How causal is quantum theory?
Does Bell’s theorem prevent the use
of causal explanations in quantum
mechanics?
Part I:
Locality, Bell’s version of locality, and its discontents
The greatest mystery in science?
• Locality = “things do not go faster than c” = “no
superluminal causal influence.”
Bell: “Now it is precisely in cleaning up intuitive ideas that one is likely to
throw the baby out with the bathwater. So the next step should be viewed with
the utmost suspicion...”
• Mystery: “nonlocality” but no more manifest signs of
superluminal influence.
• What does this “locality” mean, and why is it important?
• Is there a way to preserve locality while evading Bell’s
argument that this conflicts with QM?
The framework: local beables
Question: “For what class of theories can Bell
locality be defined?” (Classical? Hidden variable?..)
• Bell theorem has to do with (random) events
associated to spacetime regions.
• Any reasonable theory must give rise to
probabilistic predictions for events in spacetime.
Bell’s condition applies to these.
• Answer: “to all physical theories.”
Bell: “The beables must include the settings of switches and knobs on
experimental equipment, the currents in coils, and the readings of
instruments.”
Bell’s Condition: a Principle of
Common Cause
If
Then AB, BA or A{C}B
for all full specifications of the relevant past
B
A
+ Relativistic Causal Structure
C
Two clear responses to Bell
“Okay, then I will give
up on relativity.”
“Okay, then I will give
up on the PCC.”
Natural for…
Natural for…
• deBroglie-Bohm Pilot
waves
• Nelson-style mechanics
•
•
•
•
Operational QM
Collapse theories
Decoherent histories
Kent’s solution to the
LRP
Are other responses possible?
Bell’s theorem shows that:
QM is inconsistent with local theories?
Hidden variable
Separable
Free settings
Counterfactually
definite
Realistic
Antero-causal
Distant
outcomedefinite
What else could locality be?
The new “locality” will still be a condition on theories that sorted local from nonlocal.
But what does causal influence really mean?
“The philosophical treatment on the subject of causality extends over
millennia...”
Whatever the answer, locality would still have to obey
some uncontroversial criteria.
• In particular, locality must imply no-signalling.
A challenge for new accounts of
locality
“Here is the CHSH experiment. My new account of locality is [blah blah
blah]. As we can see, the predictions of QM are consistent with [blah
blah blah] so we can say that QM is local.”
• Step 1: Imagine a situation in which we observe superluminal
signalling.
• Step 2: Invent a crazy theory that could account for this and is still
consistent with [blah blah blah].
• Step 3: If [blah blah blah] does not rule out such an explanation it
is not a satisfactory account of locality.
Example: Free Settings is part of
locality
OFF ON
OFF
ON
Example: Free Settings is part of
locality
OFF ON
OFF
ON
“without such freedom I would not know how to formulate any
idea of local causality, even the modest human one
[signaling].” (J. S. Bell p.61).
“Zigzag”
Definiteness of distant outcomes: an
unwarranted assumption?
The account in QBism: for you, the measurement result in the
distant wing is indefinite.
Setting 1,1
Outcome +1,+1
A
Setting 1
Outcome +1
B
Setting ??
Outcome ??
Influences here are local because the correlation is only
realised when the agents meet.
The QBist account of locality
An Introduction to QBism with an Application to the Locality of Quantum
Mechanics, C. A. Fuchs, N. D. Mermin, R. Schack
Am. J. Phys., Vol. 82, No. 8, August 2014, 749-754, arxiv:311.5253
QBist quantum mechanics is local because its entire purpose is to enable any
single agent to organize her own degrees of belief about the contents of her own
personal experience. No agent can move faster than light: the space-time
trajectory of any agent is necessarily time-like. Her personal experience takes
place along that trajectory.
Therefore when any agent uses quantum mechanics to calculate
“[cor]relations between the manifold aspects of [her] experience”, those
experiences cannot be space-like separated. Quantum correlations, by their very
nature, refer only to time-like separated events: the acquisition of experiences by
any single agent. Quantum mechanics, in the QBist interpretation, cannot assign
correlations, spooky or otherwise, to space-like separated events, since they
cannot be experienced by any single agent. Quantum mechanics is thus
explicitly local in the QBist interpretation.
And that’s all there is to it.
The problem: crazy mechanics
For you, the measurement result in the distant wing is
indefinite.
Setting +1
Outcome +1
A
Setting +1
B
Outcome ??
Influences here are local because the correlation is only
realised when the agents meet. But wait…
The QBist account of locality
In QBism, Joe’s crazy mechanics is local because its entire purpose is to enable
any single agent to organize her own degrees of belief about the contents of her
own personal experience. No agent can move faster than light: the space-time
trajectory of any agent is necessarily time-like. Her personal experience takes
place along that trajectory.
Therefore when any agent uses crazy mechanics to calculate
“[cor]relations between the manifold aspects of [her] experience”, those
experiences cannot be space-like separated. Crazy correlations, by their very
nature, refer only to time-like separated events: the acquisition of experiences by
any single agent. Crazy mechanics, in the QBist interpretation, cannot assign
correlations, spooky or otherwise, to space-like separated events, since they
cannot be experienced by any single agent. Crazy mechanics is thus explicitly
local in the QBist interpretation.
And that’s all there is to it.
Separability vs. locality?
“[Separability] is a fundamental ontological principle which… asserts
that the contents of any two regions of spacetime separated by a
nonvanishing spatiotemporal interval constitute separable physical
systems, in the sense that (1) each possesses its own, distinct physical
state, and (2) the joint state of the two systems is wholly determined by
these separate states.” -- Howard
“We might… all along have been testing not simply local hidden variable
theories, but separable, local hidden variable theories.” -- Howard
“Our unhesitating acceptance of relativistic causal theories... involves an
assumption so basic to the thinking of most of us that we are not even
aware that we are making it…” -- Teller
“[each system in a Bell experiment] does not possess independent
properties of its own…” -- Redhead
“[i]f two systems are not separable, then there can be no interaction
between them, because they are not really two systems at all.” -- Howard
Defining Separability
A full specification of a region A is an event F such that all
other events associated to region A are either implied or
precluded by F.
Now we can define separability:
Bell locality is fine without separability
for all full specifications of the relevant past. So
there might be non-separable events. So what?
B
A
not in past
C
in past
Non-separability as inspiration to
weaken Bell’s assumptions?
Causes partially outside lightcones?
Causes and effects partially outside lightcones?
Operational events not localised in the wings?
Are other responses possible?
Bell’s theorem shows that:
QM is inconsistent with local
Hidden variable
Separable
How much of
the PCC can
we keep?
Free settings
Counterfactually
definite
Realistic
Pro-causal
Distant outcome
definite
Nonrelational
theories?
Three responses to Bell?
“Okay, then I will give
up on relativity.”
“Okay, then I will
givekeep a
“But
up on the PCC...”
remnant of the
PCC.”
Natural for…
Natural for…
Natural for…
• deBroglie-Bohm Pilot
• Operational QM
waves
• Collapse theories
• Nelson-style mechanics • Decoherent histories???
Part II:
New directions
How causal is quantum theory?
Causal reasoning in general
Conditional independence CI(A,B|C) means
Shannon entropies:
“Conditional mutual information”:
Conditional independence CI(A,B|C) is equivalent to
Causal reasoning in general: Bayesian
networks
Nodes represent random variables. Each variable depends only on its “parents”.
C
D
B
A
Alternative definition: given the values of the
variables on its parent nodes, each node is
conditionally independent of all nodes not to
the future.
Causal reasoning in general
The relation of Conditional independence is a ``semi-graphoid’’
CHSHB experiment as a Bayesian
network
A
X
B
L
Y
Work to do
• Understand causal inference.
• Reformulate quantum mechanics in analogy to
this.
• What makes the analogy especially good in
the case of QM, as opposed to other
hypothetical theories?
Echoes of causality in QM
• QM limits correlations in Bell
experiments.
no signals
quantum
local
• Quantum conditional mutual information is non-negative. The
condition that it =0 obeys the semi-graphoid axioms.
• Local operations on one system cannot increase mutual information
with a distant system.
First questions for building the analogy
• We want probabilities -> quantum states, but
probability distribution over events in
spacetime -> what?
• If quantum Bayesian networks are “more
causal” than the most general thing
imaginable – what is a general Bayesian
Network?
Finding a landscape of possibilities:
Generalising Bayesian networks
A
X
B
L
Y
What if we let nodes like L do anything that a PR box can do?
``Generalised probabilitistic theories’’ deal abstractly with preparations,
transformations and measurements. QM is one such theory. Used in
recovering QM from simple principles.
``Generalised Bayesian Networks’’ replace latent nodes with ``instruments’’
from such theories, and observable nodes with measurements on them.
Generalised Bayesian Networks:
an example
A
X
B
No conditional independences only
involving the observable nodes.
But there are constraints on p(abc).
The following is not allowed:
Y
Z
C
Surprise: in the generalised case, this restriction still holds.
Lesson: even with the most general non-signaling theories, there are
more restrictions than just conditional independences involving observed
nodes.
A new game in town
• “Characterising” quantum “correlations”.
• Quantum theory as a theory of inference?
• Generalising Baysian networks to quantum
theory and beyond.
• Taking indefiniteness seriously: another
possible approach?
Conclusions
• The is no cheap way to avoid the conclusion
that Bell’s theorem undermines any viable
relativistically causal theory.
• Freedom of settings, and reality of distant
outcomes, are part of locality.
• It is perhaps more fruitful to ask what aspects
of locality can be salvaged.
• The possibilities opened up by this are
beginning to be explored.
References
Appendices
Counterfactual definiteness an
assumption?
“Unperformed experiments have no results” – Asher Peres
Counterfactual definiteness: the results of experiments are determined by pre-existing
properties of the system being measured.
Problem: we did not use this as an independent assumption.
In fact, counterfactual definiteness is implied by local causality and the possibility of
perfect correlations.
“I disagree with counterfactual definiteness so much that I think this definition of
locality must be wrong.”
-- But what is the right definition?
Deriving factorisability
And similarly for the other two probabilities…
Lessons:
• Whatever lambda can be (e.g. whatever past region we use), it just has to be the same thing in
local causality and independence of settings to derive the theorem.
•
The proof is going to be hard to break as it is so simple and straightfoward.
“A matter of simple manipulation”
Two consequences of local causality
“Setting independence”:
“Parameter independence”:
Parameter dependence is different from superluminal signalling and does not imply it.
Parameter dependence fits best with violating relativity, if we think that an operational
intervention must be to the past of its effect.
Violating outcome independence more naturally allows us to consider theories that
are relativistic but violate the PCC.
Einstein Locality
A strong locality condition:
For any event A, there is an event C to its past that has the
same truth value for all dynamically allowed histories.
A
“Causal antecedent”
C
A third option
“A happens”:
“A does not happen”:
“It is not definite that A happened”:
New wriggle room: It is no longer clear what
constitutes a correlation or a cause.
If definiteness is just another ontological property like
the others (colour, mass etc.) this makes no difference.
But there are other possibilities.
What can’t we change?
OFF
ON
OFF
ON
This still demands a causal explanation.
What can we change?
OFF
ON
OFF
???
But what about this?
The choice: no longer demand that
the PCC covers this kind of thing.
Einstein Locality with indefiniteness
For any event A, there is an event C to its past that has the
same truth value for all dynamically allowed histories.
Things that we know are definite.
A
“Causal antecedent”
C
Superluminal signalling?
OFF ON
OFF
ON
If the decay is definite, this is
not allowed because it is
against freedom of settings.
If the decay is indefinite, it
cannot be the causal
antecedent of the output.
Conspiraton
The EPRB experiment
0
1
0
All non-signaling
correlations are
allowed
+???
+-
???
-
???
A0
A1
B0
1
-
B1