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2. Decoherence-Insensitive Quantum Communication by Optimal C*-Encoding.
- Author
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Bodmann, Bernhard G., Kribs, David W., and Paulsen, Vern I.
- Subjects
- *
MATHEMATICS , *NOISE , *ALGEBRA , *MATHEMATICAL analysis , *QUANTUM communication , *OPTICAL communications , *TELECOMMUNICATION , *BROADBAND communication systems , *ALGORITHMS - Abstract
The central issue in this paper is to transmit a quantum state in such a way that after some decoherence occurs, most of the information can be restored by a suitable decoding operation. For this purpose, we incorporate redundancy by mapping a given initial quantum state to a messenger state on a larger dimensional Hilbert space via a C* -algebra embedding. Our noise model for the transmission is a phase damping channel which admits a noiseless subsystem or decoherence-free subspace. More precisely, the transmission channel is obtained from convex combinations of a set of lowest rank yes/no measurements that leave a component of the messenger state unchanged. The objective of our encoding is to distribute quantum information optimally across the noise-susceptible component of the transmission when the noiseless component is not large enough to contain all the quantum information to be transmitted. We derive simple geometric conditions for optimal encoding and construct examples of such encodings. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
3. A New Attack on the Filter Generator.
- Author
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Rønjom, Sondre and Helleseth, Tor
- Subjects
- *
MATHEMATICS , *NONLINEAR systems , *SYSTEMS theory , *EQUATIONS , *ALGEBRA , *MATHEMATICAL analysis , *ALGORITHMS , *NUMERICAL analysis - Abstract
The filter generator is an important building block in many stream ciphers. The generator consists of a linear feedback shift register of length n that generates an m-sequence of period 2′ - 1 filtered through a Boolean function of degree d that combines bits from the shift register and creates an output bit zt at any time t. The previous best attacks aimed at reconstructing the initial state from an observed keystream, have essentially reduced the problem to solving a nonlinear system of D = (Multiple line equation(s) cannot be represented in ASCII text) (i) equations in n unknowns using techniques based on linear algebra. This attack needs about D bits of keystream and the system can be solved in complexity O (Dω), where ω can be taken to be Strassen's reduction exponent ω = log2 (7) ≈ 2.807. This paper describes a new algorithm that recovers the initial state of most filter generators after observing O(D) keystream bits with complexity O((D - n)/2) ≈ O(D), after a pre-computation with complexity O(D(log2 D)³). [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
4. The Existence of Quantum Entanglement Catalysts.
- Author
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Xiaoming Sun, Runyao Duan, and Mingsheng Ying
- Subjects
- *
CATALYSTS , *CHEMICAL inhibitors , *QUANTUM theory , *MATHEMATICS , *ALGORITHMS , *ALGEBRA , *FOUNDATIONS of arithmetic - Abstract
Without additional resources, it is often impossible to transform one entangled quantum state into another with local quantum operations and classical communication. Jonathan and Plenio (Phys. Rev. Left., vol. 83, p. 3566, 1999) presented an interesting example showing that the presence of another state, called a catalyst, enables such a transformation without changing the catalyst. They also pointed out that in general it is very hard to find an analytical condition under which a catalyst exists. In this paper, we study the existence of catalysts for two incomparable quantum states. For the simplest case of 2 × 2 catalysts for transformations from one 4 × 4 state to another, a necessary and sufficient condition for existence is found. For the general case, we give an efficient polynomial time algorithm to decide whether a k × k catalyst exists for two n × n incomparable states, where k is treated as a constant. [ABSTRACT FROM AUTHOR]
- Published
- 2005
- Full Text
- View/download PDF
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