in Sec. 2.

Another important reflection is that the discussions of Sec. 3 illustrate
nicely the well known fact about QM, i.e., that the wave-like behavior (the
superposition, quantum nonlocality and interference) is the property of each
single particle (the $\alpha$ particle, the photon and the atom discussed
here, or the electron in Tonomura’s double-slit experiment [23]), and not due
to a collective motion of, or correlation among, the particles in the beam.

The catchphrase “wave-particle duality” was used historically to describe the
apparently schizophrenic behavior of the electrons, photons and atoms. In
hindsight, though, this familiar expression left space for ambiguity and
misunderstanding. For instance, it is an entirely different story that a large
number of identical bosons form collective wavelike motions, such as the
classical electromagnetic waves (light), or Bose-Einstein condensed cold atoms
which are described in terms of a macroscopic wave function. So are all
macroscopic quantum phenomena such as superconductivity and superfluidity,
quantum Hall effects, and so on.

The wave-particle duality, a core idea of QM, is the property of each single
quantum particle 191919Indeed a less poetic but more precise expression would
be “quantum fluctuations of a particle described by the wave function.” The
words “particle” and “wave” do not have the symmetric logical roles. .

A last consideration: in this work we took it for granted that the $\alpha$
particle, the photon, and the silver atom, are all quantum particles. But what
if a large molecule such as $C_{70}$ is used instead? Is it still a quantum
particle? See [14, 24] for significant developments in our understanding of
these questions.

The quest to grasp the very essential factors which can tell quantum-
mechanical particles (the elementary particles, atomic nuclei, atoms, small
molecules, etc.) from classical ones (the center of mass of isolated
macroscopic bodies at finite body temperatures), leads us to the concept of
Quantum Ratio [9].

## Acknowledgments

The author is deeply indebted to Hans Thomas Elze, Marco Matone, Pietro
Menotti and Nobuaki Miyabe, for information and for discussions. This work is
performed under the support of the INFN special project grant GAST (Gauge and
String Theories).

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