190 research outputs found
Mutually unbiased binary observable sets on N qubits
The Pauli operators (tensor products of Pauli matrices) provide a complete
basis of operators on the Hilbert space of N qubits. We prove that the set of
4^N-1 Pauli operators may be partitioned into 2^N+1 distinct subsets, each
consisting of 2^N-1 internally commuting observables. Furthermore, each such
partitioning defines a unique choice of 2^N+1 mutually unbiased basis sets in
the N-qubit Hilbert space. Examples for 2 and 3 qubit systems are discussed
with emphasis on the nature and amount of entanglement that occurs within these
basis sets.Comment: 5 pages, 5 figures. Replacement - expanded introduction and
conclusions; added reference
Information and The Brukner-Zeilinger Interpretation of Quantum Mechanics: A Critical Investigation
In Brukner and Zeilinger's interpretation of quantum mechanics, information
is introduced as the most fundamental notion and the finiteness of information
is considered as an essential feature of quantum systems. They also define a
new measure of information which is inherently different from the Shannon
information and try to show that the latter is not useful in defining the
information content in a quantum object.
Here, we show that there are serious problems in their approach which make
their efforts unsatisfactory. The finiteness of information does not explain
how objective results appear in experiments and what an instantaneous change in
the so-called information vector (or catalog of knowledge) really means during
the measurement. On the other hand, Brukner and Zeilinger's definition of a new
measure of information may lose its significance, when the spin measurement of
an elementary system is treated realistically. Hence, the sum of the individual
measures of information may not be a conserved value in real experiments.Comment: 20 pages, two figures, last version. Section 4 is replaced by a new
argument. Other sections are improved. An appendix and new references are
adde
Experimenter's Freedom in Bell's Theorem and Quantum Cryptography
Bell's theorem states that no local realistic explanation of quantum
mechanical predictions is possible, in which the experimenter has a freedom to
choose between different measurement settings. Within a local realistic picture
the violation of Bell's inequalities can only be understood if this freedom is
denied. We determine the minimal degree to which the experimenter's freedom has
to be abandoned, if one wants to keep such a picture and be in agreement with
the experiment. Furthermore, the freedom in choosing experimental arrangements
may be considered as a resource, since its lacking can be used by an
eavesdropper to harm the security of quantum communication. We analyze the
security of quantum key distribution as a function of the (partial) knowledge
the eavesdropper has about the future choices of measurement settings which are
made by the authorized parties (e.g. on the basis of some quasi-random
generator). We show that the equivalence between the violation of Bell's
inequality and the efficient extraction of a secure key - which exists for the
case of complete freedom (no setting knowledge) - is lost unless one adapts the
bound of the inequality according to this lack of freedom.Comment: 7 pages, 2 figures, incorporated referee comment
Logical independence and quantum randomness
We propose a link between logical independence and quantum physics. We
demonstrate that quantum systems in the eigenstates of Pauli group operators
are capable of encoding mathematical axioms and show that Pauli group quantum
measurements are capable of revealing whether or not a given proposition is
logically dependent on the axiomatic system. Whenever a mathematical
proposition is logically independent of the axioms encoded in the measured
state, the measurement associated with the proposition gives random outcomes.
This allows for an experimental test of logical independence. Conversely, it
also allows for an explanation of the probabilities of random outcomes observed
in Pauli group measurements from logical independence without invoking quantum
theory. The axiomatic systems we study can be completed and are therefore not
subject to Goedel's incompleteness theorem.Comment: 9 pages, 4 figures, published version plus additional experimental
appendi
An Information-Geometric Reconstruction of Quantum Theory, I: The Abstract Quantum Formalism
In this paper and a companion paper, we show how the framework of information
geometry, a geometry of discrete probability distributions, can form the basis
of a derivation of the quantum formalism. The derivation rests upon a few
elementary features of quantum phenomena, such as the statistical nature of
measurements, complementarity, and global gauge invariance. It is shown that
these features can be traced to experimental observations characteristic of
quantum phenomena and to general theoretical principles, and thus can
reasonably be taken as a starting point of the derivation. When appropriately
formulated within an information geometric framework, these features lead to
(i) the abstract quantum formalism for finite-dimensional quantum systems, (ii)
the result of Wigner's theorem, and (iii) the fundamental correspondence rules
of quantum theory, such as the canonical commutation relationships. The
formalism also comes naturally equipped with a metric (and associated measure)
over the space of pure states which is unitarily- and anti-unitarily invariant.
The derivation suggests that the information geometric framework is directly or
indirectly responsible for many of the central structural features of the
quantum formalism, such as the importance of square-roots of probability and
the occurrence of sinusoidal functions of phases in a pure quantum state.
Global gauge invariance is seen to play a crucial role in the emergence of the
formalism in its complex form.Comment: 26 page
Invariant information and quantum state estimation
The invariant information introduced by Brukner and Zeilinger, Phys. Rev.
Lett. 83, 3354 (1999), is reconsidered from the point of view of quantum state
estimation. We show that it is directly related to the mean error of the
standard reconstruction from the measurement of a complete set of mutually
complementary observables. We give its generalization in terms of the Fisher
information. Provided that the optimum reconstruction is adopted, the
corresponding quantity loses its invariant character.Comment: 4 pages, no figure
Formulation of the uncertainty relations in terms of the Renyi entropies
Quantum mechanical uncertainty relations for position and momentum are
expressed in the form of inequalities involving the Renyi entropies. The proof
of these inequalities requires the use of the exact expression for the
(p,q)-norm of the Fourier transformation derived by Babenko and Beckner.
Analogous uncertainty relations are derived for angle and angular momentum and
also for a pair of complementary observables in N-level systems. All these
uncertainty relations become more attractive when expressed in terms of the
symmetrized Renyi entropies
Entanglement as a quantum order parameter
We show that the quantum order parameters (QOP) associated with the
transitions between a normal conductor and a superconductor in the BCS and
eta-pairing models and between a Mott-insulator and a superfluid in the
Bose-Hubbard model are directly related to the amount of entanglement existent
in the ground state of each system. This gives a physical meaningful
interpretation to these QOP, which shows the intrinsically quantum nature of
the phase transitions considered.Comment: 5 pages. No figures. Revised version. References adde
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
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