Emergence of objective properties from subjective quantum states: Environment as a witness

Transcription

1 Emergence of objective properties from subjective quantum states: Environment as a witness David Poulin Institute for Quantum Computing Perimeter Institute for Theoretical Physics Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.1

2 Quantum-Classical correspondence Interpretation of Quantum Mechanics: Why is quantum mechanics the way it is? Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.2

3 Quantum-Classical correspondence Interpretation of Quantum Mechanics: Why is quantum mechanics the way it is? Interpretation of Classical Mechanics: Why is the classical world different? Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.2

4 Quantum-Classical correspondence Interpretation of Quantum Mechanics: Why is quantum mechanics the way it is? Interpretation of Classical Mechanics: Why is the classical world different? No interferences: Consistent histories. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.2

5 Quantum-Classical correspondence Interpretation of Quantum Mechanics: Why is quantum mechanics the way it is? Interpretation of Classical Mechanics: Why is the classical world different? No interferences: Consistent histories. No superpositions: Decoherence. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.2

6 Quantum-Classical correspondence Interpretation of Quantum Mechanics: Why is quantum mechanics the way it is? Interpretation of Classical Mechanics: Why is the classical world different? No interferences: Consistent histories. No superpositions: Decoherence. Determinism: Predictability sieves. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.2

7 Quantum-Classical correspondence Interpretation of Quantum Mechanics: Why is quantum mechanics the way it is? Interpretation of Classical Mechanics: Why is the classical world different? No interferences: Consistent histories. No superpositions: Decoherence. Determinism: Predictability sieves. Objective reality. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.2

8 Quantum-Classical correspondence Interpretation of Quantum Mechanics: Why is quantum mechanics the way it is? Interpretation of Classical Mechanics: Why is the classical world different? No interferences: Consistent histories. No superpositions: Decoherence. Determinism: Predictability sieves. Objective reality. Stick to quantum mechanics, get operational answer. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.2

9 The approach My philosophy: Ask the questions correctly, you ll get the expected answer, e.g. for sufficiently large systems [X, P ] = 0 effectively. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.3

10 The approach My philosophy: Ask the questions correctly, you ll get the expected answer, e.g. for sufficiently large systems [X, P ] = 0 effectively. Joint work with Harold Ollivier and Wojciech Zurek. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.3

11 The approach My philosophy: Ask the questions correctly, you ll get the expected answer, e.g. for sufficiently large systems [X, P ] = 0 effectively. Joint work with Harold Ollivier and Wojciech Zurek. Has some similarities with Carlo Rovelli s Relational Quantum Mechanics. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.3

12 Outline Review of the decoherence program. Operational definition of objectivity. Problems remaining after decoherence. Information theoretical formulation. Main consequence of objectivity: unique preferred basis. Other consequences of objectivity. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.4

13 Schrödinger s cat e Cat alive ( e + g ) Cat alive e Cat alive + g Cat dead Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.5

14 Schrödinger s cat e Cat alive ( e + g ) Cat alive e Cat alive + g Cat dead Quantum theory allows superposition of macroscopic objects. Such superpositions however are not observed. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.5

15 Schrödinger s cat Environment induced superselection (einselection): Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.6

16 Schrödinger s cat Environment induced superselection (einselection): ( e Cat alive + g Cat dead ) Mouse alive Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.6

17 Schrödinger s cat Environment induced superselection (einselection): ( e Cat alive + g Cat dead ) Mouse alive e Cat alive Mouse dead + g Cat dead Mouse alive = Ψ Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.6

18 Schrödinger s cat Environment induced superselection (einselection): ( e Cat alive + g Cat dead ) Mouse alive e Cat alive Mouse dead + g Cat dead Mouse alive = Ψ If the mouse is not a controlable degree of freedom Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.6

19 Schrödinger s cat Environment induced superselection (einselection): ( e Cat alive + g Cat dead ) Mouse alive e Cat alive Mouse dead + g Cat dead Mouse alive = Ψ If the mouse is not a controlable degree of freedom ρ Atom+Cat = T r Mouse Ψ Ψ Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.6

20 Schrödinger s cat Environment induced superselection (einselection): ( e Cat alive + g Cat dead ) Mouse alive e Cat alive Mouse dead + g Cat dead Mouse alive = Ψ If the mouse is not a controlable degree of freedom ρ Atom+Cat = T r Mouse Ψ Ψ = 1 2 e e Cat alive Cat alive g g Cat dead Cat dead Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.6

21 Schrödinger s cat The description of the quantum systems of interest (Atom + Cat) is a classical mixture of e alive and g dead. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.7

22 Schrödinger s cat The description of the quantum systems of interest (Atom + Cat) is a classical mixture of e alive and g dead. Operationally, the interaction with an environment explains why we only experience statistical mixtures as opposed to coherent superpositions. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.7

23 Schrödinger s cat The description of the quantum systems of interest (Atom + Cat) is a classical mixture of e alive and g dead. Operationally, the interaction with an environment explains why we only experience statistical mixtures as opposed to coherent superpositions. What characterize these pointer states? Are there always pointer state? How can we identify them? Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.7

24 A toy model System S = a spin Environment E = N spin Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.8

25 A toy model System S = a spin Environment E = N spin Coupling H = j g jσ S z σ E j y, Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.8

26 A toy model System S = a spin Environment E = N spin Coupling H = j g jσ S z σ E j y, Rotation of E j along y conditioned on S. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.8

27 A toy model System S = a spin Environment E = N spin Coupling H = j g jσ S z σ E j y, Rotation of E j along y conditioned on S. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.8

28 A toy model System S = a spin Environment E = N spin Coupling H = j g jσ S z σ E j y, Rotation of E j along y conditioned on S. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.8

29 A toy model System S = a spin Environment E = N spin Coupling H = j g jσ S z σ E j y, Rotation of E j along y conditioned on S. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.8

30 A toy model Reduced density matrix: ) ρ(0) = ( z(t) = j cos(g jt/2) 1 2 ( 12 z(t) 1 ρ(t) = 2 z (t) ) Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.9

31 A toy model Reduced density matrix: ) ρ(0) = ( z(t) = j cos(g jt/2) 1 2 ( 12 z(t) 1 ρ(t) = 2 z (t) ) Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.9

32 A toy model Reduced density matrix: ) ρ(0) = ( z(t) = j cos(g jt/2) 1 2 ( 12 z(t) 1 ρ(t) = 2 z (t) ) Einselection State of S is a classical mixture of pointer states and. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.9

33 Pointer states Algebraic criterion [A, H int ] = 0. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.10

34 Pointer states Algebraic criterion [A, H int ] = 0. They have a deterministic evolution even though they interact with E. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.10

35 Pointer states Algebraic criterion [A, H int ] = 0. They have a deterministic evolution even though they interact with E. Rarely occurs in practice, e.g. H = ωσ y + j g jσ S z σ E j y. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.10

36 Pointer states Algebraic criterion [A, H int ] = 0. They have a deterministic evolution even though they interact with E. Rarely occurs in practice, e.g. H = ωσ y + j g jσ S z σ E j y. Predictability sieve: initial states which minimize entropy production S(t) = T r{ρ S (t) ln ρ S (t)}. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.10

37 Pointer states Algebraic criterion [A, H int ] = 0. They have a deterministic evolution even though they interact with E. Rarely occurs in practice, e.g. H = ωσ y + j g jσ S z σ E j y. Predictability sieve: initial states which minimize entropy production S(t) = T r{ρ S (t) ln ρ S (t)}. More realistic criterion (Quantum Brownian motion) Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.10

38 Pointer states Do we recover classicality? Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.11

39 Pointer states Do we recover classicality? Pointer states are the only one which evolve predictably, i.e. pure state pure state. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.11

40 Pointer states Do we recover classicality? Pointer states are the only one which evolve predictably, i.e. pure state pure state. Pointer states cannot be superposed. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.11

41 Pointer states Do we recover classicality? Pointer states are the only one which evolve predictably, i.e. pure state pure state. Pointer states cannot be superposed. A system with pointer states cannot be entangled with an other system. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.11

42 Pointer states Do we recover classicality? Pointer states are the only one which evolve predictably, i.e. pure state pure state. Pointer states cannot be superposed. A system with pointer states cannot be entangled with an other system. Are pointer states objective elements of reality? Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.11

43 Objectivity Operational definition: An objective property of the system of interest should be Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.12

44 Objectivity Operational definition: An objective property of the system of interest should be 1. simultaneously accessible to many observers, Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.12

45 Objectivity Operational definition: An objective property of the system of interest should be 1. simultaneously accessible to many observers, 2. who should be able to find out what it is without prior knowledge about the system and Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.12

46 Objectivity Operational definition: An objective property of the system of interest should be 1. simultaneously accessible to many observers, 2. who should be able to find out what it is without prior knowledge about the system and 3. who should arrive at a consensus about it without prior agreement. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.12

47 Objectivity Operational definition: An objective property of the system of interest should be 1. simultaneously accessible to many observers, 2. who should be able to find out what it is without prior knowledge about the system and 3. who should arrive at a consensus about it without prior agreement. This rules out direct measurements on the system. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.12

48 Objectivity A direct measurement on the system generally leads to a re-preparation. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.13

49 Objectivity A direct measurement on the system generally leads to a re-preparation. This is true even in the presence of einselection except if all observers agree on what are the pointer states, i.e. agree on what is to be measured on S. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.13

50 Objectivity A direct measurement on the system generally leads to a re-preparation. This is true even in the presence of einselection except if all observers agree on what are the pointer states, i.e. agree on what is to be measured on S. The position of a chair is not objective because we have all agree that it should be, it is forced upon us. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.13

51 Objectivity A direct measurement on the system generally leads to a re-preparation. This is true even in the presence of einselection except if all observers agree on what are the pointer states, i.e. agree on what is to be measured on S. The position of a chair is not objective because we have all agree that it should be, it is forced upon us. Only the properties of the system which can be found out indirectly have a chance of qualifying as objective. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.13

52 Objectivity Consequences of the operational definition of objectivity: Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.14

53 Objectivity Consequences of the operational definition of objectivity: An objective property of S must be encoded in its environment. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.14

54 Objectivity Consequences of the operational definition of objectivity: An objective property of S must be encoded in its environment. It is possible to learn about it indirectly. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.14

55 Objectivity Consequences of the operational definition of objectivity: An objective property of S must be encoded in its environment. It is possible to learn about it indirectly. This encoding must be redundant, i.e. accessible from disjoint subsystems of the environment. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.14

56 Objectivity Consequences of the operational definition of objectivity: An objective property of S must be encoded in its environment. It is possible to learn about it indirectly. This encoding must be redundant, i.e. accessible from disjoint subsystems of the environment. Measurements on disjoint subsystems cannot affect each other. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.14

57 Objectivity Consequences of the operational definition of objectivity: An objective property of S must be encoded in its environment. It is possible to learn about it indirectly. This encoding must be redundant, i.e. accessible from disjoint subsystems of the environment. Measurements on disjoint subsystems cannot affect each other. One s observation does not invalidate someone else s information. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.14

58 Information theory Given a joint probability of X and Y, P (x, y), we can define the marginal P (x) = y P (x, y) and the conditional P (x y) = P (x, y)/p (y). Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.15

59 Information theory Given a joint probability of X and Y, P (x, y), we can define the marginal P (x) = y P (x, y) and the conditional P (x y) = P (x, y)/p (y). Entropy H(X) = x P (x) ln P (x) Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.15

60 Information theory Given a joint probability of X and Y, P (x, y), we can define the marginal P (x) = y P (x, y) and the conditional P (x y) = P (x, y)/p (y). Entropy H(X) = x P (x) ln P (x) Conditional Entropy H(X Y ) = y P (y) x P (x y) ln P (x y) Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.15

61 Information theory Given a joint probability of X and Y, P (x, y), we can define the marginal P (x) = y P (x, y) and the conditional P (x y) = P (x, y)/p (y). Entropy H(X) = x P (x) ln P (x) Conditional Entropy H(X Y ) = y P (y) x P (x y) ln P (x y) Mutual Information I(X : Y ) = H(X) H(X Y ) Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.15

62 Information theory H(X) H(Y ) H(X Y ) I(X : Y ) H(Y X) H(X Y ) = H(X, Y ) H(Y ) = H(X) I(X : Y ) I(X : Y ) = H(X) + H(Y ) H(X, Y ) is symmetric. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.16

63 Information theory Observable σ on S: σ = {σ i } Observable τ on E: τ = {τ k } Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.17

64 Information theory Observable σ on S: σ = {σ i } Observable τ on E: τ = {τ k } Given the state of S + E, the joint probability is given by Born s rule: P (σ i, τ k ) = T r{ρ SE (σ i τ k )} Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.17

65 Information theory Observable σ on S: σ = {σ i } Observable τ on E: τ = {τ k } Given the state of S + E, the joint probability is given by Born s rule: P (σ i, τ k ) = T r{ρ SE (σ i τ k )} H(σ) unpredictability of the value of σ of S. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.17

66 Information theory Observable σ on S: σ = {σ i } Observable τ on E: τ = {τ k } Given the state of S + E, the joint probability is given by Born s rule: P (σ i, τ k ) = T r{ρ SE (σ i τ k )} H(σ) unpredictability of the value of σ of S. H(σ τ) remaining unpredictability about σ after having peeked at the environment through τ. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.17

67 Information theory Observable σ on S: σ = {σ i } Observable τ on E: τ = {τ k } Given the state of S + E, the joint probability is given by Born s rule: P (σ i, τ k ) = T r{ρ SE (σ i τ k )} H(σ) unpredictability of the value of σ of S. H(σ τ) remaining unpredictability about σ after having peeked at the environment through τ. I(σ : τ) information learned about the property σ of S given the value of τ of E. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.17

68 Information in the environment Assume that E = N j=1 Ej as in the spin model. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.18

69 Information in the environment Assume that E = N j=1 Ej as in the spin model. Denote the amount of information which can be learned about σ my interrogating m environmental subsystems Î m (σ) = max τ M m I(σ : τ) where M m is the set of all measurements on m environmental subsystems. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.18

70 Information in the environment Assume that E = N j=1 Ej as in the spin model. Denote the amount of information which can be learned about σ my interrogating m environmental subsystems Î m (σ) = max τ M m I(σ : τ) where M m is the set of all measurements on m environmental subsystems. Î N (σ) H(σ) is a prerequisite for the objective existence of σ. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.18

71 Information in the environment Î N (σ) H(σ) Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.19

72 Information in the environment Î N (σ) H(σ) It is possible to learn about σ indirectly. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.19

73 Information in the environment Î N (σ) H(σ) It is possible to learn about σ indirectly. There exists a measurement τ on E such that H(σ τ) 0. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.19

74 Information in the environment Î N (σ) H(σ) It is possible to learn about σ indirectly. There exists a measurement τ on E such that H(σ τ) 0. Evaluate in the toy model Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.19

75 Information in the environment Î N (σ) H(σ) It is possible to learn about σ indirectly. There exists a measurement τ on E such that H(σ τ) 0. Evaluate in the toy model E is made of N = 50 spin- 1 2 Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.19

76 Information in the environment Î N (σ) H(σ) It is possible to learn about σ indirectly. There exists a measurement τ on E such that H(σ τ) 0. Evaluate in the toy model E is made of N = 50 spin- 1 2 Coupling j g jσ S z σ E j y Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.19

77 Information in the environment Î N (σ) H(σ) It is possible to learn about σ indirectly. There exists a measurement τ on E such that H(σ τ) 0. Evaluate in the toy model E is made of N = 50 spin- 1 2 Coupling j g jσ S z σ E j y Interaction action a = g j t j Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.19

78 Information in the environment Î N (σ) H(σ) It is possible to learn about σ indirectly. There exists a measurement τ on E such that H(σ τ) 0. Evaluate in the toy model E is made of N = 50 spin- 1 2 Coupling j g jσ S z σ E j y Interaction action a = g j t j σ(µ) = cos(µ)σ z + sin(µ)σ x Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.19

79 Information in the environment Î N (σ) H(σ) It is possible to learn about σ indirectly. There exists a measurement τ on E such that H(σ τ) Evaluate in the toy model E is made of N = 50 spin- 1 2 Coupling j g jσ S z σ E j y Interaction action a = g j t j σ(µ) = cos(µ)σ z + sin(µ)σ x a) b) ÎN(σ) µ π/4 π/2 0 π/8 a π/4 R0.1(σ) Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.19

80 Information in the environment Î N (σ) H(σ) It is possible to learn about σ indirectly. There exists a measurement τ on E such that H(σ τ) Evaluate in the toy model E is made of N = 50 spin- 1 2 Coupling j g jσ S z σ E j y Interaction action a = g j t j σ(µ) = cos(µ)σ z + sin(µ)σ x a) b) ÎN(σ) µ π/4 π/2 0 π/8 a π/4 R0.1(σ) Not a selective criterion. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.19

81 Redundancy of information This is a manifestation of basis ambiguity. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.20

82 Redundancy of information This is a manifestation of basis ambiguity. Denote the number of disjoint subsets of E which contain a copy of this information R(σ): R(σ) = N m(σ) where m(σ) is the smallest m for which Îm(σ) ÎN(σ). Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.20

83 Redundancy of information This is a manifestation of basis ambiguity. Denote the number of disjoint subsets of E which contain a copy of this information R(σ): R(σ) = N m(σ) where m(σ) is the smallest m for which Îm(σ) ÎN(σ). R(σ) 1 is a prerequisite for the objective existence of σ. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.20

84 Redundancy of information R(σ) 1 Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.21

85 Redundancy of information R(σ) 1 There are many copies of the information about σ in the environment. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.21

86 Redundancy of information R(σ) 1 There are many copies of the information about σ in the environment. Many independent observers can learn about σ without invalidating each other s information. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.21

87 Redundancy of information R(σ) 1 R0.1(σ) There are many copies of the information about σ in the environment. Many independent observers can learn about σ without invalidating each other s information. b) c) µ = 0.23 I(σ : e) π/8 a π/4 0 0 µ π/4 π/2 0 π/8 a π/4 0 0 π/4 µ π/ m 50 Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.21

88 Redundancy of information R(σ) 1 R0.1(σ) There are many copies of the information about σ in the environment. Many independent observers can learn about σ without invalidating each other s information. b) c) µ = 0.23 I(σ : e) R(σ) 1 and ÎN(σ) H(σ) implies a unique observable. π/8 a π/4 0 0 µ π/4 π/2 0 π/8 a π/4 0 0 π/4 µ π/ m 50 Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.21

89 Objectivity from Information Combining the criteria R(σ) 1 and ÎN(σ) H(σ) Î m (σ) H(σ) for m N. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.22

90 Objectivity from Information Combining the criteria R(σ) 1 and ÎN(σ) H(σ) Î m (σ) H(σ) for m N. Theorem If Îm(σ) = H(σ) for m N, then σ is unique. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.22

91 Objectivity from Information Combining the criteria R(σ) 1 and ÎN(σ) H(σ) Î m (σ) H(σ) for m N. Theorem If Îm(σ) = H(σ) for m N, then σ is unique. More specifically, if Îm(α) = H(α) and Îm(β) = H(β) for m N, then there exists a maximally refined observable σ such that Îm(σ) = H(σ) and H(α σ) = 0 and H(β σ) = 0. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.22

92 Objectivity from Information Combining the criteria R(σ) 1 and ÎN(σ) H(σ) Î m (σ) H(σ) for m N. Theorem If Îm(σ) = H(σ) for m N, then σ is unique. More specifically, if Îm(α) = H(α) and Îm(β) = H(β) for m N, then there exists a maximally refined observable σ such that Îm(σ) = H(σ) and H(α σ) = 0 and H(β σ) = 0. Uniqueness of objective reality. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.22

93 Robustness Redundant information is naturally protected against errors. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.23

94 Robustness Redundant information is naturally protected against errors. Lost of environmental subsystems. Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.23

95 Robustness Redundant information is naturally protected against errors. Lost of environmental subsystems. Non optimal measurements on the environment, remember that Îm(σ) = max τ Mm I(σ : τ). Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.23

96 Robustness µ = 0.23 c) Redundant information is naturally protected against errors. I(σ : e) Lost of environmental subsystems. Non optimal measurements on the environment, remember that Îm(σ) = max τ Mm I(σ : τ). 0 0 π/4 π/4 8 π/2 0 m a µ 50 Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.23

97 Robustness µ = 0.23 c) Redundant information is naturally protected against errors. I(σ : e) Lost of environmental subsystems. Non optimal measurements on the environment, remember that Îm(σ) = max τ Mm I(σ : τ). Even local random measurements acquire all the information about the pointer observable. π/4 π/4 8 π/2 0 m a µ 50 Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.23

98 Conclusion Our intuitive notion of objectivity can be translated into mathematical constraints. These constraints, together with the structure of quantum theory, leads to a unique objective reality. Not only is the information about this reality easy to extract from fragments of the environment, it is impossible to ignore! Quantum Computing-Quantum Information-Quantum Gravity, February 2004 p.24