1. Quantum mechanics from classical statistics

2. Quantum mechanics can arise from classical statistics !

3. Quantum formalism for classical statistics can be useful for understanding how information propagates in probabilistic systems

4. 4 Correlations , conditional probabilities and Bell’s inequalities

5. (a) conditional probabilities and reduction of wave function

6. Schrödinger’s cat conditional probability : if nucleus decays
then cat dead with wc = 1
(reduction of wave function)

7. decoherence
not discussed here

8. reduction of wave function
either nucleus has decayed or not

9. conditional probability
two observables A and B
both with possible values +1 or -1
A : cat alive (1) or dead (-1)
B : nucleus decays (1) or not (-1)

10. conditional probability
probability to find A=1
after measurement of B=1
conditional probabilities
can be expressed in
terms of expectation value of A in eigenstate of B

11. conditional probability
Expectation value of A after measurement of B=1
can be computed by replacing after the measurement B=1 the original wave function by an eigenstate to B with eigenvalue B=1.
Reduction of wave function
Eliminate from the wave function all components that have not been measured ( e.g. B=-1) and normalize again.

12. reduction of wave function
convenient method to compute conditional probabilities
same in classical statistics : after first measurement of B : eliminate from probability distribution all parts corresponding to values of B that have not been measured, and normalize again
same for classical wave function
elimination often not unique ! exceptions are two-level systems
QM : reduction of wave function is unique only for complete measurements

13. reduction of wave function
is a technical way to describe conditional probabilities
not a unitary evolution of the quantum state

14. reduction of wave function not needed for computation of conditional probabilities
QM : compute conditional probabilities from quantum correlation function ( two state system or complete set of measurements )

15. conditional probabilities from correlation function
probabilitiy to find A=𝛼 , B=𝛽

16. conditional probabilities from correlation function

17. conditional probability and unitary evolution
conditional probabilities can be computed from standard unitary evolution of wave function by evaluating appropriate correlation functions
reduction of wave function not needed, but very useful for a simple description

18. w(t) :
not very suitable
for statement , if here and now
a pointer falls down

19. one wave function of Universe
not useful for the computation of conditional probabilities
reset initial conditions after each measurement
same setting in all probabilistic theories, classical or quantum

20. (b) EPR - paradoxon

21. Reality
Correlations are physical reality , not only expectation values or measurement values of single observables
Correlations can be non-local
( also in classical statistics ) ;
Causal processes needed only for
establishment of non-local correlations at some earlier stage.
Correlated systems cannot be separated into independent parts – The whole is more than the sum of its parts

22. EPR - paradoxon
Spin 0 decay : correlation between the two spins of the decay products is established at decay
No contradiction to causality, locality or reality,
if correlations are understood as part of reality (
for once not right… )

23. EPR - paradoxon
Recall : reduction of wave function is not a physical process !
Conditional probability for spin B up , spin A down
equals one , because of correlation

24. (c) Bell’s inequalities

25. Bell’s inequalities for classical correlation functions
Classical observables
Classical correlation function
obeys Bell’s inequalities
Measurements in quantum systems show violation of Bell’s inequalities

26. How correlations in classical statistical systems can violate Bell’s inequalities
not all observables may be classical observables
for A , B both classical observables :
correlation function relevant for measurements
may not be given by
classical correlation function

27. (i) not all observables are classical observables
example : quantum observables for classical particles

28. Quantum observables for position and momentum are compatible with coarse graining

29. (ii) measurement correlations are not classical correlations
example 1 : function observables
example 2 : conditional correlations

30. quantum correlations and coarse graining
quantum features in classical statistical systems are typically related to coarse graining
wave functions or density matrix contain only local information – much less information than overall probability distribution
coarse graining of information
classical correlation function may exist, but not be compatible with coarse graining

31. quantum correlations and coarse graining
many classical observables mapped to same local observables
equivalence class of classical observables : all observables that correspond to same local observable
local system and environment : equivalence class is system property
measurement correlation should not depend on environment, only on equivalence classes
classical correlation depends on environment ,
conditional correlation not

32. function observables and coarse graining of microphysical classical statistical ensemble
particle concept not unique
one more example for
non – commutativity in classical statistics

33. microphysical ensemble
states τ
labeled by sequences of occupation numbers or bits ns = 0 or 1
τ = [ ns ] = [0,0,1,0,1,1,0,1,0,1,1,1,1,0,…] etc.
probabilities pτ > 0

34. classical correlation
Classical local observables are functions of local occupation numbers
Their classical correlations can be computed from local probabilities
Classical correlations obey Bell’s inequalities

35. function observable

36. function observable I(x1) I(x2) I(x3) I(x4)
normalized difference between occupied and empty bits in interval
I(x1)
I(x2)
I(x3)
I(x4)

37. smooth functions
become possible for large number of bits in every interval

38. generalized function observable
normalization
classical
expectation
value
several species α

39. position
classical observable :
fixed value for every state τ

40. momentum classical observable : fixed value for every state τ
derivative observable
classical observable :
fixed value for every state τ

41. complex structure

42. classical product of position and momentum observables
commutes !
and obeys Bell’s inequalities

43. different products of observables
differs from classical product

44. Which product describes correlations of measurements ?

45. coarse graining of information for subsystems

46. density matrix from coarse graining
position and momentum observables use only
small part of the information contained in pτ ,
relevant part can be described by density matrix
subsystem described only by information
which is contained in density matrix
coarse graining of information

47. quantum density matrix
density matrix has the properties of
a quantum density matrix

48. quantum operators

49. quantum product of observables
the product
is compatible with the coarse graining
and can be represented by operator product

50. incomplete statistics
classical product
is not computable from information which
is available for subsystem !
cannot be used for measurements in the subsystem !

51. classical and quantum dispersion

52. subsystem probabilities
in contrast :

53. squared momentum quantum product between classical observables :
maps to product of quantum operators

54. non – commutativity in classical statistics
commutator depends on choice of product !

55. observables and coarse graining
In the coarse grained system P is no longer a classical observable
P contains derivatives acting on the wave function

56. measurement correlation
correlation between measurements of positon and momentum is given by quantum product
this correlation is compatible with information contained in subsystem

57. coarse graining ρ(x , x´)
from fundamental fermions at the Planck scale to atoms at the Bohr scale
p([ns])
ρ(x , x´)

58. conditional correlations

59. classical correlation
pointwise multiplication of classical observables
not available on level of local observables
definition depends on details of classical observables , while many different classical observables correspond to the same local observable
classical correlation depends on probability distribution for the local system and its environment
needed : correlation that can be computed
in terms of local observables and wave functions or density matrix !

60. classical or conditional correlation ?
Classical correlation appropriate if two measurements do not influence each other and if simultaneous probabilities for values of both observables are available.
Conditional correlation takes into account conditions after first measurement.
Two measurements of same local observable immediately after each other should yield the same value !

61. correlation ( ) = A, B : two-level observables
sum over coarse grained states 𝛼
simultaneous probabilities for values of both observables are not available for states 𝛼

62. conditional correlation
probability to find value +1 for product
of measurements of A and B
probability to find A=1
after measurement of B=1
… can be expressed in
terms of expectation value
of A in eigenstate of B

63. conditional product conditional product of observables
conditional correlation
does it commute ?

64. conditional/quantum correlation
conditional correlation in classical statistics equals quantum correlation !
using conditional correlations in classical statistics :
no contradiction to Bell’s inequalities

65. conditional correlation and anticommutators
conditional two point correlation commutes =

66. conditional three point correlation

67. conditional three point correlation in quantum language
conditional three point correlation is not commuting !

68. conditional correlations and operators
conditional correlations in classical statistics can be expressed in terms of operator products

69. non – commutativity of operator product is closely related to properties of conditional correlations !

70. Can quantum physics be described by classical probabilities ?
“ No go “ theorems
Bell , Clauser , Horne , Shimony , Holt
implicit assumption : use of classical observables and classical correlation function for correlation between measurements
Kochen , Specker
assumption : unique map from operators to classical observables

71. conclusion quantum statistics emerges from classical statistics
wave function, superposition, interference, entanglement
unitary time evolution of quantum mechanics can be described by suitable time evolution of classical probabilities
memory materials are quantum simulators
conditional correlations for measurements both in quantum and classical statistics

72. quantum mechanics from classical statistics
probability amplitude
entanglement
interference
superposition of states
fermions and bosons
unitary time evolution
transition amplitude
non-commuting operators

73. what is an atom ? quantum mechanics : isolated object
quantum field theory : excitation of complicated vacuum
classical statistics : sub-system of ensemble with infinitely many degrees of freedom

74. Quantum particle
Quantum field theory for Dirac fermions in external electromagnetic field can be described by suitable time evolution equation for classical local probabilities
Includes discrete time steps and complex structure
One particle state, non-relativistic limit yields Schroedinger equation for particle in potential
Entangled two fermion states
open : overall classical statistical probability distribution that accounts for such an evolution law for local probabilities

75. end