QIC Abstracts

 Vol.3 No.4  July 1, 2003
Infinitely entangled states (pp281-306)
        M. Keyl, D. Schlingemann and R.F. Werner
For states in infinite dimensional Hilbert spaces entanglement quantities like the entanglement of distillation can become infinite. This leads naturally to the question, whether one system in such an infinitely entangled state can serve as a resource for tasks like the teleportation of arbitrarily many qubits. We show that appropriate states cannot be obtained by density operators in an infinite dimensional Hilbert space. However, using techniques for the description of infinitely many degrees of freedom from field theory and statistical mechanics, such states can nevertheless be constructed rigorously. We explore two related possibilities, namely an extended notion of algebras of observables, and the use of singular states on the algebra of bounded operators. As applications we construct the essentially unique infinite analogue of maximally entangled states, and the singular state used heuristically in the fundamental paper of Einstein, Rosen and Podolsky.

Non-empty quantum dot as a spin-entangler (pp307-316)
        C.-L. Chou
We consider a three-port single-level quantum dot system with one input and two output leads. Instead of considering an empty dot, we study the situations that two input electrons co-tunnel through the quantum dot occupied by one or two dot electrons. We show that electron entanglement can be generated via the co-tunneling processes when the dot is occupied by two electrons, yielding non-local spin-singlet states at the output leads. When the dot is occupied by a single electron, we show that by carefully selecting model parameters non-local spin-triplet electrons can also be obtained at the output leads if the final dot electron has the same spin as that of the initial dot electron.

Shor's discrete logarithm quantum algorithm for elliptic curves (pp317-344)
        J. Proos and Ch. Zalka
We show in some detail how to implement Shor's efficient quantum algorithm for discrete logarithms for the particular case of elliptic curve groups. It turns out that for this problem a smaller quantum computer can solve problems further beyond current computing than for integer factorisation. A 160 bit elliptic curve cryptographic key could be broken on a quantum computer using around 1000 qubits while factoring the security-wise equivalent 1024 bit RSA modulus would require about 2000 qubits. In this paper we only consider elliptic curves over GF(p) and not yet the equally important ones over GF(2^n) or other finite fields. The main technical difficulty is to implement Euclid's gcd algorithm to compute multiplicative inverses modulo p. As the runtime of Euclid's algorithm depends on the input, one difficulty encountered is the ``quantum halting problem''.

Tight Bell inequality for d-outcome measurements correlations (pp345-358)
        Ll. Masanes 
In this paper we prove that the inequality introduced by Collins, Gisin, Linden, Massar and Popescu is tight, or in other words, it is a facet of the convex polytope generated by all local-realistic joint probabilities of d-outcomes. This means that this inequality is optimal. We also show that, for correlation functions generalized to deal with three-outcome measurements, the satisfyability of this inequality is a necessary and sufficient condition for the existence of a local-realistic model accounting for them.

Universal compression of ergodic quantum sources (pp359-375)
        A Kaltchenko and E-H Yang

For a real number r>0, let F(r) be the family of all stationary ergodic quantum sources with von Neumann entropy rates less than r. We prove that, for any r>0, there exists a blind, source-independent block compression scheme which compresses every source from F(r) to rn qubits per input block length~n with arbitrarily high fidelity for all large n.}As our second result, we show that the stationarity and the ergodicity of a quantum source \rho_m_{m=1}^{\infty} are preserved by any trace-preserving completely positive linear map of the tensor product form {\cal E}^{\otimes m}, where a copy of {\cal E} acts locally on each spin lattice site. We also establish ergodicity criteria for so called classically-correlated quantum sources. 

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