There is a certain type of papers whose value is that they are simple, yet very inspiring. There is no hard-core mathematical machinery behind, they rather witness a good intuition and understanding of various, seemingly unrelated problems. Such is paper “A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations” by Andrea Coladangelo . It deals with topics of high-dimensional entanglement witnessing, embezzlement of entanglement and non-closure of the quantum set of correlations, all problems whose solving used to require deep mathematical skills, such that cause dizzying in your head if you are not very familiar with C* algebras or hyperlinear profiles of groups. This paper manages to show that solving all these problems can be fairly simple once you are handy with Bell nonlocality.
One aspect of the paper is witnessing the power of Bell nonlocality [2,8] and self-testing as one of its manifestations. Roughly speaking, self-testing states that certain measurement correlations uniquely identify the state and measurements necessary to produce them (unique up to a well-defined set of transformations) [3,4]. For many years self-testing was explored as an objective: researchers were very curious about which quantum states could leave such device-independent trace. As the topic matured, self-testing became a method: knowing that a nonlocal game can be won by state belonging to a very specific equivalence class is very powerful, and can be used to provide answers to deep questions about the nature of the set of quantum correlations. Usually this information allows us to bypass the heavy machinery that would otherwise be necessary towards solving the problem.
In this paper Coladangelo uses self-testing to provide answers to three different problems: dimension witnessing, embezzlement of entanglement and non-closure of the set of quantum correlations. A certain nonlocal game is a dimension witness if a given score can only be achieved by manipulating an entangled state of at least a certain dimension . Previous progress in this field required constructing families of nonlocal games with input and output size depending on the dimension one aimed to witness. The first take-home message from Coladangelo’s paper is that there exists a simple nonlocal game, with constant and short size of inputs and outputs, such that the players’ score increases with the dimension of the quantum entangled state used. This relation is quantitative: the distance from the maximal score is directly related to the lower bound on the dimension of the entangled state. The maximal score cannot be achieved by any finite-dimensional state; it is rather a witness of infinite dimension. This leads us to the second take-home message: a rather simple nonlocal game (and accordingly Bell inequality) and its self-testing properties allows us to prove that the set of quantum correlations is not closed. This is not a new result: it has been proven by Slofstra by using representation theory of finitely-presented groups [6,7]. The self-testing proofs allowing Coladangelo to reach the same conclusion rely on basic linear algebra.
Finally, what makes the previous two results possible is that maximal score in the nonlocal game requires parties to embezzle some entanglement. What is embezzlement of entanglement? Introduced by Leung, Toner and Watrous , it is the process which allows one to transform a product state of two qubits into a maximally entangled pair. To achieve this, the parties can possess a resource, a stash of entanglement, which needs to remain unchanged during the process, but can be used as a catalyst. While it is impossible to perform this task ideally, Leung, Toner and Watrous showed it is possible if one allows the resource to remain almost unchanged. The parties steal (embezzle) a bit of entanglement and use it to transform the product state into a maximally entangled pair of qubits. Originally, this process required three parties, and in some scenarios, the verifer supervising the process had to exchange quantum messages with the parties involved in the embezzlement . Coladangelo shows that it is possible to do in a scenario with only two players, and that the process can be described in a device-independent manner: the verifier interacts with the parties only classically.
The nonlocal game is constructed in a very clever way. It consists of three parts: in the first part the optimal winning is achieved by sharing a partially entangled qubit state, in the second by sharing a maximally entangled qutrit state, and the success in the third part requires them to use the same resource in the first two parts, meaning that an entangled qubit state has to be transformed into entangled qutrit state (by local operations). For the success in the third part the players need to use entanglement embezzlement: borrow a little bit of entanglement they have in stock and use it to obtain the optimal score in the nonlocal game. The more entanglement they have in stock, the better their chances of winning the game! The dimension of the stashed entanglement is therefore related to the score in the nonlocal game.
This paper is a must read for everyone interested in certification and it is a paradigmatic example of the power of self-testing.
 A. Coladangelo, A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations, arXiv:1904.02350 https://arxiv.org/abs/1904.02350 https://doi.org/10.22331/q-2020-06-18-282.
 J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics Physique Fizika, 1, 3, 195-200, (1964) 10.1103/PhysicsPhysiqueFizika.1.195.
 N. Brunner, S. Pironio, A. Acin, N. Gisin, A. A. Méthot, and V. Scarani, Testing the dimension of Hilbert spaces, Phys. Rev. Lett. 100, 210503 (2008) 10.1103/PhysRevLett.100.210503.
 D. Leung, B. Toner, J. Watrous, Coherent state exchange in multi-prover quantum interactive proof systems, Chicago Journal of Theoretical Computer Science, 2013, 11, (2013) 10.4086/cjtcs.2013.011.
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