Adaptive recurrence quantum entanglement distillation for two-Kraus-operator channels

TL;DR

This paper introduces adaptive recurrence quantum entanglement distillation algorithms tailored for two-Kraus-operator channels, significantly enhancing efficiency and convergence speed in entanglement purification processes.

Contribution

The paper proposes two new adaptive recurrence QED algorithms with guaranteed quadratic convergence for channels with two Kraus operators, improving upon existing methods.

Findings

Algorithms achieve quadratic convergence speed.

Numerical results show significant efficiency improvements.

Algorithms are effective for phase-damping and amplitude-damping channels.

Abstract

Quantum entanglement serves as a valuable resource for many important quantum operations. A pair of entangled qubits can be shared between two agents by first preparing a maximally entangled qubit pair at one agent, and then sending one of the qubits to the other agent through a quantum channel. In this process, the deterioration of entanglement is inevitable since the noise inherent in the channel contaminates the qubit. To address this challenge, various quantum entanglement distillation (QED) algorithms have been developed. Among them, recurrence algorithms have advantages in terms of implementability and robustness. However, the efficiency of recurrence QED algorithms has not been investigated thoroughly in the literature. This paper put forth two recurrence QED algorithms that adapt to the quantum channel to tackle the efficiency issue. The proposed algorithms have guaranteed convergence for quantum channels with two Kraus operators, which include phase-damping and amplitude-damping channels. Analytical results show that the convergence speed of these algorithms is improved from linear to quadratic and one of the algorithms achieves the optimal speed. Numerical results confirm that the proposed algorithms significantly improve the efficiency of QED.

Equations

$$\mathbf{\rho }=\sum _{k=1}^K{\mathbf{C}}_k{\mathbf{\rho }}_0{\mathbf{C}}_k^{†},$$

$$\sum _{k=1}^K{\mathbf{C}}_k^{†}{\mathbf{C}}_k={\mathbb{I}}_2.$$

$$\mathbf{\rho }$$

$$⟨{\Phi }^{+}|\mathbf{\rho }|{\Phi }^{+}⟩\approx 1$$

$${\mathbf{C}}_1=\mathbf{U}\left[\begin{array}{cc}1& 0\\ 0& \sqrt{1-p}\end{array}\right]{\mathbf{V}}^{†},{\mathbf{C}}_2=\mathbf{U}\left[\begin{array}{cc}0& \eta \sqrt{p}\\ 0& \zeta \sqrt{p}\end{array}\right]{\mathbf{V}}^{†}.$$

$${\mathbf{C}}_1=\left[\begin{array}{cc}1& 0\\ 0& \sqrt{1-p}\end{array}\right],{\mathbf{C}}_2=\left[\begin{array}{cc}0& \eta \sqrt{p}\\ 0& \zeta \sqrt{p}\end{array}\right].$$

$$\mathbf{\sigma }$$

$$\check{\mathbf{\rho }}=({\mathbf{U}}_{\mathrm{A}}\otimes {\mathbf{U}}_{\mathrm{B}})\mathbf{\rho }({\mathbf{U}}_{\mathrm{A}}\otimes {\mathbf{U}}_{\mathrm{B}}{)}^{†}=F|\mu ⟩⟨\mu |+(1-F)|\nu ⟩⟨\nu |$$

$$|\mu ⟩$$

$$|\nu ⟩$$

$$F$$

$$\alpha$$

$$\gamma$$

$$\displaystyle\breve{\bm{\rho}}=\frac{\mathrm{tr}_{2,4}\big{\{}\sum_{k=1}^{\tilde{K}}(\bm{N}^{(k)}_{\mathrm{A}}\otimes\bm{N}^{(k)}_{\mathrm{B}})(\bm{P}\check{\bm{\rho}}\otimes\check{\bm{\rho}}\,\bm{P}^{\dagger})(\bm{N}^{(k)}_{\mathrm{A}}\otimes\bm{N}^{(k)}_{\mathrm{B}})^{\dagger}\big{\}}}{\mathrm{tr}\big{\{}\sum_{k=1}^{\tilde{K}}(\bm{N}^{(k)}_{\mathrm{A}}\otimes\bm{N}^{(k)}_{\mathrm{B}})(\bm{P}\check{\bm{\rho}}\otimes\check{\bm{\rho}}\,\bm{P}^{\dagger})(\bm{N}^{(k)}_{\mathrm{A}}\otimes\bm{N}^{(k)}_{\mathrm{B}})^{\dagger}\big{\}}}$$

$$\displaystyle\bm{P}=\mathbb{I}_{2}\otimes\big{(}|00\rangle\langle 00|+|10\rangle\langle 01|+|01\rangle\langle 10|+|11\rangle\langle 11|\big{)}\otimes\mathbb{I}_{2}$$

$$F^{\ast }=\underset{\{{\mathbf{N}}_{\mathrm{A}}^{(k)},{\mathbf{N}}_{\mathrm{B}}^{(k)}{\}}_{k=1}^{\tilde{K}}\in \mathcal{F}}{\max }⟨{\Phi }^{+}|\breve{\mathbf{\rho }}|{\Phi }^{+}⟩$$

$$F^{\ast }=f(F,\alpha ,\beta ,\gamma ,\delta ,\theta ).$$

$$f(F,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)\le \frac{F^2}{F^2+(1-F{)}^2},\forall F\in (\frac{1}{2},1]$$

$$f(F,\alpha ,\beta ,\gamma ,\delta ,\theta )\le \frac{F^2}{F^2+(1-F{)}^2(\frac{\gamma \delta }{\alpha \beta }{)}^2}$$

$$\breve{\mathbf{\rho }}=\frac{{\sum }_{i=1}^4{C}_i{\mathbf{\psi }}^{(i)}{\mathbf{\psi }}^{(i)†}}{{\sum }_{i=1}^4{C}_i{\mathbf{\psi }}^{(i)†}{\mathbf{\psi }}^{(i)}}$$

$${\mathbf{\psi }}^{(i)}$$

$$s_{11}{s}_{22}=s_{12}{s}_{21}$$

$$\displaystyle\bm{V}=\frac{1}{2}\left[\begin{array}[]{rrrr}1&1&1&1\\ 1&-1&1&-1\\ 1&1&-1&-1\\ 1&-1&-1&1\end{array}\right]$$

$$\displaystyle F^{*}=\frac{F^{2}}{F^{2}+(1-F)^{2}\big{(}\frac{\gamma\delta}{\alpha\beta}\big{)}^{2}}$$

$$F^{\ast }=\frac{1}{2}+\frac{\sqrt{(1-p)(1-|\eta {|}^{2}p)}}{(1-p)+(1-|\eta {|}^{2}p)}.$$

$${\mathbf{M}}_{\mathrm{B}}=\left[\begin{array}{cc}[1]\kappa & 0\\ 0& 1\end{array}\right],{\mathbf{M}}_{\bar{\mathrm{B}}}=\left[\begin{array}{cc}\sqrt{1-{\kappa }^2}& 0\\ 0& 0\end{array}\right]$$

$$\displaystyle P_{1}=\left\{\begin{array}[]{l@{\;}l}\dfrac{F_{0}^{2}\alpha^{2}\beta^{4}+(1-F_{0})^{2}\beta^{2}\gamma^{2}\delta^{2}}{2F_{0}\alpha^{2}\beta^{2}+(1-F_{0})(\alpha^{2}\gamma^{2}+\beta^{2}\delta^{2})}&\begin{array}[]{l}\mbox{for the \acs{FP}}\\ \mbox{approach}\end{array}\vspace{3mm}\\ \dfrac{4F_{0}^{2}\alpha^{4}\beta^{4}+(1-F_{0})^{2}(\alpha^{2}\gamma^{2}+\beta^{2}\delta^{2})^{2}}{4F_{0}\alpha^{4}\beta^{2}+2(1-F_{0})\alpha^{2}(\alpha^{2}\gamma^{2}+\beta^{2}\delta^{2})}&\begin{array}[]{l}\mbox{for the \acs{PP}}\\ \mbox{approach}\end{array}\end{array}\right.$$

$$\displaystyle F_{1}=\left\{\begin{array}[]{l@{\;}l}\dfrac{F_{0}^{2}}{F_{0}^{2}+(1-F_{0})^{2}(\frac{\gamma\delta}{\alpha\beta})^{2}}&\begin{array}[]{l}\mbox{for the \acs{FP}}\\ \mbox{approach}\end{array}\vspace{1.5mm}\\ \dfrac{F_{0}^{2}}{F_{0}^{2}+\frac{1}{4}(1-F_{0})^{2}(\frac{\gamma^{2}}{\beta^{2}}+\frac{\delta^{2}}{\alpha^{2}})^{2}}&\begin{array}[]{l}\mbox{for the \acs{PP}}\\ \mbox{approach}\end{array}\end{array}\right.$$

$$\displaystyle P_{k}=\frac{1}{2}\big{(}F_{k-1}^{2}+(1-F_{k-1})^{2}\big{)}$$

$$F_k=\frac{F_{k-1}^2}{F_{k-1}^2+(1-F_{k-1}{)}^2}.$$

$${\mathbf{\rho }}^{(k)}=F_{k-1}|{\Phi }^{+}⟩⟨{\Phi }^{+}|+(1-F_{k-1})|{\Psi }^{+}⟩⟨{\Psi }^{+}|.$$

$$F_k=\frac{F_{k-1}^2+\frac{1}{9}(1-F_{k-1}{)}^2}{F_{k-1}^2+\frac{2}{3}{F}_{k-1}(1-F_{k-1})+\frac{5}{9}(1-F_{k-1}{)}^2}.$$

Full text

I Introduction

Quantum entanglement shared by remote agents serves as a valuable resource for many important quantum communication operations Lloyd (2003); Horodecki et al. (2009); Ulanov et al. (2015), such as secret key distribution Ekert (1991); Koashi and Preskill (2003); Gottesman et al. (2006), dense coding Bennett and Wiesner (1992); Wang et al. (2005); Barreiro et al. (2008), and teleportation Bennett et al. (1993); Nielsen et al. (1998); Gottesman and Chuang (1999). With the assistance of entanglement, the capacity of quantum channels can be increased, particularly when the channels are very noisy Bennett et al. (1999, 2002); Shor (2004); Holevo (2012); Holevo and Shirokov. (2015). Entanglement also enables quantum relay, and therefore is a keystone of long-distance quantum communication Dür et al. (1999); Sangouard et al. (2011). To establish entanglement between two remote agents, one agent can locally generate a maximally entangled qubit pair and send one of the qubits to the other agent through a quantum channel. However, the noise inherent in the channel will contaminate the qubit during the transmission, thereby deteriorating the entanglement. To address this problem, QED algorithms Bennett et al. (1996a); Deutsch et al. (1996); Briegel et al. (1998); Opatrný and Kurizki (1999); Mundarain and Orszag (2009); Dehaene et al. (2003); Vollbrecht and Verstraete (2005); Hostens et al. (2006); Bennett et al. (1996b); Matsumoto (2003); Ambainis and Gottesman (2006); Watanabe et al. (2006) have been proposed to generate qubit pairs in the targeted maximally entangled state from many shared contaminated ones using local operations and classical communication (LOCC).

In the pioneering work Bennett et al. (1996a), two influential QED algorithms were proposed and are now known as the recurrence algorithm and the asymptotic algorithm. Recurrence algorithms Deutsch et al. (1996); Briegel et al. (1998); Opatrný and Kurizki (1999); Mundarain and Orszag (2009) operate separately on every two qubit pairs, improving the quality of entanglement in one pair at the expense of the other pair, which is then discarded. The algorithms keep repeating this operation to progressively improve the quality of entanglement in the kept qubit pairs. Asymptotic algorithms Dehaene et al. (2003); Vollbrecht and Verstraete (2005); Hostens et al. (2006) operate on a large number of qubit pairs, detecting ones that are not in the targeted state by measuring a subset of the qubit pairs, and then transforming those in the undesired state to the targeted state. Later, it was recognized that there is a duality between QED and quantum error correction (QEC) Bennett et al. (1996b) when one-way classical communication is involved. The connection between QED and QEC was further explored in scenarios involving two-way classical communication, enabling code-based QED algorithms Matsumoto (2003); Ambainis and Gottesman (2006); Watanabe et al. (2006). These algorithms operate on a few qubit pairs, look for the error syndrome using measurements specified by the error correction code, and then correct the errors to restore entanglement.

Asymptotic algorithms have revealed important theoretical insights, but these algorithms require agents that have the capability of processing a large number of qubits. Code-based algorithms require agents to have the capability of processing only a few qubits, but the number of errors that can be corrected is limited by the Hamming distance of the codewords. Designing QEC codes with large Hamming distance is challenging since the creation of information redundancy, the main mechanism adopted in classical error correction codes, is not possible in quantum codes due to the no-cloning theorem Shor (1995); Gottesman (1996); Knill (2005); Gottesman (2009). Hence, these algorithms do not apply to scenarios with strong noise in the channel. Recurrence algorithms require agents to have the capability of processing only a few qubits and can generate maximally entangled qubit pairs even in strong noise scenarios. This is because the recurrence algorithms can mitigate stronger noise by performing more rounds of distillations. In fact, the recurrence algorithm proposed in Bennett et al. (1996a) can distill contaminated qubit pairs into maximally entangled qubit pairs as long as the initial fidelity of the contaminated qubit pairs with respect to (w.r.t.) the targeted state is greater than $0.5$.111In Horodecki and Horodecki (1996); Horodecki et al. (1997), the authors proved that a state of qubit pairs is distillable if and only if its fidelity w.r.t. some maximally entangled state is greater than $0.5$. Assuming that the agents have knowledge of the channel to properly select the targeted maximally entangled state, then the sufficient distillability condition provided in Bennett et al. (1996a) is equivalent to that proposed in Horodecki and Horodecki (1996); Horodecki et al. (1997).

Building large-scale quantum circuits operating on many qubits is challenging Unruh (1995); Divincenzo (2000); Suominen (2012); even the error rates of two-qubit gates are significantly higher than those of one-qubit gates Ladd et al. (2010). In this perspective, recurrence algorithms are favorable for implementation as they require operations only on a few (typically one or two) qubits and are robust to strong noise in the quantum channel. On the other hand, since at least half of the entangled qubit pairs are discarded in each round of distillation, the efficiency of the recurrence algorithms decreases dramatically with the number of rounds.222The efficiency of QED algorithms is measured in terms of yield, which is defined as the ratio between the number of maximally entangled output qubit pairs and the number of contaminated input qubit pairs. To address this challenge, the quantum privacy amplification (QPA) algorithm was proposed in Deutsch et al. (1996), and was shown numerically to require fewer rounds of distillation than the algorithm in Bennett et al. (1996a) for contaminated qubit pairs with a specific set of initial states. However, the performance of QPA algorithm was not characterized analytically. In fact, another set of initial states was found in Opatrný and Kurizki (1999) for which the QPA algorithm was less efficient than the algorithm in Bennett et al. (1996a). In Opatrný and Kurizki (1999), the design of distillation operations was formulated into an optimization problem, which was inherently non-convex, and consequently, the optimal solution was not found.

We envision that a key enabler to designing efficient recurrence QED algorithms is to make them adaptive to quantum channel. Intuitively, compared to general algorithms, QED algorithms that adapt to channel-specific noise will better mitigate such noise and hence distill more efficiently. In fact, it has been observed that knowing the channel benefits the performance of quantum error recovery Fletcher et al. (2007), and channel-adaptive QEC schemes that outperform prior ones Fletcher et al. (2008a, b) have been designed.

In this paper, we focus on two-Kraus-operator (TKO) channels, a class that covers several typical quantum channels, e.g., phase-damping and amplitude-damping channels. The phase-damping channels describe the decoherence process of a photon traveling through a waveguide, and the amplitude-damping channels model the decay of an excited atom due to spontaneous emission (Preskill, 2015, Sec. 3.4), (Nielsen and Chuang, 2000, Sec.8.3). To achieve efficient distillation, we develop two adaptive recurrence QED algorithms which adapt to the channel by employing a remote shared-state preparation (RSSP) method.333This method is akin to remote state preparation methods Bennett et al. (2001, 2005), in which two remote agents employ LOCC to prepare a quantum state at one of the agents. The proposed RSSP method uses LOCC to prepare a state shared by both agents. The contributions of this work are

• •

characterization of the structure of TKO channels;

• •

characterization of the optimal fidelity that can be achieved by performing LOCC on two qubits pairs affected by TKO channels;

• •

design of adaptive recurrence QED algorithms which improve the convergence speed of fidelity from linear to quadratic, and one of them achieves the optimal speed.

Notations: $a$, $\bm{a}$, and $\bm{A}$ represent scalar, vector, and matrices, respectively. $\mathrm{pha}\{\cdot\}$ denotes the phase of a complex number. $(\cdot)^{\dagger}$, $\mathrm{rank}\{\cdot\}$, $\det\{\cdot\}$ and $\mathrm{tr}\{\cdot\}$, denote the Hermitian transpose, rank, determinant, and trace of a matrix, respectively. $\mathrm{tr}_{i,j}\{\cdot\}$ denotes the partial trace w.r.t. to the $i$-th and $j$-th qubits in the system. $\mathrm{span}(\cdot)$ denotes the linear space spanned by a set of vectors. $\mathbb{I}_{n}$ denotes $n\times n$ identity matrix, and $\imath$ is the unit imaginary number.

II System Model

Consider two remote agents, Alice and Bob, connected by a quantum channel and a two-way classical channel. When Alice transmits a qubit with density matrix $\bm{\rho}_{0}$, the density matrix of the qubit received by Bob is given by

[TABLE]

where the Kraus operators $\{\bm{C}_{k}\}$ representing the noisy quantum channel satisfy

[TABLE]

Since a qubit is a 2-dimensional system, the number of the Kraus operators $K\leq 2^{2}=4$.444Quantum operators on $n$-dimensional systems are $n\times n$ matrices, and hence lies in an $n^{2}$ dimensional space. Thus, from (Preskill, 2015, Sec 3.3), if a channel for such systems is represented with more than $n^{2}$ operators, there always exists an equivalent representation with no more than $n^{2}$ non-zero operators. When $K=1$, the channel is noiseless, and hence QED is not needed. For the class of TKO channels, $K=2$.

Suppose Alice and Bob wish to obtain maximally entangled qubits pairs with density matrix $\bm{\rho}_{0}=|\Phi^{+}\rangle\langle\Phi^{+}|$, where $|\Phi^{+}\rangle=\frac{1}{\sqrt{2}}\left(|00\rangle+|11\rangle\right)$. To achieve this task, Alice locally prepares qubits pairs, each with density matrix $\bm{\rho}_{0}$, then sends the second qubit in each pair through the noisy channel. Then the density matrix of the two remote qubits becomes

[TABLE]

Alice and Bob then adopt a recurrence QED algorithm outlined in Fig. 1. In each round of distillation, the agents separately operate on every two qubit pairs kept in the previous round, perform LOCC, and attempt to improve the quality of entanglement in one of the qubit pairs at the expense of the other pair, which is then discarded (agents may discard both pairs when this operation is unsuccessful). The objective of the algorithm is to generate qubit pairs with density matrix $\bm{\rho}^{*}$ close to the targeted state, i.e.,

[TABLE]

where $\langle\Phi^{+}|\,\bm{\rho}\,|\Phi^{+}\rangle$ is the fidelity of a density matrix $\bm{\rho}$ and the targeted state $|\Phi^{+}\rangle$.

III Design of Adaptive Recurrence QED Algorithm

III.1 Characterization of TKO Channels

Prior to designing adaptive QED algorithms, it is crucial to understand the effect of noisy quantum channels on the entanglement between qubits. This can be accomplished by determining the structure of the noisy quantum channels. In particular, the structure of TKO channels is provided by the following lemma.

Lemma 1** (Structure of TKO channels):**

\thlabel

lem:channel For every single-qubit TKO channel, there exist unitary matrices $\bm{U},\bm{V}\in\mathbb{C}^{2\times 2}$ and scalars $p\in[0,1]$, $\zeta\in[0,1]$, and $\eta\in\mathbb{C}$ with $|\eta|^{2}+\zeta^{2}=1$ such that the channel can be represented by

[TABLE]

Proof.

The proof is given in Appendix A. ∎

Local unitary operations $\bm{U}$ and $\bm{V}$ do not affect the amount of entanglement Vedral et al. (1997). In particular, for any $\bm{U}$ and $\bm{V}$, Bob and Alice can respectively perform local unitary operations $\bm{U}^{\dagger}$ and $\bm{V}$ and obtain an equivalent channel with $\tilde{\bm{U}}=\bm{U}^{\dagger}\bm{U}=\mathbb{I}_{2}$ and $\tilde{\bm{V}}=\bm{V}^{\dagger}\bm{V}=\mathbb{I}_{2}$. Therefore, without loss of generality, $\bm{U}$ and $\bm{V}$ are assumed to be $\mathbb{I}_{2}$ in the following analysis. In this case, the channel can be represented by

[TABLE]

Remark 1** (The effect of TKO channel on entanglement):**

From (5), $\mathrm{rank}\{\bm{C}_{2}\}=1$. Hence, $\bm{C}_{2}$ can be written as in (44). In this case, for any qubit pairs with density matrix $\bm{\sigma}_{0}$, conditional on that $\bm{C}_{2}$ operates, the density matrix of the qubit pair after the second qubit goes through the channel becomes

[TABLE]

where $\bm{\sigma}_{\mathrm{B}}=|i\rangle\langle i|$, with $|i\rangle=\eta|0\rangle+\zeta|1\rangle$. Therefore, when $\bm{C}_{2}$ operates, $\bm{\sigma}$ is a separable state, which implies that all entanglement between the two qubits is destroyed. The behavior of $\bm{C}_{2}$ is mainly characterized by parameters $p$ and $\eta$.

• •

The role of $p$: $p\in[0,1]$ is the strength of $\bm{C}_{2}$, which is proportional to the probability that $\bm{C}_{2}$ operates on a qubit. The parameter $p$ can be thought of as the severity of noise in the channel since it characterizes the extent that the channel deteriorates the entanglement. The larger the $p$, the more the entanglement is destroyed. In particular, all entanglement is preserved when $p=0$, and the opposite when $p=1$.

• •

The role of $\eta$: $\arcsin(|\eta|)\in[0,\frac{\pi}{2}]$ is the angle between the image (i.e., $\mathrm{span}(\eta|0\rangle+\zeta|1\rangle)$) and the coimage (i.e., $\mathrm{span}(|1\rangle)$) of $\bm{C}_{2}$. This characterizes the angle at which $\bm{C}_{2}$ rotates the state of a qubit, and hence indicates the type of the channel. In particular, the channel is phase-damping when $\arcsin(|\eta|)=0$, i.e., $\eta=0$, and amplitude-damping when $\arcsin(|\eta|)=\frac{\pi}{2}$, i.e., $|\eta|=1$.

The channel properties described above are summarized in Fig. 2.  $\square$

All entanglement is destroyed after the second qubit passes through a TKO channel with $p=1$. Therefore, the interesting case for QED is $p\in[0,1)$. The following theorem characterizes the structure of the density matrix $\bm{\rho}$ in this case.

Theorem 1** (Structure of $\bm{\rho}$):**

\thlabel

thm:Str_rho Consider the density matrix of a qubit pair after the second qubit passes through a TKO channel represented by (5). When $p<1$, there exist local unitary operators $\bm{U}_{\mathrm{A}}$, $\bm{U}_{\mathrm{B}}$ such that

[TABLE]

where

[TABLE]

with $\theta$ a certain constant in $[0,2\pi)$, and

[TABLE]

Proof.

The proof is given in Appendix B. ∎

Remark 2** (The role of $\bm{U}_{\mathrm{A}}$ and $\bm{U}_{\mathrm{B}}$):**

Equations (6)–(8) show that by applying the properly designed local unitary operators $\bm{U}_{\mathrm{A}}$ and $\bm{U}_{\mathrm{B}}$, the density matrix $\bm{\rho}$ can be transformed into $\check{\bm{\rho}}$ with all its eigenvectors written in the computational basis, i.e., $\{|0\rangle,|1\rangle\}$ via the Schmidt decomposition. This transformation enables the simplification for both analysis and algorithm design in the following sections. In particular, $\bm{U}_{\mathrm{A}}$ and $\bm{U}_{\mathrm{B}}$ will be employed by Alice and Bob respectively on their individual qubits before the recurrent distillation operations begin, and hence will be referred to as the pre-distillation unitary operators.  $\square$

III.2 Characterization of the Optimal Fidelity

This section proves the optimal fidelity that can be achieved by performing appropriate LOCC. Consider recurrence QED algorithms outlined in Fig. 1, which

• •

perform LOCC on two qubit pairs with density matrix $\check{\bm{\rho}}$ given by (6), and

• •

keep at most one pair.

Since operations performed by agents are local, they can be expressed as $\bm{N}^{(k)}_{\mathrm{A}}\otimes\bm{N}^{(k)}_{\mathrm{B}}$, $k\in\{1,2,\ldots,K\}$, satisfying \sum_{k=1}^{K}\big{(}\bm{N}^{(k)}_{\mathrm{A}}\big{)}^{\dagger}\bm{N}^{(k)}_{\mathrm{A}}=\sum_{k=1}^{K}\big{(}\bm{N}^{(k)}_{\mathrm{B}}\big{)}^{\dagger}\bm{N}^{(k)}_{\mathrm{B}}=\mathbb{I}_{4}. Without loss of generality, assume that agents keep the first qubit pair conditioned on the event that one of the first $\tilde{K}$ operators acts on the four qubits. Then after the LOCC, the density matrix of the first qubit pair is given by

[TABLE]

where

[TABLE]

is the permutation operator that switches the second and third qubits. With this operator, the joint density matrix $\bm{\rho}_{\mathrm{J}}=\bm{P}\check{\bm{\rho}}\otimes\check{\bm{\rho}}\bm{P}^{\dagger}$ corresponds to four qubits, where the first two belong to Alice and last two belong to Bob.

Denote $F^{*}$ as the optimal fidelity that can be achieved with initial density matrix $\check{\bm{\rho}}$ via all possible LOCC, i.e.,

[TABLE]

where ${\cal{F}}$ denotes the set of all possible LOCC.

The characterization of the optimal fidelity $F^{*}$ is challenging because it involves general TKO channels and arbitrary LOCC. These two issues are tackled by the following lemmas. \threflem:simplifybound characterizes the relationship between the $F^{*}$ for general TKO channels and that for the special case of phase-damping channels. \threflem:operation exploits the property of separable operators to determine the set of attainable density matrices of the first qubit pair.

Lemma 2** (Simplification to phase-damping):**

\thlabel

lem:simplifybound Express the optimal fidelity $F^{*}$ explicitly as a function of the density matrix parameters in (9)–(11), i.e.,

[TABLE]

If the optimal fidelity for phase-damping channels is upper bounded by

[TABLE]

then the optimal fidelity for generic TKO channels satisfies

[TABLE]

$\forall F$, $\alpha$, $\beta$, $\gamma$, $\delta$, and $\theta$ satisfying (9)–(11).

Proof.

The proof is given in Appendix C. ∎

Lemma 3** (Density matrix after arbitrary seperable operation):**

\thlabel

lem:operation For phase-damping channels, after arbitrary seperable operator acts on two qubit pairs, the density matrix of the kept qubit pair can be expressed as

[TABLE]

where $C_{1}=F^{2}$, $C_{2}=C_{3}=F(1-F)$, $C_{4}=(1-F)^{2}$,

[TABLE]

in which $s_{ij}$, $s\in\{w,x,y,z\}$, $i,j\in\{1,2\}$ are complex numbers, satisfying

[TABLE]

$\bm{v}^{(i)}$ is the $i$-th column of the unitary matrix

[TABLE]

and $\sum_{i=1}^{4}C_{i}\boldsymbol{\psi}^{(i){\dagger}}\boldsymbol{\psi}^{(i)}>0$.

Proof.

The proof is given in Appendix D. ∎

With the issues of general TKO channels and arbitrary LOCC addressed, the optimal fidelity $F^{*}$ can now be characterized.

Theorem 2** (Optimal Fidelity):**

\thlabel

thm:UBF Consider the density matrix $\check{\bm{\rho}}$, given in (6), of a pair of entangled qubits shared by agents via a TKO channel. Then the optimal fidelity of the kept qubit pair after performing LOCC is given by

[TABLE]

Proof.

The proof is given in Appendix E. ∎

Remark 3** (Key channel parameters and the optimal fidelity):**

Equation (22) describes the optimal fidelity as a function of the parameters of the density matrix. To understand how parameters of the channel affect the optimal fidelity $F^{*}$, one can substitute (9)–(11) into (22) to obtain

[TABLE]

By taking derivative of (23) w.r.t. to $p$ and $|\eta|$ respectively, it can verified that $F^{*}$ is a decreasing function of $p$ and an increasing function of $|\eta|$. An intuitive understanding of such trends can be obtained by recalling Remark 1 and Fig. 2. Operator $\bm{C}_{2}$ destroys all entanglement when it operates on a qubit, and $p$ is proportional to the probability that $\bm{C}_{2}$ operates. The larger the $p$, the less entanglement there is after qubits pass through the channel, thereby resulting in a lower $F^{*}$. The angle at which $\bm{C}_{2}$ rotates a qubit is given by $\arcsin(|\eta|)$. The larger $|\eta|$, the easier it is to detect which qubits are operated by $\bm{C}_{2}$, thereby resulting in a higher $F^{*}$. In particular, when $|\eta|=1$, $F^{*}=1$ provided that $p<1$. Therefore, for amplitude-damping channels, it is possible to design recurrence QED algorithms that generate maximally entangled qubit pairs as long as the channel does not completely destroy entanglement.  $\square$

III.3 Achieving the Optimal Fidelity

The following algorithm first adapts to the channel so that the prepared qubit pairs have density matrices with a structure invariant to the channel. Then the algorithm employs recurrent operations to progressively improve the fidelity of the kept qubit pairs. These operations are specially designed to match the prepared density matrix structure, so that the proposed algorithm achieves optimal fidelity in every round of distillation.

Algorithm **** (Adaptive recurrence QED algorithm):

.

• •

RSSP: For each qubit pair, the agents transform the density matrix into $\check{\bm{\rho}}$ using pre-distillation unitary operators $\bm{U}_{\mathrm{A}}$ and $\bm{U}_{\mathrm{B}}$.555Given a TKO channel, if $\eta=0$, $\bm{U}_{\mathrm{A}}$ and $\bm{U}_{\mathrm{B}}$ are determined by (53). Otherwise, one can obtain $\bm{\rho}$ via (3), then perform singular value decomposition (SVD) and Schmidt decomposition sequentially to get (57) and (58), and finally determine $\bm{U}_{\mathrm{A}}$ and $\bm{U}_{\mathrm{B}}$ via (66). Then Bob apply pre-distillation measurement operators

[TABLE]

on his qubit, where $\kappa=\frac{{\beta}}{{\alpha}}$. If the measurement result corresponds to $\bm{M}_{\mathrm{B}}$, Bob performs no further action; otherwise, he notifies Alice via classical communication and the agents discard the qubit pair.

• •

First round distillation: The agents take two of the kept qubit pairs, perform the following operations, and repeat these operations on all kept qubit pairs.

(i) Each agent locally performs CNOT operation, i.e., $\bm{U}=|00\rangle\langle 00|+|01\rangle\langle 01|+|10\rangle\langle 11|+|11\rangle\langle 10|$ on the two qubits at hand.

(ii) Each agent measures the target bit (i.e., the qubit in the second pair) using operators $|0\rangle\langle 0|$, $|1\rangle\langle 1|$, and transmits the measurement result to the other agent via classical communication.

(iii) If their measurement results do not agree, the agents discard the source qubit pair (i.e., the first pair). If the measurement results agree and correspond to $|1\rangle\langle 1|$, the agents keep the source qubit pair. If the measurement results agree and correspond to $|0\rangle\langle 0|$, the agents may choose to discard or keep the source qubit pair; the approach that discard or keep the qubit pair in this case is referred to as the fidelity-prioritized (FP) or probability-prioritized (PP) approach, respectively.

• •

Following rounds: Agents perform the same operations as in the first round, except that they always adopt the PP approach, i.e., keep the source qubit pair as long as the measurement results agree. Repeat this step until the fidelity of the kept qubit pairs exceeds the required threshold.  $\square$

For notational convenience, denote the fidelity of the kept qubit pairs after $n$-th rounds of iteration as ${F}_{n}$, where $F_{0}=F$. The following theorem characterizes the performance of the proposed algorithm.

Theorem 3** (Performance of the proposed algorithm):**

\thlabel

thm:dis_perf After the RSSP and first round of distillation, a qubit pair is kept with probability

[TABLE]

and fidelity

[TABLE]

In the $k$-th round ($k=2,3,4,\ldots$) of distillation, a qubit pair is kept with probability

[TABLE]

and fidelity

[TABLE]

Proof.

The proof is given in Appendix F. ∎

In the following, algorithms that adopt the FP and PP approaches in the first round of distillation are referred to as FP and PP algorithms, respectively.

Remark 4** (Convergence speed of fidelity):**

For the FP algorithm, the density matrix of the kept qubit pair after the $k$-th round of distillation is

[TABLE]

In this case, by comparing (22) with (38) or (40), it can be observed that the FP algorithm achieves the optimal fidelity in every round of distillation. This implies that the FP algorithm attains the fastest convergence speed w.r.t. the rounds of distillation.

The PP algorithm achieves a lower fidelity in the first round compared to the FP algorithm. On the other hand, (31) shows that the probability of keeping a qubit pair in the first round is higher with the PP algorithm compared to the FP algorithm by a factor more than 2. In particular, when the channel is phase-damping, i.e., $\alpha=\beta=\gamma=\delta=\frac{1}{\sqrt{2}}$, the PP algorithm doubles the probability of keeping a qubit pair without lowering the fidelity achieved in the first round.

For the first recurrence QED algorithm (will be referred to as BBPSSW algorithm in this work) proposed in Bennett et al. (1996a), the fidelity of the kept qubit pairs after $k$-th round of distillation is given by

[TABLE]

Therefore, when $F_{0}>\frac{1}{2}$, it can be shown using (41) that

[TABLE]

For the proposed algorithms, when $F_{0}>\frac{1}{2}$, it can be shown using (40) that

[TABLE]

Equation (42) shows that with the BBPSSW algorithm, the fidelity of the qubit pairs converges to $1$ linearly at rate $\frac{2}{3}$, whereas (43) shows that with the proposed algorithms, the fidelity converges to $1$ quadratically. Hence, the convergence speed of the proposed algorithms is significantly improved, i.e., from linear to quadratic, compared to the BBPSSW algorithm.  $\square$

Remark 5** (Connection to the QPA algorithm):**

When the channel is phase-damping, i.e., $\eta=0$, i) the pre-distillation unitary operators $\bm{U}_{\mathrm{A}}=\bm{U}_{\mathrm{B}}=\bm{H}$ according to (53); ii) the pre-distillation measurement operator $\bm{M}_{\mathrm{B}}=\mathbb{I}_{2}$ since $\alpha=\beta$ according to (10). In this case, both local operators employed by Alice and Bob in the RSSP are equal to the Hadamard transform $\bm{H}$, and hence the PP algorithm becomes the QPA algorithm in Deutsch et al. (1996). Therefore, the QPA algorithm is a special case of PP algorithm, which employs fixed pre-distillation operators for all channels. With such non-adaptive pre-distillation operators, the convergence of the fidelity achieved by the QPA algorithm is not guaranteed Deutsch et al. (1996). With the proposed adaptive pre-distillation operators, the fidelity achieved by both FP and PP algorithms converges quadratically for TKO channels. The proposed algorithms may be applied to more general channels, yet their convergence property for such channels remains to be characterzied.  $\square$

Remark 6** (Benefit of channel adaptation):**

In the proposed algorithms, the channel adaptation takes place in the RSSP. As shown in \threfthm:dis_perf and Remark 4, despite its simplicity of involving single-qubit operations only in the initial step, RSSP is the keystone to improve the effectiveness of distillation for TKO channels. With the BBPSSW algorithm Bennett et al. (1996a), in addition to the distillation operations, random bilateral rotations are required to restore the desired density matrix structure in every round of distillation. With the QPA algorithm, no random rotations are required, yet the density matrix structure may not be preserved for different rounds of distillation. In the proposed algorithms, the RSSP adapts to the channel so that the prepared qubit pairs have density matrices with a structure invariant to the channel. As a result, the distillation operation itself, which involves only the CNOT operation and single-qubit measurements, is sufficient to maintain the density matrix structure in every round of distillation. This feature enables simple QED algorithm with guaranteed convergence. Hence, channel adaptation also improves the implementability of QED algorithms.  $\square$

IV Numerical Results

This section provides numerical results to demonstrate the performance of the proposed algorithms. In particular, the proposed FP and PP algorithms are compared with the BBPSSW algorithm in Bennett et al. (1996a) and the QPA in Deutsch et al. (1996) for a required fidelity $F_{\mathrm{th}}=0.99$.

Fig. 3 shows the fidelity of kept qubit pairs as a function of the rounds of distillation for three types of channels, i.e., a phase-damping channel, a “mid-point” channel666This channel ($\arcsin(|\eta|)=\frac{\pi}{4}$) can be thought of as the mid-point of phase-damping channels ($\arcsin(|\eta|)=0$) and amplitude-damping channels ($\arcsin(|\eta|)=\frac{\pi}{2}$)., and an amplitude-damping channel. When the channel is phase-damping, the fidelity achieved by FP, PP, and QPA algorithms are the same, which is consistent with the observations made in Remark 4 and 5. When the channel is “mid-point” or amplitude-damping, the QPA algorithm does not achieve the required fidelity, illustrating its converge issue. The FP, PP, and the BBPSSW algorithm achieve the required fidelity on all channels, with the proposed algorithms using much less rounds of distillation. For instance, when the channel is amplitude-damping, the BBPSSW algorithm requires 24 rounds of distillation, whereas the FP and PP algorithms only require one and three rounds respectively. Since the yield is reduced by at least half after each round of distillation, the yield of the proposed algorithms are significantly higher than the classical one for all the considered channel.

Fig. 4 shows the yield777Suppose the fidelity $F_{k}$ exceeds the required threshold $F_{\mathrm{th}}$ after $K$ rounds of distillation. In the following, the agents are assumed to generate qubit pairs which go through $K-1$ and $K$ rounds of distillation with probability $\frac{F_{K}-F_{\mathrm{th}}}{F_{K}-F_{K-1}}$ and $\frac{F_{\mathrm{th}}-F_{K-1}}{F_{K}-F_{K-1}}$, respectively. This assumption assures that the average output fidelity of the algorithms is always $F_{\mathrm{th}}$, enabling a fair comparison among different scenarios. Denote the yield after $K-1$ and $K$ rounds of distillation as $Y_{K-1}$ and $Y_{k}$, respectively, then the average yield of the algorithm is given by $\frac{F_{K}-F_{\mathrm{th}}}{F_{K}-F_{K-1}}Y_{k-1}+\frac{F_{\mathrm{th}}-F_{K-1}}{F_{K}-F_{K-1}}Y_{k}$. of the distillation algorithms as a function of the noise severity $p$. While the yield of all algorithms decreases with increasing $p$, the proposed algorithms are much more resilient to noise compared to the BBPSSW algorithm. Comparing the two proposed algorithms, the FP algorithm performs better for large $p$, whereas the PP algorithm performs better for small $p$. This shows that when the noise is severe, it is beneficial to the increase the achieved fidelity at a cost of reducing the probability of keeping qubit pairs.

Finally, Fig. 5 shows the yield of the distillation algorithms as a function of the channel type parameter $|\eta|$. The QPA algorithm has the same efficiency as the PP algorithm when the channel is phase-damping, which is consistent with Remark 5. Yet the QPA algorithm does not achieve the required fidelity when $\arcsin{|\eta|}\geq 0.024\pi$. This illustrates the importance of channel adaptation. Comparing the two proposed algorithms, the FP algorithm is more efficient when the channel tends towards an amplitude-damping channel (i.e., $|\eta|$ approaches 1), and the PP algorithm is more efficient when the channel tends towards a phase-damping channel (i.e., $|\eta|$ approaches 0). This is consistent with \threfthm:Str_rho and \threfthm:dis_perf, which show that the benefit of increasing the fidelity by adopting the FP algorithm is greater when $\eta$ is close to 1 and vice versa. It can also be seen that the yield of the PP algorithm is twice of the FP algorithm when $\eta=0$. This is consistent with the observation made in Remark 4 that the PP algorithm doubles the probability of keeping a qubit pair without lowering the fidelity achieved in the first round when the channel is phase-damping.

V Conclusion

Among various types of QED algorithms, the recurrence ones require quantum operations on the minimum number of qubits and can generate maximally entangled qubit pairs even when the noise in the channel is severe. Despite their advantages, the efficiency issue of recurrence QED algorithms has not been thoroughly investigated in the literature. In this paper, we first characterize the effect of a single-qubit TKO channel on the entanglement of a qubit pair shared by the agents via this channel. We then determine the optimal fidelity that can be achieved by performing LOCC on two of such qubit pairs. Finally we propose two adaptive recurrence QED algorithms, one of which achieves the optimal fidelity. The proposed algorithms preserve the density matrix structure in every round of distillation, avoiding the need of additional random rotations. This enable simple QED algorithms with guaranteed convergence for TKO channels. In fact, the convergence speed of both algorithms are improved from linear to quadratic compared to the BBPSSW algorithm. Numerical results confirm that the proposed algorithms significantly improve the efficiency of recurrence QED algorithms. These results also indicate that the benefit of achieving the optimal fidelity is greater when the noise is severe, or the channel tends towards an amplitude-damping channel.

Acknowledgement

The authors would like to thank Aram Harrow and Peter W. Shor for the valuable discussions and suggestions.

Appendix A Proof of \threflem:channel

Consider a single-qubit TKO channel represented by $\bm{C}_{1}$, $\bm{C}_{2}$. We will first prove the theorem for the case in which $\mathrm{rank}\{\bm{C}_{2}\}=1$, then show that the case in which $\mathrm{rank}\{\bm{C}_{2}\}=2$ can be transformed into the prior case.

When $\mathrm{rank}\{\bm{C}_{2}\}=1$, SVD of $\bm{C}_{2}$ shows that there exists $|i\rangle,|j\rangle\in\mathbb{C}^{2}$, $p\in(0,1]$, and $\epsilon\in\mathbb{R}$ such that

[TABLE]

Noting that a quantum operator is invariant up to an overall phase change, $\epsilon$ can be any real number. Recall from (2) that

[TABLE]

Substituting the SVD of $\bm{C}_{1}=\bm{U}_{\mathrm{c}}\bm{D}_{\mathrm{c}}\bm{V}^{\dagger}_{\mathrm{c}}$ and (44) into (45), one can get

[TABLE]

where $\langle j|\tilde{j}\rangle=0$. Since $\bm{D}^{2}_{\mathrm{c}}$ is diagonal and $\bm{V}_{\mathrm{c}}$ is unitary,

[TABLE]

Hence, there exists $|\tilde{k}\rangle$ and $|k\rangle$ with $\langle\tilde{k}|k\rangle=0$ such that

[TABLE]

It can be verified that $\bm{C}_{1}$ in (47) and and $\bm{C}_{2}$ in (44) can be expressed in the form given in (4), with

[TABLE]

This completes the proof for the case with $\mathrm{rank}\{\bm{C}_{2}\}=1$.

Now consider the case in which $\mathrm{rank}\{\bm{C}_{2}\}=2$. Since $\bm{C}_{2}$ is full rank, $\det\{\bm{C}_{2}\}\neq 0$. Consider equation

[TABLE]

This is a second-order polynomial equation of $x$, for which the coefficient of the second-order term is $\det\{\bm{C}_{2}\}\neq 0$. Therefore, the fundamental theorem of algebra implies that (49) must have at least one solution. Denote $x_{0}$ as one of the solutions of (49). Recall from (Preskill, 2015, Sec 3.3), any single-qubit TKO channel with operators $\{\bm{C}_{k}\}$ can be equivalently represented by operators $\{\tilde{\bm{C}}_{k}\}$ satisfying

[TABLE]

where $\bm{A}$ is an arbitrary unitary matrix. In particular, let

[TABLE]

then $\det\{\tilde{\bm{C}}_{2}\}=\dfrac{\det\{-\bm{C}_{1}+x_{0}\bm{C}_{2}\}}{1+|x_{0}|^{2}}=0$. Thus $\mathrm{rank}\{\tilde{\bm{C}}_{2}\}\leq 1$. If it were the case that $\mathrm{rank}\{\tilde{\bm{C}}_{2}\}=0$, then $\bm{C}_{2}=\bm{0}$ implying that the channel has only one operator. This contradicts that the channel has two operators. Hence, $\mathrm{rank}\{\tilde{\bm{C}}_{2}\}=1$, which is the case that has been proven above.

Appendix B Proof of \threfthm:Str_rho

Since $\mathrm{rank}\{\bm{\rho}_{0}\}=1$, and the channel has only two operators, from (3), $\mathrm{rank}\{\bm{\rho}\}\leq 2$. Since density matrices are Hermitian, the spectral decomposition gives

[TABLE]

where

[TABLE]

In (51), we have used the fact that density matrices have trace 1. Without loss of generality, assume $F\in[\frac{1}{2},1]$.

If $p=0$, (3) and (5) imply that $\bm{\rho}=\bm{\rho}_{0}$. Setting $\bm{U}_{\mathrm{A}}=\bm{U}_{\mathrm{B}}=\mathbb{I}_{2}$ in (6), it is straightforward that the theorem holds. Also, if $\eta=0$, the channel is phase-damping. Then

[TABLE]

where

[TABLE]

Setting both local unitary operators in (6) to be Hadamard transform, i.e.,

[TABLE]

it is easy to see that the theorem also holds for the case of $\eta=0$. Therefore, the following analysis consider the case for which $p\in(0,1)$ and $|\eta|>0$.

We first determine the value of $F$. Set $\bm{A}=\begin{bmatrix}\kappa^{\dagger}&-\lambda^{\dagger}\\ \lambda&\kappa\end{bmatrix}$ in (50) with $\kappa,\lambda\in\mathbb{C}$ such that $|\kappa|^{2}+|\lambda|^{2}=1$. Then, it can be shown that

[TABLE]

where

[TABLE]

In (54), (a) is due to (3) together with the equivalence between $\{\bm{C}_{k}\}$ and $\{\tilde{\bm{C}}_{k}\}$, (b) is due to (50), and (c) is due to (5).

In order to make last line of (54) a spectral decomposition of $\bm{\rho}$, it is necessary to make $\bm{v}_{1}$ and $\bm{v}_{2}$ orthogonal. A sufficient condition for $\bm{v}^{\dagger}_{1}\bm{v}_{2}=0$ is given by

[TABLE]

Seting $|\psi\rangle=\frac{\bm{v}_{1}}{||\bm{v}_{1}||}$ and $|\phi\rangle=\frac{\bm{v}_{2}}{||\bm{v}_{2}||}$ in the last line of (54), one can get

[TABLE]

which is a spectral decomposition of $\bm{\rho}$. Since the spectrum of a matrix is unique, by comparing (55) with (51), one gets

[TABLE]

which proves (9).

We next show (7), (8). Perform Schmidt decomposition on the eigenvectors of $\bm{\rho}$ as

[TABLE]

where $\langle s|\tilde{s}\rangle=0$ for $s\in\{w,x,y,z\}$, and $\alpha,\beta,\gamma,\delta\in[0,1]$ satisfying

[TABLE]

Without loss of generality, assume $\beta\leq\alpha$, $\gamma\leq\delta$.

From (56), when $p\in(0,1)$, $F\in(\frac{1}{2},1)$. Substituting (5) into (3) and taking partial trace over different qubits, one can obtain the density matrices of the first and second qubits, i.e.,

[TABLE]

On the other hand, substituting (57) and (58) into (51) and taking partial trace, one can obtain alternative expression of $\bm{\rho}_{1}$ and $\bm{\rho}_{2}$ in terms of $|x\rangle$, $x\in\{a,b,c,d\}$. This together with (60) give

[TABLE]

We claim that $\gamma<\delta$ when $|\eta|>0$. If it were not the case, then $\gamma=\delta=\frac{1}{\sqrt{2}}$. Thus

[TABLE]

because $\langle c|\tilde{c}\rangle=0$. Substitute (63) into the left side of (61), one can get

[TABLE]

Since $\langle w|\tilde{w}\rangle=0$, the left side of (64) is a spectral decomposition of the right side, implying $\alpha=\beta=\frac{1}{\sqrt{2}}$. Substitute $\alpha=\beta=\gamma=\delta=\frac{1}{\sqrt{2}}$ into the left side of (62), and since $\langle x|\tilde{x}\rangle=\langle z|\tilde{z}\rangle=0$, one can get

[TABLE]

which holds only if $|\eta|=0$. This contradicts with the fact that $|\eta|>0$ and thus proves the claim.

Next construct two unitary operators as

[TABLE]

Substituting this into (51), it can be obtained that

[TABLE]

in which

[TABLE]

where the ket notations with subscript “$\mathrm{r}$” denote the rotated version of the original ones, e.g., $|y_{\mathrm{r}}\rangle=\bm{U}_{\mathrm{A}}|y\rangle$, $|\tilde{z}_{\mathrm{r}}\rangle=\bm{U}_{\mathrm{B}}|\tilde{z}\rangle$. Equation (67) gives the structure of (6), and (68) proves (7).

The following analysis focuses on proving (8). Since $\bm{U}_{\mathrm{A}}$, $\bm{U}_{\mathrm{B}}$ are unitary, (52) implies $\langle\mu|\nu\rangle=0$, which gives

[TABLE]

Substituting $|w\rangle=\bm{U}^{\dagger}_{\mathrm{A}}|0\rangle$, $|\tilde{w}\rangle=\bm{U}^{\dagger}_{\mathrm{A}}|1\rangle$, $|y\rangle=\bm{U}^{\dagger}_{\mathrm{A}}|y_{\mathrm{r}}\rangle$, and $|\tilde{y}\rangle=\bm{U}^{\dagger}_{\mathrm{A}}|\tilde{y}_{\mathrm{r}}\rangle$ into (61), one can get

[TABLE]

Since $\bm{U}_{\mathrm{A}}$, $\bm{U}_{\mathrm{B}}$ are unitary and $\langle s|\tilde{s}\rangle=0$, $\langle s_{\mathrm{r}}|\tilde{s}_{\mathrm{r}}\rangle=0$, where $s\in\{y,z\}$. Since $|0\rangle$, $|1\rangle$ and $|y_{\mathrm{r}}\rangle$, $|\tilde{y}_{\mathrm{r}}\rangle$ are two sets of orthonormal basis for two-dimensional Hilbert space, there exists $a,b\in\mathbb{C}$, $|a|^{2}+|b|^{2}=1$, such that

[TABLE]

Substitute (72) into (71), then

[TABLE]

Therefore

[TABLE]

Since $F<1$ and $\gamma<\delta$, from (76) and (77), one can get $a=0$ or $b=0$. Without loss of generality, let $b=0$, then $|a|=1$. Therefore (72) becomes

[TABLE]

where $\theta_{\mathrm{a}}=\mathrm{pha}\{a\}.$ Substituting (78) into (74) and (75) gives

[TABLE]

Since $F>\frac{1}{2}$, substituting (59) into (79) shows that

[TABLE]

Moreover, substituting (78) into (70) gives

[TABLE]

which implies

[TABLE]

On the other hand, since $\langle z_{\mathrm{r}}|\tilde{z}_{\mathrm{r}}\rangle=0$, it can be verified that $|\langle 0|z_{\mathrm{r}}\rangle|=|\langle 1|\tilde{z}_{\mathrm{r}}\rangle|$. Therefore from (80), $|\alpha\gamma\langle 0|z_{\mathrm{r}}\rangle|\leq|\beta\delta\langle 1|\tilde{z}_{\mathrm{r}}\rangle|$, with the equality holds only if $|\langle 0|z_{\mathrm{r}}\rangle|=|\langle 1|\tilde{z}_{\mathrm{r}}\rangle|=0$. This result together with (81) implies $|z_{\mathrm{r}}\rangle=e^{\imath\theta_{z}}|1\rangle$, $|\tilde{z_{\mathrm{r}}}\rangle=e^{\imath\theta_{\tilde{z}}}|0\rangle$, for some $\theta_{z},\theta_{\tilde{z}}\in[0,2\pi)$. Further noting that a quantum state is invariant up to an overall phase change, one can get

[TABLE]

where $\theta=\theta_{\tilde{z}}-\theta_{z}-2\theta_{\mathrm{a}}$. Equation (82) proves (8). Therefore the local unitary operators $\bm{U}_{\mathrm{A}}$, $\bm{U}_{\mathrm{B}}$ exhibited in (66) give (6)–(8).

Finally, we show that $\alpha$, $\beta$, $\gamma$, and $\delta$ satisfy (10) and (11). Substutite (68) and (82) into (67), then the density matrix of the second qubit $\check{\bm{\rho}}_{2}=\mathrm{tr}_{1}\{\check{\bm{\rho}}\}$ becomes

[TABLE]

Noting that unitary operations does not change the determinant of a matrix, $\det\{\check{\bm{\rho}}_{2}\}=\det\{\bm{\rho}_{2}\}$. Therefore, from (60) and (83) one can get

[TABLE]

Substituting (59) and (79) into (84), one can get

[TABLE]

This completes the proof.

Appendix C Proof of \threflem:simplifybound

Equation (13) holds trivially when $\gamma=0$ since fidelity of any qubit pair cannot exceed 1, i.e., $f(F,\alpha,\beta,\gamma,\delta,\theta)\leq 1$. Hence, it remains to consider the case for which $0<\gamma\leq\delta<1$, which will be proved by contradiction. Suppose \threflem:simplifybound is false, then $\forall F\in(\frac{1}{2},1]$,

[TABLE]

but there exists some $F_{0}$, $\alpha_{0}$, $\beta_{0}$, $\gamma_{0}$, $\delta_{0}$ and $\theta_{0}$ such that

[TABLE]

Then contradiction would arise if there exist some $\tilde{F}\in(\frac{1}{2},1]$ such that (85) does not hold. To show the the existence of such $\tilde{F}$, the RSSP method is employed to transform a given density matrix with parameters $F=\tilde{F}$, $\alpha=\beta=\gamma=\delta=\frac{1}{\sqrt{2}}$, and $\theta=0$ to another density matrix with parameters $F_{0}$, $\alpha_{0}$, $\beta_{0}$, $\gamma_{0}$, $\delta_{0}$ and $\theta_{0}$ via LOCC. In particular, consider that Alice measures her qubit using local operators

[TABLE]

and Bob measures his qubit using local operators

[TABLE]

When the measurement results correspond to $\bm{M}_{\mathrm{A}}$ and $\bm{M}_{\mathrm{B}}$, the density matrix of the qubit pair after the measurement is given by

[TABLE]

where $\check{\bm{\rho}}$ is the density matrix given in (6). Set the channel to be phase-damping, i.e. $\eta=0$, then $\alpha=\beta=\gamma=\delta=\frac{1}{\sqrt{2}}$, and $\theta=0$. Further set the channel parameter $p$ so that $F$ equals to $\tilde{F}$ given by888Equation (10)–(11) imply $0\leq\gamma\leq\beta\leq\frac{1}{\sqrt{2}}\leq\alpha\leq\delta\leq 1$ and $\alpha^{2}+\beta^{2}=\gamma^{2}+\delta^{2}=1$, showing that $\frac{\gamma\delta}{\alpha\beta}\in[0,1]$ for all valid $\alpha$, $\beta$, $\gamma$, and $\delta$. Hence, $\tilde{F}$ in (90) is in the interval $(\frac{1}{2},1]$ as long as $F_{0}\in(\frac{1}{2},1]$. This guarantees the existence of $p$.

[TABLE]

Then according to (89), the LOCC for RSSP transforms a density matrix $\check{\bm{\rho}}$ with parameters $F=\tilde{F}$, $\alpha=\beta=\gamma=\delta=\frac{1}{\sqrt{2}}$, and $\theta=0$ to another density matrix given by

[TABLE]

whose parameters are $F_{0}$, $\alpha_{0}$, $\beta_{0}$, $\gamma_{0}$, $\delta_{0}$ and $\theta_{0}$. Let $\{\bm{N}^{(k)}_{\mathrm{A}},\bm{N}^{(k)}_{\mathrm{B}}\}_{k=1}^{K}$ be local operators that achieve the optimal fidelity $f(F_{0},\alpha_{0},\beta_{0},\gamma_{0},\delta_{0},\theta_{0})$ with initial density matrix $\breve{\bm{\rho}}$. Define new local operators

[TABLE]

Then $\{\bm{L}^{(k)}_{\mathrm{A}},\bm{L}^{(k)}_{\mathrm{B}}\}_{k=1}^{K}$ are valid local operators, and achieve the same fidelity $f(F_{0},\alpha_{0},\beta_{0},\gamma_{0},\delta_{0},\theta_{0})$ with initial density matrix $\check{\bm{\rho}}$. Therefore, the optimal fidelity with initial density matrix $\check{\bm{\rho}}$ is lower bounded by

[TABLE]

This together with (86) gives

[TABLE]

With (93) the contradiction arises. This completes the proof.

Appendix D Proof of \threflem:operation

Without loss of generality, denote the seperable operator acting on two qubit pairs as $\bm{N}_{\mathrm{A}}\otimes\bm{N}_{\mathrm{B}}$, where $\bm{N}_{\mathrm{A}}$, $\bm{N}_{\mathrm{B}}$ are employed by Alice and Bob respectively. Every operator $\bm{N}$ for two qubits can be written equivalently as $\bm{N}=\bm{N}(\bm{H}^{\dagger}\otimes\bm{H}^{\dagger})(\bm{H}\otimes\bm{H}$), where $\bm{H}=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\ 1&-1\end{bmatrix}$ is the Hadamard operator. Therefore, denote $\tilde{\bm{N}}_{\mathrm{X}}=\bm{N}_{\mathrm{X}}(\bm{H}^{\dagger}\otimes\bm{H}^{\dagger})$, $X\in\{A,B\}$, then the separable operator $\bm{N}_{\mathrm{A}}\otimes\bm{N}_{\mathrm{B}}$ for two qubit pairs is equivalent to first perform $\bm{H}$ on every qubit, and then perform $\tilde{\bm{N}}_{\mathrm{A}}\otimes\tilde{\bm{N}}_{\mathrm{B}}$.

From (6), $\check{\bm{\rho}}=F|\Phi^{+}\rangle\langle\Phi^{+}|+(1-F)|\Psi^{+}\rangle\langle\Psi^{+}|$ when the channel is phase-damping. Hence after performing Hadamard operation on the qubits, the density matrix of the qubit pair becomes

[TABLE]

Therefore, the joint density matrix of two qubit pair, where the first and last two qubits belong to Alice Bob respectively, is given by

[TABLE]

where $\bm{P}$ is the permutation operator that switches the second and third qubits,

[TABLE]

From (95), after operator $\tilde{\bm{N}}_{\mathrm{A}}\otimes\tilde{\bm{N}}_{B}$ acts on the two qubit pairs, the density matrix of the first qubit pair is given by

[TABLE]

where $C_{1}=F^{2}$, $C_{2}=C_{3}=F(1-F)$, $C_{4}=(1-F)^{2}$, and \boldsymbol{\phi}^{(i)}=\big{(}\tilde{\bm{N}}_{\mathrm{A}}\otimes\tilde{\bm{N}}_{B}\big{)}|\Phi^{(i)}\rangle. Denote $|w\rangle=|00\rangle$, $|x\rangle=|01\rangle$, $|y\rangle=|10\rangle$, and $|z\rangle=|11\rangle$, and denote

[TABLE]

Then

[TABLE]

where $\bm{v}^{(i)}$ is the $i$-th column of the unitary matrix $\bm{V}$ defined in (21) and

[TABLE]

$s\in\{w,x,y,z\}$. Combining (97) and (98) gives (14) and (15).

In (99), $|ss\rangle$ is a separable state, and

[TABLE]

is a separable operator. Therefore, vectors $\bm{s}$, ${s}\in\{{w},{x},{y},{z}\}$ must also be separable. As $1\times 4$ vectors, $\bm{s}$ are separable if and only if

[TABLE]

which give (16).

Finally, the probability that $\bm{N}_{\mathrm{A}}\otimes\bm{N}_{\mathrm{B}}$ acts on the qubits must not be [math], which implies

[TABLE]

This completes the proof.

Appendix E Proof of \threfthm:UBF

We will first prove that the fidelity $F^{*}$ given in (22) is an upper bound. From \threflem:simplifybound, it is sufficient to prove the upper bound for the special case when the channel is phase-damping.

Express the LOCC performed by the agents as $\bm{N}^{(k)}_{\mathrm{A}}\otimes\bm{N}^{(k)}_{\mathrm{B}}$, $k\in\{1,2,\ldots,K\}$. Without loss of generality, assume $\bm{N}^{(1)}_{\mathrm{A}}\otimes\bm{N}^{(1)}_{\mathrm{B}}$ is one of the operators that lead to the highest fidelity. Then from \threflem:operation, conditioned on the event that $\bm{N}^{(1)}_{\mathrm{A}}\otimes\bm{N}^{(1)}_{\mathrm{B}}$ acts on the two qubit pairs, the fidelity of the kept qubit pair is given by

[TABLE]

where $C_{i}$, $\bm{v}^{(i)}$, $i\in\{1,2,3,4\}$ and $s_{kj}$, $s\in\{w,x,y,z\}$, $k,j\in\{1,2\}$ are defined in \threflem:operation. Note that from \threfthm:Str_rho, $F\in[\frac{1}{2},1]$, implying that $C_{1}\geq C_{2}=C_{3}\geq C_{4}\geq 0$. Therefore, to prove the upper bound part of \threfthm:UBF, it is sufficient to show that the following proposition is true.

Proposition 1** (Maximum fidelity):**

\thlabel

cor:FUB_1 For any $s_{kj}\in\mathbb{C}$, $s\in\{w,x,y,z\}$, $k,j\in\{1,2\}$, and $1\geq C_{1}\geq C_{2}\geq C_{3}\geq C_{4}\geq 0$, satisfying $s_{11}s_{22}=s_{12}s_{21}$ and

[TABLE]

the following inequality holds

[TABLE]

To prove the proposition above, first simplify (100) via the following lemma.

Lemma 4** (Simplify Parameters):**

\thlabel

lem:C Consider coefficients $r_{1},r_{2},\check{r}_{2},r_{3},\check{r}_{3},r_{4}\geq 0$, $\check{r}_{4}>0$ and variable $t\in[0,1]$, satisfying $r_{3}t+r_{4}>0$ and

[TABLE]

If inequality

[TABLE]

holds for $t=\check{t}\geq 0$, then it holds for all $t\in[0,\check{t}]$.

Proof.

Define function

[TABLE]

From (101), $f(0)\leq 0$. Since $r_{3}t+r_{4}>0$ and $\check{r}_{3}t+\check{r}_{4}>0$, the fact that (102) holds for $t=\check{t}$ is equivalent to $f(\check{C})\leq 0$. Moreover, since $f^{\prime\prime}(t)=r_{1}\check{r}_{3}\geq 0$, $f(t)$ is a convex function. Therefore, $f(t)\leq 0$, $\forall t\in[0,\check{t}]$, which is equivalent to (102) holds $\forall t\in[0,\check{t}]$. This completes the proof of \threflem:C. ∎

Letting $C_{4}=t$,

[TABLE]

$2C_{1}=\check{r}_{2}$, $1=\check{r}_{3}$, and $C_{1}=\check{r}_{4}$ in (100) gives the form of (102). It can be verified that

[TABLE]

Therefore, \threflem:C shows that (102) is true $\forall t\in[0,C_{3}]$ if it is true for $t=C_{3}$. This implies that to prove \threfcor:FUB_1, it is sufficient to prove (100) for the case of $C_{4}=C_{3}$. Repeating this process two more times, i.e., appling \threflem:C to (100) with $C_{3}=t$, and then with $C_{2}=t$, it can be shown that considering the case in which $C_{1}=C_{2}=C_{3}=C_{4}$ is sufficient to prove the proposition. Then, (100) simplifies to

[TABLE]

Note that

[TABLE]

where the last equality is due to the fact that $\bm{V}$ is unitary. Similarly,

[TABLE]

Then it can be obtained that

[TABLE]

where (a), (b), and (c) are due to (104), (16), and (105), respectively. This inequaltiy shows that (103) is true, which then proves \threfcor:FUB_1. This proofs that the fidelity $F^{*}$ given in (22) is an upper bound.

Finally, we use constructive method to show that fidelity $F^{*}$ given in (22) is achievable. In fact, (38) of \threfthm:dis_perf shows that the fidelity in (22) is achieved by adopting the RSSP and first round distillation of the algorithm proposed in Section III.3 and keeps a qubit pair only if measurement results correspond to $|1\rangle\langle 1|$. This completes the proof.

Appendix F Proof of \threfthm:dis_perf

First, the following lemma summarizes the effect of RSSP.

Lemma 5** (Performance of RSSP):**

\thlabel

lem:single In process of RSSP, qubit pairs are kept with probability

[TABLE]

For a kept qubit pair, its density matrix is given by

[TABLE]

where

[TABLE]

with

[TABLE]

Proof.

The qubit pairs are kept with probability

[TABLE]

and the density matrix of a kept qubit pair is given by

[TABLE]

Substituting (6)–(11) into (113) and (114), one can obtain (108)–(112). The details are omitted for brevity. ∎

From (108) and (109), after the RSSP, the joint density matrix of two qubit pairs, where the first and last two qubits belong to Alice and Bob respectively, is given by

[TABLE]

where $\bm{P}$ is the permutation operator that switches the second and third qubits, and

[TABLE]

For the first round of distillation, after both agents perform the CNOT operation, the joint density matrix of two qubit pairs becomes

[TABLE]

where

[TABLE]

From (124), if both measurement results correspond to $|1\rangle\langle 1|$, the (unnormalized) density matrix of the source qubit pair is given by

[TABLE]

Otherwise, if both measurement results correspond to $|0\rangle\langle 0|$, the (unnormalized) density matrix of the source qubit pair is given by

[TABLE]

From (134), and (135), if the agents adopt the FP approach, i.e., keep the source qubit pair only if both measurement results correspond to $|1\rangle\langle 1|$, the probability of keeping the source qubit pair is

[TABLE]

the fidelity of the kept qubit pairs is

[TABLE]

and the density matrix of the kept qubit pair is

[TABLE]

If the agents adopt the PP approach, i.e., keeping the source qubit pair if the measurement results match, the probability of preserving the source qubit pair is

[TABLE]

the fidelity of the kept qubit pairs is

[TABLE]

and the density matrix of the kept qubit pair can be written as

[TABLE]

where $G+\tilde{G}=1-F_{1}$, $|\Psi\rangle,|\tilde{\Psi}\rangle\in\mathrm{span}(|01\rangle,|10\rangle)$, and $\langle\Psi|\tilde{\Psi}\rangle=0$.

From \threflem:single, (136), (137), (139) and (140), after the RSSP and first round of distillation, a qubit pair is kept with probability

[TABLE]

and fidelity

[TABLE]

For the following rounds of distillations, one can take (138) or (141) as input, use similar analysis as in (124)–(135) and (139)–(141). This analysis will show that

[TABLE]

and the density matrix of the kept qubit pairs maintains the same structure as in (138) or (141). This competes the proof.