Effective Non-Hermitian Hamiltonian of a pre- and post-selected quantum system Lev Vaidman

Conditioned evolution Weak values and weak measurements Evolution of pre- and post-selected system Plan Two state-vector Past of a quantum particle 3-box paradox Correlations of uncorrelated pre- and post-selected particles

Unitary evolution

no click Non-unitary evolution

no click Non-unitary evolution

no click Non-unitary evolution

Collapse of the wave function

What is the evolution conditioned on nondetection? no click

What is the evolution conditioned on nondetection?

What was time evolution before the particle was detected, given that it was detected? What is the evolution conditioned on detection?

What was the interaction Hamiltonian for (weak) interaction with other systems? What is the evolution conditioned on detection?

What was the interaction Hamiltonian for (weak) interaction with other systems?

Where were the pre- and post-selected photons?

B A Asking photons where have they been POWER SPECTRUM A. Danan, D. Farfurnik, S. Bar-Ad and L. Vaidman, Phys. Rev. Lett. 111, (2013)

POWER SPECTRUM B A Asking photons where have they been Photons were on the paths they could pass

B A Asking photons where have they been POWER SPECTRUM Photons were on the paths they could pass

B A Asking photons where have they been POWER SPECTRUM Photons were on the paths they could pass

C F E POWER SPECTRUM B A Asking photons where have they been Photons were on the paths they could pass

Asking photons where have they been B C A F E POWER SPECTRUM Photons were on the paths they could not pass! How to explain this?

The two-state vector formalism

The two-state vector

The two-state vector is a complete description of a system at time t ? The two-state vector is what we can say now ( ) about the pre- and post-selected system at time t So, what can we say?

The Aharonov-Bergmann-Lebowitz (ABL) formula: described by the two-state vector: Strong measurements performed on a pre- and post-selected system

The outcomes of weak measurements are weak values Weak value of a variable C of a pre- and post-selected system described at time t by the two-state vector

Weak value of a variable C of a pre- and post-selected system described at time t by the two-state vector The outcomes of weak measurements are weak values

The weak value 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

Effective non-Hermitian Hamiltonian Y. Aharonov, S. Massar, S. Popescu, J. Tollaksen, and L. Vaidman, PRL 77, (1996 )

Asking photons where have they been B C A F E POWER SPECTRUM Photons were on the paths they could not pass! How to explain this?

B A The two-state vector formalism explanation

B A

B A POWER SPECTRUM The two-state vector formalism explanation

B C A F E D

B C A F E D

B C A F E D

B C A F E D POWER SPECTRUM

B C A F E D The two-state vector formalism explanation POWER SPECTRUM

B C A F E D The two-state vector formalism explanation

B C A F E D

B C A F E D POWER SPECTRUM K.J. Resch,, J.S. Lundeen,, A.M. Steinberg, PLA 324, 125 (2004) Experimental realization of the quantum box problem

Where is the ball? ? Aharonov and Vaidman, JPA 24, 2315 (1991) Aharon and Vaidman, PRA 77, (2008) The 3-boxes paradox Vaidman, Found. Phys. 29, 865 (1999)

The three box paradox It is in always !

The three box paradox It is always in

Two useful theorems: The three box paradox For dichotomic variables:

Correlation between separable pre- and post-selected particles Aharonov and Cohen, arXiv:

Failure of the product rule for pre- and post-selected particles

Pre- and post-selected quantum systems are described best by two-state vector and weak values of observables Evolution of systems coupled to pre- and post-selected quantum systems is described by non-Hermitian Hamiltonians Conclusions

B C A F E D The one-state vector formalism explanation

C F E POWER SPECTRUM B A Photons: Wheeler is right!

For dichotomic variables Connection between strong and weak measurements

Pointer probability distribution ! Weak measurement of Pre-selection Post-selection The outcomes of weak measurements are weak values

Pointer probability distribution Weak Measurement of 20 particles pre-selected 20 particles post-selected Robust weak measurement on a pre- and post-selected single system The system of 20 particles !