1. 4.Classical entanglement CHSH version of the Bell inequalities: These deal with correlations between a set of four classical probabilities. In particular, the either-or observable, for detecting some physical property for one particle, or the detection of for the second particle. We know that the projection of theWigner function onto any Lagrangian plane produces a classical probability distribution. So, constructing our observables from regions on such a plane should lead to measurements that satisfy the CHSH inequality.

2. In the case of a bipartite system, we can define such Lagrangian planes as, such that with conjugate variables, and, such that are symplectic transformations. Now define projection operators, and, for the variable to be in the interval a or b and the four observables, which take the value +1 In terms of the Wigner function: Projection onto the a probability. Projection on the interval a

3. Since the product of commuting observables are also observables: thth these can be measured and their expectation values satisfy = Because these are purely classical probabilities, no term with a (-4)-coefficient can be positive. This is the classic argument for the CHSH form of Bell’s inequality. Measurements of commuting variables cannot violate CHSH. with

4. This argument does not depend on whether the bipartite state is entangled,or not. What if the Wigner function itself is positive definite? Then, even projectors over noncommuting variables, such that and will lead to classical correlations, within the CHSH inequalities. Does this mean that there is no entanglement? What are the conditions for a positive Wigner function?

5. The condition for the Wigner function of a pure state to be positive is that it is a Gaussian (a generalized coherent state, or squeezed state) (Hudson: Rep. Math. Phys. 6 (1974) 249) What about a mixed state? Evidently, any probability distribution over coherent states has a positive Wigner function: The P-function with positive coefficients. But this condition is only sufficient, not necessary. WignerQ = Husimi P

6. So, probabilities obtained over intervals for squeezed states will always be classically correlated: they must satisfy the CHSH inequality. See the paper by Bell in Speakable and unspeakable in quantum mechanics Bell also produces the example of a Fock state for which such probabilities violate CHSH. Is a multidimensional squeezed state too ‘classical’ to be entangled?

7. Consider any simple, L=2, product state: Then evolve this state with the Hamiltonian, (classical, or Weyl representation) Being quadratic, this merely rotates both the p and the q coordinates (classically) in the argument of

8. Then, after a rotation, The reduced density is just a section, so

9. To show how classical an entanglement can be, choose a simple Gaussian state, the product of HO ground states: is also a Gaussian, with elliptic level curves that are also rotated. After rotation and the partial trace: or ‘‘

10. The narrowing of the Gaussian shows that the state is not pure. The Wigner function is more intuitive: Obtained by taking the Fourier transform, This broader Gaussian still integrates to one. (It could be obtained as an average over pure Gaussian Wigner functions.) Is this a freak? Nothing could be more classical for a start (positive Wigner function) and then a classical rotation produces entanglement!

11. Go back to the product state of both

12. So, the original EPR state does not violate Bell inequalities for the measurement of any four observables defined by intervals of position or momentum. But it is technically entangled. Bell leaves open the possibility that other unusual observables may lead to inequality violations for such a state. What about generalized parities: the eigenvalues of reflection operators for a given subspace?

13. The fact that the Wigner function is symmetric with respect to the origin implies that Hence, there is a finite probability of obtaining the -1 eigenvalue, if a parity measurement is performed on subsystem-1.

14. The same also holds for subsystem-2. The fact that the Wigner function is symmetric about the origin implies that all the pure states, into which can be decomposed, must have pure parity, but they are not all even.

15. Thus, we need a common basis for all these operators: the product of an even-odd basis for both subsystems, leading to the table:

16. Conclusion The original EPR state is truly quantum, i.e. correctly described as entangled, just as the Bohm version of EPR. The secret lies in the property that is measured: Generalized position measurements on the subsystems cannot distinguish this pure state from a classical distribution, but reflection eigenvalues are purely quantum.

17. Violation of CHSH: Recall that the correlation for reflection measurements on either subsystem is given by the Wigner function: We have already examined this at the origin. The decay of the Wigner function for large arguments implies that sinks from 2, its maximal classical value, obtained at the origin, to its limiting value, 1.

18. But because the expansion, leads to

19. Thus, the smoothed EPR state can be measured in ways that lead to violation of Bell’s inequalities. Banaszek and Wodkiewicz have proposed an experiment in quantum optics to achieve this. A note on classicality versus hidden variables: Bell’s inequalities set limits to the correlations of any possible classical-like theory. This is much stronger than my presentation, but the inequalities must include the classical system that corresponds directly to the quantum system under consideration.

20. 5.Decoherence: the Lindblad Equation Decoherence results through entanglement of the system under consideration with an uncontroled system, labled the environment. Generally we do not know the initial state of the environment: Further averages, beyond the implicit average in the reduced density operator.

21. A simple example: Weak scattering of many light particles. If the duration of a single scattering process is short compared to the typical time scales of the system evolving by itself: The last term accounts for the total effect of many scattering events, in which the system is dynamically insensitive, i.e. no recoil. Nonetheless, the scatterers transport information about the system! [Joos, in Giulini et. al.]

22. Consider a single scattering event from the system, if it is initially in the eigenstate. Then, if is the initial state of the environment, where is the scattering operator (S-matrix) for this configuration of the system. For a general initial state, so, the reduced density operator changes accordingly: because

23. Thus, the matrix elements of the density operator evolve as If the overlap is close to unity, then the effect of many collisions, with the rate will be: with

24. For the diagonal terms, Thus the trace of is not affected and only the offdiagonal terms decay with decoherence. Generally, the greater the difference between n and m, the faster is the decay. For the scattering off a particle, the scattering depends on its position, determined by its wave function, Then, for a single scattering event:

25. If the scattering interaction is translationally invariant, the S-matrix in the momentum representation depends on the position of the scaterrer by a phase factor: Then, for So, averaging over many scattering events:

26. The Lindblad equation has the general form: The Lindblad operators, account for the effect of the external environment on the reduced density operator, Consider the case, Then the position representation for the environmental term becomes The same form as obtained for a weakly scattering environment.

27. Hermitian Lindblad operators lead to decoherence, but no dissipation. Not so with the master equation for quantum optics: (A single cavity field mode interacting with 2-level atoms) in terms of the field mode operators, and The Hamiltonian is just the harmonic oscillator, so this is a quantum damped harmonic oscilator, allowing for emission and absorption of photons (depending on the temperature, through A).

28. 6.Linblad Equation for the Chord Function This depends on product formulae for the chord representation, where the delta-function eliminates the free side of the polygon of the original cocycle.

29. In the unitary part of the Lindblad equation there are products of two operators and of three in the open part. The problem is that common forms for are singular in the chord representation. Therefore, these will be represented by their Weyl symbol in the following formulae. In the case where the Lindblad operators are linear functions of

30. Note that the Hamiltonian is evaluated at the chord tips: In terms of the double phase space variable and

31. This is now compactified through the definition of the double phase space Hamiltonian: In the absence of dissipation, will be constants of the full classical motion, generated by in double phase space. Each of these reduced Hamiltonians generate independent motions for each chord tip. and

32. A classical canonical transformation, C, is evolved continuously by the trajectory pairs: if the evolution is generated by H(x). Mecânica quântica:

33. Nonunitary double phase space Hamiltonian ? Hamilton’s equations: Hamiltonian motion in double phase space is compatible with contraction of the centre-Wigner plane, together with expansion of the chord plane.

34. The master equation for the chord function is thus or, alternatively, that is,

35. Exemplo: Hamiltonianas quadráticas: Em geral, os movimentos de x e y estão acoplados, mas y=0 é sempre um plano invariante, onde a evolução é gerada por H’(x). Espaço simples Espaço duplo

36. Solution for a quadratic Hamiltonian: The unitary evolution of the system is simply: given in terms of the classical Poisson brackets, just as for the Wigner function. The open term, for each Lindblad operator, is

37. Then the exact general solution is simply obtained from the classical evolution, as thus,thus, Thus, the amplitude for long chords of the classically evolving chord function is dampened by the decoherence functional: taken over trajectory pairs, or a single trajectory in double phase space. The latter interpretation is mandatory, in the presence of dissipation.

38. The dissipative part of the double Hamiltonian expands, while its Fourier transform, contracts. But this is coarse-grained by in the convolution for the Wigner function:

39. Since, the classical evolution is linear, the decoherence functional is a quadratic function of the chords. Therefore, is a Gaussian in chord space, which narrows in time. Its Fourier transform is: where,

40. The width of the Gaussian decoherence window that coarsegrains the Wigner function is which equals 0 at t=0. When then this Gaussian could be the Wigner function of a pure squeezed state, so that the evolved Wigner function could be identified with a Husimi function. Then, the evolved Wigner function must be positive! The time for positivity is independent of the initial pure state. This time does depend on both the Hamiltonian and the Lindblad operators. A longer time makes all P-functions positive.

41. If the Lindblad operators, are all self-ajoint, then decoherence and diffusion, but no dissipation. Example: Evolution of the Wigner function For the “Schrödinger cat”

42. Damped Harmonic Oscillator

44. 7.Semiclasical Markovian Wigner and Chord functions the semiclassical chord function Insert the semiclassical chord function into the Lindblad the Lindblad equation and perform the integrals the method of stationary phase by the method of stationary phase:

45. Semiclassical pure state: Chords and centres are conjugate coordinates for double phase space.

46. Pictured in double phase space, both the Wigner function and the chord function are just WKB wave functions: or else:

47. Semiclassical evolution of the chord function employs the solution of the Hamilton-Jacobi equation. In terms of the original Hamiltonian, this is in the nondissipative case, or This is an ordinary H-J equation in double phase space:

48. So we can include in the double phase space Hamiltonian. For general Lindblad operators, the open term can also be evaluated by stationary phase as: Stationary phase evaluation of the commutator is

49. If we ignore the Hamiltonian motion, then we can consider the action, to be constant. Then, only the WKB amplitudes evolves as and This procedure is analogous to that leading to the Trotter formula for path integrals.

50. If is the WKB-evolution of a branch of the initial chord function, in double phase space, then decoherence functional Here the decoherence functional is and Cancelation of long chords Cancelation of quantum correlations recall that

51. The dissipative part of the double Hamiltonian expands, while its Fourier transform, contracts. But this is coarse-grained by In the convolution for the Wigner function: