Black boxes in space and time

In an FWF-funded project, our group brings the "black-box" approach of quantum information theory into space and time. In an nutshell, we study the implications of symmetries (via group representation theory) on (semi-)device-independent scenarios. Here, we describe some of the background and ideas of the project.

Black boxes: foundations and applications

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 [5, 6, 7] and semi-device-independent [8, 9] 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;
• a violation of a Bell inequality [10].

In principle, we can guarantee that there is no signalling by placing Alice and Bob at spacelike separation, such that special relativity forbids information transfer on the relevant time scales. The violation of a Bell inequality is a statistical property which 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" [10]: unknown physical systems with some specific observable statistical input-output behavior. It is remarkable that such black-box descriptions are sufficient to guarantee the functionality and security of some quantum information-theoretic protocols.

Apart from protocols, this black-box picture has proven extremely useful for the study of the foundations of quantum physics. 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 above. What possible probability tables p(a,b|x,y) (also called "behaviors" or "correlations") can we generate? Bell's theorem [10] tells us that there are correlations that can be generated via quantum physics in this causal scenario, but not via classical physics, as testified by a violation of a Bell inequality. For example, correlations that arise from certain measurements on a singlet state have this property. Despite these difference, classical and quantum correlations satisfy one common constraint. These are the no-signalling conditions [10]: Bob's choice of input y ("choice of measurement") has no impact on the marginal probabilities of Alice's outputs a ("outcomes") and vice versa:

In a relativistic universe, these conditions are crucial: if they were violated, we could construct a "Bell telephone" that allows us to transmit informtion faster than light. This is impossible in classical and quantum physics. However, quantum physics does not give us the most general non-signalling correlations that satisfy this important constraint: there are conceivable correlations that are non-signalling, but that violate Bell inequalities by more than any quantum state. An example is given by "Popescu-Rohrlich" ("PR-box") [10] correlations: Alice and Bob both choose a single bit as input, i.e. x,y∈{0,1}, and they each obtain either +1 or -1 as outputs, i.e. a,b∈{-1,+1}. The PR-box probabilities are

That is, the outcomes are either perfectly anticorrelated (if Alice and Bob choose inputs "1") or otherwise perfectly correlated. This probability table satisfies the no-signalling conditions, but it cannot be realized in a Bell scenario within quantum physics.

The situation can be illustrated as follows:

The set of classical correlations C is a strict subset of the set of quantum correlations Q, which in turn is a strict subset of the set of non-signalling correlations NS. Both C and NS are polytopes (for any fixed number of parties, inputs and outputs), but Q is not. The set of quantum correlations Q sits strangely "in between" the most restrictive (classical) and most general (non-signalling) possibilities. "Why" does nature admit the set of correlations Q, and not a smaller or larger set? Instead of simply postulating the validity of quantum theory, can we perhaps find physical principles that determine Q? The last few years have seen tremendous research efforts on this question, and some fascinating partial results, but the general problem is still open (see, however, our other research ares: "Reconstructions of Quantum Theory").

Our research project builds on these previous findings, but studies one additional and surprisingly insightful aspect: the role of space and time.

Putting the black boxes into space and time

In the context of an FWF-funded project, our group is reconsidering the "black-box" approach - not only for Bell scenarios as sketched above, but also for more general settings, such as semi-device-independent prepare-and-measure scenarios or other causal structures. As we have seen above, the inputs and outputs of black boxes are typically considered to be abstract labels, for example single bits. However, in many (if not all) experimentally relevant scenarios, the inputs are spatiotemporal quantities: for example, directions of a magnetic field, the duration of a pulse, or the the angle of a polarizer.

In other words, inputs and/or outputs of the black boxes are often quantities that break a spatiotemporal symmetry: physical systems that carry group representations. Do these symmetries impose additional constraints on the correlations? Do they yield additional information that can play a similar role as the (ultimately spatiotemporally motivated) no-signalling principle - perhaps information that we can use as well-motivated assumptions in semi-device-independent scenarios? These are some of the questions that we address in our project. Our aim is two-fold:

1. To analyze how the structures of spacetime and quantum theory mutually constrain each other. This is a foundational question motivated by pure curiosity, where we aim for insights into the logical structure of the physical world.
2. To use the additional structures and constraints, via group representation theory, to construct novel device- or semi-device-independent protocols, or to improve the analysis of assumptions that underly, for example, the analysis of experimentally relevant "nonlocality witnesses".

In preliminary work [1], we have given a proof of principle that natural assumptions on the response of a physical system to rotational symmetries allows us to certify nonlocality in ways that would not otherwise be possible. Furthermore, we have shown that a partial characterization of quantum correlations in terms of symmtries is possible, which we describe next.

Exact characterization of the (2,2,2)-quantum correlations

Consider a Bell scenario where Alice and Bob each obtain one of two possible outcomes, a,b∈{-1,+1}, and each of them can choose between two possible inputs. However, now we assume that Alice and Bob reside in d-dimensional space (where d≥2), and that their inputs are spatial directions: unit vectors on which they can act with the rotation group SO(d). For example, we could have d=3, and the choice of inputs could be given by setting a magnetic field (or its inhomogeneity) in a specific direction. Now we make one specific assumption and a non-assumption:

• We do not assume that quantum theory holds. That is, we base our proofs and calculations only on probability theory, and we demand that the probabilities respond to rotations in a consistent way.
• We assume that "the probabilities transform locally like vectors". This physics jargon means mathematically that the local outcome probabilities are affine-linear in the rotations R_A, R_B that Alice and Bob perform locally to choose their inputs.

In our paper [1], we prove that these conditions uniquely characterize the set of quantum correlations Q as depicted in the figure above -- in the special case of (2,2,2)-correlations, i.e. if we have 2 parties (Alice and Bob), each having a choice of 2 inputs and obtaining one of 2 possible outputs:

Theorem. Every correlation that can be obtained under these two assumptions is a quantum correlation, i.e. can be realized within quantum theory. Furthermore, all (2,2,2)-quantum correlations can be realized under these two assumptions (restricting Alice and Bob to two possible input directions) if we supplement Alice and Bob with shared randomness.

For example, consider the case d=2 depicted above. In this case, the input (unit vectors) can be parametrized by single angles. We can think of Alice (and Bob) choosing his (her) input by rotating a polarizer by a certain angle α (resp. β), obtaining outcome -1 if the particle passes the polarizer and +1 otherwise, yielding correlations p(a,b|α,β),  The set of probability tables that can be realized in this scenario, assuming that the probabilities transform like vectors, that Alice and Bob are restricted to two possible choices of angles, and that Alice and Bob have a supply of shared randomness, is exactly the set Q of (2,2,2)-quantum correlations.

Towards semi-device-independent protocols

This fundamental result rests on a particular assumption: that outcome probabilities transform according to a certain representation of the symmetry group. We think that assumptions of this form can, in some cases, be used as natural replacements of ad-hoc assumptions in semi-device-independent protocols.

For example, prepare-and-measure scenarios can be used to generate certified random numbers, but they typically work under certain formal assumptions like Hilbert space dimension bounds. We believe that such dimension bounds can be replaced by physically better motivated assumptions on the response of the physical systems to spatiotemporal symmetries, such as theory-independent energy bounds.

Space, time and quantum theory

How are the structures of spacetime and quantum theory related? While this is one of the motivating questions for this ongoing project, we have previously [2, 3, 4] studied further aspects of this question. Please see "General Probabilistic Theories" for more information.

References [our group]

[1] A. J. P. Garner, M. Krumm, and M. P. Müller, Semi-device-independent information processing with spatiotemporal degrees of freedom, Phys. Rev. Research 2, 013112 (2020). arXiv:1907.09274

[2] A. J. P. Garner, M. P. Müller, and O. C. O. Dahlsten, The complex and quaternionic quantum bit from relativity of simultaneity on an interferometer, Proc. R. Soc. A 473, 20170596 (2017). arXiv:1412.7112

[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. arXiv:1412.8462

[4] M. P. Müller and Ll. Masanes, Three-dimensionality of space and the quantum bit: an information-theoretic approach, New J. Phys. 15, 053040 (2013); in NJP's "Highlights of 2013" collection. arXiv:1206.0630

References [other authors]

[5] A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, Device-Independent Security of Quantum Cryptography against Collective Attacks, Phys. Rev. Lett. 98, 230501 (2007). arXiv:quant-ph/0702152

[6] J. Barrett, L. Hardy, and A. Kent, No Signaling and Quantum Key Distribution, Phys. Rev. Lett. 95, 010503 (2005). arXiv:quant-ph/0405101

[7] R. Colbeck, Quantum And Relativistic Protocols For Secure Multi-Party Computation, PhD thesis, University of Cambridge, 2006. arXiv:0911.3814

[8] M. Pawłowski and N. Brunner, Semi-device-independent security of one-way quantum key distribution, Phys. Rev. A 84, 010302(R) (2011). arXiv:1103.4105

[9] H. Li, Z. Yin, Y. Wu, X. Zou, S. Wang, W. Chen, G. Guo, and Z. Han, Semi-device-independent random-number expansion without entanglement, Phys. Rev. A 84, 034301 (2011). arXiv:1108.1480

[10] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014). arXiv:1303.2849