Abstract
We provide a sufficient condition for the monogamy inequality of multiparty quantum entanglement of arbitrary dimensions in terms of entanglement of formation. Based on the classical–classical–quantum(ccq) states whose quantum parts are obtained from the twoparty reduced density matrices of a threeparty quantum state, we show the additivity of the mutual information of the ccq states guarantees the monogamy inequality of the threeparty pure state in terms of EoF. After illustrating the result with some examples, we generalize our result of threeparty systems into any multiparty systems of arbitrary dimensions.
Introduction
Quantum entanglement is a nonclassical nature of quantum mechanics, which is a useful resource in many quantum information processing tasks such as quantum teleportation, dense coding and quantum cryptography^{1,2,3}. Because of its important roles in the field of quantum information and computation theory, there has been a significant amount of research focused on quantification of entanglement in bipartite quantum systems. Entanglement of formation(EoF) is the most wellknown bipartite entanglement measure with an operational meaning that asymptotically quantifies how many bell states are needed to prepare the given state using local quantum operations and classical communications^{4}. Although EoF is defined in any bipartite quantum systems of arbitrary dimension, its definition for mixed states is based on ‘convexroof extension’, which takes the minimum average over all purestate decompositions of the given state. As such an optimization is hard to deal with, analytic evaluation of EoF is known only in twoqubit systems^{5} and some restricted cases of higherdimensional systems so far.
In multiparty quantum systems, entanglement shows a distinct behavior that does not have any classical counterpart; if a pair of parties in a multiparty quantum system is maximally entangled, then they cannot be entanglement, not even classically correlated, with the rest parties. This restriction of sharing entanglement in multiparty quantum systems is known as the monogamy of entanglement(MoE)^{6,7}. MoE plays an important role such as the security proof of quantum key distribution in quantum cryptography^{2,8} and the Nrepresentability problem for fermions in condensedmatter physics^{9}.
Mathematically, MoE can be characterized using monogamy inequality; for a threeparty quantum state \(\rho _{ABC}\) and its twoparty reduced density matrices \(\rho _{AB}\) and \(\rho _{AC}\),
where \(E\left( \rho _{XY}\right) \) is an entanglement measure quantifying the amount of entanglement between subsystems X and Y of the bipartite quantum state \(\rho _{XY}\). Inequality (1) shows the mutually exclusive nature of bipartite entanglement \(E\left( \rho _{AB}\right) \) and \(E\left( \rho _{AC}\right) \) shared in threeparty quantum systems so that their summation cannot exceeds the total entanglement \(E\left( \rho _{A(BC)}\right) \).
Using tangle^{10} as the bipartite entanglement measure, Inequality (1) was first shown to be true for all threequbit states, and generalized for multiqubit systems as well as some cases of higherdimensional quantum systems^{11,12}. However, not all bipartite entanglement measures can characterize MoE in forms of Inequality (1), but only few measures are known so far satisfying such monogamy inequality^{13,14,15}. Although EoF is the most natural bipartite entanglement measure with the operational meaning in quantum state preparation, EoF is known to fail in characterizing MoE as the monogamy inequality in (1) even in threequbit systems; there exists quantum states in threequbit systems violating Inequality (1) if EoF is used as the bipartite entanglement measure. Thus, a natural question we can ask is ‘On what condition does the monogamy inequality hold in terms of the given bipartite entanglement measure?’.
Here, we provide a sufficient condition that monogamy inequality of quantum entanglement holds in terms of EoF in multiparty, arbitrary dimensional quantum systems. For a threeparty quantum state, we first consider the classical–classical–quantum (ccq) states whose quantum parts are obtained from the twoparty reduced density matrices of the threeparty state. By evaluating quantum mutual information of the ccq states as well as their reduced density matrices, we show that the additivity of the mutual information of the ccq states guarantees the monogamy inequality of the threeparty quantum state in terms of EoF. We provide some examples of threeparty pure state to illustrate our result, and we generalize our result of threeparty systems into any multiparty systems of arbitrary dimensions.
This paper is organized as follows. First we briefly review the definitions of classical and quantum correlations in bipartite quantum systems and recall their tradeoff relation in threeparty quantum systems. After providing the definition of ccq states as well as their mutual information between classical and quantum parts, we establish the monogamy inequality of threeparty quantum entanglement in arbitrary dimensional quantum systems in terms of EoF conditioned on the additivity of the mutual information for the ccq states. We also illustrate our result of monogamy inequality in threeparty quantum systems with some examples, and we generalize our result of entanglement monogamy inequality into multiparty quantum systems of arbitrary dimensions. Finally, we summarize our results.
Results
Correlations in bipartite quantum systems
For a bipartite pure state \({\left \psi \right\rangle }_{AB}\), its entanglement of formation (EoF) is defined by the entropy of a subsystem, \({E}_\mathbf{f}\left( {\left \psi \right\rangle }_{AB} \right) =S(\rho _A)\) where \(\rho _A={\text {tr}}_{B} {\left \psi \right\rangle }_{AB}{\left\langle \psi \right }\) is the reduced density matrix of \({\left \psi \right\rangle }_{AB}\) on subsystem A, and \(S\left( \rho \right) ={\text {tr}}\rho \ln \rho \) is the von Neumann entropy of the quantum state \(\rho \). For a bipartite mixed state \(\rho _{AB}\), its EoF is defined by the minimum average entanglement
over all possible pure state decompositions of \(\rho _{AB}=\sum _{i} p_i \psi _i\rangle _{AB}\langle \psi _i\).
For a probability ensemble \({\mathscr {E}} = \{p_i, \rho _i\}\) realizing a quantum state \(\rho \) such that \(\rho =\sum _{i}p_i\rho _i\), its Holevo quantity is defined as
Given a bipartite quantum state \(\rho _{AB}\), each measurement \(\{M^x_B\}\) applied on subsystem B induces a probability ensemble \({\mathscr {E}} = \{p_x, \rho _A^x\}\) of the reduced density matrix \(\rho _A={\text {tr}}_A\rho _{AB}\) in the way that \(p_x={\text {tr}}[(I_A\otimes M_B^x)\rho _{AB}]\) is the probability of the outcome x and \(\rho ^x_A={\text {tr}}_B[(I_A\otimes {M_B^x})\rho _{AB}]/p_x\) is the state of system A when the outcome was x. The oneway classical correlation (CC)^{16} of a bipartite state \(\rho _{AB}\) is defined by the maximum Holevo quantity
over all possible ensemble representations \({\mathscr {E}}\) of \(\rho _A\) induced by measurements on subsystem B.
The following proposition shows a tradeoff relation between classical correlation and quantum entanglement (measured by CC and EoF, respectively) distributed in threeparty quantum systems.
Proposition 1.
^{17} For a threeparty pure state \({\left \psi \right\rangle }_{ABC}\) with reduced density matrices \(\rho _{AB}={\text {tr}}_C{\left \psi \right\rangle }_{ABC}{\left\langle \psi \right }\), \(\rho _{AC}={\text {tr}}_B{\left \psi \right\rangle }_{ABC}{\left\langle \psi \right }\) and \(\rho _{A}={\text {tr}}_{BC}{\left \psi \right\rangle }_{ABC}{\left\langle \psi \right }\), we have
Classical–classical–quantum (CCQ) states
In this section, we consider a fourparty ccq states obtained from a bipartite state \(\rho _{AB}\), and provide detail evaluations of their mutual information. Without loss of generality, we assume that any bipartite state as a twoqudit state by taking d as the dimension of larger dimensional subsystem.
For a twoqudit state \(\rho _{AB}\), let us consider the reduced density matrix \(\rho _B={\text {tr}}_{A}\rho _{AB}\) and its spectral decomposition
Let
be the probability ensemble of \(\rho _A={\text {tr}}_{B}\rho _{AB}\) from the measurement \(\{{\left e_i \right\rangle }_B{\left\langle e_i \right }\}_{i=1}^{d1}\) on subsystem B of \(\rho _{AB}\), in a way that
and
Based on the eigenvectors \(\{ {\left e_j \right\rangle }_{B}\}\) of \(\rho _B\), we also consider the ddimensional Fourier basis elements \({\tilde{e}}_j \rangle _B = \frac{1}{\sqrt{d}}\sum _{k=0}^{d1}\omega _d^{jk}{\left e_k \right\rangle }_B\) for each \(j=0,\ldots ,d1\), where \(\omega _d = e^{\frac{2\pi i}{d}}\) is the dthroot of unity. Let
be the probability ensemble of \(\rho _A={\text {tr}}_{B}\rho _{AB}\) obtained by measuring subsystem B in terms of the Fourier basis \(\{{\left \tilde{e}_j \right\rangle }_B{\left\langle \tilde{e}_j \right }\}_{j=1}^{d1}\) where
and
Now we define the generalized ddimensional Pauli operators based on the eigenvectors of \(\rho _B\) as
and consider a fourqudit ccq state \(\Gamma _{XYAB}\)
for some ddimensional orthonormal bases \(\{{\left x \right\rangle }_X\}\) and \(\{{\left y \right\rangle }_Y\}\) of the subsystems X and Y, respectively. From Eqs. (8), (9), (11), (12) and (13), the reduced density matrices of \(\Gamma _{XYAB}\) are obtained as
and
where \(I_A\) and \(I_B\) are ddimensional identity operators of subsystems A and B, respectively.
Here we note that the mutual information between the classical and quantum parts of the ccq state in Eq. (14) as well as its reduced density matrices in Eqs. (15) and (16) are
and
where the detail calculation can be found in “Methods” section.
Monogamy inequality of multiparty entanglement in terms of EoF
It is known that quantum mutual information is superadditive for any ccq state of the form
that is, \({I}\left( \Xi _{XY:AB}\right) \ge {I}\left( \Xi _{X:AB}\right) +{I}\left( \Xi _{Y:AB}\right) \)^{18}. The following theorem shows that the additivity of quantum mutual information for ccq states guarantees the monogamy inequality of threeparty quantum entanglement in therms of EoF.
Theorem 1.
For any threeparty pure state \({\left \psi \right\rangle }_{ABC}\) with its twoqudit reduced density matrices \({\text {tr}}_C {\left \psi \right\rangle }_{ABC}{\left\langle \psi \right }=\rho _{AB}\) and \({\text {tr}}_B {\left \psi \right\rangle }_{ABC}{\left\langle \psi \right }=\rho _{AC}\), we have
conditioned on the additivity of quantum mutual information
and
where \(\Gamma _{XYAB}\) and \(\Gamma _{XYAC}\) are the ccq states of the form in Eq. (14) obtained by \(\rho _{AB}\) and \(\rho _{AC}\), respectively.
Conditioned on the additivity of quantum mutual information for ccq states, Theorem 1 shows that EoF can characterize the monogamous nature of bipartite entanglement shared in threeparty quantum systems, which is illustrateed in Fig. 1.
Example 1.
Let us consider threequbit GHZ state^{19},
with its reduced density matrices
The eigenvalues of \(\rho _B\) are \(\lambda _0=\lambda _1=\frac{1}{2}\) with corresponding eigenvectors \({\left e_0 \right\rangle }_B={\left 0 \right\rangle }_B\) and \({\left e_1 \right\rangle }_B={\left 1 \right\rangle }_B\) respectively. Thus the ensemble of \(\rho _A\) induced by measuring subsystem B of \(\rho _{AB}\) in terms of the eigenvectors of \(\rho _B\), that is, \(\{ M_B^0={\left 0 \right\rangle }_B{\left\langle 0 \right }, M_B^1={\left 1 \right\rangle }_B{\left\langle 1 \right }\}\) is
Because the Fourier basis elements of subsystem B with respect to the eigenvectors of \(\rho _B\) are
the ensemble of \(\rho _A\) induced by measuring subsystem B of \(\rho _{AB}\) in terms of the Fourier basis in Eq. (28) is
Now we consider the additivity of mutual information of the ccq state \(\Gamma _{XYAB}\) obtained from \(\rho _{AB}\) in Eq. (26). Due to Eq. (18), the mutual information of \(\Gamma _{XYAB}\) between XY and AB is
because \(S(\rho _A)=S(\rho _{AB})=\ln 2\) from Eqs. (26). For the mutual information of \(\Gamma _{XAB}\) between X and AB, Eq. (19) leads us to
where the second equality is from \(S(\rho _B)= \ln 2\) and
for the ensemble \({\mathscr {E}}_0\) in Eq. (27). For the mutual information of \(\Gamma _{YAB}\) between Y and AB, Eq. (20) leads us to
where the second equality is due to the ensemble \({\mathscr {E}}_1\) in Eq. (29).
From Eqs. (30), (31) and (33), we note that the mutual information of the ccq state \(\Gamma _{XYAB}\) obtained from \(\rho _{AB}\) is additive as in Eq. (23). Moreover, the symmetry of GHZ state assures that the same is also true for the reduced density matrix \( \rho _{AC}={\text {tr}}_B {\left GHZ \right\rangle }_{ABC}{\left\langle GHZ \right }\). Thus Theorem 1 guarantees the monogamy inequality of the threequbit GHZ state in Eq. (25) in terms of EoF. In fact, we have \(E_\mathbf{f}\left( {\left GHZ \right\rangle }_{A(BC)}\right) =S(\rho _A)=\ln 2\), whereas the twoqubit reduced density matrices \(\rho _{AB}\) and \(\rho _{AC}\) are separeble. Thus \(E_\mathbf{f}\left( \rho _{AB}\right) =E_\mathbf{f}\left( \rho _{AC}\right) =0\) and this implies the monogamy inequality in (22).
Let us consider another example of threequbit state.
Example 2.
Threequbit Wstate is defined as^{20}
The twoqubit reduced density matrix of \({\left W \right\rangle }_{ABC}\) on subsystem AB is obtained as
where \({\left \psi ^+ \right\rangle }_{AB}=\frac{1}{\sqrt{2}}\left( {\left 01 \right\rangle }_{AB}+{\left 10 \right\rangle }_{AB}\right) \) is the twoqubit Bell state. The onequbit reduced density matrices of \(\rho _{AB}\) are
From to the spectral decomposition of \(\rho _B\) in Eq. (36) with the eigenvalues \(\lambda _0=\frac{2}{3}, \lambda _1=\frac{1}{3}\) and corresponding eigenvectors \({\left e_0 \right\rangle }_B={\left 0 \right\rangle }_B\) and \({\left e_1 \right\rangle }_B={\left 1 \right\rangle }_B\), respectively, it is straightforward to check that the ensemble of \(\rho _A\) induced from measuring subsystem B of \(\rho _{AB}\) by the eigenvectors of \(\rho _B\) is
Because the Fourier basis of subsystem A is the same as Eq. (28), it is also straightforward to obtain the ensemble of \(\rho _A\) induced by measuring subsystem B of \(\rho _{AB}\) in terms of the Fourier basis,
where
For the mutual information of the ccq state \(\Gamma _{XYAB}\) obtained from \(\rho _{AB}\) in Eq. (35), Eq. (18) together with Eqs. (35) and (36) lead us to
For the mutual information of \(\Gamma _{XAB}\) between X and AB, Eq. (19) leads us to
where Eq. (37) impies
Due to Eq. (36), we have \(S(\rho _A)=S(\rho _B)\), therefore Eqs. (41) and (42) lead us to
For the mutual information of \(\Gamma _{YAB}\) between Y and AB, Eq. (20) leads us to
where the second equality is from the ensemble \({\mathscr {E}}_1\) in Eq. (38). Here we note that \(\tau _A^0\) and \(\tau _A^1\) in Eq. (39) have the same eigenvalues, that is \(\mu _0=\frac{3+\sqrt{5}}{6}\) and \(\mu _0=\frac{3\sqrt{5}}{6}\), therefore we have \(S\left( \tau _A^0 \right) =S\left( \tau _A^1 \right) \). From the spectral decomposition of \(\rho _A\) in Eq. (36), we have \(S\left( \rho _A\right) =\ln 3 \frac{2}{3}\ln 2\), and this turns Eq. (44) into
From Eqs. (40), (43) and (45), we have
where \(\ln 2 \approx 0.693147\), \(\ln 3 \approx 1.098612\) and \(S\left( \tau _A^0 \right) \approx 0.381264\). Thus we have
which implies the nonadditivity of mutual information fo r the ccq state \(\Gamma _{XYAB}\) obtained from \(\rho _{AB}\) in Eq. (35). We also note that the symmetry of W state in Eq. (34) would imply the nonadditivity of mutual information for the ccq state \(\Gamma _{XYAC}\) obtained from the twoqubit reduced density matrix \(\rho _{AC}\) of W state in Eq. (34).
As the additivity of mutual information in Theorem 1 is only a sufficient condition for monogamy inequality in terms of EoF, nonadditivity does not directly imply violation of Inequality (22) for the W state in Eq. (34). However, we note that \(\rho _{AB}\) in Eq. (35) is a twoqubit state, therefore its EoF can be analytically evaluated as^{5}
Moreover, the symmetry of the W state assures that the EoF of \(\rho _{AC}={\text {tr}}_B {\left W \right\rangle }_{ABC}{\left\langle W \right }\) is the same,
whereas
As Eqs. (48), (49) and (50) imply the violation of Inequality (22), W state in Eq. (34) can be considered as an example for the contraposition of Theorem 1; violation of monogamy inequality in (22) implies nonadditivity of quantum mutual information for the ccq state.
Now, we generalize Theorem 1 for multiparty quantum states of arbitrary dimension.
Theorem 2.
For any multiparty quantum state \(\rho _{A_1A_2\cdots A_n}\) with twoparty reduced density matrices \(\rho _{A_1A_i}\) for \(i=2, \cdots , n\), we have
conditioned on the additivity of quantum mutual information
where \(\Gamma _{XYA_1A_i}\) is the ccq state of the form in Eq. (14) obtained by \(\rho _{A_1A_i}\) for \(i=2, \ldots , n\).
Discussion
We have considered possible conditions for monogamy inequality of multiparty quantum entanglement in terms of EoF, and shown that the additivity of mutual information of the ccq states implies the monogamy inequality of threeparty quantum entanglement in terms of EoF. We have also provided examples of threequbit GHZ and W states to illustrate our result in threeparty case, and generalized our result into any multiparty systems of arbitrary dimensions.
Most monogamy inequalities of quantum entanglement deal with bipartite entanglement measures based on the minimization over all possible pure state ensembles. As analytic evaluation of such entanglement measure is generally hard especially in higher dimensional quantum systems more than qubits, the situation becomes far more difficult in investigating and establishing entanglement monogamy of multiparty quantum systems of arbitrary dimensions. The sufficient condition provided here deals with the quantum mutual information of the ccq states to guarantee the monogamy inequality of entanglement in terms of EoF in arbitrary dimensions. As the sufficient condition is not involved with any minimization process, our result can provide a useful methodology to understand the monogamy nature of multiparty quantum entanglement in arbitrary dimensions. We finally remark that it would be an interesting future task to investigate if the condition provided here is also necessary.
Methods
Evaluation for the quantum mutual information of the ccq states
Here we evaluate the mutual information of the ccq state in Eq. (14) as well as the reduced density matrices in Eqs. (15) and (16); the classical parts of the fourqudit ccq state \(\Gamma _{XYAB}\) in Eq. (14) is \(\Gamma _{XY}=\frac{1}{d^2}\sum _{x,y=0}^{d1}{\left x \right\rangle }_X {\left\langle x \right }\otimes {\left y \right\rangle }_Y{\left\langle y \right }\), which is the maximally mixed state in \(d^2\)dimensional quantum system, therefore its von Neumann entropy is
We also note that Eq. (17) leads us to
From the joint entropy theorem^{21,22}, we have
where the second equality is due to the unitary invariance of von Neumann entropy. Thus Eqs. (53), (54) and (55) give us the mutual information of the fourqudit ccq state \(\Gamma _{XYAB}\) with respect to the bipartition between XY and AB in Eq. (18).
For the von Neumann entropy of \(\Gamma _{XAB}\) in Eq. (15), we have
where the first equality is from the joint entropy theorem, the second equality is due to the unitary invariance of von Neumann entropy and the last equality is due to the joint entropy theorem together with \(H(\Lambda )=\sum _{i}\lambda _i \ln \lambda _i\) that is the shannon entropy of the spectrum \(\Lambda =\{\lambda _i\}\) of \(\rho _B\) in Eq. (6). Thus we can rewrite the von Neumann entropy of \(\Gamma _{XAB}\) as
Because the classical parts of \(\Gamma _{XAB}\) is the ddimensional maximally mixed state \(\Gamma _X=\frac{1}{d}\sum _{x=0}^{d1}{\left x \right\rangle }_X {\left\langle x \right }\), we have the mutual information of \(\Gamma _{XAB}\) with respect to the bipartition between X and AB as
For the von Neumann entropy of \(\Gamma _{YAB}\) in Eq. (16), we have
where the first and third equalities are due to the joint entropy theorem and the second equality is from the unitary invariance of von Neumann entropy. Thus the mutual information of \(\Gamma _{YAB}\) with respect to the bipartition between Y and AB is
Proof of Theorem 1
Let us first consider the fourqudit ccq state \(\Gamma _{XYAB}\) of the form in Eq. (14) obtained by the twoqudit reduced density matrix \(\rho _{AB}\) of \({\left \psi \right\rangle }_{ABC}\). From Eqs. (18), (19) and (20), the additivity condition of quantum mutual information for \(\Gamma _{XYAB}\) in Eq. (23) can be rewritten as
where \({\mathscr {E}}_0\) and \({\mathscr {E}}_1\) are the probability ensembles of \(\rho _{A}\) in Eqs. (7) and (10), respectively.
Because \({\mathscr {E}}_0\) and \({\mathscr {E}}_1\) can be obtained from measuring subsystem B of \(\rho _{AB}\) by the rank1 measurement \(\{{\left e_i \right\rangle }_B{\left\langle e_i \right }\}_{i=1}^{d1}\) and \(\{{\left {\tilde{e}}_j \right\rangle }_B{\left\langle {\tilde{e}}_j \right }\}_{j=1}^{d1}\), respectively, the definition of CC in Eq. (4) leads us to \({{\mathscr {J}}}^{\leftarrow }(\rho _{AB})\ge \chi ({\mathscr {E}}_0)\) and \({{\mathscr {J}}}^{\leftarrow }(\rho _{AB})\ge \chi ({\mathscr {E}}_1)\), therefore
By considering the ccq state \(\Gamma _{XYAC}\) obtained by \(\rho _{AC}\) as well as Eq. (24), we can analogously have
As the tradeoff relation of Eq. (5) in Proposition 1 is universal with respect to the subsystems, we also have \(S(\rho _A)={{\mathscr {J}}}^{\leftarrow }(\rho _{AC})+E_\mathbf{f}\left( \rho _{AB}\right) \) for the given twoqudit state \({\left \psi \right\rangle }_{ABC}\), therefore
Now Inequalities (62), (63) as well as Eq. (64) lead us to
where the second equality is due to \(\rho _{AC}=\rho _B\) and \(\rho _{AB}=\rho _C\) for threeparty pure state \({\left \psi \right\rangle }_{ABC}\).
Proof of Theorem 2
We first prove the theorem for any threeparty mixed state \(\rho _{ABC}\), and inductively show the validity of the theorem for any nparty quantum state \(\rho _{A_1A_2\cdots A_n}\). For a threeparty mixed state \(\rho _{ABC}\), let us consider an optimal decomposition of \(\rho _{ABC}\) realizing EoF with respect to the bipartition between A and BC, that is,
with \(E_\mathbf{f}\left( \rho _{A(BC)}\right) =\sum _i p_i E_\mathbf{f}\left( {\left \psi _i \right\rangle }_{A(BC)}\right) \). From Theorem 1, each pure state \({\left \psi _i \right\rangle }_{ABC}\) of the decomposition (66) satisfies \(E_\mathbf{f}\left( {\left \psi _i \right\rangle }_{A(BC)}\right) \ge E_\mathbf{f}\left( \rho ^i_{AB}\right) +E_\mathbf{f}\left( \rho ^i_{AC}\right) \) with \(\rho ^i_{AB}={\text {tr}}_C {\left \psi _i \right\rangle }_{ABC}{\left\langle \psi _i \right }\) and \(\rho ^i_{AC}={\text {tr}}_B {\left \psi _i \right\rangle }_{ABC}{\left\langle \psi _i \right }\), therefore,
For each i and the twoparty reduced density matrices \(\rho _{AB}^i\), let us consider its optimal decomposition \(\rho _{AB}^i=\sum _{j}r_{ij}{\left \mu _j^i \right\rangle }_{AB}{\left\langle \mu _j^i \right }\) realizing EoF, that is, \(E_\mathbf{f}\left( \rho _{AB}^i\right) =\sum _{j}r_{ij} E_\mathbf{f}\left( {\left \mu _j^i \right\rangle }_{AB}\right) \). Now we have
where the inequality is due to \(\rho _{AB}=\sum _i p_i \rho _{AB}^i=\sum _{i,j}p_i r_{ij}{\left \mu _j^i \right\rangle }_{AB}{\left\langle \mu _j^i \right }\) and the definition of EoF.
For each i, we also consider an optimal decomposition \(\rho _{AC}^i=\sum _{l}s_{il}{\left \nu _l^i \right\rangle }_{AC}{\left\langle \nu _l^i \right }\) such that \(E_\mathbf{f}\left( \rho _{AC}^i \right) =\sum _{l}s_{il} E_\mathbf{f}\left( {\left \nu _l^i \right\rangle }_{AC}\right) \). We can analogously have
due to \(\rho _{AC}=\sum _i p_i \rho _{AC}^i=\sum _{i,l}p_i s_{il}{\left \nu _l^i \right\rangle }_{AC}{\left\langle \nu _l^i \right }\) and the definition of EoF in Eq. (2). From Inequalities (67), (68) and (69), we have
which proves the theorem for threeparty mixed states.
For general multiparty quantum system, we use the mathematical induction on the number of parties n; let us assume Inequality (51) is true for any kparty quantum state, and consider an \(k+1\)party quantum state \(\rho _{A_1A_2\cdots A_{k+1}}\) for \(k \ge 3\). By considering \(\rho _{A_1A_2\cdots A_{k+1}}\) as a threeparty state with respect to the tripartition \(A_1\), \(A_2\cdots A_k\) and \(A_{k+1}\), Inequality (70) leads us to
As \(\rho _{A_1A_2\cdots A_k}\) in Inequality (71) is a kparty quantum state, the induction hypothesis assures that
Now Inequalities (71) and (72) lead us to the monogamy inequality in (51), which completes the proof.
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For any orthonormal basis \(\{\left{i}\right\rangle _A\}\) and probability ensemble \(\{p_i, \rho _B^i\}\), \(S\left(\sum _{i}p_i \left{i}\right\rangle {i}_A\left\langle {i}\right \otimes \rho _B^i\right)=H\left({\bf P\it }\right)+\sum _{i}p_{i}S\left(\rho _B^i\right),\) where \(H\left({\bf P\it }\right)=\sum _{i}p_i \ln p_i\) is the shannon entropy of the probability distribution \({\bf P\it }=\{p_i\}\).
Acknowledgements
This work was supported by Basic Science Research Program (NRF2020R1F1A1A010501270) and Quantum Computing Technology Development Program (NRF2020M3E4A1080088) through the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT).
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J.S.K. conceived the idea, performed the calculations and the proofs, interpreted the results, and wrote down the manuscript.
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Kim, J.S. Entanglement of formation and monogamy of multiparty quantum entanglement. Sci Rep 11, 2364 (2021). https://doi.org/10.1038/s41598021820523
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DOI: https://doi.org/10.1038/s41598021820523
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