Complementary relation between quantum entanglement and entropic uncertainty
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Yun Cao1, Dong Wang,1,2,∗, Xiao-Gang Fan1, Fei Ming1, Zhang-Yin Wang1, Liu Ye,1,∗1School of Physics & Material Science, Anhui University, Hefei 230601, China 2CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
First author contact:∗ Authors to whom any correspondence should be addressed. Received:2020-09-4Revised:2020-10-14Accepted:2020-10-26Online:2020-12-18
Abstract Quantum entanglement is regarded as one of the core concepts, which is used to describe the non-classical correlation between subsystems, and entropic uncertainty relation plays a vital role in quantum precision measurement. It is well known that entanglement of formation can be expressed by von Neumann entropy of subsystems for arbitrary pure states. An interesting question is naturally raised: is there any intrinsic correlation between the entropic uncertainty relation and quantum entanglement? Or if the relation can be applied to estimate the entanglement. In this work, we focus on exploring the complementary relation between quantum entanglement and the entropic uncertainty relation. The results show that there exists an inequality relation between both of them for an arbitrary two-qubit system, and specifically the larger uncertainty will induce the weaker entanglement of the probed system, and vice versa. Besides, we use randomly generated states as illustrations to verify our results. Therefore, we claim that our observations might offer and support the validity of using the entropy uncertainty relation to estimate quantum entanglement. Keywords：uncertainty relation;entanglement of formation;concurrence
PDF (673KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Yun Cao, Dong Wang, Xiao-Gang Fan, Fei Ming, Zhang-Yin Wang, Liu Ye. Complementary relation between quantum entanglement and entropic uncertainty. Communications in Theoretical Physics, 2021, 73(1): 015101- doi:10.1088/1572-9494/abc46f
1. Introduction
The uncertainty principle is considered as one of the features in quantum theory which is very different from that of the classical counterpart [1]. As [1] stated, the certainty of estimation for a particle’s position implies the uncertainty of the momentum estimation, and vice versa and it has led to many physical and philosophical discussions. Actually, the uncertainty relation has various mathematical expressions by means of different quantities. Apart from the standard deviation via variance [2], there is alternative expressing means by information entropy, i.e., the so-called entropic uncertainty relations (EURs) [3–7]. The main difference between EURs and other inequalities lies in that the EURs are only considered in the framework of the measurement’s probabilities.
On the other hand, the concept of entanglement was proposed by Schrödinger many decades ago, which is the amazing characteristic of quantum mechanics [8]. As an important quantum resource, entanglement is widely applied to achieve many quantum tasks including quantum teleportation [9], quantum computation [10], remote state preparation [11, 12] and so on. We would like to ask whether there is any connection between quantum entanglement and the entropic uncertainty? For the variance-based uncertainty relations it is well known that they can be used for detection of entanglement. For this, the first work which raised the question of whether EURs and entanglement are somehow connected was done, to our knowledge, in [13]. Afterwards, Gühne et al demonstrated in detail the separability conditions of the bipartite from EURs [14]. The authors in reference [14] mainly have derived criteria for separability from EURs on one part of a bipartite system. They proved EURs can be available for the witness of separable states, however might be ineffective for witnessing entangled states.
Recently, Camalet [15] derived a novel and promising monogamy inequality for any local quantum resource and entanglement. The monogamy inequality provides the intrinsic relation among three local resources: entanglement, nonuniformity, and coherence. For nonuniformity, the author has discussed in detail three types of entropies: von Neumann entropy, Rényi entropy, and Tsallis entropy. A monogamy inequality for entanglement and local nonuniformity is derived. In previous work [14, 15], they did not provide a specific expression between entropic uncertainty and entanglement. Motivated by this, the aim of this paper is to establish deeper connections between entropic uncertainty and entanglement, and put forward a concrete expression formula between them.
The remainder of this paper is organized as follows: in section 2, we review the EURs and the quantification of entanglement. In section 3, we present an inequality relation between entropic uncertainty and entanglement in pure states of any two particles. Then we discuss a class of special two-qubit pure states, namely Bell-like states. Interestingly, we derive an equality relation between entropic uncertainty and entanglement. In section 4, we take an explicit example to support our obtained conclusion, by virtue of a type of pure state. In section 5, as illustrations, we testify our results by considering some special kinds of mixed states, transformed from Werner-like states and maximally entangled mixed states by an arbitrary unitary operation. Finally, we end our paper with a brief conclusion.
2. Preliminaries
2.1. Uncertainty relation
The uncertainty relation, originally proposed by Heisenberg [1], is one of the appealing features in the regime of quantum mechanics. It provides a meaningful bound of precision for the measurement of a pair of incompatible observables, telling us that we cannot measure all the measurements on the particle of a state accurately at the same time, even if it is fully described. The uncertainty relation, which differentiates from quantum world to classical world, can be described according to a standard deviation [2, 16]$\begin{eqnarray}{\rm{\Delta }}R\cdot {\rm{\Delta }}Q\geqslant \displaystyle \frac{1}{2}\left|\left\langle \left[R,\ Q\right]\right\rangle \right|,\end{eqnarray}$where the variance ${\rm{\Delta }}R=\sqrt{\left\langle {R}^{2}\right\rangle -{\left\langle R\right\rangle }^{2}}$ where $\left\langle R\right\rangle $ refers to the expectation value of operator R, and $\left[R,\ Q\right]={RQ}-{QR}$ means the commutator. Yet, the standard deviation is not always an optimal measure for the uncertainty because the commutator is dependent of the systemic state ρ, if the expectation value ⟨[R,Q]⟩ is zero, then the right-hand side (RHS) of the inequality (1) will lead to an undesirable result. To overcome this shortcoming, Deutsch put forward the entropy-based uncertainty relations [6], which was simplified by Kraus [7], Maassen, and Uffink [4] as$\begin{eqnarray}H(R)+H(Q)\geqslant {\mathrm{log}}_{2}\displaystyle \frac{1}{c},\end{eqnarray}$where H(X) denotes the Shannon entropy $H(X)\,=-{\sum }_{\mu }{x}_{\mu }\mathrm{log}{x}_{\mu }$ with ${x}_{\mu }=\left\langle {\psi }_{\mu }\left|\rho \left|{\psi }_{\mu }\right.\right.\right\rangle $ and $X\in \left\{R,\ Q\right\}$, the parameter $c={\max }_{\mu ,\nu }\left\{{\left|\left\langle {\psi }_{\mu }| {\varphi }_{\nu }\right\rangle \right|}^{2}\right\}$ is the maximal overlap of observables and with $\left|{\psi }_{\mu }\right\rangle $ and $\left|{\varphi }_{\nu }\right\rangle $ being the eigenvectors of the observable R and Q. The merit of this relation beyond the former is that the latter does not depend on the states of the system.
More recently, the EUR in the presence of quantum memory has been proposed by Renes et al [17] and Berta et al [18], and the brand-new EUR can be mathematically expressed as$\begin{eqnarray}S\left(R\left|B\right.\right)+S\left(Q\left|B\right.\right)\geqslant {\mathrm{log}}_{2}\displaystyle \frac{1}{c}+S\left(A\left|B\right.\right),\end{eqnarray}$where $S\left(A\left|B\right.\right)$ is the conditional von Neumann entropy and its expression is $S(\rho )=-\mathrm{Tr}\left(\rho {\mathrm{log}}_{2}\rho \right)$, $S\left(A\left|B\right.\right)\,=S\left({\rho }_{{AB}}\right)-S\left({\rho }_{B}\right)$, $S\left(R\left|B\right.\right)$ is the conditional von Neumann entropy of the post-measurement state ${\rho }_{{RB}}\,={\sum }_{\mu }\left({\left|{\psi }_{\mu }\right\rangle }_{A}\left\langle {\psi }_{\mu }\right|\otimes {{\mathbb{1}}}_{{\mathbb{B}}}\right){\rho }_{{AB}}\left({\left|{\psi }_{\mu }\right\rangle }_{A}\left\langle {\psi }_{\mu }\right|\otimes {{\mathbb{1}}}_{{\mathbb{B}}}\right)$, after subsystem A is measured by R or Q, here ${{\mathbb{1}}}_{{\mathbb{B}}}$ is an identity operator in the Hilbert space of particle B. This new relation can be explained as follows: assuming there are two players, Alice and Bob, Bob firstly prepares an entangled state as ρAB in his chosen quantum state, and sends A to Alice and keeps B, then Alice performs one of the two measurement operations and informs Bob of her measured choice, Bob is able to predict the outcome of Alice’s result with the limit by the bound of equation (3). Particularly, we have that Alice’s measurement result can be accurately predicted when A and B are maximally entangled, in terms of the RHS of equation (3) being valued-zero with $S(A| B)=-\mathrm{log}d$ and ${\mathrm{log}}_{2}1/c=\mathrm{log}d$ in the above inequality. For a multi-measurement scenario, the relation can be written as [19]$\begin{eqnarray}\displaystyle \sum _{m=1}^{n}S\left({M}_{M}\left|B\right.\right)\geqslant -{\mathrm{log}}_{2}\left(\zeta \right)+\left(n-1\right)S\left(A\left|B\right.\right),\end{eqnarray}$where $\zeta ={\max }_{{{ \mathcal I }}_{n}}\left\{{\sum }_{{{ \mathcal I }}_{2}\sim {{ \mathcal I }}_{n-1}}{\max }_{{{ \mathcal I }}_{1}}\left[c\left({u}_{{{ \mathcal I }}_{1}}^{1},{u}_{{{ \mathcal I }}_{2}}^{2}\right){{\rm{\Pi }}}_{m=2}^{n-1}c\left({u}_{{{ \mathcal I }}_{m}}^{m},{u}_{{{ \mathcal I }}_{m+1}}^{m+1}\right)\right]\right\}$, as n=3 that means three measurements. Basically, EURs have attracted much attention in theoretical [20–29] and experimental aspects [30, 31], and they can be widely applied in the fields of quantum randomness [32], such as quantum cryptography [33, 34], quantum metrology [35], quantum key distribution [36, 37], and entanglement witnessing [30, 31].
2.2. Entanglement
Typically, entanglement of formation (EOF) has been defined by [38], for a two-sided measurement of the density matrix ρ for a quantum systems A and B. The density matrix can be decomposed into a set of pure states $\left|{\phi }_{i}\right\rangle $ with a certain probability xi$\begin{eqnarray}\rho =\displaystyle \sum _{i}{x}_{i}\left|{\phi }_{i}\right\rangle \left\langle {\phi }_{i}\right|.\end{eqnarray}$For each pure state, EOF can be denoted as the entropy of the subsystem A or B of the pure state as$\begin{eqnarray}E\left(\left|{\phi }_{i}\right\rangle \right)=S\left({\mathrm{Tr}}_{B}\left(\left|{\phi }_{i}\right\rangle \left\langle {\phi }_{i}\right|\right)\right)=S\left({\mathrm{Tr}}_{A}\left(\left|{\phi }_{i}\right\rangle \left\langle {\phi }_{i}\right|\right)\right),\end{eqnarray}$where ${\mathrm{Tr}}_{B}\left(\left|{\phi }_{i}\right\rangle \left\langle {\phi }_{i}\right|\right)$ and ${\mathrm{Tr}}_{A}\left(\left|{\phi }_{i}\right\rangle \left\langle {\phi }_{i}\right|\right)$ represent the reduced density matrix of A and B in pure state $\left|{\phi }_{i}\right\rangle $. Thus, entanglement of mixed states is defined as the average entanglement of the pure states of the decomposition, minimized over all decompositions of ρ, namely$\begin{eqnarray}E\left(\rho \right)=\mathop{\min }\limits_{\left\{{x}_{i},\left|{\phi }_{i}\right\rangle \right\}}\displaystyle \sum _{i}{x}_{i}E\left(\left|{\phi }_{i}\right\rangle \right).\end{eqnarray}$
Now let us introduce another measure of entanglement, which is called concurrence. Concurrence is defined by the use of so-called self-selected inversion transformations, and is a function of the state of any quantum qubit. For a two-qubit pure state $\left|\psi \right\rangle $, its concurrence can be expressed as [39]$\begin{eqnarray}C=\left|\left\langle \psi \left|\tilde{\psi }\right.\right\rangle \right|,\end{eqnarray}$where $\left|\tilde{\psi }\right\rangle =\left({\sigma }_{y}\otimes {\sigma }_{y}\left|{\psi }^{* }\right\rangle \right)$, here $\left|{\psi }^{* }\right\rangle $ is the complex conjugate of the pure state $\left|\psi \right\rangle $ and σy is the Pauli-y matrix. For a general two-qubit mixed matrix, its self-selected inversion density matrix can be given by$\begin{eqnarray}\tilde{\rho }=\left({\sigma }_{y}\otimes {\sigma }_{y}\right){\rho }^{* }\left({\sigma }_{y}\otimes {\sigma }_{y}\right),\end{eqnarray}$where, the matrix ρ* is the complex conjugate of the state ρ. Therefore, concurrence of mixed state ρ [38, 40] is as follows$\begin{eqnarray}C\left(\rho \right)=\mathop{\min }\limits_{\left\{{x}_{i},\left|{\phi }_{i}\right\rangle \right\}}\displaystyle \sum _{i}{x}_{i}C\left(\left|{\phi }_{i}\right\rangle \right).\end{eqnarray}$The minimization is taken over all possible decompositions into pure states, the analytic expression is [39]$\begin{eqnarray}C\left(\rho \right)=\max \left\{0,\ {\lambda }_{1}-{\lambda }_{2}-{\lambda }_{3}-{\lambda }_{4}\right\},\end{eqnarray}$where ${\lambda }_{n}\ \left(n\in \left\{1,\ 2,\ 3,\ 4\right\}\right)$ are the eigenvalues in decreasing order of the Hermitian matrix $R=\sqrt{\sqrt{\rho }\tilde{\rho }\sqrt{\rho }}$.
In fact, there is a functional relationship between concurrence and EOF, and this function relation can be written as$\begin{eqnarray}E\left(\rho \right)=h\left(\displaystyle \frac{1+\sqrt{1-{C}^{2}\left(\rho \right)}}{2}\right),\end{eqnarray}$where the binary entropy $h\left(x\right)=-x{\mathrm{log}}_{2}x-\left(1-x\right){\mathrm{log}}_{2}\left(1-x\right)$, and $E\left(\rho \right)\geqslant 0$.
3. Relation between entropic uncertainty and quantum entanglement for pure states
In this section, we will derive the relation between entanglement and entropic uncertainty with respect to arbitrary two-qubit pure states, and put forward a theorem and corollary to elaborate our results. To illustrate our findings in our consideration, we use three Pauli operators, which are used to measure subsystem A to obtain the states ${\rho }_{{RB}}\left(R\in \left\{{\sigma }_{x,}\ {\sigma }_{y,}\ {\sigma }_{z}\right\}\right)$, which can be written as$\begin{eqnarray}{\rho }_{{RB}}=\sum {R}_{i}\left|\psi \right\rangle \left\langle \psi \right|{{R}_{i}}^{\dagger },\end{eqnarray}$where ${R}_{1}=\left|{\psi }_{1}\right\rangle \left\langle {\psi }_{1}\right|$ and ${R}_{2}=\left|{\psi }_{2}\right\rangle \left\langle {\psi }_{2}\right|$, here $\left|{\psi }_{1}\right\rangle \ $ and $\left|{\psi }_{2}\right\rangle $ are the normalized eigenvectors of R, through a series of calculations, we obtain ${R}_{1}=\tfrac{{\mathbb{1}}+R}{2}$ and ${R}_{2}=\tfrac{{\mathbb{1}}-R}{2}$. The post-measurement states can be simplified as$\begin{eqnarray}{\rho }_{{RB}}=\displaystyle \frac{1}{2}(\left|\psi \right\rangle \left\langle \psi \right|+R\left|\psi \right\rangle \left\langle \psi \right|R).\end{eqnarray}$
For arbitrary two-qubit pure states $\left|\psi \right\rangle $, the entropic uncertainty and EOF satisfy the following complementary relation as$\begin{eqnarray}S(X\left|B\right.)+S(Y\left|B\right.)+S(Z\left|B\right.)+2E(\left|\psi \right\rangle )\geqslant 2,\end{eqnarray}$where X,Y,Z are the standard Pauli operators.
In order to prove the above theorem, we resort to the superpostion $\left|\phi \right\rangle $ with the form of$\begin{eqnarray}\left|\phi \right\rangle =\cos \theta \left|{\phi }_{1}\right\rangle +\sin \theta \left|{\phi }_{2}\right\rangle ,\end{eqnarray}$where $\left|{\phi }_{1}\right\rangle =\cos \alpha \left|00\right\rangle +\sin \alpha \left|11\right\rangle $ and $\left|{\phi }_{2}\right\rangle =\cos \beta \left|01\right\rangle \,+\sin \beta \left|10\right\rangle $ are Bell-like states. When employing σy to measure subsystem A, it is easy to obtain that the eigenvalues of states have a set of special values $\left\{0,\ 0,\ \tfrac{1}{2},\ \tfrac{1}{2}\right\}$. As a result, the entropy can be described as$\begin{eqnarray}S({\rho }_{{YB}})=1,\ S({\rho }_{B})=E(| \phi \rangle ),\end{eqnarray}$where S(ρYB) denotes the entropy measured by σy, and S(ρB) is the entropy of subsystem B. According to equation (17), we have$\begin{eqnarray}\begin{array}{rcl}U & = & S(X\left|B\right.)+S(Y\left|B\right.)+S(Z\left|B\right.)\\ & = & S\left(X\left|B\right.\right)+S\left(Z\left|B\right.\right)+1-E\left(\left|\phi \right\rangle \right).\end{array}\end{eqnarray}$With respect to the two-qubit pure state $\left|\phi \right\rangle $, we have S(ρAB)=0. Consequently, equation (3) can be written as$\begin{eqnarray}\begin{array}{rcl}S(X\left|B\right.)+S(Z\left|B\right.) & \geqslant & 1+S(A\left|B\right.),\\ S(A\left|B\right.) & = & -E(| \phi \rangle ).\end{array}\end{eqnarray}$Combining equations (17)–(19), the relation between entanglement and the entropic uncertainty for superpostion can be given by$\begin{eqnarray}U+2E(\left|\phi \right\rangle )\geqslant 2,\end{eqnarray}$which recovers our result as shown in equation (15).
Importantly, equation (15) reveals that the entropic uncertainty and EOF satisfy the complementary relation. As a matter of fact, $\left|\phi \right\rangle $ can represent the set of arbitrary two-qubit pure states, according to Schmidt decomposition. In this sense, we say our obtained result is universal regarding two-qubit pure states, verifying our theorem. As an illustration, the EUR as a function of entanglement has been plotted as figure 1, by choosing 105 randomly generated states. In terms of our result, two nontrivial conclusions can be deduced: (i) two qubits in any pure state must be entangled, when the magnitude of the entropic uncertainty is less than 2. With this in mind, we say that the uncertainty can be considered as an indicator of entanglement; (ii) the entropic uncertainty is closely anti-correlated with entanglement, indicating that the smaller entropic uncertainty shows the greater entanglement, and vice versa.
Figure 1.
New window|Download| PPT slide Figure 1.The entropic uncertainty versus EOF $E\left(\left|\psi \right\rangle \right)$ for the two-qubit pure states $\left|\psi \right\rangle $. The red line (limit) is denoted by $U=S(X\left|B\right.)+S(Y\left|B\right.)+S(Z\left|B\right.)=2-2E\left(\left|\psi \right\rangle \right)$. The figure plots the entropic uncertainty (U) along the y-axis, and the EOF (E) along the x-axis, for 105 randomly two-qubit pure states.
For any Bell-type states $\left|{\phi }_{B}\right\rangle $, we have the relation between the entropic uncertainty and EOF expressed as$\begin{eqnarray}U+2E\left(\left|{\phi }_{B}\right\rangle \right)=2.\end{eqnarray}$
We make use of complementary observations (say, three Pauli operators) to measure subsystem A of a system with any Bell-type state $\left|{\phi }_{B}\right\rangle =\cos \delta | 00\rangle +\sin \delta | 11\rangle $ (δ∈[0,π]), based on equation (14) we then attain the eigenvalues λi of operator σi⨂1: ${\lambda }_{X}={\lambda }_{Y}=\left\{0,\ 0,\ \tfrac{1}{2},\ \tfrac{1}{2}\right\};$ when ${\lambda }_{Z}\,=\left\{0,\ 0,\tfrac{1+\sqrt{1-{C}^{2}\left(\left|{\phi }_{B}\right\rangle \right)}}{2},\tfrac{1-\sqrt{1-{C}^{2}\left(\left|{\phi }_{B}\right\rangle \right)}}{2}\right\}$, here $C\left(\left|{\phi }_{B}\right\rangle \right)$ is concurrence of Bell-like states. And the eigenvalues of the reduced density matrix ${\mathrm{Tr}}_{A}[{\rho }_{{RB}}]$ is ${\lambda }_{B}\,=\left\{\tfrac{1+\sqrt{1-{C}^{2}\left(\left|{\phi }_{B}\right\rangle \right)}}{2},\tfrac{1-\sqrt{1-{C}^{2}\left(\left|{\phi }_{B}\right\rangle \right)}}{2}\right\}$. Therefore, we can derive the following relations$\begin{eqnarray}\begin{array}{rcl}S({\rho }_{{XB}}) & = & S({\rho }_{{YB}})=1,\\ S({\rho }_{{ZB}}) & = & S({\rho }_{B})=E(\left|{\phi }_{B}\right\rangle ),\end{array}\end{eqnarray}$which support the establishment of equation (21). The equality reveals that the entropic uncertainty and entanglement satisfy complementarity with regard to arbitrary Bell-type states. Furthermore, it can be harvested that the entropic uncertainty is completely inversely correlated with the twice EOF as displayed in figure 2, in the architecture of the Bell-like state’s systems.
Figure 2.
New window|Download| PPT slide Figure 2.The entropic uncertainty of Bell-type states and the EOF as a function of the state’s parameter δ. The blue solid line represents the entropic uncertainty (U) and the red solid line represents the twice EOF (2E).
4. Numerical example and discussions
To verify our result in equation (15), we consider a specific pure state ∣$\Psi$⟩ with the form of$\begin{eqnarray}\begin{array}{rcl}| \psi \rangle & = & \cos \theta \left(\cos \theta \left|00\right\rangle +\sin \theta \left|11\right\rangle \right)\\ & & +\sin \theta \left(\cos \theta \left|01\right\rangle +\sin \theta \left|10\right\rangle \right),\end{array}\end{eqnarray}$where θ∈[0,2π]. Then we make use of complementary observations (say, three Pauli operators) to measure subsystem A, and the eigenvalues of the post-measured state can be given as: ${\lambda }_{X}=\left\{0,0,\tfrac{1}{4}\left(3-\cos 4\theta \right),\tfrac{1}{2}{\cos }^{2}2\theta \right\}$, ${\lambda }_{Y}\,=\left\{\tfrac{1}{2},\tfrac{1}{2},0,0\right\}$, and ${\lambda }_{Z}=\left\{0,0,{\sin }^{2}\theta ,{\cos }^{2}\theta \right\}$. And the eigenvalues of reduced density matrix ${\rho }_{B}={\mathrm{Tr}}_{A}[{\rho }_{{RB}}]$ is ${\lambda }_{B}\,=\left\{\tfrac{1}{8}\left(4-\sqrt{2}\sqrt{7+\cos 8\theta }\right),\tfrac{1}{8}\left(4+\sqrt{2}\sqrt{7+\cos 8\theta }\right)\right\}$. As a result, the entropy can be described as$\begin{eqnarray}\begin{array}{rcl}S\left({\rho }_{{XB}}\right) & = & -\displaystyle \frac{1}{4}\left(3-\cos 4\theta \right){\mathrm{log}}_{2}\left(\displaystyle \frac{1}{4}\left(3-\cos 4\theta \right)\right)\\ & & -\displaystyle \frac{1}{2}{\cos }^{2}2\theta {\mathrm{log}}_{2}\displaystyle \frac{1}{2}{\cos }^{2}2\theta ,\\ S({\rho }_{{YB}}) & = & 1,\\ S\left({\rho }_{{ZB}}\right) & = & h\left({\sin }^{2}\theta \right).\end{array}\end{eqnarray}$
In order to show the performance of our result, we plot the uncertainty and entanglement as a function of the state’s parameter θ in figure 3. From the figure, one can directly see that the relation between entropic uncertainty and entanglement of the specific state in equation (15) is satisfied all the time.
Figure 3.
New window|Download| PPT slide Figure 3.The entropic uncertainty of state $\left|\psi \right\rangle $ and the twice EOF (2E) as a function of the state’s parameter θ.
5. Relation between entropic uncertainty and quantum entanglement for mixed states
Above, we have explored the intrinsic relation between entropic uncertainty and entanglement for an arbitrary two-qubit pure state. Then, we naturally raise another intriguing question that: what is the relation between them if the system is mixed? To answer this issue, we here discuss two special ensembles of mixed states in the following.
5.1. Werner-type states
In general, a two-qubit Werner-type state can be given by$\begin{eqnarray}{\rho }_{W}=p\left|{\phi }_{B}\right\rangle \left\langle {\phi }_{B}\right|+\displaystyle \frac{{\mathbb{1}}}{4}\left(1-p\right),\end{eqnarray}$where the parameter p is a real number in a closed interval $\left[0,\ 1\right]$. Here, the states $\left|{\phi }_{B}\right\rangle =\cos \xi | 00\rangle +\sin \xi | 11\rangle $ are Bell-type states as mentioned before. The purity of the states ρW is ${ \mathcal P }\left({\rho }_{W}\right)=\tfrac{1\,+\,3{p}^{2}}{4}$. According to equation (11), the concurrence of ρW can be calculated as [41]$\begin{eqnarray}C\left({\rho }_{W}\right)=\max \left\{0,\ {pC}\left(\left|{\phi }_{B}\right\rangle \right)-\displaystyle \frac{1-p}{2}\right\},\end{eqnarray}$where, $C\left(\left|{\phi }_{B}\right\rangle \right)$ is the concurrence of the Bell-type states $\left|{\phi }_{B}\right\rangle $. Similarly, we can acquire entropic uncertainty of Werner-type states, by employing complementary observations (three Pauli operators) to measure subsystem A. In accordance with equation (14), we have the post-measured states$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{RB}}^{W} & = & \displaystyle \frac{1}{2}\left[p\left|{\phi }_{B}\right\rangle \left\langle {\phi }_{B}\right|+\displaystyle \frac{1-p}{4}{\mathbb{1}}\right]\\ & & +\displaystyle \frac{1}{2}{\sigma }_{i}p\left|{\phi }_{B}\right\rangle \left\langle {\phi }_{B}\right|{\sigma }_{i}+\displaystyle \frac{1}{2}{\sigma }_{i}\displaystyle \frac{1-p}{4}{\mathbb{1}}{\sigma }_{i}\\ & = & \displaystyle \frac{1}{2}p\left(\left|{\phi }_{B}\right\rangle \left\langle {\phi }_{B}\right|+{\sigma }_{i}\left|{\phi }_{B}\right\rangle \left\langle {\phi }_{B}\right|{\sigma }_{i}\right)+\displaystyle \frac{1-p}{4}{\mathbb{1}}\\ & = & p{\rho }_{{RB}}+\displaystyle \frac{{\mathbb{1}}}{4}\left(1-p\right),\end{array}\end{eqnarray}$where ρRB is the post-measured states of $\left|{\phi }_{B}\right\rangle $.
In order to obtain the entropic uncertainty, the eigenvalues of the measured states ${\rho }_{{RB}}^{W}$ can be expressed as$\begin{eqnarray}\begin{array}{rcl}{\lambda }_{{XB}}^{W} & = & \left\{\displaystyle \frac{1-p}{4},\ \displaystyle \frac{1-p}{4},\ \displaystyle \frac{1+p}{4},\ \displaystyle \frac{1+p}{4}\right\},\\ {\lambda }_{{YB}}^{W} & = & \left\{\displaystyle \frac{1-p}{4},\ \displaystyle \frac{1-p}{4},\ \displaystyle \frac{1+p}{4},\ \displaystyle \frac{1+p}{4}\right\},\\ {\lambda }_{{ZB}}^{W} & = & \left\{\displaystyle \frac{1-p}{4},\ \displaystyle \frac{1-p}{4},\displaystyle \frac{1+p-2p\cos \xi }{4},\right.\\ & & \times \ \left.\displaystyle \frac{1+p+2p\cos \xi }{4}\right\},\end{array}\end{eqnarray}$and the eigenvalues of subsystem states ${\rho }_{B}={\mathrm{Tr}}_{A}({\rho }_{W})$ are$\begin{eqnarray}{\lambda }_{B}^{W}=\left\{\ \displaystyle \frac{1-p\cos 2\xi }{2},\ \displaystyle \frac{1+p\cos 2\xi }{2}\right\}.\end{eqnarray}$Then, we can obtain the entropies of the post-measured states ${\rho }_{{RB}}^{W}$ and the entropy of state ρB as$\begin{eqnarray}\begin{array}{rcl}S({\rho }_{{XB}}^{W}) & = & S({\rho }_{{YB}}^{W})=1+h\left(\displaystyle \frac{1-p}{2}\right),\\ S({\rho }_{{ZB}}^{W}) & = & -2\left(\displaystyle \frac{1-p}{4}{\mathrm{log}}_{2}\displaystyle \frac{1-p}{4}\right)\\ & & -\displaystyle \frac{1+p-2p\cos 2\xi }{4}{\mathrm{log}}_{2}\displaystyle \frac{1+p-2p\cos 2\xi }{4}\\ & & -\displaystyle \frac{1+p+2p\cos 2\xi }{4}{\mathrm{log}}_{2}\displaystyle \frac{1+p+2p\cos 2\xi }{4},\\ S({\rho }_{B}^{W}) & = & h\left(\displaystyle \frac{1-p\cos 2\xi }{2}\right),\end{array}\end{eqnarray}$respectively. Substituting the above formula into the LHS of equation (4), one can acquire the analytical expression of the entropic uncertainty.
Now, let us turn to probe the relationship between the entropic uncertainty and concurrence (i.e., entanglement). Intuitively, it seems that there is no direct connection between them from their expressions. While we provide the uncertainty and entanglement as a function of the state’s parameter ξ with different p, as illustrated in figure 4. Following the figure, we can see an interesting result that the variation of entanglement is almost opposite to the variation of entropic uncertainty. When the entropic uncertainty increases, the entanglement decreases, and vice versa. In this sense, we claim that the uncertainty and entanglement are correlated intensively in such a mixed-state framework.
Figure 4.
New window|Download| PPT slide Figure 4.The entropic uncertainty and entanglement (concurrence) of Werner-type states with respect to the state’s parameter ξ, the red solid line denotes the entropic uncertainty (U), the blue line represents concurrence (C). Graph (a): p=0.75 and graph (b): p=0.9 are set.
5.2. Maximally entangled mixed states
Maximally entangled mixed states ρMEMS can be written as [42]$\begin{eqnarray}{\rho }_{\mathrm{MEMS}}=\left(\begin{array}{cccc}f(C) & 0 & 0 & \tfrac{C}{2}\\ 0 & 1-2f(C) & 0 & 0\\ 0 & 0 & 0 & 0\\ \tfrac{C}{2} & 0 & 0 & f(C)\end{array}\right),\end{eqnarray}$with$\begin{eqnarray*}f(C)=\left\{\begin{array}{l}\tfrac{1}{3},\ \ \ \ \mathrm{if}\ \ \ C\ \lt \ \tfrac{2}{3}\\ \tfrac{C}{2},\ \ \ \ \mathrm{if}\ \ \ C\ \geqslant \ \tfrac{2}{3}\end{array}\right.\end{eqnarray*}$where C represents the concurrence of states ρMEMS. Canonically, the type of states maximize the concurrence for a given purity with ${ \mathcal P }({\rho }_{\mathrm{MEMS}})=\tfrac{1}{3}+\tfrac{{C}^{2}}{2}$ ($0\leqslant C\leqslant \tfrac{2}{3}$) and ${ \mathcal P }{({\rho }_{\mathrm{MEMS}})={C}^{2}+(1-C)}^{2}$ ($\tfrac{2}{3}\leqslant C\leqslant 1$). By using the same methods as before, we can obtain the post-measured states as$\begin{eqnarray}{\rho }_{{XB}}^{\mathrm{MEMS}}=\left(\begin{array}{cccc}\tfrac{f(C)}{2} & 0 & 0 & \tfrac{C}{4}\\ 0 & \tfrac{1-f(C)}{2} & \tfrac{C}{4} & 0\\ 0 & \tfrac{C}{4} & \tfrac{f(C)}{2} & 0\\ \tfrac{C}{4} & 0 & 0 & \tfrac{1-f(C)}{2}\end{array}\right),\end{eqnarray}$$\begin{eqnarray}{\rho }_{{YB}}^{\mathrm{MEMS}}=\left(\begin{array}{cccc}\tfrac{f(C)}{2} & 0 & 0 & \tfrac{C}{4}\\ 0 & \tfrac{1-f(C)}{2} & -\tfrac{C}{4} & 0\\ 0 & -\tfrac{C}{4} & \tfrac{f(C)}{2} & 0\\ \tfrac{C}{4} & 0 & 0 & \tfrac{1-f(C)}{2}\end{array}\right),\end{eqnarray}$$\begin{eqnarray}{\rho }_{{ZB}}^{\mathrm{MEMS}}=\left(\begin{array}{cccc}f(C) & 0 & 0 & 0\\ 0 & 1-2f(C) & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & f(C))\end{array}\right),\end{eqnarray}$whose corresponding eigenvalues are given by$\begin{eqnarray}\begin{array}{rcl}{\lambda }_{{XB}}^{\mathrm{MEMS}} & = & {\lambda }_{{YB}}^{\mathrm{MEMS}}=\left\{{\lambda }_{a},{\lambda }_{a},{\lambda }_{b},{\lambda }_{b}\right\},\\ {\lambda }_{{ZB}}^{\mathrm{MEMS}} & = & \left\{0,1-2f(C),f(C),f(C)\right\},\end{array}\end{eqnarray}$with$\begin{eqnarray*}\begin{array}{rcl} & & {\lambda }_{a}=\displaystyle \frac{1}{4}\left(1-\sqrt{1-4f\left(C\right)+4f{\left(C\right)}^{2}+{C}^{2}}\right),\\ & & {\lambda }_{b}=\displaystyle \frac{1}{4}\left(1+\sqrt{1-4f\left(C\right)+4f{\left(C\right)}^{2}+{C}^{2}}\right),\end{array}\end{eqnarray*}$and the eigenvalue of the B's reduced density matrix reads as$\begin{eqnarray}{\lambda }_{B}^{\mathrm{MEMS}}=\left\{1-f(C),\ f(C)\right\}.\end{eqnarray}$As a consequence, the explicit expression can be offered as$\begin{eqnarray}\begin{array}{rcl}S\left({\rho }_{{XB}}^{\mathrm{MEMS}}\right) & = & S\left({\rho }_{{YB}}^{\mathrm{MEMS}}\right)=-2{\lambda }_{a}{\mathrm{log}}_{2}{\lambda }_{a}-2{\lambda }_{b}{\mathrm{log}}_{2}{\lambda }_{b},\\ S\left({\rho }_{{ZB}}^{\mathrm{MEMS}}\right) & = & -2f\left(C\right){\mathrm{log}}_{2}f\left(C\right)\\ & & -\left(1-2f\left(C\right)\right){\mathrm{log}}_{2}\left(1-2f\left(C\right)\right),\\ S\left({\rho }_{B}^{\mathrm{MEMS}}\right) & = & h\left(f\left(C\right)\right),\\ U & = & S\left({\rho }_{{XB}}^{\mathrm{MEMS}}\right)+S\left({\rho }_{{YB}}^{\mathrm{MEMS}}\right)+S\left({\rho }_{{ZB}}^{\mathrm{MEMS}}\right)\\ & & -3S\left({\rho }_{B}^{\mathrm{MEMS}}\right),\end{array}\end{eqnarray}$which shows the entropic uncertainty is straightforwardly associated with the quantum entanglement C. Further, we draw the uncertainty versus the systemic entanglement (C) in figure 5. From this figure, we can obtain that (i) the relationship between the uncertainty and entanglement is monotonic. Explicitly, the uncertainty will monotonically decrease with the growing entanglement; (ii) the magnitude of the entropic uncertainty will become zero-valued, when the entanglement reaches maximum. Besides, the entropic uncertainty will maximize with its value of 8/3, if the systemic entanglement disappears.
Figure 5.
New window|Download| PPT slide Figure 5.The intrinsic relation between the entropic uncertainty and concurrence in the case of maximally entangled mixed states. The y-axis plots the entropic uncertainty (U), and the x-axis denotes the systemic concurrence (C).
6. Conclusion
In this paper, we have investigated the intrinsic relation between the entropic uncertainty relation and the entanglement. For arbitrary two-qubit pure states, we have derived an inequality between the entropy-based uncertainty and EOF, indicating the complementary relation between them. Besides, we have discussed the relationship in the Bell-type states, it has been proved that there is a complete anti-correlation between the entropic uncertainty relation and EOF, and importantly, we argue that the uncertainty can be perfectly viewed as an indicator of quantum entanglement in this scenario. Furthermore, the relationship between the uncertainty and the entanglement (concurrence) is examined for the mixed states, including the Werner-type states and maximally entangled mixed states. Basically, there are some differences between the previous article in [15] and ours, which lie in: (1) the previous paper focuses on exploring the intrinsic relation between the various entropies and entanglement. While, the concern in our paper is to investigate the inequality relation between the preparation uncertainty and the entanglement. (2) The paper in [15] derived the upper bound of the relation between entropy and entanglement. By contrast, we have deduced the lower bound of the relation between uncertainty and entanglement based on entropic uncertainty relations for two-qubit pure states in our work. (3) Besides, we also have discussed the complementary relation between entropic uncertainty relation and concurrence for special mixed states. With these in mind, we claim that we have derived new results, which are different from the previous one. We believe that our investigations would shed light on the intrinsic relationship between the entropic uncertainty and the entanglement of bipartite systems, and be nontrivial to realistic quantum-resource-based quantum information processing.
Acknowledgments
This work was supported by the National Science Foundation of China under Grant Nos. 12075001, 61601002 and 11575001, Anhui Provincial Natural Science Foundation (Grant No. 1508085QF139) and the fund from CAS Key Laboratory of Quantum Information (Grant No. KQI201701).
,1,2,∗, Xiao-Gang Fan1, Fei Ming1, Zhang-Yin Wang1, Liu Ye
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