The meaning of weak value Lev Vaidman

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

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?

Weak value as an outcome of a weak measurement

Quantum measurement of Collapse!

Weak measurement of with post-selection

Weak measurement of with post-selection

Weak measurement of with post-selection

Weak measurement of without post-selection

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

The two-state vector

The two-state vector

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

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

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

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

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

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

Experiment visibility

Experiment comparing with BC reference visibility polarizer phase modulator BC visibility

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

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.