# Characterizing quantum correlations one step at a time

This is a Perspective on "Bounding sets of sequential quantum correlations and device-independent randomness certification" by Joseph Bowles, Flavio Baccari, and Alexia Salavrakos, published in Quantum 4, 344 (2020).

By Yeong-Cherng Liang (Department of Physics and Center for Quantum Frontiers of Research & Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan).

The fact that correlations between spacelike-separated measurement outcomes observed in nature can be stronger than what is explicable by a classical, local common cause [1] has always been bewildering. For decades, this celebrated discovery due to Bell [2,1] has sparked numerous debates among physicists and philosophers. The influential work of Ekert [3], however, has given a twist to all these discussions, as it shows that Bell’s theorem has a direct bearing on the then-burgeoning field of quantum information science beyond the realm of metaphysics.

More specifically, Ekert’s work [3], and independently that of Mayers and Yao [4,5] have inspired the field of device-independent quantum information (DIQI), where only $\textit{minimal assumptions}$ about the employed devices are needed (see, e.g., [6,7]). In particular, all conclusions about the properties of these devices — be it for state preparations or measurements — are drawn directly from the correlations between measurement outcomes. Of course, the generality of the conclusion comes at a price — it is experimentally more challenging to achieve the premises needed for a nontrivial conclusion, as a Bell inequality has to be violated. Besides, one needs a decent characterization of the set of joint conditional distributions attainable by quantum theory in a Bell test.

In 2007, Navascues, Pironio, and Acin [8] (or NPA) made a breakthrough concerning the latter. Specifically, they proposed a hierarchy of semidefinite programs (SDPs) that can be used to approximate better and better the set of quantum-attainable correlations, i.e., those that can be written in the form of $P(a,b|x,y)={\rm tr}(\rho\,M^{(A)}_{a|x}\otimes M^{(B)}_{b|y})$ where $\rho$ is some shared quantum state, $M^{(A)}_{a|x}$ is the positive-operator-valued measure (POVM) element associated with the $a$-th outcome of Alice’s $x$-th measurement, while the POVM element of Bob, $M^{(B)}_{b|y}$ is defined analogously. Subsequently, this hierarchy was further shown to converge asymptotically to the set of quantum-attainable correlations when the tensor-product structure is replaced by the relaxed assumption that the observables of different parties commute (see, e.g., [9,10]).

Since then, the NPA hierarchy has played a crucial role for much of the important progress made in the field of DIQI. In particular, it has further inspired a few other hierarchies of SDPs (such as [11,12,13,14,15,15]) or more general conic programs (such as [16,17]) suitable for drawing device-independent or semi-device-independent conclusions. Importantly, all these generalizations and adaptations are meant for a standard Bell (or steering) scenario where measurements are performed only once on each subsystem. As the field of DIQI evolves, it becomes clear that various generalizations that stem from the usual Bell scenario are also of interest, both conceptually and from an application point of view.

For example, a natural generalization of the usual Bell scenario is one where each party is allowed to perform a sequence of measurements on the subsystem in their possession. Such a scenario was initially considered in the context of $\textit{hidden nonlocality}$ [18], and remained so for a long time (see, e.g., [19,20,21]). However, the work of Silva $\textit{et al.}$ [22], and subsequently that of Curchod $\textit{et al.}$ [23] have made it clear that a sequential Bell scenario is actually much richer than the usual one. Despite these exciting developments, the overall investigation of this more general Bell scenario has so far been rather limited. To some extent, this is because a comprehensive set of tools analogous to that developed by NPA was lacking.

In a recent work, Bowles, Baccari, and Salavrakos [24] developed this much-needed set of tools by adapting also the NPA hierarchy of SDPs. Their construction is intuitive. Firstly, they showed that when the Hilbert space dimension is not restricted, all POVM elements can be assumed to be projectors without loss of generality, as with the NPA hierarchy. Another important ingredient in their construction is the constraint of no-signaling from the future to the past. Together, these constraints enable them to obtain a semidefinite programming characterization of quantum correlations in a $\textit{sequential}$ Bell scenario.

They further provided a few examples to illustrate the contexts in which these new tools are relevant. One of these concerns a device-independent certification of the randomness of the output bits observed in a sequential Bell test. With these new tools, they improved over the results of [23] by certifying more than two bits of randomness by using only two measurements for Alice (rather than fourteen according to the protocol of [23]). Furthermore, they numerically verified that a monogamy relation initially proposed [22] for qubit systems also holds when the Hilbert space dimension is not restricted and that the maximal quantum violation of a sequential Bell inequality given in [19] is tight.

This contribution from Bowles, Baccari, and Salavrakos [24] is timely. Over the years, a lot of progress had been made (see, e.g., [25] and references therein) in our understanding of how spatial quantum correlations differ from those allowed in the classical world, or from the post-quantum world. The same, however, cannot be said when it comes to temporal correlations, let alone spatiotemporal correlations, where the $\textit{dynamics}$ induced by measurements must also be taken into account. However, with this new set of tools in place, the time is ripe for us to embark on a journey to a different level of understanding of quantum correlations, especially those generated via local measurements, but possibly implemented at multiple time steps, one step at a time.

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