Quantum Causal Networks

1. Introduction The Markov Evolution Law Quantum Causal Networks Local Positivity: A Lament Conclusions Quantum Causal Networks: A Quantum Informationish Approach to Causality in Quantum Theory M. S. Leifer Institute for Quantum Computing University of Waterloo Perimeter Institute Dec. 10th 2007 / Quantum Information in Quantum Gravity Workshop M. S. Leifer Quantum Causal Networks

2. Introduction The Markov Evolution Law Quantum Causal Networks Local Positivity: A Lament Conclusions Outline Introduction 1 The Markov Evolution Law 2 Quantum Causal Networks 3 Local Positivity: A Lament 4 Conclusions 5 M. S. Leifer Quantum Causal Networks

3. Introduction Disclaimer The Markov Evolution Law Religious Views Quantum Causal Networks Lessons from Quantum Information Local Positivity: A Lament Related Formalisms Conclusions Disclaimer! The author of this presentation makes no claims as to the accuracy of any speculations about quantum gravity contained within. Such speculations do not necessarily represent the views of the author, or indeed any sane person, living or dead. Any similarity to existing formalisms for theories of quantum gravity is purely coincidental. The ﬁniteness of all Hilbert spaces in this presentation does not reﬂect any views about the discreteness of spacetime. You can probably do everything with C ∗ -algebras if you prefer. This is a work in progress. M. S. Leifer Quantum Causal Networks

4. Introduction Disclaimer The Markov Evolution Law Religious Views Quantum Causal Networks Lessons from Quantum Information Local Positivity: A Lament Related Formalisms Conclusions The Church of The Smaller Hilbert Space Core Beliefs A preference for density operators, CP-maps and POVMs over state-vectors, unitary evolution and projective measurements. The latter turn out to be fairly boring in the present framework. States belong to agents, not to systems. “Other authors introduce a wave function for the whole Universe. In this book, I shall refrain from using concepts that I do not understand.” Peres. M. S. Leifer Quantum Causal Networks

9. Introduction Disclaimer The Markov Evolution Law Religious Views Quantum Causal Networks Lessons from Quantum Information Local Positivity: A Lament Related Formalisms Conclusions Three lessons from Quantum Information Quantum Theory is a noncommutative, operator-valued, 1 generalization of probability theory. We don’t have to invoke gravity in order to encounter 2 scenarios with unusual causal structures. Causal ordering of events matters a lot less than you might 3 think for describing quantum processes. M. S. Leifer Quantum Causal Networks

12. Introduction Disclaimer The Markov Evolution Law Religious Views Quantum Causal Networks Lessons from Quantum Information Local Positivity: A Lament Related Formalisms Conclusions Unusual Causality without Gravity Quantum Protocols M. S. Leifer Quantum Causal Networks

13. Introduction Disclaimer The Markov Evolution Law Religious Views Quantum Causal Networks Lessons from Quantum Information Local Positivity: A Lament Related Formalisms Conclusions Unusual Causality without Gravity Quantum Computing Y Z W X U V Quantum Circuits Measurement Based Quantum Computing M. S. Leifer Quantum Causal Networks

14. Introduction Disclaimer The Markov Evolution Law Religious Views Quantum Causal Networks Lessons from Quantum Information Local Positivity: A Lament Related Formalisms Conclusions Irrelevance of Causal Ordering The “small miracle” of time in quantum information B B A B C A A In all three causal scenarios the predictions can be calculated from the formula Tr (MA ⊗ MB ρAB ). This follows from the Choi-Jamiołkowski isomorphism and is very useful in qinfo, e.g. crypto security proofs. c.f. P(X,Y) - I want a formalism in which it is that obvious. M. S. Leifer Quantum Causal Networks

18. Introduction Disclaimer The Markov Evolution Law Religious Views Quantum Causal Networks Lessons from Quantum Information Local Positivity: A Lament Related Formalisms Conclusions Related Formalisms Hardy’s Causaloid gr-qc/0509120, arXiv:gr-qc/0608043. Consistent/Decoherent Histories - particularly Isham’s version quant-ph/9506028. Aharonov et. al.’s mutli-time states arXiv:0712.0320. Markopoulou’s Quantum Causal Histories hep-th/9904009, hep-th/0302111. M. S. Leifer Quantum Causal Networks

22. Introduction Disclaimer The Markov Evolution Law Religious Views Quantum Causal Networks Lessons from Quantum Information Local Positivity: A Lament Related Formalisms Conclusions Comparison to Quantum Causal Histories I don’t require global unitarity. I only deal with ﬁnite causal structures. No free choice of initial conditions. No free entanglement in the initial state. Note: Quantum Causal Histories could have been called Quantum Causal Networks or Quantum Bayesian Networks. M. S. Leifer Quantum Causal Networks

23. Introduction Disclaimer The Markov Evolution Law Religious Views Quantum Causal Networks Lessons from Quantum Information Local Positivity: A Lament Related Formalisms Conclusions Comparison to Quantum Causal Histories I don’t require global unitarity. I only deal with ﬁnite causal structures. No free choice of initial conditions. No free entanglement in the initial state. Note: Quantum Causal Histories could have been called Quantum Causal Networks or Quantum Bayesian Networks. M. S. Leifer Quantum Causal Networks

24. Introduction The Markov Evolution Law Conditional Density Operators Quantum Causal Networks The Partial Transpose Local Positivity: A Lament The Markov Evolution Law Conclusions Conditional Density Operators Deﬁnition A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a positive operator that satisﬁes TrB ρB|A = IA , where IA is the identity operator on HA . P(Y |X ) = 1 c.f. Y Note: A density operator determines a CDO via −1 −1 ρB|A = ρA 2 ρAB ρA 2 . 1 1 Notation: M ∗ N = M 2 NM 2 ρB|A = ρ−1 ∗ ρAB and ρAB = ρA ∗ ρAB . A c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(X )P(Y |X ). M. S. Leifer Quantum Causal Networks

30. Introduction The Markov Evolution Law Conditional Density Operators Quantum Causal Networks The Partial Transpose Local Positivity: A Lament The Markov Evolution Law Conclusions The Partial Transpose Deﬁnition Given a basis {|j A } of HA and an operator MAB = jklm Mjk ;lm |jk lm|AB ∈ L (HA ⊗ HB ), the partial transpose map on system A is given by TA Mjk ;lm |lk MAB = jm|AB . (1) jklm Notation: For a CDO ρB|A = ρTA B|A M. S. Leifer Quantum Causal Networks

31. Introduction The Markov Evolution Law Conditional Density Operators Quantum Causal Networks The Partial Transpose Local Positivity: A Lament The Markov Evolution Law Conclusions The Partial Transpose Deﬁnition Given a basis {|j A } of HA and an operator MAB = jklm Mjk ;lm |jk lm|AB ∈ L (HA ⊗ HB ), the partial transpose map on system A is given by TA Mjk ;lm |lk MAB = jm|AB . (1) jklm Notation: For a CDO ρB|A = ρTA B|A M. S. Leifer Quantum Causal Networks

32. Introduction The Markov Evolution Law Conditional Density Operators Quantum Causal Networks The Partial Transpose Local Positivity: A Lament The Markov Evolution Law Conclusions The Markov Evolution Law Theorem Let EB|A : L (HA ) → L (HB ) be a TPCP map and let be a density operator. Then for any ρAC ∈ L (HA ⊗ HC ), EB|A ⊗ IC (ρAC ) = TrA ρB|A ρAC , (2) for some ﬁxed CDO ρB|A . c.f. Classical stochastic dynamics P(Y , Z ) = X P(Y |X )P(X , Z ). M. S. Leifer Quantum Causal Networks

33. Introduction The Markov Evolution Law Conditional Density Operators Quantum Causal Networks The Partial Transpose Local Positivity: A Lament The Markov Evolution Law Conclusions The Markov Evolution Law Theorem Let EB|A : L (HA ) → L (HB ) be a TPCP map and let be a density operator. Then for any ρAC ∈ L (HA ⊗ HC ), EB|A ⊗ IC (ρAC ) = TrA ρB|A ρAC , (2) for some ﬁxed CDO ρB|A . c.f. Classical stochastic dynamics P(Y , Z ) = X P(Y |X )P(X , Z ). M. S. Leifer Quantum Causal Networks

34. Introduction The Markov Evolution Law Conditional Density Operators Quantum Causal Networks The Partial Transpose Local Positivity: A Lament The Markov Evolution Law Conclusions The Markov Evolution Law This is just a restatement of the Choi-Jamiołkowski isomorphism. Let Φ+ |jj = AA. AA j Φ+ Φ+ Then, ρB|A = EB|A ⊗ IA AA Unitaries correspond to maximally entangled CDOs. M. S. Leifer Quantum Causal Networks

35. Introduction The Markov Evolution Law Conditional Density Operators Quantum Causal Networks The Partial Transpose Local Positivity: A Lament The Markov Evolution Law Conclusions The Markov Evolution Law This is just a restatement of the Choi-Jamiołkowski isomorphism. Let Φ+ |jj = AA. AA j Φ+ Φ+ Then, ρB|A = EB|A ⊗ IA AA Unitaries correspond to maximally entangled CDOs. M. S. Leifer Quantum Causal Networks

36. Introduction Directed Acyclic Graphs The Markov Evolution Law Classical Causal Networks Quantum Causal Networks Quantum Causal Networks Local Positivity: A Lament Markov Equivalence Conclusions Directed Acyclic Graphs (DAGs) A DAG G = (V , E) models causal structure. A B C D E It is isomorphic to the Hasse diagram of a ﬁnite poset. Parents: p(v ) = {u ∈ V : (u, v ) ∈ E}. Children: c(v ) = {u ∈ V : (v , u) ∈ E}. p(D) = {B, C}, c(D) = ∅, p(C) = {A}, c(C) = {D, E} Ancestral ordering, e.g. (A, B, C, D, E) or (A, C, E, B, D) M. S. Leifer Quantum Causal Networks

40. Introduction Directed Acyclic Graphs The Markov Evolution Law Classical Causal Networks Quantum Causal Networks Quantum Causal Networks Local Positivity: A Lament Markov Equivalence Conclusions Classical Causal Networks A Classical Causal Network is a DAG G = (V , E) together with a probability distribution P(V ) that factorizes according to P(v |p(v )) P(V ) = v ∈V A B C D E P(A, B, C, D, E) = P(A)P(B|A)P(C|A)P(D|B, C)P(E|C) M. S. Leifer Quantum Causal Networks

41. Introduction Directed Acyclic Graphs The Markov Evolution Law Classical Causal Networks Quantum Causal Networks Quantum Causal Networks Local Positivity: A Lament Markov Equivalence Conclusions Quantum Causal Networks In the quantum case we have to deal with no-cloning/no-broadcasting. We cannot set CDOs independently of each other. This can be solved by putting Hilbert spaces on the edges. A (AB) (AC) B C (CE) (BD) (CD) D E Each vertex is associated with two TPCP maps A ﬁssion isometry, e.g. E(CE)(CD)|C A fusion map, e.g. FC|(AC) M. S. Leifer Quantum Causal Networks

44. Introduction Directed Acyclic Graphs The Markov Evolution Law Classical Causal Networks Quantum Causal Networks Quantum Causal Networks Local Positivity: A Lament Markov Equivalence Conclusions States on Quantum Causal Networks The state on any “spacelike slice” can be found by composing fusion and ﬁssion maps. A (AB) (AC) B C (CE) (BD) (CD) D E ρA = FA|∅ (1) ρBC = FB|(AB) ⊗ FC|(AC) E(AB)(AC)|A (ρA ) ρDE = FD|(BD)(CD) ⊗ FE|(CE) E(BD)|B ⊗ E(CD)(CE)|C (ρBC ) The states are consistent on any “foliation”. M. S. Leifer Quantum Causal Networks

48. Introduction Directed Acyclic Graphs The Markov Evolution Law Classical Causal Networks Quantum Causal Networks Quantum Causal Networks Local Positivity: A Lament Markov Equivalence Conclusions Decomposition into CDOs A (AB) (AC) B C (CE) (BD) (CD) D E ρA ∗ ρB|A ∗ ρC|A ∗ ρD|BC ∗ ρE|C is a locally ρABCDE = positive operator with the correct reduced states on all “spacelike slices”. It doesn’t depend on the choice of ancestral ordering: ρA ∗ ρC|A ∗ ρE|C ∗ ρB|A ∗ ρD|BC ρABCDE = M. S. Leifer Quantum Causal Networks

49. Introduction Directed Acyclic Graphs The Markov Evolution Law Classical Causal Networks Quantum Causal Networks Quantum Causal Networks Local Positivity: A Lament Markov Equivalence Conclusions Decomposition into CDOs A (AB) (AC) B C (CE) (BD) (CD) D E ρA ∗ ρB|A ∗ ρC|A ∗ ρD|BC ∗ ρE|C is a locally ρABCDE = positive operator with the correct reduced states on all “spacelike slices”. It doesn’t depend on the choice of ancestral ordering: ρA ∗ ρC|A ∗ ρE|C ∗ ρB|A ∗ ρD|BC ρABCDE = M. S. Leifer Quantum Causal Networks

50. Introduction Directed Acyclic Graphs The Markov Evolution Law Classical Causal Networks Quantum Causal Networks Quantum Causal Networks Local Positivity: A Lament Markov Equivalence Conclusions Markov Equivalence Because the result is so similar to the classical case, we can transcribe many known theorems with ease. Deﬁnition Two DAGs are Markov Equivalent if they support the same set of “density operators” (up to local positive maps). Theorem Two DAGs are Markov Equivalent if they have the same links and the same set of uncoupled head-to-head meetings. M. S. Leifer Quantum Causal Networks

53. Introduction Directed Acyclic Graphs The Markov Evolution Law Classical Causal Networks Quantum Causal Networks Quantum Causal Networks Local Positivity: A Lament Markov Equivalence Conclusions Markov Equivalence A B C These are equivalent A B C Uncoupled head-to-head A B C meeting A C Coupled head-to-head meeting B M. S. Leifer Quantum Causal Networks

54. Introduction The Markov Evolution Law Local Positivity Quantum Causal Networks The Universal Not Local Positivity: A Lament Local Positivity Preserving Maps Conclusions Local Positivity: A Lament Life would be easier if it were ρB|A rather than ρB|A . This stems from the fact that not all locally positive operators are positive and not all positive maps are completely positive. ∀MA , MB ≥ 0, Tr (MA ⊗ MB ρAB ) ≥ 0 doesn’t imply ∀MAB ≥ 0, Tr (MAB ρAB ) ≥ 0 Similarly, we cannot implement a map that is positive, but not completely positive. M. S. Leifer Quantum Causal Networks

57. Introduction The Markov Evolution Law Local Positivity Quantum Causal Networks The Universal Not Local Positivity: A Lament Local Positivity Preserving Maps Conclusions Example: Universal Not |0> |0> z z y y x x |1> |1> |0> |0> z z y y x x |1> |1> M. S. Leifer Quantum Causal Networks

58. Introduction The Markov Evolution Law Local Positivity Quantum Causal Networks The Universal Not Local Positivity: A Lament Local Positivity Preserving Maps Conclusions Implemeting Universal Not Passively |0> |0> z x y y x z |1> |1> |0> |0> z x y y x z |1> |1> M. S. Leifer Quantum Causal Networks

59. Introduction The Markov Evolution Law Local Positivity Quantum Causal Networks The Universal Not Local Positivity: A Lament Local Positivity Preserving Maps Conclusions So what is the problem? A B C D Whilst B and C are separate, we may perform any positive operation on them passively by just relabeling our POVM elements, obtaining any locally positive state ρBC . When we recombine them at D, we have to describe both systems in a common “reference frame”. We must ensure that ρD is positive. M. S. Leifer Quantum Causal Networks

62. Introduction The Markov Evolution Law Local Positivity Quantum Causal Networks The Universal Not Local Positivity: A Lament Local Positivity Preserving Maps Conclusions Are you down with LPP? Deﬁnition A map EB1 B2 ...Bn |A1 A2 ...Am : L HA1 ⊗ HA2 ⊗ . . . ⊗ HAm → L HB1 ⊗ HB2 ⊗ . . . ⊗ HBn is completely local positivity preserving (CLPP) if EB1 B2 ...Bn |A1 A2 ...Am ⊗ IC ρA1 A2 ...Am C (3) is locally positive for all HC and all locally positive operators ρA1 A2 ...Am C . For m = 1, n = 1 CLPP = Positive. M. S. Leifer Quantum Causal Networks

63. Introduction The Markov Evolution Law Local Positivity Quantum Causal Networks The Universal Not Local Positivity: A Lament Local Positivity Preserving Maps Conclusions Are you down with LPP? Deﬁnition A map EB1 B2 ...Bn |A1 A2 ...Am : L HA1 ⊗ HA2 ⊗ . . . ⊗ HAm → L HB1 ⊗ HB2 ⊗ . . . ⊗ HBn is completely local positivity preserving (CLPP) if EB1 B2 ...Bn |A1 A2 ...Am ⊗ IC ρA1 A2 ...Am C (3) is locally positive for all HC and all locally positive operators ρA1 A2 ...Am C . For m = 1, n = 1 CLPP = Positive. M. S. Leifer Quantum Causal Networks

64. Introduction The Markov Evolution Law Local Positivity Quantum Causal Networks The Universal Not Local Positivity: A Lament Local Positivity Preserving Maps Conclusions Are you down with LPP? I have no good classiﬁcation of CLPP maps. It would be sufﬁcient to classify CLPP maps of the form EB|A1 A2 ...Am and EB1 B2 ...Bn |A . A2 A1 A3 A B B1 B3 B2 M. S. Leifer Quantum Causal Networks

65. Introduction The Markov Evolution Law Lessons for Quantum Gravity? Quantum Causal Networks Open Questions Local Positivity: A Lament Acknowledgments Conclusions Lessons for Quantum Gravity? Correlation structure seems more intrinsic to quantum theory than causal structure. Maybe we don’t have to sum over all possible causal structures - only Markov equivalence classes. M. S. Leifer Quantum Causal Networks

66. Introduction The Markov Evolution Law Lessons for Quantum Gravity? Quantum Causal Networks Open Questions Local Positivity: A Lament Acknowledgments Conclusions Open Questions Classiﬁcation of CLPP maps. Including preparations and measurements - Quantum Inﬂuence Diagrams. M. S. Leifer Quantum Causal Networks

67. Introduction The Markov Evolution Law Lessons for Quantum Gravity? Quantum Causal Networks Open Questions Local Positivity: A Lament Acknowledgments Conclusions Acknowledgments This work is supported by: The Foundational Questions Institute (http://www.fqxi.org) MITACS (http://www.mitacs.math.ca) NSERC (http://nserc.ca/) The Province of Ontario: ORDCF/MRI M. S. Leifer Quantum Causal Networks

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