On Causal Explanations of Quantum Correlations
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1 On Causal Explanations of Quantum Correlations Rob Spekkens Perimeter Institute From XKCD comics Quantum Information and Foundations of Quantum Mechanics workshop Vancouver, July 2, 2013
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4 Simpson s Paradox P(recovery drug) > P(recovery no drug) P(recovery drug, male) < P(recovery no drug, male) P(recovery drug, female) < P(recovery no drug, female) Recovery probability drug no drug male 180/300 = 60% 70/100 = 70% female 20/100 = 20% 90/300 = 30% combined 200/400 = 50% 160/400 = 40%
5 Simpson s Paradox recovery treatment gender
6 Simpson s Paradox P(recovery do (drug)) P(recovery observe (drug) ) causation correlation
7 What formalism can we use to describe causal relations? How do we come to have knowledge of causal relations? ( we = children, scientists, machine learning systems) How do we come to have knowledge of causal relations in uncontrolled experiments?
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9 What is a Causal Model?
10 Functional Causal Model Causal Structure Causal-Statistical Parameters W X S Y T
11 Causal Model Causal Structure Causal-Statistical Parameters W X S Y T
12 Reichenbach s principle No correlation without causation! If X and Y are correlated, then X Y or X Y or X Y
13 Causal Model Causal Structure Causal-Statistical Parameters W X S Y T Parentless variables are independently distributed
14 Given a causal model, what sorts of correlations can arise? W X S Y T Causal inference algorithms seek to solve the inverse problem
15 Inferring facts about the causal structure from statistical independences
16 Given a causal model, what sorts of correlations can arise? W X Y S T Def n: and are marginally independent Denote this
17 Given a causal model, what sorts of correlations can arise? W X Y S T Def n: and are conditionally independent given C Denote this
18 chain C 6? C (? Cj) fork C 6? C (? Cj) confounded cause C 6? C ( 6? Cj) collider C? C ( 6? Cj) Pair of forks D C 6? C ( 6? Cj) (? Cj; D)
19 Markov condition: The joint distribution induced by a causal model is such that every variable X is conditionally independent of its nondescendants given its parents, X 1 X 2 X 4 X 3 X 5
20 Semi-graphoid axioms X 1 X 2 X 4 X 3 The semi-graphoid axioms then imply X 5
21 The values of the causal-statistical parameters can imply further CI relations X 1 X 2 Suppose: X 4 X 3 X 5 Then: X 3? X 1
22 ? and no other independence relations? C
23 ? P () and no other independence relations C C P () P (Cj; ) P (C) P (jc) P (j; C)? No Fine-tuning!
24 key assumption of causal discovery algorithms No fine-tuning (a.k.a. faithfulness): causal model M is not fine-tuned relative to a probability distribution P if the conditional independences that hold in P continue to hold for any variation of the parameters in M
25 (? jc) and no other independence relations? C
26 (? jc) and no other independence relations C C C C
27 ?? j C and no other independence relations? C
28 ?? j C C and no other independence relations C C
29 llowing latent variables in the causal structure Notational Convention Observed variables:,, C, Latent variables:, ¹, º,
30 Does smoking cause lung cancer? S 6? C S C? S C? Suppose you also observe S? C j T and no other independences
31 (S C T) Latent common cause for S, C and T?
32 (S C T) Latent common cause or direct causal relation (or both) between S and C?
33 (S C T) So the causal structure must be of the form
34 (S C T) Marginal independence between remaining pairs?
35 (S C T) means So the causal structure must be of the form
36 (S C T) ssume one extra piece of data: S always precedes T
37 Inferring facts about the causal structure from the strength of correlations
38 Strength of Correlations Z X Y P(X,Y,Z) can have perfect three-way correlation X Z º ¹ Y P(X,Y,Z) is bounded away from perfect three-way correlation Janzing and eth, arxiv:quant-ph/ Steudel and y, arxiv:1010:5720 Fritz, New J. Phys. 14, (2012) ranciard, Rosset, Gisin, Pironio, arxiv:
39 Strength of Correlations Z ¹ P(X; Y; Z) = 1 2 [000] [111] X º Y Z X Y
40 Strength of Correlations Inequalities on P(, X,Y) X Y P( = j0; 0) + P( = j0; 1) +P( = j1; 0) + P( 6= j1; 1) 3 where P( = jx; Y ) := P a=b P( = a; = bjx; Y ) P( 6= jx; Y ) := P a6=b P( = a; = bjx; Y )
41 X Z º Testing candidate causal structures ¹ Y Inequalities on P(X,Y,Z) Inequalities on P(,,X,Y) X Y ssumptions about causal structure Instrumental inequalities (Chap. 8 of Pearl)
42 The lesson of causal inference for ell-inequality-violating correlations Joint work with Christopher Wood See: arxiv:
43 Q: For the observed correlations P(,,X,Y) what are the independences? : The set generated by (X Y), ( Y X), ( X Y) X E Y
44 X? Y X? jy?? Y jx X Y X E Y
45 X? Y X? jy? Y jx X Y X E Y
46 X? Y X? jy? Y jx + strength of correlation X Y X E Y
47 X? Y X? jy? Y jx + strength of correlation No causal explanation without fine-tuning! X E Y
48 What are the key assumptions of ell s theorem? standard response: Realism Local causality No superdeterminism No retrocausation What is proposed here: Reichenbach s principle No fine-tuning causal model is a directed acyclic graph supplemented with conditional probabilities X Y X ¹ Y X Y
49 Causal inference from correlations on a pair of binary variables Joint work with Ciaran Lee
50 Functional Causal Models where and have at most two binary variables as parents
51 Possible functional dependences of and on their parents = º ¹ = º ¹ 1 = º¹ = º¹ 1 = ¹ = ¹ 1 = ¹ = ¹ 1 Note:
52 P(; ) = p 00 [00] + p 01 [01] + p 10 [10] + p 11 [11] [01] [00] [10] [11]
53 = º ¹ = ¹
54 = º ¹ = ¹ [01] [00] [10] [11]
![= º = [01] p 00 p 10 =](/docs-images/88/116132198/images/55-1.jpg "p 01 p 11 [00] [10]")
55 = º = [01] p 00 p 10 = p 01 p 11 [00] [10] [11]
![= ¹ = ¹ [01] p 00 p 10 =](/docs-images/88/116132198/images/56-0.jpg "p 11 p 01 [00] [10] [11]")
56 = ¹ = ¹ [01] p 00 p 10 = p 11 p 01 [00] [10] [11]
![= º ¹ = ¹ [01] p 00 p 01](/docs-images/88/116132198/images/57-0.jpg "= p 11 p 10 [00] [10]")
57 = º ¹ = ¹ [01] p 00 p 01 = p 11 p 10 [00] [10] [11]
58 = º ¹ = ¹ [01] [00] [10] [11]
59 Understanding the subset of qubit channels induced by a single qubit ancilla C See: Narang and rvind, arxiv:quant-ph/
60 Quantum ayesian Inference and Quantum Causal Models joint work with Matt Leifer See: arxiv: , arxiv:
61 Classical Quantum Joint state P(R; S) ½ Marginalization P(S) = P R P(R; S) ½ = Tr ½ Conditional state P(SjR) ½ j P S P(SjR) = 1 Tr (½ j ) = I elief propagation P(S) = P R P(SjR)P(R) ½ = Tr (½ j ½ )
62 R S P(R; S) ½ P(SjR) = P(R; S)=P(R) ½ j = ½ 1=2 ½ ½ 1=2 P(S) = P R P(SjR)P(R) ½ = Tr (½ j ½ ) ½ j 0 S P(R; S) P(SjR) = P(R; S)=P(R) ½ ½ j = ½ 1=2 ½ ½ 1=2 R P(S) = P R P(SjR)P(R) ½ = Tr (½ j ½ ) ½ T j 0
63 Conventional expression In terms of conditional states Probability distribution for X Set of states on POVM on Channel from to P(X) f½ x g fex g E! ½ X ½ jx ½ Xj ½ j Instrument fe! x g ½ Xj
64 ction of quantum channel Conventional expression ½ = E! (½ ) In terms of conditional states ½ = Tr (½ j ½ ) orn s rule Y P(Y = y) = Tr (E y ½ ) ½ Y = Tr (½ Y j ½ ) Ensemble averaging X ½ = P x P(X = x)½ x ½ = Tr X (½ jx ½ X ) Composition of channels C E!C = E!C ± E! ½ Cj = Tr (½ Cj ½ j ) State update rule Y P(Y = y)½ y = E! y (½ ) ½ Y = Tr (½ Y j ½ )
65 Quantum Causal Models X Y P (X) P (Y ) P ( ) P (j ; X) P (j ; Y ) X S Y ½ X ½ Y ½ S ½ jxs ½ jy S P (; jx; Y ) = P P (j ; X)P (j ; Y )P ( ) ½ jxy = Tr S (½ jxs ½ jy S ½ S )
66 Deriving The Set of Quantum Correlations ssume: Framework for general theories of inference [Coecke and RWS, Synthese 186, 651 (2012)] ssume: Some principles about inference Quantum Theory of ayesian inference ssume: Quantum Theory of ayesian inference ssume: Facts about causal structure The set of quantum correlations
67 Outlook: Deriving Features of Quantum Correlations across time C See: Narang and rvind, arxiv:quant-ph/
68 Outlook: Deriving Features of Quantum Correlations across space Understanding multipartite entanglement SLOCC classes See: Walter et al. arxiv:
69 Conclusions ell s theorem can be recast without the nebulous assumption of realism. Superluminal causation, superdeterminism, and retrocausation are no longer ways out of the contradiction because they require fine-tuning Quantum causal models are a promising avenue for achieving a causal explanation of ell inequality violations (and noncontextuality-inequality violations) The ideas of causal inference have applications in QF and QI E.g. understanding the space of quantum correlations (across space and time) The ideas of QF and QI have applications in causal inference E.g. Using strengths of correlations to infer causal structure
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