Approximate broadcasting of quantum correlations

TL;DR

This paper characterizes two-sided quantum discord through bilocal broadcasting and formulates optimal broadcasting fidelities as efficiently computable semidefinite programs, advancing understanding of quantum correlations.

Contribution

It provides a full operational characterization of two-sided quantum discord and introduces SDP formulations for optimal broadcasting fidelities, enabling practical computation.

Findings

Operational characterization of two-sided quantum discord.

SDP formulations for optimal broadcasting fidelities.

Evaluation of broadcasting fidelities in specific cases.

Abstract

Broadcasting quantum and classical information is a basic task in quantum information processing, and is also a useful model in the study of quantum correlations including quantum discord. We establish a full operational characterization of two-sided quantum discord in terms of bilocal broadcasting of quantum correlations. Moreover, we show that both the optimal fidelity of unilocal broadcasting of the correlations in an arbitrary bipartite quantum state and that of broadcasting an arbitrary set of quantum states can be formulized as semidefinite programs (SDPs), which are efficiently computable. We also analyze some properties of these SDPs and evaluate the broadcasting fidelities for some cases of interest.

Equations

$$D_A({\rho }_{AB})\text{mathchar}58=\underset{{\mathcal{E}}_A\in \text{QC}}{\min }(I(A\text{mathchar}58B{)}_{{\rho }_{AB}}-I(A\text{mathchar}58B{)}_{({\mathcal{E}}_A\otimes {\text{mathds}1}_B){\rho }_{AB}}),$$

$$D_{AB}({\rho }_{AB})\text{mathchar}58=\underset{{\mathcal{E}}_A,{\mathcal{F}}_B\in \text{QC}}{\min }(I(A\text{mathchar}58B{)}_{{\rho }_{AB}}-I(A\text{mathchar}58B{)}_{({\mathcal{E}}_A\otimes {\mathcal{F}}_B){\rho }_{AB}}),$$

$$\begin{array}{rl}& D_A({\rho }_{AB})\\ =& \underset{n\to \infty }{\lim }\underset{{\Lambda }_{A\to A_1\dots A_n}}{\min }\frac{1}{n}\sum _{j=1}^n\left(I(A\text{mathchar}58B{)}_{{\rho }_{AB}}-I(A_j\text{mathchar}58B{)}_{({\Lambda }_j\otimes {\text{mathds}1}_B){\rho }_{AB}}\right).\end{array}$$

$$\begin{array}{rl}& D_{AB}({\rho }_{AB})\\ =& \underset{n\to \infty }{\lim }\underset{\begin{array}{c}{\Lambda }_{A\to A_1\dots A_n}\\ {\Gamma }_{B\to B_1\dots B_n}\end{array}}{\min }\frac{1}{n}\sum _{j=1}^n\left(I(A\text{mathchar}58B{)}_{{\rho }_{AB}}-I(A_j\text{mathchar}58B_j{)}_{({\Lambda }_j\otimes {\Gamma }_j){\rho }_{AB}}\right).\end{array}$$

$$\Vert {\Lambda }_j-{\mathcal{E}}_j{\Vert }_{\diamond }\le 3{\left(\frac{\ln (2)|A{|}^6{\log }_2|A|}{n{\delta }^3}\right)}^{1/3},$$

$$\begin{array}{rl}& |I(A\text{mathchar}58B{)}_{\rho }-I(A\text{mathchar}58B{)}_{\sigma }|\\ \le & |H(A{)}_{\rho }-H(A{)}_{\sigma }|+|H(A|B{)}_{\rho }-H(A|B{)}_{\sigma }|\\ \le & 8\gamma {\log }_2|A|+\gamma {\log }_2(|A|-1)+2h_2(2\gamma )+h_2(\gamma ).\end{array}$$

$$\begin{array}{rl}& \underset{{\mathcal{E}}_A,{\mathcal{F}}_B\in \text{QC}}{\max }I(A\text{mathchar}58B{)}_{({\mathcal{E}}_A\otimes {\mathcal{F}}_B){\rho }_{AB}}\\ =& \underset{n\to \infty }{\lim }\underset{\Lambda ,\Gamma }{\max }\frac{1}{n}\sum _{j=1}^{n}I(A_j\text{mathchar}58B_j{)}_{({\Lambda }_j\otimes {\Gamma }_j){\rho }_{AB}}.\end{array}$$

$$\Vert {\Gamma }_j-{\mathcal{F}}_j{\Vert }_{\diamond }\le 3{\left(\frac{\ln (2)|B{|}^6{\log }_2|B|}{n{\delta }^3}\right)}^{1/3},$$

$$\Vert {\Lambda }_j\otimes {\Gamma }_j-{\mathcal{E}}_j\otimes {\mathcal{F}}_j{\Vert }_{\diamond }$$

$$\le 6{\left(\frac{\ln (2)d^6{\log }_{2}d}{n{\delta }^3}\right)}^{1/3}=\text{mathchar}58\epsilon ,$$

$$\Vert ({\Lambda }_j\otimes {\Gamma }_j){\rho }_{AB}-({\mathcal{E}}_j\otimes {\mathcal{F}}_j){\rho }_{AB}{\Vert }_1\le \Vert {\Lambda }_j\otimes {\Gamma }_j-{\mathcal{E}}_j\otimes {\mathcal{F}}_j{\Vert }_{\diamond }\le \epsilon .$$

$$I(A_j\text{mathchar}58B_j{)}_{({\Lambda }_j\otimes {\Gamma }_j){\rho }_{AB}}$$

$$\le$$

$$+2h_2(\epsilon )+h_2(\epsilon /2)$$

$$\le$$

$$+\frac{\epsilon }{2}{\log }_2(|A_j|-1)+2h_2(\epsilon )+h_2(\epsilon /2)$$

$$\le$$

$$=\text{mathchar}58$$

$$\frac{1}{n}\sum _{j=1}^{n}I(A_j\text{mathchar}58B_j{)}_{({\Lambda }_j\otimes {\Gamma }_j){\rho }_{AB}}$$

$$\le$$

$$\to$$

$$\underset{n\to \infty }{\lim }\underset{\Lambda ,\Gamma }{\max }\frac{1}{n}\sum _{j=1}^{n}I(A_j\text{mathchar}58B_j{)}_{({\Lambda }_j\otimes {\Gamma }_j){\rho }_{AB}}$$

$$\le$$

$$\begin{split}f_{n}(\rho_{AB})=\sup\Bigl{\{}&\frac{1}{n}\sum_{j=1}^{n}F(\rho_{AB},\operatorname{Tr}_{\backslash A_{j}B}\Lambda_{A\to A_{1}...A_{n}}(\rho_{AB}))\mathrel{\mathop{\mathchar 58\relax}}\\ &\Lambda_{A\to A_{1}\cdots A_{n}}\text{ is a quantum channel}\bigl{\}}.\end{split}$$

$$W_{\pi }|j_1,j_2,\dots ,j_n⟩=|j_{{\pi }^{-1}(1)},j_{{\pi }^{-1}(2)},\dots ,j_{{\pi }^{-1}(n)}⟩$$

$$\Lambda (\rho )=W_{\pi }(\Lambda (\rho ))W_{\pi }^{†}$$

$${\mathrm{Tr}}_{\backslash A_{j}B}{\Lambda }_{A\to A_1...A_n}({\rho }_{AB})$$

$$=$$

$$\begin{array}{rl}{f}_n({\rho }_{AB})=& \max \{F({\rho }_{AB},{\mathrm{Tr}}_{\backslash A_{1}B}{\Lambda }_{A\to A_1...A_n}({\rho }_{AB}))\text{mathchar}58\\ & {\Lambda }_{A\to A_1\cdots A_n}\text{ is a symmetric broadcasting channel}\}.\end{array}$$

$$\begin{array}{rl}{f}_n({\rho }_{AB})=\max & \frac{1}{2}\mathrm{Tr}(X_{AB}+X_{AB}^{†})\\ s.t.& \left(\begin{array}{cc}{\rho }_{AB}& X_{AB}\\ X_{AB}^{†}& {\mathrm{Tr}}_{\backslash A_{1}B}(J^{T_A}{\rho }_{AB})\end{array}\right)\ge 0,\\ & J_{A{A}_1\cdots A_n}\ge 0,{\mathrm{Tr}}_{\backslash A}{J}_{A{A}_1\cdots A_n}={\text{mathds}1}_A,\\ & J_{A{A}_1\cdots A_n}=\frac{1}{n!}\sum _{\pi \in S_n}{W}_{\pi }{J}_{A{A}_1\cdots A_n}{W}_{\pi }^{†},\end{array}$$

$${\Lambda }_{A\to A_1\cdots A_n}({\rho }_A)={\mathrm{Tr}}_A(J_{A{A}_1\cdots A_n}^{T_A}{\rho }_A).$$

$$({\Lambda }_{A\to A_1\cdots A_n}\otimes {\text{mathds}1}_B){\rho }_{AB}={\mathrm{Tr}}_A(J_{A{A}_1\cdots A_n}^{T_A}{\rho }_{AB}),$$

$${\mathrm{Tr}}_{\backslash A_{j}B}(\Lambda \otimes {\text{mathds}1}_B){\rho }_{AB}={\mathrm{Tr}}_{\backslash A_{j}B}(J_{A{A}_1\cdots A_n}^{T_A}{\rho }_{AB}).$$

$$\begin{array}{rl}{f}_n({\rho }_{AB})=\max & F({\rho }_{AB},{\hat{\rho }}_{AB})\\ s.t.& {\hat{\rho }}_{AB}={\mathrm{Tr}}_{\backslash A_{1}B}(J_{A{A}_1\cdots A_n}^{T_A}{\rho }_{AB}),\\ & J_{A{A}_1\cdots A_n}\ge 0,{\mathrm{Tr}}_{\backslash A}{J}_{A{A}_1\cdots A_n}={\text{mathds}1}_A,\\ & J_{A{A}_1\cdots A_n}=\frac{1}{n!}\sum _{\pi \in S_n}{W}_{\pi }{J}_{A{A}_1\cdots A_n}{W}_{\pi }^{†}.\end{array}$$

$$\begin{array}{rl}F(\rho ,\sigma )=\max & \frac{1}{2}\mathrm{Tr}(X+X^{†})\\ s.t.& \left(\begin{array}{cc}\rho & X\\ X^{†}& \sigma \end{array}\right)\ge 0.\end{array}$$

$$\begin{array}{rl}{f}_n({\psi }_{AB})=\max & \sqrt{\mathrm{Tr}({\hat{\rho }}_{AB}{\psi }_{AB})}\\ s.t.& {\hat{\rho }}_{AB}={\mathrm{Tr}}_{\backslash A_{1}B}(J_{A{A}_1\cdots A_n}^{T_A}{\psi }_{AB}),\\ & J_{A{A}_1\cdots A_n}\ge 0,{\mathrm{Tr}}_{\backslash A}{J}_{A{A}_1\cdots A_n}={\text{mathds}1}_A,\\ & J_{A{A}_1\cdots A_n}=\frac{1}{n!}\sum _{\pi \in S_n}{W}_{\pi }{J}_{A{A}_1\cdots A_n}{W}_{\pi }^{†},\end{array}$$

Full text

I Introduction

Copying information is a rather simple task in the classical realm, but unfortunately not in the quantum realm. It is not allowed to create an identical copy of an arbitrary unknown pure quantum state due to the no-cloning theorem Wootters and Zurek (1982); Dieks (1982). One can clone a set of pure states if and only if they are orthogonal. The no-broadcasting theorem Barnum et al. (1996) generalizes this result to mixed states, saying that a set of quantum states can be broadcast if and only if the states commute with each other.

These no-go theorems can be further extended to the setting of local broadcast for composite quantum systems. Given a bipartite quantum state $\rho_{AB}$ shared by Alice and Bob, their objective is to perform local operations only (without communication) to produce a state $\widehat{\rho}_{A_{1}A_{2}B_{1}B_{2}}=(\Lambda_{A\to A_{1}A_{2}}\otimes\Gamma_{B\to B_{1}B_{2}})\rho_{AB}$ such that $\operatorname{Tr}_{A_{1}B_{1}}\widehat{\rho}_{A_{1}A_{2}B_{1}B_{2}}=\operatorname{Tr}_{A_{2}B_{2}}\widehat{\rho}_{A_{1}A_{2}B_{1}B_{2}}=\rho_{AB}$ (see Section II for notational convention). It is shown in Piani et al. (2008) that this task can only be performed if $\rho_{AB}$ is classically correlated. Even if the task is relaxed to obtain two bipartite states with the same correlation as $\rho_{AB}$ (measured by the mutual information), it is feasible to do the task if and only if the given state $\rho_{AB}$ is classically correlated. This is called the no-local-broadcasting theorem Piani et al. (2008). Furthermore, when the local operations ar only allowed for one party (e.g., Alice), the task can be done if and only if $\rho_{AB}$ is classical on $A$ Luo and Sun (2010); Luo (2010); Piani (2017).

When the task of perfect broadcasting cannot be accomplished, it is natural to ask whether the broadcasting can be performed in an approximate fashion, and how to design the optimal broadcasting operation. We shall study the approximate broadcasting of states and correlations by utilizing semidefinite programs (SDPs). In Ref. Piani (2016) the Bose-symmetric channel is considered as unilocal broadcasting operation and an SDP is derived for this problem. Semidefinite programming optimization techniques Boyd and Vandenberghe (2004) have found many applications to the theory of quantum information and computation (see, e.g., Doherty et al. (2002); Rains (2001); Wang and Duan (2016a); Jain et al. (2011); Wang and Duan (2016b); Berta and Tomamichel (2016); Li et al. (2017); Kempe et al. (2010); Wang et al. (2016)), and also to the study of quantum correlations (see, e.g., Skrzypczyk et al. (2014); Navascués et al. (2008); Piani (2016); Napoli et al. (2016)).

Quantum discord (see Section III for definition), as an indispensable measure of quantum correlation beyond entanglement, is introduced in Ollivier and Zurek (2001) and Henderson and Vedral (2001) independently. It is argued Datta et al. (2008) that quantum discord is responsible for the quantum speed-up over classical algorithms. Quantum discord is a quite useful concept in many fields of quantum information processing, such as local broadcasting of correlations Piani et al. (2008); Brandão et al. (2015), quantum computing Knill and Laflamme (1998), quantum data hiding Piani et al. (2014), quantum data locking Boixo et al. (2011), entanglement distribution Chuan et al. (2012); Streltsov et al. (2012), common randomness distillation Devetak and Winter (2004), quantum state merging Cavalcanti et al. (2011); Madhok and Datta (2011, 2013), entanglement distillation Madhok and Datta (2013); Streltsov et al. (2011), superdense coding Madhok and Datta (2013), quantum teleportation Madhok and Datta (2013), quantum metrology Girolami et al. (2014), and quantum cryptography Pirandola (2014). Quantum discord has become an active research topic over the past few years Modi et al. (2012); Adesso et al. (2016).

The local broadcasting paradigm can provide a natural operational interpretation to quantum discord. Remarkably, the minimum average loss of mutual information resulting from local operation $\Lambda_{A\to A_{1}\cdots A_{n}}$ on $A$ for arbitrary quantum state $\rho_{AB}$ approaches the quantum discord $D_{A}(\rho_{AB})$ of $\rho_{AB}$ as $n$ goes to infinity. This result is established in Ref. Brandão et al. (2015) and it generalizes the work in Ref. Streltsov and Zurek (2013) which considers pure states $\rho_{AB}$ only. However, it remains open whether there is an analogous connection for the two-sided setting of redistributing correlations Adesso et al. (2016).

In this paper, we study the approximate broadcasting of quantum correlations in both asymptotic and non-asymptotic settings. In the asymptotic regime, we rigorously prove the conjecture in Ref. Adesso et al. (2016) and show an operational meaning of the two-sided discord in terms of bilocal broadcasting of correlations; that is, the asymptotic minimum average loss of correlation after optimal bilocal broadcasting is exactly the two-sided quantum discord of the initial state. In the non-asymptotic regime, we give an alternate derivation for the SDP characterization of the optimal unilocal broadcasting fidelity and show that the universal quantum clone machine (UQCM) can also serve as the optimal universal unilocal broadcasting operation. Moreover, the optimal state-dependent unilocal broadcasting operation for pure two-qubit states is analytically solved. Similarly, we establish the SDP for the optimal broadcasting fidelity of a finite set of quantum states.

II Preliminaries

A quantum system $A$ is associated to a Hilbert space ${\cal H}_{A}$ of dimension $|A|$ with some fixed orthonormal basis $\{|j\rangle_{A}\}_{j}$. In this work, we only deal with finite-dimensional spaces, and the spaces of systems with the same letter are always assumed to be isomorphic, for example, ${\cal H}_{A}\cong{\cal H}_{\widetilde{A}}\cong{\cal H}_{A_{1}}\cong{\cal H}_{A_{2}}$. The linear operators from ${\cal H}_{A}$ to ${\cal H}_{B}$ are always written with subscripts identifying the systems involved, for example, $X_{A\to B}$. We denote ${\cal S}(A)$ as the set of density operators Nielsen and Chuang (2011) on system $A$.

A quantum operation (or channel) ${\cal E}_{A\to B}$ with input system $A$ and output system $B$ is a completely positive (CP), trace preserving (TP) linear map from the linear operators on ${\cal H}_{A}$ to the linear operators on $B$. A quantum-to-classical channel ${\cal F}$ is a cptp map such that ${\cal F}(\cdot)=\sum_{j}\operatorname{Tr}(M_{j}\cdot)|j\rangle\!\langle j|$, where $\{M_{j}\}_{j}$ is a POVM. The set of all quantum-to-classical channels is denoted by QC. Since the subsript of an operator or operation specifies its input and output systems, we can write a product of operators or operations without the $\otimes$ symbol, and omit the identity operator or operation ${\mathds{1}}$, which would make no confusion, for example, $X_{AB}Y_{BC}\equiv(X_{AB}\otimes{\mathds{1}}_{C})({\mathds{1}}_{A}\otimes Y_{BC})$ and ${\cal E}_{B\to C}(X_{AB})\equiv({\mathds{1}}_{A}\otimes{\cal E}_{B\to C})X_{AB}$.

The Choi-Jamiołkowski matrix Jamiołkowski (1972); Choi (1975) of a quantum operation ${\cal E}_{A\to B}$ is $J_{{\cal E}}=({\mathds{1}}_{\widetilde{A}\to\widetilde{A}}\otimes{\cal E}_{A\to B})\phi_{\widetilde{A}A}$, where $\phi_{\widetilde{A}A}=\sum_{ij}|ii\rangle\!\langle jj|$ is the unnormalized maximally entangled state. The output of the channel ${\cal E}_{A\to B}$ with input $\rho_{A}$ can be recovered from $J_{{\cal E}}$ by ${\cal E}_{A\to B}(\rho_{A})=\operatorname{Tr}_{A}(J_{{\cal E}}^{T_{A}}\rho_{A})$, where $T_{A}$ denotes the partial transpose on $A$.

We use $H(\cdot)$ to denote the von Neumann entropy of quantum states, $H(A|B)\mathrel{\mathop{\mathchar 58\relax}}=H(AB)-H(B)$ the conditional quantum entropy, $I(A\mathrel{\mathop{\mathchar 58\relax}}B)\mathrel{\mathop{\mathchar 58\relax}}=H(A)+H(B)-H(AB)$ the quantum mutual information. The fidelity $F(\rho,\sigma)=\operatorname{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}$, as a measure of similarity between quantum states, can be viewed as the optimal solution to an SDP Killoran (2012); Watrous (2012). The diamond norm can be used to give the distance of two quantum operations ${\cal E},{\cal F}$, that is, $\|{\cal E}-{\cal F}\|_{\diamond}=\sup\{\|(({\cal E}-{\cal F})\otimes{\mathds{1}})X\|_{1}\mathrel{\mathop{\mathchar 58\relax}}\|X\|_{1}=1\}$, where $\|\cdot\|_{1}$ is the trace norm. In addition, we denote $[n]=\{1,\dots,n\}$, and denote by $|\cdot|$ the cardinality of a set or the dimension of a linear space.

III Asymptotic bilocal broadcasting and two-sided quantum discord

III.1 Previous results

The concept of quantum discord was introduced by Ollivier and Zurek (2001); Henderson and Vedral (2001). Here the one-sided and two-sided quantum discord of a bipartite state $\rho_{AB}$ are defined by

[TABLE]

[TABLE]

respectively.

It is shown in Brandão et al. (2015) that the one-sided quantum discord is equal to the asymptotic average loss of correlation after the optimal broadcasting operation. Consider the following scenario. Alice and Bob, apart away from each other, share a bipartite quantum state $\rho_{AB}$. The information, or correlation, shared by them is measured by quantum mutual information in what follows. The goal of Alice is to broadcast the mutual information between them to many, say $n$, recipients, using local operation only. If the state is not classical on $A$, she cannot perform the task perfectly Luo and Sun (2010); Luo (2010); Piani (2017), and the mutual information between each recipient and Bob would decrease in general. Now her task naturally becomes to design a broadcasting operation in order to minimize the average loss of mutual information. Remarkably, the minimal average loss of correlation approaches quantum discord $D_{A}$ of $\rho_{AB}$ as $n$ tends to infinity, as revealed in the following Proposition.

Proposition 1** **(Brandão et al. (2015))

Let $\rho_{AB}$ be a bipartite state and $D_{A}$ is defined by Eq. (1). Let $\Lambda_{A\to A_{1}\dots A_{n}}$ be a cptp map and $\Lambda_{j}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Tr}_{\backslash A_{j}}\circ\Lambda$. Then

[TABLE]

III.2 Operational interpretation of two-sided quantum discord

We will give an operational interpretation of two-sided quantum discord in terms of bilocal broadcasting, analogous to the case of one-sided quantum discord (see Fig. 1).

Theorem 2

Let $\rho_{AB}$ be a bipartite state, and the two-sided quantum discord $D_{AB}(\rho_{AB})$ is defined by Eq. (2). Let $\Lambda_{A\to A_{1}\dots A_{n}}$ and $\Gamma_{B\to B_{1}\dots B_{n}}$ be cptp maps, and denote $\Lambda_{j}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Tr}_{\backslash A_{j}}\circ\Lambda$ and $\Gamma_{j}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Tr}_{\backslash B_{j}}\circ\Gamma$. Then

[TABLE]

In order to prove this theorem, we need the following result.

Lemma 3** **(Brandão et al. (2015))

Let $\Lambda\mathrel{\mathop{\mathchar 58\relax}}{\cal S}(A)\to{\cal S}(A_{1}\otimes\cdots\otimes A_{n})$ be a cptp map. Denote $\Lambda_{j}\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{Tr}_{\backslash A_{j}}\circ\Lambda$, and fix a number $0<\delta<1$. Then there exsits a POVM $\{E_{k}\}_{k}$ and a set $S\subset[n]$ with $|S|\geq n(1-\delta)$ such that for all $j\in S$,

[TABLE]

with ${\cal E}_{j}(\cdot)\mathrel{\mathop{\mathchar 58\relax}}=\sum_{k}\operatorname{Tr}(E_{k}\cdot)\sigma_{j,k}$ for states $\sigma_{j,k}\in{\cal S}(A_{j})$. Here $|A|$ is the dimension of the space $A$.

We also need the continuity bound of mutual information. Let $\rho_{AB},\sigma_{AB}$ be bipartite states on system A, of dimension $|A|\geq 2$, and system B. Assume $\gamma\mathrel{\mathop{\mathchar 58\relax}}=\frac{1}{2}\|\rho_{AB}-\sigma_{AB}\|_{1}\leq\frac{1}{2}$. Due to Fannes-Audenaert inequality Fannes (1973); Audenaert (2007) and the fact that quantum operation cannot increase trace distance between two states, it holds $|H(A)_{\rho_{AB}}-H(A)_{\sigma_{AB}}|\leq\frac{1}{2}\|\rho_{A}-\sigma_{A}\|_{1}\log_{2}(|A|-1)+h_{2}(\frac{1}{2}\|\rho_{A}-\sigma_{A}\|_{1})\leq\gamma\log_{2}(|A|-1)+h_{2}(\gamma)$, where $h_{2}(x)\mathrel{\mathop{\mathchar 58\relax}}=-x\log_{2}x-(1-x)\log_{2}(1-x)$ is the binary entropy functon. Due to Alicki-Fannes inequality Alicki and Fannes (2004) (see also Winter (2016) for a tighter continuity bound for conditional entropy), it holds $|H(A|B)_{\rho}-H(A|B)_{\sigma}|\leq 8\gamma\log_{2}|A|+2h_{2}(2\gamma)$. Therefore,

[TABLE]

We are now in the position to prove Theorem 2.

Proof.

The desired statement is equivalent to that

[TABLE]

Assume the POVMs that achieve $I_{c}(A\mathrel{\mathop{\mathchar 58\relax}}B)\mathrel{\mathop{\mathchar 58\relax}}=\max_{{\cal E}_{A},{\cal F}_{B}\in\text{QC}}I(A\mathrel{\mathop{\mathchar 58\relax}}B)_{({\cal E}_{A}\otimes{\cal F}_{B})\rho_{AB}}$ are $\{M_{i}\}_{i}$ on $A$ and $\{N_{i}\}_{i}$ on $B$, then one can take $\Lambda(\cdot)=\sum_{i}\operatorname{Tr}(M_{i}\cdot)|i\rangle\!\langle i|^{\otimes n}$ and $\Gamma(\cdot)=\sum_{i}\operatorname{Tr}(N_{i}\cdot)|i\rangle\!\langle i|^{\otimes n}$. It follows that $I_{c}(A\mathrel{\mathop{\mathchar 58\relax}}B)\leq\max_{\Lambda,\Gamma}\frac{1}{n}\sum_{j=1}^{n}I(A_{j}\mathrel{\mathop{\mathchar 58\relax}}B_{j})_{(\Lambda_{j}\otimes\Gamma_{j})\rho_{AB}}$, and it remains to show $I_{c}(A\mathrel{\mathop{\mathchar 58\relax}}B)\geq\lim_{n\to\infty}\max_{\Lambda,\Gamma}\frac{1}{n}\sum_{j=1}^{n}I(A_{j}\mathrel{\mathop{\mathchar 58\relax}}B_{j})_{(\Lambda_{j}\otimes\Gamma_{j})\rho_{AB}}$.

Similarly to Lemma 3, let $\Gamma\mathrel{\mathop{\mathchar 58\relax}}{\cal S}(B)\to{\cal S}(B_{1}\otimes\cdots\otimes B_{n})$ be an arbitrary cptp map, then there exists a POVM $\{F_{k}\}_{k}$ and a set $S^{\prime}\subset[n]$ with $|S^{\prime}|\geq n(1-\delta)$ such that for all $j\in S^{\prime}$,

[TABLE]

with ${\cal F}_{j}(\cdot)\mathrel{\mathop{\mathchar 58\relax}}=\sum_{k}\operatorname{Tr}(F_{k}\cdot)\sigma^{\prime}_{j,k}$ for states $\sigma^{\prime}_{j,k}\in{\cal S}(B_{j})$. Therefore for fixed $0<\delta<1$, there exists $S^{\prime\prime}\subset[n]$ with $|S^{\prime\prime}|\geq n(1-2\delta)$ such that Eqs. (3) and (5) hold simultaneously for all $j\in S^{\prime\prime}$. Thus

[TABLE]

where $d\mathrel{\mathop{\mathchar 58\relax}}=\max\{|A|,|B|\}$.

For any state $\rho_{AB}$, by definition of the diamond norm, we have

[TABLE]

We now have

[TABLE]

where $\widetilde{{\cal E}}_{j}(\cdot)\mathrel{\mathop{\mathchar 58\relax}}=\sum_{k}\operatorname{Tr}(E_{k}\cdot)|k_{j}\rangle\!\langle k_{j}|$ and $\widetilde{{\cal F}}_{j}(\cdot)\mathrel{\mathop{\mathchar 58\relax}}=\sum_{k}\operatorname{Tr}(F_{k}\cdot)|k^{\prime}_{j}\rangle\!\langle k^{\prime}_{j}|$, and $\{|k_{j}\rangle\}_{k}$ and $\{|k^{\prime}_{j}\rangle\}_{k^{\prime}}$ are orthonormal basis of system $A_{j}$ and $B_{j}$ respectively, Eq. (7) follows from the continuity bound Eq. (LABEL:cont-bound), and Eq. (8) follows from the fact that local operations cannot increase mutual information.

Set $\delta=n^{-1/6}$, then as $n\to\infty$ one has $\delta,\varepsilon\to 0$ and $K\to I_{c}(A\mathrel{\mathop{\mathchar 58\relax}}B)$. It follows that

[TABLE]

where $d^{\prime}\mathrel{\mathop{\mathchar 58\relax}}=\max\{|A_{j}|,|B_{j}|\}_{j}$. That is,

[TABLE]

and we are done.     \sqcap$$\sqcup

IV Optimal universal and state-dependent broadcasting of correlations

We now turn to the non-asymptotic regime of the local broadcasting of quantum correlations. We first study the optimal universal unilocal broadcasting and then the optimal state-dependent unilocal broadcasting.

IV.1 Optimal universal unilocal broadcasting

We first give a general definition for the unilocal $n$-broadcasting fidelity of a bipartite state.

Definition 4

Given a bipartite state $\rho_{AB}$, the optimal unilocal $n$-broadcasting fidelity of $\rho_{AB}$ on system $A$ (see Fig. 1) is defined as the following optimal fidelity

[TABLE]

Since the set of quantum channels is compact and the fidelity function is continuous Watrous (2011), the supremum in Eq. (9) is attained. Define a unitary operator $W_{\pi}$ on systems $A_{1}\cdots A_{n}$ for each permutation $\pi\in S_{n}$, by the action

[TABLE]

for any choice of $|j_{1}\rangle,|j_{2}\rangle,\dots,|j_{n}\rangle$. A quantum channel $\Lambda_{A\to A_{1}\cdots A_{n}}$ is called a symmetric broadcasting channel, if

[TABLE]

for any $\rho\in{\cal S}(A)$ and $\pi\in S_{n}$.

We notice that for any channel $\Lambda_{A\to A_{1}\cdots A_{n}}$ and $\pi\in S_{n}$, $\Lambda(\cdot)$ and $W_{\pi}(\Lambda(\cdot))W_{\pi}^{\dagger}$ give the same average fidelity in Eq. (9), since

[TABLE]

Thus $\frac{1}{n!}\sum_{\pi\in S_{n}}W_{\pi}(\Lambda(\cdot))W_{\pi}^{\dagger}$, which is a symmetric broadcasting channel, also gives the same value. So we only need to consider the supremum over symmetric broadcasting channels. In Eq. (9), when $\Lambda$ is a symmetric broadcasting channel, the summands are all the same.

Therefore, the optimal unilocal $n$-broadcasting fidelity of a bipartite state $\rho_{AB}$ on $A$ can be rewritten as

[TABLE]

It is verified that $\Lambda_{A\to A_{1}\cdots A_{n}}$ is a symmetric broadcasting channel iff its Choi matrix $J_{\Lambda}$ satisfies $J_{\Lambda}=W_{\pi}J_{\Lambda}W_{\pi}^{\dagger}$ for any $\pi$, i.e., $J_{\Lambda}=\frac{1}{n!}\sum_{\pi\in S_{n}}W_{\pi}J_{\Lambda}W_{\pi}^{\dagger}$. Using this symmetry, we give the SDP characterization for optimal unilocal broadcasting fidelity as follows.

Theorem 5

The optimal unilocal $n$-broadcasting fidelity of $\rho_{AB}$ on $A$ is given by the optimal solution of the following SDP,

[TABLE]

where $W_{\pi}$ acts on $A_{1}\cdots A_{n}$.

Proof.

It suffices to consider the symmetric broadcasting channels only. Let $J_{AA_{1}\cdots A_{n}}$ be the Choi matrix of $\Lambda_{A\to A_{1}\cdots A_{n}}$, then for any $\rho_{A}$,

[TABLE]

By linearity, for any $\rho_{AB}$,

[TABLE]

and

[TABLE]

Now we can rewrite the optimization problem in Eq. (10) in terms of the Choi matrix of $\Lambda$ as

[TABLE]

The fidelity function $F(\rho,\sigma)$ of two states $\rho,\sigma$ is given by the optimal solution of the following SDP Killoran (2012); Watrous (2012),

[TABLE]

Combining Eqs. (12) and (13) gives the desired SDP (11).     \sqcap$$\sqcup

**Remark ** The only difference between the SDP (11) and that in Ref. Piani (2016) lies in the symmetry of the broadcasting channel, that is, $J=W_{\pi}JW_{\pi}^{\dagger}$ for any $\pi\in S_{n}$ is required in our SDP. In Ref. Piani (2016), it is required that $J=W_{\pi_{1}}JW_{\pi_{2}}^{\dagger}$ for any $\pi_{1},\pi_{2}\in S_{n}$ which makes sure that the output state lies in the symmetric subspace. These two SDPs are different generalization of perfect unilocal broadcasting. But the SDP (11) here has a more direct derivation, and it is clear that the optimal solution to SDP (11) is no less than that to the SDP in Piani (2016). Numerical experiments show that the two SDPs give the same optimal solution for some cases of $\rho_{AB}$, but we do not know how to give a rigorous proof or disproof for general case up to now.

In the SDP (11), if $(J_{AA_{1}\cdots A_{n}},X_{AB})$ is feasible solution of $f(\rho_{AB})$, then $(({\mathds{1}}_{A}\otimes U^{\otimes n})J_{AA_{1}\cdots A_{n}}({\mathds{1}}_{A}\otimes U^{\otimes n})^{\dagger},(U_{A}\otimes V_{B})X_{AB}(U_{A}\otimes V_{B})^{\dagger})$ is feasible solution of $f((U_{A}\otimes V_{B})\rho_{AB}(U_{A}\otimes V_{B})^{\dagger})$ for any local local unitaries $U_{A}$ and $V_{B}$. In other words, the unilocal broadcasting fidelity $f_{n}$ is invariant under local unitaries.

We now consider the unilocal broadcasting fidelity of a pure state $\psi_{AB}\mathrel{\mathop{\mathchar 58\relax}}=|\psi\rangle\!\langle\psi|_{AB}$, and especially the maximally entangled state, under the action of the symmetric broadcasting channel. The optimal unilocal broadcasting fidelity $f_{n}$ of a pure state $\psi_{AB}$ can be written as

[TABLE]

where $W_{\pi}$ acts on $A_{1}\cdots A_{n}$.

The corresponding dual SDP is

[TABLE]

where, again, $W_{\pi}$ acts on $A_{1}\cdots A_{n}$.

It is verified that the strong duality holds by Slater’s theorem since $J_{AA_{1}\cdots A_{n}}={\mathds{1}}/|A|^{n}$ is in the relative interior of the feasible region of SDP (14). That means the optimal solutions to SDPs (14) and (15) concide.

Proposition 6

The optimal unilocal 2-broadcasting fidelity of the maximally entangled state $\Phi_{d}\mathrel{\mathop{\mathchar 58\relax}}=|\Phi_{d}\rangle\!\langle\Phi_{d}|$ with $|\Phi_{d}\rangle=\frac{1}{\sqrt{d}}\sum_{j=0}^{d-1}|jj\rangle$ on systems $AB$ is given by

[TABLE]

Proof.

We prove this proposition by explicitly constructing feasible solutions in primal and dual problem both of which can achieve the value of $\sqrt{\frac{d+1}{2d}}$.

In the primal problem, we take

[TABLE]

where

[TABLE]

This operation is also known as the universal quantum copying machine (UQCM) Bužek and Hillery (1996, 1998).

In the dual problem, we take

[TABLE]

where $I_{0}=\sum_{i=0}^{d-1}|ii\rangle\langle ii|$.     \sqcap$$\sqcup

**Remark ** It is interesting that the optimal unilocal 2-broadcasting channel of the maximally entangled state is the same as the UQCM which comes from the global broadcasting setting. There is much progress on quantum cloning machine that has been made in the past years (see, e.g., Scarani et al. (2005); Fan et al. (2014)). For $d\otimes d$ bipartite maximally entangled state, its optimal unilocal 2-broadcasting channel is denoted as $\Upsilon_{A\rightarrow A_{1}A_{2}}^{d}$ with Choi matrix (16) and

[TABLE]

is a depolarizing channel.

Next, we will introduce a worst-case quantifier for the performance of unilocal broadcasting of a symmetric channel.

Definition 7

For any symmetric broadcasting channel $\Lambda_{A\to A_{1}\cdots A_{n}}$, we define the unilocal broadcasting power ${\cal P}(\Lambda)$ of $\Lambda$ as

[TABLE]

The unilocal broadcasting power of a symmetric broadcasting channel gives a measure of the universal unilocal broadcasting ability for symmetric broadcasting channels. The universality means it is independent of the input state. The channel with a larger value of unilocal broadcasting power is more capable of unilocal broadcasting quantum states in a universal sense.

Based on the result of optimal unilocal 2-broadcasting fidelity of maximally entangled state, we will prove that the optimal unilocal 2-broadcasting channel $\Upsilon_{A\to A_{1}A_{2}}^{d}$ for the maximally entangled state has the greatest power for unilocal 2-broadcasting.

Lemma 8

For any $d\otimes d$ pure state $|\psi\rangle$,

[TABLE]

Proof.

Consider the Schmidt decomposation $|\psi\rangle=\sum_{i}\lambda_{i}|i\rangle_{A}|i\rangle_{B}$, where $\{|i\rangle_{A}\}_{i}$ and $\{|i\rangle_{B}\}_{i}$ are some orthonormal bases. Thus,

[TABLE]

Then the second inequality in Eq. (18) follows from

[TABLE]

\sqcap$$\sqcup

Proposition 9

For any $d\otimes d$ mixed state $\rho$,

[TABLE]

Proof.

Suppose $\rho=\sum_{j}p_{j}|\psi_{j}\rangle\!\langle\psi_{j}|$ is a pure state decomposition of $\rho$ and $\hat{\rho}_{j}=\operatorname{Tr}_{A_{2}}\Upsilon_{A\to{A_{1}A_{2}}}^{d}(|\psi_{j}\rangle\!\langle\psi_{j}|)$, then we have

[TABLE]

Employing the joint concavity of fidelity, we have that

[TABLE]

where the last inequality uses the result in Lemma 8.     \sqcap$$\sqcup

Theorem 10

$\Upsilon_{A\to{A_{1}A_{2}}}^{d}$* has the strongest power for unilocal 2-broadcasting in $d\otimes d$ system, i.e.,*

[TABLE]

where the maximum is taken over all symmetric broadcasting channels.

Proof.

For any symmetric broadcasting channel $\Lambda_{A\to A_{1}A_{2}}$, we have

[TABLE]

where $\Phi_{d}$ is the maximally entangled state. The second inequality holds since $\Upsilon_{A\to{A_{1}A_{2}}}^{d}$ is the optimal unilocal 2-broadcasting channel for $\Phi_{d}$.

For the unilocal 2-broadcasting operation $\Upsilon_{A\to{A_{1}A_{2}}}^{d}$, from Proposition 9, we have that

[TABLE]

Combining Eqs. (23) and (24), it is clear that $\Upsilon_{A\to{A_{1}A_{2}}}^{d}$ maximizes the unilocal broadcasting power ${\cal P}$. Thus, it is optimal under the setting of universal unilocal 2-broadcasting.

\sqcap$$\sqcup

IV.2 Optimal unilocal broadcasting for two-qubit pure state

In the following theorem, we give analytical solution of optimal unilocal 2-broadcasting fidelity for two-qubit pure state. Since $f_{n}$ is invariant under local unitary, we only need to consider two-qubit pure state in the form of $|\psi_{\theta}\rangle=\cos\theta|00\rangle+\sin\theta|11\rangle$, $\theta\in(0,\pi/4]$ without lose of generality.

Theorem 11

For two-qubit pure state $\psi_{\theta}=|\psi_{\theta}\rangle\!\langle\psi_{\theta}|$ with $|\psi_{\theta}\rangle=\cos\theta|00\rangle+\sin\theta|11\rangle$, $\theta\in(0,\pi/4]$, its optimal unilocal 2-broadcasting fidelity is given by

[TABLE]

Proof.

We prove this theorem by explicitly constructing a feasible solution in both primal and dual problem which achieves $f_{2}(\psi_{\theta})$.

Case 1: If $\theta\in(0,\arctan(2^{-1/4})]$, in the primal problem, we construct feasible solution

[TABLE]

where

[TABLE]

In the dual problem, we construct feasible solution

[TABLE]

[TABLE]

where $x={\sqrt{2}}{p\cdot\sin^{2}\theta}$. It is easy to check that $J_{AA_{1}A_{2}}$ and $\{Y_{A},Z_{AA_{1}A_{2}}\}$ are feasible solutions to SDP (14) and (15).

Case 2: If $\theta\in(\arctan(2^{-1/4}),\frac{\pi}{4})$, in the primal problem, we construct a feasible solution

[TABLE]

where

[TABLE]

[TABLE]

In the dual problem, let us choose

[TABLE]

[TABLE]

where $x=-\frac{3}{2}\sin^{2}\theta\cos^{2}\theta$. It is also easy to check that $J_{AA_{1}A_{2}}$ and $\{Y_{A},Z_{AA_{1}A_{2}}\}$ are feasible solutions to SDP (14) and (15).     \sqcap$$\sqcup

From the above proof, we can see that the optimal unilocal 2-broadcasting channel is independent of parameter $\theta$ in the first piece, that is, $\theta\in(0,\arctan(2^{-1/4})]$. We denote this channel as $\Xi$ with Choi matrix $J_{AA_{1}A_{2}}$ (25).

We show the difference between fidelity of unilocal 2-broadcasting via channel $\Upsilon$ and $\Xi$, denoted as $f_{2,\Upsilon}(\psi_{\theta})$, $f_{2,\Xi}(\psi_{\theta})$ respectively, and the optimal unilocal 2-broadcasting fidelity $f_{2}(\psi_{\theta})$ in the following Fig. 2.

V Approximate broadcasting of a set of quantum states

V.1 Fidelity of broadcasting a set of quantum states

The no-go theorem for simultaneously broadcasting quantum states Barnum et al. (1996) says that we cannot perfectly broadcast two arbitrary noncommuting states. It is natural to ask how well we can do the task approximately. Generally, given $m$ states $\rho_{i}$ with respective prior probability $p_{i}$, how large average fidelity can we achieve when broadcasting these states via the same channel? Mathematically, assuming the given states $\rho_{i}$ are on the system $A$, we study how to optimize the $n$-broadcasting fidelity $g_{n}(\eta)$ of an ensemble $\eta\mathrel{\mathop{\mathchar 58\relax}}=\{p_{i},\rho_{i}\}_{i=1}^{m}$, which is defined as

[TABLE]

Using the idea in the derivation of Eq. (10), namely, exploiting the symmetry in the broadcasting channel $\Lambda$, we can simplify this definition. The $n$-broadcasting fidelity $g_{n}$ of an ensemble $\eta\mathrel{\mathop{\mathchar 58\relax}}=\{p_{i},\rho_{i}\}_{i=1}^{m}$ can be rewritten as

[TABLE]

Theorem 12

The $n$-broadcasting fidelity $g_{n}(\eta)$ of an ensemble $\eta=\{p_{i},\rho_{i}\}_{i=1}^{m}$ is given by the optimal solution of the SDP in (29).

Proof.

The output state on system $A_{1}$ of broadcasting $\rho_{i}$ is

[TABLE]

where $J_{AA_{1}\cdots A_{n}}$ is the Choi matrix of $\Lambda_{A\to A_{1}\cdots A_{n}}$.

By using the SDP characterization of fidelity function, we then have

[TABLE]

where $W_{\pi}$ acts on $A_{1}\cdots A_{n}$.     \sqcap$$\sqcup

VI Conclusions and discussion

In summary, we have studied the approximate broadcasting of quantum correlations from several aspects. Firstly, we extend the operational characterization of one-sided quantum discord to two-sided one, that is, the asymptotic optimal average mutual information loss after the action of two local broadcasting channels is equal to the two-sided quantum discord. Then we give an alternate derivation for the SDP characterization of the unilocal broadcasting fidelity, based on which we analyze some properties of unilocal broadcasting. We show that the universal quantum clone machine (UQCM) is also the optimal universal unilocal broadcasting operation. Moreover, the optimal state-dependent unilocal broadcasting operation for pure two-qubit states is analytically solved. Finally, we also formulate the broadcasting of a finite set of quantum states as an SDP. It would be of interest to study other topics related to broadcasting and correlations, such as the broadcasting of Gaussian state and correlation, and the relation between Gaussian quantum broadcasting and Gaussian quantum discord. One can also study the asymptotic behavior of the $n$-broadcasting fidelity of a finite set of quantum states in the large $n$ limit.

ACKNOWLEDGEMENTS

The authors are grateful to Kun Wang for useful discussion. This work was partly supported by the Australian Research Council under Grant Nos. DP120103776 and FT120100449.