In the beginning God created tensor... as a picture

Transcription

1 In the beginning God created tensor... as a picture Bob Coecke coecke@comlab.ox.ac.uk EPSRC Advanced Research Fellow Oxford University Computing Laboratory se10.comlab.ox.ac.uk:8080/bobcoecke/home en.html

2 Main result,... The quantum mechanical structure can be completely reformulated in terms of tensor structure alone.

3 Main result,... The quantum mechanical structure can be completely reformulated in terms of tensor structure alone. (in particular) quantum measurements can be described in terms of tensor structure alone, merely by making the distinct abilities to copy and delete classical & quantum data explicit.

4 Main result,... The quantum mechanical structure can be completely reformulated in terms of tensor structure alone. (in particular) quantum measurements can be described in terms of tensor structure alone, merely by making the distinct abilities to copy and delete classical & quantum data explicit. (in particular) 2 also the classical world and classical data transformations can be described in terms of tensor structure alone.

5 Samson Abramsky & BC ( 04) A categorical semantics for quantum protocols; IEEE-LiCS; quant-ph/ Peter Selinger ( 05)... mixed states & CPMs; his www BC ( 05) Kindergarten Quantum Mechanics; quant-ph/ BC ( 05) Introducing Categories to Practicing Physicists; soon BC & Dusko Pavlovic ( 06) Quantum measurements...; soon

6 Strip-tease of quantum theory,... exposing its bare bones

7 Strip-tease of quantum theory,... exposing its bare bones Mathematically: tensors without vectors,... no a priori number field cf. C, [0, 1],... no a priori sums, inner-product,... no a priori algebra of observables,...

8 Strip-tease of quantum theory,... exposing its bare bones Mathematically: tensors without vectors,... no a priori number field cf. C, [0, 1],... no a priori sums, inner-product,... no a priori algebra of observables,... Physically: combining systems as prime concept,... not probability, statistics,... not superposition, orthogonality,... not observation, measurement, tests,...

9 Strip-tease of quantum theory,... exposing its bare bones So some things vanish,... (cf. clothes, make-up)

10 Strip-tease of quantum theory,... exposing its bare bones So some things vanish,... (cf. clothes, make-up) But some things emerge,... (cf. skin, organs, and then bones) operational roots compositionality types reflecting kinds

11 Strip-tease of quantum theory,... exposing its bare bones These enable interpretation in terms of information-flow,... quantum-classical flow (measurement) quantum-quantum flow (evolution) classical-quantum flow (control) classical-classical flow (computation) and more (entanglement)

12 OUTLINE Part I: General compoundness Part II: Quantum compoundness Part III: Measurement & classicality

13 I Practising physics as a symmetric monoidal category

14 I Practising physics as a picture calculus

15 Kinds/types of systems: A, B, C,... e.g. electron, one qubit, n qubits, classical data,...

16 Kinds/types of systems: A, B, C,... e.g. electron, one qubit, n qubits, classical data,... Operations/experiments on systems: A f A, A g B, B h C,... e.g. preparation, acting force field, measurement,...

17 Kinds/types of systems: A, B, C,... e.g. electron, one qubit, n qubits, classical data,... Operations/experiments on systems: A f A, A g B, B h C,... e.g. preparation, acting force field, measurement,... Composition of operations: A g f C := A f B g C Doing nothing -operations: A 1 A A, B 1 B B, C 1 C C,...

18 Compoundness/parallel composition: A B A C f g B D A C h B D

19 Compoundness/parallel composition: A B A C f g B D A C h B D -structure captured by graphical representation: f B A A f B A g C f B A D C g C D A B h f B A C

20 Pseudo-locality of systems/operations A 1 A 2 f 1 A2 B 1 A 2 1 A1 g A 1 B 2 f 1 B2 B 1 B 2 1 B1 g f g = f g

21 Swapping systems/operations A 1 A 2 f g B 1 B 2 σ A1,A 2 σ B1,B 2 A 2 A 1 g f B 2 B 1 f g = g f

22 The pictures are not merely an illustration,... not merely an intuitive support,... but an actual formalism: diagrammatic proof algebraic proof

23 The pictures are not merely an illustration,... not merely an intuitive support,... but an actual formalism: diagrammatic proof algebraic proof they extend and formalise Dirac s bra-kets in 2-D by distinguishing between sequential & parallel modes.

24 Creating/destroying systems I := no system i.e. A I A I A A ψ π A s

25 Quantities: probabilistic weights s f := A A I f s B I Bs t = (s f) (t g) = (s t) (f g) (s f) (t g) = (s t) (f g) s s f = f = f s

26 PRACTICING PHYSICS Physical System Physical Operation PROGRAMMING Data Types Programs LOGIC & PROOF THEORY Propositions Proofs

27 PRACTICING PHYSICS Physical System Physical Operation PROGRAMMING Data Types Programs LOGIC & PROOF THEORY Propositions Proofs COOKING Vegetables, meet, fish, spices, mayonaise Growing, breeding, catching, cutting, mixing, eating

28 II Practising quantum physics as a -compact category

29 II Practising quantum physics as a pictures with U-turns

30 Copying? { A : A A A} A A f B A B A A f f B B

31 No-copying of quantum states { H : i i i } H C NO! C C C C C 1 1 ( ) ( ) (C C) (C C) ( ) ( ) Bell-states cause trouble!

32 Symmetric monoidal category with contravariant -involution adjoint f A B f B A ; involution dual A A ; Units η A : I A A with η A = σ A,A η A A I A η A 1 A (A A ) A 1 A A A I 1A η A A (A A)

33 Symmetric monoidal category with contravariant -involution adjoint f A B f B A ; involution dual A A ; Bell-states η A : I A A with η A = σ A,A η A A I A η A 1 A (A A ) A 1 A A A I 1A η A A (A A)

34 Symmetric monoidal category with contravariant -involution adjoint f A B f B A ; involution dual A A ; Bell-states η A : I A A with η A = σ A,A η A =

35 Symmetric monoidal category with contravariant -involution adjoint f A B f B A ; involution dual A A ; Bell-states η A : I A A with η A = σ A,A η A =

36 Symmetric monoidal category with contravariant -involution adjoint f A B f B A ; involution dual A A ; Bell-states η A : I A A with η A = σ A,A η A =

37 The contravariant involution arises as f : A B f : B A f * =: f

38 The covariant involution arises as f : A B f : A B f f * =:

39 From f * =: f and f f * =: follows (f )* * = f = f and analogous we can prove that (f ) = f

40 The adjoint decomposes: f = (f ) = (f )

41 The adjoint decomposes: f = (f ) = (f ) For Hilbert spaces: ( ) := transposition ( ) := complex conjugation

42 Others: Unitaries Projectors Bipartite projectors Hilbert-Schmidt inner-product Identification of projective structure Mixed states and completely positive maps

43 When setting f =: f f =: f

44 When setting f =: f f =: f we obtain f g = f g = g = g f f

45 Hyper-compositionality f g = g f

46 OUR description and proof f -1 i f i =

47 OUR description and proof Alice Alice f -1 i f i = Bob Bob

48 III Quantum measurement as an Eilenberg-Moore coalgebra

49 III Quantum measurement as a non-linear picture

50 Quantum measurement type: A M X A

51 Quantum measurement type: A M X A Classical data type: (X, δ, ɛ) X X δ X ɛ I

52 X δ X X X δ 1 X δ λ X δ ρx X X δ 1 X X X X I X ɛ 1 X X X 1 X ɛ X I = = = + commutativity + δ =δ +... (capture behaviour)

53 Proposition. Internal commutative comonoid structures over X are in 1-1 correspondence with commutative comonad structures on X : C C.

54 Proposition. Internal commutative comonoid structures over X are in 1-1 correspondence with commutative comonad structures on X : C C. Question. What are Eilenberg-Moore coalgebras for the comonad X given a classical object (X, δ, ɛ)?

55 A M X A A M 1 X M M λa X A δ 1 A X X A X A ɛ 1 A I A = =

56 λ A A M X A 1 X M I A η X 1 A X X A =

57 Thm. Self-adjoint Eilenberg-Moore coalgebras for X : FdHilb FdHilb are in 1-1 correspondence with dim(x)-outcome quantum measurements where X := i C and δ := X.

58 Thm. Self-adjoint Eilenberg-Moore coalgebras for X : FdHilb FdHilb are in 1-1 correspondence with dim(x)-outcome quantum measurements where X := i C and δ := X. Coalg-square idempotence P 2 i = P i mutual orthogonality P i P j i = 0 Coalg-triangle Completeness of spectrum Self-adjointness Orthogonality of projectors i P i = 1 H P i = P i PROJECTOR SPECTRUM

59 The classical world Thm. [Tom Fox, 1976] The category C of commutative comonoids and corresponding morphisms of a symmetric monoidal category with the forgetful functor C C, is final among all cartesian categories with a monoidal functor to C, mapping the cartesian product to the monoidal tensor. FdHilb = FSet

60 The classical world Thm. [Tom Fox, 1976] The category C of commutative comonoids and corresponding morphisms of a symmetric monoidal category with the forgetful functor C C, is final among all cartesian categories with a monoidal functor to C, mapping the cartesian product to the monoidal tensor. FdHilb = FSet ɛ-morphisms stochastic maps C C stoch C

61 A A = = A A = =

62 Full quantum teleportation Alice Bob Correction A A = = = = AX Measurement A Bell state A

63 POVMs and Naimark s theorem A A f f A A X X = auxiliary input C A A projective measurement Trace C h h A A 1 X X C C B.C. & Eric Paquette (2006) POVMs & Naimark s thm without sums.

64 Infinite behaviours? Hilb is NOT compact closed! since i i i diverges,...

65 Infinite behaviours? Hilb is NOT compact closed! since i i i diverges,... But we don t really care!

66 Infinite behaviours? Hilb is NOT compact closed! since i i i diverges,... But we don t really care! What is dimension anyway? Do there exit -Bell states mr. Popper? Capturing infinite behaviours via recursive types,...