Complete sets of cyclic mutually unbiased bases in even prime power dimensions

We present a construction method for complete sets of cyclic mutually unbiased bases (MUBs) in Hilbert spaces of even prime power dimensions. In comparison to usual complete sets of MUBs, complete cyclic sets possess the additional property of being generated by a single unitary operator. The construction method is based on the idea of obtaining a partition of multi-qubit Pauli operators into maximal commuting sets of orthogonal operators with the help of a suitable element of the Clifford group. As a consequence, we explicitly obtain complete sets of cyclic MUBs generated by a single element of the Clifford group in dimensions $2^m$ for $m=1,2,...,24$.Comment: 10 page

Structure of the sets of mutually unbiased bases with cyclic symmetry

Mutually unbiased bases that can be cyclically generated by a single unitary operator are of special interest, since they can be readily implemented in practice. We show that, for a system of qubits, finding such a generator can be cast as the problem of finding a symmetric matrix over the field $\mathbb{F}_2$ equipped with an irreducible characteristic polynomial of a given Fibonacci index. The entanglement structure of the resulting complete sets is determined by two additive matrices of the same size.Comment: 20 page

Cyclic Mutually Unbiased Bases and Quantum Public-Key Encryption

Based on quantum physical phenomena, quantum information theory has a potential which goes beyond the classical conditions. Equipped with the resource of complementary information as an intrinsic property it offers many new perspectives. The field of quantum key distribution, which enables the ability to implement unconditional security, profits directly from this resource. To measure the state of quantum systems itself for different purposes in quantum information theory, which may be related to the construction of a quantum computer, as well as to realize quantum key distribution schemes, a certain set of bases is necessary. A type of set which is minimal is given by a complete set of mutually unbiased bases. The construction of these sets is discussed in the first part of this work. We present complete sets of mutually unbiased bases which are equipped with the additional property to be constructed cyclically, which means, each basis in the set is the power of a specific generating basis of the set. Whereas complete sets of mutually unbiased bases are related to many mathematical problems, it is shown that a new construction of cyclic sets is related to Fibonacci polynomials. Within this context, the existence of a symmetric companion matrix over the finite field F_2 is conjectured. For all Hilbert spaces which have a finite dimension that is a power of two (d=2^m), the cyclic sets can be generated explicitely with the discussed methods. Results for m={1,..,600} are given. A generalization of this construction is able to generate sets with different entanglement structures. It is shown that for dimensions d=2^(2^k) with k being a positive integer, a recursive construction of complete sets exists at least for k in {0,..,11}, where for higher dimensions a direct connection to an open conjecture in finite field theory by Wiedemann is identified. All discussed sets can be implemented directly into a quantum circuit by an invented algorithm. The (unitary) equivalence of the considered sets is discussed in detail. In the second part of this work the security of a quantum public-key encryption protocol is discussed, which was recently published by Nikolopoulos, where the information of all published keys is taken into account. Lower bounds on two different security parameters are given and an attack on single qubits is introduced which is asymptotically equivalent to the optimal attack. Finally, a generalization of this protocol is given that permits a noisy-preprocessing step and leads to a higher security against the presented attack for two leaked copies of the public key and to first results for a non-optimal implementation of the original protocol

Construction of mutually unbiased bases with cyclic symmetry for qubit systems

For the complete estimation of arbitrary unknown quantum states by measurements, the use of mutually unbiased bases has been well-established in theory and experiment for the past 20 years. However, most constructions of these bases make heavy use of abstract algebra and the mathematical theory of finite rings and fields, and no simple and generally accessible construction is available. This is particularly true in the case of a system composed of several qubits, which is arguably the most important case in quantum information science and quantum computation. In this paper, we close this gap by providing a simple and straightforward method for the construction of mutually unbiased bases in the case of a qubit register. We show that our construction is also accessible to experiments, since only Hadamard and controlled-phase gates are needed, which are available in most practical realizations of a quantum computer. Moreover, our scheme possesses the optimal scaling possible, i.e., the number of gates scales only linearly in the number of qubits.Comment: 4 pages, 1 figure, minor correction

Cyclic mutually unbiased bases, Fibonacci polynomials and Wiedemann's conjecture

We relate the construction of a complete set of cyclic mutually unbiased bases, i. e., mutually unbiased bases generated by a single unitary operator, in power-of-two dimensions to the problem of finding a symmetric matrix over F_2 with an irreducible characteristic polynomial that has a given Fibonacci index. For dimensions of the form 2^(2^k) we present a solution that shows an analogy to an open conjecture of Wiedemann in finite field theory. Finally, we discuss the equivalence of mutually unbiased bases.Comment: 11 pages, added chapter on equivalenc

Exhaled breath condensate acidification in acute lung injury

AbstractLung injury in ventilated lungs may occur due to local or systemic disease and is usually caused by or accompanied by inflammatory processes. Recently, acidification of exhaled breath condensate pH (EBC-pH) has been suggested as marker of inflammation in airway disease. We investigated pH, ammonia, lactate, pCO2, HCO3−, IL-6 and IL-8 in EBC of 35 ventilated patients (AECC-classification: ARDS: 15, ALI: 12, no lung injury: 8).EBC-pH was decreased in ventilated patients compared to volunteers (5.85±0.32 vs. 7.46±0.48; P<0.0001). NH4+, lactate, HCO3−, pCO2, IL-6 and IL-8 were analyzed in EBC and correlated with EBC-pH. We observed correlations of EBC-pH with markers of local (EBC IL-6: r=−0.71, P<0.0001, EBC IL-8: r=−0.68, P<0.0001) but not of systemic inflammation (serum IL-6, serum IL-8) and with indices of severity of lung injury (Murray's Lung Injury Severity Score; r=−0.73, P<0.0001, paO2/FiO2; r=0.54, P<0.001). Among factors potentially contributing to pH of EBC, EBC-lactate and EBC-NH4+ were found to correlate with EBC-pH.Inflammation-induced disturbances of regulatory mechanisms, such as glutaminase systems may result in EBC acidification. EBC-pH is suggested to represent a marker of acute lung injury caused by or accompanied by pulmonary inflammation