Maximal Passion at a Distance
Richard Gill
Mathematical Institute, University Leiden
AND Eurandom, Eindhoven
http://www.math.leidenuniv.nl/∼gill
ETH Zürich, 12 January 2007
mathematical proof of Bell’s theorem
Bell’s 1964 inequality → CHSH inequality, 1969
experimental proof of quantum nonlocality ∗, Aspect et al 1982, ...
* “quantum nonlocality” means: “quantum induced apparent classical nonlocality”
Nature’s two ways
1
1
√
q1
√
q2
q2
q1
√
p2
√
p2
p1
p1
0
p1 + q 1 = 1 = p2 + q 2
0
p1 + q 1 = 1 = p2 + q 2
GHZ 1988 paradox, Pan et al. 2000 experiment
3 parties × 2 settings × 2 outcomes
A = 1 OR 2
B = 1 OR 2
Y = ±1
X = ±1
C = 1 OR 2
Z = ±1
CHSH, Aspect: 2 × 2 × 2 case
Overview
* Better Bell experiments! (Bell and Cirelson)
* classical polytope ⊂ quantum convex body ⊂ no-signalling polytope
* which criterion (Euclidean distance → relative entropy)
* arXiv.org:math.ST/0610115, http://www.math.leidenuniv.nl/∼gill
Passion at a Distance: Better Bell Inequalities
Case studies
1) orig Bell vs CHSH vs Hardy vs GHZ, Bell without inequalities
2) qutrits and beyond: CGLMP
3) Hardy ladder proofs
4) new CH-like inequality for binary outcomes plus undetected non-detections
Notation
Fix # parties, # settings, # outcomes
* p (x y .. | a b ..) , q (x y .. | a b ..)
probability of joint outcomes x y .. given joint settings a b ..
under classical, resp. quantum theory
* π (a b ..) probability of settings a b .. ; chosen by experimenter;
mostly: kept fixed
* p (a b .. x y..) = π (a b ..) p (x y .. | a b ..) and
q (a b .. x y..) = π (a b ..) q (x y .. | a b ..)
defines vectors pE and qE; sets { pE} and {E
q}
Key facts
No-signalling affine subspace ⊃ no-signalling polytope
⊃ quantum convex body {E
q } ⊃ classical polytope { pE}
3 completely random point
Best experiment qE for given # parties, # settings, # outcomes, solves
q(a b .. x y .. )
sup inf
q(a b .. x y .. ) log2
p
p(a b .. x y .. )
q
a b .. x y ..
X
more precisely:
sup over experimental parameters: state, measurements, joint setting probabilities
inf over classical theories
computed by missing-data maximum-likelihood (Groeneboom; programs)
Changing the range of the optimization in various ways leads to
non-locality measures for states, set-ups, . . .
The classical polytope { pE}
aka “local realism”, “the local polytope”, “local hidden variables”
∃ (X a Yb ..)ab..
such that
∀ab..
p (x y.. | ab..) x y.. = law(X a Yb ..)
Results of unperformed experiments exist, too
Counterfactual outcomes of nonmeasured observables
The quantum body {E
q}
∃ closed subspaces L ax M yb .. of Hilbert spaces H K .. such that
* ∀a (L ax )x is an orthogonal decomposition of H
* ∀b (M yb ) y is an orthogonal decomposition of K
* ..
∃ 9 ∈ H ⊗ K ⊗ .. k9k2 = 1 such that
* q (x y.. | ab..) = k5 L ax ⊗M yb ⊗··9k2
The quantum body {E
q}
∃ observables (self-adjoint operators) X a Yb ..
such that each X a commutes with each Yb , each .. , and
∃ a state 9 such that
* q (x y.. | ab..) = k5 {X a =x} ∩ {Yb =y} ∩ ·· 9k2
The classical polytope { pE}
All X a Yb .. commute
OR
There exist random variables and a probability measure . . .
* Quantum convex body {E
q}
* Classical polytope { pE}
a generalized Bell inequality
a generalized Csirelson inequality
*No-signalling
No-signalling affine subspace
polytope
GHZ paradox, Pan et al. experiment
Suppose
X a2 ≡ Yb2 ≡ Z c2 ≡ 1
Classical :
If
So
Y2Y1 = Y1Y2
then (X 1Y2 Z 2)(X 2Y1 Z 2)(X 2Y2 Z 1) = (X 1Y1 Z 1)
X 1Y2 Z 2 ≡ X 2Y1 Z 2 ≡ X 2Y2 Z 1 ≡ +1 H⇒ X 1Y1 Z 1 ≡ +1
Quantum :
But if
So
Y2Y1 = −Y1Y2
then (X 1Y2 Z 2)(X 2Y1 Z 2)(X 2Y2 Z 1) = −(X 1Y1 Z 1)
X 1Y2 Z 2 ≡ X 2Y1 Z 2 ≡ X 2Y2 Z 1 ≡ +1 H⇒ X 1Y1 Z 1 ≡ −1
This can be arranged
theoretically: GHZ
experimentally: Pan, Bouwmeester, ...
Case study 1: Comparison of famous experiments
Peres 2000; van Dam, Gill, Grünwald 2005; Groeneboom et al
eg: GHZ is potentially 9.0 times better than CHSH
but as now implemented only 9/8 times better per singlet
even assuming perfect state generation and measurement
Case study 2: 2 × 2 × d qutrits and beyond
CGLMP 2002, Acin, Gill & Gisin 2006, Gill & Zohren
Compare original form and derivation CGLMP (2002) Phys Rev Lett
quant-ph/0106024 with
X 1 ≤ Y2 and Y2 ≤ X 2 and X 2 ≤ Y1
Therefore
X 1 > Y1
implies
implies
X 1 ≤ Y1
X 1 > Y2 or Y2 > X 2 or X 2 > Y1
Therefore Pr( X 1 > Y1 ) ≤ Pr( X 1 > Y2 ) + Pr( Y2 > X 2 ) + Pr( X 2 > Y1 )
Numerics:
best state is not the maximally entangled state
best measurements are “QFT with CHSH phases” (generalized CHSH)
Using Ansatz and tricks from convexity and optimization
can compute up to d = 10 000 (see plot, next slide)
d = 10 000: Schmidt coefficients of best state (wrt relative entropy)
Conjectured best CGLMP state, measurements
|9i =
d−1
X
c j | j ji for certain c j ≥ 0, U-shaped
j=0
Alice chooses α = 0 or π/2
Bob chooses β = π/4 or −π/4
Alice, Bob apply diagonal unitaries e
i
jα
d ,
e
i
jβ
d
Alice does QFT, Bob QFT†, they measure in computational basis
Are these state and (generalized CHSH) measurements,
the best state and measurements, for CGLMP?
Proof?
d=2
Tsirel’son; Euclidean distance only
d=3
Navascues Pironio Acin quant-ph/0607119; Euclidean distance only
new upper bound, numerically attained
ie, proof to within precision d>3
only numerical evidence, increasingly sparse and/or
depending on unproven-assumption justified numerics
I believe that Stefan Zohren’s new form of CGLMP might help us
prove the optimality of the generalized CHSH measurements and obtain properties
of the optimal state . . .
CGLMP conjectures
Conjecture 1:
in 2 party × 2 setting × d outcome polytope, all interesting faces are CGLMP faces
Conjecture 2:
* optimal state is not maximally entangled
* optimal measurements are generalized CHSH
Conjecture 3:
For d → ∞ can achieve 1 ≤ 0 + 0 + 0 in
Pr( X 1 > Y1 ) ≤ Pr( X 1 > Y2 ) + Pr( Y2 > X 2 ) + Pr( X 2 > Y1 )
Polish/Russian poker: you choose Y1 or Y2; I choose X 1 or X 2; I always beat you
Case study 3: new Hardy Ladder proofs
With Marco Barbieri
Add to CGLMP inequality, same with settings 1, 2 replaced by 2, 3
and again . . .
Cancellation gives a kind of ladder or cat’s cradle
crossing pairs of adjacent measurement settings remain
intermediate horizontal rungs are deleted
eg, best ladder has length 5 and uses 10 of the 5 × 5 = 25 setting pairs
Best correlated settings:
much better experiment than CHSH
Best independent settings, so 15 superfluous pairs:
worse experiment than CHSH
Must adjust vDGG’s conjecture (# 3) :
optimal experiment with Bell singlet state is CHSH . . .
Case study 4: new CH-like CHSH inequality for detector inefficiency
With Jan-Åke Larsson
2 × 2 × 3 case — third outcome is no-detection
Define qE by adding probability of no-detection to CHSH
Relative-entropy-closest pE lies on face of classical polytope
a generalized Bell inequality
Equality constraints (affine subspace) →
inequality in “observable” probabilities only
cf. CHSH → CH inequality
Related to Larsson’s nonlinear detector-inefficiency adjusted Bell inequality
but linear
Martingale methods demonstrate insurance against memory loophole
Gill 2003, Växjö
Notes
Recall the quantum information open problems site of Reinhard Werner
Problems # 26 and # 27 are mine
see also # 1 generate all Bell inequalities
but don’t give this problem to your PhD student
Government Health Warning:
Anyone who engages in the interpretation of quantum mechanics
disappears into a black hole and is never heard of again
(R. Feynmann)
The next slides could cause serious damage to your mind
Will a succesful loophole free experiment ever be performed?
A few sensible people think no, eg, Emilio Santos
It is logically possible that quantum physics itself prevents setting up
“initial conditions” necessary for decisive experiment (cf. uncertainty relation)
John Bell agreed logically possible but unlikely, would be interested
if experimental/theoretical support were forthcoming
This “Bell’s fifth position” provides a niche for Gerard ’t Hooft to craft a
local hidden variables theory underlying quantum physics as we know it
My quantum philosophy
Detector clicks (macrosccopic events) in the past are real
The future is a wave of potentiality
its probability determined by quantum theory
Consciousness resides on the boundary between past and the future: now
As now moves relentlessly forward the past crystallizes out of it randomly
No nonlocality problem
No Schrödinger cat problem
No measurement problem (time is measurement is time — creation of space-time)
Quantum probability is ontological, not epistemological
the past is real, the wave function is real and
non-local — a function of the whole past
Quantum gravity?
There is no quantum gravity since space-time is classical (exists in the past only)
being randomly created now
Why is Nature like this?
The past is discrete, finite
The only way to have nature locally invariant under continuous rotations and shifts
is to make it random — probabilities can be invariant, continuous, . . .
This leaves us with quantum theory as the only possibility
cf. Inge Helland
Inspiration:
The math: Slava Belavkin new interpretation, eventum mechanics
The imagery: Robert M. Pirsig “Zen & the art of motor cycle maintenance”
Some slogans
WYSIATI
Copenhagen rules: there is no underlying reality
The past is particles, the future is a wave
The past is discrete, the future is continuous
natura non fecit saltum
natura abhorret a vacuo
goodbye Aristoteles, Leibniz, Kant . . . everything has a cause
mythos versus logos
Yesterday is history
Tomorrow: a mystery
Today is a gift
THE END
APPENDIX
Distance no worries for spooky particles
Stephen Pincock
ABC Science Online
Friday, 8 December 2006
A message sent using entangled, or spooky, particles of light has been beamed across
the ocean (Image: iStockphoto)
Scientists have used quantum physics to zap an encrypted message more than 140 kilometres between two Spanish islands. Professor Anton
Zeilinger from the University of Vienna and an international team of scientists used 'spooky' pulses of light to send the message. They say this is
an important step towards making international communications more secure. Zeilinger described the study this week at the Australian Institute of
Physics meeting in Brisbane. The photons they sent were linked together through a process known as quantum entanglement. This means that
their properties remained tightly entwined or entangled, even when separated by large distances, a property Einstein called spooky. The group's
achievement is important for the emerging field of quantum cryptography, which aims to use properties such as entanglement to send encrypted
messages. Research groups around the world are working in this field. But until now they have only been able to send messages relatively short
distances, limiting their usefulness. Zeilinger's team wants to be able to beam the messages to satellites in space, so they could theoretically be
relayed anywhere on the planet.
To test their system, the team went to Tenerife in the Canary Islands, where the European Space Agency operates a telescope specifically designed
to communicate with satellites. Instead of pointing the telescope at the stars, Zeilinger says, the scientists turned it to the horizontal and aimed it
towards a photon sending station 144 kilometres away on the neighbouring island of La Palma. "Very broadly speaking, we were able to establish
a quantum communication connection," he says. "We worried a lot about whether atmospheric turbulence would destroy the quantum states. But it
turned out to work much better than we feared." The results suggest it should be possible to send encrypted photons to a satellite orbiting 300 or
400 kilometres above the Earth, he says. "This is our hope. We believe that such a system is feasible." The next step is to try the system out with
an actual satellite, a project which is likely to involve the European Space Agency and others. "This is about developing quantum communications
on a grand scale," Zeilinger says. His team expects to publish its results soon.
Quantum Physics, abstract
quant-ph/0607182
From: Rupert Ursin [view email]
Date (v1): Wed, 26 Jul 2006 14:29:14 GMT
(201kb)
Date (revised v2): Thu, 27 Jul 2006 07:47:40 GMT
(374kb)
Free-Space distribution of entanglement and single photons
over 144 km
Authors: R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B.
Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Oemer, M. Fuerst, M. Meyenburg, J.
Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, A. Zeilinger
Comments: 10 pages including 2 figures and 1 table, Corrected typos.
Quantum Entanglement is the essence of quantum physics and inspires fundamental questions
about the principles of nature. Moreover it is also the basis for emerging technologies of quantum
information processing such as quantum cryptography, quantum teleportation and quantum
computation. Bell's discovery, that correlations measured on entangled quantum systems are at
variance with a local realistic picture led to a flurry of experiments confirming the quantum
predictions. However, it is still experimentally undecided whether quantum entanglement can
survive global distances, as predicted by quantum theory. Here we report the violation of the
Clauser-Horne-Shimony-Holt (CHSH) inequality measured by two observers separated by 144 km
between the Canary Islands of La Palma and Tenerife via an optical free-space link using the Optical
Ground Station (OGS) of the European Space Agency (ESA). Furthermore we used the entangled pairs
to generate a quantum cryptographic key under experimental conditions and constraints
characteristic for a Space-to-ground experiment. The distance in our experiment exceeds all
previous free-space experiments by more than one order of magnitude and exploits the limit for
ground-based free-space communication; significantly longer distances can only be reached using
air- or space-based platforms. The range achieved thereby demonstrates the feasibility of quantum
communication in space, involving satellites or the International Space Station
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© 2000 Macmillan
*+,
Two sets of linear equality constraints, one set of linear inequalities:
Normalization; No-signalling; Non-negativity
∀ab ..
∀axbb" ..
!
x y ..
!
∀ab .. x y ..
y ..
p (x y .. | ab ..) = 1
! "
#
"
p (x y .. | ab ..) =
p x y .. | ab .. etc.
y ..
p (x y .. | ab ..) ≥ 0
Why it all works:
"
"!
"
!
"
!
"!
!
1 0
0 1
0 −1
1 0
0 1
= −
=
1 0
0 −1
1 0
1 0
0 −1
van Dam, Gill & Grünwald (2005) IEEE-IT ; quant-ph/0307125
The statistical strength of nonlocality proofs
Acin, Gill & Gisin (2005) Phys Rev Lett; quant-ph/0506225
Optimal Bell tests do not use maximally entangled states
Gill (2006) in: IMS Monograph NN; math.ST/0610115
Passion at a Distance: Better Bell Inequalities
Zohren & Gill (2006) quant-ph/0612020
On the maximal violation of the CGLMP inequality for infinite dimensional states
ongoing work with Stefan Zohren (Utrecht), Marco Barbieri (Rome),
Jan-Åke Larsson (Linköping), Marek Żukowski (Gdansk), Philipp Pluch (Klagenfurt)
Peres (2000) Fortsch Phys; quant-ph/9905084
Bayesian analysis of Bell inequalities
Groeneboom, Jongbloed & Wellner (2005) math.ST/040551
Support reduction algorithm computing nonparametric function estimates mixture models
Gill (2003) Växjö II proceedings; quant-ph/0301059
Time, Finite Statistics and Bell’s Fifth Position