Parameterized Bipartite Entanglement Measure

As a distinguishing feature of quantum mechanics, quantum entanglement is an important resource schrodinger1935gegenwartige ; PhysRev.47.777 ; PhysPhys195 in quantum computation and quantum information processing nielsen2002quantum such as quantum dense coding PhysRevLett.69.2881 , clock synchronization PRL852010 , quantum teleportation PhysRevLett.70.1895 , quantum secret sharing PhysRevA.59.1829 and quantum cryptography PRL67661 . One of the problems in quantum entanglement theory is to quantify the entanglement of a bipartite system. A reasonable entanglement measure $E$ should fulfill PhysRevLett.78.2275 ; PhysRevA.57.1619 ; PhysRevA.56.R3319 ; PhysRevA.59.141 ; PRL842014 : (E1) $E\left(\rho\right)\geqslant 0$, with the equality holding iff $\rho$ is separable; (E2) $E$ is convex, i.e., $E\left(t\rho+\left(1-t\right)\sigma\right)\leqslant tE\left(\rho\right)+\left(1-t\right)E\left(\sigma\right)$ for any states $\rho$ and $\sigma$, where $t\in[0,1]$; (E3) $E$ does not increase under local operation and classical communication (LOCC), i.e., $E\left(\rho\right)\geqslant E\left(\Lambda\left(\rho\right)\right)$ for any completely positive and trace preserving (CPTP) LOCC $\Lambda$; (E4) $E$ does not increase on average under stochastic LOCC, i.e., $E\left(\rho\right)\geqslant\sum_{k}p_{k}E\left(\rho_{k}\right)$, where $p_{k}=\mathrm{Tr}A_{k}\rho A_{k}^{\dagger}$, $\rho_{k}=A_{k}\rho A_{k}^{\dagger}/p_{k}$ and $A_{k}$ are the Kraus operators such that $\sum_{k}A_{k}^{\dagger}A_{k}=I$ with $I$ the identity.

Several reasonable entanglement measures have been presented in the past few years, such as concurrence PhysRevLett.78.5022 ; PhysRevA.64.042315 , entanglement of formation PhysRevA.54.3824 ; PhysRevLett.80.2245 ; horodecki2001entanglement , negativity PhysRevA.65.032314 ; PhysRevLett.95.090503 and Rényi-$\alpha$ entropy of entanglement gour2007dual ; Kim_2010 . Generally it is formidably difficult to calculate analytically the degree of entanglement for arbitrary given states. Analytical results are often available for two-qubit states PhysRevLett.80.2245 or special higher-dimensional mixed states PhysRevLett.85.2625 ; PhysRevA.64.062307 ; PhysRevA.67.012307 for several entanglement measures PhysRevA.68.062304 ; buchholz2016evaluating . Efforts have been also made towards the analytical lower bounds of entanglement measures like the concurrence PhysRevLett.95.210501 ; Li_Guo_2009 . In PhysRevLett.95.040504 analytical lower bound of the concurrence has been derived by employing the positive partial transpose (PPT) and realignment criteria PhysRevLett.77.1413 ; PhysRevA.59.4206 ; rudolph2004computable ; rudolph2005further ; chen2003matrix . In this paper, inspired by Tsallis-$q$ entropy of entanglement raggio1995properties ; PhysRevA.81.062328 and parameterized entanglement monotone $q$-concurrence PhysRevA.103.052423 , we propose a novel parameterized entanglement measure $\alpha$-concurrence for any $0\leqslant\alpha\leqslant 1/2$. Based on PPT and realignment criteria, we also obtain an analytic lower bound of the $\alpha$-concurrence for general bipartite systems. Finally, we calculate the $\alpha$-concurrence of isotropic states and Werner states analytically.

Let $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ be an arbitrary $d\times d$ dimensional bipartite Hilbert space associated with subsystems $A$ and $B$. Any pure state on the $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ can be written as in the Schmidt form,

where $\sum_{i=1}^{r}\lambda_{i}=1$ with $\lambda_{i}>0$, $r$ is the Schmidt rank, $1\leqslant r\leqslant d$. $\left\{\ket{a_{i}}\right\}$ and $\left\{\ket{b_{i}}\right\}$ are the local bases associated with the subsystems $A$ and $B$, respectively nielsen2002quantum .

$\mathit{Definition}$. For any pure state $\ket{\psi}$ given in (1), the $\alpha$-concurrence is defined by

for any $0\leqslant\alpha\leqslant 1/2$, where $\rho_{A}=\mathrm{Tr}_{B}|\psi\rangle\langle\psi|$.

From (2), for a pure state $\ket{\psi}$ given by (1) one has

where $C_{\alpha}\left(\ket{\psi}\right)$ satisfies $0\leqslant C_{\alpha}\left(\ket{\psi}\right)\leqslant d^{1-\alpha}-1$. It is obvious that the lower bound is attained if and only if $\ket{\psi}$ is a separable state, that is, $\ket{\psi}=\ket{a_{i}b_{i}}$ for some $\ket{a_{i}}$ and $\ket{b_{i}}$. While the upper bound is achieved for the maximally entangled pure states $\ket{\Psi^{+}}=\frac{1}{\sqrt{d}}\sum_{i=1}^{d}\ket{a_{i}b_{i}}$.

For a general mixed state $\rho$ on the Hilbert space $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, the $\alpha$-concurrence is given by the convex-roof extension,

where the infimum is taken over all possible pure-state decompositions of $\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$, with $\sum_{i}p_{i}=1$ and $p_{i}>0$. Before showing that $C_{\alpha}\left(\rho\right)$ defined in (4) is indeed a bona fide entanglement measure, we first present the following lemma, see proof in Appendix A.

$\mathit{Lemma\ 1}$. The function $F_{\alpha}\left(\rho\right)=\mathrm{Tr}\rho^{\alpha}-1$ is concave, that is,

for any $0\leqslant\alpha\leqslant 1/2$, where $\set{p_{i}}$ is a probability distribution and $\rho_{i}$ are density matrices. The equality holds if and only if all $\rho_{i}$ are the same for all $p_{i}>0$.

By using the above Lemma 1, we have the following theorem.

$\mathit{Theorem\,1}$. The $\alpha$-concurrence $C_{\alpha}\left(\rho\right)$ given in (4) is a well defined parameterized entanglement measure.

$\mathit{Proof}$. We need to verify that $C_{\alpha}\left(\rho\right)$ fulfills the following four requirements.

(E1) If $\rho$ is an entangled state, then there is at least one entangled pure state $\ket{\psi}$ in any pure state decomposition of $\rho$. Thus $C_{\alpha}\left(\rho\right)>0$. Otherwise, $C_{\alpha}\left(\rho\right)=0$ for separable states.

(E2) Consider $\rho=t\rho_{1}+\left(1-t\right)\rho_{2}$. Let $\rho_{1}=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$ ($\rho_{2}=\sum_{j}q_{j}|\phi_{j}\rangle\langle\phi_{j}|$) be the optimal pure state decomposition of $C_{\alpha}\left(\rho_{1}\right)$ ($C_{\alpha}\left(\rho_{2}\right)$) with $\sum_{i}p_{i}=1$ ($\sum_{j}q_{j}=1$) and $p_{i}>0$ ($q_{j}>0$). We have

where the first inequality is due to that $\sum_{i=1}^{k}tp_{i}|\psi_{i}\rangle\langle\psi_{i}|+\sum_{j=1}^{l}\left(1-t\right)q_{j}|\phi_{j}\rangle\langle\phi_{j}|$ is also a pure state decomposition of $\rho$.

(E3) We adopt the approach given in MINTERT2005207 to show that our entanglement measure does not increase under LOCC. Denote $\vec{\lambda}_{\psi}$ ($\vec{\lambda}_{\phi}$) the Schmidt vector given by the squared Schmidt coefficients of the state $\ket{\psi}$ ($\ket{\phi}$) in the decreasing order. It has been shown that the state $\ket{\phi}$ can be prepared starting from the state $\ket{\psi}$ under LOCC if and only if $\vec{\lambda}_{\psi}$ is majorized by $\vec{\lambda}_{\phi}$ PhysRevLett.83.436 , $\vec{\lambda}_{\psi}\prec\vec{\lambda}_{\phi}$, where the majorization means that the components $\left[\lambda_{\psi}\right]_{i}$ ($\left[\lambda_{\phi}\right]_{i}$) of $\vec{\lambda}_{\psi}$ ($\vec{\lambda}_{\phi}$), listed in nonincreasing order, satisfy $\sum_{i=1}^{j}\left[\lambda_{\psi}\right]_{i}\leqslant\sum_{i=1}^{j}\left[\lambda_{\phi}\right]_{i}$ for $1<j\leqslant d$, with equality for $j=d$.

Since the entanglement cannot increase under LOCC, any entanglement measure $E$ has to satisfy that $E\left(\psi\right)\geqslant E\left(\phi\right)$ whenever $\vec{\lambda}_{\psi}\prec\vec{\lambda}_{\phi}$. This condition, known as the $\mathit{Schur\,concavity}$, is satisfied if and only if $E$, given as a function of the squared Schmidt coefficients $\lambda_{i}$’s ANDO1989163 , is invariant under the permutations of any two arguments and satisfies

for any two components $\lambda_{i}$ and $\lambda_{j}$ of $\vec{\lambda}$.

For any pure state $\ket{\psi}$ given by (1), the $\alpha$-concurrence is obviously invariant under the permutations of the Schmidt coefficients for any $0\leqslant\alpha\leqslant 1/2$. Since

for any two components $\lambda_{i}$ and $\lambda_{j}$ of the squared Schmidt coefficients of $\ket{\psi}$, we have $C_{\alpha}\left(\ket{\psi}\right)\geqslant C_{\alpha}\left(\Lambda\ket{\psi}\right)$ for any LOCC $\Lambda$ and $0\leqslant\alpha\leqslant 1/2$.

Next let $\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$ be the optimal pure state decomposition of $C_{\alpha}\left(\rho\right)$ with $\sum_{i}p_{i}=1$ and $p_{i}>0$. We obtain

for any $0\leqslant\alpha\leqslant 1/2$, where the last inequality is from the definition (4).

(E4) Let $\rho=\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$ be the optimal pure state decomposition of $C_{\alpha}\left(\rho\right)$ with $\sum_{i}p_{i}=1$ and $p_{i}>0$. Consider stochastic LOCC protocol given by Kraus operators $A_{k}$ with $\sum_{k}A_{k}^{\dagger}A_{k}=I$. We have

where $p_{k|i}=\mathrm{Tr}A_{k}|\psi_{i}\rangle\langle\psi_{i}|A_{k}^{\dagger}$ is the probability of obtaining the outcome $k$ with $\ket{\psi_{i}^{k}}=A_{k}\ket{\psi_{i}}/\sqrt{p_{k|i}}$, and $p_{k}=\mathrm{Tr}A_{k}\rho A_{k}^{\dagger}$ is the probability of obtaining the outcome $k$ with $\rho^{k}=A_{k}\rho A_{k}^{\dagger}/p_{k}$. The first inequality is due to the concavity of the Lemma 1, since $\mathrm{Tr}_{B}|\psi_{i}\rangle\langle\psi_{i}|=\sum_{k}p_{k|i}\mathrm{Tr}_{B}|\psi_{i}^{k}\rangle\langle\psi_{i}^{k}|$. The last inequality is from the definition of (4), since $\sum_{i}p_{i|k}|\psi_{i}^{k}\rangle\langle\psi_{i}^{k}|=\rho^{k}$. $\hfill\qed$

In PhysRevA.103.052423 the parameterized entanglement monotone $q$-concurrence $C_{q}\left(\ket{\psi}\right)$ for any pure state $\ket{\psi}$ defined in (1) has been introduced, $C_{q}\left(\ket{\psi}\right)=1-\mathrm{Tr}\rho_{A}^{q}$, where $q\geqslant 2$. It seems that our $\alpha$-concurrence defined by (2), $C_{\alpha}\left(\ket{\psi}\right)=\mathrm{Tr}\rho_{A}^{\alpha}-1$, is in some sense dual to the $q$-concurrence as the parameter $\alpha\in[0,1/2]$, while the parameter $q\geqslant 2$. Nevertheless, these two concurrences characterize the quantum entanglement in different aspects, even though they are both derived from the Tsallis-$q$ of entanglement PhysRevA.81.062328 . For large enough $q$, the $q$-concurrence $C_{q}(\rho)$ converges to the constant $1$ for any entangled state $\rho$, while the $\alpha$-concurrence $C_{\alpha}(\rho)$ not for any $\alpha\in[0,1/2]$. Particularly, for $\alpha=0$, the measure $C_{0}\left(\ket{\psi}\right)=r-1$ for any pure state $\ket{\psi}$ given in (1), where $r$ is the Schmidt rank of the state $\ket{\psi}$, which is solely determined by the Schmidt rank of the bipartite pure state $\ket{\psi}$. Therefore, the $\alpha$-concurrences with different $\alpha$ provide different characterizations of the feature of entanglement.

Owing to the optimization in the calculation of the entanglement measures, it is generally difficult to obtain analytical expressions of the entanglement measures for general mixed states. In this section, we derive analytical lower bounds for the $\alpha$-concurrence based on PPT and realignment criteria PhysRevLett.77.1413 ; HORODECKI19961 ; PhysRevA.59.4206 ; rudolph2004computable ; rudolph2005further ; chen2003matrix .

A bipartite state can be written as $\rho=\sum_{ijkl}\rho_{ij,kl}|ij\rangle\langle kl|$, where the subscripts $i$ and $k$ are the row and column indices for the subsystem $A$, respectively, while $j$ and $l$ are such indices for the subsystem $B$. The PPT criterion says that if the state $\rho$ is separable, then the partial transposed matrix $\rho^{\Gamma}=\sum_{ijkl}\rho_{ij,kl}|il\rangle\langle kj|$ with respect to the subsystem $B$ is non-negative, $\rho^{\Gamma}\geqslant 0$. While the realignment criterion says that the realigned matrix of $\rho$, $\mathcal{R}\left(\rho\right)=\sum_{ijkl}\rho_{ij,kl}|ik\rangle\langle jl|$, satisfies that $\|\mathcal{R}\left(\rho\right)\|_{1}\leqslant 1$ if $\rho$ is separable, where $\|X\|_{1}$ denotes the trace norm of matrix $X$, $\|X\|_{1}=\mathrm{Tr}\sqrt{XX^{\dagger}}$.

For a pure state $\ket{\psi}$ given by (1), it is straightforward to obtain that PhysRevLett.95.040504

where $\rho=|\psi\rangle\langle\psi|$. In particular, for $\alpha=1/2$, the $1/2$-concurrence becomes $C_{\frac{1}{2}}\left(\ket{\psi}\right)=\sum_{i=1}^{r}\sqrt{\lambda_{i}}-1$. One has then

for any pure state $\ket{\psi}$ on the $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$.

$\mathit{Theorem\ 2}$. For any mixed state $\rho$ on the $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, the $\alpha$-concurrence $C_{\alpha}\left(\rho\right)$ satisfies

$\mathit{Proof}$. For a pure state $\ket{\psi}$ given in (1), let us analyze the monotonicity of the following function,

for any $0\leqslant\alpha\leqslant 1/2$, where $r>1$. The first derivative of $g\left(\alpha\right)$ with respect to $\alpha$ is given by

where

Employing the Lagrange multiplies PhysRevA.67.012307 under constraints $\sum_{i=1}^{r}\lambda_{i}=1$ and $\lambda_{i}>0$, one has that there is only one stable point $\lambda_{i}=1/r$ for every $i=1,\cdots,r$, for which $G_{r\alpha}=0$ for any $0\leqslant\alpha\leqslant 1/2$. Since the second derivative at this point,

for any $0\leqslant\alpha\leqslant 1/2$, the maximum extreme value point is just the maximum value point. $g\left(\alpha\right)$ is a decreasing function for any $0\leqslant\alpha\leqslant 1/2$, since $\partial g/\partial\alpha\leqslant 0$.

We have

where $\sigma=|\psi\rangle\langle\psi|$, the second inequality is due to (10), the last inequality is due to that $\frac{r^{1-\alpha}-1}{r-1}$ is a decreasing function with respect to $r$. Assume $\rho=\sum_{i}p_{i}|\varphi_{i}\rangle\langle\varphi_{i}|$ is the optimal pure state decomposition for $C_{\alpha}\left(\rho\right)$. Then

where $\sigma_{i}=|\varphi_{i}\rangle\langle\varphi_{i}|$, the last inequality is due to the convex property of the trace norm and $\|\rho^{\Gamma}\|\geq 1$ in (9).

Similar to (III) and (III), we obtain from (9) that

for any $0\leqslant\alpha\leqslant 1/2$.

Combining (III) and (18), we complete the proof. $\hfill\qed$

In this section, we compute the $\alpha$-concurrence for isotropic states and Werner states. Let $E$ be a convex-roof extended quantum entanglement measure. Denote $S$ the set of states and $P$ the set all pure states in $S$. Let $G$ be a compact group acting on $S$ by $\left(U,\rho\right)\mapsto U\rho U^{\dagger}$. Assume that the measure $E$ defined on $P$ is invariant under the operations of $G$. One can define the projection $\mathbf{P}$:$S\rightarrow S$ by $\mathbf{P}\rho=\int dUU\rho U^{\dagger}$ with the standard (normalized) Haar measure $dU$ on $G$, and the function $\eta$ on $\mathbf{P}S$ by

Then for $\rho\in\mathbf{P}S$, we have

where $co\left(f\right)$ is the convex-roof extension of a function $f$. In other words, it is the convex hull of $f$.

We have introduced the concept of $\alpha$-concurrence and shown that the $\alpha$-concurrence is a well defined entanglement measure. Analytical lower bounds of the $\alpha$-concurrence for general mixed states have been derived based on PPT and realignment criterion. Specifically, we have derived explicit formulae for the $\alpha$-concurrence of isotropic states and Werner states. Interestingly our lower bounds are exact for isotropic states and Werner states with $d=2$. Our parameterized entanglement measure $\alpha$-concurrence gives a family of entanglement measures and enriches the theory of quantum entanglement, which may highlight further researches on the study of quantifying quantum entanglement and the related investigations like monogamy and polygamy relations in entanglement distribution, as well as the physical understanding of quantum correlations.

This work is supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 12075159 and 12171044; Beijing Natural Science Foundation (Grant No. Z190005); Academy for Multidisciplinary Studies, Capital Normal University; Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology (No. SIQSE202001), the Academician Innovation Platform of Hainan Province.