In the device-independent approach to quantum theory, introduced by Popescu and Rohrlich [1], experiments are viewed as black boxes which admit a classical input (say, the type of experiment conducted) and return a classical output (the experimental result). No assumptions are made on the inner mechanisms which led to the generation of the experimental result: in this framework, the only information we can hope to gather from several repetitions of sequential or simultaneous experiments is the probability distribution of the outputs conditioned on the inputs.

The figure bellow shows, for instance, how to model a bipartite experiment within the black-box formalism. Alice (Bob) inputs a symbol x (y) in her (his) lab, obtaining output a (b). By conducting several repetitions of this experiment, they manage to estimate the probabilities P(a,b|x,y).

The aim of this project is to understand how abstract notions from quantum theory, such as state, measurement or dimensionality, translate into statistical restrictions on P(a,b|x,y). Equivalently, we want to know what we can say about the contents of a black box, given its experimental behavior.

The first problem to consider is whether the contents of the box actually follow the laws of quantum mechanics, that is, whether there exists a quantum state and measurements giving rise to the statistics P(a,b|x,y). In 2006, we solved this problem by proposing a hierarchy of efficiently computable outer approximations to the set of quantum correlations [1].

The Navascués-Pironio-Acín (NPA) hierarchy is essentially the only systematic procedure to disprove the existence of a quantum model in a device-independent way. Not surprisingly, it quickly found application in device-independent quantum information science. It also marked the birth of Non-Commutative Polynomial Optimization theory [2], see Convex Optimization.

One interesting feature of the NPA hierarchy is that, under the assumption that quantum mechanics holds, it allows one to make claims about the quantum state contained in the box and the quantum measurements effected by the two parties [3, 4]. This procedure, known as *self-testing*, is based on the fact that certain correlations P(a,b|x,y) only admit a unique quantum representation.

In a recent work [5, 6], we explored how this framework can be used to estimate the dimension of the quantum system inside the box, namely, the maximum amount of quantum information that the system can reliably store. Suppose that the statistics P(a,b|x,y) of all boxes capable of, say, storing at most four quantum bits happen to satisfy a constraint of the form

This sort of inequalities are called “dimension witnesses”, and any experimental violation of them would signify that the system inside the considered box can store *more* than four quantum bits of information. Deriving dimension witnesses is a hard task, due to the huge number of free variables required to specify the state and measurement operators in even the simplest quantum scenarios.

The maximization of linear functions of the statistics of finite-dimensional systems belongs to a class of problems called *dimension-constrained Noncommutative Polynomial Optimization (NPO)*. Another important problem in that class is the computation of the optimal probability of success when Alice and Bob try to evaluate a bivariate function and Alice is allowed to send Bob a restricted amount of quantum information. The minimum amount of quantum information required to evaluate the function with certainty is called the *quantum communication complexity of the function*.

In [5, 6], we presented effective converging hierarchies of relaxations to attack the general dimension-constrained NPO problem. Thanks to these tools, we managed to derive new dimension witnesses for temporal and Bell-type correlation scenarios. Similarly, we solved a number of open problems in quantum communication complexity.

[1] M. Navascués, S. Pironio and A. Acín, *Bounding the set of quantum correlations*, Phys. Rev. Lett. 98, 010401 (2007).

[2] M. Navascués, S. Pironio and A. Acín, *Noncommutative Polynomial Optimization. Handbook on*

*Semidefinite, Cone and Polynomial Optimization*, M.F. Anjos and J. Lasserre (eds); Springer, 2011.

[3] T. H. Yang, T. Vértesi, J.-D. Bancal, V. Scarani and M. Navascués, *Robust and versatile black-box certification of quantum devices*, Phys. Rev. Lett. 113, 040401 (2014).

[4] J.-D. Bancal, M. Navascués, V. Scarani, T. Vértesi and T. H. Yang, *Physical characterization of quantum devices from nonlocal correlations*, Phys. Rev. A 91, 022115 (2015).

[5] M. Navascués and T. Vértesi, *Bounding the set of finite-dimensional quantum correlations,* Phys. Rev. Lett. 115, 020501 (2015).

[6] M. Navascués, A. Feix, M. Araújo and T. Vértesi, *Characterizing finite-dimensional quantum behavior*, Phys. Rev. A 92, 042117 (2015).

**Additional readings**

N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani and S. Wehner, *Bell nonlocality*, Rev. Mod. Phys. 86, 419 (2014).

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