391 research outputs found
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
Block synchronization for quantum information
Locating the boundaries of consecutive blocks of quantum information is a
fundamental building block for advanced quantum computation and quantum
communication systems. We develop a coding theoretic method for properly
locating boundaries of quantum information without relying on external
synchronization when block synchronization is lost. The method also protects
qubits from decoherence in a manner similar to conventional quantum
error-correcting codes, seamlessly achieving synchronization recovery and error
correction. A family of quantum codes that are simultaneously synchronizable
and error-correcting is given through this approach.Comment: 7 pages, no figures, final accepted version for publication in
Physical Review
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
Hudson's Theorem for finite-dimensional quantum systems
We show that, on a Hilbert space of odd dimension, the only pure states to
possess a non-negative Wigner function are stabilizer states. The Clifford
group is identified as the set of unitary operations which preserve positivity.
The result can be seen as a discrete version of Hudson's Theorem. Hudson
established that for continuous variable systems, the Wigner function of a pure
state has no negative values if and only if the state is Gaussian. Turning to
mixed states, it might be surmised that only convex combinations of stabilizer
states give rise to non-negative Wigner distributions. We refute this
conjecture by means of a counter-example. Further, we give an axiomatic
characterization which completely fixes the definition of the Wigner function
and compare two approaches to stabilizer states for Hilbert spaces of
prime-power dimensions. In the course of the discussion, we derive explicit
formulas for the number of stabilizer codes defined on such systems.Comment: 17 pages, 3 figures; References updated. Title changed to match
published version. See also quant-ph/070200
On measurement-based quantum computation with the toric code states
We study measurement-based quantum computation (MQC) using as quantum
resource the planar code state on a two-dimensional square lattice (planar
analogue of the toric code). It is shown that MQC with the planar code state
can be efficiently simulated on a classical computer if at each step of MQC the
sets of measured and unmeasured qubits correspond to connected subsets of the
lattice.Comment: 9 pages, 5 figure
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
On SIC-POVMs in Prime Dimensions
The generalized Pauli group and its normalizer, the Clifford group, have a
rich mathematical structure which is relevant to the problem of constructing
symmetric informationally complete POVMs (SIC-POVMs). To date, almost every
known SIC-POVM fiducial vector is an eigenstate of a "canonical" unitary in the
Clifford group. I show that every canonical unitary in prime dimensions p > 3
lies in the same conjugacy class of the Clifford group and give a class
representative for all such dimensions. It follows that if even one such
SIC-POVM fiducial vector is an eigenvector of such a unitary, then all of them
are (for a given such dimension). I also conjecture that in all dimensions d,
the number of conjugacy classes is bounded above by 3 and depends only on d mod
9, and I support this claim with computer computations in all dimensions < 48.Comment: 6 pages, no figures. v3 Refs added, improved discussion of previous
work. Ref to a proof of the main conjecture also adde
Std fimbriae-fucose interaction increases Salmonella-induced intestinal inflammation and prolongs colonization
Author summary The intestinal epithelium is a crucial biological interface, interacting with both commensal and pathogenic microorganisms. It’s lined with heavily glycosylated proteins and glycolipids which can act as both attachment sites and energy sources for intestinal bacteria. Fut2, the enzyme governing epithelial α1,2-fucosylation, has been implicated in the interaction between microbes and intestinal epithelial cells. Salmonella is one of the most important bacterial gastrointestinal pathogens affecting millions of people worldwide. Salmonella possesses fimbrial and non-fimbrial adhesins which can be used to adhere to host cells. Here we show that Salmonella expresses Std fimbriae in the gastrointestinal tract in vivo and exploit Std fimbriae to bind fucosylated structures in the mucus and on the intestinal epithelium. Furthermore, we demonstrate that the Std fimbriae-fucose interaction is necessary for bacterial colonization of the intestine and for triggering intestinal inflammation. These data lend new insights into bacterial adhesion-epithelial interactions which are essential for bacterial pathogenesis and key factors in determining tissue tropism and host susceptibility to infectious disease
Invariant and polynomial identities for higher rank matrices
We exhibit explicit expressions, in terms of components, of discriminants,
determinants, characteristic polynomials and polynomial identities for matrices
of higher rank. We define permutation tensors and in term of them we construct
discriminants and the determinant as the discriminant of order , where
is the dimension of the matrix. The characteristic polynomials and the
Cayley--Hamilton theorem for higher rank matrices are obtained there from
Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements
We introduce the problem of constructing weighted complex projective
2-designs from the union of a family of orthonormal bases. If the weight
remains constant across elements of the same basis, then such designs can be
interpreted as generalizations of complete sets of mutually unbiased bases,
being equivalent whenever the design is composed of d+1 bases in dimension d.
We show that, for the purpose of quantum state determination, these designs
specify an optimal collection of orthogonal measurements. Using highly
nonlinear functions on abelian groups, we construct explicit examples from d+2
orthonormal bases whenever d+1 is a prime power, covering dimensions d=6, 10,
and 12, for example, where no complete sets of mutually unbiased bases have
thus far been found.Comment: 28 pages, to appear in J. Math. Phy
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