1. School of something FACULTY OF OTHER School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES Introduction to entanglement Jacob Dunningham Paraty, August 2007

2. School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES www.quantuminfo.org October 2004 Vlatko pic 1

3. School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES www.quantuminfo.org October 2005 October 2004 Vlatko pic 19

4. School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES www.quantuminfo.org October 2005October 2006 October 2004 Vlatko pic 19~ 25

5. School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES October 2010 (projected)

6. Overview Lecture1: Introduction to entanglement: Bell’s theorem and nonlocality Measures of entanglement Entanglement witness Tangled ideas in entanglement

7. Overview Lecture1: Introduction to entanglement: Bell’s theorem and nonlocality Measures of entanglement Entanglement witness Tangled ideas in entanglement Lecture 2: Consequences of entanglement: Classical from the quantum Schrodinger cat states

8. Overview Lecture1: Introduction to entanglement: Bell’s theorem and nonlocality Measures of entanglement Entanglement witness Tangled ideas in entanglement Lecture 2: Consequences of entanglement: Classical from the quantum Schrodinger cat states Lecture 3: Uses of entanglement: Superdense coding Quantum state teleportation Precision measurements using entanglement

9. History Both speakers yesterday referred to how Schrödinger coined the term “entanglement” in 1935 (or earlier)

10. History "When two systems, …… enter into temporary physical interaction due to known forces between them, and …… separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives [the quantum states] have become entangled." Schrödinger (Cambridge Philosophical Society) Both speakers yesterday referred to how Schrödinger coined the term “entanglement” in 1935 (or earlier)

11. Entanglement Superpositions: Superposed correlations: Entanglement (pure state)

12. Entanglement Tensor Product: Separable Entangled

13. Separability Separable states (with respect to the subsystems A, B, C, D, …)

14. Separability Separable states (with respect to the subsystems A, B, C, D, …) Everything else is entangled e.g.

15. The EPR ‘Paradox’ 1935: Einstein, Podolsky, Rosen - QM is not complete Either: 1.Measurements have nonlocal effects on distant parts of the system. 2.QM is incomplete - some element of physical reality cannot be accounted for by QM - ‘hidden variables’ An entangled pair of particles is sent to Alice and Bob. The spin in measured in the z, x (or any other) direction. The measurement Alice makes instantaneously affects Bob’s….nonlocality? Hidden variables?

16. Bell’s theorem and nonlocality 1964: John Bell derived an inequality that must be obeyed if the system has local hidden variables determining the outcomes. CHSH: S = |E(a,b) - E(a, b’) + E(a’,b) + E(a’,b’)| <= 2

17. Bell’s theorem and nonlocality 1964: John Bell derived an inequality that must be obeyed if the system has local hidden variables determining the outcomes. CHSH: S = |E(a,b) - E(a, b’) + E(a’,b) + E(a’,b’)| <= 2 a b a’ b’  Alice’s axes: a and a’ Bob’s axes: b and b’

18. Bell’s theorem and nonlocality 1964: John Bell derived an inequality that must be obeyed if the system has local hidden variables determining the outcomes. CHSH: S = |E(a,b) - E(a, b’) + E(a’,b) + E(a’,b’)| <= 2 a b a’ b’  0 o (a)’ + + + + - - - - 45 o (b)’ + + + - - - - + 90 o (a’) + + - - - - + + 135 o (b’) + - - - - + + + S = +1 - (-1) +1 -1 = 2 S = +1 -(+1) +1 +1 = 2 Alice’s axes: a and a’ Bob’s axes: b and b’

19. Bell states

20. Bell’s theorem and nonlocality S = |E(a,b) - E(a, b’) + E(a’,b) + E(a’,b’)| <= 2a b a’ b’  When  =45 o, we have S = > 2 i.e no local hidden variables Without local hidden variables, e.g. for Bell states E(a,b) = cos  E(a,b’) = cos  = - sin  E(a’,b) = cos  = sin  E(a’,b’) = cos  S = | 2 cos  sin 

21. Measures of entanglement Bipartite pure states: Schmidt decomposition Positive, real coefficients

22. Measures of entanglement Bipartite pure states: Schmidt decomposition Positive, real coefficients Same coefficients Measure of mixedness Reduced density operators

23. Measures of entanglement Bipartite pure states: Schmidt decomposition Positive, real coefficients Same coefficients Measure of mixedness Reduced density operators Unique measure of entanglement (Entropy)

24. Example Consider the Bell state:

25. Example Consider the Bell state: This can be written as:

26. Example Consider the Bell state: This can be written as: Maximally entangled (S is maximised for two qubits) “Monogamy of entanglement”

27. Measures of entanglement Bipartite mixed states: Average over pure state entanglement that makes up the mixture Problem: infinitely many decompositions and each leads to a different entanglement Solution: Must take minimum over all decompositions (e.g. if a decomposition gives zero, it can be created locally and so is not entangled)

28. Measures of entanglement Bipartite mixed states: Entanglement of formation von Neumann entropy Minimum over all realisations of: Average over pure state entanglement that makes up the mixture Problem: infinitely many decompositions and each leads to a different entanglement Solution: Must take minimum over all decompositions (e.g. if a decomposition gives zero, it can be created locally and so is not entangled)

29. Entanglement witnesses An entanglement witness is an observable that distinguishes entangled states from separable ones

30. Entanglement witnesses An entanglement witness is an observable that distinguishes entangled states from separable ones Theorem: For every entangled state , there exists a Hermitian operator, A, such that Tr(A  ) =0 for all separable states,  Corollary: A mixed state, , is separable if and only if: Tr(A  )>=0

31. Entanglement witnesses An entanglement witness is an observable that distinguishes entangled states from separable ones Theorem: For every entangled state , there exists a Hermitian operator, A, such that Tr(A  ) =0 for all separable states,  Corollary: A mixed state, , is separable if and only if: Tr(A  )>=0 Thermodynamic quantities provide convenient (unoptimised) EWs

32. Covalent bonding Covalent bonding relies on entanglement of the electrons e.g. H 2 Lowest energy (bound) configuration Overall wave function is antisymmetric so the spin part is: The energy of the bound state is lower than any separable state - witness Covalent bonding is evidence of entanglement Entangled

33. Covalent bonding Covalent bonding relies on entanglement of the electrons e.g. H 2 The energy of the bound state is lower than any separable state - witness Covalent bonding is evidence of entanglement NOTE: It is not at all clear that this entanglement could be used in quantum processing tasks. You will often hear people distinguish “useful” entanglement from other sorts

34. Detecting Entanglement State tomography Bell’s inequalities Entanglement witnesses (EW)

35. Detecting Entanglement State tomography Bell’s inequalities Entanglement witnesses (EW)

36. Remarkable features of entanglement It can give rise to macroscopic effects It can occur at finite temperature (i.e. the system need not be in the ground state) We do not need to know the state to detect entanglement It can occur for a single particle

37. Remarkable features of entanglement It can give rise to macroscopic effects It can occur at finite temperature (i.e. the system need not be in the ground state) We do not need to know the state to detect entanglement It can occur for a single particle Let’s consider an example that exhibits all these features….

39. Molecule of the Year

40. Overall state: Atoms are not entangled

41. Use Entanglement Witnesses for free quantum fields e.g. Bosons Free quantum fields

42. Use Entanglement Witnesses for free quantum fields e.g. Bosons Free quantum fields “Biblical” operators - more on these later…..

43. Use Entanglement Witnesses for free quantum fields e.g. Bosons Want to detect entanglement between regions of space Free quantum fields

44. Energy Particle in a box of length L In each dimension: where

45. Energy In each dimension: where For N separable particles in a d-dimensional box of length L, the minimum energy is: Particle in a box of length L

47. Energy as an EW M spatial regions of length L/M

48. Energy as an EW M spatial regions of length L/M

49. Internal energy, temperature, and equation of state Thermodynamics

50. Ketterle’s experiments The critical temperature for BEC in an homogeneous trap is: Comparing with the onset of entanglement across the system These differ only by a numerical factor of about 2 ! Entanglement as a phase transition

51. Ketterle’s experiments Typical numbers: This gives: In experiments, the temperature of the BEC is typically: Entanglement in a BEC (even though it can be written as a product state of each particle)

52. Munich experiment A reservoir of entanglement - changes the state of the BEC Ref: I. Bloch et al., Nature 403, 166 (2000)

53. Entanglement & spatial correlations The Munich experiment demonstrates long-range order (LRO) It is tempting to think that LRO and entanglement are the same Interference term Phase coherence

54. Entanglement & spatial correlations The Munich experiment demonstrates long-range order (LRO) They are, however, related Ongoing research It is tempting to think that LRO and entanglement are the same Interference term Phase coherence A GHZ-type state is clearly entangled: BUT

55. Tangled ideas in entanglement 1. Entanglement does not depend on how we divide the system 2. A single particle cannot be ‘entangled’ 3. Nonlocality and entanglement are the same thing

56. Entanglement and subsystems Entanglement depends on what the subsystems are

57. Entanglement and subsystems Entanglement depends on what the subsystems are Entangled

58. Single particle entanglement? “Superposition is the only mystery in quantum mechanics” What about entanglement? R. P. Feynman

59. Single particle entanglement? “Superposition is the only mystery in quantum mechanics” What about entanglement? R. P. Feynman Instead of the superposition of a single particle, we can think of the entanglement of two different variables:

60. Single particle entanglement? “Superposition is the only mystery in quantum mechanics” What about entanglement? R. P. Feynman Is this all just semantics? Can we measure any real effect, e.g. violation of Bell’s inequalities? Instead of the superposition of a single particle, we can think of the entanglement of two different variables:

61. Single particle entanglement? Single photon incident on a 50:50 beam splitter:

62. Single particle entanglement? Single photon incident on a 50:50 beam splitter:

63. Single particle entanglement? Entanglement must be due to the single particle state Entangled “Bell state” Single photon incident on a 50:50 beam splitter:

64. “The term ‘particle’ survives in modern physics but very little of its classical meaning remains. A particle can now best be defined as the conceptual carrier of a set of variates... It is also conceived as the occupant of a state defined by the same set of variates... It might seem desirable to distinguish the ‘mathematical fictions’ from ‘actual particles’; but it is difficult to find any logical basis for such a distinction. ‘Discovering’ a particle means observing certain effects which are accepted as proof of its existence.” A. S. Eddington, Fundamental Theory, (Cambridge University Press., Cambridge, 1942) pp. 30-31.

65. “The term ‘particle’ survives in modern physics but very little of its classical meaning remains. A particle can now best be defined as the conceptual carrier of a set of variates... It is also conceived as the occupant of a state defined by the same set of variates... It might seem desirable to distinguish the ‘mathematical fictions’ from ‘actual particles’; but it is difficult to find any logical basis for such a distinction. ‘Discovering’ a particle means observing certain effects which are accepted as proof of its existence.” A. S. Eddington, Fundamental Theory, (Cambridge University Press., Cambridge, 1942) pp. 30-31. We need a field theory treatment of entanglement

66. Nonlocality and entanglement Nonlocality implies position distinguishability, which is not necessary for entanglement Confusion arises because Alice and Bob are normally spatially separated

67. Nonlocality and entanglement Nonlocality implies position distinguishability, which is not necessary for entanglement Confusion arises because Alice and Bob are normally spatially separated Example: This state is local, but can be considered to have entanglement PBS 1 2

68. Summary What is entanglement Bell’s theorem and nonlocality Measures of entanglement Entanglement witness in a BEC Confusing concepts in entanglement