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- 1. Quantum Theology Conditional Density Operators Conditional Independence Quantum State Pooling Conclusions The Church of the Smaller Hilbert Space (a.k.a. An Approach to Quantum State Pooling from Quantum Conditional Independence) M. S. Leifer Institute for Quantum Computing University of Waterloo Perimeter Institute March 11th 2008 / APS March Meeting M. S. Leifer The Church of the Smaller Hilbert Space
- 2. Quantum Theology Conditional Density Operators Conditional Independence Quantum State Pooling Conclusions Outline Quantum Theology 1 Conditional Density Operators 2 Conditional Independence 3 Quantum State Pooling 4 Conclusions 5 M. S. Leifer The Church of the Smaller Hilbert Space
- 3. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions Quantum Theology The Two Churches of Quantum Theory The Church of the Larger Hilbert Space The Church of the Smaller Hilbert Space Each church consists of: A moral code, i.e. a set of proof techniques. A set of core beliefs, i.e. interpretation of quantum theory. Secular theorists are free to draw their moral code from both churches. M. S. Leifer The Church of the Smaller Hilbert Space
- 4. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions Quantum Theology The Two Churches of Quantum Theory The Church of the Larger Hilbert Space The Church of the Smaller Hilbert Space Each church consists of: A moral code, i.e. a set of proof techniques. A set of core beliefs, i.e. interpretation of quantum theory. Secular theorists are free to draw their moral code from both churches. M. S. Leifer The Church of the Smaller Hilbert Space
- 5. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions Quantum Theology The Two Churches of Quantum Theory The Church of the Larger Hilbert Space The Church of the Smaller Hilbert Space Each church consists of: A moral code, i.e. a set of proof techniques. A set of core beliefs, i.e. interpretation of quantum theory. Secular theorists are free to draw their moral code from both churches. M. S. Leifer The Church of the Smaller Hilbert Space
- 6. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions Quantum Theology The Two Churches of Quantum Theory The Church of the Larger Hilbert Space The Church of the Smaller Hilbert Space Each church consists of: A moral code, i.e. a set of proof techniques. A set of core beliefs, i.e. interpretation of quantum theory. Secular theorists are free to draw their moral code from both churches. M. S. Leifer The Church of the Smaller Hilbert Space
- 7. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Larger Hilbert Space Moral Code Thou shalt purify mixed states. ρA = TrE (|ψ ψ|AE ) Thou shalt Steinspring dilate TPCP maps. † E(ρA ) = TrAER UAR |ψ ψ|AE ⊗ |0 0|R UAR Thou shalt Naimark extend POVMs. (j) (j) Tr EA ρA = TrAER PAR |ψ ψ|AE ⊗ |0 0|R M. S. Leifer The Church of the Smaller Hilbert Space
- 8. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Larger Hilbert Space Moral Code Thou shalt purify mixed states. ρA = TrE (|ψ ψ|AE ) Thou shalt Steinspring dilate TPCP maps. † E(ρA ) = TrAER UAR |ψ ψ|AE ⊗ |0 0|R UAR Thou shalt Naimark extend POVMs. (j) (j) Tr EA ρA = TrAER PAR |ψ ψ|AE ⊗ |0 0|R M. S. Leifer The Church of the Smaller Hilbert Space
- 9. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Larger Hilbert Space Moral Code Thou shalt purify mixed states. ρA = TrE (|ψ ψ|AE ) Thou shalt Steinspring dilate TPCP maps. † E(ρA ) = TrAER UAR |ψ ψ|AE ⊗ |0 0|R UAR Thou shalt Naimark extend POVMs. (j) (j) Tr EA ρA = TrAER PAR |ψ ψ|AE ⊗ |0 0|R M. S. Leifer The Church of the Smaller Hilbert Space
- 10. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Larger Hilbert Space Core Beliefs The entire universe is described by a massively entangled pure state, |Ψ U , deﬁned on an enormous number of subsystems. Quantum mechanics is a well-deﬁned dynamical theory. |Ψ U evolves unitarily according to the Schrödinger equation and that’s all there is to it! Taken seriously this leads to Everett/many worlds. M. S. Leifer The Church of the Smaller Hilbert Space
- 11. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Larger Hilbert Space Core Beliefs The entire universe is described by a massively entangled pure state, |Ψ U , deﬁned on an enormous number of subsystems. Quantum mechanics is a well-deﬁned dynamical theory. |Ψ U evolves unitarily according to the Schrödinger equation and that’s all there is to it! Taken seriously this leads to Everett/many worlds. M. S. Leifer The Church of the Smaller Hilbert Space
- 12. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Larger Hilbert Space Core Beliefs The entire universe is described by a massively entangled pure state, |Ψ U , deﬁned on an enormous number of subsystems. Quantum mechanics is a well-deﬁned dynamical theory. |Ψ U evolves unitarily according to the Schrödinger equation and that’s all there is to it! Taken seriously this leads to Everett/many worlds. M. S. Leifer The Church of the Smaller Hilbert Space
- 13. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Smaller Hilbert Space Moral Code Thou shalt not adorn your church with unnecessary ornaments. Thou shalt not purify mixed states. Thou shalt not Steinspring dilate TPCP maps. Thou shalt not Naimark extend POVMs. This talk is about what thou shouldst do instead. M. S. Leifer The Church of the Smaller Hilbert Space
- 14. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Smaller Hilbert Space Moral Code Thou shalt not adorn your church with unnecessary ornaments. Thou shalt not purify mixed states. Thou shalt not Steinspring dilate TPCP maps. Thou shalt not Naimark extend POVMs. This talk is about what thou shouldst do instead. M. S. Leifer The Church of the Smaller Hilbert Space
- 15. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Smaller Hilbert Space Core Beliefs Quantum theory is best thought of as a noncommutative generalization of classical probability theory. Classical probability distributions do not have puriﬁcations. We will lose sight of useful analogies if we purify. Taken seriously this leads to quantum logic, quantum Bayesianism, ..., any interpretation in which the structure of observables is taken as primary. M. S. Leifer The Church of the Smaller Hilbert Space
- 16. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Smaller Hilbert Space Core Beliefs Quantum theory is best thought of as a noncommutative generalization of classical probability theory. Classical probability distributions do not have puriﬁcations. We will lose sight of useful analogies if we purify. Taken seriously this leads to quantum logic, quantum Bayesianism, ..., any interpretation in which the structure of observables is taken as primary. M. S. Leifer The Church of the Smaller Hilbert Space
- 17. Quantum Theology Conditional Density Operators The Church of the Larger Hilbert Space Conditional Independence The Church of the Smaller Hilbert Space Quantum State Pooling Conclusions The Church of The Smaller Hilbert Space Core Beliefs Quantum theory is best thought of as a noncommutative generalization of classical probability theory. Classical probability distributions do not have puriﬁcations. We will lose sight of useful analogies if we purify. Taken seriously this leads to quantum logic, quantum Bayesianism, ..., any interpretation in which the structure of observables is taken as primary. M. S. Leifer The Church of the Smaller Hilbert Space
- 18. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Quantum Analog of Conditional Probability? Classical Probability Quantum Theory Sample Space: Hilbert Space: ΩX = {1, 2, . . . , n} HA Probability distribution: Density operator: P(X ) ρA Cartesian product: Tensor product: ΩX × ΩY HA ⊗ H B Joint probability: Bipartite density operator: P(X , Y ) ρAB Conditional probability: P(Y |X ) = P(X ,Y ) ? P(Y ) M. S. Leifer The Church of the Smaller Hilbert Space
- 19. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems 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 = N 2 MN 2 ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA . A c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ). M. S. Leifer The Church of the Smaller Hilbert Space
- 20. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems 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 = N 2 MN 2 ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA . A c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ). M. S. Leifer The Church of the Smaller Hilbert Space
- 21. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems 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 = N 2 MN 2 ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA . A c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ). M. S. Leifer The Church of the Smaller Hilbert Space
- 22. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems 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 = N 2 MN 2 ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA . A c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ). M. S. Leifer The Church of the Smaller Hilbert Space
- 23. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems 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 = N 2 MN 2 ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA . A c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ). M. S. Leifer The Church of the Smaller Hilbert Space
- 24. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems 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 = N 2 MN 2 ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA . A c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ). M. S. Leifer The Church of the Smaller Hilbert Space
- 25. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Example Let ρAB = |Ψ Ψ|AB be a pure state with Schmidt decomposition |Ψ ⊗ ψj p j φj = . AB A B j Then, ρB|A = |Ψ Ψ|B|A , where |Ψ ⊗ ψj = φj . B|A A B j M. S. Leifer The Church of the Smaller Hilbert Space
- 26. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Ta Da! Classical Probability Quantum Theory Sample Space: Hilbert Space: ΩX = {1, 2, . . . , n} HA Probability distribution: Density operator: P(X ) ρA Cartesian product: Tensor product: ΩX × ΩY HA ⊗ H B Joint probability: Bipartite density operator: P(X , Y ) ρAB Conditional probability: Conditional density operator: P(Y |X ) = P(X ,Y ) ρB|A = ρAB ∗ ρ−1 A P(Y ) M. S. Leifer The Church of the Smaller Hilbert Space
- 27. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions A problem with the analogy ρAB usually represents the state of two subsystems at a given time. P(X , Y ) is more ﬂexible. X and Y might refer to different subsystems. Y might represent the value of the same quantity as X , but at a later time. Y might represent the result of a measurement of the value of X . .... M. S. Leifer The Church of the Smaller Hilbert Space
- 28. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions A problem with the analogy ρAB usually represents the state of two subsystems at a given time. P(X , Y ) is more ﬂexible. X and Y might refer to different subsystems. Y might represent the value of the same quantity as X , but at a later time. Y might represent the result of a measurement of the value of X . .... M. S. Leifer The Church of the Smaller Hilbert Space
- 29. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions A problem with the analogy ρAB usually represents the state of two subsystems at a given time. P(X , Y ) is more ﬂexible. X and Y might refer to different subsystems. Y might represent the value of the same quantity as X , but at a later time. Y might represent the result of a measurement of the value of X . .... M. S. Leifer The Church of the Smaller Hilbert Space
- 30. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions A problem with the analogy ρAB usually represents the state of two subsystems at a given time. P(X , Y ) is more ﬂexible. X and Y might refer to different subsystems. Y might represent the value of the same quantity as X , but at a later time. Y might represent the result of a measurement of the value of X . .... M. S. Leifer The Church of the Smaller Hilbert Space
- 31. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Subsystems time Two classical subsystems Two quantum subsystems X Y A B ρAB = ρB|A ∗ ρA P (X, Y ) = P (Y |X)P (X) M. S. Leifer The Church of the Smaller Hilbert Space
- 32. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Dynamical CDOs time Y B Trace-preserving Classical stochastic completely-positive dynamics dynamics X A = EB|A (ρA ) P (Y ) = ΓY |X (P (X)) ρB = P (Y |X)P (X) = TrA ρB|A ∗ ρA X = P (X, Y ) = TrA (ρAB ) X M. S. Leifer The Church of the Smaller Hilbert Space
- 33. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Dynamical CDOs time Y B Trace-preserving Classical stochastic completely-positive dynamics dynamics X A P (Y ) = ΓY |X (P (X)) = EB|A (ρA ) ρB = P (Y |X)P (X) X = TrA ρTA ∗ ρA = P (X, Y ) B|A X M. S. Leifer The Church of the Smaller Hilbert Space
- 34. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Hybrid Quantum-Classical Systems X A (j) P(X = j) |j j|X ⊗ ρA ρXA = j (j) j|X ρX |j P(X = j)ρA = P(X = j) ρA = X j (j) j|X ρXA |j = P(X = j)ρA X (j) j|X ρA|X |j = ρA X M. S. Leifer The Church of the Smaller Hilbert Space
- 35. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Hybrid Quantum-Classical Systems X A (j) P(X = j) |j j|X ⊗ ρA ρXA = j (j) j|X ρX |j P(X = j)ρA = P(X = j) ρA = X j (j) j|X ρXA |j = P(X = j)ρA X (j) j|X ρA|X |j = ρA X M. S. Leifer The Church of the Smaller Hilbert Space
- 36. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Hybrid Quantum-Classical Systems X A (j) P(X = j) |j j|X ⊗ ρA ρXA = j (j) j|X ρX |j P(X = j)ρA = P(X = j) ρA = X j (j) j|X ρXA |j = P(X = j)ρA X (j) j|X ρA|X |j = ρA X M. S. Leifer The Church of the Smaller Hilbert Space
- 37. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Hybrid Quantum-Classical Systems X A (j) P(X = j) |j j|X ⊗ ρA ρXA = j (j) j|X ρX |j P(X = j)ρA = P(X = j) ρA = X j (j) j|X ρXA |j = P(X = j)ρA X (j) j|X ρA|X |j = ρA X M. S. Leifer The Church of the Smaller Hilbert Space
- 38. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Hybrid Quantum-Classical Systems X A (j) EA = j|X ρX |A |j is a POVM on HA X (j) Conversely, if EA is a POVM on HA then (j) |j j|X ⊗ EA is a valid CDO. ρX |A = j M. S. Leifer The Church of the Smaller Hilbert Space
- 39. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Hybrid Quantum-Classical Systems X A (j) EA = j|X ρX |A |j is a POVM on HA X (j) Conversely, if EA is a POVM on HA then (j) |j j|X ⊗ EA is a valid CDO. ρX |A = j M. S. Leifer The Church of the Smaller Hilbert Space
- 40. Quantum Theology Conditional Density Operators Quantum Analog of Conditional Probability Conditional Independence Dynamical Conditional Density Operators Quantum State Pooling Hybrid Quantum-Classical Systems Conclusions Preparations and Measurements A X Measurement Preparation X A ρA = TrX ρA|X ∗ ρX ρX = TrA ρX|A ∗ ρA M. S. Leifer The Church of the Smaller Hilbert Space
- 41. Quantum Theology Conditional Density Operators Classical Conditional Independence Conditional Independence Quantum Conditional Independence Quantum State Pooling Hybrid Conditional Independence Conclusions Classical Conditional Independence X Z Y H(X : Y |Z ) = H(X , Z ) + H(Y , Z ) − H(X , Y , Z ) − H(Z ) = 0 P(X |Y , Z ) = P(X |Z ) P(Y |X , Z ) = P(Y |Z ) P(X , Y |Z ) = P(X |Z )P(Y |Z ) M. S. Leifer The Church of the Smaller Hilbert Space
- 42. Quantum Theology Conditional Density Operators Classical Conditional Independence Conditional Independence Quantum Conditional Independence Quantum State Pooling Hybrid Conditional Independence Conclusions Quantum Conditional Independence A C B S(A : B|C) = S(A, C) + S(B, C) − S(A, B, C) − S(C) = 0 ρA|BC = ρA|C ρB|AC = ρB|C ⇒ ρAB|C = ρA|C ρB|C M. S. Leifer The Church of the Smaller Hilbert Space
- 43. Quantum Theology Conditional Density Operators Classical Conditional Independence Conditional Independence Quantum Conditional Independence Quantum State Pooling Hybrid Conditional Independence Conclusions Hybrid Conditional Independence X C Y S(X : Y |C) = S(X , C) + S(Y , C) − S(X , Y , C) − S(C) = 0 ρX |YC = ρX |C ρY |XC = ρY |C ρXY |C = ρX |C ρY |C M. S. Leifer The Church of the Smaller Hilbert Space
- 44. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions The Pooling Problem Classical: Alice describes a system by P(Z ), Bob by Q(Z ). If they get together, what distribution should they agree upon? Quantum: Alice describes a system by ρC , Bob by σC . If they get together, what distribution should they agree upon? Introduce an arbiter, Penelope the pooler, who’s task it is to make the decision. M. S. Leifer The Church of the Smaller Hilbert Space
- 45. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions The Pooling Problem Classical: Alice describes a system by P(Z ), Bob by Q(Z ). If they get together, what distribution should they agree upon? Quantum: Alice describes a system by ρC , Bob by σC . If they get together, what distribution should they agree upon? Introduce an arbiter, Penelope the pooler, who’s task it is to make the decision. M. S. Leifer The Church of the Smaller Hilbert Space
- 46. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions The Pooling Problem Classical: Alice describes a system by P(Z ), Bob by Q(Z ). If they get together, what distribution should they agree upon? Quantum: Alice describes a system by ρC , Bob by σC . If they get together, what distribution should they agree upon? Introduce an arbiter, Penelope the pooler, who’s task it is to make the decision. M. S. Leifer The Church of the Smaller Hilbert Space
- 47. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Diplomatic Pooling Alice Bob Penelope M. S. Leifer The Church of the Smaller Hilbert Space
- 48. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Scientiﬁc Pooling Alice Bob Penelope M. S. Leifer The Church of the Smaller Hilbert Space
- 49. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Classical Pooling Y X P (Z) P (Z) Alice Bob P (X|Z) P (Y |Z) Z P (Z) Penelope P (Z|X) P (Z|Y ) M. S. Leifer The Church of the Smaller Hilbert Space
- 50. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Simon, the supra-Bayesian Simon, the ﬁctitious know-it-all is going to update via ,Y |Z )P(Z Bayes’ rule: P(Z |X , Y ) = P(XP(X ,Y ) ) . Does Penelope have enough information to do what Simon says? Not generally, but if X and Y are conditionally independent: P(X |Z )P(Y |Z )P(Z ) P(Z |X , Y ) = P(X ,Y ) P(X )P(Y ) P(Z |X )P(Z |Y ) = P(X ,Y ) P(Z ) P(Z |X )P(Z |Y ) = NXY P(Z ) M. S. Leifer The Church of the Smaller Hilbert Space
- 51. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Simon, the supra-Bayesian Simon, the ﬁctitious know-it-all is going to update via ,Y |Z )P(Z Bayes’ rule: P(Z |X , Y ) = P(XP(X ,Y ) ) . Does Penelope have enough information to do what Simon says? Not generally, but if X and Y are conditionally independent: P(X |Z )P(Y |Z )P(Z ) P(Z |X , Y ) = P(X ,Y ) P(X )P(Y ) P(Z |X )P(Z |Y ) = P(X ,Y ) P(Z ) P(Z |X )P(Z |Y ) = NXY P(Z ) M. S. Leifer The Church of the Smaller Hilbert Space
- 52. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Simon, the supra-Bayesian Simon, the ﬁctitious know-it-all is going to update via ,Y |Z )P(Z Bayes’ rule: P(Z |X , Y ) = P(XP(X ,Y ) ) . Does Penelope have enough information to do what Simon says? Not generally, but if X and Y are conditionally independent: P(X |Z )P(Y |Z )P(Z ) P(Z |X , Y ) = P(X ,Y ) P(X )P(Y ) P(Z |X )P(Z |Y ) = P(X ,Y ) P(Z ) P(Z |X )P(Z |Y ) = NXY P(Z ) M. S. Leifer The Church of the Smaller Hilbert Space
- 53. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Quantum Pooling via indirect measurements X B Y A ρC ρC ρA|C ρB|C Alice Bob ρX|A ρY |B C ρC Penelope ρC|X ρC|Y M. S. Leifer The Church of the Smaller Hilbert Space
- 54. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Quantum supra-Bayesian Pooling If ρXY |C = ρX |C ρY |C then = ρXY |C ∗ ρC ρ−1 ρC|XY XY = ρ−1 ρX |C ρY |C ∗ ρC XY = ρ−1 ρX ρY ρC|X ρ−1 ρC|Y XY C = NXY ρC|X ρ−1 ρC|Y C M. S. Leifer The Church of the Smaller Hilbert Space
- 55. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Quantum supra-Bayesian Pooling For which ρABC is pooling always possible regardless of ρX |A , ρY |B ? It is sufﬁcient if ρAB|C = ρA|C ρB|C ρX |A ρY |B ∗ ρAB|C = TrAB ρXY |C = TrA ρX |A ∗ ρA|C TrB ρY |B ∗ ρB|C = ρX |C ρY |C . M. S. Leifer The Church of the Smaller Hilbert Space
- 56. Quantum Theology Conditional Density Operators Classical Pooling Conditional Independence Quantum Pooling via Indirect Measurements Quantum State Pooling Conclusions Quantum supra-Bayesian Pooling For which ρABC is pooling always possible regardless of ρX |A , ρY |B ? It is sufﬁcient if ρAB|C = ρA|C ρB|C ρX |A ρY |B ∗ ρAB|C = TrAB ρXY |C = TrA ρX |A ∗ ρA|C TrB ρY |B ∗ ρB|C = ρX |C ρY |C . M. S. Leifer The Church of the Smaller Hilbert Space
- 57. Quantum Theology Conditional Density Operators Moral Conditional Independence Acknowledgments Quantum State Pooling References Conclusions The Moral of the Story There is a bunch of other stuff that makes more sense in the Church of the Smaller Hilbert Space The “pretty good” measurement “Pretty good” error correction Results on steering entangled states Entanglement in time Quantum sufﬁcient statistics Causality ...but the Church of the Larger Hilbert Space has some pretty nifty proofs too. So which one is right? M. S. Leifer The Church of the Smaller Hilbert Space
- 58. Quantum Theology Conditional Density Operators Moral Conditional Independence Acknowledgments Quantum State Pooling References Conclusions The Moral of the Story There is a bunch of other stuff that makes more sense in the Church of the Smaller Hilbert Space The “pretty good” measurement “Pretty good” error correction Results on steering entangled states Entanglement in time Quantum sufﬁcient statistics Causality ...but the Church of the Larger Hilbert Space has some pretty nifty proofs too. So which one is right? M. S. Leifer The Church of the Smaller Hilbert Space
- 59. Quantum Theology Conditional Density Operators Moral Conditional Independence Acknowledgments Quantum State Pooling References Conclusions Blind Men and the Elephant by J. G. Saxe It was six men of Indostan To learning much inclined, Who went to see the Elephant (Though all of them were blind), That each by observation Might satisfy his mind M. S. Leifer The Church of the Smaller Hilbert Space
- 60. Quantum Theology Conditional Density Operators Moral Conditional Independence Acknowledgments Quantum State Pooling References Conclusions Blind Men and the Elephant by J. G. Saxe The First approached the Elephant, And happening to fall Against his broad and sturdy side, At once began to bawl: quot;God bless me! but the Elephant Is very like a wall!quot; The Second, feeling of the tusk, Cried, quot;Ho! what have we here So very round and smooth and sharp? To me ’tis mighty clear This wonder of an Elephant Is very like a spear!quot; M. S. Leifer The Church of the Smaller Hilbert Space
- 61. Quantum Theology Conditional Density Operators Moral Conditional Independence Acknowledgments Quantum State Pooling References Conclusions Blind Men and the Elephant by J. G. Saxe And so these men of Indostan Disputed loud and long, Each in his own opinion Exceeding stiff and strong, Though each was partly in the right, And all were in the wrong! Moral: So oft in theologic wars, The disputants, I ween, Rail on in utter ignorance Of what each other mean, And prate about an Elephant Not one of them has seen! M. S. Leifer The Church of the Smaller Hilbert Space
- 62. Quantum Theology Conditional Density Operators Moral Conditional Independence Acknowledgments Quantum State Pooling References 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 The Church of the Smaller Hilbert Space
- 63. Quantum Theology Conditional Density Operators Moral Conditional Independence Acknowledgments Quantum State Pooling References Conclusions References Conditional Density Operators: M. S. Leifer, Phys. Rev. A 74, 042310 (2006). arXiv:quant-ph/0606022. M. S. Leifer (2006) arXiv:quant-ph/0611233. Conditional Independence: M. S. Leifer and D. Poulin, Ann. Phys., in press. arXiv:0708.1337 Quantum State Pooling: M. S. Leifer and R. W. Spekkens, in preparation. R. W. Spekkens and H. M. Wiseman, Phys. Rev. A 75, 042104 (2007). arXiv:quant-ph/0612190. Quantum Theology: The book with this title is unrelated to this talk. M. S. Leifer The Church of the Smaller Hilbert Space

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