5,642 research outputs found
Classical and quantum evaluation codes at the trace roots
Producción CientíficaWe introduce a new class of evaluation linear codes by evaluating polynomials at the roots of a suitable trace function. We give conditions for self-orthogonality of these codes and their subfield-subcodes with respect to the Hermitian inner product. They allow us to construct stabilizer quantum codes over several finite fields which substantially improve the codes in the literature. For the binary case, we obtain records at http://codetables.de/. Moreover, we obtain several classical linear codes over the field with 4 elements which are records at http://codetables.de/.This work was supported in part by the Spanish MINECO/FEDER (Grants No. MTM2015-65764-C3-2-P and MTM2015-69138-REDT), in part by the University Jaume I (Grant No. P1-1B2015-02), in part by The Danish Council for Independent Research (Grant No. DFF--4002-00367), and in part by RYC-2016-20208 (AEI/FSE/UE)
Alternative fidelity measure for quantum states
We propose an alternative fidelity measure (namely, a measure of the degree
of similarity) between quantum states and benchmark it against a number of
properties of the standard Uhlmann-Jozsa fidelity. This measure is a simple
function of the linear entropy and the Hilbert-Schmidt inner product between
the given states and is thus, in comparison, not as computationally demanding.
It also features several remarkable properties such as being jointly concave
and satisfying all of "Jozsa's axioms". The trade-off, however, is that it is
supermultiplicative and does not behave monotonically under quantum operations.
In addition, new metrics for the space of density matrices are identified and
the joint concavity of the Uhlmann-Jozsa fidelity for qubit states is
established.Comment: 12 pages, 3 figures. v2 includes minor changes, new references and
new numerical results (Sec. IV
Mutually unbiased phase states, phase uncertainties, and Gauss sums
Mutually unbiased bases (MUBs), which are such that the inner product between
two vectors in different orthogonal bases is a constant equal to 1/sqrt{d),
with d the dimension of the finite Hilbert space, are becoming more and more
studied for applications such as quantum tomography and cryptography, and in
relation to entangled states and to the Heisenberg-Weil group of quantum
optics. Complete sets of MUBs of cardinality d+1 have been derived for prime
power dimensions d=p^m using the tools of abstract algebra. Presumably, for non
prime dimensions the cardinality is much less. Here we reinterpret MUBs as
quantum phase states, i.e. as eigenvectors of Hermitean phase operators
generalizing those introduced by Pegg & Barnett in 1989. We relate MUB states
to additive characters of Galois fields (in odd characteristic p) and to Galois
rings (in characteristic 2). Quantum Fourier transforms of the components in
vectors of the bases define a more general class of MUBs with multiplicative
characters and additive ones altogether. We investigate the complementary
properties of the above phase operator with respect to the number operator. We
also study the phase probability distribution and variance for general pure
quantum electromagnetic states and find them to be related to the Gauss sums,
which are sums over all elements of the field (or of the ring) of the product
of multiplicative and additive characters. Finally, we relate the concepts of
mutual unbiasedness and maximal entanglement. This allows to use well studied
algebraic concepts as efficient tools in the study of entanglement and its
information aspectsComment: 16 pages, a few typos corrected, some references updated, note
acknowledging I. Shparlinski adde
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