By Topic

• Abstract

The mathematical formalism that stands behind the information theoretic description of a quantum channel represent a flexible framework, which allows for one to study the quantum information conveying capabilities of noisy quantum links. In this work we reveal that more efficient quantum communication is possible over a PD (partially degradable [13], [14]) channel in comparison to standard degradable quantum channels.

The definition of conjugate degradable quantum channels was introduced by Bradler et al. [1]. They showed that for a conjugate degradable channel the complex conjugated environment state ${E}^{\prime}$ can be simulated from the channel output $B$. In their examples [1], the connection between environment states $E$ and ${E}^{\prime}$ was a complex conjugation. For a PD channel the connection between $E$ and ${E}^{\prime}$ is a degrading CPTP (Completely Positive Trace Preserving) map [13]. Currently, we have no result for the rates of quantum communication over a PD channel that can be achieved by polar coding. The polar coding [4] has been already studied in the quantum setting, and results have been obtained on the rates of classical and quantum communication over degradable and anti-degradable channels [6]– [7][8], [12], [13]. On the other hand, up to this date the possibilities of partial degradability are still unrevealed. Here we show that if a quantum channel is both degradable and has the PD property, then polar codes can produce better performance in comparison to degradable channels. As we have found, the rate of quantum communication with polar codes can be exceeded in comparison to the currently known results on degradable channels [7], [8]. By exploiting the PD property, quantum communication will be possible in a regime nearer to the channel's quantum capacity [9]– [10][11] in comparison to polar codes that were developed for degradable channels.

This paper is organized as follows. First, we define the codeword sets for a PD channel, and then we give the rates of the quantum communication. In the Supplemental Material, we study the performance of the code.

For a PD channel, we use the following description (see Fig. 1). The channel between Alice and Bob is denoted by ${\cal N}_{AB}$. The input and output of ${\cal N}_{AB}$ are denoted by $A$ and $B$. Since the channel is degradable, Bob can simulate the environment's channel ${\cal N}_{A{E}^{\prime}}$ by applying the degrading map ${\cal D}^{B\to{E}^{\prime}}$ on $B$, ${\cal N}_{A{E}^{\prime}}={\cal D}^{B\to{E}^{\prime}}\circ{\cal N}_{AB}$. For the error-probabilities of a degradable channel ${\cal N}_{AB}$, the relation $p_{A{E}^{\prime}}\geq p_{AB}$ holds [1]– [2][3], [5]. Since the channel is PD, the output $E$ of the channel between Alice and the environment, called the complementary channel ${\cal N}_{AE}$, can be used to simulate the degraded environment state ${E}^{\prime}$. Applying the degrading map ${\cal D}^{E\to{E}^{\prime}}$ on $E$, the result is ${E}^{\prime}$. From the PD property, an important corollary follows: the output $E$ of the complementary channel ${\cal N}_{AE}$ is not equivalent to the output ${E}^{\prime}$ of channel ${\cal N}_{A{E}^{\prime}}$, where ${\cal N}_{A{E}^{\prime}}={\cal N}_{AE}\circ{\cal D}^{E\to{E}^{\prime}}$.

Fig. 1. A PD channel. The environment state $E$ is outputted by the complementary channel ${\cal N}_{AE}$. The degraded environment state will be referred to as ${E}^{\prime}$, which is the output of channel ${\cal N}_{A{E}^{\prime}}={\cal N}_{AE}\circ{\cal D}^{E\to{E}^{\prime}}$, where ${\cal D}^{E\to{E}^{\prime}}$ is the degradation map. The environment state ${E}^{\prime}$ contains less valuable information than state $E$, which makes it possible to use a PD channel with higher quantum communication rates in comparison to a degradable channel.

The polar coding scheme investigated for a PD channel ${\cal N}$, follows the basic polar coding scheme of [8] for degradable channels. The encoding of quantum information will use the amplitude and phase information. The amplitude “good” and “bad” codewords are depicted by ${\cal G}\left({{\cal N}_{amp},\beta}\right)$, ${\cal B}\left({{\cal N}_{amp},\beta}\right)$, and similarly for the phase encoding ${\cal G}\left({{\cal N}_{phase},\beta}\right)$, ${\cal B}\left({{\cal N}_{phase},\beta}\right)$. From these sets further polar codeword sets can be constructed. The density matrices of the frozen bits $\varsigma_{A_{1}},\ldots,\varsigma_{A_{n-l}}$ belong to the set TeX Source$${\cal G}\left({{\cal N}_{phase},\beta}\right)\cap{\cal B}\left({{\cal N}_{ampl},\beta}\right)\eqno{\hbox{(1)}}$$ or TeX Source$${\cal G}\left({{\cal N}_{ampl},\beta}\right)\cap{\cal B}\left({{\cal N}_{phase},\beta}\right).\eqno{\hbox{(2)}}$$ The frozen bits from the set of TeX Source$${\cal B}\left({{\cal N}_{ampl},\beta}\right)\cap{\cal B}\left({{\cal N}_{phase},\beta}\right)\eqno{\hbox{(3)}}$$ require pre-shared entanglement. We extend the results on a degradable channel for a PD channel.

#### Theorem 1

The frozen bits of the polar encoder ${\cal E}$ for a PD channel do not require pre-shared entanglement.

#### Proof

The proof trivially follows from the previously derived results in [8]. Since our channel ${\cal N}$ is degraded, the rate of entanglement consumption between encoder ${\cal E}$ and Bob's decoder ${\cal D}$ is zero [8], i.e., $\lim\limits_{n\to\infty}{{1}\over{n}}\left\vert{\cal B}\right\vert=0$.

#### Corollary 1. (On the Net Rate for a PD Channel)

Since for a PD channel the rate of entanglement consumption is zero, the net rate will be equal to the rate of quantum communication $R_{Q}\left({\cal N}\right)$.

#### Polar Code Sets for a PD Channel

To derive the $R_{Q}\left({\cal N}\right)$ rate of quantum communication for a PD channel ${\cal N}$, we define two sets that are “good” for a PD channel but “bad” for a degradable channel, ${\cal P}^{\prime}_{1}$ and ${\cal P}_{2}^{\prime}$ as follows:TeX Source$${\cal P}^{\prime}_{1}\subseteq{\cal P}_{1},\quad{\cal P}^{\prime}_{2}\subseteq{\cal P}_{2}\eqno{\hbox{(4)}}$$ and the amplitude and phase frozen sets for a PD channel as TeX Source$${\cal P}_{1}\backslash{\cal P}_{1}^{\prime},\quad{\cal P}_{2}\backslash{\cal P}_{2}^{\prime},\eqno{\hbox{(5)}}$$ where TeX Source\eqalignno{{\cal P}_{1}=&\,\left({{\cal G}\left({{\cal N}_{amp},\beta}\right)\cap{\cal B}\left({{\cal N}_{phase},\beta}\right)}\right)^{degr},&{\hbox{(6)}}\cr{\cal P}_{2}=&\,\left({{\cal B}\left({{\cal N}_{amp},\beta}\right)\cap{\cal G}\left({{\cal N}_{phase},\beta}\right)}\right)^{degr}&{\hbox{(7)}}} are the amplitude or phase frozen sets for a degradable channel, and $\left\vert{S_{bad}}\right\vert=n-\left({m+\Delta}\right)=n-m-\left\vert{{\cal P}_{1}^{\prime}}\right\vert$, where TeX Source$$\Delta=\left\vert{{\cal P}_{1}}\right\vert-\left\vert{{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right\vert=\left\vert{{\cal P}_{1}^{\prime}}\right\vert,\eqno{\hbox{(8)}}$$ along with TeX Source$$R_{Q}\left({{\cal N}_{AE}}\right)-R_{Q}\left({{\cal N}_{A{E}^{\prime}}}\right)=\lim\limits_{n\to\infty}{{1}\over{n}}\left({\left\vert{{\cal P}_{1}^{\prime}}\right\vert}\right).\eqno{\hbox{(9)}}$$ From the quantity of (8) trivially follows, that if ${\cal D}^{E\to{E}^{\prime}}$ is equal to a complex conjugation ${\cal C}$, then $\Delta=0$; for this case the performance of the code is equivalent to the codes that of degradable channels.

For a PD channel with degrading map ${\cal D}^{E\to{E}^{\prime}}$ it results in a new “good”, denoted by $S_{in}^{conj.PD}$, and defined as:TeX Source\eqalignno{S_{in}^{conj.PD}=&\, S_{in}^{degr}{\cal P}\cup{\cal P}_{\infty}^{\prime}\cr=&\,\left({{\cal G}\left({{\cal N}_{amp},\beta}\right)\cap{\cal G}\left({{\cal N}_{phase},\beta}\right)}\right)^{conj.PD},\qquad&{\hbox{(10)}}} with TeX Source$$\left\vert{S_{in}^{conj.PD}}\right\vert=\left\vert{S_{in}^{degr}}\right\vert+\Delta=m+\Delta,\eqno{\hbox{(11)}}$$ where $m<m+\Delta$, and $\left\vert{S_{in}^{conj.PD}}\right\vert+\left\vert{S_{bad}}\right\vert=n=2^{k}$, and $k$ is the level of the polar structure [3]. In comparison to a degradable channel, for a PD channel, the density matrices of the frozen bits $\varsigma_{A_{1}},\ldots,\varsigma_{A_{n-l}}$ will be selected from the sets ${\cal P}_{1}\backslash{\cal P}_{1}^{\prime}$ and ${\cal P}_{2}\backslash{\cal P}_{2}^{\prime}$, instead of ${\cal P}_{1}$ and ${\cal P}_{2}$. The bad set is defined as in the case of degradable channels ${\cal B}={\cal B}\left({{\cal N}_{amp},\beta}\right)\cap{\cal B}\left({{\cal N}_{phase},\beta}\right)$. Set ${\cal B}$ will not be used as frozen bits, since the channel is degradable.

#### Theorem 2. (On the Rates of Polar Coding for a PD Channel)

For the non-empty set ${\cal P}_{1}^{\prime}$, $\left\vert{{\cal P}_{1}^{\prime}}\right\vert>0$, the $R_{Q}$ rate of quantum communication for a PD channel with set $S_{in}^{conj.PD}$ is higher than the rate of quantum communication that can be obtained for a degradable channel with $S_{in}^{degr}$.

#### Proof

The defined sets ${\cal P}_{1}^{\prime}$ and ${\cal P}_{2}^{\prime}$ for a PD channel are also disjoint [7], [8], [12], thus TeX Source$$\left\vert{{\cal P}_{1}^{\prime}\cup{\cal P}_{2}^{\prime}}\right\vert=\left\vert{{\cal P}_{1}^{\prime}}\right\vert+\left\vert{{\cal P}^{\prime}_{2}}\right\vert,\eqno{\hbox{(12)}}$$ and since the elements of set ${\cal P}_{2}$ are not transmitted over the channel (Bob synthesizes ${\cal P}_{2}$ from the corresponding subset $\Omega_{{\cal P}_{1}}$ of ${\cal P}_{1}$ by using Hadamard operations in the decoding process):TeX Source$$\lim\limits_{n\to\infty}{{1}\over{n}}\left\vert{\left({{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\right)}\right\vert=0\eqno{\hbox{(13)}}$$ and TeX Source$$\lim\limits_{n\to\infty}{{1}\over{n}}\left\vert{{\cal P}^{\prime}_{2}}\right\vert=0,\eqno{\hbox{(14)}}$$ along with relations $\left\vert{\left({{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\right)\cap{\cal G}\left({{\cal N}_{amp},\beta}\right)}\right\vert=0$, $\left\vert{{\cal P}^{\prime}_{2}\cap{\cal G}\left({{\cal N}_{amp},\beta}\right)}\right\vert=0$, $\left\vert{{\cal B}\left({{\cal N}_{amp},\beta}\right)\cap{\cal G}\left({{\cal N}_{amp},\beta}\right)}\right\vert=0$, which follows from the fact that ${\cal P}^{\prime}_{2}\subseteq{\cal P}_{2}$ and TeX Source$$\left({{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\right)\subseteq{\cal B}\left({{\cal N}_{amp},\beta}\right),\eqno{\hbox{(15)}}$$ where TeX Source$${\cal B}\left({{\cal N}_{amp},\beta}\right)=\left[n\right]\backslash{\cal G}\left({{\cal N}_{amp},\beta}\right)\eqno{\hbox{(16)}}$$ and TeX Source\eqalignno{&\left\vert{\left({{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\right)\cap\left({S_{in}^{conj.PD}\cup{\cal B}}\right)}\right\vert=0,&{\hbox{(17)}}\cr &{\cal P}^{\prime}_{1}\cup{\cal P}^{\prime}_{2}\subseteq{\cal G}\left({{\cal N}_{amp},\beta}\right),&{\hbox{(18)}}} and TeX Source$$\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)\cup\left({{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\right)\subseteq{\cal G}\left({{\cal N}_{amp},\beta}\right),\eqno{\hbox{(19)}}$$ see (13) and (14). For a PD channel ${\cal P}^{\prime}_{1}\subseteq{\cal G}\left({{\cal N}_{amp},\beta}\right)$ and $\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)\subseteq{\cal G}\left({{\cal N}_{amp},\beta}\right)$, which demonstrates that the defined codewords sets ${\cal P}_{1}^{\prime}$, ${\cal P}^{\prime}_{2}$, ${\cal P}_{1}\backslash{\cal P}_{1}^{\prime}$ and ${\cal P}_{2}\backslash{\cal P}_{2}^{\prime}$ are pairwise disjoint, where TeX Source\eqalignno{&\null\lim\limits_{n\to\infty}{{1}\over{n}}\left\vert{{\cal G}\left({{\cal N}_{amp},\beta}\right)}\right\vert\cr &~~+\lim\limits_{n\to\infty}{{1}\over{n}}\left\vert{\left[n\right]\backslash\left({\left({{\cal P}^{\prime}_{1}\cup{\cal P}^{\prime}_{2}}\right)\cup\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)\cup\left({{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\right)}\right)}\right\vert=1, &{\hbox{(20)}}} and TeX Source\eqalignno{&\left\vert{{\cal G}\left({{\cal N}_{amp},\beta}\right)\backslash{\cal G}\left({{\cal N}_{amp},\beta}\right)\cap{\cal G}\left({{\cal N}_{phs},\beta}\right)}\right\vert+\left\vert{\left({{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\right)}\right\vert\cr &\quad~~+\left\vert{\left[n\right]\backslash\left({\left({{\cal P}^{\prime}_{1}\cup{\cal P}^{\prime}_{2}}\right)\cup\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)\cup\left({{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\right)}\right)}\right\vert\leq n&{\hbox{(21)}}} with TeX Source\eqalignno{&\left({\left[n\right]\backslash\left({{\cal P}^{\prime}_{1}}\right)}\right)\subseteq\left({{\cal G}\left({{\cal N}_{amp},\beta}\right)\cap{\cal G}\left({{\cal N}_{phase},\beta}\right)}\right)^{conj.PD}\cr &\hskip9.8em\cup\left({\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)}\right).&{\hbox{(22)}}} Using the codeword construction defined for the PD channel, the $R_{Q}\left({\cal N}\right)$ rate can be achieved over the channel ${\cal N}_{AB}$ can be expressed as:TeX Source\eqalignno{& R_{Q}\left({\cal N}\right)=\lim\limits_{n\to\infty}{{1}\over{n}}\left({\left\vert{S_{in}^{degr}\cup{\cal P}^{\prime}_{1}\cup{\cal P}^{\prime}_{2}}\right\vert}\right)\cr &\quad=\lim\limits_{n\to\infty}{{1}\over{n}}\left\vert{\left({{\cal G}\left({{\cal N}_{ampl},\beta}\right)\cap{\cal G}\left({{\cal N}_{phs},\beta}\right)}\right)^{degr}\cup{\cal P}^{\prime}_{1}\cup{\cal P}^{\prime}_{2}}\right\vert.} Assuming $\beta<0.5$, the following relation holds for the Holevo information of channel ${\cal N}_{AB}$ and ${\cal N}_{A{E}^{\prime}}$ of ${\cal N}$:TeX Source\eqalignno{\chi\left({{\cal N}_{AB}}\right)=&\,\lim\limits_{n\to\infty}{{1}\over{n}}\left\vert{{\cal G}\left({{\cal N}_{amp},\beta}\right)\cup{{{\cal P}^{\prime}}}_{2}}\right\vert,&{\hbox{(23)}}\cr\chi\left({{\cal N}_{AE}}\right)=&\,\lim\limits_{n\to\infty}{{1}\over{n}}\left({\left\vert{{\cal P}_{1}}\right\vert+\left\vert{{\cal P}_{2}}\right\vert}\right)&{\hbox{(24)}}\cr\chi\left({{\cal N}_{A{E}^{\prime}}}\right)=&\,\lim\limits_{n\to\infty}{{1}\over{n}}\left({\left\vert{\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)}\right\vert+\left\vert{\left({{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\right)}\right\vert}\right),&{\hbox{(25)}}} where $\chi\left({{\cal N}_{AB}}\right)$, $\chi\left({{\cal N}_{AE}}\right)$ and $\chi\left({{\cal N}_{A{E}^{\prime}}}\right)$ are the Holevo information of the channels ${\cal N}_{AB}$, ${\cal N}_{AE}$ and ${\cal N}_{A{E}^{\prime}}$ of ${\cal N}$. Combing this result with (13) and (14), one will get TeX Source$$\chi\left({{\cal N}_{A{E}^{\prime}}}\right)=\lim\limits_{n\to\infty}{{1}\over{n}}\left({\left\vert{\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)}\right\vert}\right)\eqno{\hbox{(26)}}$$ and the rate of quantum communication is (Note: no maximization and regularization needed in the first line of(27), since in terms of polar coding the quantum capacity is symmetric and it is additive for any degradable channel [1], [5], [6], [8]) TeX Source\eqalignno{R_{Q}\left({\cal N}\right)=&\,\chi\left({{\cal N}_{AB}}\right)-\chi\left({{\cal N}_{A{E}^{\prime}}}\right)\cr\noalign{\vskip-3pt}=&\,\lim\limits_{n\to\infty}{{1}\over{n}}\left({\left\vert{{\cal G}\left({{\cal N}_{amp},\beta}\right)}\right\vert-\left\vert{\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)}\right\vert}\right).&{\hbox{(27)}}} The result obtained in (27) can be rewritten as follows:TeX Source$$R_{Q}\left({\cal N}\right)\!=\!\lim\limits_{n\to\infty}{{1}\over{n}}\left({\left\vert{S_{in}^{degr}}\right\vert\!+\!\left\vert{{\cal P}_{1}^{\prime}}\right\vert}\right)\!=\!\lim\limits_{n\to\infty}{{1}\over{n}}\!\left({\left\vert{S_{in}^{conj.PD}}\right\vert}\right).\eqno{\hbox{(28)}}$$ From the polar encoding scheme, it follows that for $\beta<0.5$:TeX Source$$\sqrt{F\left({S_{in}^{conj.PD}\cup\left({{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\right)}\right)}<2^{-n^{\beta}},\eqno{\hbox{(29)}}$$ and for the fidelity parameters of ${\cal P}_{1}$:TeX Source$$\sqrt{F\left({\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)}\right)}\geq 1-2^{-n^{\beta}}.\eqno{\hbox{(30)}}$$ If (29) and (30) hold, then $\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)\cap{\cal B}\ne\emptyset$. Using the defined sets it follows that TeX Source\eqalignno{&\left({{\cal B}\left({{\cal N}_{amp},\beta}\right)\cap{\cal B}\left({{\cal N}_{phase},\beta}\right)}\right)\cr\noalign{\vskip-3pt}&\subseteq\left({S_{in}^{conj.PD}\cup\left({{\cal B}\left({{\cal N}_{amp},\beta}\right)\cap{\cal B}\left({{\cal N}_{phase},\beta}\right)}\right)}\right)=\emptyset.&{\hbox{(31)}}} After some steps of reordering, we get that TeX Source\eqalignno{&\left({\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)}\right)\cap\left({{\cal B}\left({{\cal N}_{amp},\beta}\right)\cap{\cal B}\left({{\cal N}_{phase},\beta}\right)}\right)\cr\noalign{\vskip-3pt}&\subseteq\left({\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)}\right)\cap\left({S_{in}^{conj.PD}\cup\left({\matrix{{\cal B}\left({{\cal N}_{amp},\beta}\right)\hfill\cr\noalign{\vskip-3pt}\cap{\cal B}\left({{\cal N}_{phase},\beta}\right)\hfill}}\right)}\right)=\emptyset.} and TeX Source$$\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)\cap\left({S_{in}^{conj.PD}\cup\left({{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\right)}\right)=\emptyset.\eqno{\hbox{(32)}}$$ Since channel ${\cal N}_{A{E}^{\prime}}$ is degraded [8], TeX Source$$\lim\limits_{n\to\infty}{{1}\over{n}}\left\vert{\cal B}\right\vert=0.\eqno{\hbox{(33)}}$$ These results also mean that the codeword sets $S_{in}^{degr}$, $S_{in}^{conj.PD}$, ${{{\cal P}^{\prime}}}_{1}\subseteq{\cal P}_{1}$, ${{{\cal P}^{\prime}}}_{2}\subseteq{\cal P}_{2}$ and ${\cal B}$ are disjoint sets with relation TeX Source\eqalignno{&\left\vert{S_{in}^{degr}\cup{{{\cal P}^{\prime}}}_{1}\cup{{{\cal P}^{\prime}}}_{2}\cup\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)\cup\left({{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\right)}\right\vert\cr\noalign{\vskip-3pt}&\qquad\qquad\qquad=\left\vert{S_{in}^{conj.PD}\cup\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)\cup\left({{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\right)}\right\vert\cr\noalign{\vskip-3pt}&\qquad\qquad\qquad=\left\vert{S_{in}^{conj.PD}\cup\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)}\right\vert,&{\hbox{(34)}}} hence for a PD channel ${\cal N}$ we get:TeX Source$$R_{Q}\left({\cal N}\right)=\lim\limits_{n\to\infty}{{1}\over{n}}\left({\left\vert{S_{in}^{degr}\cup{{{\cal P}^{\prime}}}_{1}}\right\vert}\right)=\lim\limits_{n\to\infty}{{1}\over{n}}\left({\left\vert{S_{in}^{conj.PD}}\right\vert}\right).$$ It can be rewritten as follows:TeX Source\eqalignno{R_{Q}\left({\cal N}\right)=&\,\lim\limits_{n\to\infty}{{1}\over{n}}\left({\left\vert{S_{in}^{degr}}\right\vert+\left\vert{{\cal P}_{1}^{\prime}}\right\vert}\right)\cr=&\,\lim\limits_{n\to\infty}{{1}\over{n}}\left({\left\vert{{\cal G}\left({{\cal N}_{amp},\beta}\right)}\right\vert-\left\vert{\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)}\right\vert}\right)\cr=&\,\lim\limits_{n\to\infty}{{1}\over{n}}\left({\left\vert{S_{in}^{conj.PD}}\right\vert}\right).&{\hbox{(35)}}} The available codewords for quantum communication over a PD channel ${\cal N}_{AB}$ will be TeX Source$$\left\vert{S_{in}^{conj.PD}}\right\vert=\left\vert{S_{in}^{degr}}\right\vert+\left\vert{{\cal P}_{1}^{\prime}}\right\vert,\eqno{\hbox{(36)}}$$ which concludes our proof. These results show that for the non-empty set ${\cal P}_{1}^{\prime}$, the results on rate of quantum communication $R_{Q}\left({\cal N}\right)$ for a PD channel ${\cal N}$ exceed the result obtained for a degradable channel with $\left\vert{S_{in}^{degr}}\right\vert$.

#### 2. Summary

Since for a degradable channel ${\cal N}$, only set $S_{in}^{degr}=\left({{\cal G}\left({{\cal N}_{ampl},\beta}\right)\cap{\cal G}\left({{\cal N}_{phs},\beta}\right)}\right)^{degr}$ can be used for quantum communication, it will result in quantum data rate TeX Source\eqalignno{R_{Q}\left({\cal N}\right)\!=&\,\!\lim\limits_{n\to\infty}{{1}\over{n}}\left({\left\vert{S_{in}^{degr}}\right\vert}\right)\cr\!=&\,\!\lim\limits_{n\to\infty}{{1}\over{n}}\left\vert{\left({{\cal G}\left({{\cal N}_{amp},\beta}\right)\cap{\cal G}\left({{\cal N}_{phase},\beta}\right)}\right)^{degr}}\right\vert.&{\hbox{(37)}}} The codewords that can transmit amplitude information are depicted by ${\cal G}\left({{\cal N}_{amp},\beta}\right)$. The set that can transmit phase is denoted by ${\cal G}\left({{\cal N}_{phs},\beta}\right)$. For a degradable channel ${\cal N}$ only the set $\left\vert{S_{in}^{degr}}\right\vert=m$ can be used for quantum communication. For a degradable channel ${\cal N}$, valuable information can be leaked only from the set ${\cal P}_{1}$ [7], [8] which results in $\left({S_{in}^{degr}\cup{\cal P}_{1}\cup{\cal P}_{2}}\right)\backslash\left({{\cal P}_{1}\cup{\cal P}_{2}}\right)$, where $\lim\limits_{n\to\infty}{{1}\over{n}}\left\vert{{\cal P}_{2}}\right\vert=0$, i.e., $\left({S_{in}^{degr}\cup{\cal P}_{1}}\right)\backslash\left({{\cal P}_{1}}\right)$. The set ${\cal P}_{1}$ represents the information of channel ${\cal N}_{AE}$, between Alice and the environment. For these sets $R_{Q}\left({{\cal N}_{AE}}\right)>0$ and $R_{Q}\left({{\cal N}_{AB}}\right)=0$. As we have found, this is not the case, if the degradable channel is also partially degradable. While for a degradable channel, only the set $S_{in}^{degr}$ can be used for quantum communication, the situation will change for a PD channel, since this property causes changes in the information of complementary channel. In this case, the achievable codeword set for quantum communication is TeX Source\eqalignno{S_{in}^{conj.PD}=&\, S_{in}^{degr}\cup{{{\cal P}^{\prime}}}_{1}\cup{{{\cal P}^{\prime}}}_{2}\cr=&\,\left({{\cal G}\left({{\cal N}_{ampl},\beta}\right)\cap{\cal G}\left({{\cal N}_{phs},\beta}\right)}\right)^{degr}\cup{{{\cal P}^{\prime}}}_{1}\cup{{{\cal P}^{\prime}}}_{2}\cr=&\,\left({{\cal G}\left({{\cal N}_{ampl},\beta}\right)\cap{\cal G}\left({{\cal N}_{phs},\beta}\right)}\right)^{conj.PD},&{\hbox{(38)}}} where ${{{\cal P}^{\prime}}}_{1}\subseteq{\cal P}_{1}$, ${{{\cal P}^{\prime}}}_{2}\subseteq{\cal P}_{2}$ can be used for quantum communication, which results in quantum data rate TeX Source$$R_{Q}\!\left({\cal N}\right)\!=\!\lim\limits_{n\to\infty}{{1}\over{n}}\!\left(\!{\left\vert{S_{in}^{degr}\cup{{{\cal P}^{\prime}}}_{1}}\right\vert}\!\right)\!=\!\lim\limits_{n\to\infty}{{1}\over{n}}\!\left({\left\vert{S_{in}^{conj.PD}}\right\vert}\right).\eqno{\hbox{(39)}}$$ For a PD channel the extended set $S_{in}^{conj.PD}=S_{in}^{degr}\cup{{{\cal P}^{\prime}}}_{1}\cup{{{\cal P}^{\prime}}}_{2}$ with $\vert{S_{in}^{conj.PD}}\vert=\vert{S_{in}^{degr}}\vert+\Delta=m+\Delta$ will be available for quantum communication, where ${{{\cal P}^{\prime}}}_{1}\subseteq{\cal P}_{1}$, ${{{\cal P}^{\prime}}}_{2}\subseteq{\cal P}_{2}$, and $\lim\limits_{n\to\infty}{{1}\over{n}}\vert{{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\vert=0$ and $\lim\limits_{n\to\infty}{{1}\over{n}}\vert{{{{\cal P}^{\prime}}}_{2}}\vert=0$.

The extended set results in higher quantum communication rates in comparison to a degradable one. The amount of information in the degraded environment state ${E}^{\prime}$ outputted by channel ${\cal N}_{A{E}^{\prime}}={\cal N}_{AE}\circ{\cal D}^{E\to{E}^{\prime}}$ is less than in the environment state $E$, outputted by channel ${\cal N}_{AE}$. This property results in better quantum communication rates over PD quantum channels in comparison to a quantum channels that has no the PD property. The results allow to implement quantum protocols over quantum links with higher performance than it is currently available by the standard quantum codes.

## Footnotes

This work was supported in part by the grant TAMOP-4.2.2.B-10/1-2010-0009 and in part by COST Action under MP1006.

## References

No Data Available

## Cited By

No Data Available

None

## Multimedia

No Data Available
This paper appears in:
No Data Available
Issue Date:
No Data Available
On page(s):
No Data Available
ISSN:
None
INSPEC Accession Number:
None
Digital Object Identifier:
None
Date of Current Version:
No Data Available
Date of Original Publication:
No Data Available

Comment Policy