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 *eliahuco@post.tau.ac.il Together with INRIM ICNFP 2013 02.09.13

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.

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

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:

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.

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

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

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

Why Weak Measurement? ? ? n

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:

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↑

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

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

14 Uncertainty

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

Which Path Measurement Followed by Interference

Non-Locality & Causality 22 Non-Locality & Causality

A Quantum Experiment with Causality γ 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: 1206.6224

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

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!

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: 1206.6224

Collaboration with INRiM 32 Group of: Quantum Information, Metrology and Foundations Led by Prof. Marco Genovese http://www.inrim.it/res/qm/index.shtml

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%

The Preliminary Experimental Setup

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

Correspondence & Hermiticity 36 Correspondence & Hermiticity

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

Challenging The Correspondence Principle 38 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

Challenging The Correspondence Principle 39 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

40 Spectral Analysis

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)

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

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

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.

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.

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

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

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

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, http://arxiv.org/abs/1308.5614

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 !

“Why do we have 5 fingers???” Questions eliahuco@post.tau.ac.il arXiv.org “Why do we have 5 fingers???”