1. Revisiting Backwards Causality With The Help Of Weak Measurements
Eliahu Cohen1*, Boaz Tamir2,3, Avshalom C. Elitzur3, Yakir Aharonov1,3
1School of Physics and Astronomy, Tel Aviv University, Tel-Aviv 69978, Israel
2Faculty of Interdisciplinary Studies, Bar-Ilan University, Ramat-Gan, Israel
3Iyar, The Israeli Institute for Advanced Research, Rehovot, Israel
Together with INRIM
ICNFP 2013

2. Preface
A new phase in quantum theory and experimentation is increasingly spreading among laboratories worldwide, namely the Two-State-Vector Formalism (TSVF) and Weak Measurements (WM), are now maturing after ¼ century of germination.
For this reason my compressed talk will present several surprising findings, all related through TSVF.
You are therefore most welcome to later examine the original papers and experimental results.

3. Outline Short Review Challenging some conventions in QM:
TSVF (future is here and now)
Weak measurements (enable us to see it)
Challenging some conventions in QM:
Uncertainty
Non-Locality+Causality (PLEASE stay with me)
Correspondence+Hermiticty
Spectral analysis (Hope you’re still here)
Bearings on quantum information
Summary

4. ABL
In their 1964 paper Aharonov, Bergmann and Lebowitz introduced time-symmetric quantum probabilities.
By performing both pre- and post-selection ( and
respectively) they were able to form a symmetric formula for the probability of measuring the eigenvalue cj of the observable c:

5. TSVF
This idea was later broadened into a new formalism of quantum mechanics: the Two-State-Vector Formalism (TSVF).
TSVF suggests that at every moment, probabilities, interactions, and measurements’ results are determined by two state-vectors which evolve, one from the past and the other from the future, towards the present.
This is a hidden-variables theory, in that it completes quantum mechanics, but a very subtle one as we shall later see.
Equivalence to orthodox formalism of QM.

6. The TSVF – New Account Of Time
Tuesday β
Monday
Uncertainty?
space
time
Sunday α

7. efficient detectors (very low momentum uncertainty)
Strong Measurement ?
efficient detectors (very low momentum uncertainty)
Stern-Gerlach magnet

8. inefficient detectors (high momentum uncertainty)
Weak Measurement - I
inefficient detectors (high momentum uncertainty) ? ?
Stern-Gerlach magnet

9. Why Weak Measurement? ? ? n

10. Weak Values
The “Weak Value” of a pre- and post-selected (PPS) ensemble:
It can be shown that when weakly measuring a PPS ensemble, the pointer is displaced by the weak value:

11. Weak Measurement - II
The Weak measurement can be described by the Hamiltonian:
In order to get blurred results we choose a pointer with zero expectation and standard deviation.
This way, when measuring a single spin we get most results within the wide range , but when summing up the results, most of them appear in the narrow range agreeing with the strong results when choosing
N/2↑

12. B. Tamir, E. Cohen, A. Priel, forthcoming
Weak Evolution
B. Tamir, E. Cohen, A. Priel, forthcoming

13. Weak-Strong Equivalence
Measurement Paradox
Available information
B. Tamir, E. Cohen, A. Priel, forthcoming

14. 14
Uncertainty

15. Which Path Measurement Followed by Interference
Yakir Aharonov, Eliahu Cohen, Avshalom C. Elitzur 15
50%
100% - λ2/2
λ2/2 R2 L1 Lf L2
R 1
R f L0
Also for one particle

16. Which Path Measurement Followed by Interference

17. Non-Locality & Causality22
Non-Locality & Causality

18. A Quantum Experiment with Causalityγ γ β β
Last minute choice!
Evening γ β α γ γ α β γ
50-50% β β
50-50% β γ α β γ α β γ
space
time β
Morning γ
50-50%
Aharonov, Cohen, Grossman, Elitzur
arXiv:

19. No pre-established spins can exist for every possible pair of choices
J.S Bell’s Proof
Alice and Bob can freely choose at the last moment
the spin orientation to be measured. α α γ β γ β
Correlations or anti-correlations will emerge
depending on the relative angle between magnets
Conclusion:
No pre-established spins can exist for every possible pair of choices

20. A Quantum Experiment with Causality
501,312
498,688
1=↑ 2=↓ 3=↓ 4=↑ 5=↓ 6=↑ 7=↓ 8=↓ 9=↓ …n=↑ I II
501,312
498,688
1=↑ 2=↓ 3=↓ 4=↑ 5=↓ 6=↓ 7=↑ 8=↓ 9=↓ …n=↑
The spins “knew” Bob’s specific choices and their results but couldn’t tell us!

21. Correlations
“Horizontal” correlations (past-past) suggest the influence of the future state vector which creates a new kind of Bell inequalities.
“Vertical” correlations (past-future) suggest either the advantage of time-symmetric formalisms or a complex network of “noise adjustments”.
Aharonov, Cohen, Grossman, Elitzur
arXiv:

22. Collaboration with INRiM32
Group of: Quantum Information, Metrology and Foundations
Led by Prof. Marco Genovese

23. Gedanken Setup Rare case ←+ε 1β 2β Singa Photon ← ↑ mostly
↑1%
←99%
↑99%
Rare case
←+ε
Idler photon
Singa Photon α β -β
α+θ 2β
Preselected ←
Preselected ↑
←1% 1β ↑
mostly
Trans.
coef. 1%
Trans
Coef.. 1%

24. The Preliminary Experimental Setup

25. Current Experimental Setup
Gedanken setup is temporarily not practical due to the experimental complexity of coordinating two pairs of SPDC-generated photons.
Also the preliminary setup was given up due to its inability to project weak correlations and odd values.
So currently we are studying a third setup composed of a Sagnac interferometer where photons are acquiring small polarization biases before being strongly detected.
Results in 2014

26. Correspondence & Hermiticity36
Correspondence & Hermiticity

27. The Weak potential 37
We generalize the concept of weak measurement to the broader “weak interaction.”
It can be shown that the Hamiltonian
, when particle 1 is pre- and post- selected, results, to first order in , in the weak potential :
Y. Aharonov, E. Cohen, S. Ben-Moshe, Unusual interactions of pre-and post-selected particles, forthcoming in ICNFP 2012 proceedings

28. Challenging The Correspondence Principle38
Let
We pre- and post-select :
and , where ,
The weak values can then be calculated:
and
-x0
+x0
Y. Aharonov, E. Cohen, S. Ben-Moshe, Unusual interactions of pre-and post-selected particles, forthcoming in ICNFP 2012 proceedings

29. Challenging The Correspondence Principle39
We argue that, using the idea of weak interaction, this weird result gets a very clear physical meaning:
When interacting with another oscillator
through , the latter changes its momentum rather then its position:
Y. Aharonov, E. Cohen, S. Ben-Moshe, Unusual interactions of pre-and post-selected particles, forthcoming in ICNFP 2012 proceedings

30. 40
Spectral Analysis

31. Super-Weak Values Y. Aharonov , D. Albert and L. Vaidman, PRL (1988)
Y. Aharonov, E. Cohen, A.C. Elitzur, forthcoming
Y. Aharonov , D. Albert and L. Vaidman, PRL (1988)

32. The Three Boxes Paradox
Aharonov Y., Rohrlich D., “Quantum Paradoxes”, Wiley-VCH (2004)

33. Anomalous Momentum Exchanges
1=↑ 2=↓ 3=↓ 4=↑ 5=↓ 6=↑ 7=↓ 8=↓ 9=↓ …n=↑
1=↓ 2=↓ 3=↑ 4=↑ 5=↓ 6=↑ 7=↓ 8=↓ 9=↓ …n=↑
Strong: 90%x:10%z
Strong: 90%x:-10%z z -z
Weak: z N
> 
< 
n↑> n↓
n↓> n↑
Y. Aharonov, E. Cohen, A.C. Elitzur, Anomalous Momentum Exchanges Revealed by Weak Quantum Measurement: Odd but Real, forthcoming

34. Phantom Particles
A new way of describing quantum mechanics in-between two strong measurements, using weak values.
We study a gedanken experiment related to quantum entanglement of high angular momentum in order to demonstrate an interaction of a particle with another remote particle whose weak position is odd.

35. Weak Information
Mathematical structure of the weak measurement Hamiltonian, as well as the above theoretical predictions, led us to investigate a few applications in quantum information theory.

36. Eavesdropping
Instead of strongly measuring Alice’s photons, Eve can use weak measurements.
I showed that even by using a simple technique of two consecutive weak measurements performed along orthogonal axes, Eve can enlarge the number of correct bits she discovers.
Similar techniques (e.g. weak measurement at 45°) can yield even better results.
However, the code-makers are not alone – new cryptography scheme based on the “future” paper is in process. Information is shown to propagate from future to past in a secure way.
E. Cohen, Does weak measurement threaten quantum cryptography?, submitted to PRL

37. A. Aharonov, E. Cohen, B. Tamir, A.C. Elitzur
QKD & Cryptography
Alice and Bob can use the above scheme in order to create a secret key.
Intriguingly, this key is sent from future to past.
When preparing this key using a mixture of strong and weak measurements, the scheme is shown to secure.
A. Aharonov, E. Cohen, B. Tamir, A.C. Elitzur

38. QSD & Tomography
Describing probability distributions using weak measurements
B. Tamir, E. Cohen, A. Priel, forthcoming

39. B. Tamir, E. Cohen, S. Masis, http://arxiv.org/abs/1308.5614
Cross-Correlations
The quantum analog of a classical “match filter” which allows noise reduction
B. Tamir, E. Cohen, S. Masis,

40. Summary
Weak measurements enable us to see and feel the TSVF. They also present the uniqueness of quantum mechanics with its full glory.
By using them we overcome the uncertainty principle in a subtle way and enjoy both which-path measurement and interference.
Causality, correspondence principle and other conventions are likewise challenged.
Weak values, as strange as they are, have physical meaning.
Many applications to quantum information.
Research continues: We hope to find new theoretical predictions and investigate experimental results by ICNFP 2014 !

42. “Why do we have 5 fingers???”
Questions
arXiv.org
“Why do we have 5 fingers???”