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Register a new userWitnessing a Poincare recurrence with Mathematica
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The often elusive Poincare recurrence can be witnessed in a completely separable system. For such systems, the problem of recurrence reduces to the classic mathematical problem of simultaneous Diophantine approximation of multiple numbers. The latter problem then can be somewhat satisfactorily solved by using the famous Lenstra-Lenstra-Lov{a}sz (LLL) algorithm, which is implemented in the Mathematica built-in function verbLatticeReduce. The procedure is illustrated with a harmonic chain. The incredibly large recurrence times are obtained exactly. They follow the expected scaling law very well.
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The existence in the physical QCD vacuum of nonzero gluon condensates, such as $<g^2F^2>$, requires dominance of gluon fields with finite mean action density. This naturally allows any real number value for the unit ``topological charge $q$ characterising the fields approximating the gluon configurations which should dominate the QCD partition function. If $q$ is an irrational number then the critical values of the $theta$ parameter for which CP is spontaneously broken are dense in $mathbb{R}$, which provides for a mechanism of resolving the strong CP problem simultaneously with a correct implementation of $U_{rm A}(1)$ symmetry. We present an explicit realisation of this mechanism within a QCD motivated domain model. Some model independent arguments are given that suggest the relevance of this mechanism also to genuine QCD.
This Mathematica 7.0/8.0 package upgrades and extends the quantum computer simulation code called QDENSITY. Use of the density matrix was emphasized in QDENSITY, although that code was also applicable to a quantum state description. In the present version, the quantum state version is stressed and made amenable to future extensions to parallel computer simulations. The add-on QCWAVE extends QDENSITY in several ways. The first way is to describe the action of one, two and three- qubit quantum gates as a set of small ($2 times 2, 4times 4$ or $8times 8$) matrices acting on the $2^{n_q}$ amplitudes for a system of $n_q$ qubits. This procedure was described in our parallel computer simulation QCMPI and is reviewed here. The advantage is that smaller storage demands are made, without loss of speed, and that the procedure can take advantage of message passing interface (MPI) techniques, which will hopefully be generally available in future Mathemati
This book is an attempt to help students transform all of the concepts of quantum mechanics into concrete computer representations, which can be constructed, evaluated, analyzed, and hopefully understood at a deeper level than what is possible with more abstract representations. It was written for a Masters and PhD lecture given yearly at the University of Basel, Switzerland. The goal is to give a language to the student in which to speak about quantum physics in more detail, and to start the student on a path of fluency in this language. On our journey we approach questions such as: -- You already know how to calculate the energy eigenstates of a single particle in a simple one-dimensional potential. How can such calculations be generalized to non-trivial potentials, higher dimensions, and interacting particles? -- You have heard that quantum mechanics describes our everyday world just as well as classical mechanics does, but have you ever seen an example where such behavior is calculated in detail and where the transition from classical to quantum physics is evident? -- How can we describe the internal spin structure of particles? How does this internal structure couple to the particles motion? -- What are qubits and quantum circuits, and how can they be assembled to simulate a future quantum computer?
The purpose of an entanglement witness experiment is to certify the creation of an entangled state from a finite number of trials. The statistical confidence of such an experiment is typically expressed as the number of observed standard deviations of witness violations. This method implicitly assumes that the noise is well-behaved so that the central limit theorem applies. In this work, we propose two methods to analyze witness experiments where the states can be subject to arbitrarily correlated noise. Our first method is a rejection experiment, in which we certify the creation of entanglement by rejecting the hypothesis that the experiment can only produce separable states. We quantify the statistical confidence by a p-value, which can be interpreted as the likelihood that the observed data is consistent with the hypothesis that only separable states can be produced. Hence a small p-value implies large confidence in the witnessed entanglement. The method applies to general witness experiments and can also be used to witness genuine multipartite entanglement. Our second method is an estimation experiment, in which we estimate and construct confidence intervals for the average witness value. This confidence interval is statistically rigorous in the presence of correlated noise. The method applies to general estimation problems, including fidelity estimation. To account for systematic measurement and random setting generation errors, our model takes into account device imperfections and we show how this affects both methods of statistical analysis. Finally, we illustrate the use of our methods with detailed examples based on a simulation of NV centers.
The ability to coherently control mechanical systems with optical fields has made great strides over the past decade, and now includes the use of photon counting techniques to detect the non-classical nature of mechanical states. These techniques may soon be used to perform an opto-mechanical Bell test, hence highlighting the potential of cavity opto-mechanics for device-independent quantum information processing. Here, we propose a witness which reveals opto-mechanical entanglement without any constraint on the global detection efficiencies in a setup allowing one to test a Bell inequality. While our witness relies on a well-defined description and correct experimental calibration of the measurements, it does not need a detailed knowledge of the functioning of the opto-mechanical system. A feasibility study including dominant sources of noise and loss shows that it can readily be used to reveal opto-mechanical entanglement in present-day experiments with photonic crystal nanobeam resonators.
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