Comment on "Semiquantum-key distribution using less than four quantum states"

Comment on Phys. Rev. A 79, 052312 (2009), http://pra.aps.org/abstract/PRA/v79/i5/e05231

On the Tits alternative for PD(3)PD(3) groups

We prove the Tits alternative for an almost coherent $PD(3)$ group which is not virtually properly locally cyclic. In particular, we show that an almost coherent $PD(3)$ group which cannot be generated by fewer than four elements always contains a rank 2 free group.Comment: 16 pages, minor corrections, to appear in Ann. Fac. Sci. Toulouse Mat

Security Against Collective Attacks of a Modified BB84 QKD Protocol with Information only in One Basis

The Quantum Key Distribution (QKD) protocol BB84 has been proven secure against several important types of attacks: the collective attacks and the joint attacks. Here we analyze the security of a modified BB84 protocol, for which information is sent only in the z basis while testing is done in both the z and the x bases, against collective attacks. The proof follows the framework of a previous paper (Boyer, Gelles, and Mor, 2009), but it avoids the classical information-theoretical analysis that caused problems with composability. We show that this modified BB84 protocol is as secure against collective attacks as the original BB84 protocol, and that it requires more bits for testing.Comment: 6 pages; 1 figur

Roots of torsion polynomials and dominations

We show that the nonzero roots of the torsion polynomials associated to the infinite cyclic covers of a given compact, connected, orientable 3-manifold M are contained in a compact part of the complex plane a priori determined by M. This result is applied to prove that when M is closed, it dominates at most finitely many Sol manifolds.Comment: This is the version published by Geometry & Topology Monographs on 29 April 200

Attacks against a Simplified Experimentally Feasible Semiquantum Key Distribution Protocol

A semiquantum key distribution (SQKD) protocol makes it possible for a quantum party and a classical party to generate a secret shared key. However, many existing SQKD protocols are not experimentally feasible in a secure way using current technology. An experimentally feasible SQKD protocol, "classical Alice with a controllable mirror" (the "Mirror protocol"), has recently been presented and proved completely robust, but it is more complicated than other SQKD protocols. Here we prove a simpler variant of the Mirror protocol (the "simplified Mirror protocol") to be completely non-robust by presenting two possible attacks against it. Our results show that the complexity of the Mirror protocol is at least partly necessary for achieving robustness.Comment: 9 page

On definite strongly quasipositive links and L-space branched covers

We investigate the problem of characterising the family of strongly quasipositive links which have definite symmetrised Seifert forms and apply our results to the problem of determining when such a link can have an L-space cyclic branched cover. In particular, we show that if $\delta_n = \sigma_1 \sigma_2 \ldots \sigma_{n-1}$ is the dual Garside element and $b = \delta_n^k P \in B_n$ is a strongly quasipositive braid whose braid closure $\widehat b$ is definite, then $k \geq 2$ implies that $\widehat b$ is one of the torus links $T(2, q), T(3,4), T(3,5)$ or pretzel links $P(-2, 2, m), P(-2,3,4)$. Applying Theorem 1.1 of our previous paper we deduce that if one of the standard cyclic branched covers of $\widehat b$ is an L-space, then $\widehat b$ is one of these links. We show by example that there are strongly quasipositive braids $\delta_n P$ whose closures are definite but not one of these torus or pretzel links. We also determine the family of definite strongly quasipositive $3$-braids and show that their closures coincide with the family of strongly quasipositive $3$-braids with an L-space branched cover.Comment: 62 pages, minor revisions, accepted for publication in Adv. Mat

Branched covers of quasipositive links and L-spaces

Let $L$ be a oriented link such that $\Sigma_n(L)$, the $n$-fold cyclic cover of $S^3$ branched over $L$, is an L-space for some $n \geq 2$. We show that if either $L$ is a strongly quasipositive link other than one with Alexander polynomial a multiple of $(t-1)^{2g(L) + (|L|-1)}$, or $L$ is a quasipositive link other than one with Alexander polynomial divisible by $(t-1)^{2g_4(L) + (|L|-1)}$, then there is an integer $n(L)$, determined by the Alexander polynomial of $L$ in the first case and the Alexander polynomial of $L$ and the smooth $4$-genus of $L$, $g_4(L)$, in the second, such that $n \leq n(L)$. If $K$ is a strongly quasipositive knot with monic Alexander polynomial such as an L-space knot, we show that $\Sigma_n(K)$ is not an L-space for $n \geq 6$, and that the Alexander polynomial of $K$ is a non-trivial product of cyclotomic polynomials if $\Sigma_n(K)$ is an L-space for some $n = 2, 3, 4, 5$. Our results allow us to calculate the smooth and topological 4-ball genera of, for instance, quasi-alternating quasipositive links. They also allow us to classify strongly quasipositive alternating links and $3$-strand pretzel links.Comment: 49 pages, 7 figures, minor corrections and improved exposition, accepted for publication by the Journal of Topolog