Transcript N - CERN Indico

Revisiting Backwards Causality With
The Help Of Weak Measurements
1
E L I A H U C O H E N 1*, B O A Z T A M I R 2,3,
A V S H A L O M C . E L I T Z U R 3, Y A K I R A H A R O N O V 1,3
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
*[email protected]
ICNFP 2013
02.09.13
Preface
2
 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
3
 Short Review


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
4
 In their 1964 paper Aharonov, Bergmann and Lebowitz
introduced time-symmetric quantum probabilities.
 By performing both pre- and post-selection (  (t ') and
(t '') respectively) they were able to form a symmetric
formula for the probability of measuring the
eigenvalue cj of the observable c:
P (c j ) 
 (t ) c j

i
c j  (t )
(t ) ci ci  (t )
2
2
TSVF
5
 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
6
Tuesday
β
[ i ,  j ]  2i ijk k
Monday
Uncertainty?
time
Sunday
α
space
Strong Measurement
7
?
efficient detectors
(very low momentum uncertainty)
Stern-Gerlach magnet
Weak Measurement - I
8
inefficient detectors
(high momentum uncertainty)
?
?
Stern-Gerlach magnet
Why Weak Measurement?
9
s
ns
s


0
s
ns
ns
[ i ,  j ]  2i ijk k
?
n
?
Weak Values
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 The “Weak Value” of a pre- and post-selected (PPS)
ensemble:
A
w

 fin A  in
 fin  in
 It can be shown that when weakly measuring a PPS
ensemble, the pointer is displaced by the weak value:
 fin (Qd )  e
(i / ) A
w
Pd
 in (Qd )   in (Qd  A w )
Weak Measurement - II
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 The Weak measurement can be described by the Hamiltonian:
H (t ) 

N
g (t ) AsPd
 In order to get blurred results we choose a pointer with zero
expectation and
 

N
standard deviation.
 This way, when measuring a single spin we get most results

  , but when summing up the N/2↑
within the wide range
N
results, most of them appear in the narrow range 
N / 2  N / 2
agreeing with the strong results when choosing    .
Weak Evolution
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B. Tamir, E. Cohen, A. Priel, forthcoming
Weak-Strong Equivalence
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Measurement Paradox
Available information
B. Tamir, E. Cohen, A. Priel, forthcoming
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Which Path Measurement Followed by Interference
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Yakir Aharonov, Eliahu Cohen, Avshalom C. Elitzur
50%
50%
Lf
Rf
L2
R2
λ2/2
100%
L1
- λ2/2
R1
50%
50%
L0
L
1
w

R f  L1 L 0
R f L0

0.5
1 ,
0.5
Which Path Measurement Followed by Interference
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22
A Quantum Experiment with Causality
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γ
time
Morning
Last minute choice!
γ 50-50% γ
α
β
β 50-50% α
β
γ
α
β
α
γ
50-50%
space
Aharonov, Cohen, Grossman, Elitzur
arXiv: 1206.6224
β
Evening
J.S Bell’s Proof
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Alice and Bob can freely choose at the last moment
the spin orientation to be measured.
γ
α
β
α
γ
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
β
A Quantum Experiment with Causality
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498,688
I
1=↑ 2=↓ 3=↓ 4=↑ 5=↓ 6=↑ 7=↓ 8=↓ 9=↓ …n=↑
II
501,312
498,688
1=↑ 2=↓ 3=↓ 4=↑ 5=↓ 6=↓ 7=↑ 8=↓ 9=↓ …n=↑
501,312
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
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Group of: Quantum Information, Metrology and Foundations
Led by Prof. Marco Genovese
http://www.inrim.it/res/qm/index.shtml
Gedanken Setup
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Rare case
ε+←
↑
mostly
β1%↑
99%←
99%↑
1%←
1%↑
Trans
Coef.1% .
β2
β2
β
β
Idler photon
α
α
99%←
Trans.
coef. 1%
β1
Singa Photon
β-
β-
β2
1%←
99%↑
The Preliminary Experimental Setup
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Current Experimental Setup
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 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.
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The Weak potential
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 We generalize the concept of weak measurement to
the broader “weak interaction.”
 It can be shown that the Hamiltonian
H (1, 2)  H1 (1)  H 2 (2)  V (1, 2)  H 0 (1, 2)  V (1, 2)
, when
particle 1 is pre- and post- selected, results, to first
order in  , in the weak potential :
VW (2)  
 20 V  10
 2 1
Y. Aharonov, E. Cohen, S. Ben-Moshe, Unusual interactions of pre-and post-selected particles,
forthcoming in ICNFP 2012 proceedings
Challenging The Correspondence Principle
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 Let
1
H  ( x2  p2 )
2
1/ 4
2



exp[

(
x

x
)
/ 2)]
 We pre- and post-select : i
0
1/ 4
2
and  f   exp[( x  x0 ) / 2)] , wherex0  1 ,
-x0
[
x
i
, p]   0
x0
x0
+x0
 The weak values can then be calculated:
x  0 and pw  ix0 .
w
Y. Aharonov, E. Cohen, S. Ben-Moshe, Unusual interactions of pre-and post-selected particles,
forthcoming in ICNFP 2012 proceedings
.
Challenging The Correspondence Principle
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 We argue that, using the idea of weak interaction,
this weird result gets a very clear physical meaning:
2
 When interacting with another oscillator  t  exp( p )
through Hint   p1 p2 g (t ) , the latter changes its
momentum rather then its position:
exp(i   p1 p2 g (t )dt ) exp( p2 2 )  exp( x0 p2 ) exp( p2 2 ) 
 exp( x0 p2 ) exp( p2 2 )  exp[( p2   x0 / 2)2 ]exp( 2 x0 2 / 4)
Y. Aharonov, E. Cohen, S. Ben-Moshe, Unusual interactions of pre-and post-selected particles,
forthcoming in ICNFP 2012 proceedings
40
Super-Weak Values
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z
w
 1
1
 2 /2
Y. Aharonov, E. Cohen, A.C. Elitzur, forthcoming
Y. Aharonov , D. Albert and L. Vaidman, PRL (1988)
The Three Boxes Paradox
42
 fin
1

(A  B C )
3
A
 in
i  i i
W
1
B
W
1
C
W
1

(A  B  C )
3
Aharonov Y., Rohrlich D., “Quantum Paradoxes”, Wiley-VCH (2004)
 1
Anomalous Momentum Exchanges
43
 fin   cos  x  sin  z
1=↑ 2=↓ 3=↓ 4=↑ 5=↓ 6=↑ 7=↓ 8=↓ 9=↓ …n=↑
Strong:
90%x:-10%z
n↑> n↓
z
<
>
Weak: z
1=↓ 2=↓ 3=↑ 4=↑ 5=↓ 6=↑ 7=↓ 8=↓ 9=↓ …n=↑
n↓> n↑
z
N
N
Strong:
90%x:10%z
 in  cos  x  sin  z
Y. Aharonov, E. Cohen, A.C. Elitzur, Anomalous Momentum Exchanges Revealed by
Weak Quantum Measurement: Odd but Real, forthcoming
Phantom Particles
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 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.
 f    m 1    m
Lz
w
 m
 i    m    m ,  2   2  1,   
Weak Information
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 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
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 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
QKD & Cryptography
47
 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
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Describing probability distributions using weak measurements
B. Tamir, E. Cohen, A. Priel, forthcoming
Cross-Correlations
49
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
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 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 !
  !
Questions
[email protected]
arXiv.org
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