Optimal Robust Self-Testing by Binary Nonlocal XOR Games

Self-testing a quantum apparatus means verifying the existence of a certain quantum state as well as the effect of the associated measuring devices based only on the statistics of the measurement outcomes. Robust (i.e., error-tolerant) self-testing quantum apparatuses are critical building blocks for quantum cryptographic protocols that rely on imperfect or untrusted devices. We devise a general scheme for proving optimal robust self-testing properties for tests based on nonlocal binary XOR games. We offer some simplified proofs of known results on self-testing, and also prove some new results

Self-testing of binary observables based on commutation

We consider the problem of certifying binary observables based on a Bell inequality violation alone, a task known as self-testing of measurements. We introduce a family of commutation-based measures, which encode all the distinct arrangements of two projective observables on a qubit. These quantities by construction take into account the usual limitations of self-testing and since they are "weighted" by the (reduced) state, they automatically deal with rank-deficient reduced density matrices. We show that these measures can be estimated from the observed Bell violation in several scenarios and the proofs rely only on standard linear algebra. The trade-offs turn out to be tight and, in particular, they give non-trivial statements for arbitrarily small violations. On the other extreme, observing the maximal violation allows us to deduce precisely the form of the observables, which immediately leads to a complete rigidity statement. In particular, we show that for all $n \geq 3$ the $n$-partite Mermin-Ardehali-Belinskii-Klyshko inequality self-tests the $n$-partite Greenberger-Horne-Zeilinger state and maximally incompatible qubit measurements on every party. Our results imply that any pair of projective observables on a qubit can be certified in a truly robust manner. Finally, we show that commutation-based measures give a convenient way of expressing relations among more than two observables.Comment: 5 + 4 pages. v2: published version; v3: formatting errors fixe

Robust self-testing of many-qubit states

We introduce a simple two-player test which certifies that the players apply tensor products of Pauli $\sigma_X$ and $\sigma_Z$ observables on the tensor product of $n$ EPR pairs. The test has constant robustness: any strategy achieving success probability within an additive $\varepsilon$ of the optimal must be $\mathrm{poly}(\varepsilon)$-close, in the appropriate distance measure, to the honest $n$-qubit strategy. The test involves $2n$-bit questions and $2$-bit answers. The key technical ingredient is a quantum version of the classical linearity test of Blum, Luby, and Rubinfeld. As applications of our result we give (i) the first robust self-test for $n$ EPR pairs; (ii) a quantum multiprover interactive proof system for the local Hamiltonian problem with a constant number of provers and classical questions and answers, and a constant completeness-soundness gap independent of system size; (iii) a robust protocol for delegated quantum computation.Comment: 36 pages. Improves upon and supersedes our earlier submission arXiv:1512.0209

Bell nonlocality

Bell's 1964 theorem, which states that the predictions of quantum theory cannot be accounted for by any local theory, represents one of the most profound developments in the foundations of physics. In the last two decades, Bell's theorem has been a central theme of research from a variety of perspectives, mainly motivated by quantum information science, where the nonlocality of quantum theory underpins many of the advantages afforded by a quantum processing of information. The focus of this review is to a large extent oriented by these later developments. We review the main concepts and tools which have been developed to describe and study the nonlocality of quantum theory, and which have raised this topic to the status of a full sub-field of quantum information science.Comment: 65 pages, 7 figures. Final versio

The quantum commuting model (Ia): The CHSH game and other examples: Uniqueness of optimal states

We present in this paper that the CHSH game admits one and only one optimal state and so remove all ambiguity of representations. More precisely, we use the well-known universal description of quantum commuting correlations as state space on the universal algebra for two player games, and so allows us to unambigiously compare quantum strategies as states on this common algebra. As such we find that the CHSH game leaves a single optimal state on this common algebra. In turn passing to any non-minimal Stinespring dilation for this unique optimal state is the only source of ambiguity (including self-testing): More precisely, any state on some operator algebra may be uniquely broken up into its minimal Stinespring dilation as an honest representation for the operator algebra followed by its vector state. Any other Stinespring dilation however arises simply as an extension of the minimal Stinespring dilation (i.e., as an embedding of the minimal Hilbert space into some random ambient one). As such this manifests the only source of ambiguity appearing in most (but not all!) traditional self-testing results such as for the CHSH game as well as in plenty of similar examples. We then further demonstrate the simplicity of our arguments on the Mermin--Peres magic square and magic pentagram game. Most importantly however, we present this article as an illustration of operator algebraic techniques on optimal states and their quotients, and we further pick up the results of the current article in another following one (currently under preparation) to derive a first robust self-testing result in the quantum commuting model

Near-optimal quantum strategies for nonlocal games, approximate representations, and BCS algebras

Quantum correlations can be viewed as particular abstract states on the tensor product of operator systems which model quantum measurement scenarios. In the paradigm of nonlocal games, this perspective illustrates a connection between optimal strategies and certain representations of a finitely presented $*$-algebra affiliated with the nonlocal game. This algebraic interpretation of quantum correlations arising from nonlocal games has been valuable in recent years. In particular, the connection between representations and strategies has been useful for investigating and separating the various frameworks for quantum correlation as well as in developing cryptographic primitives for untrusted quantum devices. However to make use of this correspondence in a realistic setting one needs mathematical guarantees that this correspondence is robust to noise. We address this issue by considering the situation where the correlations are not ideal. We show that near-optimal finite-dimensional quantum strategies using arbitrary quantum states are approximate representations of the affiliated nonlocal game algebra for synchronous, boolean constraint systems (BCS), and XOR nonlocal games. This result robustly extends the correspondence between optimal strategies and finite-dimensional representations of the nonlocal game algebras for these prominent classes of nonlocal games. We also show that finite-dimensional approximate representations of these nonlocal game algebras are close to near-optimal strategies employing a maximally entangled state. As a corollary, we deduce that near-optimal quantum strategies are close to a near-optimal quantum strategy using a maximally entangled state. A boolean constraint system $B$ is $pp$-definable from another boolean constraint system $B'$ if there is a $pp$-formula defining $B$ over $B'$. There is such a $pp$-formula if all the constraints in $B$ can be defined via conjunctions of relations in $B'$ using additional boolean variables if needed. We associate a finitely presented $*$-algebra, called a BCS algebra, to each boolean constraint system $B$. We show that $pp$-definability can be interpreted algebraically as $*$-homomorphisms between BCS algebras. This allows us to classify boolean constraint languages and separations between various generalized notions of satisfiability. These types of satisfiability are motivated by nonlocal games and the various frameworks for quantum correlations and state-independent contextuality. As an example, we construct a BCS that is $C^*$-satisfiable in the sense that it has a representation on a Hilbert space $H$ but has no tracial representations, and thus no interpretation in terms of commuting operator correlations