
Quantum correlations and generalized
probabilistic theories: an introduction
Every Wednesday (Mittwoch), 14.15  15.45,
Institute for Quantum Optics and Quantum Information
Seminar Room, Boltzmanngasse 3.
Exception: No lecture on Dec. 5,
but instead on Dec. 4, 17:15  18:45.
Note that as a preparation for the oral exam, I will
prepare a few exercises (calculations) that you can
do alone or together before the exam. (You will not have
to hand them in at any point, but you will have to
complete and understand them.) The current version can be
downloaded here. Please
check regularly for updates!
Office hours / Sprechstunde: 1 hour after each lecture (15
Min break between lecture and office hours), office of
Markus Müller, room 3.15, IQOQI.
The goal of this lecture is to provide the theoretical
foundations of the operational approach to quantum theory,
which is the basis of Quantum Information Theory and the
related research field of Quantum Foundations. The main
emphasis is on correlations: quantum theory admits
“stronger” correlations than classical physics (namely
those that violate Bell inequalities), but, surprisingly,
even stronger correlations are conceivable (socalled “PR
box correlations”). We will first see how such
correlations can be described mathematically, and how the
violation of Bell inequalities can be used for
technological applications (e.g. for the certification of
randomness). Then we will see that quantum theory is only
a special case of a larger class of generalized
probabilistic theories (with physical properties different
from quantum theory), and we will derive the Hilbert space
formalism (with its operators, complex numbers etc.) from
simple physical principles.
Contact: Markus
Müller
As shown theoretically by John Bell, and confirmed in
numerous experiments, quantum theory admits correlations
that are impossible classically, in the sense that they
violate a Bell inequality. Much later, Tsirelson as well
as Popescu and Rohrlich demonstrated a fact which seems
surprising at first sight: there are conceivable
correlations  now called "PRboxes"  which violate Bell
inequalities even stronger than any quantum state, while
still respecting the "nosignalling principle" necessary
to comply with relativity. This has initiated a research
program, aiming at bounding the set of quantum
correlations in terms of simple physical principles.
From a more general perspective, quantum information
theory has demonstrated that quantum theory is only a
special case of a large class of "generalized
probabilistic theories" (GPTs), with different physical
predictions such as superstrong nonlocality or
higherorder interference. This is comparable to the
earlier insight that Lorentz transformations are just a
special case of a large class of "theories of geometry".
This lecture gives an introduction to GPTs and some
classical results from the last few years by the quantum
information community. Possible applications range from
deviceindependent cryptography to the design of
experimental tests of quantum theory, approaches to
quantum gravity, and simple operational explanations for
"why" we have the strange formalism of quantum mechanics
with its complex numbers, operators, and Hilbert spaces.
This is a great overview article on the general research
direction: S. Popescu, Nonlocality
beyond quantum mechanics, Nature Physics 10,
264270 (2014).
Furthermore, Asher Peres has written a great book (Quantum
Theory: Concepts and Methods, Kluwer 2002)
covering many of the topics of this lecture (and actually
much more interesting stuff). Google for it!
More good books  have a look:
* K. Kraus, States, Effects, and Operations,
Lecture Notes in Physics, Springer Verlag, 1983.
* A. S. Holevo, Probabilistic
and Statistical Aspects of Quantum Theory,
NorthHolland, 1982.
Finally, Rob Spekkens is regularly
giving a great course on quantum foundations at Perimeter
Institute. You can watch the videos
online! Some of the lectures cover some material in
much more depth than I can do here  see, for example,
Lecture 8, which contains a great presentation that PRbox
correlations cannot be simulated classically (in contrast
to "Bertlmann's socks"type correlations). Rob's lectures
have a more philosophical flavor, and also much more
conceptual clarity than what I can offer in my lecture
(I'm instead putting more emphasis on recent mathematical
results in this field).
List of lectures, and additional
links
1. Overview and perspective on the course; the
BellCHSH inequality. (3.10.2018, download PDF)
What is research on "quantum foundations" all about? Bell
scenarios; classical and quantum behaviors
(=correlations).
Here are some papers for further reading. Boris Tsirelson
was the first to consider general behaviors; for example
* L. A. Khalfin, B. S. Tsirelson, Quantum
and quasiclassical analogs of Bell inequalities,
Symposium on the Foundations of Modern Physics (ed. Lahti
et al.; World Sci. Publ.), 441460 (1985).
* B. Tsirelson, Quantum
Belltype inequalities, Hadronic Journal
Supplement 8, 329345 (1993).
I forgot to mention that there is also some motivation to
consider GPTs (or related mathematical structures) in the
context of quantum gravity. For example, see this paper
for how "almost quantum" correlations might be relevant in
the context of a "histories" approach to QG, and see Caslav
Brukner's and Ognyan Oreshkov's framework for quantum
correlations with no causal order.
And here is Reinhold Bertlmann  I apologize for the
talk about dirty socks, this is all completely fictitious:
Wikipedia:
Reinhold Bertlmann
2. Nosignalling, PRboxes, convex geometry,
Bell's Theorem. (10.10.2018, download PDF)
The "PopescuRohrlich box" correlations appear already in
Tsirelson's work, but were rediscovered in this paper
* S. Popescu and D. Rohrlich, Quantum
Nonlocality as an Axiom, Found. Phys. 24(3),
379385 (1993).
Here is a standard references (book) on convex geometry
(we will work with convex geometry later, when we derive
quantum theory from postulates):
* R. Webster, Convexity,
Oxford University Press (1994).
Here is another paper where the nosignalling conditions
appear; it will become important later:
* J. Barrett, Information
processing in generalized probabilistic theories,
Phys. Rev. A 75, 032304 (2007).
Correlation really is different from causation, yet
another example:
Number
people who drowned by falling into a swimmingpool
correlates with Number of films Nicolas Cage appeared in
To learn more about causality, see this
excellent book by Judea Pearl: Causality:
Models, Reasoning and Inference. There is also a
brandnew popularscientific book that explains the main
findings: J.
Pearl and D. Mackenzie, The Book of Why: The New Science
of Cause and Effect.
3. Implausible consequences of superstrong
nonlocality: collapse of communication complexity.
(17.10.2018, download PDF)
The result is from this paper (which appeared on the arxiv
in 2005, but was published only 8 years later):
* W. van Dam, Implausible
consequences of superstrong nonlocality,
Natural Computing 12(1), 912 (2013).
Brassard and coauthors have generalized this to the case
where the PRboxes are not perfect, and the BellCHSH
violation is not 4, but 3.3 (still larger than the quantum
bound of 2.82):
* G. Brassard, H. Buhrman, N. Linden, A. A. Méthot, A.
Tapp, and F. Unger, Limit on
Nonlocality in Any World in Which Communication
Complexity Is Not Trivial, Phys. Rev. Lett. 96,
250401 (2006).
"The" standard book on communication complexity can be
found here:
* E. Kushilevitz and N. Nisan, Communication
Complexity, Cambridge University Press
(2008).
See also this little document: E. Kushilevitz, Communication
Complexity.
The claimed bound on the quantum communication complexity
of the innerproduct function is in the following paper:
* R. Cleve, W. van Dam, M. Nielsen, and A. Tapp, Quantum
Entanglement and the Communication Complexity of the
Inner Product Function, Lect. Notes Comput.
Sci. 1509, 6174 (1998).
4. Nosignalling and nonlocality imply
irreducible randomness in physics. (24.10.2018,
download PDF)
The main idea is old, and specific formulations of it have
come up several times in several different forms. The
short introduction is inspired by this talk by Toni Acín:
* A. Acin, Randomness
and quantum nonlocality (talk at QCRYPT
2012, Singapore).
A very strong recent result, saying that random
predictions of quantum theory cannot be improved (under
assumptions similar to those mentioned in the lecture) is
this one:
* R. Colbeck and R. Renner, No extension of
quantum theory can have improved predictive power,
Nature Communications 2, 411 (2011).
There is lots of material on deviceindependent
cryptography; see for example the paper above by Barrett,
Hardy and Kent. The specific result proven in the
lecture (that there cannot be hidden nonsignalling states
improving the predictions of measurements on a maximally
entangled state) is a special case of the result in this
paper:
* S. Pironio, Randomness
vs. nonlocality in a nosignalling world,
Journal of Physics: Conference Series 67,
012017 (2007).
5. Principles bounding the set of quantum
correlations. Example: macroscopic locality.
(31.10.2018, download PDF)
Here is the paper introducing macroscopic locality:
* M. Navascués and H. Wunderlich, A
glance beyond the quantum model, Proc. R.
Soc. A 466 (2010).
The definition of the set of "almost quantum
correlations" (which agrees with Q^(1+AB) for correlations
on two parties only) is here. It's not too
difficult, have a look:
* M. Navascués, Y. Guryanova, M. J. Hoban, and A. Acín, Almost
quantum correlations, Nat. Comm 6,
6288 (2015).
By the way, Miguel Navascués is a group leader colleague here at
IQOQI.
The relation to the "consistent histories" approach to
quantum gravity and the path integral is shown here:
* F. Dowker, J. Henson, and P. Wallden, A
histories perspective on characterising quantum
nonlocality, New J. Phys. 16, 033033
(2014).
6. Further example of possible beyondquantum
physics: higherorder interference. (7.11.2018,
download PDF)
Sorkin's measuretheoretic definition can be found in this
paper:
* R. D. Sorkin, Quantum
mechanics as quantum measure theory, Mod.
Phys. Lett. A 9, 31193128 (1994).
A first experimental test of higherorder interference is
described here:
* U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G.
Weihs, Ruling
Out MultiOrder Interference in Quantum Mechanics,
Science 329, 418 (2010).
The NMR test for higherorder interference is in this
paper:
* D. K. Park, O. Moussa, and R. Laflamme, Three
path interference using nuclear magnetic resonance: a
test of the consistency of Born's rule, New J.
Phys. 14, 113025 (2012).
Finally, the fivepath interferometer test is published
here  very nice to read, have a look:
* T. Kauten, R. Keil, T. Kaufmann, C. Pressl, Č.
Brukner, and G. Weihs, Obtaining
tight bounds on higherorder interferences with a
5path interferometer, New J. Phys. 19,
033017 (2017).
See also this
popularscientific article. As you can see, absence
of thirdorder interference is usually sold as
"correctness of the Born rule"; but, as we will see, this
is not strictly correct. As soon as we describe states of
physical systems by density matrices, with the usual
interpretation of convex combinations as probabilistic
mixtures, the Born rule follows trivially and cannot be
wrong. Rather, these experiments test deviations from the
state space of quantum theory.
Higherorder interference can be formulated in the
framework of generalized probabilistic theories as
described in the following paper:
* C. Ududec, H. Barnum, and J. Emerson, Three Slit
Experiments and the Structure of Quantum Theory,
Found. Phys. 41, 396405 (2011),
and it can be used as one of four postulates to derive the
Hilbert space formalism of quantum theory:
* H. Barnum, M. P. Müller, and C. Ududec, Higherorder
interference and singlesystem postulates
characterizing quantum theory, New J. Phys. 16,
123029 (2014).
It turns out that absence of thirdorder interference
constrains the possible correlations of a theory:
* J. Henson, Bounding
quantum contextuality with lack of thirdorder
interference, Phys. Rev. Lett. 114,
220403 (2015).
7. Quantum operations and generalized
probabilistic theories, part 1. (14.11.2018, slides 1 of 2, more on quantum
operations, slides 2 of
2)
Here is another nice introduction to generalized
probabilistic theories:
* P. Janotta and H. Hinrichsen, Generalized
Probabilistic Theories: What determines the structure
of quantum theory?, J. Phys. A:
Math. Theor. 47, 323001 (2014).
To see how mathematicians talk about the same structures,
have a look here:
* B. Mielnik, Generalized
quantum mechanics, Commun. Math. Phys. 37,
221256 (1974).
A nice introduction, though with a bit of a different
notation, is in Jon Barrett's paper:
* J. Barrett, Information
processing in generalized probabilistic theories,
Phys. Rev. A 75, 032304 (2007).
To learn more about the (intricate) geometry of the set
of quantum states, have a look at the book by Bengtsson
and Zyczkowski,
* I. Bengtsson and K. Zyczkowski, Geometry
of Quantum States, Cambridge University Pres, 2006,
or at this paper, which is where I've found picture that
I've used in my slides:
* I. Bengtsson, S. Weis, and K. Zyczkowski, Geometry of the
set of mixed quantum states: An apophatic approach,
Geometric Methods in Physics, XXX Workshop 2011, Trends on
Mathematics, 175197, Springer, Basel, 2013.
The best reference for quantum operations and all related
examples is the book by Nielsen and Chuang:
* M. A. Nielsen and I. L. Chuang, Quantum
Computation and Quantum Information,
Cambridge University Press, Cambrige, 2000.
8. Generalized probabilistic theories, part 2:
formal definition of states and transformations.
(21.11.2018, download PDF)
The following paper by Jon Barrett gives a nice
introduction to generalized probabilistic theories:
* J. Barrett, Information
processing in generalized probabilistic theories,
Phys. Rev. A 75, 032304 (2007).
Hardy's 2001 paper does not only introduce the framework
of GPTs (in some slightly different notation), but also
can give you a hint on what we are going later in the
lecture  namely, reconstruct quantum theory from simple
principles (later more):
L. Hardy, Quantum Theory From Five Reasonable Axioms,
arXiv:quantph/0101012
A good resource for Wigner's Theorem (quickly explained in
this
Wikipedia entry), containing a proof, is Bargmann's
paper:
* V. Bargmann, Note
on Wigner's Theorem on Symmetry Operations,
J. Math. Phys. 5(7), 862868 (1964).
9. Generalized probabilistic theories, part 3:
measurements. (28.11.2018, download PDF)
For very good explanations and nice pictures of the states
and effects of a gbit, see again the paper by Janotta and
Hinrichsen mentioned above. More about POVMs
(positive operatorvalued measures) in the quantum case,
and their physical meaning, is explained in the quantum
information book by Nielsen
and Chuang, also mentioned above.
In this lecture, I do not talk about postmeasurement
states, but one can do that (for example, Pfister's Master
thesis mentioned above is all about postmeasurement
states). As it turns out, the first general definition of
a "quantum instrument", giving all possibilities what
postmeasurement states can look like in quantum theory,
was defined in full generality for (even
infinitedimensional) generalized probabilistic theories!
See the following paper:
* E. B. Davies and J. T. Lewis, An
Operational Approach to Quantum Probability,
Commun. Math. Phys. 17, 239260 (1970).
The lecture ended with a section on continuous reversible
time evolution and uncertainty relations. The state spaces
considered there are such that the sets of normalized
states are the pnorm unit balls. These state spaces have
also been considered in the following paper, where it was
shown that the maximal CHSH Bell violation (in quantum
theory, the Tsirelson bound) can be derived from the shape
of that state space only.
* G. ver Steeg and S. Wehner, Relaxed
uncertainty relations and information processing,
Quantum Information and Computation 9,
08010832 (2009).
10. Generalized probabilistic theories, part 4:
composite systems. (4.12.2018, download PDF)
The parameters N and K have first been introduced by
Wootters and Hardy:
* W. K. Wootters, Quantum
mechanics without probability amplitudes,
Found. Phys. 16, 391405 (1986).
* L. Hardy, Quantum
Theory From Five Reasonable Axioms,
arXiv:quantph/0101012.
The paper by Hardy also contains more material on local
tomography. To learn more about quaternionic quantum
mechanics, see
Matthew Graydon's Master thesis. There is also the book
by Stephen L. Adler on "Quaternionic Quantum
Mechanics and Quantum Fields".
Preview on further lectures (subject to change):
"Boxworld"
A generalized nocloning theorem
Deriving quantum theory from simple informationtheoretic
principles

