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 (so-called “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 "PR-boxes" -- which violate Bell inequalities even stronger than any quantum state, while still respecting the "no-signalling 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 higher-order 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 device-independent 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, 264-270 (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, North-Holland, 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 PR-box 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 Bell-CHSH 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 quasi-classical analogs of Bell inequalities, Symposium on the Foundations of Modern Physics (ed. Lahti et al.; World Sci. Publ.), 441-460 (1985).
* B. Tsirelson, Quantum Bell-type inequalities, Hadronic Journal Supplement 8, 329-345 (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. No-signalling, PR-boxes, convex geometry, Bell's Theorem. (10.10.2018, download PDF)
The "Popescu-Rohrlich 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), 379-385 (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 no-signalling 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 swimming-pool 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 popular-scientific 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), 9-12 (2013).
Brassard and coauthors have generalized this to the case where the PR-boxes are not perfect, and the Bell-CHSH 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 inner-product 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, 61-74 (1998).

4. No-signalling 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 non-locality (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 device-independent cryptography; see for example the paper above by Barrett, Hardy and Kent. The specific result proven in the lecture (that there cannot be hidden non-signalling 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. non-locality in a no-signalling 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 non-locality, New J. Phys. 16, 033033 (2014).

6. Further example of possible beyond-quantum physics: higher-order interference. (7.11.2018, download PDF)
Sorkin's measure-theoretic definition can be found in this paper:
* R. D. Sorkin, Quantum mechanics as quantum measure theory, Mod. Phys. Lett. A 9, 3119-3128 (1994).
A first experimental test of higher-order interference is described here:
* U. Sinha, C. Couteau, T. Jennewein, R. Laflamme, and G. Weihs, Ruling Out Multi-Order Interference in Quantum Mechanics, Science 329, 418 (2010).
The NMR test for higher-order 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 five-path 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 higher-order interferences with a 5-path interferometer, New J. Phys. 19, 033017 (2017).

See also this popular-scientific article. As you can see, absence of third-order 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.
Higher-order 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, 396-405 (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, Higher-order interference and single-system postulates characterizing quantum theory, New J. Phys. 16, 123029 (2014).

It turns out that absence of third-order interference constrains the possible correlations of a theory:
* J. Henson, Bounding quantum contextuality with lack of third-order 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, 221-256 (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, 175-197, 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:quant-ph/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), 862-868 (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 operator-valued 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 post-measurement states, but one can do that (for example, Pfister's Master thesis mentioned above is all about post-measurement states). As it turns out, the first general definition of a "quantum instrument", giving all possibilities what post-measurement states can look like in quantum theory, was defined in full generality for (even infinite-dimensional) generalized probabilistic theories! See the following paper:
* E. B. Davies and J. T. Lewis, An Operational Approach to Quantum Probability, Commun. Math. Phys. 17, 239-260 (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 p-norm 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, 0801-0832 (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, 391-405 (1986).
* L. Hardy, Quantum Theory From Five Reasonable Axioms, arXiv:quant-ph/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):
A generalized no-cloning theorem
Deriving quantum theory from simple information-theoretic principles