Quantum cryptography and the generation of random numbers are two of the most exciting applications of quantum information theory (QIT). Ideally, we would like to base the security of such protocols on as few assumptions as possible. Device-independent and semi-device-independent protocols offer a kind of "gold standard" towards this goal. For example, there are protocols that allow two parties (say, Alice and Bob) to establish a provably secure cryptographic key if only two assumptions are satisfied:

  • the no-signalling principle: there is no instantaneous information transfer between Alice and Bob;
  • violation of a Bell inequality.

In principle, we can guarantee that there is no signalling by placing Alice and Bob far enough apart, such that special relativity forbids information transfer on the relevant time scales. The violation of a Bell inequality is a statistical property that Alice and Bob can observe in their recorded data. No assumptions on the devices have to be made (they could have been bought from the eavesdropper) - we do not even have to assume the validity of quantum theory! In principle, all components of the experiment are modelled as "black boxes": unknown physical systems with some specific observable statistical input-output behavior.

In addition to these applications, the black-box picture of quantum physics has proven extremely fruitful in the study of the foundations of quantum theory. For example, we can ask about the set of correlations that any physical theory admits in a causal scenario as the one in the figure. What possible probability tables p(a,b|x,y) can we generate? Bell inequalities tell us that there are probability tables that can be generated within quantum physics, but not within classical physics. More generally, ...

 

TO BE CONTINUED.

 

To illustrate one example of (1), we have shown in [1] that relativity of simultaneity enforces part of the structure of the quantum bit: if we use a Bloch ball with d degrees of freedom as our starting point (our description of a “generalized bit”), and work under some natural background assumptions, then relativity on a two-armed interferometer enforces d to be either 3 (the standard quantum bit over the complex numbers) or 5 (the quaternionic qubit). In a nutshell, this is because the temporal order of transformations performed on both arms is observer-dependent (see the picture above), so both orders must in the end lead to identical measurement outcome probabilities, which turns out to be the case only for d=3 and d=5.

Research in the sense of (2) has some conceptual overlap with other recent developments in the general context of quantum gravity (very broadly construed), where colleauges discuss, for example, the idea to derive spatiotemporal structures from entanglement entropy in a “dual” field theory [5], or they try to derive Einstein’s equations from the dynamics of entanglement entropy [6]. The main difference to these approaches is that here we work in a more directly operational framework and also start with more basic thought experiments that do not rely on particular features of field-theoretic models. In this sense, our research does not go at all as deeply into high-energy physics as the approaches just mentioned, but it relies on fewer assumptions and can thus yield valuable complementary insights.

Along these lines, we have shown in [2] that a certain interplay between probability and geometry (roughly similar to the Stern-Gerlach example above) is only possible if the dimension of space is three.  In [3], we have shown that a standard argumentation in theoretical physics can be reversed: usually we postulate the Lorentz or Poincaré symmetry group of spacetime, and derive from this that there must be certain degrees of freedom of quantum particles (“spin”) and measurement devices with certain properties (“Stern-Gerlach devices”). But as we show in [3] (see the picture below and also the video abstract in the reference), one can argue the other way around: assuming that there are measurement devices with certain universality properties, and asking for the “minimal dictionary” that translates between different observers’ descriptions of local quantum physics (as in the communication scenario in the picture), this yields in turn the Lorentz group, and the fact that quantum systems carry representations of SU(2).

We think that this research can help to illuminate the fundamental relation between quantum theory and spacetime, based on what we already know about physics. Currently, our main research question is whether properties of spacetime constrain the observable correlations in Bell scenarios or the order of interference in multislit experiments,  even if we do not assume all of quantum mechanics. If this is indeed the case, then there are important consequences for experimental tests of quantum theory, and possible applications in the device-independent framework of quantum information.

[1] A. J. P. Garner, M. P. Müller, and O. C. O. Dahlsten, The quantum bit from relativity of simultaneity on an interferometer, arXiv:1412.7112.

[2] M. P. Müller and Ll. Masanes, Three-dimensionality of space and the quantum bit: an information-theoretic approach, New J. Phys. 15, 053040 (2013). arXiv:1206.0630

[3] P. A. Höhn and M. P. Müller, An operational approach to spacetime symmetries: Lorentz transformations from quantum communication, New J. Phys. 18, 063026 (2016). In NJP’s “Highlights of 2016” collection; see also the video abstract. arXiv:1412.8462

 

Further reading:

[4] W. K. Wootters, The acquisition of information from quantum measurements, PhD thesis, University of Texas at Austin, 1980.

[5] S. Ryu and T. Takayanagi, Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence, Phys. Rev. Lett. 96, 181602 (2006). arXiv:hep-th/0603001

[6] T. Jacobson, Entanglement Equilibrium and the Einstein Equation, Phys. Rev. Lett. 116, 201101 (2016). arXiv:1505.04753

[7] S. Popescu, Nonlocality beyond quantum mechanics, Nat. Phys. 10, 264 (2014).

[8] T. Rudolph, Quantum causality: information insights, Nat. Phys. 8, 860 (2012).

[9] O. Oreshkov, F. Costa, and Č. Brukner, Quantum correlations with no causal order, Nat. Comm. 3, 1092 (2012). arXiv:1105.4464

[10] A. Kent, Quantum Tasks in Minkowski Space, Class. Quantum Grav. 29, 224013 (2012). arXiv:1204.4022

[11] J. Barrett, Information processing in generalized probabilistic theories, Phys. Rev. A 75, 032304 (2007). arXiv:quant-ph/0508211