Quantum Physics

Complexity class MA is a class of yes/no problems for which the answer `yes\' has a short certificate that can be efficiently checked by a classical randomized algorithm. We prove that MA has a natural complete problem: stoquastic k-SAT. This is a quantum-mechanical analogue of the satisfiability problem such that k-bit clauses are replaced by k-qubit projectors with non-negative matrix elements. Complexity class AM is a generalization of MA in which the certificate may include a short conversation between Prover and Verifier. We prove that AM also has a natural complete problem: stoquastic Local Hamiltonian with a quenched disorder. The problem is to evaluate expectation value of the ground state energy of disordered local Hamiltonian with non-positive matrix elements.

Newton\'s first law of motion - and the very meaning of inertia - has been described as either completely obvious (D\'Alembert) or a \'logician\'s nightmare\' (ex-editor of the American Journal of Physics). Sometimes the simplest things in physics are the most subtle. The first law will be described in historical context, explaining a connection with the ancient Greeks’ distinction between natural and violent motion and with Descartes\' natural philosophy. You will also learn why it still requires careful handling and what it tells us about time in physics. \'Time and Motion\', Harvey Brown, time, motion, relative, Copernicus, Ptolemy, Galileo, Copernicanism, Descartes, inertia, Newton, standard of time, Fitzgerald, duration, inertial frame

I will discuss various different ways of quantifying the differences between two quantum observables (POVMs). Each of these approaches gives rise to a notion of approximately measuring one observable by means of measuring some other observable. This will be illustrated in the case of position and momentum by studying the question which POVMs on phase space can reasonably be said to represent a joint approximate determination of these observables. A new, universally valid trade-off relation for the associated inaccuracies will be rigorously formulated. I will sketch the proof which is an adaptation of some interesting techniques and properties of covariant phase space observables used recently by R Werner in a related project. Recommended reading (optional): quant-ph/0405184 (R Werner), quant-ph/0609185 (PB et al), and also for further background information quant-ph/0309091 (M Hall), quant-ph/0310070 (M Ozawa), quant-ph/9803051 (DM Appleby).

We describe a protocol for distilling maximally entangled bipartite states between random pairs of parties (``random entanglement'') from those sharing a tripartite W state, and show that this may be done at a higher rate than distillation of bipartite entanglement between specified pairs of parties (``specified entanglement''). Specifically, the optimal distillation rate for specified entanglement for the W has been previously shown to be the asymptotic entanglement of assistance of 0.92 EPR pairs per W, while our protocol can distill 1 EPR pair per W between random pairs of parties, which we conjecture to be optimal. We further extend this to a more general class of W-like states and show by increasing the number of parties in the protocol that there exist states with fixed lower-bounded distillable random entanglement for arbitrarily small specified entanglement. [Work done in collaboration with Benjamin Fortescue. Preprint available at http://arxiv.org/abs/quant-ph/0607126 ]

Quantum mechanics is a non-classical probability calculus -- but hardly the most general one imaginable. In this talk, I'll discuss some familiar non-classical properties of quantum-probabilistic models that turn out to be features of {em all} non-classical models. These include a generic no-cloning theorem obtained in recent work with Howard Barnum, Jon Barrett and Matt Leifer.

Entanglement is one of the most studied features of quantum mechanics and in particular quantum information. Yet its role in quantum information is still not clearly understood. Results such as (R. Josza and N. Linden, Proc. Roy. Soc. Lond. A 459, 2011 (2003)) show that entanglement is necessary, but stabilizer states and the Gottesman-Knill theorem (for example) imply that it is far from sufficient. I will discuss three aspects of entanglement. First, a quantum circuit with a "vanishingly small" amount of entanglement that admits an apparent exponential speed-up over the classical case. Second, I will discuss techniques for lower-bounding the amount of entanglement in bipartite quantum states. Finally, I will discuss the role of entanglement in quantum metrology. Specifically, I will show that entangling ancillas can make no difference to the accuracy of a quantum parameter estimation, regardless of the nature of the coupling Hamiltonian. I will conclude by discussing strategies for improving the scaling of quantum parameter estimation.

Traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n. But using ideas from computational learning theory, I'll show that "for most practical purposes" one can learn a quantum state using a number of measurements that grows only linearly with n. I'll discuss applications of this result in experimental physics and quantum computing theory, as well as possible implications for the foundations of quantum mechanics. quant-ph/0608142

My field is the foundations of quantum mechanics, in particular Bohmian mechanics, a non-relativistic theory that is empirically equivalent to standard quantum mechanics while solving all of its paradoxes in an elegant and simple way, essentially by assuming that particles have trajectories. Bohmian mechanics possesses a straightforward generalization to relativistic space-time, be it flat or curved, if one assumption against the spirit of relativity is granted: the existence of a "time foliation", i.e., a physical object mathematically represented by a slicing of space-time into spacelike 3-surfaces, which evolves according to a Lorentz-invariant law. On the basis of this kind of theory, describing particles in a background 4-geometry, I propose an extension in which the space-time geometry is dynamically generated, as in general relativity. Whether my model is empirically equivalent to any known type of quantum gravity I don't know. In this model, there is a Lorentzian metric on configuration-space-time, evolving according to the higher-dimensional analog of the Einstein field equation. The 4-metric is obtained from the configuration-space-time metric and the actual particle configuration. Thus, this Bohm-like model generates (up to diffeomorphisms) a 4-metric and particle world lines from a given wave function.

In this talk I will present several new results from joint work with Dmitry Gavinsky, Oded Regev and Ronald de Wolf, relating to the model of one-way communication and the simultaneous model of communication. I will describe several separations between various resources (entanglement versus event coin, quantum communication versus classical communication), showing in particular that quantum communication cannot simulate a public coin and that entanglement can be much more powerful than a public coin, even if communication is quantum. I will also present a characterization of the quantum fingerprinting technique.

How should we think about quantum computing? The usual answer to this question is based on ideas inspired by computer science, such as qubits, quantum gates, and quantum circuits. In this talk I will explain an alternate geometric approach to quantum computation. In the geometric approach, an optimal quantum computation corresponds to "free falling" along the minimal geodesics of a certain Riemannian manifold. This reformulation opens up the possibility of using tools from geometry to understand the strengths and weaknesses of quantum computation, and perhaps to understand what makes certain physical operations difficult (or easy) to synthesize.