Codes for Simultaneous Transmission of Quantum and Classical Information

We consider the characterization as well as the construction of quantum codes that allow to transmit both quantum and classical information, which we refer to as `hybrid codes'. We construct hybrid codes $[\![n,k{: }m,d]\!]_q$ with length $n$ and distance $d$, that simultaneously transmit $k$ qudits and $m$ symbols from a classical alphabet of size $q$. Many good codes such as $[\![7,1{: }1,3]\!]_2$, $[\![9,2{: }2,3]\!]_2$, $[\![10,3{: }2,3]\!]_2$, $[\![11,4{: }2,3]\!]_2$, $[\![11,1{: }2,4]\!]_2$, $[\![13,1{: }4,4]\!]_2$, $[\![13,1{: }1,5]\!]_2$, $[\![14,1{: }2,5]\!]_2$, $[\![15,1{: }3,5]\!]_2$, $[\![19,9{: }1,4]\!]_2$, $[\![20,9{: }2,4]\!]_2$, $[\![21,9{: }3,4]\!]_2$, $[\![22,9{: }4,4]\!]_2$ have been found. All these codes have better parameters than hybrid codes obtained from the best known stabilizer quantum codes.Comment: 6 page

Existence of SRB Measures for A Class of Partially Hyperbolic Attractors in Banach spaces

In this paper, we study the existence of SRB measures for infinite dimensional dynamical systems in a Banach space. We show that if the system has a partially hyperbolic attractor with nontrivial finite dimensional unstable directions, then it has an SRB measure.Comment: arXiv admin note: text overlap with arXiv:1508.0330

SRB Measures for A Class of Partially Hyperbolic Attractors in Hilbert spaces

In this paper, we study the existence of SRB measures and their properties for infinite dimensional dynamical systems in a Hilbert space. We show several results including (i) if the system has a partially hyperbolic attractor with nontrivial finite dimensional unstable directions, then it has at least one SRB measure; (ii) if the attractor is uniformly hyperbolic and the system is topological mixing and the splitting is H\"older continuous, then there exists a unique SRB measure which is mixing; (iii) if the attractor is uniformly hyperbolic and the system is non-wondering and and the splitting is H\"older continuous, then there exists at most finitely many SRB measures; (iv) for a given hyperbolic measure, there exist at most countably many ergodic components whose basin contains an observable set