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arXiv:1307.6839 · Physical Review A

Bloch-sphere colorings and Bell inequalities

Adrian Kent, Damián Pitalúa-García

Published Dec 18, 2014• 7 citations
arXivHTMLPDF

Abstract

We consider the quantum and local hidden variable (LHV) correlations obtained by measuring a pair of qubits by projections defined by randomly chosen axes separated by an angle $\ensuremath{\theta}$. Local hidden variables predict binary colorings of the Bloch sphere with antipodal points oppositely colored. We prove Bell inequalities separating the LHV predictions from the singlet quantum correlations for $\ensuremath{\theta}\ensuremath{\in}(0,\frac{\ensuremath{\pi}}{3})$. We raise and explore the hypothesis that, for a continuous range of $\ensuremath{\theta}>0$, the maximum LHV anticorrelation is obtained by assigning to each qubit a coloring with one hemisphere black and the other white.

DOI
10.1103/physreva.90.062124

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We consider the quantum and local hidden variable (LHV) correlations obtained by measuring a pair of qubits by projections defined by randomly chosen axes separated by an angle \ensuremathθ\ensuremath{\theta}\ensuremathθ. Local hidden variables predict binary colorings of the Bloch sphere with antipodal points oppositely colored. We prove Bell inequalities separating the LHV predictions from the singlet quantum correlations for \ensuremathθ\ensuremath∈(0,\ensuremathπ3)\ensuremath{\theta}\ensuremath{\in}(0,\frac{\ensuremath{\pi}}{3})\ensuremathθ\ensuremath∈(0,3\ensuremathπ​). We raise and explore the hypothesis that, for a continuous range of \ensuremathθ>0\ensuremath{\theta}>0\ensuremathθ>0, the maximum LHV anticorrelation is obtained by assigning to each qubit a coloring with one hemisphere black and the other white.
Bell's seminal work showed that no local hidden variable (LHV) model can fully reproduce the quantum correlations of a two-qubit singlet state. His argument and later developments by Clauser et al. effectively rely on gaps between the anticorrelations achievable by classical models and quantum theory for projective measurements along randomly chosen axes separated by a fixed angle. However, the size of these gaps has to date remained unknown. Here we numerically determine the LHV models maximizing anticorrelations for random axes separated by any fixed angle, by mapping the problem onto ground state configurations of fixed-range spin models. We identify angles where this gap is largest and thus best suited for Bell tests. These findings enrich the understanding of Bell non-locality as a physical resource in quantum information theory and quantum cryptography.
This PhD thesis contains a general introduction and three main chapters. Chapter 2 investigates Bell inequalities that generalize the CHSH and Braunstein-Caves inequalities. Chapter 3 shows a derivation of an upper bound on the success probability of a class of quantum teleportation protocols, denoted as port-based teleportation, from the no-cloning theorem and the no-signalling principle. Chapter 4 introduces the principle of quantum information causality. Chapter 2 considers the predictions of quantum theory and local hidden variable theories (LHVT) for the correlations obtained by measuring a pair of qubits by projections defined by randomly chosen axes separated by a given angle θ. The predictions of LHVT correspond to binary colourings of the Bloch sphere with antipodal points oppositely coloured. We show a Bell inequality for all θ, which generalizes the CHSH and the Braunstein-Caves inequalities in the sense that the measurement choices are not restricted to be in a finite set, but are constrained only by the angle θ. We motivate and explore the hypothesis that for a continuous range of θ > 0, the maximum correlation (anticorrelation) is obtained by assigning to one qubit the colouring with one hemisphere black and the other white, and assigning the same (reverse) colouring to the other qubit. We describe numerical tests that are consistent with this hypothesis and bound the range of θ. Chapter 3 shows a derivation of an upper bound on the success probability of port-based teleportation from the no-cloning theorem and the no-signalling principle. Chapter 4 introduces the principle of quantum information causality, a quantum version of the information causality principle. The quantum information causality principle states the maximum amount of quantum information that a transmitted quantum system can communicate as a function of its dimension, independently of any quantum physical resources previously shared by the communicating parties. These principles reduce to the no-signalling principle if no systems are transmitted. We present a new quantum information task, the quantum information causality game, whose success probability is upper bounded by the new principle, and show that an optimal strategy to perform it combines the quantum teleportation and superdense coding protocols with a task that has classical inputs.
Abstract We propose a novel class of local deterministic models to explain the correlations observed in entangled singlet quantum states. In these models, hidden variables influence both the entangled pair preparation and the measurement settings, thereby relaxing the measurement independence assumption used in Bell’s theorem. Our model accurately reproduces the expected quantum results and demonstrates that freely chosen measurement settings do not necessarily preclude the possibility of hidden variables inducing correlations between the settings and outcomes. This approach reveals an intricate relationship between local determinism and quantum mechanics, showing that local hidden variable theories can be consistent with the predictions of quantum theory. This result calls into question the conventional interpretation, based on Bell’s inequalities, that local theories are fundamentally incompatible with quantum mechanics.
The predictions of quantum mechanics cannot be resolved with a completely classical view of the world. In particular, the statistics of space-like separated measurements on entangled quantum systems violate a Bell inequality. We put forward a computational perspective on a broad class of Bell tests that study correlators, or the statistics of joint measurement outcomes. We associate particular maps, or functions to particular theories. The violation of a Bell inequality then implies the ability to perform some functions, or computations that classical, or more generally, local hidden variable (LHV) theories cannot.
One of the most striking nonclassical features of quantum mechanics is in the correlations it predicts between spatially separated measurements. In local hidden variable theories, correlations are constrained by Bell inequalities, but quantum correlations violate these. However, experimental imperfections lead to loopholes whereby LHV correlations are no longer constrained by Bell inequalities, and violations can be described by LHV theories. For example, loopholes can emerge through selective detection of events. In this Letter, we introduce a clean, operational picture of multiparty Bell tests, and show that there exists a nontrivial form of loophole-free postselection. Surprisingly, the same postselection can enhance quantum correlations, and unlock a connection between nonclassical correlations and nonclassical computation.