## Quantum Causality / Quantum Theory and Spacetime

Our group is part of an international research collaboration on “Quantum Causality” [8], funded by the John Templeton Foundation. This collaboration also involves Časlav Brukner’s group at IQOQI, the Perimeter Institute for Theoretical Physics in Canada, the University of Oxford, the University of Pavia, Hong Kong University, and the University of Queensland. We are thinking about the role of causality in the quantum world, and what it means, for example, to implement superpositions of causal orders [9]. The goal of this collaboration is to obtain new ideas and tools that may be relevant in the search for a quantum theory of gravity, but also to develop new applications for quantum information processing.

One major research question of our group in this context is to study how spacetime and, in a broad sense, its causal structure is related to the structure of quantum theory. Below we describe this idea and some of its possible implications.

When students learn quantum mechanics, they are often told the rules by example of the *Stern-Gerlach experiment* and certain versions of it. They learn that the spin of an electron is a two-level quantum system (a quantum bit, or “qubit”), and so a measurement yields one of two outcomes (“up” or “down”). This outcome can be interpreted as nature’s answer to a question that we decide to ask: namely, *what is the spin in z-direction?*, where z is the axis defined by the inhomogeneity of the magnetic field that we have decided to set up in the laboratory.

While this example is certainly physically important and pedagogically useful, at second thought it involves a puzzling mishmash of two a priori very distinct physical concepts: on the one hand, the notion of *information or probability*, represented by the quantum state (we find one of two outcomes, as in a bit; the outcome is not deterministic) – and on the other hand, the notion of *space or geometry* (the gradient of the magnetic field as a vector). Having grown up with paradigmatic experiments like this, most physicists will typically not find this in any way puzzling. Isn’t quantum theory, after all, a fundamental theory of the world, and our world is geometric in nature? So why be surprised that the two are somehow tied together? Isn’t a qubit just *by definition* a representation of SU(2)?

We think that this standard way of thinking is wrong – or, at the very least, it prevents us from learning an important lesson from nature. It is again quantum information theory that shows this in a particularly clear way.

First, quantum information theory teaches us how to disentangle the mishmash: we can consider a quantum bit simply as a physical realization of an abstract binary alternative. Every qubit has the same structure, no matter in what way it is realized. We can obtain one of two possible answers or “outcomes” by choosing a Hilbert space basis (such as

{|0>, |1>} or {|+>, |->}) and performing a measurement. So far, this is a purely abstract probabilistic or information-theoretic description which does not assume any of the structure of spacetime as we know it (apart from the ability to have a temporal order of observations).

But something really interesting happens in situations like the one above: probability and geometry get tied together, and the “glue” that holds them together is group representation theory. That is, the choice of measurement is somehow set by the choice of a direction vector (“quantization axis”); the geometry of physical space is directly related to the geometry of the abstract probabilistic state space of the qubit. This is nothing that we should have expected, had we *only* known the information-theoretic structure of QT.

Second, QIT gives us some hints that this behavior is in fact *characteristic* of QT. This phenomenon can neither happen in classical physics, nor can it happen in most other “general probabilistic theories” like “generalized nonsignaling theory” [11] (which are neither classical nor quantum): the state space must admit spatial rotations as linear symmetries, and this is simply not true for generic state spaces. Moreover, Minkowski spacetime introduces many more constraints than just the “ability to rotate your coordinate system” [10], and these will in the end constrain the probabilities of detector clicks in our laboratories and the performance of information-theoretic tasks.

This suggests that the structures of spacetime and of QT are fundamentally related. And it motivates to analyze this relationship in a mathematically rigorous way, and possibly exploit it for applications.

How can this be done? There are two lines of research that suggest themselves:

- Relax some assumptions of QT, and work in a more general framework (for example, general probabilistic theories [11] or non-signalling correlations [7]). Assume that a certain relation between space(time) and detector click probabilities holds true (in a way that we know is true in our world). Then, prove that this already implies certain structural aspects of QT.
- Assume the validity of QT as we know it, together with some operational relations between QT and spacetime (like in the Stern-Gerlach experiment). Then, prove that this already implies some structural aspects of Minkowski space.

Both lines of research have the goal to improve our understanding of the structure of our physical laws, but in two different ways. Program (1) is mainly targeted at questions that researchers is QIT or Quantum Foundations find interesting: why the quantum? How can we classify the quantum correlations within the set of non-signalling correlations? Can we use functional aspects of spacetime geometry in the “device-independent” paradigm of QIT for applications like cryptography?

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