Let p be an odd prime and let r be the smallest generator of the multiplicative group Z p ∗ . We show that there exists a correlation of size Θ ( r 2 ) that self-tests a maximally entangled state of local dimension p − 1 . The construction of the correlation uses the embedding procedure proposed by Slofstra ( Forum of Mathematics, Pi. ( 2019 )). Since there are infinitely many prime numbers whose smallest multiplicative generator is in the set { 2 , 3 , 5 } (D.R. Heath-Brown The Quarterly Journal of Mathematics ( 1986 ) and M. Murty The Mathematical Intelligencer ( 1988 )), our result implies that constant-sized correlations are sufficient for self-testing of maximally entangled states with unbounded local dimension.
Let p be an odd prime and let r be the smallest generator of the multiplicative group Z p ∗ . We show that there exists a correlation of size Θ ( r 2 ) that self-tests a maximally entangled state of local dimension p − 1 . The construction of the correlation uses the embedding procedure proposed by Slofstra ( Forum of Mathematics, Pi. ( 2019 )). Since there are infinitely many prime numbers whose smallest multiplicative generator is in the set { 2 , 3 , 5 } (D.R. Heath-Brown The Quarterly Journal of Mathematics ( 1986 ) and M. Murty The Mathematical Intelligencer ( 1988 )), our result implies that constant-sized correlations are sufficient for self-testing of maximally entangled states with unbounded local dimension.
One of the most fundamental and counterintuitive features of quantum me- chanics is entanglement, which is central to many demonstrations of the quantum advantage. Studying quantum correlations generated by local measurements on an entangled physical system is one of the direct ways to gain insights into en- tanglement. The focus of this dissertation is to get better understanding of the hardness of determining if a given correlation is quantum, which is also known as the membership problem of quantum correlations. Previous work has shown that the general membership problem is computationally undecidable. Where does the hardness come from? Is it just because the size of a quantum correlation (i.e., the number of real values in the description of the correlation) can be arbitrarily large? We would like to understand the role played by the varying sizes of correlations in the hardness of the membership problem. It has been shown that certain quantum correlations require the measured quantum system to be maximally entangled with a certain dimension. This is a unique phenomenon of quantum correlations and it is known as self-testing. The first step towards answering the hardness of the membership problem of quantum correlations is to get deeper understandings about self-testing, and more specifically, about the size of a correlation that can self-test a maximally entangled state of arbitrarily large dimension. If correlations of a fixed size can self- test entangled states of unbounded dimension, this phenomenon is a strong evidence suggesting that deciding membership of fixed-sized correlations can be very hard. We first show that there exists an infinite subset of the set of all the prime numbers such that, for each prime p in this set, a maximally entangled state of local dimension (p − 1) can be self-tested by a correlation of a fixed size. Since this set is infinite, this result implies that constant-sized correlations are sufficient to self-test maximally entangled states of unbounded dimension. Building on the self-testing result, we show that the varying sizes of correlations are not the only root of the hardness. Specifically, we show that the membership problem of fixed finite-sized correlations is still computationally undecidable when the fixed size is sufficiently large. That is, the hardness of the membership problem of quantum correlations is independent of the varying sizes of correlations. In fact, the hardness arises from the fact that the structure of some set of correlations of a particular size is so complicated that no finite description of this set can allow a Turing machine to decide if a correlation is quantum or not.
Self-testing is a powerful certification of quantum systems relying on measured, classical statistics. This paper considers self-testing in bipartite Bell scenarios with small number of inputs and outputs, but with quantum states and measurements of arbitrarily large dimension. The contributions are twofold. Firstly, it is shown that every maximally entangled state can be self-tested with four binary measurements per party. This result extends the earlier work of Mančinska-Prakash-Schafhauser (2021), which applies to maximally entangled states of odd dimensions only. Secondly, it is shown that every single binary projective measurement can be self-tested with five binary measurements per party. A similar statement holds for self-testing of projective measurements with more than two outputs. These results are enabled by the representation theory of quadruples of projections that add to a scalar multiple of the identity. Structure of irreducible representations, analysis of their spectral features and post-hoc self-testing are the primary methods for constructing the new self-tests with small number of inputs and outputs.
Completely determining the relationship between quantum correlation sets is a long-standing open problem, known as Tsirelson's problem. Following recent progress by Slofstra [arXiv:1606.03140 (2016), arXiv:1703.08618 (2017)] only two instances of the problem remain open. One of them is the question of whether the set of finite-dimensional quantum correlations is strictly contained in the set of infinite-dimensional ones (i.e. whether Cq=Cqs). The usual formulation of the question assumes finite question and answer sets. In this work, we show that, when one allows for either infinite answer sets (and finite question sets) or infinite question sets (and finite answer sets), there exist correlations that are achievable using an infinite-dimensional quantum strategy, but not a finite-dimensional one. For the former case, our proof exploits a recent result [Nat. Comm. 8, 15485 (2017)], which shows self-testing of any pure bipartite entangled state of arbitrary local dimension d, using question sets of size 3 and 4 and answer sets of size d. For the latter case, a key step in our proof is to show a novel self-test, inspired by [Nat. Comm. 8, 15485 (2017)], of all bipartite entangled states of any local dimension d, using question sets of size O(d), and answer sets of size 4 and 3 respectively.
Abstract Bell nonlocality as a resource for device-independent certification schemes has been studied extensively in recent years. The strongest form of device-independent certification is referred to as self-testing, which given a device, certifies the promised quantum state as well as quantum measurements performed on it without any knowledge of the internal workings of the device. In spite of various results on self-testing protocols, it remains a highly nontrivial problem to propose a certification scheme of qudit–qudit entangled states based on violation of a single d -outcome Bell inequality. Here we address this problem and propose a self-testing protocol for the maximally entangled state of any local dimension using the minimum number of measurements possible, i.e., two per subsystem. Our self-testing result can be used to establish unbounded randomness expansion, log2d log 2 d perfect random bits, while it requires only one random bit to encode the measurement choice.
Abstract Self-testing was originally introduced as a device-independent method of certification of entangled quantum states and local measurements performed on them. Recently, in (Baccari et al 2020 Phys. Rev. Lett. 125 260507) the notion of state self-testing has been generalized to entangled subspaces and the first self-testing strategies for exemplary genuinely entangled subspaces have been given. The main aim of our work is to pursue this line of research and to address the question how ‘large’ (in terms of dimension) are genuinely entangled subspaces that can be self-tested, concentrating on the multiqubit stabilizer formalism. To this end, we first introduce a framework allowing to efficiently check whether a given stabilizer subspace is genuinely entangled. Building on it, we then determine the maximal dimension of genuinely entangled subspaces that can be constructed within the stabilizer subspaces and provide an exemplary construction of such maximally-dimensional subspaces for any number of qubits. Third, we construct Bell inequalities that are maximally violated by any entangled state from those subspaces and thus also any mixed states supported on them, and we show these inequalities to be useful for self-testing. Interestingly, our Bell inequalities allow for identification of higher-dimensional face structures in the boundaries of the sets of quantum correlations in the simplest multipartite Bell scenarios in which every observer performs two dichotomic measurements.
Abstract Bell nonlocality—the existence of quantum correlations that cannot be explained by classical means—is certainly one of the most striking features of quantum mechanics. Its range of applications in device-independent protocols is constantly growing. Many relevant quantum features can be inferred from violations of Bell inequalities, including entanglement detection and quantification, and state certification applicable to systems of arbitrary number of particles. A complete characterisation of nonlocal correlations for many-body systems is, however, a computationally intractable problem. Even if one restricts the analysis to specific classes of states, no general method to tailor Bell inequalities to be violated by a given state is known. In this work we provide a general construction of Bell expressions tailored to the graph states of any prime local dimension. These form a broad class of multipartite quantum states that have many applications in quantum information, including quantum error correction. We analytically determine their maximal quantum values, a number of high relevance for device-independent applications of Bell inequalities. Importantly, the number of expectation values to determine in order to test the violation of our inequalities scales only linearly with the system size, which we expect to be the optimal scaling one can hope for in this case. Finally, we show that these inequalities can be used for self-testing of multi-qutrit graph states such as the well-known four-qutrit absolutely maximally entangled state AME(4,3).