Codes for Simultaneous Transmission of Quantum and Classical Information

Abstract: 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.