# Consistent Quantum Causes

Robert B. Griffiths
Department of Physics
Carnegie Mellon University
Pittsburgh, PA 15213 Electronic address<EMAIL_ADDRESS>
(Version of 23 March 2023)

###### Abstract

The fact that projectors on a quantum Hilbert space do not in general commute
has made it difficult to extend into the quantum domain certain well-developed
and powerful ideas (think of directed acyclic graphs) for studying ordinary
classical causes. An approach is proposed which uses the single framework rule
of consistent histories quantum theory to identify intuitively plausible
microscopic quantum causes in the case of a simple gedanken experiment, and
certain extensions thereof, including the correlations that violate Bell
inequalities. The approach is simpler, both formally and intuitively, than
that developed under the heading of “Quantum Causal Models,” and might be
extended into a much more satisfactory quantum realization of important
classical ideas of causation.

###### Contents

  1. I Introduction
  2. II Causes
  3. III  Hilbert Space Quantum Mechanics
  4. IV  Beamsplitters and Interference
  5. V Spin Half
  6. VI  Generalizations
    1. VI A  Projective Measurements
    2. VI B Einstein-Podolsky-Rosen-Bohm (EPRB)
    3. VI C  Further Generalizations
  7. VII  Quantum Causal Models
  8. VIII  Summary and Conclusion

##  I Introduction

Identifying causes is useful not only in everyday life, but also in the
physics laboratory where the the macroscopic outcomes of measurements are
often interpreted as caused by invisible particles whose properties and motion
cannot be directly detected, and are governed by quantum mechanical rather
than classical laws. Causal thinking is required both to understand what is
going on, and also when designing equipment, and testing it to see if it is
working properly. Consequently any claim to a satisfactory understanding of
causality in the quantum domain must be able to deal in a sensible way with
laboratory phenomena, especially since they provide the empirical support for
quantum theory.

Efforts to understand causes in the everyday classical domain have resulted in
the development of some fairly sophisticated stochastic modeling to represent
causal structures using directed acyclic graphs.111Standard references are [1,
2]; for a compact introduction see Sec. 3 of [3]. Attempts to extend these
ideas into the quantum domain, starting with the pioneering work of Wood and
Spekkens[4] in connection with the correlations associated with Bell
inequalities, encountered serious difficulties. Efforts to deal with them led
to the development of what are called _quantum causal models_ (QCMs), which
are discussed below in Sec. VII. However, the QCM approach appears incapable
of dealing with simple laboratory situations of the kind discussed in Sec. IV.
This, along with its mathematical and conceptual complexity, suggests the need
for something better.

The present analysis is based on principles associated with the use of
projectors on the quantum Hilbert space to represent physical properties in
the manner proposed by von Neumann [5]. The fact that two projectors do not in
general commute is a central feature of what might be called _Hilbert space
quantum mechanics_ , which, along with some extensions used in the consistent
histories (CH) approach, is the subject of Sec. III. This follows some
elementary remarks, sufficient for the purposes of this paper, relating
probabilities, correlations, and causes, in Sec. II.

The simple gedanken experiments discussed in Sec. IV using Fig. 1 illustrate
some of what is needed for a satisfactory quantum theory of microscopic causes
and their effects. An analysis using the analogous case of a spin-half
particle in Sec. V shows how paying attention to alternative incompatible
refinements of Hilbert-space projectors, together with the single-framework
rule of CH, provides a consistent and relatively simple way of understanding
the causal relationships in Sec. IV. Sections IV and V form the heart of the
present paper.

The idea of how microscopic quantum properties cause outcomes of spin-half
measurements is easily extended, Sec. VI A, to the case of general projective
measurements, thus providing a reply to Bell’s tirade [6] against the way
measurements were (and still are) presented in textbooks. The extension of the
discussion in Sec. V to two spin-half particles, Sec. VI B, provides a simple
solution to the correlation paradox associated with Bell inequalities. While a
general theory of quantum causes lies outside the scope of the present paper,
Sec. VI C indicates some features of a plausible approach.

Following the discussion of the QCM approach in Sec. VII, a short summary in
Sec. VIII concludes the paper.

##  II Causes

A probabilistic model that uses standard (Kolmogorov) probabilities employs a
_sample space_ of mutually exclusive possibilities. In the case of stochastic
(random) time development the sample space consists of what we shall call
_histories_ , one and only one of which occurs during a particular
experimental run. To these a dynamical law assigns non-negative probabilities
that sum to $1$. In the quantum case a history consists of events, each
represented by a projector on the quantum Hilbert space, at a succession of
times. Using discrete times is adequate, and as in classical stochastic
processes, is mathematically simpler than the continuous case.

Given this setup, one can use standard probabilistic reasoning to calculate
correlations between events at different times. Let $F$ be an event at an
earlier time $t_{1}$ and $G$ one at a later time $t_{2}$. In the simplest case
one can call $F$ the _cause_ of $G$, and $G$ an _effect_ produced by $F$
provided the probabilities satisfy two conditions:

$\displaystyle\Pr(G\,|\,F)=\Pr(F,G)/\Pr(F)=1,$
$\displaystyle\Pr(F\,|\,G)=\Pr(F,G)/\Pr(G)=1.$ (1)

The first tells us that if the earlier $F$ occurred the later $G$ was
inevitable, and the second that if the later $G$ occurred, one can be certain
that it was preceded by $F$. We will only be concerned with the simplest
situations, and probabilities a bit less than $1$ will not trouble the working
physicist.

What at first seems odd is that the conditions in (1) remain the same if $F$
and $G$ are interchanged. Surely causes must _precede_ their effects, or we
would not call them causes. One can view this as arising from the fact that
while the fundamental microscopic laws of both classical and quantum physics
do not single out a direction of time, we as macroscopic beings live in a
world of increasing entropy in which we can remember the past but not the
future. Thus if “cause” is to be a useful part of reasoning about the world,
the requirement that $F$ precede $G$ in time well help extend our intuition,
based on everyday experience, into the mysterious quantum world.

But there are examples in which two events at different times satisfy (1), and
yet one would not want to call $F$ the cause of $G$. In particular, the
correlation between them may result from a yet earlier _common cause_ $C$. For
example, Charlie may send identical signals, both $0$ or both $1$, to two
parties Alice and Bob in different directions. That Alice receives the same
signal as Bob, but at an earlier time since she is closer to Charlie, hardly
means that her detecting it causes what Bob receives. Distinguishing what one
might call a “pseudo-cause” from a genuine cause is an important aspect of
causal modeling. One way of dealing with it is to imagine an _intervention_
which affects one signal rather than the other. E.g., Eve intercepts the one
sent to Bob and interchanges $0$ and $1$ before sending it on to its final
destination, thus changing the correlation. It helps to keep in mind the
slogan that “Correlations are Not (necessarily) Causes.”

##  III  Hilbert Space Quantum Mechanics

With apologies to von Neumann, who invented the term to refer to sophisticated
ideas in functional analysis, for our purposes a _Hilbert space_ is simply a
finite-dimensional complex linear vector space equipped with the usual inner
product. In Sec. III.5 of his difficult book [5], von Neumann identifies a
_projector_ , an operator $P=P^{\dagger}=P^{2}$, as the proper mathematical
representation of a quantum property. It is the counterpart of an _indicator_
function $J=J^{*}=J^{2}$ defined on a classical phase space: a function which
takes the value $1$ at those points that represent some physical property, and
$0$ elsewhere. For example, the phase-space points of a harmonic oscillator
with energy less than some value $E$ lie inside an ellipse centered at the
origin, and the corresponding indicator $J$ takes the value $1$ inside this
ellipse and $0$ outside.

Classical indicators can conveniently represent certain logical constructions,
as in Venn diagrams. Thus the complementary property “NOT $J$” is represented
by the indicator $I-J$, where $I$, the identity indicator, is equal to $1$
everywhere. The product $JK$ of two indicators represents the property “$J\
\mbox{\small AND}\ K$”, equal to $1$ at all points where both properties hold.
Similarly, the quantum complement of $P$ is $I-P$, with $I$ the identity
operator. However $PQ$ represents the property “$P\ \mbox{\small AND}\ Q$” if
and only if $P$ and $Q$ _commute_ : $PQ=QP$. Otherwise the product, in either
order, is not a projector, and cannot represent a physical property.

In many ways the noncommutation of projectors marks the boundary between
classical and quantum physics. It underlies various _uncertainty relations_ ,
and is at the heart of numerous conceptual headaches and quantum paradoxes.
Von Neumann was aware of this, and along with Birkhoff[7] invented a quantum
logic to try and make sense of “$P\ \mbox{\small AND}\ Q$” in the noncommuting
case. It is so complicated that it has been of little help in understanding
quantum mechanics and resolving its conceptual difficulties. The much simpler
Consistent Histories222 A compact overview of CH will be found in [8], while
[9] is a detailed treatment that remains largely up-to-date. For CH as a form
of quantum logic, see [10] and Ch. 16 of [9]. How CH principles resolve
quantum paradoxes is discussed in Chs. 20 to 25 of [9], and in [11] and [12].
(CH) approach treats “$P\mbox{\small AND}\ Q$” as _meaningless_ in the
noncommuting case, so one never has to discuss its meaning, and limits quantum
reasoning to collections of commuting projectors called _frameworks_. One
starts with a _projective decomposition of the identity_ (PDI), the
counterpart of a _sample space_ in standard (Kolmogorov) probability theory, a
collection $\\{P_{j}\\}$ of mutually orthogonal projectors that sum to the
identity:

$I=\sum_{j}P_{j},\quad P_{j}=(P_{j})^{\dagger}=(P_{j})^{2},\quad
P_{j}P_{k}=\delta_{jk}P_{j}.$ (2)

The corresponding _event algebra_ consists of all the projectors which are
members of, or sums of members of the PDI. The CH term “framework” can refer
to either the PDI or the corresponding event algebra. Its _single framework
rule_ states that any sort of logical reasoning must be carried out using a
collection of commuting projectors; reasoning which makes simultaneous use of
noncommuting projectors is invalid. Note that any collection ${\mathcal{C}}$
of commuting projectors generates, via complements and products, a PDI, and
thus an associated framework, so the single framework rule is in this sense
not very restrictive. What is ruled out is the application of classical
reasoning to situations in which two or more projectors do not commute, the
source of an endless number of unresolved quantum paradoxes.

Here is a simple example that will help illustrate these ideas. Let
$|0\rangle$ and $|1\rangle$ be the two lowest energy eigenstates, with
energies $\epsilon_{0}$ and $\epsilon_{1}$, of an atom. Suppose the energy is
not greater than $\epsilon_{1}$. This fact can be represented by the projector

$R=[0]+[1],$ (3)

using the notation $[\psi]=|\psi\rangle\langle\psi|$ for the projector
corresponding to the normalized ket $|\psi\rangle$. Does the truth of $R$
imply that the energy is either $\epsilon_{0}$ or $\epsilon_{1}$? No, for any
superposition of $|0\rangle$ and $|1\rangle$ also lies in the subspace on
which $R$ projects, and so possesses the property $R$, and most such
superpositions do not have well-defined energies. However, if one uses the
framework corresponding to the PDI

$\\{[0],\,[1],\,I-R\\},$ (4)

then in this framework $R$ implies that the energy is $\epsilon_{0}$ or
$\epsilon_{1}$. There are other incompatible frameworks, but (4) is the one
needed in order to discuss the energy.

The use of PDIs at a single time is easily extended to a family of histories
using the fact that a sequence of quantum properties (projectors) at a
succession of times is an element of a _history Hilbert space_ :333As first
pointed out by Isham[13].  a tensor product of one-time Hilbert spaces.

##  IV  Beamsplitters and Interference

Figure 1: (a) Photon source $S$ and beamsplitter $BS_{1}$ followed by two
detectors at unequal distances. (b) Add mirrors and a second beamsplitter
$BS_{2}$ to form a Mach-Zehnder interferometer. $BS_{2}$ can be removed, as
sketched to the right of the dashed line.

The first gedanken experiment is shown in Fig. 1(a). A photon is sent into a
beamsplitter and emerges as a coherent superposition of amplitudes in the two
output ports,

$|\psi_{t}\rangle=\alpha|a_{t}\rangle+\beta|b_{t}\rangle,$ (5)

at time $t$. Here $|a_{t}\rangle$ is a wave packet traveling along the $a$
path; $|b_{t}\rangle$ the $b$ path. The two paths lead to detectors $D_{a}$
and $D_{b}$ located at unequal distances from the beamsplitter. In a
particular run of the experiment the photon will be detected in either $D_{a}$
or $D_{b}$, but never in both. Given a large number of runs, the fraction in
which $D_{a}$ triggers will be approximately $|\alpha|^{2}$; those in which
$D_{b}$ triggers $|\beta|^{2}$. Someone with experience in the laboratory will
tend to think that on a particular run in which $D_{a}$ is triggered, the
photon after emerging from the beamsplitter followed path $a$ and was absent
from path $b$, while in a run in which $D_{b}$ detects the photon it followed
path $b$ and was absent from $a$. But since since the photon (or other quantum
particle, such as a neutron, in an analogous experiment) is invisible, it
would seem useful to do additional tests to corroborate these ideas. Here are
some possibilities.

If during certain runs the $a$ path is blocked by some macroscopic absorbing
object—think of this as an intervention—$D_{a}$ never triggers, while the
fraction of runs with a $D_{b}$ detection remains unchanged. Or replace the
block by a mirror which deflects the photon off the $a$ path towards a third
detector $D^{*}_{a}$. Its average rate is that of $D_{a}$, whereas the rate of
$D_{b}$ remains unchanged. These observations support the idea that some sort
of random process occurs at $BS_{1}$, where the photon chooses
(anthropomorphic language) either the $a$ or the $b$ path, and then follows it
till it reaches the corresponding detector. In addition, if the photon is
emitted at a well-defined time, and the time of detection measured with
sufficient precision, the time to reach detector $D_{a}$ is less than that
required to reach $D_{b}$.

This seems very odd from a textbook perspective in which random processes only
occur when something is measured, resulting in the collapse of a wavefunction.
Let us explore that idea. Since detector $D_{a}$ is closer to $BS_{1}$ than
$D_{b}$, one might suppose that when the $\alpha|a\rangle$ part of
$|\psi_{t}\rangle$ in (5) reaches detector $D_{a}$ and the photon is detected
there, this instantly removes the amplitude traveling towards $D_{b}$, which
will not be triggered on this particular run. Such a nonlocal effect seems
strange, and may be hard to reconcile with special relativity. An additional
difficulty arises in runs in which $D_{a}$ does _not_ detect the particle.
Does such nondetection result in the amplitude collapsing onto path $b$ at the
instant of nondetection? Or does the later detection by $D_{b}$ (retro)cause
an earlier collapse on path $a$ that prevents detection by $D_{a}$?

Might it be that some sort of _decoherence_ process, caused by an interaction
of the photon with the environment, either at the beam splitter or due to gas
particles on the way to the detector, is at work, so that the coherent state
(5) is no longer is the correct quantum description when the two paths have
separated by a macroscopic distance? Decoherence removes quantum interference,
so can be checked with the alternative experimental arrangement shown in Fig.
1(b). Here a second beamsplitter, $BS_{2}$, has been added, so that along with
the first beamsplitter $BS_{1}$ the result is a Mach-Zehnder interferometer.
Let us suppose that the beamsplitters have been chosen in such a way that the
photon always emerges on path $c$, leading to a detection by $D_{c}$, never
$D_{d}$. Decoherence along the $a$ or $b$ path inside the interferometer would
eliminate the interference, so can be excluded if the interferometer is
functioning properly.

Motivated by Wheeler’s delayed choice paradox[14], let us suppose that
$BS_{2}$ can be moved out of the way, the situation sketched at the far right
of Fig. 1(b), in which case we are back at something that resembles Fig. 1(a),
but now the two paths cross. That nothing special happens when they cross can
be checked, as before, by blocking the paths at various points before they
cross, and by moving $D_{c}$ further away and checking the timing. Thus one
concludes (as did Wheeler) that the photon detected by $D_{c}$ was earlier
following path $a$ before reaching the crossing point, and likewise it was
following path $b$ in a run in which it was detected by $D_{d}$. Having
considered the case with $BS_{2}$ absent, we must now try and understand what
happens when it is present. One can check that blocking either the $a$ or the
$b$ path inside the interferometer not only reduces, by the expected amount,
the number of times either $D_{c}$ or $D_{d}$ triggers, but in addition
sometimes $D_{d}$ triggers instead of $D_{c}$. What is even more interesting
is to insert variable phase shifters in each of the arms. While these can
change the ratio of the $D_{d}$ to $D_{c}$ counts, it remains fixed if _both_
phases are changed by the _same_ amount. Thus the photon must in some sense
have been simultaneously present in _both_ arms of the interferometer. Very
odd given that with $BS_{2}$ absent we had good reason to believe the photon
was in either the $a$ arm or the $b$ arm after emerging from $BS_{1}$. Can
removing $BS_{2}$, as against leaving it in place, influence what happened
_before_ the photon arrived at the crossing point? Providing coherent answers
to questions of this sort is what one should expect from an adequate theory of
quantum causes.

##  V Spin Half

It is useful to consider a spin-half analog of the situation in Fig. 1 of Sec.
IV, since the mathematics of a two-dimensional Hilbert space, which is all
that is needed to discuss spin orientation, is simpler than what is needed to
describe the different spatial positions of the photon in Fig. 1, and
simplicity is important for understanding quantum systems at the most
fundamental level. Think of a spin-half silver atom moving along a straight
line until it enters a Stern-Gerlach (SG) measuring device with a field and
field gradient in the $z$ direction, so that it measures $S_{z}$, yielding one
of the two values $+1/2$ and $-1/2$ in units to $\hbar$. Let $S_{z}=+1/2$
correspond to the $a$ path in Fig. 1(a), and its extension into the $c$ path
in Fig. 1(b), and $S_{z}=-1/2$ be the counterpart of the $b$ and $d$ paths.
The SG device replaces the pair of detectors $D_{a}$ and $D_{b}$ in Fig. 1(a)
or $D_{c}$ and $D_{d}$ in part (b).

The beamsplitters in Fig. 1 are replaced by limited regions of uniform
magnetic field chosen so that as the atom moves through the region its spin
precesses by the desired amount: a unitary transformation on its $d=2$
dimensional Hilbert space. Given a source of spin-polarized atoms, e.g., the
upper beam emerging from a SG apparatus, an appropriate uniform field allows
the preparation of particles with a spin polarization $S_{v}=+1/2$ in any
spatial direction $v$. Similarly, if the final SG measurement apparatus is
preceded by a suitable region of uniform field, the combination allows the
measurement of any spin component $S_{w}$, provided passage through the region
of uniform field maps $S_{w}=+1/2$ to $S_{z}=+1/2$, and $S_{w}=-1/2$ to
$S_{z}=-1/2$. That this analog of $BS_{2}$ in Fig. 1(b) is functioning
properly can be checked, at least in certain cases, by the less convenient
process of rotating the final SG magnet.

Suppose that in this way Alice uses the analog of $BS_{1}$ to prepare a state
with $S_{x}=+1/2$ which travels on to Bob, whose SG measures $S_{z}$, the
analog of Fig. 1(b) with $BS_{2}$ absent. The results are random: in roughly
half the runs the outcome corresponds to $S_{z}=+1/2$, and the other half to
$S_{z}=-1/2$. What can one say about the spin state during the the time
interval when the particle is moving through the central field-free region on
its way from Alice’s preparation to Bob’s measurement? Alice might say
$S_{x}=+1/2$, since she has carried out a lot of checks to test whether her
apparatus is working properly, perhaps sending the beam into an SG with field
gradient in the $x$ direction. Bob is equally careful in checking his
apparatus, and is sure that $S_{z}$ was $-1/2$ in those runs where it
indicated this value, and $+1/2$ in the others. But there is no room in the
spin-half Hilbert space, no projector, that can represent the combination
“$S_{x}=+1/2\ \mbox{\small AND}\ S_{z}=-1/2$” or “$S_{x}=+1/2\ \mbox{\small
AND}\ S_{z}=+1/2$”. We seem to have a dispute between two parties. What shall
we to do?

Let us send Bob off on vacation. Alice is perfectly capable of designing and
building both the preparation and the measurement apparatus, and, as with any
competent experimenter, is conversant with the relevant theory. Thus for any
given run she knows both the preparation and the measurement outcome, and
faces the problem of what these data tell her about the spin state at an
intermediate time. She begins with the spin-half identity operator $I$, which
is consistent with any data but totally uninformative. This she can refine
using either of two different incompatible PDIs or frameworks:

${\mathcal{F}}_{x}:\\{[x+],[x-]\\}\quad{\mathcal{F}}_{z}:\\{[z+],[z-]\\},$ (6)

where $[x+]$ is the projector on the state $S_{x}=+1/2$; similarly for $[x-]$,
$[z+]$, and $[z-]$. If Alice uses ${\mathcal{F}}_{x}$ she can conclude, on the
basis of her knowledge of the preparation, that at this intermediate time the
property was $[x+]$, $S_{x}=+1/2$. By using ${\mathcal{F}}_{z}$ she can infer
that in a run in which the measurement outcome corresponds to
$S_{z}=+1/2$—recall that she has carefully calibrated and checked the
measurement device ahead of time—that on this run $S_{z}$ was $+1/2$ and not
$-1/2$. Similarly a measurement outcome corresponding to $S_{z}=-1/2$ leads
her to conclude that $S_{z}$ earlier had this value. Thus the conditions in
(1) are satisfied, and Alice can properly identify the intermediate time
$S_{z}$ value as the _cause_ of the later measurement outcome. In addition,
having passed the final exam in an up-to-date introductory quantum mechanics
course, Alice knows the rules needed to reach sensible conclusions in a regime
where quantum projectors do not commute, and that assuming classical unicity
will only lead to paradoxes such as Kochen-Specker and Bell nonlocality. (The
latter is discussed in Sec. VI B below.)

While the above discussion identifies the central issue, a number of details
needed for a more complete description have been omitted. These would include
the use of quantum histories (Sec. III), a proper quantum description, at
least in principle, of the measurement apparatus and its macroscopic outcomes,
and assigning probabilities to histories in a closed quantum system in a
manner satisfying appropriate consistency conditions. For these we refer the
reader to [11] and references given there. The fundamental point remains the
same: Alice, in order to identify the _cause_ of the $S_{z}$ measurement
outcome, must _choose_ to use ${\mathcal{F}}_{z}$ rather than
${\mathcal{F}}_{x}$ or some other framework. There is nothing irrational about
this choice, or similar choices made every day in the laboratory by
researchers who regularly interpret their results in terms of the microscopic
causes their apparatus was designed to detect.

##  VI  Generalizations

###  VI A  Projective Measurements

The ideas just discussed in the case of a spin-half particle can immediately
be generalized to that of _projective measurements_ , in which a particular
collection of quantum properties of a microscopic system, represented by a
PDI, is associated in a one-to-one fashion with a set of distinct macroscopic
outcomes (“pointer positions” in the archaic language of quantum foundations).
That is, there is a one-to-one association between the microscopic cause
represented by the projector $P_{j}$ and the later pointer at position $j$.
This correspondence can be checked through a set of calibration runs in which
particles with a particular microscopic property are repeatedly sent into the
measurement apparatus, and lead to the expected macroscopic outcomes. An
appropriate quantum analysis using histories then justifies thinking of the
previous microscopic property as the cause of the later. In this sense a
projective measurement actually measures something, a property of the measured
system just before the measurement took place. Bell’s tirade in [6] can be
answered by associating what he called “beables” with quantum properties
represented by Hilbert space projectors. Textbooks assert that incompatible
properties, thus corresponding to incompatible PDIs, cannot be simultaneously
measured, which is true. What they ought to say is that this impossibility
arises from the fact that such joint properties do not exist, and even skilled
experimenters cannot measure what isn’t there.

The reader is referred to [11], along with Chs. 17 and 18 of [9], for a
general discussion of projective measurements, as well as how to think about
POVMs (positive-operator-valued measurements), where identifying causes is a
more subtle matter. The infamous _measurement problem_ , unitary time
evolution leading to macroscopic coherent superpositions (as in Schrödinger’s
cat) which cannot be given a sensible physical interpretation, dissolves once
one adopts Born’s proposal (with which von Neumann concurred, and which is
fundamental to the CH approach) that Schrödinger’s wave is a means of
calculating probabilities, rather than a unique description of what is going
on in a physical system. The point is that there are various different
possible and mutually incompatible quantum frameworks for describing
measurement processes, and if one wants to discuss what goes on in the
laboratory it is reasonable to adopt one that makes physical sense to the
experimenter.

###  VI B Einstein-Podolsky-Rosen-Bohm (EPRB)

An example that illustrates the general case of a projective measurement as
discussed above arises in the famous Einstein-Podolsky-Rosen paradox [15],
here considered in Bohm’s spin-half version [16], which is easily translated
into a discussion of polarizations in the case of two photons produced by down
conversion, as measured in tests of Bell inequalities. Two spin-half particles
$S_{a}$ and $S_{b}$ are prepared in a singlet state and sent off in different
directions to Alice and Bob who measure particular spin components. The
outcomes can be understood as caused by the corresponding properties of the
individual particles just before the measurements took place. The experimental
results (using photons) have by now amply confirmed the straightforward
quantum mechanical predictions.

A fundamental mistake behind derivations of Bell inequalities, such as CHSH,
is the idea that one can combine statistics from different runs in which
incompatible properties are measured. Suppose Alice measures $S_{x}$ on
certain runs, and $S_{z}$ on others. It is tempting to suppose that on a run
in which $S_{x}$ was measured, this particle also possessed a value of $S_{z}$
which would have been revealed had $S_{z}$ been measured instead. Such
counterfactual reasoning assumes classical unicity, and fails when dealing
with incompatible quantum properties. When invalid classical reasoning of this
sort is abandoned, derivations of Bell inequalities fall apart, and mysterious
nonlocal influences vanish; see [12] for details.

From a causal perspective, if Alice measures $S_{x}$ and Bob measures $S_{z}$,
to consider a particular example, the respective measurement outcomes, say
$+1/2$ and $-1/2$ can be assigned to the corresponding causes, the properties
$S_{ax}=+1/2$, and $S_{bz}=-1/2$, just before the measurements took place. If
the particles have been traveling through a field-free region, these
properties can be traced back to a common cause right after the entangled
state was first created. One could also introduce interventions by letting
either or both of the particles pass through regions of uniform magnetic field
that rotate the spin polarization, in order to check that the correlations do,
indeed, arise from a common cause.

###  VI C  Further Generalizations

There is no reason why the CH approach that works in the case of projective
measurements and POVMs cannot be extended to more complicated situations. The
use of appropriate histories to model quantum stochastic time development
would seem to be sufficient for analyzing causes in most situations of
physical interest, or at least there is no reason to think otherwise until
counterexamples have been identified. For a closed quantum system
probabilities can be assigned to a family of histories using the extended Born
rule as long as certain _consistency conditions_ are satisfied.444The earliest
formulation of these conditions [17] should be replaced by the correct version
due to Gell-Mann and Hartle [18]: the vanishing of off-diagonal elements in
their decoherence functional, or orthogonality of the chain operators in Ch.
10 of [9]. The correct form is given in [8]. Note that a (quantum!) measuring
apparatus can itself be part of such a closed system. Since such a family
satisfies the usual conditions of a classical stochastic model, the general
methods used to analyze causes in the classical case can at once be applied to
the quantum domain. A possible difficulty, worth exploring through specific
models, could arise in the case of interventions used to distinguish genuine
causes from mere correlations, since one would need to pay some attention to
consistency conditions and the single framework rule in the extended histories
framework needed to accommodate the intervention.

The situation with open quantum systems is more complicated, and deserves
further exploration. An approach in which the environment is modeled by a
separate quantum system, which together with the one of interest forms a
closed quantum system to which CH rules can be applied, is an obvious
possibility. Note that there is no difficulty, at least in principle, in
making a measuring apparatus part of a closed system; see [11] and Chs. 17 and
18 in [9] for the basic idea.

##  VII  Quantum Causal Models

The pioneering work of Wood and Spekkens[4] showed that a straightforward
attempt to apply certain ideas of causal modeling which had shown considerable
success in ordinary (classical) situations, such as identifying causes of
disease, ran into serious difficulties when applied to the EPRB situation
discussed in Sec. VI B. Subsequent work attempting to arrive at more
satisfactory results led to the study of _Quantum Causal Models_ (QCM) in an
effort to understand correlations and causes in a quantum context. Items [19,
20, 21] are selected from a rather substantial literature on the topic, and
provide some references to other work. Neither the mathematics nor the
associated concepts are easy to understand; perhaps [20] is the most
accessible for someone unfamiliar with this approach.

So far as this author can tell, it is unable to address the simple laboratory
situations discussed above in Sections IV and V, as it lacks means to deal
with noncommuting Hilbert-space projectors. When it comes to EPRB
correlations, Shrapnel [20] describes the QCM approach as a “sledgehammer”,
which is certainly apt when it is compared with the simplicity of the analysis
in Sec. VI B, where the measurement outcomes are caused by microscopic
properties of the particles just before the measurements take place, using
devices designed for just that purpose, and carefully tested to ensure their
reliability. No sledgehammer is needed when microscopic descriptions use
quantum Hilbert space projectors.

Two reasons among others may have led to the complexity of the QCM approach.
The first is an inadequate understanding of what quantum measurements measure,
as illustrated by Spekkens’ toy model [22] which makes no mention of Hilbert-
space quantum properties. The second is what Spekkens refers to in [23] as the
_Leibniz principle_ , that in a situation with specific macroscopic inputs and
outputs, there is a unique ontological, thus microscopic, description
applicable at an intermediate time. Clearly this is classical unicity as
opposed to the quantum pluricity used by Alice in the example discussed in
Sec. V.

None of this is to say that the QCM approach is without value. However, it
would seem worthwhile for its practitioners to pay attention to Hilbert space
projectors, noncommutation, PDIs, and the quantum histories used in CH, in
order to achieve a more useful, and hopefully much simpler, connection between
the mathematical formulation and laboratory physics. There is much that needs
to be understood about the quantum physics of systems at successive times, and
the identification of causes consistent with fundamental quantum principles
could play a very useful role in such investigations.

##  VIII  Summary and Conclusion

The analysis of quantum causes in this paper is based upon certain key ideas,
foremost among them the notion that microscopic causes useful in understanding
the results of laboratory experiments are physical properties represented
mathematically by projectors on the quantum Hilbert space, Sec. III. A key
difference between projectors and the corresponding indicator functions on a
classical phase space is that when projectors do not commute their product is
not a projector, and thus cannot represent a physical property. Hence in
Hilbert space quantum mechanics one encounters a _pluricity_ of incompatible
descriptions, each based upon a particular collection of commuting projectors,
or equivalently, the corresponding PDI. This contrasts with the _unicity_ of
classical phase space, where no such incompatibility arises.

Ignoring noncommutation leads to all sorts of insoluble quantum paradoxes,
including Wheeler’s delayed choice, Sec. IV. The consistent histories
approach, illustrated in the spin-half example in Sec. V, avoids such
paradoxes through its single-framework rule that specifies that logical
reasoning about a quantum system must be based upon a single PDI, even though
the physicist may very well employ different, incompatible PDI’s when
analyzing different aspects of an experiment.

The examples in Sec. IV and V involve a two-state projective measurement, but
the idea is easily extended to a general projective measurement, Sec. VI A,
resolving Bell’s complaint [6] by identifying “beables”, his terminology, with
properties represented by Hilbert space projectors. In particular the
correlations in the Einstein-Podolksy-Rosen-Bohm paradox, Sec. VI B, can be
explained very simply as resulting from a common cause, with no need for the
complicated “sledgehammer” developed under the heading of Quantum Causal
Models, Sec. VII. A much simpler way of consistently embedding ideas from the
classical theory of causes (DAGs and all that) in a quantum context should be
possible using quantum histories, Sec. VI C, but this remains to be developed.

In closing it is worth remarking that the CH approach, while it is in certain
ways counterintuitive—perhaps necessary, given that modern quantum mechanics
was developed precisely because classical intuition failed—is much simpler
than the earlier quantum logic of Birkhoff and von Neumann, and can be taught
to undergraduate students in an introductory quantum mechanics course. It may
not resolve all the conceptual difficulties of quantum theory, but its success
in dealing with a number of quantum paradoxes, as in Chs. 20 to 25 of [9],
suggests it is worth further exploration.

## Acknowledgements

The author expresses his appreciation to Carnegie-Mellon University and its
Physics Department for continuing support of his activities as an emeritus
faculty member.

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