Optimal local filtering operation for enhancing quantum entanglement

The phenomenon of the existence of non-classical correlations between spatially separated quantum systems is known as entanglement, which is firstly noted by well known EPR paradox Einstein1935 and named by Schrödinger Schrodinger1935 . Together with nonlocality Bell64 , entanglement is the fundamental property of quantum mechanics. Those remarkable properties of quantum mechanics have fundamentally changed human’s view of the world and taught us to think the unthinkable. The combination of quantum mechanical theory and computation theory provides a new paradiagram for computing, which is known as quantum computation and has exponential advantages over its classical counterpart Nielsen2000 . Quantum entanglement is acted as the indispensable resource in many important protocols of quantum computation and information tasks RH09 . For example, the important protocols quantum dense coding and quantum teleportation are based on the resource of maximally entangled states PhysRevLett.69.2881 ; PhysRevLett.70.1895 . Protocols are also proposed to generate maximal entanglement from general entangled states ZS17 and to extend transmission distance via efficient strategy of quantum repeaters with respect to the transmission rate and the cost of local operation and classical information (LOCC)  PhysRevA.97.012325 . Therefore, quantum entanglement do not only has theoretical importance on quantum mechanics but also has practical significance in quantum computation and information.

It is a well-known fact that separable quantum states do not contain any entanglement and can not be used for quantum advantage protocols. On the contrary, singlet states have widely used in advantage quantum protocols and known as maximally entanglement. And there are other states containing some amount of quantum entanglement between separable states and singlet states. Therefore, it is of practical significance to measure the quantity of entanglement for a specified quantum system. Several measures of quantum entanglement have been proposed in last three decades. For example, distillable entanglement Plenio2007 , entanglement of formation (EoF) PhysRevA.54.3824 and concurrence PhysRevLett.78.5022 . For the two-qubit system, EoF and concurrence are acted as the similar measure of entanglement but with different scale, which are related by the binary entropy function.

In this work, we consider the concurrence as the measure for quantum entanglement of a general two-qubit system. Analytically, the concurrence $C(\rho)$ of a general two-qubit state $\rho$ is equal to the following equation,

where $\lambda_{i}$ are eigenvalues of the operator $R\equiv\sqrt{\sqrt{\rho}\tilde{\rho}\sqrt{\rho}}$ in decreasing order and $\tilde{\rho}=(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})$. Then, the EoF of $\rho$ can be calculated via the relation $EoF(\rho)=h\left(\frac{1+\sqrt{1-C(\rho)^{2}}}{2}\right)$ where $h(x)\equiv-x\log_{2}x-(1-x)\log_{2}(1-x)$ is known as Shannon’s binary entropy function PhysRevLett.80.2245 .

Due to the necessity of maximally entangled resource in the quantum information protocols, it is of practical significance to create maximally entangled states from less entangled resources, which is knowns as quantum entanglement distillation PhysRevA.67.022310 . It is well known that quantum entanglement is a resource that could not be created via local operation and classical communication (LOCC). However, it is possible to enhance quantum entanglement by local operations, say generating maximally entangled state from less entangled resources by local operations. One of such local operations is filtering operation. Suppose Alice and Bob share a two-qubit state $\rho_{AB}$. The filtering operation $F$ applies on any side, say Alice, is an operator on the subsystem with restriction $F^{\dagger}F\leq I$. The filtering operation evolutes the joint system as following,

Local filtering operations are widely used in quantum information processing protocols. The filtering operation is useful for teleportation applications and the optimal solution with respect to the fidelity were proposed PhysRevLett.90.097901 . The distillable key rate for quantum key distribution can be increased by local filtering operations PhysRevA.102.032415 . The filtering operation has been successfully realized by physical experiments PhysRevA.65.052319 ; ChaoLiu2022 .

Verstraete et. al. PhysRevA.64.010101 investigated the effect of local filtering operation on two-qubit state and found the following quantitative relation,

which provides the dynamic change of entanglement over general local filtering operation. However, it is still an open question to analytically find the optimal filtering operation which can maximally enhance the quantum entanglement with respect to concurrence for a given quantum state.

In this paper, we consider the effect of local operations on general two-qubit system and further investigate the open problem remained in the work of Verstraete et. al.  PhysRevA.64.010101 . In Sec. II, we try to find an analytical expression of the optimal local filtering operation with respect to the maximal entanglement increase measured by concurrence. In Sec. III, we make extension to consider the effect of local general measurements for quantum entanglement and give numerically evidence to the well-known fact that local measurement can not increase the expectation of quantum entanglement.

In this section, we investigate the optimal filtering operation for generating entanglement from a general two-qubit system and further consider the effect of local measurement.

Consider the scenario that separated parties Alice and Bob share a two-qubit quantum system. Suppose $\rho_{AB}$ is the general state of the system, which can be represented with respect to Pauli operators as follows,

where $\vec{a},\vec{b}$ are vectors in $\mathbb{R}^{3}$ with elements $a_{j}=tr(\rho_{AB}(\sigma_{j}\otimes I))$ and $b_{k}=tr(\rho_{AB}(I\otimes\sigma_{k}))$ for $k,j\in\{1,2,3\}$, $\vec{\sigma}$ is vector of Pauli operators, and $T_{j_{1}j_{2}}=tr(\rho_{AB}(\sigma_{j_{1}}\otimes\sigma_{j_{2}}))$ are elements of a $3\times 3$ real matrix $T$. Note that $\rho_{A}\equiv tr_{B}(\rho_{AB})=\frac{1}{2}(I+\vec{a}\cdot\vec{\sigma})$ is the state of Alice’s partial system and $\vec{a}$ contains all the information for the system. And similarly $\rho_{B}\equiv tr_{A}(\rho_{AB})$ is that of Bob’s system. It is obvious that the norm of vectors $\vec{a}$ and $\vec{b}$ are no greater than 1. The property $tr(\rho_{AB}^{2})\leq 1$ of a density operator naturally results the constraint $\|\vec{a}\|^{2}+\|\vec{b}\|^{2}+\|T\|^{2}\leq 3$ where $\|T\|^{2}\equiv\sum_{j_{1},j_{2}=1}^{3}T_{j_{1}j_{2}}^{2}$ entropy2306728 .

Suppose Alice applies a filtering operation $F$ on her system. The filtering operator $F$ has polar decomposition of the form $F=U_{F}K_{F}$ where $K_{F}\equiv\sqrt{F^{\dagger}F}$ and $U_{F}$ is a unitary based on the singular value decomposition of $F$. To be specific, suppose $F=\sum_{k}f_{k}|\alpha_{k}\rangle\langle\beta_{k}|$ is the singular value decomposition of $F$, then $U_{F}=\sum_{k}\frac{f_{k}}{|f_{k}|}|\alpha_{k}\rangle\langle\beta_{k}|$ where we set $\frac{f_{k}}{|f_{k}|}=1$ for the cases with zero singular values.

Let $F=\frac{1}{2}\sum_{k=0}^{3}x_{k}\sigma_{k}=\frac{1}{2}(x_{0}I+\vec{x}\cdot\vec{\sigma})$ with $\vec{x}=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}$ and $x_{j}\equiv tr(K_{F}\sigma_{j})$ for $j=0,1,2,3$. It is trivial to calculate that the eigenvalues of $K_{F}$ are $\frac{1}{2}(x_{0}+\|\vec{x}\|)$ and $\frac{1}{2}(x_{0}-\|\vec{x}\|)$. Note that $K_{F}$ is a positive operator and the filtering operation is constrained by $F^{\dagger}F\leq I$. The parameters have to satisfy the constraints $0\leq\frac{1}{2}(x_{0}-\|\vec{x}\|)\leq\frac{1}{2}(x_{0}+\|\vec{x}\|)\leq 1$, which is equivalent to the following

with $\|\vec{x}\|\leq 1$.

According to Verstraete’s result PhysRevA.64.010101 , the problem of find optimal filtering operation is equivalent to maximizing the ratio $\frac{|det(F)|}{tr((F^{\dagger}F\otimes I)\rho_{AB})}$ over all possible filtering operator $F$. The determinant of filtering operator $F$ is

Note that $F^{\dagger}F=K_{F}^{\dagger}K_{F}=K_{F}^{2}$. Via some trivial calculation, we get the probability of observing the filtering operation as follows,

Let $\vec{\omega}\equiv\frac{\vec{x}}{x_{0}}$ and $\omega\equiv\|\vec{\omega}\|$ denote the corresponding vector norm. The ratio can be equivalently written as

where the equality holds when $\vec{\omega}=-\frac{\omega}{a}{\vec{a}}$ and we have taken the notation $a\equiv\|\vec{a}\|$. To optimize over the possible filtering parameters, we define the function $f(\omega)\equiv\frac{1-\omega^{2}}{1+\omega^{2}-2a\omega}$. The derivatives of the function is $f(\omega)^{\prime}=\frac{2(a\omega^{2}-2\omega+a)}{(1+\omega^{2}-2a\omega)^{2}}$. It is trivial to find that the equation $f(\omega)^{\prime}=0$ holds at points $\omega_{+}=\frac{1+\sqrt{1-a^{2}}}{a}$ and $\omega_{-}=\frac{1-\sqrt{1-a^{2}}}{a}$ . Note that the variable $\omega\leq 1$ due to the constraint of Eq. (5). Thus, the function have a sole valid extreme point $\omega_{-}=\frac{1-\sqrt{1-a^{2}}}{a}$ with the corresponding maximum value $f(\omega_{-})=\frac{1}{\sqrt{1-a^{2}}}$. Therefore, the ratio that a filtering operation can increase the concurrence of the joint system has the upper bound as follows,

Note that the degree of mixture of the state $\rho_{A}$ is characterized by its purity $P(\rho_{A})\equiv tr(\rho_{A}^{2})=\frac{1}{2}(1+a^{2})$ JPAMT43.5.055302 . Obviously, the closer the state $\rho_{A}$ to be pure state, the more entanglement of the joint system can be promoted.

Now, we consider the format of the optimal filtering operation that can derive the maximal concurrence. Recall that any filtering operation is described by four parameters $x_{0}\in\mathbb{R}$ and $\vec{x}\in\mathbb{R}^{3}$. According to the aforementioned discussion, the optimal filtering operation should satisfy the constraints $\omega=\frac{1-\sqrt{1-a^{2}}}{a}$ and $\vec{\omega}=-\frac{\omega}{a}\vec{a}$ where we have taken the notation $\vec{\omega}\equiv\frac{\vec{x}}{x_{0}}$. It is obvious to conclude that a filtering operation $F$ can derive the optimal concurrence if the parameters satisfy the following relation

And the probability to observe the optimal operation is given by

The aforementioned discussion shows that the filtering operation can derive the maximum concurrence if the vector $\vec{x}$ is along vector $\vec{a}$ of the local state while the scaling parameter $x_{0}$ affect the probability of observing the operation. To construct an optimal filtering strategy, we need to maximize the scaling parameter $x_{0}$ while the constraint in Eq. (9) holds. Note that a filtering operation $F$ is valid if the condition in Eq. (5) holds. Namely, $x_{0}\geq\|\vec{x}\|\geq 0$ and $x_{0}+\|\vec{x}\|\leq 2$. Further we can derive the constraint of $x_{0}$ as follows,

Therefore, the parameters for describing the optimal filtering apparatus are $x_{0}=1+\sqrt{\frac{1-a}{1+a}}$ and $\vec{x}=-(1-\sqrt{\frac{1-a}{1+a}})\frac{\vec{a}}{a}$. And the corresponding probability is $1-a$. In the above discussion, we analyze the filtering operation of the polar decomposition form $F=U_{F}K_{F}$. Since local unitary operation has no effect on quantum entanglement, we only need to consider the positive operation $K_{F}$.

It is obvious that the local Bloch vector of the qubit state plays a key role for determining the performance of the optimal Filtering operation. The optimal Filtering operation can lead to entanglement increase with a ratio which grows with the norm of the local Bloch vector. On the contrary, the success probability of the optimal Filtering operation decrease with the Bloch vector norm. Note that the percentage of the entanglement increase is $\frac{1}{\sqrt{1-a^{2}}}-1$. We depict the performance in Fig. (1).

We have proved the following theorem.

Note that the density operator $\rho_{A}=\frac{1}{2}(I+\vec{a}\cdot\vec{a})$ of the partial system have eigenvalues $\frac{1}{2}(1\pm a)$ with corresponding projectors $\frac{1}{2}(I\pm\frac{\vec{a}}{a}\cdot\vec{\sigma})$. It is trivial to get $\rho_{A}^{-\frac{1}{2}}=\frac{\sqrt{1-a}+\sqrt{1+a}}{\sqrt{2}\sqrt{1-a^{2}}}(I-\frac{1-\sqrt{1-a^{2}}}{a}\frac{\vec{a}}{a}\cdot\vec{\sigma})$ which is proportional to $F_{optimal}$. Namely, $F_{optimal}=\sqrt{\frac{1-a}{2}}\rho_{A}^{-\frac{1}{2}}$. In some literatures, the operator $\rho_{A}^{-\frac{1}{2}}$ is used as a quantum operator. However, the eigenvalue of $\rho_{A}^{-\frac{1}{2}}$ is greater that 1 for mixed state $\rho_{A}$ and thus it can not act as a quantum operation. To be a valid quantum operator, a coefficient should be added such that $\mu\rho_{A}^{-\frac{1}{2}}$ with $\mu\leq\sqrt{\frac{1-a}{2}}$.

Further we consider the scenario that both Alice and Bob take local filtering operations. Suppose Bob’s filtering operator is described by the operator $E=\frac{1}{2}\sum_{j=0}^{3}y_{j}\cdot\sigma_{j}=\frac{1}{2}(y_{0}I+\vec{y}\cdot\vec{\sigma})$. The filtering operations on both side can increase the entanglement measured by concurrence with ratio $\frac{|det(F)|\cdot|det(E)|}{tr((F^{\dagger}F\otimes E^{\dagger}E)\rho_{AB})}$. It is not obvious to optimize the ratio over all valid filtering operators $F$ and $E$ simultaneously. However, we can consider that there is a tiny gap in time to apply the operations. Alice applies the optimal filtering operation $F_{optimal}$ on her system and increase the concurrence by ratio $\frac{1}{\sqrt{1-a^{2}}}$, which does not affect the Bloch vector $\vec{b}$ of Bob’s system. Nearly simultaneously, Bob applies the optimal filtering operation $E_{optimal}$ which is only determined by the Bloch vector $\vec{b}$ on his side and further increase the concurrence with ratio $\frac{1}{\sqrt{1-b^{2}}}$. We have taken the notation $b\equiv\|\vec{b}\|$. Eventually, the concurrence of the joint system is increased by ratio $\frac{1}{\sqrt{(1-a^{2})(1-b^{2})}}$ via local filtering operations $F_{optimal}$ and $E_{optimal}$ on their subsystems, respectively. Thus, we can conclude the following lemma.

In this section, we investigate the concurrence change under local measurement and try to answer the question of finding optimal quantum measurement to enhance quantum entanglement. We evaluate the performance of a measurement strategy via the expectation of the post-measurement concurrence which is denoted as $EC(\mathcal{M})$ for measurement apparatus $\mathcal{M}$.

Suppose measurement apparatus $\mathcal{M}\equiv\{M_{k}\}$ and $\mathcal{N}\equiv\{N_{j}\}$ are performed on the subsystem of Alice and Bob, respectively. Trivial analysis shows that the expectation of post-measurement concurrence is as follows,

which indicates that the local measurements $\mathcal{M}\equiv\{M_{k}\}$ and $\mathcal{N}\equiv\{N_{j}\}$ are independent with respect to the effect on entanglement. Thus, we only need to consider the performance on one side.

We consider dichotomic measurement $\mathcal{M}\equiv\{M_{0},M_{1}\}$ on Alice system. Suppose $M_{0}^{\dagger}M_{0}=\sum_{k=0}^{1}m_{k}^{2}|m_{k}\rangle\langle m_{k}|$ is the spectral decomposition of the operator $M_{0}^{\dagger}M_{0}$. Due to the completeness of quantum measurement, we have $M_{1}^{\dagger}M_{1}=\sum_{k=0}^{1}(1-m_{k}^{2})|m_{k}\rangle\langle m_{k}|$. Note that $|det(M_{0})|=\sqrt{det(M_{0}^{\dagger}M_{0})}=|m_{0}m_{1}|$ and similarly $|det(M_{1})|=\sqrt{(1-m_{0}^{2})(1-m_{1}^{2})}$. Thus, the expectation of concurrence is

Let $m_{0}\equiv\sin{\theta}$ and $m_{1}\equiv\sin{\phi}$ with $\theta,\phi\in[0,\frac{\pi}{2}]$. Then, it follows that

Obviously, $EC(\mathcal{M})=0$ when $\theta=\frac{\pi}{2}+\phi$, namely $\mathcal{M}$ is a projective measurement in the basis $\{|m_{0}\rangle,|m_{1}\rangle\}$, which coincides with the fact that projective measurement breaks entanglement in joint system. And $EC(\mathcal{M})=C$ when $\theta=\phi$, namely, the measurement operator is identity. It coincides with the fact that local operations can not create entanglement and therefore the best local operation for keeping quantum entanglement is the identity operation to leave it along.

Although the results are well known, our analytical work gives more rigorous support to the facts.

In this article, we considered the problem of quantitatively increasing the entanglement of a bipartite quantum state shared between two separated parties. Firstly, we investigated the optimal local filtering operation for increasing the most entanglement, measured by the concurrence of the joint system. With some non-trivial calculations, we derived the upper bound of the entanglement increase ratio and the analytical expression of the optimal operation for reaching the upper bound ratio as well as the success probability of the operation. We found that the optimal settings are completely determined by the corresponding local Bloch vector. Specifically, the entanglement increase ratio increases with the length of the Bloch vector while the success probability decreases with it. Secondly, we considered the effect of the general measurement by extending the result of the filtering operation. We found that any local measurement could not increase the expectation of entanglement. Although it is a well-known fact, our result gives more rigorous analytical evidence .