Rüdiger Schack

Classical model for bulk-ensemble NMR quantum computation

We present a classical model for bulk-ensemble NMR quantum computation: the quantum state of the NMR sample is described by a probability distribution over the orientations of classical tops, and quantum gates are described by classical transition probabilities. All NMR quantum computing experiments performed so far with three quantum bits can be accounted for in this classical model. After a few entangling gates, the classical model suffers an exponential decrease of the measured signal, whereas there is no corresponding decrease in the quantum description. We suggest that for small numbers of quantum bits, the quantum nature of NMR quantum computation lies in the ability to avoid an exponential signal decrease.

Information-theoretic characterization of quantum chaos

Hypersensitivity to perturbation is a criterion for chaos based on the question of how much information about a perturbing environment is needed to keep the entropy of a Hamiltonian system from increasing. We demonstrate numerically that hypersensitivity to perturbation is present in the following quantum maps: the quantum kicked top, the quantum bakers map, the quantum lazy bakers map, and the quantum sawtooth and cat maps.

Quantum chaos in open systems: a quantum state diffusion analysis

Except for the universe, all quantum systems are open, and according to quantum state diffusion theory, many systems localize to wavepackets in the neighbourhood of phase space points. This is due to decoherence from the interaction with the environment, and makes the quasiclassical limit of such systems both more realistic and simpler in many respects than the more familiar quasiclassical limit for closed systems. A linearized version of this theory leads to the correct classical dynamics in the macroscopic limit, even for nonlinear and chaotic systems. We apply the theory to the forced, damped Duffing oscillator, comparing the numerical results of the full and linearized equations, and argue that this can be used to make explicit calculations in the decoherent histories formalism of quantum mechanics.

Explicit product ensembles for separable quantum states

Abstract We present a general method for constructing pure-product-state representations for density operators of N quantum bits. If such a representation has non-negative expansion coefficients, it provides an explicit separable ensemble for the density operator. We derive the condition for separability of a mixture of the Greenberger—Horne—Zeilinger state with the maximally mixed state.

Chaos for Liouville probability densities

Using the method of symbolic dynamics, we show that a large class of classical chaotic maps exhibits exponential hypersensitivity to perturbation, i.e., a rapid increase with time of the information needed to describe the perturbed time evolution of the Liouville density, the information attaining values that are exponentially larger than the entropy increase that results from averaging over the perturbation. The exponential rate of growth of the ratio of information to entropy is given by the Kolmogorov-Sinai entropy of the map. These findings generalize and extend results obtained for the bakers map [R. Schack and C. M. Caves, Phys. Rev. Lett. 69, 3413 (1992)].

Shifts on a finite qubit string: A class of quantum baker's maps

Abstract. We present two complementary ways in which Saracenos symmetric version of the quantum bakers map can be written as a shift map on a string of quantum bits. One of these representations leads naturally to a family of quantizations of the bakers map.

Hypersensitivity and chaos signatures in the quantum baker's maps

Classical chaotic systems are distinguished by their sensitive dependence on initial conditions. The absence of this property in quantum systems has led to a number of proposals for perturbation-based characterizations of quantum chaos, including linear growth of entropy, exponential decay of fidelity, and hypersensitivity to perturbation. All of these accurately predict chaos in the classical limit, but it is not clear that they behave the same far from the classical realm. We investigate the dynamics of a family of quantizations of the bakers map, which range from a highly entangling unitary transformation to an essentially trivial shift map. Linear entropy growth and fidelity decay are exhibited by this entire family of maps, but hypersensitivity distinguishes between the simple dynamics of the trivial shift map and the more complicated dynamics of the other quantizations. This conclusion is supported by an analytical argument for short times and numerical evidence at later times.

Classical limit in terms of symbolic dynamics for the quantum baker's map

We derive a simple closed form for the matrix elements of the quantum bakers map that shows that the map is an approximate shift in a symbolic representation based on discrete phase space. We use this result to give a formal proof that the quantum bakers map approaches a classical Bernoulli shift in the limit of a small effective Plancks constant.

Quantum-state diffusion with a moving basis: Computing quantum-optical spectra.

Quantum state diffusion (QSD) as a tool to solve quantum-optical master equations by stochastic simulation can be made several orders of magnitude more efficient if states in Hilbert space are represented in a moving basis of excited coherent states. The large savings in computer memory and time are due to the localization property of the QSD equation. We show how the method can be used to compute spectra and give an application to second harmonic generation.

Homer Nodded: Von Neumann’s Surprising Oversight

We review the famous no-hidden-variables theorem in von Neumann’s 1932 book on the mathematical foundations of quantum mechanics (Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932). We describe the notorious gap in von Neumann’s argument, pointed out by Hermann (Abhandlungen der Fries’schen Schule 6:75–152, 1935) and, more famously, by Bell (Rev Modern Phys 38:447–452, 1966). We disagree with recent papers claiming that Hermann and Bell failed to understand what von Neumann was actually doing.