A very brief introduction to quantum computing and quantum information theory for mathematicians

This is a very brief introduction to quantum computing and quantum information theory, primarily aimed at geometers. Beyond basic definitions and examples, I emphasize aspects of interest to geometers, especially connections with asymptotic representation theory. Proofs of most statements can be found in standard references

Why the Tsirelson bound?

Wheeler's question 'why the quantum' has two aspects: why is the world quantum and not classical, and why is it quantum rather than superquantum, i.e., why the Tsirelson bound for quantum correlations? I discuss a remarkable answer to this question proposed by Pawlowski et al (2009), who provide an information-theoretic derivation of the Tsirelson bound from a principle they call 'information causality.'Comment: 17 page

An experimental test of non-local realism

Most working scientists hold fast to the concept of 'realism' - a viewpoint according to which an external reality exists independent of observation. But quantum physics has shattered some of our cornerstone beliefs. According to Bell's theorem, any theory that is based on the joint assumption of realism and locality (meaning that local events cannot be affected by actions in space-like separated regions) is at variance with certain quantum predictions. Experiments with entangled pairs of particles have amply confirmed these quantum predictions, thus rendering local realistic theories untenable. Maintaining realism as a fundamental concept would therefore necessitate the introduction of 'spooky' actions that defy locality. Here we show by both theory and experiment that a broad and rather reasonable class of such non-local realistic theories is incompatible with experimentally observable quantum correlations. In the experiment, we measure previously untested correlations between two entangled photons, and show that these correlations violate an inequality proposed by Leggett for non-local realistic theories. Our result suggests that giving up the concept of locality is not sufficient to be consistent with quantum experiments, unless certain intuitive features of realism are abandoned.Comment: Minor corrections to the manuscript, the final inequality and all its conclusions do not change; description of corrections (Corrigendum) added as new Appendix III; Appendix II replaced by a shorter derivatio

Multi-Prover Commitments Against Non-Signaling Attacks

We reconsider the concept of multi-prover commitments, as introduced in the late eighties in the seminal work by Ben-Or et al. As was recently shown by Cr\'{e}peau et al., the security of known two-prover commitment schemes not only relies on the explicit assumption that the provers cannot communicate, but also depends on their information processing capabilities. For instance, there exist schemes that are secure against classical provers but insecure if the provers have quantum information processing capabilities, and there are schemes that resist such quantum attacks but become insecure when considering general so-called non-signaling provers, which are restricted solely by the requirement that no communication takes place. This poses the natural question whether there exists a two-prover commitment scheme that is secure under the sole assumption that no communication takes place; no such scheme is known. In this work, we give strong evidence for a negative answer: we show that any single-round two-prover commitment scheme can be broken by a non-signaling attack. Our negative result is as bad as it can get: for any candidate scheme that is (almost) perfectly hiding, there exists a strategy that allows the dishonest provers to open a commitment to an arbitrary bit (almost) as successfully as the honest provers can open an honestly prepared commitment, i.e., with probability (almost) 1 in case of a perfectly sound scheme. In the case of multi-round schemes, our impossibility result is restricted to perfectly hiding schemes. On the positive side, we show that the impossibility result can be circumvented by considering three provers instead: there exists a three-prover commitment scheme that is secure against arbitrary non-signaling attacks

How much contextuality?

The amount of contextuality is quantified in terms of the probability of the necessary violations of noncontextual assignments to counterfactual elements of physical reality.Comment: 5 pages, 3 figure

Decoherence of matter waves by thermal emission of radiation

Emergent quantum technologies have led to increasing interest in decoherence - the processes that limit the appearance of quantum effects and turn them into classical phenomena. One important cause of decoherence is the interaction of a quantum system with its environment, which 'entangles' the two and distributes the quantum coherence over so many degrees of freedom as to render it unobservable. Decoherence theory has been complemented by experiments using matter waves coupled to external photons or molecules, and by investigations using coherent photon states, trapped ions and electron interferometers. Large molecules are particularly suitable for the investigation of the quantum-classical transition because they can store much energy in numerous internal degrees of freedom; the internal energy can be converted into thermal radiation and thus induce decoherence. Here we report matter wave interferometer experiments in which C70 molecules lose their quantum behaviour by thermal emission of radiation. We find good quantitative agreement between our experimental observations and microscopic decoherence theory. Decoherence by emission of thermal radiation is a general mechanism that should be relevant to all macroscopic bodies.Comment: 5 pages, 4 figure

No extension of quantum theory can have improved predictive power

According to quantum theory, measurements generate random outcomes, in stark contrast with classical mechanics. This raises the question of whether there could exist an extension of the theory which removes this indeterminism, as suspected by Einstein, Podolsky and Rosen (EPR). Although this has been shown to be impossible, existing results do not imply that the current theory is maximally informative. Here we ask the more general question of whether any improved predictions can be achieved by any extension of quantum theory. Under the assumption that measurements can be chosen freely, we answer this question in the negative: no extension of quantum theory can give more information about the outcomes of future measurements than quantum theory itself. Our result has significance for the foundations of quantum mechanics, as well as applications to tasks that exploit the inherent randomness in quantum theory, such as quantum cryptography.Comment: 6 pages plus 7 of supplementary material, 3 figures. Title changed. Added discussion on Bell's notion of locality. FAQ answered at http://perimeterinstitute.ca/personal/rcolbeck/FAQ.htm

Extension of Information Geometry to Non-statistical Systems: Some Examples

Our goal is to extend information geometry to situations where statistical modeling is not obvious. The setting is that of modeling experimental data. Quite often the data are not of a statistical nature. Sometimes also the model is not a statistical manifold. An example of the former is the description of the Bose gas in the grand canonical ensemble. An example of the latter is the modeling of quantum systems with density matrices. Conditional expectations in the quantum context are reviewed. The border problem is discussed: through conditioning the model point shifts to the border of the differentiable manifold.Comment: 8 pages, to be published in the proceedings of GSI2015, Lecture Notes in Computer Science, Springe

Quantum Storage of Photonic Entanglement in a Crystal

Entanglement is the fundamental characteristic of quantum physics. Large experimental efforts are devoted to harness entanglement between various physical systems. In particular, entanglement between light and material systems is interesting due to their prospective roles as "flying" and stationary qubits in future quantum information technologies, such as quantum repeaters and quantum networks. Here we report the first demonstration of entanglement between a photon at telecommunication wavelength and a single collective atomic excitation stored in a crystal. One photon from an energy-time entangled pair is mapped onto a crystal and then released into a well-defined spatial mode after a predetermined storage time. The other photon is at telecommunication wavelength and is sent directly through a 50 m fiber link to an analyzer. Successful transfer of entanglement to the crystal and back is proven by a violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality by almost three standard deviations (S=2.64+/-0.23). These results represent an important step towards quantum communication technologies based on solid-state devices. In particular, our resources pave the way for building efficient multiplexed quantum repeaters for long-distance quantum networks.Comment: 5 pages, 3 figures + supplementary information; fixed typo in ref. [36

Conclusive quantum steering with superconducting transition edge sensors

Quantum steering allows two parties to verify shared entanglement even if one measurement device is untrusted. A conclusive demonstration of steering through the violation of a steering inequality is of considerable fundamental interest and opens up applications in quantum communication. To date all experimental tests with single photon states have relied on post-selection, allowing untrusted devices to cheat by hiding unfavourable events in losses. Here we close this "detection loophole" by combining a highly efficient source of entangled photon pairs with superconducting transition edge sensors. We achieve an unprecedented ~62% conditional detection efficiency of entangled photons and violate a steering inequality with the minimal number of measurement settings by 48 standard deviations. Our results provide a clear path to practical applications of steering and to a photonic loophole-free Bell test.Comment: Preprint of 7 pages, 3 figures; the definitive version is published in Nature Communications, see below. Also, see related experimental work by A. J. Bennet et al., arXiv:1111.0739 and B. Wittmann et al., arXiv:1111.076