Separations of probabilistic theories via their information processing capabilities

1. Introduction Framework Cloning and Broadcasting The de Finetti Theorem Teleportation Conclusions Separations of probabilistic theories via their information processing capabilities H. Barnum1 J. Barrett2 M. Leifer2,3 A. Wilce4 1 Los Alamos National Laboratory 2 Perimeter Institute 3 Institute for Quantum Computing University of Waterloo 4 Susquehanna University June 4th 2007 / OQPQCC Workshop M. Leifer et. al. Separations of Probabilistic Theories

2. Introduction Framework Cloning and Broadcasting The de Finetti Theorem Teleportation Conclusions Outline Introduction 1 Review of Convex Sets Framework 2 Cloning and Broadcasting 3 The de Finetti Theorem 4 Teleportation 5 Conclusions 6 M. Leifer et. al. Separations of Probabilistic Theories

3. Introduction Framework Motivation Cloning and Broadcasting Frameworks for Probabilistic Theories The de Finetti Theorem Types of Separation Teleportation Conclusions Why Study Info. Processing in GPTs? Axiomatics for Quantum Theory. What is responsible for enhanced info processing power of Quantum Theory? Security paranoia. Understand logical structure of information processing tasks. M. Leifer et. al. Separations of Probabilistic Theories

7. Introduction Framework Motivation Cloning and Broadcasting Frameworks for Probabilistic Theories The de Finetti Theorem Types of Separation Teleportation Conclusions Examples Security of QKD can be proved based on... Monogamy of entanglement. The quot;uncertainty principlequot;. Violation of Bell inequalities. Informal arguments in QI literature: Cloneability ⇔ Distinguishability. Monogamy of entanglement ⇔ No-broadcasting. These ideas do not seems to require the full machinery of Hilbert space QM. M. Leifer et. al. Separations of Probabilistic Theories

10. Introduction Framework Motivation Cloning and Broadcasting Frameworks for Probabilistic Theories The de Finetti Theorem Types of Separation Teleportation Conclusions Generalized Probabilistic Frameworks C*-algebraic Generic Theories Quantum Classical M. Leifer et. al. Separations of Probabilistic Theories

11. Introduction Framework Motivation Cloning and Broadcasting Frameworks for Probabilistic Theories The de Finetti Theorem Types of Separation Teleportation Conclusions Specialized Probabilistic Frameworks p3 Quantum Theory Classical Probability p2 p1 M. Leifer et. al. Separations of Probabilistic Theories

12. Introduction Framework Motivation Cloning and Broadcasting Frameworks for Probabilistic Theories The de Finetti Theorem Types of Separation Teleportation Conclusions Types of Separation C*-algebraic Generic Theories Quantum Classical Classical vs. Nonclassical, e.g. cloning and broadcasting. All Theories, e.g. de Finetti theorem. Nontrivial, e.g. teleportation. M. Leifer et. al. Separations of Probabilistic Theories

15. Introduction States Framework Effects Cloning and Broadcasting Informationally Complete Observables The de Finetti Theorem Tensor Products Teleportation Dynamics Conclusions Review of the Convex Sets Framework A traditional operational framework. Preparation Transformation Measurement Goal: Predict Prob(outcome|Choice of P, T and M) M. Leifer et. al. Separations of Probabilistic Theories

16. Introduction States Framework Effects Cloning and Broadcasting Informationally Complete Observables The de Finetti Theorem Tensor Products Teleportation Dynamics Conclusions State Space Deﬁnition The set V of unnormalized states is a compact, closed, convex cone. Convex: If u, v ∈ V and α, β ≥ 0 then αu + βv ∈ V . Finite dim ⇒ Can be embedded in Rn . Deﬁne a (closed, convex) section of normalized states Ω. Every v ∈ V can be written uniquely as v = αω for some ω ∈ Ω, α ≥ 0. Extreme points of Ω/Extremal rays of V are pure states. M. Leifer et. al. Separations of Probabilistic Theories

21. Introduction States Framework Effects Cloning and Broadcasting Informationally Complete Observables The de Finetti Theorem Tensor Products Teleportation Dynamics Conclusions Examples Classical: Ω = Probability simplex, V = conv{Ω, 0}. Quantum: V = {Semi- + ve matrices}, Ω = {Denisty matrices }. Polyhedral: Ω V M. Leifer et. al. Separations of Probabilistic Theories

22. Introduction States Framework Effects Cloning and Broadcasting Informationally Complete Observables The de Finetti Theorem Tensor Products Teleportation Dynamics Conclusions Effects Deﬁnition The dual cone V ∗ is the set of positive afﬁne functionals on V . V ∗ = {f : V → R|∀v ∈ V , f (v ) ≥ 0} ∀α, β ≥ 0, f (αu + βv ) = αf (u) + βf (v ) Partial order on V ∗ : f ≤ g iff ∀v ∈ V , f (v ) ≤ g(v ). ˜ ˜ Unit: ∀ω ∈ Ω, 1(ω) = 1. Zero: ∀v ∈ V , 0(v ) = 0. ˜˜ ˜ ˜ Normalized effects: [0, 1] = {f ∈ V ∗ |0 ≤ f ≤ 1}. M. Leifer et. al. Separations of Probabilistic Theories

26. Introduction States Framework Effects Cloning and Broadcasting Informationally Complete Observables The de Finetti Theorem Tensor Products Teleportation Dynamics Conclusions Examples ˜˜ Classical: [0, 1] = {Fuzzy indicator functions}. Quantum: [0, 1] ∼ {POVM elements} via f (ρ) = Tr(Ef ρ). ˜˜= Polyhedral: ˜˜ V∗ [0, 1] ˜ 1 Ω V ˜ 0 M. Leifer et. al. Separations of Probabilistic Theories

27. Introduction States Framework Effects Cloning and Broadcasting Informationally Complete Observables The de Finetti Theorem Tensor Products Teleportation Dynamics Conclusions Observables Definition An observable is a finite collection (f1 , f2 , . . . , fN ) of elements of [0, 1] that satisfies N fj = u. ˜˜ j=1 Note: Analogous to a POVM in Quantum Theory. Can give more sophisticated measure-theoretic definition. M. Leifer et. al. Separations of Probabilistic Theories

30. Introduction States Framework Effects Cloning and Broadcasting Informationally Complete Observables The de Finetti Theorem Tensor Products Teleportation Dynamics Conclusions Informationally Complete Observables An observable (f1 , f2 , . . . , fN ) induces an afﬁne map: ψf : Ω → ∆N ψf (ω)j = fj (ω). Deﬁnition An observable (f1 , f2 , . . . , fN ) is informationally complete if ∀ω, µ ∈ Ω, ψf (ω) = ψf (µ). Lemma Every state space has an informationally complete observable. M. Leifer et. al. Separations of Probabilistic Theories

33. Introduction States Framework Effects Cloning and Broadcasting Informationally Complete Observables The de Finetti Theorem Tensor Products Teleportation Dynamics Conclusions Tensor Products Definition Separable TP: VA ⊗sep VB = conv {vA ⊗ vB |vA ∈ VA , vB ∈ VB } Definition ∗ ∗ ∗ Maximal TP: VA ⊗max VB = VA ⊗sep VB Definition A tensor product VA ⊗ VB is a convex cone that satisfies VA ⊗sep VB ⊆ VA ⊗ VB ⊆ VA ⊗max VB . M. Leifer et. al. Separations of Probabilistic Theories

36. Introduction States Framework Effects Cloning and Broadcasting Informationally Complete Observables The de Finetti Theorem Tensor Products Teleportation Dynamics Conclusions Dynamics Definition The dynamical maps DB|A are a convex subset of the affine maps φ : VA → VB . ∀α, β ≥ 0, φ(αuA + βvA ) = αφ(uA ) + βφ(vA ) Dual map: φ∗ : VB → VA ∗ ∗ [φ∗ (fB )] (vA ) = fB (φ(vA )) ˜ ˜ Normalization preserving affine (NPA) maps: φ∗ (1B ) = 1A . ∗ Require: ∀f ∈ VA , vB ∈ VB , φ(vA ) = f (vA )vB is in DB|A . M. Leifer et. al. Separations of Probabilistic Theories

40. Introduction Framework Distinguishability Cloning and Broadcasting Cloning The de Finetti Theorem Broadcasting Teleportation Conclusions Distinguishability Deﬁnition A set of states {ω1 , ω2 , . . . , ωN }, ωj ∈ Ω, is jointly distinguishable if ∃ an observable (f1 , f2 , . . . , fN ) s.t. fj (ωk ) = δjk . Fact The set of pure states of Ω is jointly distinguishable iff Ω is a simplex. M. Leifer et. al. Separations of Probabilistic Theories

41. Introduction Framework Distinguishability Cloning and Broadcasting Cloning The de Finetti Theorem Broadcasting Teleportation Conclusions Distinguishability Deﬁnition A set of states {ω1 , ω2 , . . . , ωN }, ωj ∈ Ω, is jointly distinguishable if ∃ an observable (f1 , f2 , . . . , fN ) s.t. fj (ωk ) = δjk . Fact The set of pure states of Ω is jointly distinguishable iff Ω is a simplex. M. Leifer et. al. Separations of Probabilistic Theories

42. Introduction Framework Distinguishability Cloning and Broadcasting Cloning The de Finetti Theorem Broadcasting Teleportation Conclusions Cloning Definition An NPA map φ : V → V ⊗ V clones a state ω ∈ Ω if φ(ω) = ω ⊗ ω. ˜ Every state has a cloning map: φ(µ) = 1(µ)ω ⊗ ω = ω ⊗ ω. Definition A set of states {ω1 , ω2 , . . . , ωN } is co-cloneable if ∃ an affine map in D that clones all of them. M. Leifer et. al. Separations of Probabilistic Theories

45. Introduction Framework Distinguishability Cloning and Broadcasting Cloning The de Finetti Theorem Broadcasting Teleportation Conclusions The No-Cloning Theorem Theorem A set of states is co-cloneable iff they are jointly distinguishable. Proof. N ⊗ ωj is cloning. If J.D. then φ(ω) = j=1 fj (ω)ωj If co-cloneable then iterate cloning map and use IC observable to distinguish the states. M. Leifer et. al. Separations of Probabilistic Theories

48. Introduction Framework Distinguishability Cloning and Broadcasting Cloning The de Finetti Theorem Broadcasting Teleportation Conclusions The No-Cloning Theorem Universal cloning of pure states is only possible in classical theory. C*-algebraic Generic Theories Quantum Classical M. Leifer et. al. Separations of Probabilistic Theories

49. Introduction Framework Distinguishability Cloning and Broadcasting Cloning The de Finetti Theorem Broadcasting Teleportation Conclusions Reduced States and Maps Definition Given a state vAB ∈ VA ⊗ VB , the marginal state on VA is defined by ˜ ∗ ∀fA ∈ VA , fA (vA ) = fA ⊗ 1B (ωAB ). Definition Given an affine map φBC|A : VA → VB ⊗ VC , the reduced map φ : VA → VB is defined by ˜ ∗ ∀fB ∈ VB , vA ∈ VA , fB (φB|A (vA )) = fB ⊗ 1C φBC|A (vA ) . M. Leifer et. al. Separations of Probabilistic Theories

50. Introduction Framework Distinguishability Cloning and Broadcasting Cloning The de Finetti Theorem Broadcasting Teleportation Conclusions Reduced States and Maps Definition Given a state vAB ∈ VA ⊗ VB , the marginal state on VA is defined by ˜ ∗ ∀fA ∈ VA , fA (vA ) = fA ⊗ 1B (ωAB ). Definition Given an affine map φBC|A : VA → VB ⊗ VC , the reduced map φ : VA → VB is defined by ˜ ∗ ∀fB ∈ VB , vA ∈ VA , fB (φB|A (vA )) = fB ⊗ 1C φBC|A (vA ) . M. Leifer et. al. Separations of Probabilistic Theories

51. Introduction Framework Distinguishability Cloning and Broadcasting Cloning The de Finetti Theorem Broadcasting Teleportation Conclusions Broadcasting Deﬁnition A state ω ∈ Ω is broadcast by a NPA map φA A |A : VA → VA ⊗ VA if φA |A (ω) = φA |A (ω) = ω. Cloning is a special case where outputs must be uncorrelated. Deﬁnition A set of states is co-broadcastable if there exists an NPA map that broadcasts all of them. M. Leifer et. al. Separations of Probabilistic Theories

54. Introduction Framework Distinguishability Cloning and Broadcasting Cloning The de Finetti Theorem Broadcasting Teleportation Conclusions The No-Broadcasting Theorem Theorem A set of states is co-broadcastable iff it is contained in a simplex that has jointly distinguishable vertices. Quantum theory: states must commute. Universal broadcasting only possible in classical theories. Theorem The set of states broadcast by any afﬁne map is a simplex that has jointly distinguishable vertices. M. Leifer et. al. Separations of Probabilistic Theories

57. Introduction Framework Distinguishability Cloning and Broadcasting Cloning The de Finetti Theorem Broadcasting Teleportation Conclusions The No-Broadcasting Theorem distinguishable Ω co-broadcastable M. Leifer et. al. Separations of Probabilistic Theories

58. Introduction Framework Introduction Cloning and Broadcasting Exchangeability The de Finetti Theorem The Theorem Teleportation Conclusions The de Finetti Theorem A structure theorem for symmetric classical probability distributions. In Bayesian Theory: Enables an interpretation of “unknown probability”. Justiﬁes use of relative frequencies in updating prob. assignments. Other applications, e.g. cryptography. M. Leifer et. al. Separations of Probabilistic Theories

61. Introduction Framework Introduction Cloning and Broadcasting Exchangeability The de Finetti Theorem The Theorem Teleportation Conclusions Exchangeability Let ω (k ) ∈ ΩA1 ⊗ ΩA2 ⊗ . . . ⊗ ΩAk . ω (k ) A1 A2 A3 A4 Ak marginalize ω (k +1) A1 A2 A3 A4 Ak Ak +1 ω (k +2) A1 A2 A3 A4 Ak Ak +1 Ak +2 M. Leifer et. al. Separations of Probabilistic Theories

62. Introduction Framework Introduction Cloning and Broadcasting Exchangeability The de Finetti Theorem The Theorem Teleportation Conclusions Exchangeability Let ω (k ) ∈ ΩA1 ⊗ ΩA2 ⊗ . . . ⊗ ΩAk . ω (k ) A5 A3 A2 A1 A4 marginalize ω (k +1) A2 A6 Ak A5 A3 A4 ω (k +2) A9 Ak A7 Ak +1 A2 A1 Ak M. Leifer et. al. Separations of Probabilistic Theories

63. Introduction Framework Introduction Cloning and Broadcasting Exchangeability The de Finetti Theorem The Theorem Teleportation Conclusions Exchangeability Let ω (k ) ∈ ΩA1 ⊗ ΩA2 ⊗ . . . ⊗ ΩAk . ω (k ) Ak Ak −1 A4 A3 A1 marginalize ω (k +1) A7 A5 A2 A3 A6 A4 ω (k +2) Ak A5 Ak +2 A2 Ak +1 A1 A3 M. Leifer et. al. Separations of Probabilistic Theories

64. Introduction Framework Introduction Cloning and Broadcasting Exchangeability The de Finetti Theorem The Theorem Teleportation Conclusions The de Finetti Theorem Theorem All exchangeable states can be written as p(µ)µ⊗k dµ ω (k ) = (1) ΩA where p(µ) is a prob. density and the measure dµ can be any induced by an embedding in Rn . M. Leifer et. al. Separations of Probabilistic Theories

65. Introduction Framework Introduction Cloning and Broadcasting Exchangeability The de Finetti Theorem The Theorem Teleportation Conclusions The de Finetti Theorem Proof. Consider an IC observable (f1 , f2 , . . . , fn ) for ΩA . {fj1 ⊗ fj2 ⊗ . . . ⊗ fjk } is IC for Ω⊗k . A The prob. distn. it generates is exchangeable - use classical de Finetti theorem. Prob(j1 , j2 , . . . , jk ) = P(q)qj1 qj2 . . . qjk dq ∆N Verify that all q’s are of the form q = ψf (µ) for some µ ∈ ΩA . M. Leifer et. al. Separations of Probabilistic Theories

69. Introduction Framework Introduction Cloning and Broadcasting Exchangeability The de Finetti Theorem The Theorem Teleportation Conclusions The de Finetti Theorem Have to go outside framework to break de Finetti, e.g. Real Hilbert space QM. C*-algebraic Generic Theories Quantum Classical M. Leifer et. al. Separations of Probabilistic Theories

70. Introduction Framework Cloning and Broadcasting Quantum Teleportation The de Finetti Theorem Generalized Teleportation Teleportation Conclusions Teleportation ρ σj Bell Measurement |Φ+ = |00 + |11 ρ M. Leifer et. al. Separations of Probabilistic Theories

71. Introduction Framework Cloning and Broadcasting Quantum Teleportation The de Finetti Theorem Generalized Teleportation Teleportation Conclusions Conclusive Teleportation ρ ? YES NO |Φ+ ? |Φ+ = |00 + |11 ρ M. Leifer et. al. Separations of Probabilistic Theories

72. Introduction Framework Cloning and Broadcasting Quantum Teleportation The de Finetti Theorem Generalized Teleportation Teleportation Conclusions Generalized Conclusive Teleportation µA ? VA ⊗ VA ⊗ VA YES NO fAA ? µA ωA A M. Leifer et. al. Separations of Probabilistic Theories

73. Introduction Framework Cloning and Broadcasting Quantum Teleportation The de Finetti Theorem Generalized Teleportation Teleportation Conclusions Generalized Conclusive Teleportation Theorem If generalized conclusive teleportation is possible then VA is ∗ afﬁnely isomorphic to VA . Not known to be sufﬁcient. Weaker than self-dual. Implies ⊗ ∼ D. = M. Leifer et. al. Separations of Probabilistic Theories

77. Introduction Framework Cloning and Broadcasting Quantum Teleportation The de Finetti Theorem Generalized Teleportation Teleportation Conclusions Examples ˜˜ Classical: [0, 1] = {Fuzzy indicator functions}. Quantum: [0, 1] ∼ {POVM elements} via f (ρ) = Tr(Ef ρ). ˜˜= Polyhedral: ˜˜ V∗ [0, 1] ˜ 1 Ω V ˜ 0 M. Leifer et. al. Separations of Probabilistic Theories

78. Introduction Framework Cloning and Broadcasting Quantum Teleportation The de Finetti Theorem Generalized Teleportation Teleportation Conclusions Generalized Conclusive Teleportation Teleportation exists in all C ∗ -algebraic theories. C*-algebraic Generic Theories Quantum Classical M. Leifer et. al. Separations of Probabilistic Theories

79. Introduction Framework Summary Cloning and Broadcasting Open Questions The de Finetti Theorem References Teleportation Conclusions Summary Many features of QI thought to be “genuinely quantum mechanical” are generically nonclassical. Can generalize much of QI/QP beyond the C ∗ framework. Nontrivial separations exist, but have yet to be fully characterized. M. Leifer et. al. Separations of Probabilistic Theories

80. Introduction Framework Summary Cloning and Broadcasting Open Questions The de Finetti Theorem References Teleportation Conclusions Open Questions Finite de Finetti theorem? Necessary and sufﬁcient conditions for teleportation. Other Protocols Full security proof for Key Distribution? Bit Commitment? Which primitives uniquely characterize quantum information? M. Leifer et. al. Separations of Probabilistic Theories

81. Introduction Framework Summary Cloning and Broadcasting Open Questions The de Finetti Theorem References Teleportation Conclusions References H. Barnum, J. Barrett, M. Leifer and A. Wilce, “Cloning and Broadcasting in Generic Probabilistic Theories”, quant-ph/0611295. J. Barrett and M. Leifer, “Bruno In Boxworld”, coming to an arXiv near you soon! M. Leifer et. al. Separations of Probabilistic Theories

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Separations of probabilistic theories via their information processing capabilities