Introduction to entanglement

1. 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 Vlatko pic October 2004 1 www.quantuminfo.org

3. School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES Vlatko pic October 2005 October 2004 1 9 www.quantuminfo.org

4. School of Physics and Astronomy FACULTY OF MATHEMATICAL AND PHYSICAL SCIENCES Vlatko pic October 2005 October 2006 October 2004 1 9 ~ 25 www.quantuminfo.org

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: Entangled Separable

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: • Measurements have nonlocal effects on distant parts of the system. • 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 0o (a)’ + + + + - - - - 45o (b)’ + + + - - - - + 90o (a’) + + - - - - + + 135o (b’) + - - - - + + + b  a’ b’ Alice’s axes: a and a’ Bob’s axes: b and b’ S = +1 - (-1) +1 -1 = 2 S = +1 -(+1) +1 +1 = 2

19. Bell states

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

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

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

23. Measures of entanglement • Bipartite pure states: Schmidt decomposition Positive, real coefficients Reduced density operators Same coefficients Measure of mixedness 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: • 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) Entanglement of formation von Neumann entropy Minimum over all realisations of:

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 and 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 and 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. H2 Lowest energy (bound) configuration Overall wave function is antisymmetric so the spin part is: Entangled The energy of the bound state is lower than any separable state - witness Covalent bonding is evidence of entanglement

33. Covalent bonding • Covalent bonding relies on entanglement of the electrons e.g. H2 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 The energy of the bound state is lower than any separable state - witness Covalent bonding is evidence of entanglement

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. Molecule of the Year Overall state: Atoms are not entangled

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

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

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

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

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

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

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