A Bloch-Sphere-Type Model for Two Qubits in the Geometric Algebra of a 6-D Euclidean Vector Space

Geometric algebra is a mathematical structure that is inherent in any metric vector space, and defined by the requirement that the metric tensor is given by the scalar part of the product of vectors. It provides a natural framework in which to represent the classical groups as subgroups of rotation groups, and similarly their Lie algebras. In this article we show how the geometric algebra of a six-dimensional real Euclidean vector space naturally allows one to construct the special unitary group on a two-qubit (quantum bit) Hilbert space, in a fashion similar to that used in the well-established Bloch sphere model for a single qubit. This is then used to illustrate the Cartan decompositions and subalgebras of the four-dimensional special unitary group, which have recently been used by J. Zhang, J. Vala, S. Sastry and K. B. Whaley [Phys. Rev. A 67, 042313, 2003] to study the entangling capabilities of two-qubit unitaries.Comment: 14 pages, 2 figures, in press (Proceedings of SPIE Conference on Defense & Security

On the swap-distances of different realizations of a graphical degree sequence

One of the first graph theoretical problems which got serious attention (already in the fifties of the last century) was to decide whether a given integer sequence is equal to the degree sequence of a simple graph (or it is {\em graphical} for short). One method to solve this problem is the greedy algorithm of Havel and Hakimi, which is based on the {\em swap} operation. Another, closely related question is to find a sequence of swap operations to transform one graphical realization into another one of the same degree sequence. This latter problem got particular emphases in connection of fast mixing Markov chain approaches to sample uniformly all possible realizations of a given degree sequence. (This becomes a matter of interest in connection of -- among others -- the study of large social networks.) Earlier there were only crude upper bounds on the shortest possible length of such swap sequences between two realizations. In this paper we develop formulae (Gallai-type identities) for these {\em swap-distance}s of any two realizations of simple undirected or directed degree sequences. These identities improves considerably the known upper bounds on the swap-distances.Comment: to be publishe

Subsystem Pseudo-pure States

A critical step in experimental quantum information processing (QIP) is to implement control of quantum systems protected against decoherence via informational encodings, such as quantum error correcting codes, noiseless subsystems and decoherence free subspaces. These encodings lead to the promise of fault tolerant QIP, but they come at the expense of resource overheads. Part of the challenge in studying control over multiple logical qubits, is that QIP test-beds have not had sufficient resources to analyze encodings beyond the simplest ones. The most relevant resources are the number of available qubits and the cost to initialize and control them. Here we demonstrate an encoding of logical information that permits the control over multiple logical qubits without full initialization, an issue that is particularly challenging in liquid state NMR. The method of subsystem pseudo-pure state will allow the study of decoherence control schemes on up to 6 logical qubits using liquid state NMR implementations.Comment: 9 pages, 1 Figur

Reflection Symmetries for Multiqubit Density Operators

For multiqubit density operators in a suitable tensorial basis, we show that a number of nonunitary operations used in the detection and synthesis of entanglement are classifiable as reflection symmetries, i.e., orientation changing rotations. While one-qubit reflections correspond to antiunitary symmetries, as is known for example from the partial transposition criterion, reflections on the joint density of two or more qubits are not accounted for by the Wigner Theorem and are well-posed only for sufficiently mixed states. One example of such nonlocal reflections is the unconditional NOT operation on a multiparty density, i.e., an operation yelding another density and such that the sum of the two is the identity operator. This nonphysical operation is admissible only for sufficiently mixed states.Comment: 9 page

Design of Strongly Modulating Pulses to Implement Precise Effective Hamiltonians for Quantum Information Processing

We describe a method for improving coherent control through the use of detailed knowledge of the system's Hamiltonian. Precise unitary transformations were obtained by strongly modulating the system's dynamics to average out unwanted evolution. With the aid of numerical search methods, pulsed irradiation schemes are obtained that perform accurate, arbitrary, selective gates on multi-qubit systems. Compared to low power selective pulses, which cannot average out all unwanted evolution, these pulses are substantially shorter in time, thereby reducing the effects of relaxation. Liquid-state NMR techniques on homonuclear spin systems are used to demonstrate the accuracy of these gates both in simulation and experiment. Simulations of the coherent evolution of a 3-qubit system show that the control sequences faithfully implement the unitary operations, typically yielding gate fidelities on the order of 0.999 and, for some sequences, up to 0.9997. The experimentally determined density matrices resulting from the application of different control sequences on a 3-spin system have overlaps of up to 0.99 with the expected states, confirming the quality of the experimental implementation.Comment: RevTeX3, 11 pages including 2 tables and 5 figures; Journal of Chemical Physics, in pres

Selective coherence transfers in homonuclear dipolar coupled spin systems

Mapping the physical dipolar Hamiltonian of a solid-state network of nuclear spins onto a system of nearest-neighbor couplings would be extremely useful for a variety of quantum information processing applications, as well as NMR structural studies. We demonstrate such a mapping for a system consisting of an ensemble of spin pairs, where the coupling between spins in the same pair is significantly stronger than the coupling between spins on different pairs. An amplitude modulated RF field is applied on resonance with the Larmor frequency of the spins, with the frequency of the modulation matched to the frequency of the dipolar coupling of interest. The spin pairs appear isolated from each other in the regime where the RF power (omega_1) is such that omega_weak << omega_1 << omega_strong. Coherence lifetimes within the two-spin system are increased from 19 us to 11.1 ms, a factor of 572.Comment: 4 pages. Paper re-submitted with minor changes to clarify that the scheme demonstrated is not an exact mapping onto a nearest neighbor system. However, this is the first demonstration of a controlled evolution in a subspace of an extended spin system, on a timescale that is much larger than the dipolar dephasing tim

Multivariate calibration approach for quantitative determination of cell-line cross contamination by intact cell mass spectrometry and artificial neural networks

Cross-contamination of eukaryotic cell lines used in biomedical research represents a highly relevant problem. Analysis of repetitive DNA sequences, such as Short Tandem Repeats (STR), or Simple Sequence Repeats (SSR), is a widely accepted, simple, and commercially available technique to authenticate cell lines. However, it provides only qualitative information that depends on the extent of reference databases for interpretation. In this work, we developed and validated a rapid and routinely applicable method for evaluation of cell culture cross-contamination levels based on mass spectrometric fingerprints of intact mammalian cells coupled with artificial neural networks (ANNs). We used human embryonic stem cells (hESCs) contaminated by either mouse embryonic stem cells (mESCs) or mouse embryonic fibroblasts (MEFs) as a model. We determined the contamination level using a mass spectra database of known calibration mixtures that served as training input for an ANN. The ANN was then capable of correct quantification of the level of contamination of hESCs by mESCs or MEFs. We demonstrate that MS analysis, when linked to proper mathematical instruments, is a tangible tool for unraveling and quantifying heterogeneity in cell cultures. The analysis is applicable in routine scenarios for cell authentication and/or cell phenotyping in general

Quantum Codes for Controlling Coherent Evolution

Control over spin dynamics has been obtained in NMR via coherent averaging, which is implemented through a sequence of RF pulses, and via quantum codes which can protect against incoherent evolution. Here, we discuss the design and implementation of quantum codes to protect against coherent evolution. A detailed example is given of a quantum code for protecting two data qubits from evolution under a weak coupling (Ising) term in the Hamiltonian, using an ``isolated'' ancilla which does not evolve on the experimental time scale. The code is realized in a three-spin system by liquid-state NMR spectroscopy on 13C-labelled alanine, and tested for two initial states. It is also shown that for coherent evolution and isolated ancillae, codes exist that do not require the ancillae to initially be in a (pseudo-)pure state. Finally, it is shown that even with non-isolated ancillae quantum codes exist which can protect against evolution under weak coupling. An example is presented for a six qubit code that protects two data spins to first order.Comment: Reformatted single spaced with figures incorporated into text (18 pages, 6 figures, PDF only, submitted to J. Chem. Phys.

Trabecular and Cortical Bone and Ossified Vessel Alterations in Rat Tibiae with the Onset and Progression of Type 2 Diabetes Mellitus in a Novel, Transgenic Rat Model

Type 2 Diabetes Mellitus (T2DM) is a metabolic disorder of systemic complications including increased fracture risk and microvascular pathology, suggesting a potential link between the two. PURPOSE: We determined how the onset and progression of T2DM affected bone and marrow vasculature in the University of California Davis T2DM transgenic rat model. METHODS: Forty-eight male T2DM rats were divided accordingly: pre-diabetes (12wks), diabetes onset (14wks), early-stage diabetes (20wks; 4wks post-onset), and late-stage diabetes (22wks; 12wks post-onset) matched with four healthy control (CTL; Sprague Dawley) groups. Body mass(g), HbA1c(%), and fasted blood glucose(g/dL) were measured at sacrifice. Tibiae were scanned via µCT (15µm) to assess trabecular volume-to-total volume ratio (BV/TV, %), trabecular thickness (Tb.Th, µm), trabecular number (Tb.N, /mm), trabecular separation (Tb.Sp µm), and density (mgHA/ccm) in the proximal metaphysis. Cortical thickness (Ct.Th, µm) and density (mmHg/ccm) were measured at the mid-shaft, and cortical porosity (%) was calculated (1-Ct.BV/TV). Ossified vessel volume (OsVV, %), ossified vessel thickness (OsV.Th, µm), and OsV density (mgHA/ccm) were analyzed in the diaphyseal marrow, representing conversion of blood vessels into bone-like tissue. A General Linear Model determined significance at p\u3c0.05, a priori. RESULTS: Body mass (455-622g vs. 342-435g) and HbA1c (5-12% vs. 5%) was higher (p\u3c0.05) in the T2DM vs. CTL groups, respectively. Blood glucose rose (p\u3c0.05) in early- (113±9g/dL vs. 71±7g/dL) and late- (244±10g/dL vs. 68±2g/dL) stage diabetes vs. CTL. Trabecular BV/TV was lower (p\u3c0.05) in pre- (4±1% vs. 9±2%) and late-stage (5±2% vs. 8±2%) diabetes vs. CTL, from reduced (p\u3c0.05) Tb.N in pre- (2.5±0.1/mm vs. CTL, 3.8±0.2/mm) and late-stage (2.1±0.3/mm vs. CTL, 2.6±0.4/mm), and reduced (p\u3c0.05) Tb.Th in late-stage (56±3µm vs. CTL, 67±4µm) diabetes. Trabecular separation increased (p\u3c0.05) in pre-(407±23µm vs. CTL, 263±15µm) and late-stage (482±85µm vs. CTL, 406±85µm). Trabecular density and Ct.Th, density, and porosity did not differ. OsVV was lower (p\u3c0.05) in early-stage diabetes (1.7±0.2% vs. CTL, 4.7±1.5%), OsV.Th was higher (p\u3c0.05) in pre-(69±14µm vs. CTL, 56±13µm) and late-stage (80±10µm vs. CTL, 59±13µm) diabetes, and OsV density was higher (p\u3c0.05) in late-stage diabetes (918±17mgHA/ccm vs. CTL, 891±31mgHA /ccm).CONCLUSION: T2DM developed in the transgenic rat model (i.e., increases in HbA1c, and blood glucose). Cortical bone parameters were not altered. Trabecular bone declined in pre- and late-stage diabetes, via reduced trabecular number and thickness. Ossified vessels were thicker at these stages. Thus, the observed trabecular bone and vascular pathologies coincided in the tibia with the onset and progress of T2DM