17,372 research outputs found
Unextendible Mutually Unbiased Bases from Pauli Classes
We provide a construction of sets of (d/2+1) mutually unbiased bases (MUBs)
in dimensions d=4,8 using maximal commuting classes of Pauli operators. We show
that these incomplete sets cannot be extended further using the operators of
the Pauli group. However, specific examples of sets of MUBs obtained using our
construction are shown to be strongly unextendible; that is, there does not
exist another vector that is unbiased with respect to the elements in the set.
We conjecture the existence of such unextendible sets in higher dimensions
(d=2^{n}, n>3) as well. Furthermore, we note an interesting connection between
these unextendible sets and state-independent proofs of the Kochen-Specker
Theorem for two-qubit systems. Our construction also leads to a proof of the
tightness of a H_{2} entropic uncertainty relation for any set of three MUBs
constructed from Pauli classes in d=4.Comment: 22 pages, v2: minor changes, references added; published versio
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 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
Solution to the King's Problem in prime power dimensions
It is shown how to ascertain the values of a complete set of mutually
complementary observables of a prime power degree of freedom by generalizing
the solution in prime dimensions given by Englert and Aharonov [Phys. Lett.
A284, 1-5 (2001)].Comment: 16 pages, 6 tables. A typo in an inequality on the line preceding
Eqn.(4)has been correcte
Systems of mutually unbiased Hadamard matrices containing real and complex matrices
We use combinatorial and Fourier analytic arguments
to prove various non-existence results on systems of real and com-
plex unbiased Hadamard matrices. In particular, we prove that
a complete system of complex mutually unbiased Hadamard ma-
trices (MUHs) in any dimension cannot contain more than one
real Hadamard matrix. We also give new proofs of several known
structural results in low dimensions
All Mutually Unbiased Product Bases in Dimension Six
All mutually unbiased bases in dimension six consisting of product states
only are constructed. Several continuous families of pairs and two triples of
mutually unbiased product bases are found to exist but no quadruple. The
exhaustive classification leads to a proof that a complete set of seven
mutually unbiased bases, if it exists, cannot contain a triple of mutually
unbiased product bases.Comment: 32 pages, 3 figures, identical to published versio
Geometrical approach to mutually unbiased bases
We propose a unifying phase-space approach to the construction of mutually
unbiased bases for a two-qubit system. It is based on an explicit
classification of the geometrical structures compatible with the notion of
unbiasedness. These consist of bundles of discrete curves intersecting only at
the origin and satisfying certain additional properties. We also consider the
feasible transformations between different kinds of curves and show that they
correspond to local rotations around the Bloch-sphere principal axes. We
suggest how to generalize the method to systems in dimensions that are powers
of a prime.Comment: 10 pages. Some typos in the journal version have been correcte
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