How causal is quantum theory Does Bells theorem

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 C + Relativistic Causal Structure

Two clear responses to Bell “Okay, then I will give up on relativity. ” “Okay, then I will give up on the PCC. ” Natural for… • de. Broglie-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? ED RSENvariable NO Hidden Separable Free settings Counterfactually SENED NOR definite Antero-causal NED NORSE Realistic 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]. As we can see, the predictions of QM are consistent with [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]. • Step 3: If [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

Example: Free Settings is part of locality 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… SIGNALS INAL M SUPERLU

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? ELP TH DOES NO Causes and effects partially outside lightcones? IGNALS S L A N I M SUPERLU Operational events not localised in the wings? S GNAL I S L A N I M SUPERLU LY ST SIL U J ’S T I AND

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… • de. Broglie-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 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 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 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 +? ? ? +- ? ? ? A 0 A 1 B 0 1 - B 1