# Unitary evolution and elements of reality in consecutive quantum
measurements

D. Sokolovski1,3<EMAIL_ADDRESS>1 Departmento de Química-Física,
Universidad del País Vasco, UPV/EHU, Leioa, Spain  3 IKERBASQUE, Basque
Foundation for Science, Plaza Euskadi 5, 48009 Bilbao, Spain

###### Abstract

Probabilities of the outcomes of consecutive quantum measurements can be
obtained by construction probability amplitudes, thus implying unitary
evolution of the measured system, broken each time a measurement is made. In
practice, the experimenter needs to know all past outcomes at the end of the
experiment, and that requires the presence of probes carrying the
corresponding records. In this picture a composite system+probes can be seen
to undergo an unbroken unitary evolution until the end of the trial, where the
state of the probes is examined. For these two descriptions to agree one
requires a particular type of coupling between a probe and the system, which
we discuss in some details. With this in mind, we consider two different ways
to extend the description of a quantum system’s past beyond what is actually
measured and recorded. One is to look for quantities whose values can be
ascertained without altering the existing probabilities. Such “elements of
reality” can be found, yet they suffer from the same drawback as their EPR
counterparts. The probes designed to measure non-commuting operators frustrate
each other if set up to work jointly, so no simultaneous values of such
quantities can be established consistently. The other possibility is to
investigate the system’s response to weekly coupled probes. Such weak probes
are shown either to reduce to a small fraction the number of cases where the
corresponding values are still accurately measured, or lead only to the
evaluation of the system’s probability amplitudes, or their combinations. It
is difficult, we conclude, to see in quantum mechanics anything other than a
formalism for predicting the likelihoods of the recorded outcomes of actually
performed observations.

###### pacs:

03.65.Ta, 03.65.AA, 03.65.UD

##  I Introduction

In FeynL  Feynman gave a brief yet surprisingly thorough description of
quantum behaviour. Quantum systems are intrinsically stochastic, calculation
of probabilities must rely on complex valued probability amplitudes, and it is
unlikely that one will be able to get a further insight into the mechanism
behind the formalism. One may ask two separate questions about the view
expressed in FeynL . Firstly, is it consistent? There have been recent
suggestions INC1  that quantum mechanics may be self-contradictory, and that
its flaws can be detected from within the theory, i.e., by considering certain
though experiments. In DSwig  we the argued that the proposed “contradictions”
are easily resolved if Feynman’s description is adopted. Secondly, one can ask
if the rules can be explained further? There have been proposals of “new
physics” based on such concepts as time symmetry, weak measurements, and weak
values (see 2time1  \- 2time3 , and Refs. therein). Recently, we have shown
the weak values to be but Feynman’s probability amplitudes, or their
combinations DSw1 , DSw . The ensuing paradoxes occur if the amplitudes are
used inappropriately, e.g., as a proof of the system’s presence at a
particular location DSpar , a practice known for quite some time to be unwise
(see Bohr1  and Ch.6 pp.144-145 of FeynC ). It is probably fair to say that
Feynman’s conclusions have not been successfully challenged to date, and we
will continue to rely on them in what follows.
The approach of FeynL  is particularly convenient for describing situations
where several measurements are made one after another on the same quantum
system. Such consecutive or sequential measurements have been studied by
various authors over a number of years CONSEC1 \- CONSEC5 , and we will
continue to study them here. The simplest case involves just two observations,
of which the first prepares the measured system in a known state, and the
second yields the value of the measured quantity “in that state.” Adding
intermediate measurements between these two significantly changes the
situation, as it brings in a new type of interference the measurements can now
destroy. Below we will discuss two particular issues which arise in the
analysis of such sequential measurements. One is the break down of the unitary
evolution of the measured system, which occurs each time a measurement is
made. Another is the possibility of extending the description of the system
beyond what is actually being measured. This can be done, e.g., by looking for
“elements of reality”, i.e., the properties or values which can be ascertained
without changing anything else in the system’s evolution. This can also be
done by studying a system’s response to weekly coupled inaccurate measuring
devices. It is not our purpose here to dispute the findings made by the
authors using alternative approaches (see, for example, 2time2 ). Rather we we
want to see how the above issues can be addressed in conventional quantum
mechanics, as presented in FeynL .
The rest of the paper is organised as follows. In Sect.II we recall the basic
rules and discuss the broken unitary evolution of the measured system, In
Sect. III we note that, in order to be able to gather the statistics, the
experimenter would need the records of the previous outcomes. The system’s
broken evolution can then be traded for an unbroken unitary evolution of a
composite {system + the probes which carry the records}. In Sect. IV we
discuss two different (and indeed well known) types of the probes. In Sect. V
we discuss the quantities whose additional measurements would not alter the
likelihoods of all other outcomes. However, like their EPR counterparts, these
“elements of reality” cannot be observed simultaneously. In Sect. VI we
illustrate this on a simple two-level example. In Sect. VII we look at what
would happen in an attempt to measure two of such quantities jointly. Sect.
VIII asks if something new can be learnt about the system by minimising the
perturbation incurred by the probes. Sect. IX contains a summary of our
conclusions.

##  II Feynman’s rules of quantum motion. Broken unitary evolutions

Consider a system (S) with which the theory associates $N$-dimensional Hilbert
space $\mathcal{H}_{S}$. The $L+1$ quantities $\hat{Q}^{\ell}$ to be measured
at the times $t_{0}...<t_{\ell}...<t_{L}$ are represented by Hermitian
operators $\hat{Q}^{\ell}$ acting in $\mathcal{H}_{S}$, each with
$M_{\ell}\leq N$ distinct real valued eigenvalues $Q^{\ell}_{m_{\ell}}$

$\displaystyle\hat{Q}^{\ell}=\sum_{m_{\ell}=1}^{M_{\ell}}\sum_{n_{\ell}}^{N}\Delta(Q^{\ell}_{m_{\ell}}-{\langle}q^{\ell}_{n_{\ell}}|\hat{Q}^{\ell}|q^{\ell}_{n_{\ell}}{\rangle})|q^{\ell}_{n_{\ell}}{\rangle}{\langle}q^{\ell}_{n_{\ell}}|\equiv\sum_{m_{\ell}=1}^{M_{\ell}}Q^{\ell}_{m_{\ell}}\hat{\pi}^{\ell}_{m_{\ell}},$
(1)

where $|q^{\ell}_{n_{\ell}}{\rangle}$,
(${\langle}q^{\ell}_{n_{\ell}}|q^{\ell}_{n^{\prime}_{\ell}}{\rangle}=\delta_{n_{\ell}n^{\prime}_{\ell}}$,
$n_{\ell}=1,...N$) are the measurement bases, $\Delta(X-Y)=1$ if $X=Y$, and
$0$ otherwise, and $\hat{\pi}_{m_{\ell}}$ is the projector onto the eigen-
subspace, corresponding to an eigenvalue $Q^{\ell}_{m_{\ell}}$. The first
operator $\hat{Q}^{0}$ is assumed to have only non-degenerate eigenvalues,
i.e.,
$\hat{Q}^{0}=\sum_{n_{0}=1}^{N}Q^{0}_{n_{0}}|q^{0}_{n_{0}}{\rangle}{\langle}q^{0}_{n_{0}}|$.
This is needed to initialise the system, in order to proceed with the
calculation .
The possible outcomes of the experiment are, therefore, the sequences of the
observed values $Q^{L}_{m_{L}}..Q^{0}_{n_{0}}$, and one wishes to predict the
probabilities (frequencies) with which a particular real path
$\\{Q^{L}_{m_{L}}...\leftarrow Q^{\ell}_{m_{\ell}}...\leftarrow
Q^{0}_{n_{0}}\\}$ would occur after many trials. Following FeynL , one can
obtain these obtained by constructing first the system’s virtual paths
$\\{q^{L}_{n_{L}}...\leftarrow q^{\ell}_{n_{\ell}}...\leftarrow
q^{0}_{n_{0}}\\}$, connecting the corresponding states in $\mathcal{H}_{S}$,
and ascribing to each path a probability amplitude (we use $\hbar=1$)

$\displaystyle A(q^{L}_{n_{L}}...\leftarrow q^{\ell}_{n_{\ell}}...\leftarrow
q^{0}_{n_{0}})=\prod_{\ell=0}^{L-1}{\langle}q^{\ell+1}_{n_{\ell+1}}|\hat{U}_{S}(t_{\ell+1},t_{\ell})|q^{\ell}_{n_{\ell}}{\rangle},$
(2)

where
$\hat{U}_{S}(t_{\ell+1},t_{\ell})=\exp[-i\int_{t_{\ell}}^{t_{\ell+1}}\hat{H}_{S}(t^{\prime})dt^{\prime}]$
is the system’s evolution operator [time ordered product is assumed if the
system’s hamiltonian operators $\hat{H}_{S}(t^{\prime})$ do not commute at
different times, $[\hat{H}_{S}(t^{\prime}),\hat{H}_{S}(t^{\prime\prime})]\neq
0$.] We will assume that all Hermitian operators
$\hat{Q}^{\ell}=(\hat{Q}^{\ell})^{\dagger}$ can be measured in this way. We
will also allow for all unitary evolutions,
$\hat{U}^{\dagger}_{S}(t_{\ell+1},t_{\ell})=\hat{U}^{-1}_{S}(t_{\ell+1},t_{\ell})$.
Combining the virtual paths according to the degeneracies of the intermediate
eigenvalues $Q^{\ell}_{m_{\ell}}$, $1\leq\ell\leq L-1$, yields elementary
paths, endowed with both the amplitudes

$\displaystyle A(q^{L}_{n_{L}}...\leftarrow Q^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})=\sum_{n_{1}...n_{L-1}=1}^{N}\prod_{\ell=1}^{L-1}\Delta(Q^{\ell}_{m_{\ell}}-{\langle}q_{n_{\ell}}|\hat{Q}^{\ell}|q_{n_{\ell}}{\rangle})A(q^{L}_{n_{L}}...\leftarrow
q^{\ell}_{n_{\ell}}...\leftarrow q^{0}_{n_{0}}).\quad$ (3)

and the probabilities,

$\displaystyle p(q^{L}_{n_{L}}...\leftarrow Q^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})=|A(q^{L}_{n_{L}}...\leftarrow Q^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})|^{2}$ (4)

We note that the amplitudes in Eq.(3) depend only on the projectors
$\hat{\pi}^{\ell}_{m_{\ell}}$ in Eq.(2) , and not on the corresponding
eigenvalues $Q^{\ell}_{m_{\ell}}$. To stress this, we are able to write

$\displaystyle A(q^{L}_{n_{L}}...\leftarrow Q^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})=A(q^{L}_{n_{L}}...\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}}).$ (5)

Finally, summing $p(q^{L}_{n_{L}}...\leftarrow
Q^{\ell}_{m_{\ell}}...\leftarrow q^{0}_{n_{0}})$ over the degeneracies of the
last operator $\hat{Q}^{L}$, yields the desired probabilities for the real
paths,

$\displaystyle P(Q^{L}_{m_{L}}...\leftarrow Q^{\ell}_{m_{\ell}}...\leftarrow
Q^{0}_{n_{0}})=\sum_{n_{L}}^{N}\Delta(Q^{L}_{m_{L}}-{\langle}q^{L}_{n_{L}}|\hat{Q}^{L}|q^{L}_{n_{L}}{\rangle})p(q^{L}_{n_{L}}...\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}}).$ (6)

Note that there is no interference between the paths leading to different
(i.e., orthogonal) final states $|q^{L}_{n_{L}}{\rangle}$, even if they
correspond to the same eigenvalue $Q^{L}_{m_{L}}$ FeynL . This is necessary,
since an additional ($L+2$)-nd measurement of an operator
$\hat{Q}^{L+1}=\sum_{n_{L}=1}^{N}Q^{L+1}_{n_{L+1}}|q^{L}_{n_{L+1}}{\rangle}{\langle}q^{L}_{n_{L+1}}|$
immediately after $t=t_{L}$ would destroy any interference between the paths
ending in different $|q^{L}_{n_{L}}{\rangle}$s at $t=t_{L}$. Since future
measurements are not supposed to alter the results already obtained, one never
adds the amplitudes for the final orthogonal states FeynL . Note that the same
argument cannot be repeated for the past measurements at $t_{\ell}<t_{L}$.
It may be convenient to cast Eq.(6) in an equivalent form,

$\displaystyle P(Q^{L}_{m_{L}}...\leftarrow Q^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})={\langle}\Phi(\hat{\pi}^{L}_{m_{L}}...\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}}|\Phi(\hat{\pi}^{L}_{m_{L}}...\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}}){\rangle},\quad$ (7)

where

$\displaystyle|\Phi(\hat{\pi}^{L}_{m_{L}}...\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}}{\rangle}=\prod_{\ell=1}^{L}\hat{\pi}^{\ell}_{m_{\ell}}(t_{\ell},t_{0})|q^{0}_{n_{0}}{\rangle},\quad\hat{\pi}^{\ell}_{m_{\ell}}(t_{\ell},t_{0})\equiv\hat{U}_{S}^{-1}(t_{\ell},t_{0})\hat{\pi}^{\ell}_{m_{\ell}}\hat{U}_{S}(t_{\ell},t_{0}),\quad$
(8)

and a unitary evolution of the initial state $|q^{0}_{n_{0}}{\rangle}$ with
the system’s evolution operator $\hat{U}_{s}$ is seen to be interrupted at
each $t=t_{\ell}$. check that the probabilities in Eq.(6) sum, as they should,
to unity.
It is worth bearing in mind the Uncertainty Principle which, we recall, states
that FeynL  “one cannot design equipment in any way to determine which of two
alternatives is taken, without, at the same time, destroying the pattern of
interference”. In particular, this means that if two or more virtual paths in
Eq.(2) are allowed to interfere, it must be absolutely impossible to find out
which one was followed by the system. Moreover, one will not even able to say
that, in a given trial, one of them was followed, while the others were not
(see Bohr1  and Ch.6 pp.144-145 of FeynC ).
With the basic rules laid out, and an example given in Fig.1, we will turn to
practical realisations of an experiment involving several consecutive
measurements of the kind just described.

Figure 1: Three measurements, $L=3$, are made on a three-level system, $N=3$.
The first one, yields an outcome $Q^{0}_{1}$ and prepares the system in a
state $|q^{0}_{1}{\rangle}$. Two other operators have degenerate eigenvalues,
$\hat{Q}^{1}|q^{1}_{1,2}{\rangle}=Q^{1}_{1}|q^{1}_{1,2}{\rangle}$,
($M_{1}=2$), and
$\hat{Q}^{2}|q^{2}_{2,3}{\rangle}=Q^{2}_{2}|q^{1}_{2,3}{\rangle}$,
($M_{2}=2$). a) Nine virtual paths in Eq.(3). b) Four real paths (i.e., the
observed sequences $Q^{2}_{m_{2}}\leftarrow Q^{1}_{m_{1}}\leftarrow
Q^{0}_{1}$), $m_{1},m_{2}=1,2$. Different colours are used to relate the
virtual paths to the observed outcomes.

##  III The need for records. Unbroken unitary evolutions

In an experiment, described in Sect. II, there are $N\times M_{1}\times
M_{2}.....\times M_{L}$ possible sequences of observed outcomes. At the end of
each trial, the experimenter identifies the real path followed by the system,
$path=\\{Q^{L}_{m_{L}}...\leftarrow Q^{\ell}_{m_{\ell}}...\leftarrow
Q^{0}_{n_{0}}\\}$, and increases by $1$ the count in the corresponding part of
his inventory, $K(path)\to K(path)+1$. After $K>>1$ trials, the ratios
$K(path)/K$ will approach the probabilities in Eq.(6), from which all the
quantities of interest, such as averages or correlations, can be obtained
later.
There is one practical point. In order to identify the path, an Observer must
have readable records of all past outcomes once the experiment is finished,
i.e., just after $t=t_{L}$. There are two reasons for that. Firstly, quantum
systems are rarely visible to the naked eye, so something accessible to the
experimenter’s senses is clearly needed. Secondly, and more importantly, the
condition of the system changes throughout the process [cf. Eq.(8)], and its
final state simply cannot provide all necessary information. In other words,
one requires $L$ probes which copy the system’s state at $t=t_{\ell}$,
$\ell=0,1,...L$ and retain this information till the end of the trial. It is
easy to see what such probes must do. The experiment begins by coupling the
first probe to a previously unobserved system at $t=t_{0}$. To proceed with
the calculation, we may assume that just $t_{0}$ the initial state of a
composite $system+probes$ is

$\displaystyle|\Psi_{S+Probes}(0){\rangle}=|q^{0}_{n_{0}}{\rangle}|D^{0}(n_{0}){\rangle}...|D^{\ell}(0){\rangle}...|D^{L}(0){\rangle}\equiv|q^{0}_{n_{0}}{\rangle}|\Psi_{Probes}(0){\rangle}$
(9)

where $|D^{\ell}(0){\rangle}$ is the initial state of the $\ell$-th probe
which, if found changed into $|D^{\ell}(m_{\ell}){\rangle}$,
${\langle}D^{\ell}(m_{\ell})|D^{\ell}(m^{\prime}_{\ell}){\rangle}=\delta_{m_{\ell}m^{\prime}_{\ell}}$,
would tell the experimenter that the outcome at $t=t_{\ell}$ was
$Q^{\ell}_{m_{\ell}}$. Note that the first probe $D^{0}$ has already been
coupled to a previously unobserved system and produced a reading $n_{0}$, thus
preparing the system in a state $|q^{0}_{n_{0}}{\rangle}$.
The composite would undergo unitary evolution with an (yet unknown) evolution
operator $\hat{U}_{S+Probes}(t_{L},t_{0})$. The rules of the previous Section
still apply, albeit in a larger Hilbert space, and with only two ($L=1$)
measurements, of which the first one prepares the entire composite in the
state (9). For simplicity, we let the last operator have non-degenerate
eigenvalues, $M_{L}=N$. By (6), the probability to have an outcome
$Q^{L}_{n_{L}}...\leftarrow Q^{\ell}_{m_{\ell}}...\leftarrow Q^{0}_{n_{0}}$ is

$\displaystyle\tilde{P}(Q^{L}_{n_{L}}...\leftarrow
Q^{\ell}_{m_{\ell}}...\leftarrow
Q^{0}_{n_{0}})=\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
(10)
$\displaystyle\sum_{n^{\prime}_{L}=1}^{N}\left|{\langle}q^{L}_{n^{\prime}_{L}}|{\langle}\Psi_{Probes}(n_{L},...m_{\ell}...n_{0})|\hat{U}_{S+Probes}(t_{L},t_{0})|\Psi_{S+Probes}(0){\rangle}\right|^{2}$

where

$\displaystyle|\Psi_{Probes}(n_{L},...m_{\ell}...n_{0}){\rangle}=|D^{L}(n_{L}){\rangle}\prod_{\ell=1}^{L-1}|D^{\ell}(m_{\ell}){\rangle}|D^{0}(n_{0}){\rangle}$
(11)

We want the probabilities in Eq.(10) (the ones the experimenter measures) and
the probabilities in Eq.(6) (the ones the theory predicts) to agree. Consider
again the scenarios $\\{q^{L}_{n_{L}}...\leftarrow
Q^{\ell}_{n_{\ell}}...\leftarrow q^{0}_{n_{0}}\\}$ in Eq.(3). In the absence
of the probes they lead to the same final state, $|q^{L}_{n_{L}}{\rangle}$,
interfere, and cannot be told apart, according to the Uncertainty Principle.
If we could use the probes to turn these scenarios into exclusive alternatives
FeynL , e.g., by directing them to different (orthogonal) final states in the
larger Hilbert space, Eq.(6) for the system subjected to $L+1$ measurements
would follow. In other words, we will be able to trade a broken evolution in a
smaller space $\mathcal{H}_{S}$ [cf. Eq.(8)] for an uninterrupted unitary
evolution in a larger Hilbert space $\mathcal{H}_{S+Probes}$. For this we need
an evolution operator $\hat{U}_{S+Probes}(t_{L},t_{0})$ such that

$\displaystyle{\langle}q^{L}_{n^{\prime}_{L}}|\hat{U}_{S+Probes}(t_{L},t_{0})|\Psi_{S+Probes}(0){\rangle}=\delta_{n_{L}^{\prime}n_{L}}\times\quad\quad\quad\quad\quad$
(12)
$\displaystyle\sum_{m_{1}....m_{L-1}=1}^{M_{1}...M_{L-1}}A_{S}(q^{L}_{n_{L}}...\leftarrow
Q^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})|\Psi_{Probes}(n_{L},...m_{\ell}...n_{0}){\rangle},$

where the orthogonal states
$|\Psi_{Probes}(n_{L},...m_{\ell}...n_{0}){\rangle}$ play the role of “tags”,
by which previously interfering paths $\\{q^{L}_{n_{L}}...\leftarrow
Q^{\ell}_{m_{\ell}}...\leftarrow q^{0}_{n_{0}}\\}$ can now be distinguished.
For the reader worried about the collapse of the wave function we note that
the same probabilities can be obtained in two different ways. Either the
evolution of the wave function of the system only is broken every time an
instantaneous measurement is made, as happens in Eq.(8), or the evolution of
the system + the probes continues until the end of the experiment as in
Eq.(12).
Finally, we note the difference between producing all $L$ records, but not
using or having no access to some of them, and not producing some of the
records at all. There is also a possibility of destroying, say, the $\ell$-th
record by making a later measurement on a composite {the system+ the $\ell$-th
probe} DSm , DSob . In this case the composite becomes the new measured
system, and the rules of the previous Section still apply.

Figure 2:  The measurements in Fig.1 seen from a different prospective. Just
after $t=t_{2}$, the experimenter needs to compare three records, in order to
determine which of the real paths in Fig.1b was actually taken. This
information is encoded in the final conditions of the three probes,
$D^{\ell}$, $\ell=0,1,2$. The composite $\\{system+probes\\}$ undergoes an
unbroken unitary evolution for $t_{0}\leq t\leq t_{2}$. There are nine virtual
paths ending in distinguishable states of the composite. The same colours are
used to indicate which of the nine path probabilities should be added to
obtain likelihoods of the four real scenarios in Fig.1b.

##  IV Two kinds of probes

We note next that does not really matter for the theory how exactly the
records are produced, as long as the interference between the virtual paths is
destroyed, and Eq.(12) is satisfied. The states $|D^{\ell}{\rangle}$ in Eq.(9)
may equally refer to devices, to Observer’s or Observers’ memories, or to the
notes the Observers have made in the course of a trail DSm , DSob . We will
assume for simplicity that the probes have no own dynamics, an retain their
states after having interacted with the measured system,

$\displaystyle\hat{H}_{S+Probes}=\hat{H}_{S}+\hat{H}_{int},\quad\hat{H}_{Probes}=0.$
(13)

Several interactions which have the desired effect are, in fact, well known,
and we will discuss them next. There are at least two types of probes
consistent with Eq.(12). They require different treatments, and we will
consider them separately.

###  IV.1 Discrete gates

For the $\ell$-th probe consider a register of $M_{\ell}$ two-level sub-
systems, each prepared in its lower states $|1_{m_{\ell}}{\rangle}$

$\displaystyle|D^{\ell}(0){\rangle}=\prod_{m_{\ell}=1}^{M_{\ell}}|1_{m_{\ell}}{\rangle}.$
(14)

The probe, designed to measure a quantity
$\hat{Q}^{\ell}=\sum_{m_{\ell}=1}^{M_{\ell}}Q^{\ell}_{m_{\ell}}\hat{\pi}^{\ell}_{m_{\ell}}$,
is coupled to the system via

$\displaystyle\hat{H}_{int}^{\ell}=-(\pi/2)\sum_{m_{\ell}=1}^{M_{\ell}}\hat{\pi}^{\ell}_{m_{\ell}}\hat{\sigma}^{\ell}_{x}(m_{\ell})\delta(t-t_{\ell}),$
(15)

where $\hat{\sigma}^{\ell}_{x}(m_{\ell})$ is the Pauli matrix, which acts on
the $m_{\ell}$-th sub-system in the usual way,
$\hat{\sigma}^{\ell}_{x}(m_{\ell})|1_{m_{\ell}}{\rangle}=|2_{m_{\ell}}{\rangle}$.
Since the individual terms in Eq.(15) commute, the evolution operator of the
$\\{System+\ell-{th}\quad Probe\\}$ over a short interval
$[t_{\ell}-\epsilon,t_{\ell}+\epsilon]$, $\epsilon\to 0$ is

$\displaystyle\hat{U}^{\ell}_{int}(t_{\ell})=\exp\left[i(\pi/2)\sum_{m_{\ell}=1}^{M_{\ell}}\hat{\pi}^{\ell}_{m_{\ell}}\hat{\sigma}^{\ell}_{x}(m_{\ell})\right]=i\sum_{m_{\ell}=1}^{M_{\ell}}\hat{\pi}^{\ell}_{m_{\ell}}\sigma_{x}(m_{\ell}).$
(16)

The probe entangles with the system in the required way,

$\displaystyle\hat{U}^{\ell}_{int}(t_{\ell})|\psi_{S}{\rangle}|D^{\ell}(0){\rangle}=i\sum_{m_{\ell}=1}^{M_{\ell}}\hat{\pi}^{\ell}_{m_{\ell}}|\psi_{S}{\rangle}|D^{\ell}(m_{\ell}){\rangle},$
(17)

where in $|D^{\ell}(m_{\ell}){\rangle}$ is obtained from
$|D^{\ell}(0){\rangle}$ by flipping the state of the $m_{\ell}$-th sub-system,

$\displaystyle|D^{\ell}(m_{\ell}){\rangle}\equiv|2_{m_{\ell}}{\rangle}\prod_{k_{\ell}\neq
m_{\ell}}|1_{k_{\ell}}{\rangle}.$ (18)

We note that, whatever the state $|\psi_{S}{\rangle}$, one of the subsystems
will change its condition (the system will be found somwhere). We note also
that in each trial only one subsystem will be affected (the system is never
found simultaneously in two or more places). The full evolution operator is,
therefore, given by

$\displaystyle\hat{U}_{S+Probes}(t_{L},t_{0})=\hat{U}^{\ell}_{int}(t_{L})\prod_{\ell=0}^{L-1}\hat{U}_{S}(t_{\ell+1},t_{\ell})\hat{U}^{\ell}_{int}(t_{\ell})$
(19)

where, as before, we we assumed $M_{0}=M_{L}=N$,
$\hat{\pi}_{n_{0}}=|q_{n_{0}}{\rangle}{\langle}q_{n_{0}}|$, and
$\hat{\pi}_{n_{L}}=|q_{n_{L}}{\rangle}{\langle}q_{n_{L}}|$.
The experimenter prepares all probes in the states (14) and, once the
experiment is finished, only needs to check which sub-system of $D^{\ell}$,
say, the $m_{\ell}$-th, has changed its state. This will tell him/her that the
value of $\hat{Q}^{\ell}$ at $t=t_{\ell}$ was $Q^{\ell}_{m_{\ell}}$. As a
simple example, Fig.3 shows an outcome of five measurements made on a four-
state system. There the first probe, capable of distinguishing between all
four states prepares the system in a state $|q^{0}_{n_{0}{\rangle}}$. The
second probe cannot tell apart the third and the second states, so
$\hat{\pi}^{1}_{1}=|q^{1}_{1}{\rangle}{\langle}q^{1}_{1}|$,
$\hat{\pi}^{1}_{2}=|q^{1}_{2}{\rangle}{\langle}q^{1}_{2}|$, and
$\hat{\pi}^{1}_{3}=|q^{1}_{3}{\rangle}{\langle}q^{1}_{3}|+|q^{1}_{4}{\rangle}{\langle}q^{1}_{4}|$,
and so on. The sequence of the measured valued obtained by inspecting the
probes at $t>t_{4}$ is $Q^{4}_{4}\leftarrow Q^{3}_{2}\leftarrow
Q^{2}_{2}\leftarrow Q^{1}_{3}\leftarrow Q^{0}_{1}$. After many trials the
sequence will be observed with a probability $P(Q^{4}_{4}\leftarrow
Q^{3}_{2}\leftarrow Q^{2}_{2}\leftarrow Q^{1}_{3}\leftarrow
Q^{0}_{1})=|A_{S}(q^{4}_{4}\leftarrow\hat{\pi}^{3}_{2}\leftarrow\hat{\pi}^{2}_{2}\leftarrow\hat{\pi}^{1}_{3}\leftarrow
q^{0}_{1})|^{2}$ [cf. Eq.(5)]

Figure 3: Five consecutive measurements of the quantities
$\hat{Q}^{0},\hat{Q}^{1}...\hat{Q}^{4}$ are made on a four-level system
($N=4)$. Some of the eigenvalues are degenerate. Each probe consists of
$M_{\ell}\leq 4$ [cf. Eq.(15)] two-level sub-systems. a) Initially sub-systems
of all probes are prepared in their lower states $|1{\rangle}$. b) At the end
of a trial some these states are found changed, and a record
$\\{Q^{4}_{4}\leftarrow Q^{3}_{2}\leftarrow Q^{2}_{2}\leftarrow
Q^{1}_{3}\leftarrow Q^{0}_{1}\\}$ is produced.

###  IV.2 Von Neumann’s pointers

In classical mechanics one can measure the value of a dynamical variable
$Q(x,p)$ at $t_{0}$ by coupling the system to a “pointer”, a heavy one-
dimensional particle with position $f$ and momentum $\lambda$. The full
Hamiltonian is given by $H_{S}(x,p)+\lambda Q(x,p)\delta(t-t_{0})$, and at
$t=t_{0}$ the pointer is rapidly displaced by $\delta f=Q(x(t_{0}),p(t_{0}))$,
which providies the desired reading. What happens to the system, depends on
the pointer’s momentum, which remains unchanged by the interaction. If
$\lambda=0$, the system continues of its way unperturbed. If $\lambda\neq 0$,
the system experiences a sudden kick, whereby its position and momentum are
changed by $\Delta x=\lambda\partial_{p}Q(x(t_{0}),p(t_{0}))$ and $\Delta
p=\lambda\partial_{x}Q(x(t_{0}),p(t_{0}))$, respectively.
The quantum version of the pointer vN  employs a coupling
$\hat{H}_{int}=g(t)\hat{\lambda}\hat{Q}$, where $\hat{\ell}$,
${\langle}f^{\prime}|\hat{\lambda}|f{\rangle}=-i\delta(f-f^{\prime})\partial_{f}$
is the pointer’s momentum operator, and $\hat{Q}=\sum_{m}Q_{m}\hat{\pi}_{m}$
is the (system’s) operator to be measured. The function $g(t)=1/\tau$ can be
chosen constant for the duration of the measurement $\tau$, $t_{0}\leq t\leq
t_{0}+\tau$, and zero otherwise. It tends to a Dirac delta $\delta(t-t_{0})$
for an instantaneous (impulsive) measurement, where $\tau\to 0$. For a system
whose state $|\psi_{S}{\rangle}$ lies in the eigen sub-space of a projector
$\hat{\pi}_{m}$, the action of $\hat{H}_{int}$ results in a spatial shift of
the pointer’s initial state $|G(0){\rangle}$ by $Q_{m}$,

$\displaystyle\exp(-i\hat{H}_{int}\tau)|\psi_{s}{\rangle}|G(0){\rangle}=|\psi_{s}{\rangle}\int
G(f-Q_{m})|f{\rangle}df\equiv|\psi_{s}{\rangle}|G(m){\rangle}$ (20)

With $L+1$ pointers employed to measure $L+1$ quantities $\hat{Q}^{\ell}$, the
initial state of the composite can be chosen to be [cf. Eq.(9)]

$\displaystyle|\Psi_{S+Pointers}(0){\rangle}=|q^{0}_{n_{0}}{\rangle}|G^{0}(n_{0}){\rangle}...|G^{\ell}(0){\rangle}...|G^{L}(0){\rangle},$
(21)

where the initial pointer states can be, e.g., identical Gaussians of a width
$\Delta f$, all centred at the origin,

$\displaystyle{\langle}f_{\ell}|G^{\ell}(0){\rangle}=C\exp(-f_{\ell}^{2}/\Delta
f^{2})\equiv G^{\ell}(f_{\ell}),\quad C_{1}=[2/\pi\Delta f^{2}]^{1/4},$ (22)

except for the first probe, where we would need a narrow Gaussian,
$|G^{0}(f_{0})|^{2}\to\delta(f)$ in order to prepare the system in
$|q^{0}_{n_{0}}{\rangle}$. If all couplings are instantaneous, for the
amplitude in Eq.(3) [with $L=2$, and $|q^{L}_{n_{L}}{\rangle}$ replaced by a
state of the composite,
$|q^{L}_{n_{L}}{\rangle}\otimes|\overline{f}{\rangle}$,
$|\overline{f}{\rangle}\equiv|f_{0}{\rangle}\otimes|f_{1}{\rangle}...\otimes|f_{L}{\rangle}$]
one finds

$\displaystyle
A_{S+Pointers}(\overline{f},q^{L}_{n^{\prime}_{L}}\leftarrow\Psi_{0})={\langle}\overline{f}|{\langle}q^{L}_{n^{\prime}_{L}}|\hat{U}_{S+Pointers}(t_{L},t_{0})|\Psi_{S+Pointers}(0){\rangle}=\quad\quad\quad\quad\quad\quad\quad\quad\quad$
(23) $\displaystyle\delta_{n_{L}^{\prime}n_{L}}\times
G^{L}(f_{L}-Q_{n_{L}})\sum_{m_{1}...m_{L-1}=1}^{M_{1}...M_{L-1}}\prod_{\ell=1}^{L-1}G^{\ell}(f_{\ell}-Q_{m_{L}})G^{0}(f_{0}-Q_{n_{0}})A_{S}(q^{L}_{n_{L}}...\leftarrow
Q^{\ell}_{m_{\ell}}...\leftarrow q^{0}_{n_{0}})\quad\quad$

where, again, $A_{S}(q^{L}_{n_{L}}...\leftarrow
Q^{\ell}_{n_{\ell}}...\leftarrow Q^{0}_{n_{0}})$ is the system’s amplitude
(3).
If one wants his/her measurements to be accurate, the pointers need to be set
to zero with as little uncertainty as possible. This uncertainty is determined
by the Gaussian’s width $\Delta f$, and sending it to zero we have [since
$G^{\ell}(f_{\ell})^{2}\to\delta(f_{\ell})$ and
$G^{\ell}(f_{\ell})G^{\ell}(f_{\ell}-X)=0$ for any $X\neq 0$]

$\displaystyle
P_{Pointers}(f_{L}...f_{1},f_{0})\equiv\sum_{n_{L}}|A_{S+Pointers}(\overline{f},q^{L}_{n_{L}}\leftarrow\Psi_{0})|^{2}=$
(24)
$\displaystyle\sum_{m_{1}...m_{L}=1}^{M_{1}...M_{L}}\prod_{\ell=0}^{L}\delta(f_{\ell}-Q^{\ell}_{m_{\ell}})P_{S}(Q^{L}_{m_{L}}...\leftarrow
Q^{\ell}_{m_{\ell}}...\leftarrow Q^{0}_{n_{0}}),$

where $P_{S}(Q^{L}_{m_{L}}...\leftarrow Q^{\ell}_{m_{\ell}}...\leftarrow
Q^{0}_{n_{0}})$ is the probability, computed with the help of Eq.(7).

Figure 4:  The measurements shown in Fig.3, this time made by employing five
accurate von Neumann pointers. a) Initially the pointers are set to zero. b)
At the end of the trial one finds each pointer shifted by the corresponding
eigenvalue. As in Fig.3, an outcome $\\{Q^{4}_{4}\leftarrow
Q^{3}_{2}\leftarrow Q^{2}_{2}\leftarrow Q^{1}_{3}\leftarrow Q^{0}_{1}\\}$ is
recorded.

Equation (24) is clearly the desired result, which deserves a brief
discussion. In each trial the pointers’ readings may take only discrete values
$Q^{\ell}_{m_{\ell}}$, and the observed sequences occur with the
probabilities, predicted for the system by Feynman’s rules of Sect.II.
However, unlike in the classical case, this information comes at the cost of
perturbing the system’s evolution. Indeed, writing $G^{\ell}(0)=\int
G^{\ell}(\lambda_{\ell})\exp(i\lambda_{\ell}f_{\ell})d\lambda_{\ell}$ and
proceeding as before, one obtains terms like
$\prod_{\ell=1}^{L-1}\exp(-i\lambda_{\ell}\hat{Q}^{\ell})\hat{U}_{S}(t_{\ell},t_{\ell-1})$,
where $\exp(-i\lambda_{\ell}\hat{Q}^{\ell})$ represents the “kick”, produced
on the system by the $\ell$-th pointer at $t_{\ell}$. As in the classical
case, we can get rid of the kick by ensuring that the pointer’s momentum
$\lambda_{\ell}$ is approximately zero. But, by the Heisenberg’s uncertainty
principle (see, e.g., FeynL ) , this will make the uncertainty in the initial
pointer position very large. Accuracy and perturbation go hand in hand, and
the measured values do not “pre-exist” measurements, but are produced in the
course of it Merm . Notably, one can still predict the probabilities by not
mentioning the pointers at all, and analysing instead an isolated system,
whose unitary evolution is broken each time the coupling takes place.
Secondly, and importantly, von Neumann pointers have many states, and only few
of them are actually used. This suggests that the pointers and the probes of
the previous Section could, in principle, be replaced by much more complex
devices, with only a few states of their vast Hilbert spaces coming into play.
For example, there is nothing in quantum theory which forbids using printers,
which print the observed values on a piece of paper. If an experiment which
measures spin’s component is set properly, the machine will print only ”up” or
”down” with the predicted frequencies, and would never digress into French
romantic poetry.

##  V The past of a quantum system. Elements of reality.

The stock of an experiment, described so far, is taken just after the last
measurement at $t=t_{L}$. This is the “present” moment, the times
$t_{0},...t_{L}$ are relegated to the “past”, and the “future” is yet unknown.
Possible pasts are defined by the choice of the measured quantities
$\hat{Q}^{\ell}$, and of the times $t_{\ell}$ at which the impulsive
measurements are performed. The $N\times M_{1}\times M_{2}.....\times M_{L}$
possible outcomes $\\{Q^{L}_{m_{L}}...\leftarrow
Q^{\ell}_{m_{\ell}}...\leftarrow Q^{0}_{n_{0}}\\}$ occur with probabilities
$P(Q^{L}_{m_{L}}...\leftarrow Q^{\ell}_{m_{\ell}}...\leftarrow Q^{0}_{n_{0}})$
which the theory aims to predict. There are clearly gaps in the description of
the system between successive measurements at $t_{\ell}$ and $t_{\ell+1}$. One
way to fill them (without adding new measurements, which would change the
problem) is to look for quantities whose values can be ascertained at some
$t_{\ell}<t^{\prime}<t_{\ell+1}$ without altering the existing probabilities
$P(Q^{L}_{m_{L}}....\leftarrow Q^{\ell+1}_{m_{\ell+1}}\leftarrow
Q^{\ell}_{m_{\ell}}...\leftarrow Q^{0}_{n_{0}})$. Or, to put it slightly
differently, to ask what can be measured without destroying the interference
between the virtual paths [cf. (2)] which contribute to the amplitudes
$A_{S}(Q^{L}_{m_{L}}...\leftarrow Q^{\ell+1}_{m_{\ell+1}}\leftarrow
Q^{\ell}_{m_{\ell}}...\leftarrow Q^{0}_{n_{0}})$ in Eq.(3).
There is a well known analogy. EPR-like scenarios EPR1  are often used to
question the manner in which quantum theory describes the physical world. In a
nutshell the argument goes is as follows. Alice and Bob, at two separate
locations, share an entangled pair of spins. Alice can ascertain that Bob’s
spin has any desired direction, while apparently unable to influence it due to
the restrictions imposed by special relativity. Hence, all possible values of
the spin’s projections can exist simultaneously, i.e., be in some sense real.
If quantum mechanics insists that different projections cannot have well
defined values at the same time, it must be incomplete. We are not interested
here in the details of this important ongoing discussion (for a overview see
EPR1  and Refs. therein), or the implications relativity theory may have for
elementary quantum mechanics Peres . Rather we want to make use of the
Criterion of Reality (CR) used by the authors of EPR  to determine what should
be considered “real”. This criterion reads: “If, without in any way disturbing
a system, we can predict with certainty (i.e., with probability equal to
unity) the value of a physical quantity, then there exists an element of
reality corresponding to that quantity.” EPR .

Consider again an experiment in which $L+1$ measurements are made on the
system at $t=t_{\ell}$, $\ell=0,1...L$, while the system’s condition at some
$t^{\prime}$ between, say, $t_{\ell}$ and $t_{\ell+1}$ remains unknown. To
fill this gap, one may use the CR criterion just cited, and look for any
information about the system, which can be obtained without altering the
existing statistical ensemble. Thus, one needs a variable $\hat{Q}^{\prime}$
whose measurement at $t_{\ell}<t^{\prime}<t_{\ell+1}$ results is

$\displaystyle A_{S}(q^{L}_{n_{L}}...\leftarrow
Q^{\ell}_{m_{\ell+1}}\leftarrow Q^{\prime}_{m^{\prime}}\leftarrow
Q^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})=A_{S}(q^{L}_{n_{L}}...\leftarrow
Q^{\ell+1}_{m_{\ell+1}}\leftarrow Q^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}}).$ (25)

There are at least two kinds of quantities that satisfy this condition. To the
first kind belong operators of the type

$\displaystyle\hat{Q}^{-}(t^{\prime})=\sum_{m_{-}}Q^{-}_{m_{-}}\hat{\pi}^{-}_{m_{-}}\equiv\hat{U}_{S}(t^{\prime},t_{\ell})\hat{Q}^{\ell}\hat{U}^{-1}_{S}(t^{\prime},t_{\ell})=\sum_{m_{\ell}}Q^{\ell}_{m_{\ell}}\hat{\pi}_{m_{\ell}}(t_{\ell},t^{\prime}),$
(26)

where
$\hat{\pi}_{m_{\ell}}(t_{\ell},t^{\prime})=\hat{U}_{S}(t^{\prime},t_{\ell})\hat{\pi}_{m_{\ell}}\hat{U}^{-1}_{S}(t^{\prime},t_{\ell})$
is the projector $\hat{\pi}_{m_{\ell}}$ evolved backwards in time from
$t^{\prime}$ to $t_{\ell}$. To the second kind belong the quantities

$\displaystyle\hat{Q}^{+}(t^{\prime})=\sum_{m_{+}}Q^{+}_{m_{+}}\hat{\pi}^{+}_{m_{+}}\equiv\hat{U}^{-1}_{S}(t_{\ell+1},t^{\prime})\hat{Q}^{\ell+1}\hat{U}_{S}(t_{\ell+1},t^{\prime})\equiv\sum_{m_{\ell+1}}Q^{\ell+1}_{m_{\ell+1}}\hat{\pi}_{m_{\ell+1}}(t_{\ell+1},t^{\prime})$
(27)

where
$\hat{\pi}_{m_{\ell+1}}(t_{\ell+1},t^{\prime})=\hat{U}^{-1}_{S}(t_{\ell+1},t^{\prime})\hat{\pi}_{m_{\ell+1}}\hat{U}_{S}(t_{\ell+1},t^{\prime})$
is the projector $\hat{\pi}_{m_{\ell+1}}$ evolved forwards in time from
$t^{\prime}$ to $t_{\ell+1}$. Indeed, since
$\hat{\pi}_{m_{\ell}}\hat{\pi}_{m^{\prime}_{\ell}}=\hat{\pi}_{m_{\ell}}\delta_{m_{\ell}m^{\prime}_{\ell}}$
and
$\hat{\pi}_{m_{\ell+1}}\hat{\pi}_{m^{\prime}_{\ell+1}}=\hat{\pi}_{m_{\ell+1}}\delta_{m_{\ell+1}m^{\prime}_{\ell+1}}$,
we have

$\displaystyle\hat{\pi}_{m_{\ell+1}}\hat{U}_{S}(t_{\ell+1},t^{\prime})\hat{\pi}_{m_{\ell}}(t_{\ell},t^{\prime})\hat{U}_{S}(t^{\prime},t_{\ell})\hat{\pi}_{m_{\ell}}=\hat{\pi}_{m_{\ell+1}}\hat{U}_{S}(t_{\ell+1},t^{\prime})\hat{\pi}_{m_{\ell+1}}(t_{\ell+1},t^{\prime})\hat{U}_{S}(t^{\prime},t_{\ell})\hat{\pi}_{m_{\ell}}$
(28)
$\displaystyle=\hat{\pi}_{m_{\ell+1}}\hat{U}_{S}(t_{\ell+1},t_{\ell})\hat{\pi}_{m_{\ell}},\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

and [cf. Eq.(6)]

$\displaystyle P(Q^{L}_{m_{L}}...\leftarrow Q^{\ell+1}_{m_{\ell+1}}\leftarrow
Q^{\ell}_{m_{\ell}}...\leftarrow q^{0}_{n_{0}})=P(Q^{L}_{m_{L}}...\leftarrow
Q^{\ell+1}_{m_{\ell+1}}\leftarrow Q^{-}_{m_{\ell}}(t^{\prime})\leftarrow
Q^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})=\quad\quad\quad\quad\quad\quad$ (29) $\displaystyle
P(Q^{L}_{m_{L}}...\leftarrow Q^{\ell+1}_{m_{\ell+1}}\leftarrow
Q^{+}_{m_{\ell+1}}(t^{\prime})\leftarrow Q^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}}).\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$

There is, of course, a simple explanation. The states
$\hat{U}_{S}(t^{\prime},t_{\ell})|q^{\ell}_{n_{\ell}}{\rangle}$ form an
orthogonal basis for measuring $\hat{Q}^{-}(t^{\prime})$, and the system in
$|q^{\ell}_{n_{\ell}}{\rangle}$ at $t_{\ell}$ can only go to
$|\hat{U}_{S}(t^{\prime},t_{\ell})|q^{\ell}_{n_{\ell}}{\rangle}$ at
$t^{\prime}$, as all other matrix elements of
$\hat{U}_{S}(t^{\prime},t_{\ell})$ vanish. Similarly, the system in
$\hat{U}^{-1}_{S}(t_{\ell+1},t^{\prime})|q^{\ell}_{n_{\ell+1}}{\rangle}$ at
$t^{\prime}$ can only go to $|q^{\ell}_{n_{\ell+1}}{\rangle}$ at $t_{\ell+1}$.
The presence of the operators $\hat{U}_{S}^{-1}(t^{\prime},t_{\ell})$ and
$\hat{U}_{S}(t_{\ell+1},t^{\prime})$ in Eqs. (26) and (27) ensures that
Eq.(25) holds, and Eqs.(29) follow.
The problem is as follows. By using the CR, we appear to be able to say that
at $t=t^{\prime}$ a quantity $\hat{Q}^{-}(t^{\prime})$ has a definite value
$Q^{-}_{m_{\ell}}$ if the value of $\hat{Q}^{\ell}$ at $t=t_{\ell}$ was
$Q^{\ell}_{m_{\ell}}$. Similarly, it would appear that
$\hat{Q}^{+}(t^{\prime})$ also has a definite value $Q^{+}_{m_{\ell+1}}$ if
the value of $\hat{Q}^{\ell+1}$ at $t=t_{\ell+1}$ is $Q^{\ell+1}_{m_{\ell+1}}$
Since in general $\hat{Q}^{-}(t^{\prime})$ and $\hat{Q}^{+}(t^{\prime})$ do
not commute, $[\hat{Q}^{-}(t^{\prime}),\hat{Q}^{+}(t^{\prime})]\neq 0$, and
quantum mechanics forbids ascribing simultaneous values to non-commuting
quantities, we seem to have a contradiction,
Fortunately, the contradiction is easily resolved. At the end of the
experiment one needs to have all the relevant records, and to produce these
records an additional probe must be coupled at $t=t^{\prime}$. Measuring
$\hat{Q}^{-}(t^{\prime})$, or $\hat{Q}^{+}(t^{\prime})$ requires different
probes, which affect the system differently, and produce different statistical
ensembles. The values $Q^{-}_{m_{\ell}}$ and $\hat{Q}^{+}_{m_{\ell+1}}$ do not
pre-exist their respective measurements Merm  DSprob1 , and appear as a result
of a probe acting on a system. The caveat is the same as in Bohr’s answer Bohr
to the authors of EPR . There are no practical means of ascertaining these
conflicting values simultaneously. Next we give a simple example

Figure 5:  a) A measurement of an operator
$\hat{B}=\sum_{i=1}^{2}|b_{i}{\rangle}B_{i}{\langle}b_{i}|$ prepares the
system ($\hat{H}_{S}=0$) in a state $|b_{1}{\rangle}$, and is followed by a
measurement of $\hat{C}=\sum_{i=1}^{2}|c_{i}{\rangle}C_{i}{\langle}c_{i}|$. b)
An additional measurement of $\hat{B}$ at $t_{0}<t^{\prime}<t_{1}$ yields an
outcome $B_{1}$, and finds the system in the state $|b_{1}{\rangle}$ with
certainty. c) An additional measurement of $\hat{C}$ at the same $t^{\prime}$
yields $C_{i}$ with certainty, if the last outcome is also $C_{i}$. The
probabilities are unchanged, $P(C_{i}\leftarrow B_{1})=P(C_{i}\leftarrow
B_{1}\leftarrow B_{1})=P(C_{i}\leftarrow C_{i}\leftarrow B_{1})$, and it would
appear that at $t=t^{\prime}$ the system has well defined values of non-
commuting operators $\hat{B}$ and $\hat{C}$. Figure 6: a) In the case shown in
Fig.5b, at $t^{\prime}<t^{\prime\prime}<t_{1}$ one can add a measurement of
$\hat{C}$, still leaving the probabilities unchanged, $P(C_{i}\leftarrow
B_{1})=P(C_{i}\leftarrow C_{i}\leftarrow B_{1}\leftarrow B_{1})$. It would
appear that in a transition $\\{C_{1}\leftarrow B_{1}\\}$ intermediate values
$B_{1}$ and $C_{1}$ co-exist for any $t^{\prime}$ and $t^{\prime\prime}$, such
that $t^{\prime\prime}>t^{\prime}$. b) The above is no longer true if
$\hat{C}$ is measured before $\hat{B}$. The transition between the cases a)
and b) is discussed in Sect. VII.

##  VI A two-level example

Consider a two-level system, (a qubit), $N=2$, prepared by the first
measurement of an operator

$\displaystyle\hat{Q}^{0}\equiv\hat{B}=\sum_{i=1}^{2}|b_{i}{\rangle}B_{i}{\langle}b_{i}|$
(30)

in a state $|b_{1}{\rangle}$ at $t=t_{0}$. The second measurement (we have
$L=1$) yields one of the eigenvalues of an operator

$\displaystyle\hat{Q}^{1}\equiv\hat{C}=\sum_{i=1}^{2}|c_{i}{\rangle}C_{i}{\langle}c_{i}|,\quad[\hat{B},\hat{C}]\neq
0,$ (31)

With only two dimensions involved, all eigenvalues are non-degenerate. If for
simplicity we put $\hat{H}_{S}=0$, $\hat{U}_{S}(t_{1},t_{0})=1$ [see Fig.5a],
one can easily verify that at any $t_{0}<t^{\prime}<t_{1}$ the value of
$\hat{B}$ is $B_{1}$ [see Fig.5b], or that $\hat{C}$ has the same value it
will have at $t=t_{1}$ [see Fig.5c]. Moreover, its is easy to ascertain that
if the first and the last outcomes are $B_{1}$ and $C_{i}\\}$, the values of
$\hat{B}$ at $t=t^{\prime}$ and of $\hat{C}$ $t=t^{\prime\prime}$ are $B_{1}$
and $C_{i}$, as long as $t_{0}<t^{\prime}<t^{\prime\prime}<t_{1}$ [see
Fig.6a]. Indeed, according to (3) we have

$\displaystyle P(C_{1}\leftarrow C_{1}\leftarrow B_{1}\leftarrow
B_{1})=|{\langle}c_{1}|c_{1}{\rangle}{\langle}c_{1}|b_{1}{\rangle}{\langle}b_{1}|b_{1}{\rangle}|^{2}=|{\langle}c_{1}|b_{1}{\rangle}|^{2}=P(C_{1}\leftarrow
B_{1}).$ (32)

The former is no longer true if $\hat{C}$ is measured before $\hat{B}$. A
final value $C_{1}$ can now be reached via four real paths $\\{C_{1}\leftarrow
B_{i}\leftarrow C_{j}\leftarrow B_{1}\\}$ shown in Fig.6b, and the
probabilities no longer agree with those Eq.(32),

$\displaystyle\tilde{P}(C_{1}\leftarrow
B_{1})\equiv\sum_{i,j=1}^{2}P(C_{1}\leftarrow B_{i}\leftarrow C_{j}\leftarrow
B_{1})\neq P(C_{1}\leftarrow B_{1}).$ (33)

The transition between the two regimes occurs at
$t^{\prime}=t^{\prime\prime}$, when an attempt is made to measure two non-
commuting quantities at the same time. The rules of Sect. II imply that such
measurements are not possible in principle, since $\hat{B}$ and $\hat{C}$ do
not have a joint set of eigenstates which could inserted into Eq.(2). However,
ultimately one is interested in the records available at the end of the
experiment. Next we will look at the readings the probes would produce, should
they be set up to measure non-commuting $\hat{B}$ and $\hat{C}$
simultaneously.

##  VII Joint measurement of non-commuting variables

We want to consider two measurements made on the system in Fig.6 which can
overlap in time at least partially. No longer instantaneous, both measurement
will last $\tau$ seconds, start at $t^{\prime}$ and $t^{\prime\prime}$ ,
$t^{\prime},t^{\prime\prime}>t_{0}$, respectively, and finish before
$t=t_{1}$, $t^{\prime}+\tau,t^{\prime\prime}+\tau<t_{1}$. The degree to which
the measurements overlap will be controlled by a parameter $\beta$,

$\displaystyle\beta=(t^{\prime\prime}-t^{\prime})/\tau,$ (34)

so that for $\beta=1$ the measurement of $\hat{B}$ precedes that of $\hat{C}$,
$\beta=0$ corresponds to simultaneous measurements of both $\hat{B}$ and
$\hat{C}$, and for $\beta=-1$ $\hat{C}$ is measured first. Next we consider
the two kind of probes introduced in Sect.IV separately.

###  VII.1  C-NOT  gates as a meters

For our two level example of Sect. VI we can further simplify the probe
described in Sect.IV. Since we only need to distinguish between two of the
system’s conditions, a two-level probe, whose state either changes or remains
the same, is all that is required. We will need two such probes, $D^{\prime}$
and $D^{\prime\prime}$, two sets of states

$\displaystyle|D^{\ell}(1){\rangle}=|1^{\ell}{\rangle},\quad|D^{\ell}(2){\rangle}=|2^{\ell}{\rangle},\quad\ell=,^{\prime},^{\prime\prime}$
(35)

four projectors

$\displaystyle\hat{\pi}^{\prime}_{1}=|b_{1}{\rangle}{\langle}b_{1}|,\quad\hat{\pi}^{{}^{\prime}}_{2}=|b_{2}{\rangle}{\langle}b_{2}|,\quad\hat{\pi}^{{}^{\prime\prime}}_{1}=|c_{1}{\rangle}{\langle}c_{1}|,\quad\hat{\pi}^{{}^{\prime\prime}}_{2}=|c_{2}{\rangle}{\langle}c_{2}|.$
(36)

and two couplings

$\displaystyle\hat{H}^{\prime}_{int}=-(\pi/2)\tau^{-1}\hat{\pi}^{{}^{\prime}}_{2}\hat{\sigma}^{{}^{\prime}}_{x},\quad\hat{H}^{{}^{\prime\prime}}_{int}=-(\pi/2)\tau^{-1}\hat{\pi}^{{}^{\prime\prime}}_{2}\hat{\sigma}^{{}^{\prime\prime}}_{x},$
(37)

where $\hat{\sigma}_{x}^{\ell}|D^{\ell}(1){\rangle}=|D^{\ell}(2){\rangle}$. In
what follows we will put $\tau=1$. The probes are prepared in the respective
states $|D^{\prime}(1){\rangle}$ and $|D^{\prime\prime}(1){\rangle}$, and
after finding the system in $|c_{2}{\rangle}$ at $t_{2}$ their state is given
by

$\displaystyle|\Phi_{Probes}(t_{2}){\rangle}\equiv\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
(38)
$\displaystyle{\langle}c_{1}|\exp[-i|\beta|\hat{H}_{int}^{{}^{\prime\prime}}]\otimes\exp[-i(1-|\beta|)(\hat{H}_{int}^{{}^{\prime\prime}}+\hat{H}_{int}^{{}^{\prime}})]\otimes\exp[-i|\beta|\hat{H}_{int}^{{}^{\prime}}]|b_{1}{\rangle}|D^{\prime}(1){\rangle}|D^{{}^{\prime\prime}}(1){\rangle}.\quad\quad$

if $t^{\prime\prime}>t^{\prime}$, while for $t^{\prime\prime}<t^{\prime}$ the
order of operators in (38) is reversed. It is easy to check that
($\ell=,^{\prime},^{\prime\prime}$)

$\displaystyle\exp[-i|\beta|\hat{H}_{int}^{\ell}]=\left\\{\hat{\pi}^{\ell}_{1}+\hat{\pi}^{\ell}_{2}[\cos(\pi|\beta|/2)+i\sin(\pi|\beta|/2)\hat{\sigma}^{\ell}_{x}]\right\\}$
(39)

so that for $|\beta|=1$ the r.h.s. of Eq.(39) reduces to
$\hat{\pi}^{\ell}_{1}+i\hat{\pi}^{\ell}_{2}\sigma^{\ell}_{x}$. The action of
the coupling is, therefore, that of a quantum (C)ontrolled-NOT gate CNOT ,
which flips the probe’s (target) state if the system’s (control) state is
$|b_{2}{\rangle}$ or $|c_{2}{\rangle}$, and leaves the probes’s condition
unchanged if it is $|b_{1}{\rangle}$ or $|c_{1}{\rangle}$.
There are four possible outcomes, $(1^{\prime},1^{\prime\prime})$,
$(1^{\prime},2^{\prime\prime})$, $(2^{\prime},1^{\prime\prime})$, and
$(2^{\prime},2^{\prime\prime})$, and four corresponding probabilities,

$\displaystyle
P(i^{\prime},j^{\prime\prime})=|{\langle}D^{\prime}(i)|{\langle}D^{\prime\prime}(j)|\Phi_{Probes}(t_{2}){\rangle}|^{2}/\sum_{k,l=1}^{2}|{\langle}D^{\prime}(k)|{\langle}D^{\prime\prime}(l)|\Phi_{Probes}(t_{2}){\rangle}|^{2},\quad
i,j=1,2.\quad\quad$ (40)

The matrix elements are easily evaluated (for details see Appendix B), and the
results are shown in Fig.7.

Figure 7: The values of $\hat{B}=\hat{\sigma}_{x}$ and
$\hat{C}=\hat{\sigma}_{y}$ are measured jointly [cf. Eqs.(36)-(38)] for a two-
level system ($\hat{H}_{S}=0$), initially polarised along the $x$-axis,
$|b_{1}{\rangle}=|\uparrow_{x}{\rangle}$, $B_{1}=1$, and later found polarised
along the $y$-axis, $|c_{1}{\rangle}=|\uparrow_{y}{\rangle}$, $C_{1}=1$. Four
probabilities in Eq.(40) are plotted vs. $\beta$ in Eq.(35). For $\beta=1$ the
measurement of $\hat{B}$ precedes that of $\hat{C}$, for $\beta=0$, $\hat{B}$
and $\hat{C}$ are measured simultaneously, and for $\beta=-1$, $\hat{C}$ is
measured first.

If $\beta=1$, there is no overlap, and $\hat{B}$ is measured before $\hat{C}$.
Dividing $\tau$ into $K$ sub-intervals and sending $K\to\infty$ we have an
identity [cf. Eq.(39)]

$\displaystyle\exp[-i\hat{H}_{int}^{{}^{\prime}}]|b_{1}{\rangle}=\left[\exp(-i\hat{H}_{int}^{{}^{\prime}}/K])\right]^{K}|b_{1}{\rangle}=(\hat{\pi}^{\prime}_{1})^{K}|b_{1}{\rangle}=|b_{1}{\rangle},$
(41)

and the state of the first probe remains unchanged. For $\beta=0$ both probes
act simultaneously. Now the use of the Trotter’s formula yields

$\displaystyle\exp[-i(\hat{H}_{int}^{{}^{\prime}}+\hat{H}_{int}^{{}^{\prime\prime}})]|b_{1}{\rangle}=\lim_{K\to\infty}\left(1+i\pi\hat{\pi}^{{}^{\prime}}_{2}\hat{\sigma}^{{}^{\prime}}_{x}/2K+i\pi\hat{\pi}^{{}^{\prime\prime}}_{2}\hat{\sigma}^{{}^{\prime\prime}}_{x}/2K\right)^{K}|b_{1}{\rangle}.$
(42)

Equation (42) contains scenarios where both probes change their states, and
since $\hat{\pi}^{{}^{\prime}}\otimes\hat{\pi}^{{}^{\prime\prime}}\neq 0$, the
evolution of one of them must affect what happens to the other. Now all four
probabilities in Eq.(40) have non-zero values, although it is still more
likely that both probes will remain in their initial states (cf. Fig.7).
Finally, for $\beta=-1$, $\hat{C}$ is measured before $\hat{B}$ as if both
measurements were instantaneous, and all four paths in Fig.6b are equally
probable. A different result is obtained if the two measurements are of von
Neumann type, as we will discuss next .

###  VII.2 Von Neumann meters

Consider the same problem but with the two-level probes replaced by two von
Neumann pointers with positions $f^{\prime}$ and $f^{\prime\prime}$,
respectively. As before, the interaction with each pointer lasts $\tau$
seconds, so the two Hamiltonians are

$\displaystyle\hat{H}^{\prime}_{int}=-i\partial_{f^{\prime}}\hat{B}/\tau,\quad\hat{H}^{\prime\prime}_{int}=-i\partial_{f^{\prime\prime}}\hat{C}/\tau.$
(43)

If the pointers are prepared in identical Gaussian states (22)
${\langle}f^{\ell}|G^{\ell}{\rangle}=G(f^{\ell}),\quad\ell=,^{\prime},^{\prime\prime}$
the probability distribution of their final positions is given by

$\displaystyle P(f^{\prime},f^{\prime\prime})=\left|\int
dy^{\prime}dy^{\prime\prime}G(f^{\prime}-y^{\prime})G(f^{\prime\prime}-y^{\prime\prime})\Phi(y^{\prime},y^{\prime\prime})\right|^{2}\equiv|\Psi(f^{\prime},f^{\prime\prime})|^{2}$
(44)

where (we measure $f$ in units of $g_{0}$ and put $\tau=1$)

$\displaystyle\Phi(f^{\prime},f^{\prime\prime})\equiv{\langle}c_{1}|{\langle}f^{\prime}|{\langle}f^{\prime\prime}|\exp[-|\beta\partial_{f^{\prime\prime}}\hat{C}]\otimes\exp[-(1-2|\beta|)(\partial_{f^{\prime}}\hat{B}+\partial_{f^{\prime\prime}}\hat{C})]$
(45)
$\displaystyle\otimes\exp[-|\beta|\partial_{f^{\prime\prime}}\hat{B}]|0^{\prime}{\rangle}|0^{\prime\prime}{\rangle}|b_{1}{\rangle}$

where ${\langle}f^{\prime}|0^{\prime}{\rangle}=\delta(f^{\prime})$,
${\langle}f^{\prime\prime}|0^{\prime\prime}{\rangle}=\delta(f^{\prime\prime})$,
and $t^{\prime}\leq t^{\prime\prime}$. (For $t^{\prime}>t^{\prime\prime}$ we
interchange $\hat{B}$ with $\hat{C}$.)
Consider first the case $\beta=0$ where the measurements coincide. The
amplitude $\Phi(f^{\prime},f^{\prime\prime})$ has several general properties.
Firstly, it cannot be a smooth finite function of $y^{\prime}$ and
$y^{\prime\prime}$, or the integral in (44) would vanish in the limit of
narrow Gaussians, $\Delta f\to 0$, due to the normalisation of
$|G^{\prime}{\rangle}$ and $|G^{\prime\prime}{\rangle}$ [cf. Eq.(22)]. It
must, therefore, have $\delta$-singularities DSerg , DSpit . Secondly, using
the Trotter’s formula Trot , we have

$\displaystyle\exp[-(\partial_{f^{\prime}}\hat{B}+\partial_{f^{\prime\prime}}\hat{C})]=\lim_{K\to\infty}\left[\exp(-\partial_{f^{\prime}}\hat{B}/K)\otimes\exp(-\partial_{f^{\prime\prime}}\hat{C}/K)\right]^{K}$
(46)

It is readily seen that each time the product in the r.h.s. of Eq.(46) is
applied, the pointers are displaced by $B_{1}/K$ or $B_{2}\tau/K$ and
$C_{1}/K$ or $C_{2}/K$, respectively. If $B_{1,2},C_{1,2}=\pm 1$, one has a
quantum random walk, where the pointers are shifted by an equal amount $1/K$
ether to the right or to the left. Since the largest possible displacement is
$1$, $\Phi(f^{\prime},f^{\prime\prime})$ must vanish outside a square $-1\leq
f^{\prime},f^{\prime\prime}\leq 1$. One also notes that for $f^{\prime}=\pm
1$, the maximum of $\Phi(\pm 1,f^{\prime\prime})$ is reached for
$f^{\prime\prime}=0$, since there are (let $K$ be an even number)
$K!/(K/2)!(K/2)!$ walks, each contributing to
$\Phi(f^{\prime},f^{\prime\prime})$ the same amount $1/2^{2K+1}$. Similarly, a
maximum of $\Phi(f^{\prime},\pm 1)$ is reached for $f^{\prime}=0$.

Figure 8:  The values of $\hat{B}=\hat{\sigma}_{x}$ and
$\hat{C}=\hat{\sigma}_{y}$ are measured simultaneously ($\beta=0$, $\Delta
f=0.05$) for a two-level system ($\hat{H}_{S}=0$), initially polarised along
the $x$-axis, $|b_{1}{\rangle}=|\uparrow_{x}{\rangle}$, $B_{1}=1$, and later
found polarised along the $y$-axis, $|c_{1}{\rangle}=|\uparrow_{y}{\rangle}$,
$C_{1}=1$.

Figure 9:  The same probability as in Fig.9, but for different degrees of
overlap between the measurements. The value of $\beta$ in Eq.(34) is indicated
in each panel. The cases $\beta=1$ and $\beta=-1$ correspond to the scenarios
a) and b) shown in Fig.4, respectively.

A detailed combinatorial analysis is complicated, but the location of the
singularities can be determined as was done in DSnj . As explained in the
Appendix B, the amplitude $\Phi(y^{\prime},y^{\prime\prime})$ in Eq.(45) can
also be written as ($f\equiv\sqrt{f^{\prime 2}+f^{\prime\prime 2}}$,
$\cos(\theta)=f^{\prime}/f$)

$\displaystyle\Psi(f^{\prime},f^{\prime\prime})=(2\pi)^{-1/2}\Delta
f{\langle}c_{1}|b_{1}{\rangle}\int_{0}^{\infty}d\lambda\lambda\exp(-\lambda^{2}\Delta
f^{2}/4)\times$ (47) $\displaystyle\left[\cos(\lambda)J_{0}(\lambda
f)-\sqrt{2}i\sin(\lambda)J_{1}(\lambda f)\sin(\theta-\pi/4)\right]$

where $J_{k}$ is the Bessel function of the first kind of order $k$. From
Eq.(44) one notes that in the limit $\Delta f\to 0$,
$P(f^{\prime},f^{\prime\prime})$ in Eq.(44) will become singular, and that its
singularities will coincide with those of $\Phi(f^{\prime},f^{\prime\prime})$
in Eq.(45). Since abram

$\displaystyle J_{k}(\lambda f\to\infty)\to(2/\pi\lambda f)^{1/2}\cos(\lambda
f-k\pi/2-\pi/2),$ (48)

the integral in Eq.(47) will diverge at large $\lambda$’s provided the
oscillations of $\cos(\lambda)$ and $\sin(\lambda)$ are cancelled by those of
$J_{0}(\lambda f)$ and $J_{1}(\lambda f)$, i.e. for $\sqrt{f^{\prime
2}+f^{\prime\prime 2}}=1$. As a result, we find the pointer readings of two
simultaneous accurate measurements of $\hat{B}$ and $\hat{C}$ (with
$B_{1,2}=C_{1,2}=\pm 1$) distributed along the perimeter of a unit circle as
shown in Fig.9. Figure 9 shows the distribution of the pointer’s readings for
different degrees of overlap between the two measurements.
There is, therefore, an important difference between employing discreet probes
and von Neumann pointers. In the previous example shown in Fig.7 one could
(although should not) assume, e.g., that for $\beta=0$
$P(1^{\prime},2^{\prime\prime})$ yields the probability for $\hat{B}$ and
$\hat{C}$ to have the values $1$ and $-1$, if measured simultaneously. Figures
9 and 9 show this conclusion to be inconsistent. As $\beta$ decreases from $1$
to $-1$, the pointer’s readings are not restricted to $\pm 1^{\prime}$, $\pm
1^{\prime\prime}$. Rather, for $\beta=0$ they are along the perimeters of a
unit circle (cf. Figs.9 and 9), which, if taken at face value, implies that
the probability in question is zero. We already noted that the theory in
Sect.II is unable to prescribe probabilities of simultaneous values of non-
commuting operators. In practice this means that the probes, capable of
performing the task in a consistent manner, simply cannot be constructed.

##  VIII The past of a quantum system. Weakly perturbing probes and the
Uncertainty Principle

It remains to see what information can be obtained from the measurements,
designed to perturb the measured system as little as possible. If such a
measurement were an attempt to distinguish between interfering scenarios,
without destroying interference between them, it would contradict the
Uncertainty Principle cited in Sect.II. As before, we treat the two types of
probes separately.

###  VIII.1 Weak discrete gates

We start by reducing the coupling strength, so the interaction Hamiltonians
(15) become

$\displaystyle\hat{H}_{int}^{\ell}=-\gamma\sum_{m_{\ell}=1}^{M_{\ell}}\hat{\pi}^{\ell}_{m_{\ell}}\hat{\sigma}_{x}(m_{\ell})\delta(t-t_{\ell}),\quad\gamma<\pi/2.$
(49)

In FeynC  Feynman described a double-slit experiment where photons, scattered
by the the passing electron, allowed one to know through which of the two
slits the electron has travelled. With every electron duly detected, their
distribution on the screen, $P_{\text{Slit-unknown}}(x)$, does not exhibit an
interference pattern. With no photons present, the pattern is present in the
distribution $P_{\text{Slit-known}}(x)$. If the intensity of light (i.e., the
number of photons) is decreased, some of the electrons pass undetected. The
total distribution on the screen is, therefore, “a mixture of the two curves”
FeynC . $P(x)=aP_{\text{Slit-unknown}}(x)+bP_{\text{Slit-known}}(x)$, where
$a$ and $b$ are some constants,.
Something very similar happens if an extra discrete probe $D^{\prime}$ is
added to measure
$\hat{Q}^{\prime}=\sum_{m^{\prime}=1}^{M^{\prime}}Q^{\prime}_{m^{\prime}}\hat{\pi}^{\prime}_{m^{\prime}}$
at $t=t^{\prime}$, $t_{\ell}<t^{\prime}<t_{\ell+1}$. As before [cf Eq.(16)] ,
we find

$\displaystyle\hat{U}_{int}^{\prime}(t^{\prime})=\exp\left[i\gamma\sum_{m^{\prime}=1}^{M^{\prime}}\hat{\pi}^{\prime}_{m^{\prime}}\hat{\sigma}^{\prime}_{x}(m^{\prime})\right]=\cos(\gamma)+i\sin(\gamma)\sum_{m^{\prime}l=1}^{M^{\prime}}\hat{\pi}^{\prime}_{m^{\prime}}\hat{\sigma}^{\prime}_{x}(m^{\prime}),$
(50)

where the cosine term accounts for the possibility that the systems passes the
check undetected. Replacing $P(x)$ in Feynman’s example with the probability
$P(q^{L}_{n_{L}})$ of detecting the system in a final state
$|q^{L}_{n_{L}}{\rangle}$ at $t_{L}$. With a values of $\hat{Q}^{\prime}$
detected in every run, $\gamma=\pi/2$, one has a distribution

$\displaystyle
P_{\text{Q'-known}}(q^{L}_{n_{L}})=\sum_{m^{\prime}=1}^{M^{\prime}}\left|A_{S}(q^{L}_{n_{L}}...\leftarrow\hat{\pi}^{\ell+1}_{m_{\ell+1}}\leftarrow\hat{\pi}^{\prime}_{m^{\prime}}\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})\right|^{2}$ (51)

where $A_{S}$ is the amplitude in Eq.(5). If the probe is uncoupled,
$\gamma=0$, the distribution is

$\displaystyle
P_{\text{Q'-unknown}}(q^{L}_{n_{L}})=\left|\sum_{m^{\prime}=1}^{M^{\prime}}A_{S}(q^{L}_{n_{L}}...\leftarrow\hat{\pi}^{\ell+1}_{m_{\ell+1}}\leftarrow\hat{\pi}^{\prime}_{m^{\prime}}\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})\right|^{2}$ (52)

With $0<\gamma<\pi/2$ the outcomes fall into two groups, those where the probe
$D^{\prime}$ remains in its initial state, and those where the state of one of
its sub-systems has been flipped. The two alternatives are exclusive FeynH ,
and the total distribution is indeed a mixture of the two curves. As
$\gamma\to 0$ one has

$\displaystyle
P(q^{L}_{n_{L}})=(1-\gamma^{2})P_{\text{Q'-unknown}}(q^{L}_{n_{L}})+\gamma^{2}P_{\text{Q'-known}}(q^{L}_{n_{L}})+o(\gamma^{2}).$
(53)

In other words, in the vast majority of cases the system remains undetected,
and the interference is preserved [cf. Eq.(52)]. In the few remains cases it
is detected, and the interference is destroyed. The Uncertainty Principle
FeynL  is obeyed to the letter: probabilities are added where records allow
one to distinguish between the scenarios; otherwise one sums the amplitudes.
One has, however, to admit that nothing really new has been learned from this
example, as both possibilities simply illustrate the rules of Sect. II.
The above analysis is easily extended to include more extra measurements,
whether impulsive or not. Since to the first order in the coupling constant
$\gamma$ weak probes act independently of each other, the r.h.s. of Eq.(53)
would contain additional terms $P_{\text{Q''-known}}(q^{L}_{n_{L}})$,
$P_{\text{Q{'''}-known}}(q^{L}_{n_{L}})$, etc.

###  VIII.2 Weak von Neumann pointers

Next we add to $L+1$ accurate impulsive von Neumann pointers an extra “weak”
pointer, designed to measure
$\hat{Q}^{\prime}=\sum_{m^{\prime}=1}^{M^{\prime}}Q^{\prime}_{m^{\prime}}\hat{\pi}^{\prime}_{m^{\prime}}$
at $t=t^{\prime}$ between $t_{\ell}$ and $t_{\ell+1}$. The new coupling, given
by

$\displaystyle\hat{H}^{\prime}_{int}=-i\gamma\partial_{f^{\prime}}\hat{Q}^{\prime},$
(54)

will perturb the system only slightly in the limit $\gamma\to 0$. To see what
happens in this limit, one can replace $f^{\prime}$ by $\gamma f^{\prime}$,
$\hat{H}_{int}^{\ell}$ in Eq.(54) by $-i\partial_{f^{\prime}}\hat{Q}^{\ell}$,
and the pointer’s initial state $G^{\prime}(f^{\prime})$ by
$\gamma^{1/2}G^{\prime}(\gamma f^{\prime})$. Now as $\gamma\to 0$ the
pointer’s initial states become very broad, while the coupling remains
unchanged. For a Gaussian pointers (22), considered here, this means replacing
$\Delta f^{\prime}$ with $\Delta f^{\prime}/\gamma$, i.e., making the
measurement highly inaccurate. This makes sense. The purpose of a pointer is
to destroy interference between the system’s virtual paths, [cf. Eq.(24)],
which it is clearly unable to do if the coupling vanishes. Accordingly, with
the pointer’s initial position highly uncertain, its final reading are also
spread almost evenly between $-\infty$ and $\infty$. Measured in this manner,
the value of $\hat{Q}^{\prime}$ remain indeterminate, as required by the
Uncertainty Principle.
This could be the end of our discussion, except for one thing. It is still
possible to use the broad distribution of a pointer’s readings in order to
evaluate averages, which could, in principle, remain finite in the limit
$\Delta f^{\prime}\to\infty$. Maybe this can tell us something new about the
system’s condition at $t=t^{\prime}$. Note, however, that whatever information
is extracted in this manner should not not contradict the Uncertainty
Principle, or the whole quantum theory would be in trouble FeynL .
From Eq.(22) it is already clear that any average of this type will be
expressed in terms of the amplitudes
$A(q^{L}_{n_{L}}...\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})$. The simplest average is the pointer’s mean position. If the
outcomes of the accurate measurements are
$Q^{L}_{n_{L}}...Q^{\ell}_{m_{\ell}}...Q^{0}_{n_{0}}$, for the mean reading of
the weakly coupled pointer we obtain (see Appendix C)

$\displaystyle{\langle}f^{\prime}(Q^{L}_{n_{L}}...Q^{\ell}_{m_{\ell}}...Q^{0}_{n_{0}}){\rangle}\approx\text{Re}\left[\frac{\sum_{m^{\prime}=1}^{M^{\prime}}Q^{\prime}_{m^{\prime}}A_{S}(q^{L}_{n_{L}}....\leftarrow\hat{\pi}^{\ell+1}_{m_{\ell+1}}\leftarrow\hat{\pi}^{\prime}_{m^{\prime}}\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})}{\sum_{m^{\prime}=1}^{M^{\prime}}A_{S}(q^{L}_{n_{L}}....\leftarrow\hat{\pi}^{\ell+1}_{m_{\ell+1}}\leftarrow\hat{\pi}^{\prime}_{m^{\prime}}\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})}\right].$ (55)

If the measured operator is one of the projectors, say,
$\hat{Q}^{\ell}=\hat{\pi}^{\ell}_{m^{\prime}}$, this reduces to

$\displaystyle{\langle}f^{\prime}(Q^{L}_{n_{L}}...Q^{\ell}_{m_{\ell}}...Q^{0}_{n_{0}}){\rangle}\approx\text{Re}\left[\frac{A_{S}(q^{L}_{n_{L}}....\leftarrow\hat{\pi}^{\ell+1}_{m_{\ell+1}}\leftarrow\hat{\pi}^{\prime}_{m}\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})}{\sum_{m^{\prime}=1}^{M^{\prime}}A_{S}(q^{L}_{n_{L}}....\leftarrow\hat{\pi}^{\ell+1}_{m_{\ell+1}}\leftarrow\hat{\pi}^{\prime}_{m^{\prime}}\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})}\right]$ (56)

We note that, as in the previous example, different pointers do not affect
each other to the leading order in the small parameter $\gamma$.
The quantities in the l.h.s. of Eqs.(55) and Eqs.(56) are the standard
averages of the probes’ variables. In the l.h.s. of these equations one finds
probability amplitudes for the system’s entire paths,
$\\{q^{L}_{n_{L}}....\leftarrow\hat{\pi}^{\ell+1}_{m_{\ell+1}}\leftarrow\hat{\pi}^{\prime}_{m^{\prime}}\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}}\\}$. The values of these amplitudes can be deduced from the
probes’s probabilities DSamp . The problem is, these values offer no insight
into the condition of the system at $t=t_{\ell}$. In the double-slit case, to
conclude that a particle “…goes through one hole or the other when you are not
looking is to produce an error in prediction” FeynC . In our case, one cannot
say that the value of $\hat{Q}^{\prime}$ was or was not a particular
$Q^{\prime}_{m^{\prime}}$. The Uncertainty Principle prevails again, this time
by letting one only gain information not sufficient for determining the
condition of unobserved system at $t=t^{\prime}$. A more detailed discussion
of this point can be found, e.g., in DSw1 , DSw .

##  IX Summary and conclusions

A very general way to describe quantum mechanics is to say that it is a theory
which prescribes probability amplitudes to sequences of events, and then
predicts the probability of a sequence by taking an absolute square of the
corresponding amplitude FeynL . Where several ($L+1$) consecutive measurements
are made on the same system ($S$), a sequence of interest is that of the
measured values $\hat{Q}^{\ell}_{m_{\ell}}$, endowed with a (system’s)
probability amplitude
$A_{S}(q^{L}_{n_{L}}...\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})$ in Eq.(5).
This is not, however, the whole story. To test the theory’s predictions, an
experimenter (Observer) must keep the records the measured values, in order to
collect the statistics once the experiment is finished. This is more than a
mere formality. The system, whose condition changes after each measurement,
cannot itself store this information. Hence the need for the probes, material
objects, whose conditions must be directly accessible to the Observer at the
end. One can think of photons FeynL , FeynC  devices with or without dials, or
Observer’s own memories of the past outcomes DSm , DSob . The probes must be
prepared in suitable initial condition $|\Psi_{Probes}(0){\rangle}$ and be
found in one of the orthogonal states
$|\Psi_{Probes}(n_{L},...m_{\ell}...n_{0}){\rangle}$ later, with an amplitude
$A_{S+Probes}(\Psi_{Probes}(n_{L},...m_{\ell}...n_{0}),q^{L}_{n_{L}}\leftarrow\Psi_{Probes(0)},q^{0}_{n_{0}})$.
To be consistent, the theory must construct the amplitudes $A_{S}$ and
$A_{S+Probes}$ using exactly the same rules, and ensure that
$|A_{S+Probes}|^{2}=|A_{S}|^{2}$. In other words, the experimenter should see
a record occurring with a frequency the theory predicts for an isolated (no
probes) system, going through its corresponding conditions. This requires the
existence of a suitable coupling between the system and the probe. Its choice
is not unique, and for a system with a finite dimensional Hilbert space
studied here, two different kinds of probes were discussed in Sect. IV. The
first one is a discrete gate, using the interaction in Eq.(15), while the
second is the original von Neumann pointer vN .
Now one can obtain the same probability $P=|A_{S+Probes}|^{2}=|A_{S}|^{2}$ by
considering a unitary evolution of a composite $\\{System+Probes\\}$ until the
moment the Observer examines his records at the end of the experiment. Or one
can consider such an evolution of the system only, but broken every time a
probe is coupled to it. For a purist intent on identifying quantum mechanics
with unitary evolution (see, e.g., the discussion in Kast ), the first way may
seem preferable. Yet there is no escaping the final collapse of the
composite’s wave function when the stock is taken at the end of the
experiment.
It is often simpler to discuss measurements in terms of the measured system’s
amplitude, leaving out, bit not forgetting, the probes. The rules formulated
in Sect. II readily give an answer to any properly asked question, but offer
no clues regarding a question which has not been asked operationally. One may
try to extend the description of a quantum system’s past by looking for
additional quantities whose values could be ascertained without changing the
probabilities of the measured outcomes. In general, this is not possible. To
find the value of a quantity $\hat{Q}^{\prime}$ at a $t^{\prime}$ between to
successive measurements, $t_{\ell}<t^{\prime}<t_{\ell+1}$, one needs to
connect an extra probe. This would destroy interference between the system’s
paths and change other probabilities, leaving the question ”what was the value
of $\hat{Q}^{\prime}$ if was not measured?” without an answer.
There are two seeming exceptions to this rule. If $\hat{Q}^{\prime}$ is
obtained by evolving backwards in time the previously measured
$\hat{Q}^{\ell}$ [cf. Eq.(26)], call it $\hat{Q}^{-}$ , its value is certain
to equal that of $\hat{Q}^{\ell}$, and all other probabilities will remain
unchanged, [cf. Eq.(28)]. Similarly, the value of a $\hat{Q}^{+}$ [cf.
Eq.(27)], obtained by forward evolution of the next measured operator
$\hat{Q}^{\ell+1}$, will also agree with that of $\hat{Q}^{\ell+1}$. It would
be tempting to assume that these values represent some observation-free
“reality”, were it not for the fact that they cannot be ascertained
simultaneously. The two measurements require different probes, each affecting
the system in a particular way. Th probes frustrate each other if employed
simultaneously. It is hardly surprising that different evolution operators
$\hat{U}_{S+Probes}(t_{L},t_{0})$, in Eq.(10) may lead to different outcomes.
One notes also that measuring these two quantities one after another would
also leave all other probabilities intact, but only if $\hat{Q}^{-}$ is
measured first, as shown in Fig.6a. Changing this order results in a
completely different statistical ensemble, shown in Fig.6b. The rules of Sect.
II say little about what happens if the measurements coincide, except that if
$\hat{Q}^{-}$ and $\hat{Q}^{+}$ do not have common eigenstates, Eq.(5) cannot
be applied. One can still analyse the behaviour of the two probes at different
degrees of the overlap, to explain why it is impossible to reach consistent
conclusions about the simultaneous values of $\hat{Q}^{-}$ and $\hat{Q}^{+}$.
For example, if two discrete gates are used, Fig.(7) appears to offer four
joint probabilities of having the values $B,C=\pm 1$. If discrete probes are
replaced by von Neumann pointers, the readings shown in Fig.9 suggest that
joint values of $\hat{B}$ and $\hat{C}$ should lie on the perimeter of a unit
circle, in a clear contradiction with the previous conclusion.
Another way to explore the system’s past beyond what has been established by
accurate measurements, is to study its response to a weakly perturbing probe,
set up to measure some $\hat{Q}^{\prime}$ at an intermediate time
$t^{\prime}$. In this limit, the two types probes produce different effects
but, in accordance with the Uncertainty Principle, reveal nothing new that can
be added to the rules formulated in Sect.N II. If the coupling of an
additional discrete probe is reduced, trials are divided into two groups. In a
(larger) number of cases, the system remains undetected, and interference
between its virtual paths passing through different eigenstates of
$\hat{Q}^{\prime}$ remains intact. In a (smaller) fraction of cases the value
of $\hat{Q}^{\prime}$ is accurately determined, and the said interference is
destroyed completely. Individual readings of a weak von Neumann pointer extend
of a range much wider than the region which contains the values of
$\hat{Q}^{\prime}$, and are in this sense practically random. Its mean
position (reading) allows one to learn something about the probability
amplitude in Eq.(56), or a combination of such amplitudes as in Eq.(55). The
problem is that even after obtaining the values of these amplitudes (and this
can be done in practice DSamp ), one still does not know the value of
$\hat{Q}^{\prime}$, for the same reason he/she cannot know the slit chosen by
an unobserved particle in a double-slit experiment. Quantum probability
amplitudes simply do no have this kind of predictive power FeynL , FeynC .

In summary, quantum mechanics can consistently be seen as a formalism for
calculating transition amplitudes by means of evaluating matrix elements of
evolution operators. In such a “minimalist” approach (see also DSmin ), the
importance of a wave function, represented by an evolving system’s state, is
reduced to that of a convenient computational tool. In the words of Peres
Peres  (see also Fuchs ) “… there is no meaning to a quantum state before the
preparation of the physical system nor after its final observation (just as
there is no ‘time’ before the big bang or after the big crunch).” This is,
however, not a universally accepted view. For example, the authors of 2time1 ,
2time2 , 2time3  propose a time-symmetric formulation of quantum mechanics,
employing not one but two evolving quantum states. We will examine the
usefulness of such an approach in future work.

##  X Acknowledgements:

Support of the Basque Government, Grant No. IT986-16, and of MINECO, the
Ministry of Science and Innovation of Spain, Grant
PGC2018-101355-B-100(MCIU/AEI/FEDER,UE) is gratefully acknowledged.

##  Appendix A  Evaluation of the probabilities in Eq.(40)

For the eigenstates of the probes’s operators $\hat{\sigma}^{\ell}_{x}$ we
have

$\displaystyle|D^{\ell}(\lambda^{\ell}){\rangle}=[|1^{\ell}{\rangle}+\lambda^{\ell}|2^{\ell}{\rangle})]/\sqrt{2},\quad\hat{\sigma}^{\ell}_{x}|D^{\ell}(\lambda^{\ell}){\rangle}=\lambda^{\ell}|D^{\ell}(\lambda^{\ell}){\rangle},\quad\lambda^{\ell}=\pm
1,\quad\ell=^{\prime},^{\prime\prime}.\quad\quad$ (57)

The probes’ are prepared in an initial state
$\Phi_{Probes}(t_{0})=|1^{\prime}{\rangle}|1^{\prime\prime}{\rangle}=\sum_{\lambda^{\prime},\lambda^{\prime\prime}=\pm}|D^{\prime}(\lambda^{\prime}){\rangle}|D^{\prime\prime}(\lambda^{\prime\prime}){\rangle}/2$,
and after post-selecting the system in $|c_{1}{\rangle}$, their final state
(38) is given by

$\displaystyle|\Phi_{Probes}(t_{2}){\rangle}=\sum_{\lambda^{\prime},\lambda^{\prime\prime}=\pm}U_{\lambda^{\prime}\lambda^{\prime\prime}}|D^{\prime}(\lambda^{\prime}){\rangle}|D^{\prime\prime}(\lambda^{\prime\prime}){\rangle}/2,$
(58)

where

$\displaystyle
U_{\lambda^{\prime}\lambda^{\prime\prime}}={\langle}c_{1}|\exp(i\pi\lambda^{\prime\prime}|\beta|\hat{\pi}_{2}^{{}^{\prime\prime}}/2)\otimes\exp[i\pi(1-|\beta|)(\lambda^{\prime}\hat{\pi}_{2}^{{}^{\prime}}+\lambda^{\prime\prime}\hat{\pi}_{2}^{{}^{\prime\prime}})/2]\otimes\exp(i\pi\lambda^{\prime}|\beta|\hat{\pi}_{2}^{{}^{\prime}}/2)|b_{1}{\rangle}.\quad\quad\quad$
(59)

Operators in (58) can be diagonalised either analytically or numerically, and
the matrix elements in Eq.(40),

$\displaystyle{\langle}j^{\prime}|{\langle}j^{\prime\prime}|\Phi_{Probes}(t_{2}){\rangle}=\sum_{\lambda^{\prime},\lambda^{\prime\prime}=\pm
1}U_{\lambda^{\prime}\lambda^{\prime\prime}}{\langle}j^{\prime}|D^{\prime}(\lambda^{\prime}){\rangle}{\langle}j^{\prime\prime}|D^{\prime\prime}(\lambda^{\prime\prime}){\rangle}/2,\quad
j^{\prime},j^{\prime\prime},=1,2,$ (60)

are easily evaluated.

##  Appendix B  Derivation of Eq.(47)

The two-level system (a spin-$1/2$), with $\hat{H}_{S}=0$, is pre- and post-
selected in the states $|\uparrow_{\nu}{\rangle}$ and
$|\uparrow_{\mu}{\rangle}$, $\mu,\nu=x,y,z$, $\mu\neq\nu$. respectively. Two
pointers are employed to measure $\hat{\sigma}_{\nu}$ and $\hat{\sigma}_{\mu}$
simultaneously, $\beta=0$. (Extension to measurements along non-orthogonal
axes is trivial.) The final distribution of the pointer’s positions (readings)
is, therefore, given by

$\displaystyle P(f_{1},f_{2})=|A(f_{1},f_{2})|^{2}\equiv\left|\it
K(f_{1},f_{2},\lambda_{1},\lambda_{2})G(\lambda_{1})G(\lambda_{2})d\lambda_{1}\lambda_{2}\right|^{2},$
(61)

where $|G_{1,2}{\rangle}=\int
G(\lambda_{1,2})|\lambda_{1,2}{\rangle}d\lambda_{1,2}$ and

$\displaystyle
K(f_{1},f_{2},\lambda_{1},\lambda_{2})={\langle}f_{1}|\lambda_{1}{\rangle}{\langle}f_{2}|\lambda_{2}{\rangle}{\langle}\uparrow_{\mu}|\exp(-i\hat{\lambda}_{1}\hat{\sigma}_{\nu}-i\hat{\lambda}_{2}\hat{\sigma}_{\mu})|\uparrow_{\nu}{\rangle},$
(62)

We recall that

$\displaystyle{\langle}\uparrow_{\mu}|\exp(-i\hat{\lambda}_{1}\hat{\sigma}_{\nu}-i\hat{\lambda}_{2}\hat{\sigma}_{\mu})|\uparrow_{\nu}{\rangle}={\langle}\uparrow_{\mu}|\cos(\lambda)-i\sin(\lambda)[\cos(\varphi)\hat{\sigma}_{\nu}+\sin(\varphi)\hat{\sigma}_{\mu}]|\uparrow_{\nu}{\rangle}\quad\quad$
(63)
$\displaystyle={\langle}\uparrow_{\mu}|\uparrow_{\nu}{\rangle}\\{\cos(\lambda/2)-\sin(\lambda/2)[(i+1)\exp(i\varphi)+(i-1)\exp(-i\varphi)]/2\\},\quad\quad$

where $\lambda\equiv\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}$, and
$\varphi=\arccos(\lambda_{1}/\lambda)$. Defining
$f\equiv\sqrt{f_{1}^{2}+f_{2}^{2}}$, $\theta=\arccos(f_{1}/f)$, we obtain
${\langle}f_{1}|\lambda_{1}{\rangle}{\langle}f_{2}|\lambda_{2}{\rangle}=(2\pi)^{-1}\exp(if_{1}\lambda_{1}+if_{2}\lambda_{2})\equiv(2\pi)^{-1}\exp[i\lambda
f\cos(\varphi-\theta)]=(2\pi)^{-1}\exp[i\lambda f\sin(\xi)]$, where
$\xi=\pi/2+\theta-\varphi$.
Furthermore,

$\displaystyle\exp[i\lambda f\sin(\xi)]=\sum_{k=-\infty}^{\infty}J_{k}(\lambda
f)\exp(ik\xi)=\sum_{k=-\infty}^{\infty}i^{k}J_{k}(\lambda f)\exp(ik\theta-
ik\varphi)$ (64)

where $J_{k}$ is the Bessel function of the first kind of urder $k$ abram .
Finally, for a Gaussian $G$ in Eq.(22) we have
$G(\lambda_{1})G(\lambda_{2})=\Delta f/\sqrt{2\pi}\exp(-\lambda^{2}\Delta
f^{2}/4)$, and performing in the cylindrical coordinates, $\int
d\lambda_{1}d\lambda_{2}=\int_{0}^{\infty}\lambda
d\lambda\int_{0}^{2\pi}d\varphi$ the integration over $d\varphi$ , yields
Eq.(47).

##  Appendix C Derivation of Eq.(55)

Consider $L+1$ impulsive von Neumann pointers accurately measuring the
quantities $\hat{Q}^{\ell}$. $\hat{Q}^{0}$ and $\hat{Q}^{L}$ have non-
degenerate eigenvalues, the first measurement yields an outcome
$Q^{0}_{n_{0}}$ and leaves the system in a state $|q^{0}_{n_{0}}{\rangle}$.
The last measurement yields $Q^{L}_{n_{L}}$ and leaves the system in
$|q^{L}_{n_{L}}{\rangle}$. An extra weakly coupled pinter is added at
$t=t^{\prime}$, $t_{\ell}<t^{\prime}<t_{\ell+1}$ to measure
$\hat{Q}^{\prime}=\sum_{m^{\prime}=1}^{M^{\prime}}Q^{\prime}_{m^{\prime}}\hat{\pi}^{\prime}_{m^{\prime}}$.
Just after $t_{L}$ the state of the pointers is given by

$\displaystyle{\langle}f_{0}...f^{\prime}...f_{L}|\Psi_{Pointers}(t_{L}){\rangle}=\sum_{m_{1}...m^{\prime}...m_{L-1}=1}^{M_{1}...M^{\prime}...M_{L-1}}G^{\prime}(f^{\prime}-Q^{\prime}_{m^{\prime}})G(f_{L}-Q^{L}_{n_{L}})G(f_{0}-Q^{0}_{n_{0}})$
(65)
$\displaystyle\times\prod_{\ell=1}^{L-1}G(f_{\ell}-Q^{\ell}_{m_{\ell}})A_{S}(q^{L}_{n_{L}}....\leftarrow\hat{\pi}^{\ell+1}_{m_{\ell+1}}\leftarrow\hat{\pi}^{\prime}_{m^{\prime}}\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})$

where
$G^{*}(f_{\ell}-Q^{\ell}_{m_{\ell}})G(f_{\ell}-Q^{\ell}_{m^{\prime}_{\ell}})=\delta_{m_{\ell},m^{\prime}_{\ell}}\delta(f_{\ell}-Q^{\ell}_{m_{\ell}})$,
$\ell=0,...L$, and $G^{\prime}(f^{\prime})$ is very broad, so that
$G^{\prime}(f^{\prime}-Q^{\prime}_{m^{\prime}})\approx
G^{\prime}(f^{\prime})$. For the distribution of the readings we have

$\displaystyle
P_{Pointers}(f_{0}...f^{\prime}...f_{L})={\langle}f_{0}...f^{\prime}...f_{L}|\Psi_{Pointers}(t_{L}){\rangle}=\quad\quad\quad\quad\quad\quad$
(66)
$\displaystyle\sum_{m_{1}...m_{L-1}=1}^{M_{1}...M_{L-1}}\prod_{\ell=0}^{L}\delta(f_{\ell}-Q^{\ell}_{m_{\ell}})\sum_{\mu^{\prime},\nu^{\prime}=1}^{M^{\prime}}G^{\prime*}(f^{\prime}-Q^{\prime}_{\mu^{\prime}})G^{\prime}(f^{\prime}-Q^{\prime}_{\nu^{\prime}})\quad\quad\quad\quad\quad\quad$
$\displaystyle
A^{*}_{S}(q^{L}_{n_{L}}....\leftarrow\hat{\pi}^{\ell+1}_{m_{\ell+1}}\leftarrow\hat{\pi}^{\prime}_{\mu^{\prime}}\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})A_{S}(q^{L}_{n_{L}}....\leftarrow\hat{\pi}^{\ell+1}_{m_{\ell+1}}\leftarrow\hat{\pi}^{\prime}_{\nu^{\prime}}\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})$

and the (unnormalised) distribution of the weak pointer’s readings given that
the other outcomes are $Q^{L}_{n_{L}}...Q^{\ell}_{m_{\ell}}...Q^{0}_{m_{0}}$
is given by ($\epsilon\to 0$)

$\displaystyle
W_{Pointer}(f^{\prime}|Q^{L}_{n_{L}}...Q^{\ell}_{m_{\ell}}...Q^{0}_{n_{0}})=\int_{Q^{L}_{n_{L}}-\epsilon}^{Q^{L}_{n_{L}}+\epsilon}df_{L}...\int_{Q\ell_{m_{\ell}}-\epsilon}^{Q^{\ell}_{m_{\ell}}+\epsilon}df_{\ell}...\int_{Q^{0}_{n_{0}}-\epsilon}^{Q^{0}_{n_{0}}+\epsilon}df_{0}P_{Pointers}(f_{0}...f^{\prime}...f_{L})\quad\quad$
(67)
$\displaystyle\approx\sum_{\mu^{\prime},\nu^{\prime}=1}^{M^{\prime}}[|G^{\prime}(f^{\prime})|^{2}-\partial_{f^{\prime}}G^{\prime*}(f^{\prime})G^{\prime}(f^{\prime})Q^{\prime}_{\mu^{\prime}}-\partial_{f^{\prime}}G^{\prime}(f^{\prime})G^{\prime*}(f^{\prime})Q^{\prime}_{\nu^{\prime}}]\times\quad\quad\quad\quad\quad$
$\displaystyle
A^{*}_{S}(q^{L}_{n_{L}}....\leftarrow\hat{\pi}^{\ell+1}_{m_{\ell+1}}\leftarrow\hat{\pi}^{\prime}_{\mu^{\prime}}\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}})A_{S}(q^{L}_{n_{L}}....\leftarrow\hat{\pi}^{\ell+1}_{m_{\ell+1}}\leftarrow\hat{\pi}^{\prime}_{\nu^{\prime}}\leftarrow\hat{\pi}^{\ell}_{m_{\ell}}...\leftarrow
q^{0}_{n_{0}}),$

where we expanded the (broad)
$G^{\prime}(f^{\prime}-Q^{\prime}_{\mu^{\prime}})$ to the first order in its
(small) derivatives. For a Gaussian pointer $G^{\prime}(f^{\prime})$ is given
by Eq.(22), $G^{\prime*}(f)=G^{\prime}(f^{\prime})$, and $\int
f^{\prime}[|G^{\prime}(f^{\prime})|^{2}df^{\prime}=0$. Defining

$\displaystyle{\langle}f^{\prime}(Q^{L}_{n_{L}}...Q^{\ell}_{m_{\ell}}...Q^{0}_{n_{0}}){\rangle}\equiv\frac{\int
f^{\prime}W_{Pointer}(f^{\prime}|Q^{L}_{n_{L}}...Q^{\ell}_{m_{\ell}}...Q^{0}_{m_{0}})df^{\prime}}{\int
W_{Pointer}(f^{\prime}|Q^{L}_{n_{L}}...Q^{\ell}_{m_{\ell}}...Q^{0}_{m_{0}})df^{\prime}},$
(68)

and integrating by parts, yields Eq.(55).

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