Mutually Unbiased Bases in Infinite Dimension Hilbert Spaces
While there has been investigation into mutually unbiased bases in infinite dimension Hilbert space, their existence remains an open question. It is conjectured that in a continuous Hilbert space, two orthonormal bases and are said to be mutually unbiased if
For the generalized position and momentum eigenstates and, the value of k is
The existence of mutually unbiased bases in a continuous Hilbert space remains open for debate, as further research in their existence is required before any conclusions can be reached.
Position states and momentum states are eigenvectors of Hermitian operators and, respectively. Weigert and Wilkinson were first to notice that also a linear combination of these operators have eigenbases, which have some features typical for the mutually unbiased bases. An operator has eigenfunctions proportional to with and the corresponding eigenvalues . If we parametrize and as and, the overlap between any eigenstate of the linear combination and any eigenstate of the position operator (both states normalized to the Dirac delta) is constant, but dependent on :
where and stand for eigenfunctions of and .
Read more about this topic: Mutually Unbiased Bases
Famous quotes containing the words mutually, unbiased, bases, infinite, dimension and/or spaces:
“For the support of this declaration we mutually pledge to each other our lives, our fortunes, & our sacred honour.”
—Thomas Jefferson (1743–1826)
“There is not a more disgusting spectacle under the sun than our subserviency to British criticism. It is disgusting, first, because it is truckling, servile, pusillanimous—secondly, because of its gross irrationality. We know the British to bear us little but ill will—we know that, in no case do they utter unbiased opinions of American books ... we know all this, and yet, day after day, submit our necks to the degrading yoke of the crudest opinion that emanates from the fatherland.”
—Edgar Allan Poe (1809–1845)
“The information links are like nerves that pervade and help to animate the human organism. The sensors and monitors are analogous to the human senses that put us in touch with the world. Data bases correspond to memory; the information processors perform the function of human reasoning and comprehension. Once the postmodern infrastructure is reasonably integrated, it will greatly exceed human intelligence in reach, acuity, capacity, and precision.”
—Albert Borgman, U.S. educator, author. Crossing the Postmodern Divide, ch. 4, University of Chicago Press (1992)
“Whatever we have got has been by infinite labour, and search, and ranging through every corner of nature; the difference is that instead of dirt and poison, we have rather chosen to fill our hives with honey and wax, thus furnishing mankind with the two noblest of things, which are sweetness and light.”
—Jonathan Swift (1667–1745)
“Authority is the spiritual dimension of power because it depends upon faith in a system of meaning that decrees the necessity of the hierarchical order and so provides for the unity of imperative control.”
—Shoshana Zuboff (b. 1951)
“Every true man is a cause, a country, and an age; requires infinite spaces and numbers and time fully to accomplish his design;—and posterity seem to follow his steps as a train of clients.”
—Ralph Waldo Emerson (1803–1882)