Characterization of Quantum Correlations with Local Dimension Constraints and Its Device-Independent Applications

The future progress of semi-device-independent quantum information science depends crucially on our ability to bound the strength of the nonlocal correlations achievable with finite-dimensional quantum resources. In this work, we characterize quantum nonlocality under local dimension constraints via a complete hierarchy of semidefinite programming relaxations. In the bipartite case, we find that the first level of the hierarchy returns nontrivial bounds in all cases considered, allowing us to study nonlocality scenarios with four measurement settings on one side and twelve on the other in a normal desktop. In the tripartite case, we apply the hierarchy to derive a Bell-type inequality that can only be violated when each of the three parties has local dimension greater than 2, hence certifying three-dimensional tripartite entanglement in a device-independent way. Finally, we show how the new method can be trivially modified to detect nonseparable measurements in two-qubit scenarios.

Popular Summary

In cryptography, the fight between code makers and code breakers has been going on for thousands of years. Quantum cryptography at its inception a few decades ago appeared to hold the promise for perfectly secure communication schemes, given that the laws of quantum mechanics must be followed even by the most capable code breakers. This turned out to be true only in theory. In practice, however, devices that the communicating parties use to perform a cryptography task, such as secret key distribution, proved to be one of the weakest points in earlier protocols that required a priori knowledge of the inner workings of the devices: The requirement could be a severe limiting factor to broad applications, and functional defects of the devices could become loopholes that code breakers could exploit.

Device-independent (DI) quantum communication has emerged recently with the ultimate goal to remove this weakness. The key point is that it treats the devices used as black boxes, containing an unknown quantum setup, ultimately producing classical statistics. These statistics are known as quantum correlations and are the object of our analysis. Many practical settings, such as the trapped-ion platform, have been proposed for implementations of DI protocols. These settings usually come with their own constraints. One such constraint is a bound on their physical dimensionality. Up to now, there has been very little concrete knowledge of which nonlocal quantum correlations—the basis for quantum device-independent schemes—are attainable and can be exploited in DI protocols where some of the devices are under a dimensionality constraint (semi-device-independent protocols). In this paper, we meet this current and important need by characterizing nonlocal quantum correlations with a new numerical method.

Our method, which combines ideas from entanglement theory, noncommutative algebraic geometry, and convex optimization, allows the user to impose local dimension constraints in Bell-type nonlocality scenarios where the number of parties, measurements, and outcomes relevant to the scenario are given as free parameters to the method—rendering the method very general. We test the efficiency of our tool by solving a number of problems in device-independent quantum information theory, for example, proving that three-dimensional tripartite entanglement—a form of nonlocal correlation—can be certified in a device-independent way.

Our method should enable systematic analysis of current and future device-independent quantum communication protocols with dimensionality constraints.

Figure 1

Pictorial representation of the state constraints: The diamonds (heads) denote Alice’s and Bob’s state spaces; the circles represent rank-one projectors. $N$ circles joined by a line (legs) must be understood as living in the symmetric space of $N$ particles. In the figure, Alice’s projectors $|u^1⟩⟨u^1|$, $|u^2⟩⟨u^2|$ are represented by legs of lengths 3, 2, respectively. Also, positivity has been imposed under the partial transpose of one of the circles of Bob’s first leg and Bob’s second leg.

Figure 2

A possible relaxation to bound the convex hull of Eq. (10) when Alice’s measurements are not trusted. Here, just Bob and Charlie have expanded bodies, represented by a triangle. The number (3) indicates the order of the moment relaxation.

Figure 3

Pictorial representation of the second relaxation used to compute the maximal violation of $M_{4,12}$ in qubit systems.

Figure 4

Picturing the scenario of entangled measurements in bounded Hilbert spaces.