Minimum Dimension of a Hilbert Space Needed to Generate a Quantum Correlation

Abstract

Consider a two-party correlation that can be generated by performing local measurements on a bipartite quantum system. A question of fundamental importance is to understand how many resources, which we quantify by the dimension of the underlying quantum system, are needed to reproduce this correlation. In this Letter, we identify an easy-to-compute lower bound on the smallest Hilbert space dimension needed to generate a given two-party quantum correlation. We show that our bound is tight on many well-known correlations and discuss how it can rule out correlations of having a finite-dimensional quantum representation. We show that our bound is multiplicative under product correlations and also that it can witness the non-convexity of certain restricted-dimensional quantum correlations.In what ranks as one of the most important achievements of modern physics, it was shown by John Bell in 1964 that some correlations generated within the framework of quantum mechanics can be nonlocal, in the sense that the statistics generated by quantum mechanics cannot always be reproduced by a local hidden-variable model [1,2]. Over the last 40 years, there have been significant efforts in trying to verify this fact experimentally. The first such experimental data [3] was published in 1972 and this remains an active area of research [4]. Moreover, as a central concept in quantum physics and quantum information theory, fully understanding quantum entanglement and nonlocality still remains a very interesting and important problem with far-reaching applications. Indeed, profound relationships between quantum nonlocality and other fundamental quantum concepts or phenomena such as entanglement measures [5,6], entanglement distillation [7,8], and teleportation [9] have been identified. Meanwhile, for many tasks, e.g. in cryptography [10,11], it has been realized that due to quantum nonlocality, quantum strategies enjoy remarkable advantages over their classical counterparts.However, even though quantum nonlocal effects can lead to interesting and often surprising advantages in some applications, this does not paint the full picture. After all, for practical applications, it is just as important to understand the amount of quantum resources required for these advantages to manifest. For instance, if there is an exponential blowup in the amount of resources required, then whatever advantage gained by employing quantum mechanics may not be useful in practice. Quantifying the amount of quantum resources needed to perform a certain task is the central focus of this Letter.We study quantum nonlocality from the viewpoint of two-party quantum correlations that arise from a Bell experiment. A two-party Bell experiment is performed between two parties, Alice and Bob, whose labs are set up in separate locations. Alice (resp. Bob) has in her possession a measurement apparatus whose possible settings are labelled by the elements of a finite set X (resp. Y ) and the possible measurement outcomes are labelled by a finite set A (resp. B). After repeat...

Device-independent characterizations of a shared quantum state independent of any Bell inequalities

“…Concerning the first characterization, we provide examples for which it is tight and where the shared pure quantum states are actually pinned down completely. Second, we show that it implies lower bounds on both the dimension and the amount of entanglement of the underlying quantum state, which are device-independent tasks that have drawn much attention recently [5,8,9]. We then show that the second characterization allows us to exclude the pure state that can produce a given Bell correlation from being particular states.…”

“…The corresponding statistics of the measurement outcomes is called a Bell correlation. It has been shown that the dimension and the entanglement of the underlying quantum state can be quantified in a device-independent way using only the Bell correlation data [5,[8][9][10]. In fact, some quantum states can even be pinned down completely by their violations of particular Bell inequalities, but this is only known to be possible for some special cases [1][2][3][4][15][16][17].…”

“…We often hope the dimension of ρ is as small as possible due to the fact that quantum dimensionality is a precious resource. Interestingly, for an arbitrary Bell correlation, it is known that this quantum state with minimal dimension can always be pure [9]. Conveniently, a pure state can be described using its Schmidt decomposition, where the Schmidt coefficients completely capture its quantum properties.…”

“…Relation to device-independent dimension test.-Device-independent lower bounds on the dimension of a quantum state used in a Bell setting is a very interesting problem that has attracted much attention recently [5,9]. Recall that for any Bell correlation, a quantum state with the minimal size that produces this correlation can always be pure [9].…”

“…However, sometimes we still want to draw nontrivial conclusions on the quantum properties of the involved system. This sounds like a challenging, or even impossible task, but it has been shown to be possible in many cases [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. These kinds of tasks are called device-independent as their application assumes only the correctness of quantum mechanics as a valid description of nature, and is independent of the internal workings of the devices used.…”