Small sets of complementary observables

Two observables are called complementary if preparing a physical object in an eigenstate of one of them yields a completely random result in a measurement of the other. We investigate small sets of complementary observables that cannot be extended by yet another complementary observable. We construct explicit examples of the unextendible sets up to dimension $16$ and conjecture certain small sets to be unextendible in higher dimensions. Our constructions provide three complementary measurements, only one observable away from the ultimate minimum of two observables in the set. Almost all of our examples in finite dimension allow to discriminate pure states from some mixed states, and shed light on the complex topology of the Bloch space of higher-dimensional quantum systems

Minimal-memory realization of pearl-necklace encoders of general quantum convolutional codes

Quantum convolutional codes, like their classical counterparts, promise to offer higher error correction performance than block codes of equivalent encoding complexity, and are expected to find important applications in reliable quantum communication where a continuous stream of qubits is transmitted. Grassl and Roetteler devised an algorithm to encode a quantum convolutional code with a "pearl-necklace encoder." Despite their theoretical significance as a neat way of representing quantum convolutional codes, they are not well-suited to practical realization. In fact, there is no straightforward way to implement any given pearl-necklace structure. This paper closes the gap between theoretical representation and practical implementation. In our previous work, we presented an efficient algorithm for finding a minimal-memory realization of a pearl-necklace encoder for Calderbank-Shor-Steane (CSS) convolutional codes. This work extends our previous work and presents an algorithm for turning a pearl-necklace encoder for a general (non-CSS) quantum convolutional code into a realizable quantum convolutional encoder. We show that a minimal-memory realization depends on the commutativity relations between the gate strings in the pearl-necklace encoder. We find a realization by means of a weighted graph which details the non-commutative paths through the pearl-necklace. The weight of the longest path in this graph is equal to the minimal amount of memory needed to implement the encoder. The algorithm has a polynomial-time complexity in the number of gate strings in the pearl-necklace encoder.Comment: 16 pages, 5 figures; extends paper arXiv:1004.5179v

On the bulk-skin temperature difference and its impact on satellite remote sensing of sea surface temperature

Satellite infrared sensors only observe the temperature of the skin of the ocean rather than the bulk sea surface temperature (SST) traditionally measured from ships and buoys. In order to examine the differences and similarities between skin and bulk temperatures, radiometric measurements of skin temperature were made in the North Atlantic Ocean from a research vessel along with coincident measurements of subsurface bulk temperatures, radiative fluxes, and meteorological variables. Over the entire 6-week data set the bulk-skin temperature differences (AT) range between -1.0 and 1.0 K with mean differences of 0.1 to 0.2 K depending on wind and surface heat flux conditions. The bulk-skin temperature difference varied between day and night (mean differences 0.11 and 0.30 K, respectively) as well as with different cloud conditions, which can mask the horizontal variability of SST in regions of weak horizontal temperature gradients. A coherency analysis reveals strong correlations between skin and bulk temperatures at longer length scales in regions with relatively weak horizontal temperature gradients. The skin-bulk temperature difference is pararneterized in terms of heat and momentum fluxes (or their related variables) with a resulting accuracy of 0.11 K and 0.17 K for night and daytime. A recommendation is made to calibrate satellite derived SST's during night with buoy measurements and the additional aid of meteorological variables to properly handle AT variations

On the connection between mutually unbiased bases and orthogonal Latin squares

We offer a piece of evidence that the problems of finding the number of mutually unbiased bases (MUB) and mutually orthogonal Latin squares (MOLS) might not be equivalent. We study a particular procedure which has been shown to relate the two problems and generates complete sets of MUBs in power-of-prime dimensions and three MUBs in dimension six. For these cases, every square from an augmented set of MOLS has a corresponding MUB. We show that this no longer holds for certain composite dimensions.Comment: 6 pages, submitted to Proceedings of CEWQO 200

SIC~POVMs and Clifford groups in prime dimensions

We show that in prime dimensions not equal to three, each group covariant symmetric informationally complete positive operator valued measure (SIC~POVM) is covariant with respect to a unique Heisenberg--Weyl (HW) group. Moreover, the symmetry group of the SIC~POVM is a subgroup of the Clifford group. Hence, two SIC~POVMs covariant with respect to the HW group are unitarily or antiunitarily equivalent if and only if they are on the same orbit of the extended Clifford group. In dimension three, each group covariant SIC~POVM may be covariant with respect to three or nine HW groups, and the symmetry group of the SIC~POVM is a subgroup of at least one of the Clifford groups of these HW groups respectively. There may exist two or three orbits of equivalent SIC~POVMs for each group covariant SIC~POVM, depending on the order of its symmetry group. We then establish a complete equivalence relation among group covariant SIC~POVMs in dimension three, and classify inequivalent ones according to the geometric phases associated with fiducial vectors. Finally, we uncover additional SIC~POVMs by regrouping of the fiducial vectors from different SIC~POVMs which may or may not be on the same orbit of the extended Clifford group.Comment: 30 pages, 1 figure, section 4 revised and extended, published in J. Phys. A: Math. Theor. 43, 305305 (2010

Gas/particle partitioning and global distribution of polycyclic aromatic hydrocarbons – A modelling approach

The global atmospheric distribution and long-range transport (LRT) potential of three polycyclic aromatic hydrocarbons (PAH) - anthracene, fluoranthene and benzo[a]pyrene - are studied. The model used is a global aerosol-chemistry-transport-model, which is based on an atmospheric general circulation model. The model includes an in-built dynamic aerosol model coupled to two-dimensional surface compartments. Several parameterisations of gas/particle partitioning and different assumptions of degradation in the aerosol particulate phase were tested. PAHs are mostly distributed in the source regions but reach the Arctic and the Antarctic. The Canadian Arctic is predicted to be significantly less affected by mid-latitude PAH emissions than the European Arctic. Re-volatilisation is significant for semivolatile PAHs. Accumulation of semivolatile PAHs in polar regions, however, is not indicated. The model study suggests that gas/particle partitioning in air drastically influences the atmospheric cycling, the total environmental fate (e.g. compartmental distributions) and the LRT potential of the substances studied. A parameterisation which calculates the gas/particle partitioning assuming absorption into organic matter and adsorption to black carbon (soot) agrees best with the observations at remote sites. The study provides evidence that the degradation in the particulate phase must be slower than that in the gas-phase. The predicted concentrations of the semivolatile PAHs anthracene and fluoranthene in near-ground air at remote sites in mid and high northern latitudes are in line with measured concentrations, if adsorption of the substances to soot combined with absorption in particulate organic matter is assumed to determine gas/particle partitioning, but cannot be explained by adsorption alone (Junge-Pankow parameterisation of gas/particle partitioning). The results suggest that PAHs absorbed in the organic matrix of particulate matter is shielded from the gas-phase. (C) 2009 Elsevier Ltd. All rights reserved. [References: 42

Quantum Capacity Approaching Codes for the Detected-Jump Channel

The quantum channel capacity gives the ultimate limit for the rate at which quantum data can be reliably transmitted through a noisy quantum channel. Degradable quantum channels are among the few channels whose quantum capacities are known. Given the quantum capacity of a degradable channel, it remains challenging to find a practical coding scheme which approaches capacity. Here we discuss code designs for the detected-jump channel, a degradable channel with practical relevance describing the physics of spontaneous decay of atoms with detected photon emission. We show that this channel can be used to simulate a binary classical channel with both erasures and bit-flips. The capacity of the simulated classical channel gives a lower bound on the quantum capacity of the detected-jump channel. When the jump probability is small, it almost equals the quantum capacity. Hence using a classical capacity approaching code for the simulated classical channel yields a quantum code which approaches the quantum capacity of the detected-jump channel

Affine Constellations Without Mutually Unbiased Counterparts

It has been conjectured that a complete set of mutually unbiased bases in a space of dimension d exists if and only if there is an affine plane of order d. We introduce affine constellations and compare their existence properties with those of mutually unbiased constellations, mostly in dimension six. The observed discrepancies make a deeper relation between the two existence problems unlikely.Comment: 8 page

Constructing Mutually Unbiased Bases in Dimension Six

The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. In dimension six, the required observables only exist if it is possible to identify six mutually unbiased complex 6x6 Hadamard matrices. Prescribing a first Hadamard matrix, we construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and we never find more than two that are mutually unbiased. This result adds considerable support to the conjecture that no seven mutually unbiased bases exist in dimension six.Comment: As published version. Added discussion of the impact of numerical approximations and corrected the number of triples existing for non-affine families (cf Table 3