1. The meaning of weak value
Lev Vaidman

2. The meaning of weak value
What is “a value of a variable”?
Weak value controversy
Weak value as an outcome of weak measurements
Weak value as a property of a single system
Experimental results

3. A value of a physical observable
is classical
Classical values always have definite values
has a definite value eigenvalue
is quantum
The system is pre-selected
is a statistical expectation value
is quantum
has no definite value
The system is pre-selected
post-selected
is quantum
The system is pre-selected
and
The system is described by the weak value
Is the weak value statistical as the expectation value or definite as the eigenvalue?

8. Weak value as an outcome
of a weak measurement

9. Quantum measurement of
Collapse!

10. Weak measurement of with post-selection

11. Weak measurement of with post-selection

12. Weak measurement of with post-selection

13. Weak measurement of without post-selection

14. Weak value as an outcome of a weak measurement
on condition of post-selection
Weak value is a conditional expectation value

15. The two-state vector

16. The two-state vector

17. The weak value as a property of a single system
The two-state vector is a complete description of a pre- and post-selected system in a particular time
If the pre- and post-selected system is coupled to other systems through C, then its coupling at time t is described (completely) by the weak value

18. The meaning of “a value of ”
is quantum
The system is pre-selected
and
post-selected
weak value
is classical
classical value
The system is pre-selected
is quantum
eigenvalue
The system is pre-selected
is quantum
expectation value
is quantum
The system is pre-selected
and
post-selected
weak value
= expectation value

19. The meaning of “a value of ”
is quantum
The system is pre-selected
and
post-selected
weak value
The system is pre-selected
is quantum
eigenvalue
is quantum
The system is pre-selected
expectation value

20. Comparing states of external system after
eigenvalue
weak value
expectation value
Bures angle distance

21. System: charged particle, variable: electric field at the origin
eigenvalue
expectation value
weak value

22. Pre- and post-selection superposition
Example: spin 1 particle
Eigenstate eigenvalue
Superposition
Pre- and post-selection superposition
with weak value

23. Experiment
visibility

24. Experiment comparing with BC reference visibility polarizer phase
modulator BC
visibility

25. B Experiment comparing with with BC reference visibility polarizer
phase
modulator B BC
visibility

29. Summary
Weak value is a property of a single pre- and post-selected system
(It is not a property of a measuring procedure)
An ensemble is needed only to test that it is true
A pre and post-selected system interacts with external systems during infinitesimal time as if the weak value is an eigenvalue.
It is significantly different from coupling of pre-selected only system with expectation value equal to the weak value.