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

Figure 1
Fig. 1. A PD channel. The environment state Formula$E$ is outputted by the complementary channel Formula${\cal N}_{AE}$. The degraded environment state will be referred to as Formula${E}^{\prime}$, which is the output of channel Formula${\cal N}_{A{E}^{\prime}}={\cal N}_{AE}\circ{\cal D}^{E\to{E}^{\prime}}$, where Formula${\cal D}^{E\to{E}^{\prime}}$ is the degradation map. The environment state Formula${E}^{\prime}$ contains less valuable information than state Formula$E$, 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 Formula${\cal N}$, 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 Formula${\cal G}\left({{\cal N}_{amp},\beta}\right)$, Formula${\cal B}\left({{\cal N}_{amp},\beta}\right)$, and similarly for the phase encoding Formula${\cal G}\left({{\cal N}_{phase},\beta}\right)$, Formula${\cal B}\left({{\cal N}_{phase},\beta}\right)$. From these sets further polar codeword sets can be constructed. The density matrices of the frozen bits Formula$\varsigma_{A_{1}},\ldots,\varsigma_{A_{n-l}}$ belong to the set FormulaTeX Source$${\cal G}\left({{\cal N}_{phase},\beta}\right)\cap{\cal B}\left({{\cal N}_{ampl},\beta}\right)\eqno{\hbox{(1)}}$$ or FormulaTeX 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 FormulaTeX 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 Formula${\cal E}$ 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 Formula${\cal N}$ is degraded, the rate of entanglement consumption between encoder Formula${\cal E}$ and Bob's decoder Formula${\cal D}$ is zero [8], i.e., Formula$\lim\limits_{n\to\infty}{{1}\over{n}}\left\vert{\cal B}\right\vert=0$.

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 Formula$R_{Q}\left({\cal N}\right)$.

Polar Code Sets for a PD Channel

To derive the Formula$R_{Q}\left({\cal N}\right)$ rate of quantum communication for a PD channel Formula${\cal N}$, we define two sets that are “good” for a PD channel but “bad” for a degradable channel, Formula${\cal P}^{\prime}_{1}$ and Formula${\cal P}_{2}^{\prime}$ as follows:FormulaTeX 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 FormulaTeX Source$${\cal P}_{1}\backslash{\cal P}_{1}^{\prime},\quad{\cal P}_{2}\backslash{\cal P}_{2}^{\prime},\eqno{\hbox{(5)}}$$ where FormulaTeX 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 Formula$\left\vert{S_{bad}}\right\vert=n-\left({m+\Delta}\right)=n-m-\left\vert{{\cal P}_{1}^{\prime}}\right\vert$, where FormulaTeX 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 FormulaTeX 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 Formula${\cal D}^{E\to{E}^{\prime}}$ is equal to a complex conjugation Formula${\cal C}$, then Formula$\Delta=0$; for this case the performance of the code is equivalent to the codes that of degradable channels.

For a PD channel with degrading map Formula${\cal D}^{E\to{E}^{\prime}}$ it results in a new “good”, denoted by Formula$S_{in}^{conj.PD}$, and defined as:FormulaTeX 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 FormulaTeX Source$$\left\vert{S_{in}^{conj.PD}}\right\vert=\left\vert{S_{in}^{degr}}\right\vert+\Delta=m+\Delta,\eqno{\hbox{(11)}}$$ where Formula$m<m+\Delta$, and Formula$\left\vert{S_{in}^{conj.PD}}\right\vert+\left\vert{S_{bad}}\right\vert=n=2^{k}$, and Formula$k$ 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 Formula$\varsigma_{A_{1}},\ldots,\varsigma_{A_{n-l}}$ will be selected from the sets Formula${\cal P}_{1}\backslash{\cal P}_{1}^{\prime}$ and Formula${\cal P}_{2}\backslash{\cal P}_{2}^{\prime}$, instead of Formula${\cal P}_{1}$ and Formula${\cal P}_{2}$. The bad set is defined as in the case of degradable channels Formula${\cal B}={\cal B}\left({{\cal N}_{amp},\beta}\right)\cap{\cal B}\left({{\cal N}_{phase},\beta}\right)$. Set Formula${\cal B}$ 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 Formula${\cal P}_{1}^{\prime}$, Formula$\left\vert{{\cal P}_{1}^{\prime}}\right\vert>0$, the Formula$R_{Q}$ rate of quantum communication for a PD channel with set Formula$S_{in}^{conj.PD}$ is higher than the rate of quantum communication that can be obtained for a degradable channel with Formula$S_{in}^{degr}$.

Proof

The defined sets Formula${\cal P}_{1}^{\prime}$ and Formula${\cal P}_{2}^{\prime}$ for a PD channel are also disjoint [7], [8], [12], thus FormulaTeX 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 Formula${\cal P}_{2}$ are not transmitted over the channel (Bob synthesizes Formula${\cal P}_{2}$ from the corresponding subset Formula$\Omega_{{\cal P}_{1}}$ of Formula${\cal P}_{1}$ by using Hadamard operations in the decoding process):FormulaTeX 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 FormulaTeX Source$$\lim\limits_{n\to\infty}{{1}\over{n}}\left\vert{{\cal P}^{\prime}_{2}}\right\vert=0,\eqno{\hbox{(14)}}$$ along with relations Formula$\left\vert{\left({{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\right)\cap{\cal G}\left({{\cal N}_{amp},\beta}\right)}\right\vert=0$, Formula$\left\vert{{\cal P}^{\prime}_{2}\cap{\cal G}\left({{\cal N}_{amp},\beta}\right)}\right\vert=0$, Formula$\left\vert{{\cal B}\left({{\cal N}_{amp},\beta}\right)\cap{\cal G}\left({{\cal N}_{amp},\beta}\right)}\right\vert=0$, which follows from the fact that Formula${\cal P}^{\prime}_{2}\subseteq{\cal P}_{2}$ and FormulaTeX Source$$\left({{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\right)\subseteq{\cal B}\left({{\cal N}_{amp},\beta}\right),\eqno{\hbox{(15)}}$$ where FormulaTeX 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 FormulaTeX 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 FormulaTeX 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 Formula${\cal P}^{\prime}_{1}\subseteq{\cal G}\left({{\cal N}_{amp},\beta}\right)$ and Formula$\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)\subseteq{\cal G}\left({{\cal N}_{amp},\beta}\right)$, which demonstrates that the defined codewords sets Formula${\cal P}_{1}^{\prime}$, Formula${\cal P}^{\prime}_{2}$, Formula${\cal P}_{1}\backslash{\cal P}_{1}^{\prime}$ and Formula${\cal P}_{2}\backslash{\cal P}_{2}^{\prime}$ are pairwise disjoint, where FormulaTeX 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 FormulaTeX 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 FormulaTeX 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 Formula$R_{Q}\left({\cal N}\right)$ rate can be achieved over the channel Formula${\cal N}_{AB}$ can be expressed as:FormulaTeX 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 Formula$\beta<0.5$, the following relation holds for the Holevo information of channel Formula${\cal N}_{AB}$ and Formula${\cal N}_{A{E}^{\prime}}$ of Formula${\cal N}$:FormulaTeX 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 Formula$\chi\left({{\cal N}_{AB}}\right)$, Formula$\chi\left({{\cal N}_{AE}}\right)$ and Formula$\chi\left({{\cal N}_{A{E}^{\prime}}}\right)$ are the Holevo information of the channels Formula${\cal N}_{AB}$, Formula${\cal N}_{AE}$ and Formula${\cal N}_{A{E}^{\prime}}$ of Formula${\cal N}$. Combing this result with (13) and (14), one will get FormulaTeX 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]) FormulaTeX 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:FormulaTeX 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 Formula$\beta<0.5$:FormulaTeX 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 Formula${\cal P}_{1}$:FormulaTeX 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 Formula$\left({{\cal P}_{1}\backslash{\cal P}_{1}^{\prime}}\right)\cap{\cal B}\ne\emptyset$. Using the defined sets it follows that FormulaTeX 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 FormulaTeX 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 FormulaTeX 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 Formula${\cal N}_{A{E}^{\prime}}$ is degraded [8], FormulaTeX 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 Formula$S_{in}^{degr}$, Formula$S_{in}^{conj.PD}$, Formula${{{\cal P}^{\prime}}}_{1}\subseteq{\cal P}_{1}$, Formula${{{\cal P}^{\prime}}}_{2}\subseteq{\cal P}_{2}$ and Formula${\cal B}$ are disjoint sets with relation FormulaTeX 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 Formula${\cal N}$ we get:FormulaTeX 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:FormulaTeX 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 Formula${\cal N}_{AB}$ will be FormulaTeX 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 Formula${\cal P}_{1}^{\prime}$, the results on rate of quantum communication Formula$R_{Q}\left({\cal N}\right)$ for a PD channel Formula${\cal N}$ exceed the result obtained for a degradable channel with Formula$\left\vert{S_{in}^{degr}}\right\vert$.

2. Summary

Since for a degradable channel Formula${\cal N}$, only set Formula$S_{in}^{degr}=\left({{\cal G}\left({{\cal N}_{ampl},\beta}\right)\cap{\cal G}\left({{\cal N}_{phs},\beta}\right)}\right)^{degr}$ can be used for quantum communication, it will result in quantum data rate FormulaTeX 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 Formula${\cal G}\left({{\cal N}_{amp},\beta}\right)$. The set that can transmit phase is denoted by Formula${\cal G}\left({{\cal N}_{phs},\beta}\right)$. For a degradable channel Formula${\cal N}$ only the set Formula$\left\vert{S_{in}^{degr}}\right\vert=m$ can be used for quantum communication. For a degradable channel Formula${\cal N}$, valuable information can be leaked only from the set Formula${\cal P}_{1}$ [7], [8] which results in Formula$\left({S_{in}^{degr}\cup{\cal P}_{1}\cup{\cal P}_{2}}\right)\backslash\left({{\cal P}_{1}\cup{\cal P}_{2}}\right)$, where Formula$\lim\limits_{n\to\infty}{{1}\over{n}}\left\vert{{\cal P}_{2}}\right\vert=0$, i.e., Formula$\left({S_{in}^{degr}\cup{\cal P}_{1}}\right)\backslash\left({{\cal P}_{1}}\right)$. The set Formula${\cal P}_{1}$ represents the information of channel Formula${\cal N}_{AE}$, between Alice and the environment. For these sets Formula$R_{Q}\left({{\cal N}_{AE}}\right)>0$ and Formula$R_{Q}\left({{\cal N}_{AB}}\right)=0$. 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 Formula$S_{in}^{degr}$ 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 FormulaTeX 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 Formula${{{\cal P}^{\prime}}}_{1}\subseteq{\cal P}_{1}$, Formula${{{\cal P}^{\prime}}}_{2}\subseteq{\cal P}_{2}$ can be used for quantum communication, which results in quantum data rate FormulaTeX 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 Formula$S_{in}^{conj.PD}=S_{in}^{degr}\cup{{{\cal P}^{\prime}}}_{1}\cup{{{\cal P}^{\prime}}}_{2}$ with Formula$\vert{S_{in}^{conj.PD}}\vert=\vert{S_{in}^{degr}}\vert+\Delta=m+\Delta$ will be available for quantum communication, where Formula${{{\cal P}^{\prime}}}_{1}\subseteq{\cal P}_{1}$, Formula${{{\cal P}^{\prime}}}_{2}\subseteq{\cal P}_{2}$, and Formula$\lim\limits_{n\to\infty}{{1}\over{n}}\vert{{\cal P}_{2}\backslash{\cal P}_{2}^{\prime}}\vert=0$ and Formula$\lim\limits_{n\to\infty}{{1}\over{n}}\vert{{{{\cal P}^{\prime}}}_{2}}\vert=0$.

The extended set results in higher quantum communication rates in comparison to a degradable one. The amount of information in the degraded environment state Formula${E}^{\prime}$ outputted by channel Formula${\cal N}_{A{E}^{\prime}}={\cal N}_{AE}\circ{\cal D}^{E\to{E}^{\prime}}$ is less than in the environment state Formula$E$, outputted by channel Formula${\cal N}_{AE}$. 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.

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Laszlo Gyongyosi

Laszlo Gyongyosi received the M.Sc. degree with honors in computer science, in 2008, and the Ph.D. degree with highest distinction in quantum information from the Budapest University of Technology and Economics (BUTE), in 2013. He is a Research Associate in Quantum Computations and Communications at the Hungarian Academy of Sciences and BUTE. His current research interests include quantum Shannon theory, quantum computation and communication, and quantum cryptography. He is the co-author of the book “Advanced Quantum Communications” (Wiley-IEEE Press, New Jersey, USA), and he is a Lecturer of Quantum Information at the Faculty of Electrical Engineering and Informatics at BUTE.

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